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Examining the portfolio balance theory of foreign exchange rates: A time series analysis

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Examining the portfolio balance theory of foreign exchange rates: A time series analysis

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Content EXAMINING THE PORTFOLIO BALANCE THEORY OF FOREIGN EXCHANGE RATES: A TIME SERIES ANALYSIS by Randa Azar A D i s s e r t a t i o n P re se n ted to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In P a r t i a l F u l f i l l m e n t of the Requiremnets f o r the Degree DOCTOR OF PHILOSOPHY ( Economics) September 1983 UMI Number: DP23320 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dissertation Publishing UMI DP23320 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106-1346 UNIVERSITY O F S O U T H E R N CALIFORNIA T H E G R A D U A T E S C H O O L U N IV E R S IT Y PA R K L O S A N G E L E S , C A L IF O R N IA 9 0 0 0 7 ppa £ c - '83 This dissertation, written by RANDA AZAR under the direction of h.zx.... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R OF P H I L O S O P H Y Dean Dates DISSERTATION COMMITTEE airman ~ c / ii i ACKNOWLEDGEMENTS The com pletion of t h i s t h e s i s i s the r e s u l t of the su p p o rt and encouragement of many people. F i r s t and fo rem ost, I want to thank my a d v is e r , P ro fe ss o r Gerald N ic k elsb u rg , fo r h is advice and guidance t h a t were in v a lu a b le in the development of t h i s stud y. The number of hours he devoted to d i s c u s s in g id ea s with me have been in s tr u m e n ta l in enhancing my u n d e rsta n d in g of the v a rio u s a s p e c ts of t h i s work and a re deeply a p p r e c ia te d , not to mention th e time he spent re a d in g and c o r r e c t i n g a number of e a r l i e r d r a f t s . I owe s p e c i a l thanks to P ro f e s s o r J e f f r e y Nugent f o r h is advice and encouragement thro ug ho ut my g rad u a te work. His comments and th e hours he c o n tr i b u te d m e tic u lo u sly read ing v a rio u s d r a f t s of t h i s t h e s i s a re g r a t e f u l l y acknowledged. I a ls o want to thank P ro fe ss o r Arthur L a ffe r fo r the i m o tiv a tio n and comments he provided me in p u rsuing t h i s j r e s e a r c h . I I have b e n e f i te d from d i s c u s s i o n s w ith P r o f e s s o r Lee L i l l a r d and my c o lle a g u e s Joseph H irschberg and Akbar J a z a y e r i , whose w i l l i n g n e s s to l i s t e n and o f f e r t h e i r advice i s deeply a p p r e c ia te d . I a ls o want to thank Vivian Smith fo r her ty p in g of p a r t s o f t h i s t h e s i s . iii Finally, I want to thank my mother for her support and i encouragement and for instilling in me the desire to achieve. i ! i ABSTRACT The primary goal of t h i s study i s to e m p i r ic a ll y examine the p o r t f o l i o balance approach to f o r e ig n exchange d e te r m in a tio n . While th e i m p l i c a t i o n s of the p o r t f o l i o balance theory have a lre a d y been probed, major d e f i c i e n c i e s s t i l l remain. E x is tin g e m p ir ic a l a n a ly se s have e i t h e r co n sid e re d th e p o r t f o l i o model as an a g g re g a te reduced form, th e re b y , n e g le c t i n g th e a p p r o p r i a te m o delling of the b e h a v io ra l f o r c e s and economic p ro c e s s e s t h a t underly the observed movements in exchange r a t e s , or, not a d m itte d jenough d i v e r s i f i c a t i o n to c a p tu re the essence of and gain jin sig h t about how p o r t f o l i o d i v e r s i f i c a t i o n in both c u r r e n c i e s and ty p es of a s s e t s i n f lu e n c e s exchange r a t e s . These d e f i c i e n c i e s a re overcome in t h i s study by f o rm u la tin g a gen eral dynamic p o r t f o l i o model t h a t may in clu d e any number of a s s e t s denominated in d i f f e r e n t c u r r e n c i e s and, in p a r t i c u l a r , gold. Assuming r a t i o n a l e x p e c ta ti o n s and o p tim iz in g behavior on th e p a r t of i n v e s t o r s , the a s s e t demand e q u a tio n s are d e riv e d u sing dynamic o p tim iz a tio n te c h n iq u e s . For the e m p iric a l a n a l y s i s we r e s t r i c t the p o r t f o l i o to th r e e a s s e t s , two V c u r r e n c i e s and gold. Furtherm ore, s in c e i t i s not p o s s ib l e to d e riv e closed form s o l u t i o n s f o r the a s s e t demand f u n c t i o n s in th e g e n eral model, we r e s t r i c t o u r s e lv e s to the case of a q u a d r a t i c o b j e c t i v e f u n c t i o n . Two i n t e r r e l a t e d e m p iric a l s t r a t e g i e s a re implemented. F i r s t , we e x p lo re the i m p l i c a ti o n s of th e p o r t f o l i o balance approach by e s tim a t i n g a s t r u c t u r e - f r e e m u l t i v a r i a t e time s e r i e s model. This te c h n iq u e , proposed by C h risto p h e r Sims, uses e m p iric a l evidence to observe th e r e l a t i o n s h i p s between exchange r a t e s , i n t e r e s t r a t e s , money, p r i c e s , and gold w ith o u t imposing any a p r i o r i s t r u c t u r e . Second, the s t r u c t u r a l model i s analyzed by e s t im a t i n g an e q u a tio n f o r exchange r a t e s , d e riv e d from the a s s e t demand f u n c t i o n s and m arket c l e a r i n g c o n d it i o n s , sim u lta n e o u sly with the s t o c h a s t i c p ro c e s s e s and c o n s t r a i n t s f a c i n g the i n v e s t o r s . Since the only r e s t r i c t i o n s imposed a re t h a t i n v e s t o r s ' f o r e c a s t s of the r e l e v a n t f u t u r e v a r i a b l e s are c o n s i s t e n t w ith t h e i r a c tu a l p ro c e s s e s , and t h a t m arkets c l e a r , the ad hoc n a tu re of e x i s t i n g p o r t f o l i o balance models of fo r e ig n exchange r a t e s i s overcome. Based on th e r e s u l t s from the two econom etric models, the m ain tain ed h y po theses concerning th e process of f o r e ig n exchange d e te rm in a tio n a re t e s t e d . V I TABLE OF CONTENTS ACKNOWLEDGEMENTS ............................................................................................ i i ABSTRACT.......................................................................................................................iv Chapter p.age I . INTRODUCTION .............................................................................................. 1 Recent Developments in Exchange Rate T h eo ries . 7 The Role of G o l d .....................................................................21 Scope of The S t u d y ................................................................24 I I . THE PORTFOLIO MODEL......................................................................28 The General Model ................................... 31 R a tio n al E x p e c ta tio n s E q u ilib riu m and The Dynamic P o r t f o l i o Model .................................... 39 E m pirical S t r a t e g i e s fo r E stim a tin g Dynamic P o r t f o l i o Models ......................................................... 43 I I I . AN ALTERNATIVE STRATEGY FOR ANALYZING EXCHANGE RATE DYNAMICS............................................. 50 M e t h o d o l o g y ...............................................................................51 The D a t a . . . . . . . . . . . . . . . . . . . 57 T e s tin g Lag Length And S t r u c t u r a l S t a b i l i t y . 61 Decomposition Of V ariance .............................................. 71 Sim u latio n of System Response to Random Shocks 79 Granger C a u s a lity and Econometric Exogeneity . 90 IV. ESTIMATION OF THE DYNAMIC PORTFOLIO ALLOCATION MODEL.........................................................................................93 A sset E q u ilib riu m in The Case of L inear U t i l i t y 94 M e t h o d o l o g y ..............................................................................112 R e s u l t s ........................................................................................121 V. SUMMARY AND CONCLUSIONS...........................................................132 VX1 Appendix es page A. DERIVATION OF ASSET DEMAND EQUATIONS ........................... 1 4 0 B. STATISTICAL TABLES ......................................................................... 158 C. CHARTS FROM SIMULATING SYSTEM RESPONSE TO RANDOM S H O C K S ....................................................................... 172 REFERENCES............................................................................................................... 208 viii LIST OF TABLES Table P..a&e 1 . T ests f o r S t r u c t u r a l S t a b i l i t y ............................................... 68 2. Decomposition of the Exchange Rate F o re c a s t E rro r V a r i a n c e .........................................................................................74 3. C o n tr ib u tio n of Exchange Rate In n o v a tio n s to The V ariance of The Model V a r ia b le s . . .......................75 4. Decomposition of the V ariance of P r ic e of Gold . . 78 5. T e s ts of E x o g e n e i t y ......................................................................92 6. R egression E stim a te s of the Exchange Rate Equation 122 7. T e s t s 3 on Model Lag L en g th :(1974-1980) . . . . . . 159 8. T e s ts f o r S t r u c t u r a l S t a b i l i t y : 1974-1976 vs. 1977-1980 ........................................................................ 160 9. T ests f o r S t r u c t u r a l S t a b i l i t y : 1974-1977 vs. 1 978-1980 161 10. T ests f o r S t r u c t u r a l S t a b i l i t y : 1974-1979 vs. 1 979-1980 ............................................................................................ 162 11. CANADA - T r ia n g u la r i z e d I n n o v a tio n s in : 163 12. FRANCE - T r ia n g u la r i z e d In n o v a tio n s in : 164 13. GERMANY - T r ia n g u la r i z e d In n o v a tio n s in : ...................... 165 1,4. SWITZERLAND - T r ia n g u la r i z e d In n o v a tio n s in : . . . 166 15. K-Step Ahead Standard E rro r of F o re c a s t ...................... 167 16. R egression E stim a te s fo r The P o r t f o l i o Model - C a n a d a .........................................................................168 ix 17. R egression E stim a te s f o r The P o r t f o l i o Model - F r a n c e ...................................................... 169 18. R egression E stim a te s f o r The P o r t f o l i o Model - G e r m a n y ............................................................................... 170 19. R egression E stim a te s f o r The P o r t f o l i o Model - S w itz e rla n d .................................. 171 X LIST OF CHARTS Chart page 1. Responses to U.S. Money S h o c k s ......................................................82 2. Responses to Domestic Money Shocks ..................................... 83 3. Responses to I n t e r e s t D i f f e r e n t i a l Shocks . . .. 84 4. Responses to P ric e of Gold S h o c k s ............................................85 5. CANADA - Responses of P ric e of G o l d ..................................... 86 6. CANADA - Responses of The Exchange R a t e .........................173 7. FRANCE - Responses of The Exchange R a t e .........................174 8. GERMANY - Responses of The Exchange Rate ...................... 175 9. SWITZERLAND- Responses of The Exchange Rate . . . 176 10. FRANCE - Responses of P ric e of G o l d ...................................177 11. GERMANY - Responses of P ric e of Gold ...................... , . 178 12. SWITZERLAND- Responses of P ric e of G o l d .................179 13. CANADA - Responses to U.S. Money Shocks . . . . . 180 14. CANADA - Responses to Domestic Money Shocks . . . 181 15. CANADA - Responses to I n t e r e s t D i f f e r e n t i a l s . . . 182 16. CANADA - Responses to U.S. P r ic e S h o c k s .........................183 17. CANADA - Responses to Domestic P r ic e Shocks . . . 184 18. CANADA - Responses to P ric e of Gold Shocks . . . . 185 19. CANADA - Responses to Exchange Rate Shocks . . . . 186 XI 20. FRANCE - Responses to U.S. Money Shocks ..................... 187 21 . FRANCE - Responses to Domestic Money Shocks . . . 188 22. FRANCE - Responses to I n t e r e s t D i f f e r e n t i a l s . . . 189 • no C M FRANCE - Responses to U.S. P ric e Shocks ..................... 190 24. FRANCE - Responses to Domestic P r ic e Shocks . . . 191 25. FRANCE - Responses to P ric e of Gold Shocks . . . . 192 26. FRANCE - Responses to Exchange Rate Shocks . . . . 193 27. GERMANY - Responses to U.S. Money Shocks ..................... 194 28. GERMANY - Responses to Domestic Money Shocks . . . 195 29. GERMANY - Responses to I n t e r e s t D i f f e r e n t i a l s . . 196 30. GERMANY - Responses to U.S. P ric e Shocks ..................... 197 31 . GERMANY - Responses to Domestic P r ic e Shocks . . . 198 32. GERMANY - Responses to P r ic e of Gold Shocks . . . 199 33. GERMANY - Responses to Exchange Rate Shocks . . . 200 34. SWITZERLAND - Responses to U.S. Money Shocks . . . 201 35. SWITZERLAND - Responses to Domestic Money Shocks . 202 36. SWITZERLAND - Responses to I n t e r e s t D i f f e r e n t i a l s 203 37. SWITZERLAND - Responses to U.S. P ric e Shocks . . . 204 38. SWITZERLAND - Responses to Domestic P r ic e Shocks . 205 39. SWITZERLAND - Responses to P ric e of Gold Shocks 206 40. SWITZERLAND - Responses to Exchange Rate Shocks 207 1 Chapter I INTRODUCTION The i s s u e s in v o lv e d in th e d e te rm in a tio n of f o r e ig n exchange r a t e s have a t t r a c t e d a g r e a t deal of a t t e n t i o n in th e l a s t decade. The economic l i t e r a t u r e encompasses both new and r e v i s e d v e r s i o n s of old approaches f o r e x p la in in g th e behavior of f o r e i g n exchange r a t e s under f l e x i b l e r a t e regim es. While much of the e m p iric a l re s e a r c h has provided su p p o rt f o r the " a s s e t ” view which m a in ta in s t h a t exchange r a t e s r e f l e c t th e r e l a t i v e v alue of two ( n a t i o n a l ) a s s e t s , much work remains to be completed in a n a ly z in g the u n d e rly in g economic p r o c e s s e s and b e h a v io ra l f a c t o r s t h a t a f f e c t th e r e l a t i v e demands of such a s s e t s and, t h e r e f o r e , t h e i r r e l a t i v e p r i c e s . Such an u n d e rs ta n d in g of the u n d e rly in g f o r c e s i s im p o rta n t to r e s e a r c h e r s , monetary a u t h o r i t i e s , and i n v e s t o r s . I t may be used to design more powerful t e s t s f o r many m ain tained h yp o th eses concerning th e f o r e i g n exchange m arket, to make b e t t e r f o r e c a s t s , to draw i n f e r e n c e s w ith a g r e a t e r degree of co n fid e n ce , and to su g g est more e f f e c t i v e p o l i c i e s fo r s u p p o rtin g exchange r a t e s . 2 The a s s e t view of the exchange r a t e d e te r m in a tio n p ro cess evolved from the Monetary Approach to the Balance of Payments which was p io n eered by Harry Johnson. The o r i g i n s of t h i s approach can be tr a c e d back not only to the widely c i t e d t h e o r e t i c a l work of Robert M undell(56), but even f u r t h e r back to th e e a r l i e r a n a l y s i s of c l a s s i c a l econom ists such as Wheatly, Hume, Ricardo and C a sse l. S u p p o rte rs of the Monetary Approach contended t h a t e a r l i e r models based on t r a d e flow s, such as th e Keynesian models, had f a i l e d in e x p la in i n g th e movements in exchange r a t e s a f t e r the breakdown in th e B re tto n Woods System in th e e a r l y 1 9 7 0 's. The reaso n c i t e d f o r t h i s f a i l u r e l i e s in th e f a c t t h a t an exchange r a t e r e f l e c t s th e r e l a t i v e value of two n a ti o n a l monies, and not the r e l a t i v e p r ic e of two n a ti o n a l o u tp u ts . I t i s th e demand f o r and supply of th ese two monies t h a t w i l l u l t i m a t e l y determ ine t h e i r r e l a t i v e v a lu e . Rather than emphasize the r o l e of the supply of money and th e p u rch a sin g power p a r i t y h y p o th e s is as the e a r l i e r monetary models did, the a s s e t view emphasizes p o r t f o l i o b a la n ce . These changing n o tio n s about the d e te rm in a n ts of f o r e i g n exchange r a t e s a re b e s t i l l u s t r a t e d in a s e r i e s of papers by Dornbusch"* which begins w ith a monetary p u rch a sin g power p a r i t y model and ev o lv es i n t o a ^ See Rudiger Dornbusch(1 2 ), (1 3 ), (1 4 ), (15) and (17) 3 p o r t f o l i o balance model as r e s t r i c t i o n s a re r e la x e d along th e way. While the i m p l i c a t i o n s of the " a s s e t ” or p o r t f o l i o balance theory have a lre a d y been probed, major d e f i c i e n c i e s s t i l l remain. E x is ti n g a n a ly s e s have e i t h e r c o n sid e re d th e p o r t f o l i o problem as an a g g re g a te reduced form type of model, n e g le c t i n g to model a p p r o p r i a t e l y the b e h a v io ra l f o r c e s and economic p r o c e s s e s t h a t underly the observed movements in exchange r a t e s , or, did not admit enough d i v e r s i f i c a t i o n to c a p tu r e the essence of and g ain i n s i g h t i n t o how p o r t f o l i o d i v e r s i f i c a t i o n w ith r e s p e c t to both currency and o th e r a s s e t s i n f lu e n c e th e exchange r a t e m arket. These d e f i c i e n c i e s a re overcome in t h i s study by f o r m u la tin g an e m p ir ic a l g e n eral dynamic p o r t f o l i o model t h a t may in c lu d e any number of a s s e t s denominated in any number of d i f f e r e n t c u r r e n c i e s . The h i s t o r i c a l r e l a t i o n s h i p between monies and gold prompted us to in c lu d e gold in th e p o r t f o l i o . Since e x p e c ta ti o n s about r e t u r n s on d i f f e r e n t a s s e t s play the major r o l e in p o r t f o l i o c h o ic e , an a p p r o p r i a te method of m o d ellin g th e way i n v e s t o r s form t h e i r p r e d i c t i o n s i s n e c e s s a r y . Under the r a t i o n a l e x p e c ta ti o n s assum ption, exchange r a t e dynamics can be e m p i r ic a ll y analyzed w ith o u t being s u b j e c t to th e c r i t i c i s m l e v e l e d 4 a g a i n s t e x i s t i n g eco no m etric models of exchange r a t e s . The d e riv e d a s s e t demand f u n c t i o n s w i l l depend on th e p a ra m eters of the a c t u a l s t o c h a s t i c p ro c e s s e s governing r a t e s of r e t u r n , a s s e t sto c k s and th e p r i c e of gold and w i l l , th u s, vary with changes in th e economic environm ent go verning a s s e t m ark e ts. In a d d i t i o n i t makes th e econom etric problem much sim p ler s in c e i t p ro v id e s th e n e ce ssa ry i d e n t i f y i n g r e s t r i c t i o n s f o r model e s t im a t i o n . Some l i m i t a t i o n s , though, are imposed. For th e econom etric model we r e s t r i c t th e p o r t f o l i o to th r e e a s s e t s , two c u r r e n c i e s and gold. F urtherm ore, sin c e i t i s not p o s s i b l e to d e riv e c lo sed form s o l u t i o n s f o r the a s s e t demand f u n c t i o n s in th e g e n eral model, we r e s t r i c t o u r s e l v e s to th e case of a q u a d r a t i c o b j e c t i v e f u n c t i o n . The s t r u c t u r a l model i s examined by e s t im a t i n g a s s e t demand f u n c t i o n s d e riv e d from the p o r t f o l i o s e l e c t i o n problem sim u lta n e o u s ly with the s t o c h a s t i c p r o c e s s e s and c o n s t r a i n t s f a c i n g th e i n v e s t o r s . The ad hoc n a tu re of e x i s t i n g p o r t f o l i o balance models of f o r e i g n exchange r a t e s i s overcome sin c e the only r e s t r i c t i o n s imposed a re t h a t i n v e s t o r s 1 f o r e c a s t s of th e r e l e v a n t f u t u r e v a r i a b l e s a re c o n s i s t e n t w ith t h e i r a c tu a l p r o c e s s e s , and t h a t m arkets c l e a r . 5 The i m p l i c a t i o n s of the p o r t f o l i o balan ce approach are f u r t h e r e x plored by e s t im a t i n g a s t r u c t u r e - f r e e v e c to r a u t o r e g r e s s i v e model. Vector a u t o r e g r e s s i v e models were f i r s t used by Sims(69) as a way of u sin g e m p ir ic a l evidence to study r e l a t i o n s h i p s among economic v a r i a b l e s w ith o u t imposing any a p r i o r i s t r u c t u r e on th e s e r e l a t i o n s h i p s . By a n a ly z in g th e data i n i t i a l l y in t h i s manner, we examine the r e l a t i o n s h i p s between exchange r a t e s , i n t e r e s t r a t e s , money, p r i c e s , and gold. We are then a b le to i n f e r which of the f a c t o r s c o n sid e re d have the g r e a t e s t i n f lu e n c e on exchange r a t e s , and what kind of c a u sa l r e l a t i o n s h i p s e x i s t among th e v a r i a b l e s examined. This h e lp s in m inim izing s p e c i f i c a t i o n b ia s in any econom etric model. The r e s u l t s from t h i s a n a l y s i s a re a ls o used in t e s t i n g some assum ptions in tr o d u c e d i n th e s t r u c t u r a l model, as w e ll as p ro v id in g in f o rm a tio n about th e p ro c e s s e s assumed exogenous to the model. For example, for those v a r i a b l e s assumed to follow an a u t o r e g r e s s i v e p ro c e ss , the data a n a l y s i s su g g e s ts th e a p p r o p r i a te la g o r d e r. The two econom etric models a re a p p lie d to th e r a t e of exchange of the German, Swiss, French and Canadian c u r r e n c i e s v i s - a - v i s th e U.S. d o l l a r f o r the perio d fo llo w in g th e Sm ithsonian Agreement in 1973. The r e s u l t s o b ta in e d from e s t im a t i o n of both the v e c to r a u t o r e g r e s s i v e 6 model and th e s t r u c t u r a l p o r t f o l i o model provide us w ith in fo rm a tio n about th e a t t i t u d e of i n v e s t o r s in th e f o r e i g n exchange m arket, how the u n d e rly in g p r o c e s s e s a f f e c t th e movements in exchange r a t e s , and in p a r t i c u l a r , how the gold m arket i n f l u e n c e s f o r e i g n exchange r a t e s under f l e x i b l e r a t e regim es. The rem ainder of t h i s c h a p t e r i s o rg a n iz e d i n t o th r e e s e c t i o n s . F i r s t , we p r e s e n t a summary of the r e c e n t developments in f o r e i g n exchange t h e o r i e s and dynamic eco no m etric m o delling , and our r e s e a r c h s t r a t e g y . The d is c u s s i o n p ro v id es a p e r s p e c t i v e on th e a s s e t approach from the h i s t o r i c a l developments and th e e m p iric a l evidence, to the d e f i c i e n c i e s e n c o u n te re d in m o d ellin g and t e s t i n g t h i s view. The i s s u e s d is c u s s e d o f f e r the r a t i o n a l e f o r the methodology adopted in t h i s stu d y . S e c tio n 1.2 in c lu d e s a b r i e f review of th e r o l e of gold in i n t e r n a t i o n a l f i n a n c i a l p o r t f o l i o s and o f f i c i a l r e s e r v e s which prov ided th e m o tiv a tio n f o r i t s i n c l u s i o n in th e a n a l y s i s . F i n a l l y , an o u t l i n e of th e s t r u c t u r e of the rem aining c h a p t e r s i s - p r o v i d e d . 1.1 RECENT DEVELOPMENTS IN EXCHANGE RATE THEORIES 7 There a re two b a s ic frameworks w i t h in which t h e o r i e s of the l a s t decade developed: th e Keynesian f lo w - e q u ilib r iu m framework, and th e a s s e t market e q u ilib r iu m view. The fundam ental d i f f e r e n c e between th e s e two views i s t h a t Keynesian models emphasize the r o l e the c u r r e n t account p lay s in d e te rm in in g exchange r a t e s , w hile " a s s e t ” models emphasize a s s e t a r b i t r a g e as th e key note to e x p la in i n g exchange r a t e dynamics. The l a t t e r models r e p r e s e n t a much more r e c e n t approach which was s e t f o r t h w ith the Monetary Approach to exchange r a t e d e te r m in a tio n , an e x te n s io n to the Monetary Approach to Balance of Payments Theory. This approach encompasses a wide v a r i e t y of models ra n g in g from those t h a t emphasize a monetary i n t e r p r e t a t i o n of exchange r a t e dynamics to th ose t h a t adopt a p o r t f o l i o balance th e o ry . All b a s i c a l l y a s s ig n a major r o l e to the f i n a n c i a l s e c t o r , th e re b y , s u p p la n tin g th e Keynesian emphasis on t r a d e . The monetary or a s s e t view may be new, but th e c e n t r a l themes a re n o t. C a p ita l m o b ility , the pu rch asin g power p a r i t y d o c t r i n e , and th e r o l e of e x p e c ta ti o n s were a l l co n cep ts developed e a r l i e r by econom ists s t a r t i n g w ith W heatley, Ricardo, Hume, M arsh all, W ic k se ll, Cassel and Keynes. These th r e e co n cep ts w i l l be b r i e f l y e x p la in e d 8 b e fo re we go on to d e s c r ib e the a s s e t view. Keynesian models w i l l not be d is c u s s e d sin c e i t i s not our purpose to t e s t them a g a i n s t monetary models, but to examine the e m p ir ic a l r e g u l a r i t i e s in exchange r a t e s and to f i n d .out i f they a re s u p p o r tiv e of the p o r t f o l i o balance view. The p u rch a sin g power p a r i t y (PPP) h y p o th e s is simply im p lie s t h a t th e e q u ilib riu m exchange r a t e i s equal to th e r a t i o of dom estic t o f o r e i g n p r i c e s . Jacob Frenkel a s s e r t a i n s t h a t i t i s not th e same as commodity a r b i t r a g e because "one may imagine a s i t u a t i o n in which a l l tr a d e d goods p o sse ss th e same p r i c e , when e x p re ss e d in common c u rre n cy , but th e exchange r a t e i s in d i s e q u i l i b r i u m "(25, p. 2 0 2 ). Indeed, r e l a t i v e a s s e t sto c k s , and more im p o r ta n tly a s s e t e q u ilib r iu m , are what the p u rch a sin g power p a r i t y c o n d itio n i m p l i e s . P r i c e s and exchange r a t e s a re d eterm ined sim u lta n e o u s ly by monetary e q u il i b r i u m . Stock e q u ilib r iu m i s , and was, the g e n e r a l l y acc ep ted d e te rm in a n t of exchange r a t e s as i n d i c a t e d by the fo llo w in g s ta te m e n t made by Keynes in 1924, and c i t e d by Jacob F r e n k e l . What th en, has determ ined th e v alue of the f ra n c ? F i r s t , the q u a n t i t y , p r e s e n t and p r o s p e c t i v e , of the f r a n c s in c i r c u l a t i o n . Second, th e amount of p u rc h a sin g power which i t s u i t s th e p u b lic to ho ld in t h a t shape. 9 ( Keynes, C o lle c te d W ritin g s , v o l. 4 ( A T ra c t on Monetary Reform) T (London, 1971) P* x v i i i ) In a d d i t i o n to sto c k e q u ilib r iu m as a d e te rm in a n t of exchange r a t e s , th e a s s e t view b rin g s i n t o p e r s p e c t iv e the r o l e of e x p e c t a t i o n s . Changes in i n d i v i d u a l e x p e c ta ti o n s r e g a r d in g f u t u r e economic v a r i a b l e s a re i n c o r p o r a te d in a number of a s s e t view s t u d i e s in a tte m p tin g to e x p la in th e v o l a t i l i t y in exchange r a t e s . The e a r l i e r monetary models were based on monetary e q u ilib riu m and th e p u rch a sin g power p a r i t y d o c t r i n e . A t y p i c a l monetary model such as J e f f r e y A. F r a n k e l 's ( 2 4 ) can be developed b a s i c a l l y from t h i s h y p o t h e s is and t h a t of monetary e q u ilib riu m in each of th e c o u n t r i e s in v o lv e d . The b a s ic model e q u a tio n s would then be: ( i ) th e p u rch a sin g power p a r i t y c o n d itio n : (1 .1 ) e = P/P* ( i i ) monetary e q u ilib riu m which i s a t t a i n e d when money demand, a f u n c t i o n of income and th e o p p o rtu n ity c o s t of h o ld in g money, i s equal to th e money s u p p l y : M s = f (Y, r , _____) .P e i s the exchange r a t e (1 .2 ) where 10 p i s th e dom estic p r i c e l e v e l M s i s th e nominal money supply r i s th e nominal r a t e of - i n t e r e s t Y i s dom estic p h y s ic a l o u tp u t and r e f e r s to f o r e i g n country v a r i a b l e s . S u b s t i t u t i n g e q u a tio n (1 .2 ) i n to e q u a tio n (1 .1 ) and assuming an i d e n t i c a l demand f u n c t i o n f o r the f o r e ig n co u n try , the fo llo w in g e q u a tio n i s d e riv e d in term s of the exchange r a t e , the money s u p p l i e s and th e dom estic and f o r e i g n argum ents of the money demand f u n c t i o n s , namely r e a l income and nominal i n t e r e s t r a t e s : E quation (1 .3 ) then y i e l d s th e reduced form e q u a tio n fo r exchange r a t e s to be s u b j e c t e d to e m p ir ic a l t e s t i n g . The p u rch a sin g power p a r i t y c o n d itio n i s g e n e r a l l y b e lie v e d t o hold in th e lo n g - r u n . However, i t has been dem onstrated in s e v e r a l s t u d i e s i n c l u d i n g th o se of Dornbusch and Krugman(18) and I s a r d (42) t h a t i t f a i l s to be tr u e in th e s h o r t - r u n , where a s s e t a r b i t r a g e becomes th e dominant f a c t o r in d e te rm in in g th e l e v e l of exchange r a t e s . The a s s e t view was i n i t i a l l y d e p ic te d in term s of a ( 1 .3 ) 9 • • • )/f(Y,r ) 11 "monetary" model w ith the p u rch a sin g power p a r i t y assum ption r e l a x e d . The assum ption of p e r f e c t a s s e t s u b s t i t u t a b i l i t y kept th e focus on th e supply of and demand f o r money. Many econom ists, such as Dornbusch(1 3 ), K o u ri(4 4 ), and M ussa(58), i n d i c a t e d t h a t both th e monetary and a s s e t models a re only "p ed ago gical s i m p l i f i c a t i o n s " of a more g e n e ra l a s s e t model, where d i f f e r e n t n a ti o n a l monies a re viewed to be j u s t d i f f e r e n t ty p e s of r is k y a s s e t s w ith d i f f e r e n t currency denom inations t h a t can be h e ld in a f i n a n c i a l p o r t f o l i o . A number of econom ists, however, draw sharp d i s t i n c t i o n s between th e monetary and a s s e t ap pro ach es. They hold t h a t in o rd er to p ro p e rly c h a r a c t e r i z e p o r t f o l i o d i v e r s i f i c a t i o n , dom estic and f o r e i g n a s s e t s should be viewed as im p e rf e c t s u b s t i t u t e s w ith u n c e r t a i n r e t u r n s . The theory of p o r t f o l i o s e l e c t i o n p ro v id es th e r e l e v a n t method f o r d e r i v i n g th e demand fo r th e s e a s s e t s w ith d i f f e r e n t currency denom inations under such c o n d it i o n s . P o r t f o l i o p r e f e r e n c e s and e x p e c ta ti o n s about f u t u r e y i e l d s on th e s e s u b s t i t u t a b l e a s s e t s can le a d to a d ju s tm e n ts in th e p o r t f o l i o h o ld in g s of a g en ts in d i f f e r e n t c o u n t r i e s , r e s u l t i n g in c a p i t a l movements and f l u c t u a t i o n s in exchange r a t e s . In a p a r a l l e l development, th e d i f f e r e n t monetary and a s s e t models a re being extended to in c lu d e th e r a t i o n a l 12 e x p e c ta ti o n s assum ption. The models by B arro ( 6 ) , B i l s o n ( 8 ) , and Hodrick(4’l ), a l l y i e l d a reduced form e q u a tio n f o r the exchange r a t e t h a t depends upon th e c u r r e n t and exp ected f u t u r e v a lu e s of some exogenous v a r i a b l e s . The r a t i o n a l e x p e c ta ti o n s assum ption i s used to r e l a t e the exp ected f u t u r e v a lu e s of the exogenous v a r i a b l e s to th e v a lu e s p r e d i c t e d by the model. In o rd er to invoke th e r a t i o n a l e x p e c t a t i o n s assum ption in t h i s sen se, i t becomes n e c e ssa ry to s p e c if y the s t o c h a s t i c p ro c e s s e s t h a t g e n e ra te th e exogenous v a r i a b l e s . In a s i m i l a r c o n te x t Dornbusch (15) in tr o d u c e s th e r a t i o n a l e x p e c ta ti o n s h y p o th e s is in o rd er to e x p la in u n a n t i c i p a t e d changes in exchange r a t e s . He c o n s id e r s th e e f f e c t of "news", which i s given by th e d i f f e r e n c e between a c t u a l and a n t i c i p a t e d v a r i a b l e s , by r e g r e s s i n g u n a n t i c i p a t e d d e p r e c i a t i o n in exchange r a t e s on news about th e c u r r e n t acc o u n t, c y c l i c a l demand f a c t o r s , and i n t e r e s t r a t e s . All of the 'monetary* and a s s e t models g e n e r a l l y p r e s e n t a framework d e s c r i b i n g how monetary or a s s e t e q u il i b r i u m can be r e s t o r e d . The d i f f e r e n t frameworks adopt e i t h e r a sto c k or a flow e q u ilib riu m approach. As i n d i c a t e d in K o u ri(4 4 ), th e d i f f e r e n c e between th e two ap pro aches i s an e m p ir ic a l one. Some models focu s upon sto c k e q u il i b r i u m and c o n c e n tr a te on th e mechanisms through which i n c i p i e n t 13 c a p i t a l flows a re e l i m in a te d . One such mechanism works through changing r e a l money b a la n c e s v ia exchange r a t e - i n d u c e d p r i c e l e v e l v a r i a t i o n s . Changes in r e a l money b a la n c e s induce f u r t h e r a d ju s tm e n ts in a g e n t s ’ p o r t f o l i o h o ld in g s c r e a t i n g o f f s e t t i n g c a p i t a l flows e q u iv a le n t to the n e c e ssa ry sto c k a d ju s tm e n ts needed to r e s t o r e e q u il i b r i u m . K o u ri’ s paper (44) shows th e r o l e of momentary or s h o r t - r u n a s s e t e q u il i b ri u m , and th e p ro ce ss of a s s e t accu m u latio n in th e l o n g - r u n , in d e te rm in in g exchange r a t e s . The second type of model t h a t d e s c r ib e s th e p ro ce ss through which a s s e t m arkets c l e a r fo c u s e s upon flow e q u ilib r iu m and views th e e q u ilib r iu m r a t e of exchange as t h a t which ■ makes th e c a p i t a l account balan ce the c u r r e n t account in th e balance of payments. In t h i s type of a n a l y s i s , a p a r t i a l e q u ilib r iu m framework i s u s u a lly chosen due to the speed of r e a c t i o n in th e c a p i t a l account compared to th e c u r r e n t acc o u n t. However, as Kouri p o in ts o u t, a l o g i c a l i n c o n s is te n c y would a r i s e i f one assumes i n s t a n t a n e o u s p o r t f o l i o e q u ilib r iu m because the balan ce of payments e q u a tio n i s an p o st i d e n t i t y and not an ejc a n te e q u ilib riu m c o n d it i o n . The way around t h i s problem i s to assume t h a t t r a n s a c t i o n c o s t s p r o h i b i t in s ta n t a n e o u s p o r t f o l i o a d ju s tm e n ts , or t h a t i n v e s t o r s have p e r f e c t 14 f o r e s i g h t . A t y p i c a l a s s e t market model of t h i s s o r t would s p e c if y a demand f u n c t i o n f o r r e a l b a la n c e s, a nominal money supply f u n c tio n made up of dom estic c r e d i t and f o r e i g n r e s e r v e s or an exogenously determ ined money supply, and a law of motion f o r the exchange r a t e as a f u n c t i o n of the balance of payments of a sm all country vs. a l a r g e country or the r e s t of the w orld. More r e c e n t l y , exchange r a t e th eo ry has been extended to r e i n t r o d u c e the c u r r e n t account as an im p o rta n t d e te rm in a n t of exchange r a t e s such as in Dornbusch and F is c h e r (1 7 ) and, as a consequence, u t i l i z e s th e flow e q u ilib riu m framework. The v a rio u s exchange r a t e models have been e m p i r i c a l l y examined u sin g d a ta from e p is o d e s of f l e x i b l e exchange r a t e regim es. I f we d iv id e them i n to t h r e e c a t e g o r i e s : m o n e ta ry -p u rc h a s in g power p a r i t y models, p o r t f o l i o b alan ce models, and t r a d e models, we f i n d t h a t ample evidence has been produced in su p p o rt of the p o r t f o l i o balance model and th e view t h a t the c u r r e n t account m a t t e r s in e x p la in i n g f l u c t u a t i o n s i n exchange r a t e s . The evid en ce, however, has not been c o n s i s t e n t w ith th e monetary - p u rch a sin g power p a r i t y model. The r e s u l t s from e a r l i e r s t u d i e s by ^ Dornbusch, in "Exchange Rate Economics: Where Do We S t a n d ? " (15) , review s th e evidence f o r the v a r io u s v e r s i o n s of the monetary model and h i s f i n d in g s r e f l e c t th e inadequacy of the s t r i c t l y m o n etary -p u rc h asin g power p a r i t y models in e x p la in i n g f l u c t u a t i o n s in exchange r a t e s . F u r th e r evidence i s c i t e d in B i l s o n 's paper 15 B i l s o n ( 7 ) , F r a n k e l( 2 4 ) , F r e n k e l( 2 5 ) , and Hodrick(41) show t h a t the change in th e e q u ilib r iu m exchange r a t e can be b a s i c a l l y decomposed i n t o v a r i a t i o n s stemming from fou r f a c t o r s : changes in th e r e l a t i v e r a t e s of growth in th e money s u p p l i e s of th e two c o u n t r i e s , changes in th e r e l a t i v e income growth r a t e s , changes in th e d i f f e r e n c e in i n f l a t i o n a r y e x p e c ta ti o n s , and v a r i a t i o n s in th e i n t e r e s t r a t e d i f f e r e n t i a l . The f i r s t t h r e e f a c t o r s would have lo n g -te rm e f f e c t s on th e exchange r a t e , w hile the l a s t component would have a s h o r t - t e r m e f f e c t . The i n f lu e n c e of the u n a n t i c i p a t e d components in th e s e f a c t o r s on u n a n t i c i p a t e d movements in exchange r a t e s were i n v e s t i g a t e d by DornbuschC15). His f i n d i n g s can be summarized by the fo llo w in g q u o t a t i o n : The e m p ir ic a l a n a l y s i s confirm s t h a t u n a n t i c i p a t e d r e a l and f i n a n c i a l d i s t u r b a n c e s b r in g about unexpected movements in th e exchange r a t e . Whether the s i z e of exchange r a t e movements sta n d s in r e a s o n a b le r e l a t i o n to th e d i s t u r b a n c e rem ains an open q u e s t io n . C le a rly the answer d e p e n d s ...o n th e s t r u c t u r a l p a r a m e t e r s . . . ( 1 5 , p. 163) In th e same paper, Dornbusch a ls o examined th e r o l e of p o r t f o l i o e f f e c t s in th e c o n te x t of im p e rf e c t a s s e t s u b s t i t u t a b i l i t y . By examining th e v a r i a n c e s and c o v a r ia n c e s of r e a l r e t u r n s on d i f f e r e n t c u r r e n c i e s , he "Recent Developments i n Monetary Models of Exchange Rate D e t e r m in a t io n " ( 9 ). 16 concluded t h a t " th e r e s u l t i n g p o r t f o l i o s h i f t s or c a p i t a l flows account f o r some of the u n a n t i c i p a t e d a p p r e c i a t i o n in the Deutschmark." (15, p. 166) In any c ase , th e evidence should be viewed w ith c a u tio n sin c e most of th e econom etric p ro c e d u re s employed a re s u b j e c t to s e r i o u s o b j e c t i o n s t h a t have been r a i s e d by econom ists such as L u ca s(4 8 ), Sims(69) and Leamer(47) in th e e v a l u a t i o n of e x i s t i n g s t r a t e g i e s f o r a n a ly z in g macroeconomic m odels. The main o b j e c t i o n r a i s e d i s th e use of a p r i o r i s p e c i f i c a t i o n r e s t r i c t i o n s on th e model s t r u c t u r e . Such r e s t r i c t i o n s , i f not su p p o rted by the d a ta , may produce b ia s e d e s t i m a t e s t h a t may le a d to th e drawing of i n c o r r e c t i n f e r e n c e s , and models w ith poor p r e d i c t i v e a b i l i t y . S t a t i s t i c a l models used in e x p la in i n g the dynamics of exchange r a t e s o f te n e s t im a t e a reduced form of th e model d e riv e d from a s s e t demand e q u a tio n s based on supposed b e h a v io r a l s t r u c t u r a l r e l a t i o n s h i p s . The f a c t i s , though, t h a t th e demand e q u a tio n s (assumed to be s t a b l e ) a re s p e c i f i e d on an ad hoc b a s i s . To the e x te n t t h a t the a p r i o r i r e s t r i c t i o n s used in s p e c i f y i n g th e a s s e t demand f u n c t i o n s a re i n c o r r e c t , th e i n f e r e n c e s drawn from such models a re b ia s e d . The i s s u e of model s p e c i f i c a t i o n stems from the g e n e ra l " r a t i o n a l e x p e c t a t i o n s " school c r i t i q u e , i n i t i a t e d by 17 L u ca s(4 8 ), of econom etric p o lic y e v a l u a t i o n p ro c e d u re s t h a t in v o lv e e s t im a t i n g economic r e l a t i o n s h i p s or d e c i s i o n r u l e s t h a t a re based on a g e n t s 1 b ehav ior in th e m arket, w ith o u t e i t h e r s p e c i f y i n g th e o b j e c t i v e f u n c t i o n and c o n s t r a i n t s of the a g e n ts or a tte m p tin g to i d e n t i f y th e e f f e c t of t h e i r p a ra m eters on th e p a ra m e te rs of the reduced form e q u a tio n s . Often i t i s th e s e l a t t e r " re d u c e d ” form e q u a tio n s t h a t a re t r e a t e d as th e s t u c t u r a l e q u a tio n s in econ om etric models. I t i s g e n e r a l l y a cc ep ted t h a t agents* b eh av ior v a r i e s w ith changes in th e environm ent th ere b y a f f e c t i n g t h e i r d e c i s i o n s . T h e re fo re , in i n t e r p r e t i n g time s e r i e s d a ta , models which assume t h a t a g g re g a te p a ra m eters a re i n v a r i a n t cannot be e xp ected to pro vide good p r e d i c t i o n s under a new environm ent. F urtherm ore, th e p r i o r r e s t r i c t i o n s imposed on such models, such as th e n e u t r a l i t y of money or the assum ption of i d e n t i c a l money demand f u n c t i o n s in d i f f e r e n t c o u n t r i e s , may not be j u s t i f i e d by e m p ir ic a l e v id e n ce . This problem does not go u n n o tic e d by econom ists working on exchange r a t e t h e o r i e s as th e p rev io u s q u o t a t i o n from Dornbusch i n d i c a t e s . However, l i t t l e has been done to c o r r e c t t h i s d e f i c ie n c y . A second r e l a t e d problem in v o lv e s th e f a c t t h a t most of the e m p iric a l s t u d i e s based on p e rio d s of f l e x i b l e exchange r a t e s presume the models to be s t r u c t u r a l l y s t a b l e . A ctual 18 e v e n ts d u rin g th e s e p e rio d s o f t e n imply th e p o s s i b i l i t y of s t r u c t u r a l changes. In th e case of Germany, fo r example, where a l a r g e number of th e s e s t u d i e s q u ote evidence based on th e b e h av io r of the mark, the o n -a g a in , o f f - a g a i n c o n t r o l s in th e 1970s n e c e s s i t a t e t e s t i n g f o r p o s s i b l e changing s t r u c t u r e . ^ A f i n a l is s u e has to do w ith s p e c i f y i n g models of f o r e i g n exchange r a t e s u sing a p o s s ib ly over s im p li f y i n g r e s t r i c t i o n on th e number of a s s e t s in a g e n t s ' p o r t f o l i o s . D esp ite the f a c t t h a t some models c o n s id e r monies simply as one type of r is k y a s s e t s (money a s s e t s ) t h a t can be h e ld in an e f f i c i e n t l y d i v e r s i f i e d p o r t f o l i o , they tend to s im p lif y th e a s s e t view d r a s t i c a l l y by assuming t h a t a g e n t s ' p o r t f o l i o s c o n s i s t of only two n a t i o n a l c u r r e n c i e s (K o u ri(4 4 ), Mussa(58), and Dornbusch(13) and ( 1 5 ) ) . The i n c l u s i o n of a t h i r d a s s e t , whether a t h i r d c u rre n cy , s to c k s , bonds or a non-monetary a s s e t such as gold or o i l s to c k s , could le a d to a r i c h e r c h a r a c t e r i z a t i o n and, t h e r e f o r e , to a b e t t e r u n d e rs ta n d in g of th e r e l a t i o n s h i p s ( c a u s a l r e l a t i o n s h i p s in p a r t i c u l a r ) among th e v a r i a b l e s in th e model. Indeed as K ouri(44, p. 280-281) p u ts i t : 3 These s t u d i e s in c lu d e Hodrick (41) where problems w ith i n t e r p r e t i n g e s t i m a t e s were a t t r i b u t e d to in c r e a s e d c o n t r o l s in Germany; B ils o n ( 8 ) , and Branson (1.1)* 19 The essen ce of th e m onetary, or a s s e t m arket, approach to exchange r a t e s . . . i s t h a t the exchange r a t e , as a r e l a t i v e p r ic e of monies, i s viewed as one of th e p r i c e s t h a t e q u i l i b r a t e s th e i n t e r n a t i o n a l m arkets f o r v a rio u s f i n a n c i a l a s s e t s . T h e re fo re , th e behavio r of the s u p p l i e s of v a rio u s monies and o th e r f i n a n c i a l a s s e t s , as w e ll as th e b e hav ior of the demands f o r th e s e a s s e t s , have to be examined in o rd er to e x p la in th e b e h av io r of exchange r a t e s . In consumer theo ry with two commodities th e commodities can only be s u b s t i t u t e s , t h a t i s , th e c r o s s - s u b s t i t u t i o n e f f e c t in th e S lu tsk y e q u a tio n w i l l always be p o s i t i v e . This may not be n e c e s s a r i l y so when a t h i r d commodity i s added to the consumption s e t of the consumer. S i m i l a r l y , w ith th r e e a s s e t s th e changing p r o b a b i l i t i e s of f u t u r e e v e n ts need not imply s u b s t i t u t i o n between any p a r t i c u l a r p a i r of a s s e t s . In th e end th e choice of in c lu d in g more a s s e t s in th e p o r t f o l i o should be an e m p ir ic a l q u e s t io n . S p e c i f i c a t i o n s of e m p ir ic a l models should be t e s t e d and a naly zed i n t i a l l y to see whether they a re r i c h enough to e x p la in th e dynamics in exchange r a t e s or whether a s i g n i f i c a n t p o r t io n of the v a r i a t i o n in th e exchange r a t e rem ains u n e x p la in e d by the in c lu d e d v a r i a b l e s . T h e re fo re , in i t s e l f , th e s i m p li f y i n g assum ption of two a s s e t s draws no o b j e c t i o n a s long as s p e c i f i c a t i o n s e a rc h e s have been perform ed. A d m itted ly , th e th eo ry of s p e c i f i c a t i o n s e a r c h e s have only r e c e n t l y been developed. This may e x p la in why p r e t e s t i n g was not commonly perform ed. The n o v e lty of 20 s p e c i f i c a t i o n s e a r c h e s , however, does not overshadow t h e i r im portance and, in o rd er to reduce s p e c i f i c a t i o n b i a s , e m p ir ic a l s t u d i e s should u n d e rta k e such s e a rc h e s b e fo re p ro ceedin g in a n a ly z in g e m p ir ic a l ev id e n ce . C u r r e n tly , much r e s e a r c h i s being u n d e rta k en aiming a t s u g g e s tin g and im plem enting a l t e r n a t i v e s t r a t e g i e s f o r the fo r m u la tio n of economic m odels. Of i n t e r e s t to us a re the s t r a t e g i e s used in s p e c i f y i n g dynamic models and th e time s e r i e s p r o c e s s e s t h a t g e n e ra te the exogenous v a r i a b l e s . D e s c r ip tio n and im p lem en tatio n of such s t r a t e g i e s can be found in work by S h i l l e r ( 6 7 ) , Leamer(47), Sim s(69), and S a rg e n t and S im s(65 ). These s t u d i e s focu s on e s t i m a t i n g l i n e a r d i s t r i b u t e d l a g m odels. The i n t e r e s t in such models l i e s in the f a c t t h a t , as p o in te d out in S a r g e n t ( 6 3 ) , d i s t r i b u t e d la g models of second or h ig h e r o r d e r s can a c c u r a t e l y g e n e ra te c o r r e l a t i o n s over time t h a t a re e x h i b i t e d by most economic v a r i a b l e s . Sims and S a rg e n t use u n r e s t r i c t e d v e c t o r a u t o r e g r e s s i v e models f o r a n a ly z in g c o r r e l a t i o n s among economic v a r i a b l e s using no p r i o r in f o r m a tio n . Learner and S h i l l e r use B ayesian d i s t r i b u t e d l a g e s t i m a t i o n p ro c e d u re s . In Learner's work p r i o r in f o r m a tio n on th e exp ected v a lu e s and c o v a r ia n c e s of the i n d i v i d u a l la g c o e f f i c i e n t s a re used, w hile S h i l l e r uses p r i o r s on th e smoothness of the p r o c e s s e s r a t h e r than th e i n d i v i d u a l c o e f f i c i e n t s . 21 The advantage of u sin g such p ro ce d u res w ith o u t imposing any a p r i o r i s t r u c t u r e on th e r e l a t i o n s h i p s among economic v a r i a b l e s i s t h a t d ata evidence alo ne i s used to form r e s t r i c t i o n s and s p e c if y a model s t r u c t u r e . Such a s y s te m a tic approach to i d e n t i f y i n g and f o r m u la tin g models t h a t summarize the d ata c a p tu r e s more a c c u r a t e l y the e m p iric a l r e g u l a r i t i e s and i n t e r a c t i o n among th e model v a r i a b l e s . C a u s a lity t e s t s a re used to prov id e a d d i t i o n a l in fo rm a tio n on th e type of r e l a t i o n s h i p s t h a t may e x i s t among th e v a r i a b l e s , i . e . whether c a u sa l or p urely c o r r e l a t e d . 1.2 THE ROLE OF GOLD An i n v e r s e r e l a t i o n s h i p between th e p r i c e of gold and th e perform ance of th e d o l l a r was b e lie v e d to a r i s e only under a f ix e d exchange r a t e regime based on a gold s ta n d a rd or a D o lla r Reserve Currency System w ith the p r i c e of gold f i x e d in d o l l a r s . The B r e tto n Woods agreem ent was based on a Reserve Currency System, w ith the U.S. d o l l a r as th e r e s e r v e c u rre n c y . The U nited S t a t e s had th e o b l i g a t i o n to m a in ta in th e p r ic e of gold a t $32 an ounce w h ile o th e r n a ti o n s were o b lig e d to m a in ta in th e v a lu e s of t h e i r c u r r e n c i e s a t a f i x e d p a r i t y with th e d o l l a r . A U.S. balan ce of payments d e f i c i t would put downward p r e s s u r e on 22 the d o l l a r , and th e U.S. would have to use i t s r e s e r v e s to m a in ta in th e v alu e of the d o l l a r . T his i s e x a c tl y what happened in th e e a r ly 1970s. A growing U.S. e x t e r n a l d e f i c i t t h r e a t e n e d to ex h au st th e r e s e r v e s of the U nited S t a t e s and f i n a l l y le d t o the d e v a lu a tio n of the d o l l a r from $32 to $42 per ounce of gold. A fter th e c o ll a p s e of the B r e tto n Woods agreem ent, i t was h e ld t h a t t h i s in v e r s e r e l a t i o n s h i p should no lo n g e r e x i s t . But, gold, once c a l l e d a "b arb a ro u s r e l i c " by Keynes, remained a "barom eter of w orld a n x i e t i e s , " and c o u n t r i e s m a in ta in e d t h e i r gold r e s e r v e s f o r tim es of c r i s i s . While gold h o ld in g s as a p e rc e n ta g e of th e o f f i c i a l r e s e r v e s of IMF member c o u n t r i e s did not change much, once th e s e c o u n t r i e s sto pp ed v a lu in g gold a t th e old p r i c e of $32 an ounce, the worth of th e s e h o ld in g s i n c r e a s e d more than te n f o l d . In 1979 and 1980 i t grew to tw ice as much as t h e i r currency r e s e r v e s . In June of 1980, gold v a lu e d a t m arket p r i c e s amounted to 94, 79 and 67% of o f f i c i a l r e s e r v e s of the U nited S t a t e s , S w itz e rla n d and West Germany r e s p e c t i v e l y . These gold r e s e r v e s were not i d l e , but r a t h e r were p la y in g an i n c r e a s i n g r o l e in i n t e r n a t i o n a l s e t t l e m e n t s and s t a b i l i z a t i o n p o lic y . The f a i l u r e in denying gold a monetary f u n c t i o n was dem on strated in th e l a t e 70s by the European Monetary System, which r e q u i r e d i t s members to 23 {convert 25% of t h e i r gold r e s e r v e s i n t o ecus (European cu rre n cy u n i t s ) , and by th e U nited S t a t e s , which used i t s gold r e s e r v e s to h e lp cover i t s t r a d e d e f i c i t . This f a i l u r e can be s a id to have been f u e l l e d by r i s i n g i n f l a t i o n and i n t e r n a t i o n a l t e n s i o n , and s in c e gold has proven to be a b e t t e r s t o r e of v a lu e , major bankers and i n d i v i d u a l s were a c q u i r in g more of i t . The gold c r i s i s in l a t e 1979 su p p o rts t h i s view f u r t h e r w ith two major i n t e r n a t i o n a l c r i s e s o c c u rin g j u s t one month b e fo re the d ram a tic surge in th e p r i c e of gold a g a i n s t a l l major c u r r e n c i e s : the ta k in g of American h o s ta g e s by I r a n i a n m i l i t a n t s , and th e S o v ie t in v a s io n of A fg h a n is ta n . The above e x p o s i t i o n i l l u s t r a t e s t h a t gold p la y s an im p o rta n t r o l e in i n v e s t o r s 1 p o r t f o l i o s as w e ll as in th e o f f i c i a l r e s e r v e s of c e n t r a l banks. Given s t a b l e economic, s o c i a l and p o l i t i c a l c o n d i t i o n s , i n v e s t o r s tend to hold more paper c u r r e n c i e s due to th e h ig h e r r a t e of r e t u r n . However, once such s t a b i l i t y i s t h r e a t e n e d , i n v e s t o r s tend to f l e e paper c u r r e n c i e s to th e s e c u r i t y of r e a l a s s e t s w ith gold having th e g r e a t e s t a t t r a c t i o n f o r th e same q u i t e f a m i l i a r re a so n s t h a t made i t th e most a t t r a c t i v e commodity money. i 24 1 .3 SCOPE OF THE STUDY In t h i s stu d y , th e dynamics of the f o r e i g n exchange m arket 'are examined from two d i f f e r e n t , y e t r e l a t e d , p e r s p e c t i v e s . In both models, th e q u e s t i o n s a d d re ss e d concern th e d e te r m in a n ts of exchange r a t e s . More s p e c i f i c a l l y , we aim to f i n d out to what e x t e n t th e b e h a v io r a l b a s i s of p o r t f o l i o b alan ce th eo ry can e x p la in th e observed f l u c t u a t i o n s in exchange r a t e s , and what r o l e the gold m arket p la y s in t h i s p ro c e s s . The t h e o r e t i c a l b a s i s used in th e a n a l y s i s in v o lv e s f o r m u la tin g a dynamic a s s e t model based on th e assum ptions t h a t i n v e s t o r s d is p la y o p tim iz in g b e h av io r and form ’’r a t i o n a l e x p e c t a t i o n s ” in choosing among d i f f e r e n t a s s e t s , monetary and o th e r w is e . Because of th e l a r g e sh are t h a t gold h o ld s in i n t e r n a t i o n a l p o r t f o l i o s and o f f i c i a l r e s e r v e s , i t i s s p e c i f i c a l l y in c lu d e d in th e l i s t of a s s e t s . By i n tr o d u c in g th e assum ption of r a t i o n a l i t y on th e p a r t of i n v e s t o r s , t h i s study overcomes th e ad hoc n a tu r e of th e e x i s t i n g p o r t f o l i o models f o r exchange r a t e s . This i s so s in c e no a p r i o r i assum ptions a re made about th e a c t u a l p r o c e s s e s , except t h a t th e s e s t o c h a s t i c p r o c e s s e s a re c o n s i s t e n t w ith i n v e s t o r s ’ f o r e c a s t s of the r e l e v a n t f u t u r e v a r i a b l e s . The p a ra m e te rs of th e d e c i s i o n r u l e s d e riv e d from the dynamic o p t i m i z a t i o n problem are f u n c t i o n s 25 of p a ra m eters of a g e n t s ' o b j e c t i v e f u n c t i o n s , the s t o c h a s t i c p ro c e s s e s governing th e exogenous v a r i a b l e s in th e model, and th e a g e n t s ' c o n s t r a i n t s . Sim ultaneous e s t i m a t i o n of th e s t o c h a s t i c p r o c e s s e s , c o n s t r a i n t s , and r u l e s of choice , w ith m arket c l e a r i n g c o n d it i o n s and th e r a t i o n a l e x p e c ta ti o n s h y p o th e s is p r o v id in g th e c r o s s - e q u a t i o n r e s t r i c t i o n s , le a d s to th e i d e n t i f i c a t i o n of th e d i f f e r e n t p a ra m e te rs . I d e n t i f i c a t i o n of the p a ra m e te rs as such would make i t p o s s i b l e to p r e d i c t how a g e n t s ' demand f u n c t i o n s would change along w ith i i A l t e r a t i o n s in t h e i r environm ent, and y e t have c o nfid ence t h a t th e f i n d i n g s a re not b ia s e d . This method i s not s u b j e c t to L u c a s 's c r i t i q u e of dynamic econ om etric p ro c e d u re s which in th e c o n te x t of p o r t f o l i o s e l e c t i o n can be i n t e r p r e t e d as f o l lo w s . I f p r i v a t e a g e n ts have some degree of r a t i o n a l i t y , th e a g e n t s ' r u l e s of c h o ic e, i . e . , t h e i r dynamic a s s e t demand f u n c t i o n s , should vary c o n s i s t e n t l y with changes in th e c o n s t r a i n t s f a c i n g them. In r e a l i t y , t h i s kind of b e h av io r on th e p a r t of i n v e s t o r s in a s s e t m arkets cannot escape d e t e c t i o n . The main problem in th e use of such te c h n iq u e s in m o d e llin g i s t h a t i t i s not o f t e n p o s s i b l e to d e r iv e e x ac t c lo s e d form s o l u t i o n s fo r the d e c is io n r u l e s . Unless th e o b j e c t i v e f u n c t i o n a l has a q u a d r a t i c form, th e com putations 26 jare e x ce ed in g ly co m p lica te d and s t a b l e s o l u t i o n s in g e n eral do not e x i s t . In few c ase s, and only under sev ere r e s t r i c t i o n s on th e form of the o b j e c t i v e f u n c t i o n and d i s t r i b u t i o n of exogenous v a r i a b l e s in th e system (which may be no b e t t e r an appro xim ation of th e t r u e model than th e q u a d r a t i c c a s e ) , can a s o l u t i o n be d e riv e d ( M erton(53), and Hakansson(33)) . For th e econom etric model , we so lv e f o r the a s s e t demand f u n c t i o n s by r e s t r i c t i n g th e o b j e c t i v e f u n c t i o n a l to a q u a d r a t i c form. We use th e demand f u n c t i o n s t o determ ine th e e q u ilib riu m r a t e of exchange between two c u r r e n c i e s when i n v e s t o r s may hold e i t h e r of th e s e c u r r e n c i e s and gold in t h e i r p o r t f o l i o s . P r i o r to e s t i m a t i n g th e r e s t r i c t e d s t r u c t u r a l model we e s t im a te an u n r e s t r i c t e d m u l t i v a r i a t e time s e r i e s model of f l e x i b l e exchange r a t e s f o llo w in g th e work done by S im s(69). The model v a r i a b l e s in c lu d e th o se i n d i c a t e d in th e p o r t f o l i o model to have an i n f lu e n c e on exchange r a t e s . The e s tim a te d v e c to r a u t o r e g r e s s i o n s a re t e s t e d f o r s t a b i l i t y over tim e, then they are used to s im u la te the system response to random shocks. In a d d i t i o n , f o r e c a s t s of a l l the model v a r i a b l e s a re g e n e ra te d , and th e f o r e c a s t e r r o r v a r i a n c e s a re decomposed i n t o o rth o g o n a l in n o v a tio n s . The in f o r m a tio n from th e s im u la tio n s and deco m position of 27 the f o r e c a s t e r r o r v a r i a n c e s i s im p o rta n t sin c e i t i n d i c a t e s w hether the d a ta , w ith o u t any a p r i o r i r e s t r i c t i o n s , su p p o rts th e m a in ta in e d h y p o th e se s used to f o r m u la te p o r t f o l i o models, and p ro v id e s th e r e q u i s i t e s p e c i f i c a t i o n se arc h and p r e t e s t i n g f o r th e r e s t r i c t e d s t r u c t u r a l model. The o r g a n i z a t i o n of th e rem ainder of t h i s study i s as f o llo w s . In Chapter I I , a g e n e ra l f o r m u la tio n of the dynamic p o r t f o l i o problem w ith cu rrency d i v e r s i f i c a t i o n and u n c e r t a i n t y about r a t e s of r e t u r n ( t h a t i s , w ith im p e r f e c t s u b s t i t u t i o n among a s s e t s ) i s developed, and i t s p r o p e r t i e s a re a n a ly z e d . In Chapter I I I , an u n r e s t r i c t e d dynamic model of exchange r a t e s u sin g s t r u c t u r e - f r e e v e c to r a u t o r e g r e s s i o n s i s e s tim a t e d . Follow ing p r e lim in a r y t e s t s of the model s p e c i f i c a t i o n co n ce rn in g th e la g le n g th and s t r u c t u r a l s t a b i l i t y , the e s tim a te d model i s used to c a rry out a time s e r i e s f a c t o r a n a l y s i s on th e model v a r i a b l e s . In Chapter IV, a p a r t i c u l a r sim ple f o r m u la tio n of the s t r u c t u r a l p o r t f o l i o model i s used to d e riv e e x p l i c i t l y the r e g r e s s i o n model. The c h a p te r a ls o c o n ta in s a d e s c r i p t i o n of th e e s t i m a t i o n methodology and an a n a l y s i s of the r e s u l t s . F i n a l l y , Chapter V c o n ta in s th e summary and c o n c lu s io n s drawn from t h i s stu d y . 28 Chapter I I THE PORTFOLIO MODEL In t h i s c h a p t e r a dynamic a s s e t m arket model fo r th e d e te r m in a tio n of exchange r a t e s i s developed w ith the f o llo w in g main f e a t u r e s : - The a s s e t demand e q u a tio n s a re d e riv e d from a p o r t f o l i o s e l e c t i o n problem u sin g dynamic microeconomic o p t i m i z a t i o n in o rd e r to p ro p e rly c h a r a c t e r i z e th e e m p ir ic a l exchange r a t e dynamics. The use of dynamic o p t i m i z a t i o n t e c h n iq u e s e n a b le s i | one to i n t e r p r e t f o r e i g n exchange time s e r i e s in a I manner which allow s a g e n t s ' b e h av io r and d e c i s i o n I I r u l e s t o be i n f lu e n c e d by the c o n s t r a i n t s f a c i n g them . - The system of demand e q u a t io n s d e riv e d i s in g e n e ra l form in term s of n a s s e t s . This g r a n t s th e f l e x i b i l i t y to study the e f f e c t s of any type and any number of a s s e t s on exchange r a t e s . A main goal of t h i s r e s e a r c h i s to i n v e s t i g a t e th e r o l e of gold, a n on-m onetary, n o n - i n t e r e s t b e a rin g a s s e t , in th e f o r e i g n exchange m arket. The i n c l u s i o n of 29 non-monetary a s s e t s in a p o r t f o l i o model of exchange r a t e s w i l l b e t t e r i l l u s t r a t e a fundam ental a s p e c t of the p o r t f o l i o view, namely, a s s e t s u b s t i t u t a b i l i t y . Since the exchange r a t e between two n a t i o n a l monies i s a s s e r t e d to be simply the r e l a t i v e p r i c e of two monies, such a r e l a t i v e p r i c e w i l l be g r e a t l y i n f lu e n c e d by changes in th e r e l a t i v e demands f o r a l l a s s e t s . - The r a t i o n a l e x p e c t a t i o n s h y p o t h e s is i s used as an e q u ilib r iu m c o n d itio n which p ro v id e s th e n e c e ssa ry i d e n t i f i c a t i o n r e s t r i c t i o n s on th e model p a ra m e te rs . In g e n e r a l, p o r t f o l i o models prov ide a t h e o r e t i c a l framework f o r d e s c r i b i n g th e d e c i s i o n making p ro c e ss through which i n v e s t o r s s e l e c t t h e i r a s s e t h o ld in g s from a d i v e r s i t y of a s s e t s w ith u n c e r t a i n r e t u r n s . I n v e s t o r s a re u s u a lly assumed to be o p tim iz in g a g e n ts who aim a t maximizing t h e i r ex pected u t i l i t y given t h e i r i n i t i a l w e alth . I n v e s t o r s ’ e x p e c ta ti o n s r e g a r d i n g th e f u t u r e r a t e s of r e t u r n on th e v a rio u s a s s e t s , as w e ll as o th e r p e r t i n e n t in fo r m a tio n , would e n te r i n t o th e d e c i s i o n r u l e s . In p a r t i c u l a r , an i n c r e a s e i n th e ex p ec te d r a t e of r e t u r n on j an a s s e t r e l a t i v e to o th e r a s s e t s in th e p o r t f o l i o , o th e r J th in g s c o n s t a n t , i n c r e a s e s th e sh are of t h a t a s s e t in th e j optim al p o r t f o l i o . In p o r t f o l i o models of f o r e i g n exchange 30 r a t e s d i f f e r e n t c u r r e n c i e s se rv e as th e a s s e t s t h a t may be i n c lu d e d in a d i v e r s i f i e d p o r t f o l i o , and exchange r a t e s a re the r e l a t i v e p r i c e s of th e d i f f e r e n t "money1 ' a s s e t s . Hence, th e r e l a t i v e r a t e s of r e t u r n on d i f f e r e n t c u r r e n c i e s , or on a s s e t s denominated in d i f f e r e n t c u r r e n c i e s , would i n f l u e n c e th e r e s p e c t i v e exchange r a t e s v ia t h e i r e f f e c t on th e r e l a t i v e demand f o r th ese c u r r e n c i e s . Other demand f a c t o r s and th e s to c k s of th ese c u r r e n c i e s w i l l a ls o be of i n f lu e n c e . The p o r t f o l i o model developed h e re p ro v id e s a theo ry of the behavior of economic a g e n ts in choosing t h e i r p o r t f o l i o a s s e t h o ld in g s u sin g a g e n e ra l form of th e u t i l i t y f u n c t i o n . In th e s e c t i o n t h a t f o llo w s , th e model w i l l be d e s c r ib e d , and th e assu m p tio n s used w i l l be s t a t e d . S e c tio n 2 .2 w i l l c o n ta in an a n a l y s i s of e q u ilib r iu m in th e f o r e i g n exchange m arket, as w e ll as of the p r o p e r t i e s of t h i s model, w h ile e m p ir ic a l s t r a t e g i e s f o r t e s t i n g th e s e p r o p e r t i e s w i l l be d is c u s s e d in S e c tio n 2 . 3 . 31 2.1 THE GENERAL MODEL A r e p r e s e n t a t i v e a g ent i s assumed to choose a co ntin gen cy p lan f o r h i s p o r t f o l i o a s s e t h o ld in g s as i f he were maximizing h i s ex p ec te d u t i l i t y of l i f e time consumption where h i s l i f e t i m e i s assumed to i n f i n i t e . The a s s e t s i n t h i s p o r t f o l i o can be of v a rio u s cu rren cy den om in atio ns, monetary and o th e r w is e . The o b j e c t i v e f u n c t i o n i s < OQ (2 .1 ) Max E0 lim L 0fc U(Ct ) {ct ,T t f N ^ oo t=0 where U (.) i s th e one p e rio d u t i l i t y f u n c t i o n assumed to be m o n o to n ic a lly i n c r e a s i n g , s t r i c t l y concave and tw ice d i f f e r e n t i a b l e . C le a rly t h i s i m p lie s r i s k a v e r s io n on th e p a r t of th e i n v e s t o r sin c e U! >0 and U"<0. i s an a p p r o p r i a t e d is c o u n t f a c t o r CK/3<1. Given i n i t i a l a s s e t h o ld in g s T_-| , th e ag en t chooses th e consumption l e v e l in each p e rio d , Ct , to s a t i s f y the fo llo w in g budget c o n s t r a i n t : (2 .2 ) C = Y + R ' T - J ' ( T - T )- K( T -T „) t t t t t t t-1 t t-1 where Ct r e p r e s e n t s th e amount of a composite good th e ag en t consumes in p e rio d t Y +. n o n - c a p i t a l stream of income C 1 2 n T^ = (Tt Tt . . T t ) ’ an nx1 v e c to r of a s s e t s h e ld from t to t+1 where th e i th a s s e t i s denominated in country i ' s c u r rency 32 X 2 n Rt = (r r ..r )' an nxl v e c to r of e a rn in g s w ith t t t th e i t h ei ement r e p r e s e n t i n g th e th e r e t u r n on a u n i t of th e i a s s e t in r e a l terms n J t = J+- **J t an nx1 v e c to r of> a s s e t p r i c e s e x p re s s e d in r e a l term s K (.) c o s t of ad ju stm en t f u n c t i o n Et i s th e m ath em atical e x p e c t a t i o n s o p e r a t o r u sin g a l l a v a i l a b l e in f o r m a tio n in th e t th p e rio d A ll of the above v a r i a b l e s a re norm alized in such a way t h a t th e p r i c e of a u n i t of the consumption good in dom estic cu rren cy i s one. When an a s s e t i s a f o r e i g n bond, i t s p r i c e w i l l simply be th e exchange r a t e between th e dom estic and f o r e i g n c u rre n c y . The above budget c o n s t r a i n t s t a t e s t h a t i n each p e rio d th e agent a l l o c a t e s h i s w e alth which c o n s i s t s of h i s a s s e t h o ld in g s from th e p rev io u s p e rio d , e a r n in g s from h i s a s s e t h o ld in g s , and an exogenous stream of income, among consumption and in v estm en t in r is k y a s s e t s . In t h i s c o n te x t, r i s k r e f e r s to th e u n c e r t a i n r e t u r n on th e a s s e t s . In th e case of th o se a s s e t s denominated in a f o r e i g n c u rre n c y , th e u n c e r t a i n t y about th e r e t u r n a r i s e s from two s o u r c e s : u n c e r t a i n t y about th e r a t e of r e t u r n , and u n c e r t a i n t y about th e r a t e of exchange between th e dom estic and f o r e i g n c u r r e n c i e s . In th e case where p u rc h a sin g power p a r i t y h o ld s, one a s p e c t of th e r i s k a s s o c i a t e d w ith the u n c e r t a i n t y about f u t u r e exchange r a t e s 33 w i l l be red u ced, namely th e p u rc h a sin g power r i s k of m o n ie s .2 * T h e re fo re , th e u n c e r t a i n t y i s only with r e s p e c t to r e a l r a t e s of r e t u r n . In f o r m u la tin g t h i s model s e v e r a l assum ptions a r e made. F i r s t , we assume t h a t a g e n ts cannot i n f lu e n c e th e p r i c e s nor the r a t e s of r e t u r n on f o r e i g n a s s e t s i n c l u d i n g gold, and t h e r e f o r e , th e s e a re t r u l y exogenous. The p r i c e s and r a t e s of r e t u r n on dom estic a s s e t s a r e m arket d ete rm in e d . This assum ption i s s i m i l a r to th e sm all country assum ption o f t e n used i n exchange r a t e models. Moreover, a g e n ts view th e r e t u r n s on th e v a rio u s a s s e t s , as w e ll as a s s e t p r i c e s and exchange r a t e s , to fo llo w given p r o c e s s e s t h a t a re not a f f e c t e d by t h e i r h o ld in g d e c i s i o n s . I f we l e t J.J. and Rj. be the f i r s t 2n e le m e n ts of a v e c to r a u t o r e g r e s s i v e p ro c e ss X ^. then t h e i r laws of motion can be d e s c r ib e d by ( 2 .3 ) «(L )X t = Ut q where a(L ) = I - «L - . . . - «nL •1 ^ a i s a pxp m a trix of c o n s t a n t s k L i s th e l a g o p e r a t o r . * * For f u r t h e r d i s c u s s i o n of f o r e ig n exchange r i s k , see Fama and F a r b e r ( 2 3 ) . 34 The a u t o r e g r e s s i v e s t o c h a s t i c p ro c e ss d e s c r ib e d by (2 .3 ) i s assumed to be a q th o rd er p ro c e ss of mean e x p o n e n tia l order l e s s th an (\/~/3 ) “ " * . The o th e r p-2n e le m e n ts of X ^. a re v a r i a b l e s t h a t h e lp in p r e d i c t i n g J t and The assum ption on th e mean e x p o n e n tia l o rd er of the p ro c e s s i s a n e c e ssa ry one f o r i n s u r i n g th e i n v e r t i b i l i t y and, t h e r e f o r e , the s t a b i l i t y of the p r o c e s s , w hile b eing a l e s s s t r i n g e n t assum ption than t h a t r e q u i r i n g th e r o o t s of th e m a trix polynom ial a ( L ) to be o u t s i d e th e u n i t c i r c l e . An a d d i t i o n a l assum ption i s t h a t E [Ut ] = 0 and E [U U'] = O t t t where i s a p o s i t i v e s e m i - d e f i n i t e m a tr ix f o r a l l t . A s i m p l i f y i n g assum ption i s added which s t i p u l a t e s t h a t a g e n ts from country 1 p urch ase f o r e i g n a s s e t s t o h o ld in t h e i r p o r t f o l i o s w h ile a g e n ts from th e f o r e i g n c o u n t r i e s do not h o ld a s s e t s denominated in country 1 fs c u rre n c y . This assum ption should not i n f l u e n c e th e f i n d i n g s s in c e i t i s c o n s i s t e n t , and no more r e s t r i c t i v e than th e e x o g en e ity of f o r e i g n v a r i a b l e s or th e sm all country assum ption. Let us i n i t i a l l y so lv e th e problem f o r th e case where a d ju s tm e n t c o s t s do not e x i s t , i . e . K (.)= 0 . In o rd e r to o b t a i n a maximum, th e E uler c o n d i t i o n s , which would be 35 o b ta in e d by d i f f e r e n t i a t i n g e q u a tio n (2 .1 ) p a r t i a l l y w ith r e s p e c t to Tt must be s a t i s f i e d , so t h a t f o r the i t h a s s e t = 0 (2 .4 ) ^ U«(C ) ( J - R ) t t t t + Et | g t+1 U'(Ct + 1 ) J t + , s u b j e c t t o th e boundary c o n d it i o n s given by i n i t i a l a s s e t h o l d in g s , T_^ , and th e t r a n s v e r s a l i t y c o n d itio n where no b e q u e a th a l of w e alth a t death i s allo w ed, t h a t i s , = 0 : (2 .5 ) -BN U '(C n) ( J n - R„ ) = 0 Before p ro ce ed in g t o so lv e t h i s problem we can v e r i f y t h a t i n t e r i o r maxima would be o b ta in e d given the assu m p tio ns on th e u t i l i t y f u n c t i o n to be s t r i c t l y concave and tw ice d i f f e r e n t i a b l e . We should a ls o c l a r i f y th e meaning of the dynamic p o r t f o l i o a l l o c a t i o n problem. Our b a s ic assum ption was t h a t th e i n v e s t o r has to choose a c o nting ency plan f o r h i s a s s e t h o ld in g s < T+- > over an < *t=0 N -perio d h o r iz o n , where N i s n ot f i x e d . At t=0, the i n v e s t o r chooses CQ and T q knowing R q , J q ^ but not knowing what t h e i r f u t u r e v a lu e s w i l l tu rn out to be. The next p e rio d R-| and w i l l be known and based on t h i s in f o r m a tio n th e i n v e s t o r w i l l choose T^. However, a t t=0__T^ w i l l not be known f o r c e r t a i n because i t w i l l be c o n tin g e n t upon v a r i a b l e s unknown a t th e t h a t tim e. Hence we say t h a t th e i n v e s t o r w i l l choose a co n tin g en cy p lan in which an 36 o ptim al d e c i s i o n i s s e l e c t e d in each p e rio d based on in f o r m a tio n a v a i l a b l e up to t h a t p e rio d such as th e outcomes from p re v io u s d e c i s i o n s and e x p e c t a t i o n s about f u t u r e v a r i a b l e s - The dynamic d e c i s i o n r u l e s given by (2 .4 ) d e f i n e n e q u a t io n s , one f o r each of the a s s e t s , where the i t h e q u a tio n can be i n t e r p r e t e d as f o llo w s . The f i r s t term i s th e c o s t of buying a u n i t of the i t h a s s e t i n term s of foreg on e c u r r e n t consumption, and t h i s should be eq ual to th e ex pected ( d is c o u n te d ) f u t u r e consumption b e n e f i t from the a p p r e c i a t i o n in th e v a lu e of the a s s e t . Another way of i n t e r p r e t i n g th e dynamic p o r t f o l i o a l l o c a t i o n problem in a s t o c h a s t i c sense was su g g e ste d by Grossman and S h i l l e r ( 3 2 ) . N otice t h a t th e E u ler e q u a tio n s may a ls o be w r i t t e n as : (2.6) E BU»(Ct+ 1 ) J t+1 U '( C t ) ( J_- R J. = 1 t t Hence th e o ptim al p o r t f o l i o a l l o c a t i o n r e q u i r e s t h a t the e x p ec te d p ro d u c t of th e u n c e r t a i n r e t u r n and u n c e r t a i n m arg in al r a t e of s u b s t i t u t i o n in consumption i s u n i ty . T h e re fo re , under u n c e r t a i n t y , th e e x p e c te d r e t u r n on an a s s e t does not n e c e s s a r i l y have to be equal to th e s u b j e c t i v e r a t e of time p r e f e r e n c e , nor should i t be the 37 same f o r a l l a s s e t s . Equation (2 .6 ) im p lie s t h a t th e w eig hted e x p e c t a t i o n of r e t u r n s with th e m argin al r a t e s of s u b s t i t u t i o n as th e w e ig h ts , t h a t i s th e ex pected u t i l i t y from f u t u r e r e t u r n s , should be the same f o r a l l a s s e t s . In o rd e r to d e r i v e t h i s co nting en cy plan we have to so lv e e x p l i c i t l y e i t h e r (2 .4 ) or ( 2 .6 ) s u b j e c t to ( 2 . 5 ) . T his r e q u i r e s form ing a d d i t i o n a l assum p tion s r e g a r d i n g th e form of the u t i l i t y f u n c t i o n , the s t o c h a s t i c p r o p e r t i e s and s t a t i s t i c a l independence of the model v a r i a b l e s . Most dynamic p o r t f o l i o models s p e c if y th e u t i l i t y f u n c t i o n to be one of th o se b e lo n g in g to th e c l a s s o f H yperbolic A bsolute Risk A version (HARA) u t i l i t y f u n c t i o n s . This c l a s s i n c l u d e s among o t h e r s th e c o n s ta n t a b s o l u t e r i s k a v e r s i o n , c o n s ta n t r e l a t i v e r i s k a v e r s io n , q u a d r a t i c , l o g a r i th m i c , and e x p o n e n tia l u t i l i t y f u n c t i o n s as s p e c i a l c a s e s . The main a t t r a c t i o n of the HARA u t i l i t y f u n c t i o n s d e r i v e s from the f a c t t h a t i t i s th e only c l a s s of f u n c t i o n s which y i e l d l i n e a r demand c u rv e s . Both M erton(54) and M ilne(55) p ro v id e s e p e r a t e p ro o fs of t h i s p r o p e r ty . Let us, t h e r e f o r e , c o n s id e r a sim ple form of th e s e u t i l i t y f u n c t i o n s and proceed w ith s o l v i n g th e model in o rd er to study both th e i m p l i c a t i o n s and problem s t h a t may a r i s e in i n t e r t e m p o r a l p o r t f o l i o models w ith c u rren cy d i v e r s i f i c a t i o n and o b se rv e what f u r t h e r s i m p l i f i c a t i o n s 38 may be n e c e ssa ry f o r o b t a i n i n g a s o l u t i o n . Take, for example, the c o n s ta n t r e l a t i v e r i s k a v e r s io n u t i l i t y f u n c tio n U(C^) = 1/y Cy. The i^*1 E uler e q u a tio n w i l l be S u b s t i t u t i n g f o r Cj. and from e q u a tio n (2 .2 ) y i e l d s : V R tT t- 'i'V W r 1 ♦ E tj^t+l[Y t+l+ ^+lT t+l- - w w v ] f ’ 1 " 0 T his i s a n o n - l i n e a r , second o r d e r , s t o c h a s t i c d i f f e r e n c e e q u a tio n . The n o n - l i n e a r i t y a r i s e s from ta k in g th e powers of the term s i n s i d e th e b r a c k e t s as w e ll as from th e second term which in v o lv e s t a k i n g e x p e c t a t i o n s of th e p ro d u c ts of random v a r i a b l e s , th e J ’ s, R 's and T 's . Although i n v e s t o r s view a s s e t p r i c e s to be exogenous th ey a re , n e v e r t h e l e s s , market d eterm ined and, t h e r e f o r e , a re f u n c t i o n s of p a s t a s s e t h o l d in g s . This sim ple e x e r c i s e i l l u s t r a t e s t h a t f o r th e c o n s t a n t r e l a t i v e r i s k a v e r s io n u t i l i t y f u n c t i o n , or indeed any u t i l i t y f u n c t i o n r e p r e s e n t i n g s t r i c t l y convex p r e f e r e n c e s , no e x a c t s o l u t i o n s f o r th e d e c i s i o n r u l e s can be d e riv e d w ith o u t imposing s e v e re r e s t r i c t i o n s on th e s t o c h a s t i c p r o c e s s e s g ov erning t h e b e h av io r of th e exogenous model 39 v a r i a b l e s (see M erto n (53 ), (5 4 ), and H akansson(33) ) . The assu m p tio ns r e q u i r e d would reduce t h i s problem to an u n i n t e r e s t i n g one where the dependence between exchange r a t e s and p a s t a s s e t h o ld in g s would be e l i m in a te d , and c o n se q u e n tly , we would be unable to study th e e f f e c t s of p o r t f o l i o a d ju s tm e n ts on f o r e i g n exchange r a t e s u sing eco n o m etric a n a l y s i s . D e sp ite the f a c t t h a t a s s e t demand f u n c t i o n s cannot be d e riv e d f o r the g e n e ra l p o r t f o l i o model, one can s t i l l a n aly ze e m p i r i c a l l y th e b e h av io r of exchange r a t e s in th e c o n te x t of dynamic p o r t f o l i o th e o r y . In s e c t i o n 2 . 3 , th e a l t e r n a t i v e methods used i n s i m i l a r s i t u a t i o n s w i l l be d i s c u s s e d and th e p a r t i c u l a r s t r a t e g y chosen h e re f o r p ro ce ed in g w ith e s t i m a t i n g a model fo r exchange r a t e s and gold w i l l be d e s c r i b e d . F i r s t , though, we w i l l a n aly ze in th e next s e c t i o n th e p r o p e r t i e s of th e th e g e n e ra l model and i t s i m p l i c a t i o n s c o n ce rn in g e q u il i b r i u m in th e f o r e i g n exchange m arket. 2 .2 RATIONAL EXPECTATIONS EQUILIBRIUM AND THE DYNAMIC PORTFOLIO MODEL So f a r we have only focu sed on t h e p o r t f o l i o s e l e c t i o n problem and on th e d e r i v a t i o n of the demand f u n c t i o n s f o r th e a s s e t s denominated in d i f f e r e n t c u r r e n c i e s . In o rd er to study th e e f f e c t of p o r t f o l i o d i v e r s i f i c a t i o n and 40 im p e r f e c t s u b s t i t u t a b i l i t y among d i f f e r e n t c u r r e n c i e s on exchange r a t e s we have to d e s c r i b e e q u il i b r i u m in th e f o r e i g n exchange m ark et. From th e b alan ce of payments i d e n t i t y we know t h a t the a c t u a l change i n f o r e i g n a s s e t h o ld in g s i s equal to th e c u r r e n t a c c o u n t, whereas th e d e s i r e d changes in f o r e i g n a s s e t h o ld in g s i s d e riv e d from th e p o r t f o l i o s e l e c t i o n problem. In a two co un try s e t t i n g , an e q u il i b r i u m exchange r a t e w i l l be t h a t which e q u a te s th e a c t u a l and d e s i r e d change in th e f o r e i g n c u rre n cy h o ld in g s . Hence, changes in th e f a c t o r s t h a t appear i n th e a s s e t demand f u n c t i o n s cause the exchange r a t e to f l u c t u a t e i n o rd e r to e q u i l i b r a t e the demand ( d e s i r e d change ) and th e supply ( c u r r e n t account ) of the f o r e i g n c u rre n c y . For example, a c o n t r a c t i o n in th e dom estic money supply sends dom estic i n t e r e s t r a t e s upwards in d u c in g an i n c i p i e n t c a p i t a l in flo w . This w i l l cause an a p p r e c i a t i o n in th e v a lu e of the dom estic c u rre n c y . I f the speed of ad ju stm en t in th e a s s e t m ark ets i s h ig h e r than t h a t in th e goods m ark e ts, th e exchange r a t e may even o v e rsh o o t g iv in g r i s e to e x p e c t a t i o n s of a su bsequ ent d e p r e c i a t i o n in th e dom estic cu rren cy once th e e f f e c t of the monetary c o n t r a c t i o n ( and h ig h e r exchange r a t e s ) on th e t r a d e b alan ce i s r e a l i z e d . On th e o th erh a n d , easy monetary p o lic y would g iv e r i s e to 41 e x p e c t a t i o n s of h ig h e r r a t e s of i n f l a t i o n le a d in g to an i n c r e a s e in th e demand f o r r e a l a s s e t s such as gold. This r e s u l t s i n an ou tflo w of c a p i t a l , which i s needed to pay fo r th e d e s i r e d i n c r e a s e in gold h o ld in g s , and a d e p r e c i a t i o n of th e do m estic c u rre n c y . In o rd e r to t e s t th e i m p l i c a t i o n s of p o r t f o l i o th e o ry of exchange r a t e s we have to model e x p l i c i t l y i n v e s t o r s ' e x p e c t a t i o n s about a s s e t p r i c e s and r a t e s of r e t u r n . This i s n e c e s s a ry sin c e th e a s s e t demand f u n c t i o n s depend on th e s e e x p e c t a t i o n s . By assuming t h a t a g e n ts form t h e i r e x p e c t a t i o n s r a t i o n a l l y , two aims a re se rv e d . The r a t i o n a l e x p e c t a t i o n s assum ption p ro v id e s th e n e c e ssa ry i d e n t i f y i n g r e s t r i c t i o n s f o r c a r r y i n g out an e m p ir ic a l a n a l y s i s of the model. More i m p o r t a n t l y , i t a llo w s us to study how changes in th e exogenous p r o c e s s e s go vern ing th e f o r e i g n a s s e t p r i c e s and r a t e s of r e t u r n i n f l u e n c e exchange r a t e s . This i s p o s s i b l e s in c e we d e r i v e th e a s s e t demand f u n c t i o n s from a m icroeconomic o p t i m i z a t i o n problem, given th e assum ption of r a t i o n a l e x p e c t a t i o n s , which im p l i e s t h a t a g e n ts maximize t h e i r o b j e c t i v e f u n c t i o n s s u b j e c t to th e t r u e 1 s t o c h a s t i c p r o c e s s e s of th e economy. By s u b s t i t u t i n g th e! i optim al p r e d i c t o r s f o r th e c o n d i t i o n a l e x p e c t a t i o n s of th e f u t u r e v a lu e s of th e exogenous v a r i a b l e s t h a t e n t e r th e demand f u n c t i o n s f o r th e a s s e t s , th e s e demand f u n c t i o n s 42 depend on th e p a ra m e te rs of th e exogenous p r o c e s s e s . Indeed, th e c o e f f i c i e n t s i n th e exchange r a t e e q u a tio n depend in a n o n - l i n e a r f a s h io n on t h e s e p a ra m e te rs and, t h e r f o r e , a re not i n v a r i a n t to changes in th e economic environm ent d e s c r ib e d by the exogenous s t o c h a s t i c p r o c e s s e s . This p r o p e r ty makes our exchange r a t e model f r e e of th e r a t i o n a l e x p e c t a t i o n s school c r i t i q u e of e co n o m etric p ro c e d u re s d is c u s s e d in Chapter I . We should no te h e re t h a t th e r a t i o n a l e x p e c t a t i o n s h y p o t h e s is i s not used as a " p e r f e c t f o r e s i g h t " assum ption but as an e q u il i b r i u m c o n d i t i o n ; i n v e s t o r s know the a c t u a l d i s t r i b u t i o n s of th e exogenous v a r i a b l e s , but not t h e i r a c t u a l p a th s over tim e. The d i s t i n c t i o n between th e two was p in p o in te d by Lucas where he e x p la in e d t h a t r a t i o n a l e x p e c t a t i o n s e q u il i b r i u m in a s s e t m ark ets im p lie s t h a t p r i c e s sh ou ld r e f l e c t f u l l y a l l a v a i l a b l e i n f o rm a tio n , and t h a t As Muth [61] made c l e a r , t h i s h y p o t h e s is ( l i k e u t i l i t y m axim ization ) i s not " b e h a v i o r a l " : i t does not d e s c r i b e the way a g e n ts t h i n k about t h e i r environm ent, how they l e a r n , p ro c e ss i n f o r m a tio n , and so f o r t h . I t i s r a t h e r a p ro p e r ty l i k e l y to be (a p p ro x im a te ly ) p o sse sse d by th e outcome of t h i s u n s p e c i f i e d p ro c e ss of l e a r n i n g and a d a p tin g (49 , p . 1429). T h e re fo re , when th e r a t i o n a l e x p e c t a t i o n s assum ption i s used as an e q u il i b r i u m c o n d it i o n i n a s s e t models, i t means: The m arket c l e a r i n g p r i c e f u n c t i o n . . . im p lied by consumer b e h avio r i s assumed to be th e same as 43 th e p r i c e f u n c t i o n . * . on which consumer d e c i s i o n s a re based (49 , p . 1431). Moreover, by assuming t h a t a s s e t p r i c e s and r a t e s of r e t u r n fo llo w given a u t o r e g r e s s i v e schemes o f f i n i t e la g le n g t h we avoid a problem t h a t o f te n a r i s e s in r a t i o n a l e x p e c t a t i o n s models c o n c e rn in g th e e x i s t e n c e of m u l t i p l e e q u i l i b r i a . This s p e c i f i c a t i o n of th e s t o c h a s t i c p r o c e s s e s e n s u r e s th e u niq u en ess o f th e s o l u t i o n . In an e m p i r ic a l s e t t i n g , we have to v e r i f y t h a t the assumed p r o c e s s e s c lo s e l y approxim ate th e a c t u a l path of th e s e v a r i a b l e s in o rd e r to j u s t i f y th e claim t h a t p r e d i c t o r s of th e exogenous v a r i a b l e s a r e indeed o p tim a l. We would a l s o need to v e r i f y t h a t th e e q u ilib r iu m r a t e of exchange im p lie d by th e model i s c o n s i s t e n t w ith th e p r o c e s s e s on which i n v e s t o r s based t h e i r e x p e c a tio n s . These i s s u e s a r e d e a l t w ith in Chapter IV where a s im p if ie d v e r s i o n of the model i s d e riv e d and e s t im a t e d u s in g th e s t r a t e g y d e s c r ib e d in th e next s e c t i o n . 2.3 EMPIRICAL STRATEGIES FOR ESTIMATING DYNAMIC PORTFOLIO MODELS We have seen e a r l i e r how dynamic p o r t f o l i o models t h a t in v o lv e c u rren cy d i v e r s i f i c a t i o n , and which assume i n v e s t o r s to have convex p r e f e r e n c e s , cannot be used to d e r i v e c lo s e d form demand f u n c t i o n s f o r th e v a r io u s a s s e t s c o n s id e re d a s long as a s s e t p r i c e s and r e t u r n s a r e allow ed 44 to vary randomly or be determ ined in th e a s s e t m a rk e ts . The re a so n f o r t h i s i s t h a t t h e r e a re a t th e p r e s e n t time no known methods f o r s o lv in g a n a l y t i c a l l y n o n - l i n e a r s t o c h a s t i c d i f f e r e n c e e q u a t io n s . R e se a rc h e rs have used a l t e r n a t i v e p ro c e d u re s f o r a n a ly z in g e m p i r i c a l l y such models, t h r e e of which a re : ( i ) O btain a num erical s o l u t i o n t o th e s t o c h a s t i c E uler e q u a t io n s . T his p a r t i c u l a r avenue i s u s e f u l when i t i s d e s i r a b l e to o b t a i n v a lu e s f o r the d e c i s i o n v a r i a b l e s . ' I t i s of l i m i t e d v a lu e , however, in e x p l a i n i n g how exchange r a t e s a r e determ in ed , or in p o lic y a n a l y s i s . ( i i ) E stim a te the model p a ra m e te rs w ith o u t s o l v i n g th e s t o c h a s t i c E u ler e q u a tio n s u sin g an i n s t r u m e n t a l v a r i a b l e s te c h n iq u e . This i s th e G e n e r a liz e d Method of Moments E s tim a to r s developed by H ansen(34). This method makes use of th e o r t h o g o n a l i t y c o n d it i o n s t h a t a r i s e in th e model in d e f i n i n g th e i n s t r u m e n t a l v a r i a b l e s to be used in th e e s t i m a t i o n p ro c e d u re . A GM M e s t i m a t o r of th e model p a ra m e te rs i s one t h a t s e t s l i n e a r co m bin atio ns of the sample c ro s s p ro d u c ts as c lo s e to zero as p o s s i b l e . ( i i i ) O b tain a c h a r a c t e r i z a t i o n of th e s o l u t i o n to th e o p t im i z a ti o n problem by s i m u la t in g th e system by 45 g e n e r a t i n g random v a lu e s or u sin g e s t i m a t e s of the p r o c e s s e s t h a t govern th e exogenous v a r i a b l e s . Of th e s e t h r e e s t r a t e g i e s , only th e second i s a p p r o p r i a t e f o r th e problem a t hand because th e main goal of t h i s study i s to i n t e r p r e t th e ob served b e h av io r of f o r e i g n exchange r a t e s under a f l e x i b l e r a t e regime and to see w hether or not p o r t f o l i o th eo ry can p ro v id e an e x p la n a t i o n of such b e h a v io r. By s i m u l a t i n g th e model, or s o lv in g i t n u m e ric a lly we cannot a c h ie v e t h i s . Although th e G e n e ra liz e d Method of Moments seems to be a very prom ising avenue, i t i s a very c o m p lica te d method t h a t has been a p p li e d so f a r only by Hansen and S i n g l e t o n ( 3 7 ) . Given th e o p tio n s a v a i l a b l e , we have chosen to implement two d i f f e r e n t s t r a t e g i e s f o r th e e m p i r ic a l a n a l y s i s . The f i r s t s t r a t e g y in v o lv e s e s t i m a t i n g a time s e r i e s model of exchange r a t e s w ith o u t imposing any a p r i o r i s t r u c t u r a l r e s t r i c t i o n s . Sims($9) and S a rg e n t and Sims(65 ) proposed t h i s methodology as a way of i n v e s t i g a t i n g th e type of I t r e l a t i o n s h i p s among economic v a r i a b l e s by i n s p e c t i n g t h e i r c r o s s - c o r r e l a t i o n s over tim e. In a d d i t i o n to Sims and S a rg e n t, t h i s s t r a t e g y has been a p p l i e d by N ick elsb urg (6 T ) and E c k s te in ( 2 1 ) as a sup plem en tal way of t e s t i n g e m p i r i c a l l y c e r t a i n h y p o th e s e s t h a t cannot be t e s t e d u sin g c o n v e n tio n a l econ om etric t e c h n iq u e s . 46 The second s t r a t e g y i s t o e s t i m a t e a s p e c i f i c f o r m u la tio n of th e p o r t f o l i o model to be d e s c rib e d below. The eviden ce from th e two a n a l y s e s i s th e n compared and used to t e s t th e f o llo w in g two h y p o th e s e s : ( i ) t h a t th e b e h a v io r a l b a s i s of p o r t f o l i o balance th e o ry p ro v id e s an e x p a l n a t i o n of the observed f l u c t u a t i o n s in exchange r a t e s in th e r e c e n t e p iso d e of f l o a t i n g r a t e s which s t a r t e d in th e e a r l y 70s. ( i i ) t h a t f l u c t u a t i o n s in th e p r i c e of gold a re c l o s e l y r e l a t e d t o th o se in th e v a lu e of th e d o l l a r and may, th u s , be used to shed l i g h t on th e f o r c e s d r i v i n g exchange r a t e s . Given t h a t s o l u t i o n s t o s t o c h a s t i c dynamic o p t i m i z a t i o n problems can be o b ta in e d f o r a q u a d r a t i c o b j e c t i v e f u n c t i o n , we can study th e p o r t f o l i o a l l o c a t i o n problem f o r the i n t e r e s t i n g case of a l i n e a r u t i l i t y f u n c tio n w ith a q u a d r a t i c c o s t of a d ju s tm e n t.^ G e n e ra lly , p o r t f o l i o s e l e c t i o n th e o ry r e l i e s h e a v i ly on th e assum ption of r i s k a v e r s i o n f o r e x p l a i n i n g th e phenomena of p o r t f o l i o d i v e r s i f i c a t i o n . However, major i n v e s t o r s ( b a s i c a l l y s p e c u l a t o r s ) in f o r e i g n exchange m ark ets who have an ^ U n f o r tu n a te ly , we can not d e riv e a s o l u t i o n f o r any q u a d r a t i c o b j e c t i v e f u n c t i o n a l , and in p a r t i c u l a r a q u a d r a t i c u t i l i t y f u n c t i o n s in c e we w i l l s t i l l fac e th e problem of having to so lv e n o n - l i n e a r , second o r d e r , s t o c h a s t i c d i f f e r e n c e e q u a t io n s . 47 i n f lu e n c e on f o r e i g n exchange r a t e s need n ot be r i s k a v e r s e , t h a t i s , have no d im in is h in g m arg in al u t i l i t y of w e a lth or c o n s u m p t i o n . ^ A c tu a lly , s p e c u l a t o r s a re o f t e n d e p ic te d i n th e l i t e r a t u r e on fo rw ard exchange r a t e s and th e e f f i c i e n c y of f o r e i g n exchange m ark ets a s p r o f i t m ax im iz ers. ^ I t can be e a s i l y shown how th e problem of maximizing e x p ec te d u t i l i t y , when th e u t i l i t y f u n c t i o n i s l i n e a r , red u c es to a problem of maximizing e x p e c te d w e a lth , or to t h a t e f f e c t , stream of income. In such a case, d i v e r s i f i c a t i o n would be b ro u g h t about by o th e r f a c t o r s , a p o s s i b l e one being c o s t s of p o r t f o l i o a d ju s tm e n ts . This would not only i n c lu d e t r a n s a c t i o n c o s t s , but a ls o in f o r m a tio n c o s t s a s w e ll as some premium f o r f o r e i g n c u rren cy exposure. K ouri(44) has su g g e ste d i n c l u d i n g t r a n s a c t i o n c o s t s i n p o r t f o l i o models of exchange r a t e s u sin g a flow e q u il i b r i u m framework to r u l e out i n s t a n t a n e o u s p o r t f o l i o a d ju s tm e n ts s in c e th e s e would be i n c o n s i s t e n t w ith the f a c t t h a t th e b alan ce of payments ^ L ucas(49, p . 1441) made th e f o llo w in g i n t e r e s t i n g remark c o n c e rn in g " r i s k a v e r s i o n " : "In t h i s m u l t i p e r i o d c o n te x t, th e term ‘ r i s k a v e r s i o n ' i s perhaps m is le a d in g , s in c e the c u r v a tu r e of U [ t h e u t i l i t y fu n c tio n ] a l s o governs th e i n t e r t e m p o r a l s u b s t i t u t a b i l i t y of consum ption. With tim e a d d i t i v e u t i l i t y , t h e r e i s no way to d i s e n t a n g l e th e s e c o n c e p tu a lly d i s t i n c t a s p e c t s of p r e f e r e n c e s . " 7 Examples of such d e p i c t i o n of s p e c u l a t o r s may be found in D ornbusch(13) and B i l s o n ( 9 ) . 48 e q u a tio n i s an jgjc p o s t i d e n t i t y and n o t an £& a n te e q u il i b r i u m c o n d itio n and, t h e r e f o r e , th e d e s i r e d change in a s s e t h o ld in g s i s n ot n e c e s s a r i l y equal to th e a c t u a l change. Thus, we a re prompted to r e l a x th e p re v io u s assum ption t h a t no t r a n s a c t i o n c o s t s e x i s t . The p ro ce d u re used i n s o lv in g th e q u a d r a t i c program ing problem has been developed by Hansen and S a rg e n t(3 5 ) and (36) f o r th e e s t i m a t i o n of m u l t i - v a r i a t e l i n e a r r a t i o n a l e x p e c t a t i o n s tim e s e r i e s models. I t has been employed by S a rg e n t(6 2 ) and Meese(52) in th e d e r i v a t i o n and e s t i m a t i o n of dynamic f a c t o r demand e q u a t io n s , w hile E c k s te in (2 1 ) a p p li e d i t in e s t i m a t i n g a g r i c u l t u r a l supply f u n c t i o n s and a n a ly z in g la n d a l l o c a t i o n d e c i s i o n s . In e sse n ce a l l t h r e e s t u d i e s i l l u s t r a t e an a p p r o p r i a t e s t r a t e g y fo r m o d e llin g dynamic economic r e l a t i o n s h i p s . The e m p ir ic a l r e s u l t s in S a r g e n t ’ s study were m o derately s u p p o r t i v e of th e dynamic f o r m u la tio n of th e demand f o r employment. The Meese r e s u l t s were not d e c i s i v e in t h a t th e r e s t r i c t i o n s in h i s model were not s t r o n g l y su p p o rte d by th e d a ta . The most f a v o r a b l e r e s u l t s were th o se of E c k s te in , a lth o u g h they I i n d i c a t e d t h a t a r i c h e r dynamic s p e c i f i c a t i o n of the technology was needed. The r e s u l t s c l e a r l y i n d i c a t e t h a t th e q u a d r a t i c o b j e c t i v e f u n c t i o n need not be th e c o r r e c t s p e c i f i c a t i o n of 49 the te c h n o lo g y , c o s t of a d ju s tm e n t, or p r e f e r e n c e s . This by i t s e l f , though, in no way d im in is h e s th e v alu e of t h i s s t r a t e g y towards m o d e llin g and s p e c i f y i n g s t r u c t u r a l r e l a t i o n s h i p s . Moreover, s in c e we w i l l an aly ze th e d ata w ith o u t any p r i o r s t r u c t u r a l r e s t r i c t i o n s as w e ll as c a rry out th e n e c e ssa ry t e s t s o f c a u s a l i t y a l a Granger to v e r i f y the e x o g en e ity of a s s e t p r i c e s and i n t e r e s t r a t e w ith r e s p e c t to exchange r a t e s . The ev id ence from both a n a ly s e s can be used i n t e s t i n g th e i m p l i c a t i o n s of p o r t f o l i o th e o ry of exchange r a t e s w h ile overcoming th e problems t h a t may a r i s e from th e sim ple s p e c i f i c a t i o n of the dynamics in f o r e i g n exchange r a t e m a rk e ts . In Chapter I I I , th e s t r u c t u r e - f r e e time s e r i e s model of exchange r a t e s w i l l be e s t im a t e d . In a d d i t i o n , the n e c e ssa ry p r e t e s t i n g of th e assu m p tio ns used in th e s t r u c t u r a l p o r t f o l i o model w i l l be perform ed. In p a r t i c u l a r , th e s t a b i l i t y of th e s t o c h a s t i c p r o c e s s e s over time w i l l be i n v e s t i g a t e d b e fo re they a re a c t u a l l y e s tim a te d j o i n t l y w ith th e exchange r a t e e q u a tio n in Chapter IV where they w i l l be t r e a t e d as though they a re s t r u c t u r a l l y s t a b l e . 50 Chapter I I I AN ALTERNATIVE STRATEGY FOR ANALYZING EXCHANGE RATE DYNAMICS In t h i s c h a p t e r exchange r a t e dynamics a re an aly ze d u s in g a m u l t i v a r i a t e time s e r i e s model which i s n ot c o n s t r a i n e d by any a p r i o r i s t r u c t u r e . The m otive behind t h i s i s to i n v e s t i g a t e th e i m p l i c a t i o n s of the p o r t f o l i o balan ce approach to exchange r a t e s by exam ining, in th e time domain, th e r e l a t i o n s h i p s between exchange r a t e s and v a r i a b l e s t h a t a c c o rd in g to p o r t f o l i o th eo ry should play a r o l e in th e d e te r m in a tio n of exchange r a t e s . R e s u lts of the a n a l y s i s u sing d a ta f o r d i f f e r e n t c o u n t r i e s a re compared in o rd er to c a p tu r e th e r e g u l a r i t i e s in th e dynamics of f o r e i g n exchange r a t e s t h a t a re not c o u n t r y - s p e c i f i c . R e ac tio n s t o s p e c i f i c p o l i c i e s and c o n t r o l s may be i d e n t i f i e d , and i n f e r e n c e s can th u s be drawn r e g a r d i n g v a r io u s h y p o th e s e s about th e d e te r m in a tio n of f o r e i g n exchange r a t e s . In p a r t i c u l a r , we examine the [ i n t e r a c t i o n between th e gold and f o r e i g n exchange m arkets w ith o u t making any assu m p tio ns about th e u n d e rly in g b e h a v io r a l f o r c e s . In a d d i t i o n , the a n a l y s i s w i l l encompass th e n e c e ssa ry p r e t e s t i n g of th e s p e c i f i c a t i o n of 51 the s t r u c t u r a l model developed in Chapter I I . In th e next s e c t i o n we w i l l d i s c u s s th e methodology ad opted, follow ed by a d e s c r i p t i o n of th e d a ta in S e c tio n 2 . 2 . The r e s u l t s i of th e v a rio u s s t a g e s o f the a n a l y s i s w i l l be d is c u s s e d in s e p a r a t e s e c t i o n s . 3.1 METHODOLOGY Follow ing Sims (6 9 ), a v e c to r a u t o r e g r e s s i v e model i s s e l e c t e d f o r th e a n a l y s i s of th e time s e r i e s d a ta . The e ssen ce of time s e r i e s a n a l y s i s , in g e n e r a l , i s to decompose s t o c h a s t i c economic p r o c e s s e s i n t o components, which when s t u d i e d s e p a r a t e l y , o f f e r b e t t e r i n s i g h t i n to the dynamics o f th e o r i g i n a l p r o c e s s e s . This i s p o s s i b l e under c e r t a i n c ir c u m s ta n c e s . In p a r t i c u l a r , i t i s p o s s ib l e to implement such a s t r a t e g y fo r s t a t i o n a r y time s e r i e s . The assum ption of s t a t i o n a r i t y in time s e r i e s a n a l y s i s i s analogous to assum ption s used in eco n o m etric a n a l y s i s r e g a r d in g th e a sy m p to tic p r o p e r t i e s of random v a r i a b l e s , w ith o u t which a sy m p to tic d i s t r i b u t i o n th e o ry cannot be used I nor i n f e r e n c e s be drawn. In p a r t i c u l a r , a s t a t i o n a r y s t o c h a s t i c p ro c e ss i s a l s o a s t a b l e , or co n v erg en t p ro c e s s . Although many economic s e r i e s can h a rd ly be t r e a t e d as s t a t i o n a r y s t o c h a s t i c p r o c e s s e s , a p p r o p r i a te 52 t r a n s f o r m a t i o n s of th e s e r i e s , such as t a k i n g d i f f e r e n c e s , lo g a r ith m s , moving a v e ra g e s , or r a t e s of growth, may be t r e a t e d as such. In m u l t i v a r i a t e time s e r i e s m odels, a g e n e r a l i z a t i o n of Q Wold’ s decom p ositio n theorem can be employed to tra n s fo rm a v e c to r of given s e r i e s to a v e c to r of m u tu ally s t o c h a s t i c a l l y u n c o r r e l a t e d w hite n o ise p r o c e s s e s . Wold’ s d ecom position theorem s t a t e s t h a t , given any v e c to r time s e r i e s , X^., whose elem en ts a re j o i n t l y c o v a ria n c e s t a t i o n a r y with z e ro means, i t can be decomposed i n t o two o rth o g o n a l components, a l i n e a r l y d e t e r m i n i s t i c p r o c e s s , Nt which i s a f u n c t i o n of p a s t v a lu e s of th e elem en ts of Xt and an i n d e t e r m i n i s t i c p a r t , r t , o f te n c a l l e d th e jin n o v atio n s i n th e elem ents of Xfc. This can be w r i t t e n a s : Xt = C(L)Tt + Nt given Xj. as an m elem ent v e c t o r , C(L) as an mxm m a trix polynom ial in th e l a g o p e r a to r L such t h a t : C O C ( L) = I + H c i L1 1 = 0 o See Hanan, E . J . : M u ltip le Time S e r i e s . New York: W e s le y ,1970. 53 jand r t a s a v e c to r w h ite no ise p ro c e ss w ith m e lem en ts, i . e . , ^ t ^ ^ i j t } su °h t h a t : = 0 f o r a * 1 1 I 0 f o r a l l T, U j E [ Yi , t Y j , t - T ] = I ( 0 T i 0 , i= j The c o v a ria n c e s t a t i o n a r i t y assum ption r e q u i r e s t h a t the r o o t s of !C( L)!=0 l i e o u t s i d e th e u n i t c i r c l e and t h a t C(L) i s i n v e r t i b l e . I f we focus on th e i n d e te rm in i s t i c component of the | 1 v e c t o r time s e r i e s o n ly , i . e . Zfc= Xt - Nt , then th e moving average r e p r e s e n t a t i o n (MAH) Z t = C ( L ) r t can be tra n s fo rm e d to a v e c to r a u t o r e g r e s s i v e r e p r e s e n t a t i o n (VAR) fo r Zt i . e. B(L)Zt = f t where B(L)=C(L)- ^ i s a l s o a m a trix polynom ial in L of i n f i n i t e o r d e r . j The models e s t im a t e d in t h i s C h a p te r a re of the { a u t o r e g r e s s i v e ty p e . In a d d i t i o n to th e exchange r a t e , th e J v a r i a b l e s in c lu d e d in th e a n a l y s i s a r e : f o r e i g n money supply, dom estic money supply, s h o r t - t e r m i n t e r e s t 54 d i f f e r e n t i a l , index of f o r e i g n consumer p r i c e s , index of dom estic consumer p r i c e s , and th e p r i c e of gold. The time s e r i e s v e c t o r , Y, i s th e n : Y = (MUS HD INT PUS PD PG EX) where the f i r s t s ix elem en ts of Y co rre sp o n d to th e s ix v a r i a b l e s l i s t e d above r e s p e c t i v e l y , and th e l a s t elem ent, EX, c o rre sp o n d s to th e exchange r a t e . P o r t f o l i o b alan ce th eory p ro v id e s th e r a t i o n a l e f o r the choice of the above v a r i a b l e s . Data from fo u r c o u n t r i e s , Canada, France, Germany, and S w itz e r la n d a re used f o r th e dom estic v a r i a b l e s , w hile U.S. d a ta i s used f o r th e f o r e i g n ones. With f i n i t e samples th e r a t e of dampening of the s t o c h a s t i c p r o c e s s e s must be f a s t enough to allow fo r s u f f i c i e n t d e g re e s of freedom to e s t im a t e the model p a ra m e te rs and to t e s t h y p o th e s e s . However, i n s t e a d of assuming a p r i o r i knowledge of the system l a g le n g t h , each model i s e s t im a t e d u sin g d i f f e r e n t l a g l e n g t h s and th e s h o r t e r l a g s a re t e s t e d as a r e s t r i c t i o n on th e lo n g e r l a g i models. On th e n u ll h y p o th e s is t h a t th e s h o r t e r l a g i s j I a c c e p t a b l e , th e l i k e l i h o o d r a t i o s t a t i s t i c i s | a s y m p to t i c a ll y d i s t r i b u t e d as c h i - s q u a r e . As Sims(69) i n d i c a t e d , th e s e l i k e l i h o o d r a t i o s t a t i s t i c s , and th e 55 s i n g l e e q u a tio n t e s t s t a t i s t i c s which a re a s y m p to t i c a ll y d i s t r i b u t e d F, a re b ia s e d a g a i n s t th e n u ll h y p o th e s is in f i n i t e samples due to t h e i r s e n s i t i v i t y to n o n -n o rm ality of r e s i d u a l s . This b ia s i s even g r e a t e r f o r the l i k e l i h o o d r a t i o th an i t i s f o r th e s e F s t a t i s t i c s . For t h i s re a so n , an a d d i t i o n a l B ayesian based t e s t i s employed. In a d d i t i o n , w hether or not th e e s tim a te d r e s i d u a l s c o n s t i t u t e a w h ite n o ise p ro c e ss i s checked by i n s p e c t i n g th e sample a u t o c o r r e l a t i o n s of th e s e r e s i d u a l s and th e s t a t i s t i c a l s t a b i l i t y of th e model i s t e s t e d to v e r i f y t h a t the p ro c e s s e s s a t i s f y th e c o v a ria n c e s t a t i o n a 'r i t y assum ption. I t would be i n a c c u r a t e to i n t e r p r e t a u t o r e g r e s s i v e p a ra m e te rs in th e same manner by which o rd in a ry r e g r e s s i o n p a ra m e te rs a re i n t e r p r e t e d . The r e a so n s a re tw o -fo ld . F i r s t , a s tr o n g c o r r e l a t i o n may e x i s t among th e lag ged v a r i a b l e s . Second, i f th e co n tin u o u s s t o c h a s t i c p r o c e s s e s go verning th e data a re ex p ec te d to be smooth, th e w e ig h ts a t t a c h e d to s u c c e s s iv e l a g s of a c e r t a i n v a r i a b l e would ten d to o s c i l l a t e . I t i s p o s s i b l e though to a n a ly z e and i n t e r p r e t th e i n t e r a c t i v e b e h av io r of th e time s e r i e s under study by th e i n s p e c t i o n of th e moving average (MA) r e p r e s e n t a t i o n of th e model. I f the model i s i n v e r t i b l e , the M A r e p r e s e n t a t i o n would c o n ta in th e same in f o r m a tio n about th e s t o c h a s t i c b e h av io r of th e system, and i f the 56 system f o llo w s a smooth p ath , i t would r e s u l t i n smooth M A c o e f f i c i e n t s . T ransform ing th e model i n t o i t s M A r e p r e s e n t a t i o n i s no easy t a s k s in c e i t i n v o lv e s i n v e r t i n g a m a tr ix polynom ial of VAR c o e f f i c i e n t s . For our pu rpo ses, though, i t would be s u f f i c i e n t to s im u la te the M A p a r t of the p ro c e s s . R e c a llin g th e assu m p tio n s co n ce rn in g th e e r r o r term, r ^ i s a v e c to r w h ite n o ise p ro c e ss w ith z e ro mean and d iag o n a l v a ria n c e c o v a ria n c e m a tr ix , t h a t i s , th e i n d i v i d u a l e r r o r term s a re s e r i a l l y u n c o r r e l a t e d over time and a c r o s s e q u a t io n s . In t h i s case, th e elem en ts of a re o rth o g o n a l in n o v a tio n s in each elem ent of the time s e r i e s v e c t o r Z^.. However, in g e n e r a l, th e e s tim a te d c o v a ria n c e m a trix o r t h o g o n a l iz e d . Note t h a t th e " n o n - o r th o g o n a l" in n o v a t i o n s a re the o n e - s te p ahead f o r e c a s t e r r o r s or th e s ta n d a r d e r r o r of e s t i m a t e s of th e c o rre sp o n d in g components of Z^- Once o r t h o g o n a l i z a t i o n of i n n o v a tio n s i s done, th e M AR of th e system can be s im u la te d by t r a c i n g th e sy s te m ’ s re sp o n se to t y p i c a l random shocks e q u i v a l e n t t o o rth o g o n a l in n o v a tio n s in th e system v a r i a b l e s . Another way w ith which to a n a ly z e the i n t e r a c t i v e b e h av io r of th e system i s to g e n e r a te the d e co m p o sitio n of the v a r ia n c e of the f o r e c a s t e r r o r s , over d i f f e r e n t i s not d iag o n a l and must be i n i t i a l l y 57 f o r e c a s t i n g h o r i z o n s , i n t o components due to o rth o g o n a l in n o v a tio n s i n th e d i f f e r e n t model v a r i a b l e s . As th e f o r e c a s t h o riz o n goes to i n f i n i t y the f o r e c a s t v a r ia n c e goes to th e v a r i a b l e ’ s v a ria n c e and, t h e r e f o r e , we can i n f e r the c o n t r i b u t i o n of the v a rio u s i n n o v a tio n s to th e system v a r i a b l e v a r i a n c e . In t h i s way, we conduct a f a c t o r a n a l y s i s of our v e c to r of v a r i a b l e s f o r each co u n try . 3 .2 THE DATA The v e c to r a u t o r e g r e s s i o n s a re e s t im a t e d u s in g d a ta fo r jCanada, Germany, France and S w itz e r la n d . These c o u n t r i e s were chosen because they have a r e l a t i v e l y f l e x i b l e exchange r a t e v i s - a - v i s th e U.S. d o l l a r , and because c a p i t a l and exchange r a t e c o n t r o l s a re r e l a t i v e l y fewer compared w ith most o th e r c o u n t r i e s . Moreover, th e s e c o u n t r i e s have w e ll developed f i n a n c i a l m arkets which allow dom estic i n v e s t o r s to choose from among a s s e t s of v a rio u s c u rre n cy d en o m in atio n s. Data a v a i l a b i l i t y and th e above a s p e c t s make th e s e c o u n t r i e s i n t e r e s t i n g c a s e s in so f a r as a llo w in g us t o study th e e f f e c t of p o r t f o l i o a d ju s tm e n ts , and more s p e c i f i c a l l y th e gold m arket, on exchange r a t e s in a f l e x i b l e r a t e m i l i e u . By choosing th e s e c o u n t r i e s we may be p o s s ib ly v i o l a t i n g th e sm all country assum ption. However, to r e l a x t h i s assum ption would make th e model 58 q u i t e c o m p lica te d , w hereas, c o u n t r i e s t h a t s a t i s f y t r u e l y th e sm all cou ntry assum ption do not have, in g e n e r a l , a f l o a t i n g exchange r a t e or f r e e c a p i t a l m a rk e ts. This should be kept in mind when i n t e r p r e t i n g th e r e s u l t s . Monthly o b s e r v a t i o n s a re used c o v e rin g th e p e rio d from Janu ary 1974 to December 1980. This p e rio d was chosen as th e most r e c e n t e p iso d e of f l e x i b l e exchange r a t e s which s t a r t e d in March 1973. We o m itte d th e f i r s t nine months of t h i s e p is o d e (th o s e in 1973) from our sample as p r e lim in a r y t e s t s i n d i c a t e d l a c k of homogeneity between th e pre and p o st 1974 p a r t s of th e sample. One can lo o k a t th e s e months as a t r a n s i t i o n p e rio d over which i n v e s t o r s were " l e a r n i n g " about th e new c u rrency system, and g a th e r in g in f o r m a tio n n e c e ssa ry f o r form ing e x p e c t a t i o n s t h a t a re of major i n f l u e n c e on t h e i r in v e s tm e n ts . Since our i n t e r e s t i s to examine th e p o r t f o l i o m otive as a d e te rm in a n t of f o r e i g n exchange r a t e s , where t h i s m otive i s b e lie v e d t o be dominant in th e s h o r t - r u n , though perhaps not in th e v e r y - s h o r t - r u n , we choose to use monthly ( o b s e r v a t i o n s . Moreover, whenever p o s s i b l e our d a ta a re in monthly average form. For our p u rp o se s, a monthly average i s b e t t e r than th e end of month f i g u r e as i t f i l t e r s out components of very s h o r t - r u n d u r a t i o n which are s p e c i f i c to th e day chosen as th e e n d -o f-m o n th . The p r a c t i c e of 59 f i l t e r i n g out components t h a t a re of no i n t e r e s t w ith r e g a rd s to th e i s s u e s being a d d re ss e d i s common in time s e r i e s s t a t i s t i c a l a n a l y s i s . The p r i n c i p l e i s th e .same as t h a t of removing t r e n d s where one i s i n v e s t i g a t i n g s h o r t - r u n phenomena, or removing th e se a s o n a l components when one i s s tu d y in g lo n g -te rm b e h a v io r. This p r a c t i c e d i s t i l l s out components which may dominate in s i z e th e b e h av io r of an economic s e r i e s so t h a t o th e r im p o rta n t components a re cam ouflaged. In a d d i t i o n , we remove th e tr e n d and s e a s o n a l components from a l l our s e r i e s . ! Narrowly d e fin e d money, M 1 , was u t i l i z e d f o r the money supply measure. The d ata was tak e n from th e F e d e ra l Reserve B u l l e t i n f o r the U .S ., and from th e IMF • I n t e r n a t i o n a l F i n a n c i a l S t a t i s t i c s ' t a p e s f o r Canada, Germany, France and S w itz e r la n d . For th e q u e s t i o n s being a d d re s s e d in t h i s stu d y , M 1 i s th e most a p p r o p r i a t e measure s in c e i t i n c l u d e s th e h ig h ly l i q u i d and n e g o t i a b l e components of th e money supply which u s u a l ly compose s p e c u l a t i v e b a la n c e s . The excluded components t h a t appear in o t h e r money m easures such as sa v in g s and tim e d e p o s i t s , and n o n - n e g o t i a b l e c e r t i f i c a t e s of d e p o s i t , do not play a s i g n i f i c a n t r o l e in s h o r t - t e r m p o r t f o l i o a d ju s tm e n ts . Since th e s e ex clu ded components a re r e l a t i v e l y l a r g e in s i z e , t h e i r movements may overshadow changes in s p e c u l a t i v e b alan ce s . | 60 ! For the i n t e r e s t r a t e s e r i e s , we chose th e d a y -to -d a y money r a t e s s in c e th e s e r e f l e c t th e s h o r t - t e r m movements in r a t e s of r e t u r n on d i f f e r e n t c u r r e n c i e s much b e t t e r than th e 3-month T reasu ry b i l l r a t e s . The F e d e ra l Funds r a t e was used f o r th e U.S. r a t e of i n t e r e s t , w h ile th e c h a r t e r e d banks d a y -to -d a y lo a n s r a t e was used f o r Canada. Call-money r a t e was used f o r th e o th e r c o u n t r i e s . The Swiss and Canadian i n t e r e s t r a t e s e r i e s were o b ta in e d from Morgan G a u r a n t e e 's World F i n a n c i a l M a r k e ts . The r e s t of the d ata s e r i e s were o b ta in e d from th e IFS ta p e s . The exchange r a t e s e r i e s s e l e c t e d i s th e c o n v ersio n f a c t o r used to tr a n s fo r m d ata on t r a d e and o t h e r flow s from dom estic cu rren cy to U.S. d o l l a r s and v ic e v e r s a , which appear in th e IFS t a p e s . In th e next Chapter we s h a l l need ito use d ata on t r a d e and f o r e i g n a s s e t s e x p re ss e d in U.S. d o l l a r s f o r e s t i m a t i n g th e s t r u c t u r a l model. Since th e d a ta i s a v a i l a b l e in dom estic c u rre n c y , no i n c o n s i s t e n c i e s w i l l a r i s e by u sin g t h i s p a r t i c u l a r exchange r a t e to tra n s fo rm i t . I t i s c l e a r from the above d i s c u s s i o n t h a t r e s u l t s based on a l t e r n a t i v e c h o ic e s f o r the m easures of our model v a r i a b l e s should be e x p e c te d to d i f f e r . The main reason being t h a t d i f f e r e n t m easures c a p tu r e d i f f e r e n t a s p e c t s of the v a r i a b l e s and would, t h e r e f o r e , convey d i f f e r e n t 61 in f o r m a tio n . Thus, th e choice of a d e f i n i t i o n i s an i 7 e m p i r ic a l m a t t e r . I t should be based on th e kind of I q u e s t i o n s t h a t a re b eing a d d re s s e d . 3.3 TESTING LAG LENGTH AND STRUCTURAL STABILITY I !a11 s e r i e s were i n i t i a l l y r e g r e s s e d a g a i n s t a c o n s t a n t , a i l i n e a r tr e n d and e le v e n se a s o n a l dummies. The r e s i d u a l s (from th e s e r e g r e s s i o n s were then used f o r e s t i m a t i n g th e jtime s e r i e s m odels. We d e tre n d e d and d e s e a s o n a l iz e d th e d a ta to i s o l a t e th e d e t e r m i n i s t i c components of th e s e r i e s ! Ifrom th e i n d e t e r m i n i s t i c ones s in c e t h e r e i s no re a so n to I ■expect th e s e two d i f f e r e n t components to follow th e same 'a u to r e g r e s s i v e p r o c e s s . Moreover, i n s p e c t i o n of the i a u t o c o r r e l a t i o n s and p a r t i a l a u t o c o r r e l a t i o n s of the o r i g i n a l s e r i e s i n d i c a t e d t h a t they cannot be t r e a t e d as c o v a ria n c e s t a t i o n a r y . However, a f t e r e l i m i n a t i n g th e tr e n d and s e a s o n a l components, we can j u s t i f y the a s s e r t i o n t h a t the r e s i d u a l s e r i e s a re c o v a ria n c e s t a t i o n a r y . The p a r t i c u l a r p ro ce d u re adopted f o r d e tr e n d in g and d e s e a s o n a l i z i n g th e d a ta removes only the t r e n d in mean and average se a s o n a l f a c t o r s and l e a v e s much more in f o r m a tio n ■in th e time s e r i e s when compared w ith a l t e r n a t i v e methods such as u sin g n o n - l i n e a r tr e n d s and d e s e a s o n l i z i n g t e c h n iq u e s t h a t t r a c e th e s e r i e s very c l o s e l y . With th ese 62 l a t t e r a l t e r n a t i v e te c h n iq u e s one may not be su re whether the se a s o n a l a d ju s tm e n t p roced ure simply removed th e se a so n a l components or i t has d e s tr o y e d some in f o r m a tio n in the o r i g i n a l s e r i e s u n r e l a t e d to s e a s o n a l i t y . ^ T e s ts on la g l e n g t h in v o lv e d e s t i m a t i n g th e v e c to r a u t o r e g r e s s i o n s w ith l a g l e n g t h s of one, two, fo ur and e i g h t u sin g th e method of seem ingly u n r e l a t e d e q u a t io n s . jWith th e s h o r t e r l a g models t e s t e d as a r e s t r i c t i o n of the lo n g e r ones, the l i k e l i h o o d r a t i o test^® a c c e p te d a l a g le n g t h of two f o r Canada, France, Germany, and S w itz e r la n d . The l i k e l i h o o d r a t i o s t a t i s t i c s f o r th e two vs. fo ur l a g s no dels a r e x 2 98)=105.46, 1 0 7 . 7 3 , 8 9 . 1 2 , and 108.20 w ith m arg in al s i g n i f i c a n c e l e v e l s of about . 2 8 , .2 3, . 7 0 , and 9 This im p o rta n t q u e s t i o n has been r a i s e d i n s e v e r a l s t u d i e s , fo r example, N erlove(60) and Granger and Newbol d (31, pp.35-36) where the l a t t e r su g g e st t h a t th e tr e n d , s e a s o n a l and i r r e g u l a r components should be hand led , as f a r as p o s s i b l e , s im u lta n e o u s ly in one model to d e s c r i b e as a c c u r a t e l y as p o s s i b l e th e b e h av io r of the time s e r i e s under stu d y . For our a n a l y s i s , th e high degree of c o r r e l a t i o n between th e d e t e r m i n i s t i c components of th e time s e r i e s r e s u l t e d in an i l l - c o n d i t i o n e d d e sig n m a trix which prompted th e J p r e f i l t e r i n g of the d a ta . ; 1 0 A c tu a lly a v a r i a n t of the l i k e l i h o o d r a t i o t e s t i s used where T-k r a t h e r than T i s used as sample s i z e , k being th e number of r e g r e s s i o n c o e f f i c i e n t s in each e q u a tio n . The s t a t i s t i c i s ( T - k ) ( logiDR I-lo g iD U i), where DR i s th e v a r i a n c e c o v a ria n c e m a trix of r e s i d u a l s of the r e s t r i c t e d model, and DU i s th e analogous m a tr ix f o r th e u n r e s t r i c t e d model. The s t a t i s t i c i s a s y m p t o t i c a l l y X ( r ) , r being th e number r e s t r i c t i o n s . '.22 f o r th e above c o u n t r i e s r e s p e c t i v e l y . ^ Due to th e b ia s of the l i k e l i h o o d r a t i o t e s t a g a i n s t th e n u ll h y p o t h e s is a su pp lem ental t e s t was used. Sawa(6 6 ) d eveloped a B ay esian In fo rm a tio n C r i t e r i o n (BIC) f o r d i s c r i m i n a t i n g among r e g r e s s i o n models w ith a l t e r n a t i v e f o r m u l a t i o n s . The B ayesian l o s s f u n c t i o n i s based on an i n f o r m a tio n measure of d i s t a n c e between two p r o b a b i l i t y d e n s i ty f u n c t i o n s . The BIC m easure i s an u n b iased e s t i m a t e of th e r i s k f u n c t i o n . The model w ith th e s m a l l e s t BIC measure i s , t h e r e f o r e , a s s o c i a t e d w ith th e lo w e st r i s k and i ’ !is th e one t h a t i s a c c e p te d . Except f o r th e model for France which had th e s m a l l e s t BIC measure a s s o c i a t e d w ith a la g l e n g t h of two, a l l th e e s t im a t e d models were a c c e p t in g of l a g one. As m entioned e a r l i e r , p o s s i b l e m u l t i c o l l i n e a r i t y among th e lag g e d v a r i a b l e s could le a d to im p re c ise i n d i v i d u a l p aram eter e s t i m a t e s . T e s ts i n v o lv i n g i n d i v i d u a l c o e f f i c i e n t s a re not r e l i a b l e , and t and F s t a t i s t i c s can not be p ro p e rly used f o r t e s t i n g th e s i g n i f i c a n c e of c o e f f i c i e n t s of c e r t a i n l a g s . I f a n y th in g , though, th e F s t a t i s t i c w i l l be b ia s e d a g a i n s t th e n u l l h y p o t h e s is and i t A m arg in al s i g n i f i c a n c e l e v e l below .10 would l e a d to th e r e j e c t i o n of the n u ll h y p o th e s is a t th e 90 % c o n fid e n c e l e v e l , which means t h a t we can not a c c e p t th e s h o r t e r l a g l e n g t h as a r e s t r i c t i o n on th e lo n g e r la g m odel. 64 would be u n l i k e l y t h a t s i n g l e - e q u a t i o n t e s t s which are d i s t r i b u t e d a s y m p t o t i c a l l y F would l e a d to choosing a c e r t a i n l a g le n g t h when th e t r u e o rd e r i s l o n g e r . To su p p o rt th e f i n d i n g t h a t a r a t h e r parsim on io us model w ith l a g o r d e r of one or two w i l l s u f f i c i e n t l y r e p r e s e n t th e dynamics of the model th e s i n g l e e q u a tio n F s t a t i s t i c s a re i n s p e c t e d . The f i n d i n g s su p p o rt our c o n fir m a tio n t h a t a jlag l e n g t h of two i s a p p r o p r i a t e . The r e s u l t s a re r e p o r t e d i n T able 7 Appendix B which i n c l u d e s th e d e t a i l e d s t a t i s t i c a l t a b l e s f o r the VAR a n a l y s i s . For f u r t h e r evidence on th e a p p r o p r i a t e l a g l e n g t h , th e r e s i d u a l s of th e VAR models were checked f o r any p o s s i b l e d e p a r t u r e from w h ite n o ise b e h a v io r. The use of seemingly u n r e l a t e d e q u a t io n s te c h n iq u e f o r e s t i m a t i o n produces in p ssen ce f i t t e d e q u a t io n s w ith r e s i d u a l s as c lo s e to w h ite n o is e as p o s s i b l e . However, a u t o c o r r e l a t i o n s of r e s i d u a l s a t r e l a t i v e l y l a r g e l a g s may p o in t out any v i o l a t i o n s of jwhite n o ise b e h av io r ( see Box and P i e r c e ( 1 0 ) ). T h e o r e t i c a l l y , no s e r i a l a u t o c o r r e l a t i o n should e x i s t in the r e s i d u a l s . I f th e e s tim a te d model i s th e t r u e one, then th e sample r e s i d u a l s would have in d e p e n d e n tly % ^ d is trib u te d a u t o c o r r e l a t i o n s w ith s ta n d a r d d e v i a t i o n N , N b e in g th e sample s i z e ."'2 D urbin(20) showed t h a t th e 12 Anderson(4) p ro v id ed th e proof f o r t h i s f o r samples t h a t are m o d era te ly l a r g e . 65 s ta n d a r d d e v i a t i o n of th e e s tim a te d a u t o c o r r e l a t i o n s may i. indeed be much s m a lle r than N 2 f o r sm all l a g s , t h e r e f o r e , u n d e r e s t i m a t in g th e s i g n i f i c a n c e of th e s e a u t o c r r e l a t i o n s . T e s ts of s i g n i f i c a n c e f o r th e sample a u t o c o r r e l a t i o n s were c a r r i e d f o r th e f i t t e d models w ith l a g l e n g t h s of one and two. Durbin-Watson and th e B o x -P ierce Q s t a t i s t i c f o r f i r s t and h ig h e r o rd er a u t o c o r r e l a t i o n were a ls o i n s p e c t e d . No i n d i c a t i o n of s i g n i f i c a n t a u t o c o r r e l a t i o n of r e s i d u a l s was found f o r th e second o r d e r v e c t o r a u t o r e g r e s s i v e m odels. The Durbin-Watson s t a t i s t i c s were c lo s e to two in v a lu e f o r a l l e q u a tio n s , w h ile th e Q - s t a t i s t i c s were found to be r e l a t i v e l y sm all a t a m arg in al s i g n i f i c a n c e l e v e l of a t l e a s t .30 fo r a l l e q u a tio n s e x ce p t f o r th e e q u a tio n f o r p r i c e s in th e U.S. T his was due to some r e s i d u a l c o r r e l a t i o n a t l a g 12. Since s e a s o n a l i t y was p r e v io u s ly removed from th e d a ta , we assumed t h i s t o be due to sample s i z e f a c t o r s and a c c e p te d a VAR of l e n g t h 2 . ^ T e s tin g f o r model homogeneity was done by s p l i t t i n g th e sample a t d i f f e r e n t i n t e r v a l s and t e s t i n g f o r e q u a l i t y between th e two s e t s of a u t o r e g r e s s i o n c o e f f i c i e n t s c o rre s p o n d in g to th e two subsam ples. The t e s t s i n d i c a t e d ^3 T his choice i s even f u r t h e r su p p o rte d by i n s p e c t i o n of a u t o c o r r e l a t i o n s and p a r t i a l a u t o c o r r e l a t i o n s of th e i n d i v i d u a l s e r i e s . The most s u i t a b l e model was a f i r s t o rd er a u t o r e g r e s s i v e one in f i r s t d i f f e r e n c e s of the v a r i a b l e s which red u c es t o an a u t o r e g r e s s i v e model w ith two l a g s . 66 t h a t only th e p e rio d from January 1974 to O ctober 1979 can be t r e a t e d as c o v a ria n c e s t a t i o n a r y . That i s , th e model p a ra m e te rs e s tim a te d over th e p e rio d 1974 to 1980 cannot be c o n s id e r e d tim e i n v a r i a n t . 1 ii Chow t e s t s ' n were done f o r t e s t i n g th e s t a b i l i t y over time of a s i n g l e e q u a tio n , as w e ll as l i k e l i h o o d r a t i o I t e s t s f o r the s t a b i l i t y of th e system as a whole. Again, h e re , due to n o n -n o rm a lity of r e s i d u a l s , th e t e s t s t a t i s t i c may be b ia s e d , so we lo ok ed a t both the s i n g l e - e q u a t i o n and l i k e l i h o o d r a t i o t e s t s i n making any i n f e r e n c e c o n ce rn in g ithe s t r u c t u r a l s t a b i l i t y over tim e. When th e p e rio d of s p l i t was such t h a t i n s u f f i c i e n t d e g re e s of freedom were l e f t in one of the sam ples, t e s t i n g was done f o r the e q u a l i t y of c o e f f i c i e n t s of the VAR e s tim a te d over the whole sample and th o s e e s t im a t e d over th e sub-sam ple w ith s u f f i c i e n t d e g re es of freedom. The r e s u l t s f o r the two mid sample s p l i t s in Jan u ary of 1977 and 1978, and f o r th e s p l i t in November of 1979 a re summarized in T able 1 and T ab les 8 , 9 and 10 in Appendix B. I I 1 il The e x a c t f o r m u la tio n of the t e s t f o llo w s th e framework s e t by F r a n k l in M. F i s h e r ( 2 3 ) . The o r i g i n a l paper by Chow i s " T e sts of E q u a lity Between S u b s e ts Of C o e f f i c i e n t s In Two L in e a r R e g re ss io n s, 1 1 E c o n o m e tric a . 28 ( 1960), 591-605 . 67 I For both mid sample s p l i t s no s i g n i f i c a n t d i f f e r e n c e jappears to e x i s t between th e two p e rio d s f o r th e m odels as ! 'a whole. The s i n g l e e q u a tio n t e s t s , though, show t h a t a d i f f e r e n c e between th e two p a r t s of the sample a p p e a rs in th e e q u a tio n f o r dom estic money fo r both S w itz e r la n d and France w ith th e s p l i t i n 1977 (T able 8 ), and f o r Germany and S w itz e r la n d f o r th e s p l i t in 1978 (T ab le 9 ) , and i n th e e q u a tio n f o r th e p r i c e of gold in t h e model fo r Canada f o r the s p l i t in 1978. This a p p a r e n t s t r u c t u r a l s h i f t may be e x p la in e d by a s e r i e s of monetary p o lic y changes, and w i l l be used t o i d e n t i f y demand f a c t o r s . A f te r y e a r s of a high r a t e of growth of th e money supp ly, and c o n se q u e n tly high r a t e s of i n f l a t i o n a monetary squeeze was imposed i n a l l of th e s e c o u n t r i e s s t a r t i n g i n 1 9 7 8 . Other p o lic y changes w ith consequences a f f e c t i n g th e jmonetary s e c t o r were a l s o implemented. In S w itz e r la n d , s e v e re r e s t r i c t i o n s on c a p i t a l in v e s tm e n ts in th e Swiss f r a n c were imposed, a f f e c t i n g n e g a t iv e i n t e r e s t r a t e s on f o r e i g n i n v e s tm e n ts . These r e s t r i c t i o n s were l a t e r eased and n e g a t iv e i n t e r e s t r a t e s d is a p p e a re d by th e end of 1 9 7 9 . In Germany, th e p o lic y of r e s t r i c t i n g f o r e i g n in v e s tm e n t, which has been in e f f e c t th ro u g h o u t th e 7 0 s, was r e v e r s e d in e a r l y 1980. C o n tr o ls on movements of c a p i t a l have been in e f f e c t in France th ro u g h o u t th e sample p e rio d e x c e p t f o r 68 TABLE 1 T e s ts o f S t r u c t u r a l S t a b i l i t y PERIOD CANADA FRANCE GERMANY SWITZERLAND 1974_ 197 6 X2 (91 ) x 2 <91) x?<91) X2 C 91) vs. = 85.85 = 97.26 = 86.09 = 88.40 1977-1980 ( .6 3 ) (.3 0 ) (.6 2 ) (.5 5 ) 1974-1977 X2 (91) x2 (91) x 2(91 ) X 2 (91) vs. ' ' = 74.51 = 88.77 = 96.50 = 87.13 1978-1980 ( .8 9 ) ( .5 4 ) (.3 2 ) (.5 9 ) 1974_1979 X2 ^ 91) X2 (91) x 2 (91) X2 (91) vs. =255.63 =218,21 =204.06 =18 9 .3 1979-1980 (0) (10-11 ) (10- 9 ) (10~ 8 ) The l i k e l i h o o d r a t i o s t a t i s t i c s a re d i s t r i b u t e d x w hile the f i g u r e s in p a r a n t h e s i s a re th e m arg in al s i g n i f i c a n c e l e v e l s . A m arg in al s i g n i f i c a n c e l e v e l below .10 im p lie s ith a t th e n u ll h y p o t h e s is i s r e j e c t e d a t th e 90% c i n f i d e n c e l e v e l . s e c u r i t y t r a d i n g s , w h ile r e s i d e n t s must c o n v e r t f o r e i g n ! exchange proceeds from th e s a l e of s e c u r i t i e s w i t h i n a two month p e rio d . Moreover, th e e s t a b l is h m e n t of th e European Monetary System ( EMS ) in March of 1979 r e v iv e d th e SNAKE agreem ent which s t i p u l a t e d t h a t EMS members must m a in ta in q u a s i - f i x e d exchange r a t e s w ith 4.5% m argins of f l u c t u a t i o n s , and i n s t i t u t e d th e European cu rre n cy u n i t 69 (ecu) which i s a w eig h ted b a s k e t of a l l EEC member c u r r e n c i e s . The a l l too f a m i l i a r l a c k of monetary autonomy a r i s i n g from f i x e d r a t e regim es fo llo w ed , but i n t h i s case th e q u a s i - f i x e d regime r e s u l t e d i n a l e s s s e v e re l a c k of autonomy. F i n a l l y , U.S. ad o p tio n of exchange r a t e management to s t a b i l i z e th e d o l l a r in 1978 and 1979 was su p p o rte d by both Germany and S w i t z e r l a n d . . All of th e s e f a c t o r s can be viewed to have caused th e h e t e r o g e n e i t y in th e money p r o c e s s e s . For th e sample s p l i t i n 1979, model homogeneity was i s t r o n g l y r e j e c t e d f o r a l l c o u n t r i e s w ith m arg in al s i g n i f i c a n c e l e v e l s t h a t a re not s t a t i s t i c a l l y d i f f e r e n t i ifrom z e r o . T est r e s u l t s f o r i n d i v i d u a l e q u a tio n s (T able 1 0 ) show t h a t th e s t r u c t u r a l s h i f t i s c o n c e n tr a te d in th e e q u a t io n s f o r gold and to a l e s s e r degree in th e e q u a tio n s f o r i n t e r e s t r a t e d i f f e r e n t i a l . The November 1979 d a te was chosen f o r th e sample s p l i t to c o rre sp o n d t o th e gold c r i s i s f u e l l e d by p o l i t i c a l f e a r s c o n ce rn in g the [Afghanistan in v a s io n and th e h o sta g e c r i s i s in I r a n , and economic news c o n c e rn in g th e U.S. T reasury s to p p in g i t s i monthly a u c t io n s of gold in O ctober. The ensu in g | i i s p e c u l a t i v e f e v e r r e f l e c t e d a fundam ental change in th e demand f o r gold. I t became c l e a r t h a t they re g a rd t h i s m etal as som ething which r e t a i n s i t s v a lu e when paper 70 c u r r e n c i e s a r e s u s p e c t. This p e r c e p t i o n was r e i n f o r c e d by p e r s i s t e n t high l e v e l s of i n f l a t i o n , th e t i g h t supply of gold and th e absence of r e c e n t major d i s c o v e r i e s of new gold. G o ld 's a p p r e c i a t i o n a g a i n s t th e d o l l a r , as w e ll as o t h e r m ajor c u r r e n c i e s , made th e gold f l u r r y in 1979 c o n sp ic u o u s ly d i f f e r e n t from p re v io u s ones. The a p p a re n t i n s t a b i l i t y in i n t e r e s t r a t e d i f f e r e n t i a l s may be e x p la in e d by th e advent of the new l e a d e r s h i p of Volcker f o r the F e d e ra l Reserve System in th e U .S ., and th e p o lic y sw itch from o p e r a t i n g thro ug h i n t e r e s t r a t e s t o s e t t i n g i n t e r m e d i a t e t a r g e t s f o r th e money base c r e a t i n g r e p e r c u s s i o n s in th e i n t e r n a t i o n a l f i n a n c i a l m a rk e ts . This change l e a d to g r e a t e r f l u c t u a t i o n s in i n t e r e s t r a t e s . S p e c u la ti v e f o r c e s t r a n s l a t e d t h i s change and th e announced f i g h t a g a i n s t i n f l a t i o n i n t o h ig h e r i n t e r e s t r a t e s w ith re c o rd h ig h s r e a c h in g 19% in December of 1980. To sum up, the model homogeneity t e s t s su g g e st t h a t th e imodel e q u a tio n s e x h i b i t s t a b l e b e h av io r over th e p e rio d from 1974 to 1979 and t h i s p e rio d w i l l be used i n the s i m u la t io n of the M A r e p r e s e n t a t i o n and th e f a c t o r a n a l y s i s to f o llo w . No f u r t h e r sample s p l i t s were perform ed u sin g 1974-1979 as th e p e rio d over which homogeneity i s to be t e s t e d s in c e th e sample s i z e would be too sm all to w a rra n t any r e l i a b l e r e s u l t s . I t would be 71 i n t e r e s t i n g t o study th e b eh avio r of t h i s model when a v a r i a b l e m easuring i n v e s t o r s ’ c o n fid e n c e in th e w orld economy i s added. However, t h a t i s beyond th e scope of t h i s stu d y . 3.4 DECOMPOSITION OF VARIANCE The r e s u l t s from a tim e s e r i e s f a c t o r a n a l y s i s on th e f o r e c a s t e r r o r v a r i a n c e s a re p r e s e n te d h e re . The method used in t h e f a c t o r i z a t i o n of th e c o v a ria n c e m a tr ix , > of the e s t im a t e d r e s i d u a l s from th e second o rd e r VAR model i s th e C holeski decom position where a lower t r i a n g u l a r m a trix S i s computed, such t h a t X)=SS*. S u b seq u e n tly , th e r e s i d u a l s a re tra n s fo rm e d i n t o o rth o g o n a l form by i p r e m u l t i p l i c a t i o n of the v e c to r by the i n v e r s e of th e f a c t o r S. The te c h n iq u e i s e q u i v a l e n t to e s t i m a t i n g a m o d ifie d a u t o r e g r e s s i v e system u sin g th e method of seem ingly u n r e l a t e d e q u a tio n s where th e e q u a tio n c o rre s p o n d in g t o th e i t h elem ent of th e v e c to r tim e s e r i e s Zt has in a d d i t i o n to a l l th e lag g e d v a r i a b l e s on t h e r i g h t hand s i d e the contemporaneous v a lu e s of a l l th e p re c e d in g i-1 e le m e n ts. The r e s u l t i n g c o v a ria n c e m a tr ix of r e s i d u a l s w i l l be d ia g o n a l. I t should be noted h e re t h a t th e o rd e r in which th e elem ents of Z^ a re l i s t e d i s i m p o r ta n t. The f i r s t elem ent 72 i s allow ed to have a contemporaneous e f f e c t on a l l o th e r v a r i a b l e s in th e system , w hile i t s e l f being i n i t i a l l y u n a f f e c t e d by contemporaneous i n n o v a tio n s t o th e r e s t of the elem ents th us behaving as a p re d e te rm in e d v a r i a b l e . C onsequently, when a v a r i a b l e i s p lac ed f i r s t i t s in n o v a tio n s would, in g e n e r a l , be e xp ected to e x p la in a r e l a t i v e l y g r e a t e r f r a c t i o n of v a ria n c e as compared to i t s c o n t r i b u t i o n when p la c e d f u r t h e r down in th e o r d e r i n g . In th e l a t t e r c a se , th e number of r e g r e s s o r s i s l a r g e . This c au se s th e u n e x p la in e d v a r ia n c e to be s m a ll e r , t h e r e f o r e , a llo w in g th e i n n o v a tio n in th e v a r i a b l e to c o n t r i b u t e a sm a lle r f r a c t i o n in e x p la in i n g i t s own movements. The o rd e r chosen should match the p r e d i c t i v e v a lu e each component p o s s e s s e s fo r i t s e l f and th e o t h e r components. Economic th eo ry as w e ll as e x p e r im e n ta tio n should be the guide to th e most a p p r o p r i a t e o r d e r i n g . Four d i f f e r e n t o r d e r i n g s of th e ele m e n ts of were ex perim en ted w ith in c a r r y i n g th e d eco m p o sitio n of v a r i a n c e . The c o n te n tio n in p o r t f o l i o b a la n c e th eo ry i s t h a t changes in a s s e t p r i c e s r e p r e s e n t movements towards th e l e v e l s t h a t would c l e a r the a s s e t m a rk e ts . T h e re fo re , changes in a s s e t s to c k s as w ell as f a c t o r s i n f l u e n c i n g i n v e s t o r s ' demand w i l l le a d to a s s e t p r i c e a d ju s tm e n ts . In th e c o n te x t of th e p r e s e n t model, t h i s t r a n s l a t e s to say ing 73 t h a t money in n o v a t i o n s , and in n o v a t i o n s in i n t e r e s t r a t e s jin th e case of th e monetary a u t h o r i t i e s 1 a d o p tio n of an I l i n t e r e s t r a t e p o lic y , account l a r g e l y f o r th e v a r i a b i l i t y in a s s e t p r i c e s . Based on th e d i s c u s s i o n above co n ce rn in g th e cho ice of an a p p r o p r i a t e o r d e r i n g , a s s e t p r i c e s were put l a s t , w h ile s to c k s and i n t e r e s t r a t e s were put f i r s t . I n te r c h a n g i n g th e p o s i t i o n s f o r p r i c e of gold and exchange r a t e s did n o t a f f e c t th e components of v a r ia n c e due to th e o th e r v a r i a b l e s . S t r a n g e l y , both seem to have improved p r e d i c t i v e power when p la c e d l a s t which may be i n t e r p r e t e d to mean t h a t the c o r r e l a t i o n between them i s not due to a cause e f f e c t one, r a t h e r c o n firm in g th e e x i s t e n c e of a jcommon c a u s a l f a c t o r . i When a s s e t p r i c e s were p lac ed f i r s t in th e o r d e r i n g t h e i r p r e d i c t i v e v a lu e in o th e r e q u a tio n s did n ot i n c r e a s e by any s i g n i f i c a n t amount. T his c o nfirm s th e low p r e d i c t i v e v a lu e of exchange r a t e s f o r th e o th e r v a r i a b l e s land j u s t i f i e s p la c in g i t a t th e bottom of the o r d e r i n g . However, an i n t e r e s t i n g change noted was t h a t th e i n c r e a s e i n g o l d 's c o n t r i b u t i o n to th e v a r i a b i l i t y in th e money supply in th e U.S. matched c l o s e l y th e drop in th e U.S. money supply induced v a r i a t i o n in th e p r i c e of gold when t h e i r o r d e r i n g s a re r e v e r s e d . 74 Based on th e s e e x p e rim e n ts th e o rd er chosen p la c e s money sto c k s f i r s t , fo llo w ed by the i n t e r e s t r a t e d i f f e r e n t i a l , consumer p r i c e i n d i c e s , p r i c e of gold, and th e exchange r a t e . The d e co m p o sitio n s of v a r i a n c e based on t h i s o r d e r i n g a re shown in t a b l e s 2 and 3 f o r th e 24 month ahead f o r e c a s t s , w h ile th e d e co m p o sitio n s u sin g from 1 to 24 m onth-ahead f o r e c a s t s a re p r e s e n te d i n Appendix B T ab les 11 to 14. TABLE 2 D ecom position of the Exchange Rate F o r e c a s t E rro r V a ria n ce Decom position of the 24-Month Ahead F o r e c a s t E rro r V aria n ce of Exchange R ates due to In n o v a tio n s in : MUS M D INT PUS PD PGOLD EX CANADA FRANCE GERMANY SWITZERLAND .27 .31 .0 2 .07 .04 C M O • .15 • u> 0 0 .32 .23 .03 .14 .09 .0 2 .16 .14 .0 2 .2 0 .1 0 .23 .0 8 .2 2 .27 .09 .03 .06 .0 8 .0 2 .45 The f i g u r e s above r e p r e s e n t th e c o n t r i b u t i o n of each of the 7 v a r i a b l e s to th e v a r i a n c e of th e exchange r a t e s . For example, 31% of th e v a r i a n c e i n th e C anadian/U .S. d o l l a r exchange r a t e i s due to in n o v a t i o n s in U.S. money. 75 TABLE 3 C o n t r ib u t i o n of Exchange Rate I n n o v a tio n s to The V ariance of The Model V a r ia b l e s The C o n t r ib u t i o n of Fo reign Exchange Rate In n o v a tio n s to th e 24-Month Ahead F o r e c a s t E rro r V ariance in : MUS MD INT PUS PD PGOLD EX CANADA .03 .16 .05 .09 .25 on o • .38 FRANCE .06 .03 .08 .01 .02 .03 .16 GERMANY .01 .01 .02 .02 .03 .01 .22 SWITZERLAND .03 .35 .13 .24 .22 .11 .45 The f i n d i n g s a re only m ild ly s u p p o r t i v e of th e p o r t f o l i o view of the d e te r m in a n ts of f o r e i g n exchange r a t e s . Monies a re the s i n g l e most im p o rta n t f a c t o r i n f l u e n c i n g exchange r a t e s . Only in th e case of the Deutschmark do i n n o v a tio n s in th e i n t e r e s t r a t e d i f f e r e n t i a l account f o r any i s i g n i f i c a n t f r a c t i o n of th e v a r ia n c e in exchange r a t e . Moreover, th e combined e v id en ce f o r the fo ur c o u n t r i e s c o n s id e re d i n d i c a t e s t h a t a s i g n i f i c a n t p o r t io n of th e v a r ia n c e in exchange r a t e s i s l e f t u n e x p la in e d by the v a r i a b l e s c o n s id e r e d . The i n f l u e n c e of gold on exchange r a t e s i s d e t e c t e d f o r the Canadian d o l l a r and Deutschmark only , and t h i s 76 i n f l u e n c e i s q u i t e weak. However, th e f i n d in g t h a t U.S. money and p r i c e s a re th e most im p o rta n t d e te r m i n a n ts of the i p r i c e of gold s u g g e s ts t h a t i n v e s t o r s choose between gold and U.S. d o l l a r s in form ing t h e i r o ptim al p o r t f o l i o depending on t h e i r e x p e c t a t i o n s of th e f u t u r e economic env iro nm ent. Except fo r the French f r a n c , f l u c t u a t i o n s in exchange r a t e s can m ainly be accounted f o r by t h e i r own in n o v a t i o n s , where they c o n t r i b u t e from 22 to 45% of th e v a r i a n c e . This r e p r e s e n t s th e p a r t of exchange r a t e s which can not be f o r e c a s t e d by th e v a r i a b l e s c o n s id e re d and which r e f l e c t s th e i n f l u e n c e of o t h e r u n in clu d ed v a r i a b l e s and pure i n n o v a tio n s in exchange r a t e s . For F rance, th e combined e f f e c t of dom estic and U.S. money i n n o v a tio n s p ro v id e th e s t r o n g e s t e x p la n a to r y power of over one h a l f of the v a r ia n c e in th e French f r a n c / d o l l a r r a t e . Honey i n n o v a tio n s have a s i g n i f i c a n t i n f l u e n c e on th e o th e r exchange r a t e s as w e ll, a c c o u n tin g f o r 16, 33, and 36% of th e v a r i a t i o n s in the Deutschmark, Canadian d o l l a r , and Swiss f r a n c , r e s p e c t i v e l y , w ith U.S. money a c c o u n tin g f o r the b ig g e r s h a r e . S u r p r i s i n g l y , i n t e r e s t r a t e d i f f e r e n t i a l s which a re g e n e r a l l y assumed to have a g r e a t i n f lu e n c e on exchange r a t e s in th e s h o r t - r u n , e x p la in only a sm all p e rc e n ta g e of the v a r ia n c e in th e Canadian 77 d o l l a r , Swiss f r a n c , and French f r a n c . Only in th e case of |the Deutschmark do i n n o v a t i o n s in i n t e r e s t r a t e s appear to induce p o r t f o l i o - a d j u s t m e n t s t h a t acco u n t f o r up to 20% of the v a r i a t i o n s in i t s r a t e of exchange w ith r e s p e c t to th e U.S. d o l l a r . S i m i l a r l y , U.S. and dom estic p r i c e i n n o v a t i o n s have no t s i g n i f i c a n t e f f e c t on th e exchange r a t e s s t u d i e d e x c e p t f o r the Deutschmark. German p r i c e in n o v a t i o n s p ro v id e th e s t r o n g e s t e x p la n a to r y power of c lo s e to one f o u r t h of the v a r ia n c e in th e D e u ts c h m a rk /d o lla r r a t e . I n n o v a tio n s in t h e p r i c e of gold appear to have a s h o r t - t e r m e f f e c t only on th e Canadian d o l l a r . They e x p la in a p p ro x im a te ly 15% of th e v a r i a t i o n in th e Canadian exchange r a t e , but e x p la in v i r t u a l y no v a r i a t i o n in th e o th e r t h r e e exchange r a t e s . Looking a t th e gold m ark et, th e p r i c e of gold i s mainly in f lu e n c e d by U.S. money supply and p r i c e s . The U.S. money supply e x p la in s from one t h i r d t o over one h a l f of th e t o t a l v a r i a t i o n in th e p r i c e of gold, w h ile i n n o v a t i o n s in p r i c e s c o n t r i b u t e around one f i f t h of th e v a r i a n c e . G old’ s own i n n o v a t i o n s a cc o u n t from 3 to 8% of i t s v a r i a n c e . These r e s u l t s c l e a r l y i n d i c a t e t h a t u n a n t i c i p a t e d changes in U.S. money and p r i c e s cause f l u c t u a t i o n s in th e gold m a rk e t. A p o s s i b l e i n t e r p r e t a t i o n i s t h a t i n v e s t o r s r e a c t 7 8 T A B L E 4 Decom position of th e V ariance of P r ic e of Gold D ecom position of th e 24-Month Ahead F o r e c a s t E rro r V arian ce of P r ic e of Gold due to In n o v a tio n s in : MUS M D INT PUS PD PGOLD EX CANADA .44 .01 .24 .17 .05 .06 .03 FRANCE .54 .05 .08 .21 .05 .03 .03 GERMANY .35 .07 .05 .21 .23 .08 .01 SWITZERLAND .51 .03 .02 .19 .04 .08 .11 to i n n o v a t i o n s in "world" money and p r i c e s by i n i t i a l l y a d j u s t i n g t h e i r gold h o ld in g s , and e v e n t u a l l y t h e i r o th e r a s s e t s , c o n firm in g th e view t h a t gold i s h e ld m ainly as a s t o r e of value when th e p u rc h a si ng power of maj or c u r r e n c i e s i s u n c e r t a i n . There a re many p o s s i b l e re a s o n s a s t o why U.S. money supply has such a big i n f l u e n c e on gold. One such reason i s th e t r a d i t i o n a l t i e between th e d o l l a r and gold which p e r s i s t e d d e s p i t e e f f o r t s to dem onetize gold Both the d o l l a r and gold a re s t i l l used as m ajor r e s e r v e a s s e t s , and gold r e s e r v e s have been used in f o r e i g n exchange management by the U.S. and o t h e r c o u n t r i e s . A second re a so n has to 79 do w ith monetary changes in th e U.S. which have i n f lu e n c e d i n v e s t o r s ’ e x p e c t a t i o n s r e g a r d i n g t h e d o l l a r and probably le a d to p o r t f o l i o a d ju s tm e n ts between th e d o l l a r and gold. F i n a l l y , th e f a c t t h a t gold s to c k s a r e not in c lu d e d in th e a n a l y s i s may have a b e a r in g on th e r e s u l t s , s in c e a l a r g e f r a c t i o n of th e s e s to c k s a re h e ld as r e s e r v e s by c e n t r a l banks. 3.5 SIMULATION OF SYSTEM RESPONSE 10 RANDOM SHOCKS The moving averag e r e p r e s e n t a t i o n of the system i s sim u la te d by g e n e r a t i n g th e dynamic re s p o n se of the model v a r i a b l e s to a s e t of shocks t o th e system . The s p e c i f i c shocks used in g e n e r a t i n g th e im pulse r e s p o n s e s a re o r t h o g o n a l iz e d in n o v a t i o n s t o each of th e system v a r i a b l e s . I P l o t s of re s p o n s e s over 36 s t e p s ahead a re shown in C h a rts 1 through 40 in Appendix C. C harts 5 through 12 show the r e s p o n s e s c o rre s p o n d in g to th e exchange r a t e s and th e p r i c e of gold g e n e ra te d to a shock in each of th e system v a r i a b l e s . In a d d i t i o n to comparing th e s t r e n g t h of the c o r r e l a t i o n s between exchange r a t e s and a s s e t m arket f a c t o r s over tim e, C h a rts 13 through 40 e n ab le us to o b serv e the s ig n s of th e s e c o r r e l a t i o n s t h a t a r e im p lie d by th e d a ta . Each of th e s e c h a r t s shows a s e t of r e s p o n s e s by a l l th e model components g e n e ra te d t o a shock in only one 8 0 v a r i a b l e of th e system . The im pu lses a re s c a le d to allow p l o t t i n g th o se p e r t a i n i n g to d i f f e r e n t v a r i a b l e s on th e i same p l o t . S c a lin g i s done by d i v i d i n g th e r e s p o n s e s of each v a r i a b l e by th e sta n d a rd d e v i a t i o n of r e s i d u a l s in t h a t v a r i a b l e . By p l o t t i n g t o g e t h e r th e r e s p o n s e s of d i f f e r e n t v a r i a b l e s to a s p e c i f i c shock, we may be a b le to d e t e c t th e c h a n n e ls of i n f lu e n c e s t h a t would h e lp us i n t e r p r e t any d e v i a t i o n s in th e im p lied b e h a v io r of the d ata from th e t h e o r e t i c a l p r e d i c t i o n s and th u s p o s s ib ly r e c o n c i l e them. The in f o r m a tio n conveyed by th e im pulse resp o n se c h a r t s , summarized in t e x t C harts 1 thro ug h 4 about th e r e a c t i o n s of exchange r a t e s to exogenous shocks in a s s e t m a rk e ts , a re in c o n c e rt w ith th e r e s u l t s o b ta in e d from th e decom p osition of v a r ia n c e in th e p re v io u s s e c t i o n . The Deutschmark emerges as th e only c u rre n cy whose f l u c t u a t i o n s may be e x p la in e d by the i n t e r e s t r a t e s d i f f e r e n c e s view. Money s to c k s ( in p a r t i c u l a r U.S. money ) have th e most prolonged i n f l u e n c e on exchange r a t e s , w h ile i n t e r e s t d i f f e r e n t i a l s produce no such s i g n i f i c a n t e f f e c t . With r e s p e c t to th e gold m ark et, no c a u s e - e f f e c t r e l a t i o n s h i p a p p e a rs t o e x i s t between gold and f o r e i g n exchanges, w h ile U.S. money and p r i c e s e x e r t th e b i g g e s t i n f l u e n c e on th e p r i c e of gold l jwhich c on firm s th e view t h a t gold i s p e rc e iv e d by i n v e s t o r s 8 1 to be a s t o r e of v alu e in i n f l a t i o n a r y tim es or when economic c o n d i t i o n s a re u n s t a b l e . For i l l u s t r a t i o n p urp oses we p r e s e n t h e re th e re s p o n s e s of gold in th e Canadian model only (C h a rt 5 ) . E xam ination of C harts 1 thro u g h 4 y i e l d s th e fo llo w in g o b s e r v a t i o n s . The most s i g n i f i c a n t e f f e c t on exchange r a t e s i s due to in n o v a t i o n s in exchange r a t e s , w hile th e most p e r s i s t e n t e f f e c t i s due to shocks in U.S. money. The r e a c t i o n s to exchange r a t e shocks decay r a p i d l y i l l u s t r a t i n g th e speed w ith which f o r e i g n exchange m ark ets a d j u s t . Indeed, re s p o n s e s of exchange r a t e s to any kind of shock a re dampened w i t h in th e f i r s t y e ar which i n d i c a t e s t h a t f o r e i g n exchange m ark ets a re s t a b l e w ith r e s p e c t to ! our v a r i a b l e s and th e time frame of t h i s a n a l y s i s . An u n a n t i c i p a t e d in c r e a s e in U.S. money low ers th e v a lu e of th e d o l l a r r e l a t i v e to o th e r c u r r e n c i e s . The f a l l in exchange r a t e s i s alm ost i n s t a n t a n e o u s w ith the r i s e in U.S. money. The only e x c e p tio n i s th e Canadian d o l l a r where i t s r a t e of exchange w ith th e U.S. d o l l a r f a l l s f o r 1 p e rio d o n ly , a f t e r which i t r i s e s s i g n i f i c a n t l y above i t s t r e n d . This may be an i n d i c a t i o n of th e s e n s i t i v i t y and high speed of a d ju s tm e n t of th e Canadian d o l l a r to U.S. money i n n o v a t i o n s . C on versely , an i n c r e a s e in dom estic money sends th e exchange r a t e up w ith a more p ro longed v i x n o i w -cm xoa • t / > * c o-< w p w z o ' c w n ' s 82 CHART 1 RESPONSES TO US. MONEY SHOCKS - 0 . 3 - - 0 . 7 - 2 4 6 8 12 10 1 8 20 22 2 6 1 4 16 28 3 4 24 30 32 36 MONTHS fHE FOLLOWING SYMBOLS REPRESENT THE RESPONSES OF THE • DEUTSCHMARK ! a CANADIAN DOLLAR 6 FRENCH FRANC X SWISS FRANC I w?fooiw < n z o i n —-»v>m3oo o-« ttntflZCDwwx 83 CHART 2 RESPONSES TO DOMESTIC MONEY SHOCKS 0 - 8 - 0 . 3 - 00 -02 - - 0 . 3 - - 0 . 4 - 32 34 20 22 26 30 24 2 8 MONTHS THE F0LL0W1NC STM80LS REPRESENT THE NESPONSES OF THE• DEUTSCHMARK n CANADIAN DOLLAR A FRENCH FRANC X SWISS FRANC CHART 3 84 RESPONSES TO INTEREST DIFFERENTIAL SHOCK 8 10 12 14 16 18 2 0 22 24 26 2 8 3 0 3 2 34 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSES OF THE* • DEUTSCHMARK a CANADIAN DOLLAR A FRENCH FRANC X SWISS FRANC Q r o o “no — O —< • W R I A Z O ' C W 'T l X CHART 4 RESPONSES TO PRICE OF GOLD SHOCKS o . 20 0 . 10 - 0 .0 5 - - 0 0 5 - - 0 - 20 - 34 30 36 20 22 26 2 8 32 MONTHS THE FOLLOW 1NC STMBQlS REPRESENT [HE RESPONSES OF T H E ■ . DEUTSCHMARK O CANADIAN DOLLAR A FRENCH FRANC X SWISS FRANC or on m o m o —xt m o w n w i o 'o w r i a CHART 5 86 CANADA RESPONSES' OF PRTCE- OF1 GOLD 10- 9 - 8 - 6- 5 - 3 - 2 - - 2 - - 3 - - 4 - - 5 - 2 4 6 8 10 12 t 4 16 18 20 22 24 26 28 3 0 36 3 2 34 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSE OF P R IC E OF GOLD TO SHOCKS TO* + U.S. MONEY « DOMESTIC MONEY □ INTEREST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES o PRICE OF COLD X EXCHANGE RATE 87 e f f e c t than t h a t due to U.S. money shocks in th e case of Canada and F rance. This s u g g e s ts t h a t i n n o v a tio n s in U.S. money le a d to p o r t f o l i o a d ju s tm e n ts and a s h o r t - t e r m e f f e c t on th e French f r a n c and th e Canadian d o l l a r , w hile in n o v a tio n s in dom estic money may have p e r s i s t e n t r e a l e f f e c t s t h a t i n f l u e n c e th e exchange r a t e f o r lo n g e r p e r i o d s . An e x c e p tio n to th e above i s th e b e h a v io r of th e Deutschmark where i t s r a t e of exchange v i s - a - v i s th e d o l l a r f a l l s i n i t i a l l y , and does not i n c r e a s e u n t i l a f t e r alm ost e ig h t e e n months. Economic th e o ry s u g g e s ts th e fo llo w in g i n t e r p r e t a t i o n . The v a lu e of th e d o l l a r would f a l l in re sp o n se to an i n c r e a s e in U.S. money su p p ly , w h ile an i n c r e a s e in th e demand f o r U.S. money le a d s to an a p p r e c i a t i o n in th e d o l l a r . Data on money s u p p l i e s may not n e c e s s a r i l y r e f l e c t th e exogenous v a r i a t i o n s in s to c k s of money but may w ell r e f l e c t changes in t h e demand f o r money. This i s th e usu al i d e n t i f i c a t i o n problem of demand and su p p ly . Hence, we may i n f e r t h a t th e money in n o v a tio n s a re in n o v a tio n s in th e s to c k s of money e x cep t in th e l a s t case where the D eu tsch m ark 's v alu e r i s e s in resp o n se to an in n o v a tio n in dom estic demand f o r money. However, in th e absence of c l e a r i d e n t i f y i n g c r i t e r i a t h i s i s only a su g g e ste d i n t e r p r e t a t i o n . 8 8 ! Shocks to th e i n t e r e s t r a t e d i f f e r e n t i a l have a l n e g l i g i b l e e f f e c t on exchange r a t e s e x c e p t fo r Germany. An u n a n t i c i p a t e d i n c r e a s e in U.S. i n t e r e s t r a t e s r e l a t i v e to r a t e s on th e Deutschmark, sends th e d o l l a r up. A p o s s i b l e re a so n as to why such an e f f e c t i s ob served only f o r the Deutschmark i s t h a t th e d o l l a r and th e Deutschmark command th e b i g g e s t s h a r e s in i n t e r n a t i o n a l p o r t f o l i o s , and th e e f f e c t of i n t e r e s t a r b i t r a g e i s t h e r e f o r e more pronounced in t h i s c a se . An u n a n t i c i p a t e d i n c r e a s e in do m estic i n f l a t i o n r e l a t i v e to t h a t in th e U nited S t a t e s l e a d s to a d e c re a s e in th e exchange r a t e ( t h a t i s , a r i s e in th e v a lu e of dom estic c u r r e n c i e s ) in Canada, France and S w it z e r l a n d , where th e i [in c re a s e in th e v a lu e of th e Canadian d o l l a r i s tem porary and n e g l i g i b l e . A f a l l in th e D e u ts c h m a rk /d o lla r r a t e i s ob served f o r only one p e r i o d , a f t e r . w h i c h i t r i s e s . C harts 13, 20, 27, and 34 may p ro v id e an e x p l a n a t i o n f o r t h i s u n p r e d ic te d b e h a v io r. P r ic e i n n o v a tio n s a re fo llo w ed by lo n g e r c o n t r a c t i o n a r y p e rio d s in money in France and S w itz e r la n d as compared to th e r e a c t i o n in Germany. French and Swiss a u t h o r i t i e s may be more s e n s i t i v e to i n f l a t i o n \ I and could be u sin g money as an a n t i - i n f l a t i o n a r y m easure, [ t h e r e f o r e , a t th e same time p o stp o n in g and re d u c in g th e e x t e n t to which t h e i r cu rren cy would have d e c l in e d . 89 However, s in c e we have not observed t h i s feedb ack in th e o r d e r in g e x p e rim e n ts , the so urce of t h i s c o u n te r ev id ence i s unknown. F i n a l l y , in n o v a tio n s in th e p r i c e of gold do not appear to have much i n f l u e n c e on th e f o r e i g n exchanges, a lth o u g h one can d e t e c t an i n i t i a l sm all i n c r e a s e in exchange r a t e s (a f a l l in th e v a lu e of th e d o l l a r ) . As f o r th e gold m ark et, th e p r i c e of gold a p p e a rs to be most s e n s i t i v e to U.S. money in n o v a t i o n s . Indeed, U.S. money i n n o v a tio n s have alm ost as g r e a t an i n f l u e n c e on th e p r i c e of gold as on U.S. p r i c e s (se e c h a r t s 13, 20, 27 and 34 in Appendix C ), w h ile the e f f e c t of U.S. p r i c e in n o v a t i o n s on th e p r i c e of gold a r e alm ost as s i g n i f i c a n t . Again t h i s c on firm s t h a t gold i s viewed as a s t o r e of v alue where i n f l a t i o n a r y p r e s s u r e s le a d to p o r t f o l i o s h i f t s from i U.S. d o l l a r s to gold th e re b y r a i s i n g i t s p r i c e . U n a n tic ip a te d i n c r e a s e s in th e i n t e r e s t d i f f e r e n t i a l between th e U.S. and d om estic r a t e s have sm a ll, and s h o r t - l i v e d d e p r e s s in g contemporaneous e f f e c t on th e p r i c e of g o ld . This i n d i c a t e s a sw itc h from gold to h ig h e r y i e l d i n g d o l l a r denominated a s s e t s . This i s not observed in th e case of Canada, where th e i n c r e a s e in p r i c e of gold i s alm ost as l a r g e as t h a t due to U.S. p r i c e i n n o v a t i o n s , f and as pro lon ged as t h a t due to U.S. money i n n o v a t i o n s . jF in a lly , exchange r a t e i n n o v a tio n s have an e x tre m e ly sm all e f f e c t on th e p r i c e of gold to be c o n s id e r e d s i g n i f i c a n t . Yet, we can d e t e c t a s l i g h t r i s e in th e p r i c e of gold in r e s p o n se to an u n a n t i c i p a t e d f a l l in th e v a lu e of th e U.S. d o l l a r r e l a t i v e to th e Canadian d o l l a r and French f r a n c . We may conclude from th e above a n a l y s i s t h a t i t i s th e s to c k s of money r a t h e r than th e r e l a t i v e r a t e s of r e t u r n t h a t appear to i n f l u e n c e exchange r a t e s . Perhaps th e f i n d i n g s would be more s u p p o r t i v e of th e p r e d i c t i o n s of p o r t f o l i o th e o ry i f weekly d a ta i s used s in c e one would e x p ec t any changes in a s s e t m ark ets to be q u ic k ly d i s s i p a t e d . [3 . 6 GRANGER CAUSALITY AND ECONOMETRIC EXOGENEITY R e ca ll t h a t i n t e r e s t r a t e s , a s s e t s to c k s and th e p r i c e of gold were assumed to be exogenous in our s t r u c t u r a l model f o r exchange r a t e s in Chapter I I . B efore e s t i m a t i n g a sim ple v e rs io n of t h i s model, we need to v e r i f y t h i s a ssu m p tio n . In a d d i t i o n , t e s t i n g d i r e c t l y f o r e x o g en e ity may shed some l i g h t on th e feedb ack h y p o t h e s is advanced above. This can be done u sin g an eco n o m etric t e s t due to Sim s(6 8 ). A v a r i a b l e i s c o n s id e r e d exogenous in eco n o m etric models i f i t i s not Granger caused by any of th e endogeneous 91 v a r i a b l e s of the model. A v a r i a b l e Y does not Granger cause a n o th e r v a r i a b l e X i f X can be e q u a lly w ell p r e d i c t e d from th e p a s t of X alo n e as from th e p a s t of X and Y t o g e t h e r . Sims dem o n stra te d th e e q u iv a le n c e of Granger c a u s a l i t y and econ om etric e x o g e n e ity and d e l i n e a t e d two e q u i v a l e n t and t e s t a b l e i m p l i c a t i o n s of e x o g e n e ity . B a s i c a l l y , i f th e main i n t e r e s t i s to v e r i f y th e e x o g e n e ity of a v a r i a b l e X to an endogeneous p ro c e s s d e s c r i b i n g th e b e h a v io r of some v a r i a b l e Y, then t h i s can be ach iev e d by r e g r e s s i n g Y on p a s t , p r e s e n t and f u t u r e v a lu e s o f X. I f th e c o e f f i c i e n t s on th e f u t u r e v a lu e s of X a re not s i g n i f i c a n t l y d i f f e r e n t from z e ro , th en th e e vidence would not c o n t r a d i c t s p e c i f y i n g X as an exogenous v a r i a b l e in a r e g r e s s i o n of Y on p a s t and c u r r e n t v a lu e s of X. Each of th e exchange r a t e s were r e g r e s s e d on the c o rre sp o n d in g i n t e r e s t d i f f e r e n t i a l s , U.S. money and th e p r i c e of g o ld . Four l a g s and l e a d s , in a d d i t i o n to th e c u r r e n t v a lu e , were used f o r th e r i g h t hand s i d e v a r i a b l e s . Four l a g s o f th e exchange r a t e s were a ls o added to the r e g r e s s o r s s in c e f i r s t and h ig h e r o rd e r s e r i a l c o r r e l a t i o n were i n i t i a l l y d e t e c t e d . No r e s i d u a l s e r i a l c o r r e l a t i o n i i was d e t e c t e d a f t e r w a r d s . The t e s t s r e s u l t s i n d i c a t e t h a t th e exchange r a t e s f o r Canada, F ra n ce , Germany and S w itz e r la n d do not Granger 92 TABLE 5 T e s ts o f Exogeneity CANADA FRANCE GERMANY SWITZERLAND F (4 ,3 6 ) F (4 ,3 6 ) F ( 4 ,36) F (4 ,3 6 ) I n t e r e s t 1 .24 1 .77 1 .42 1 .05 D i f f e re n - ( .31 ) ( .15) ( .24) ( .39) t i a l .'Price of . 33 .62 .61 .34 Sold ( .85) ( .65) (.6 5 ) ( .84) ,u.s. 1 .17 .80 .35 .47 'loney ( .34) ( .53) (.8 4 ) ( .76) jThe F s t a t i s t i c s c o rre sp o n d to t e s t i n g th e s i g n i f i c a n c e of |the fo u r l e a d s f o r each of the above v a r i a b l e s . M arginal s i g n i f i c a n c e l e v e l s a re r e p o r t e d in p a r a n t h e s i s . iA m a rg in a l s i g n i f i c a n c e l e v e l above .10 im p lie s t h a t we Jcannot r e j e c t th e h y p o t h e s is t h a t th e fo u r l e a d s a r e equal ito z e ro a t th e 90% c o n fid e n c e l e v e l . I cause the p r i c e of g o ld , U.S. money and i n t e r e s t j d i f f e r e n t i a l s between t h e U.S. and dom estic i n t e r e s t r a t e s . jHence, we may t r e a t them as e c o n o m e t r i c a ll y exogenous to th e exchange r a t e p r o c e s s e s . 93 Chapter IV ESTIMATION OF THE DYNAMIC PORTFOLIO ALLOCATION MODEL In t h i s Chapter the dynamic p o r t f o l i o model developed in Chapter I I i s e x p l i c i t l y so lv e d f o r th e case of th e l i n e a r u t i l i t y f u n c t i o n w ith a q u a d r a t i c c o s t of a d ju s tm e n t. Then a p a r t i a l e q u il i b r i u m model fo r th e f o r e i g n exchange r a t e m arket i s p r e s e n t e d . The eco n o m etric model d e riv e d c o n s i s t s o f th e f o llo w in g e q u a t i o n s : 1 . An exchange r a t e e q u a tio n d e riv e d from th e m arket c l e a r a n c e c o n d it i o n and a s s e t demand e q u a t io n s . 2. The exogenous s t o c h a s t i c p r o c e s s e s g ov ern in g th e i n t e r e s t r a t e s , p r i c e of gold, and o th e r i n f o r m a tio n v a r i a b l e s . Follow ing a d i s c u s s i o n of th e methodology in v o lv e d in e s t i m a t i n g th e model, th e r e s u l t s based on d ata f o r Canada, France, Germany, and S w itz e r la n d a re p r e s e n te d and a n a ly z e d . 94 4.1 ASSET EQUILIBRIUM JJj THE CASE OF LINEAR UTILITY The re a s o n s f o r s tu d y in g th e case of th e l i n e a r u t i l i t y f u n c t i o n a r e tw o - fo ld . The f i r s t i s to t e s t th e i m p l i c a t i o n s of p o r t f o l i o b a la n ce th e o ry of exchange r a t e s and t o i n v e s t i g a t e (a) whether i n v e s t o r s ’ b e h a v io r in f o r e i g n exchange m ark e ts can be d i s t i n g u i s h e d from t h a t of th o se who a c t as i f they were r i s k n e u t r a l ; (b) whether t r a n s a c t i o n c o s t s play a s i g n i f i c a n t r o l e in p o r t f o l i o a d ju s tm e n t s ; (c) and how, i f a t a l l gold a f f e c t s f o r e i g n exchange m a rk e ts . The second re a so n i s p e d a g o g ic a l. I t s e r v e s to i l l u s t r a t e how dynamic r a t i o n a l e x p e c t a t i o n s models a lo n g th e l i n e s o f Hansen and S a rg e n t c a p tu r e the e f f e c t of changes in th e environm ent o th e rw is e u n d e te c te d i n p o r t f o l i o models. As m entioned e a r l i e r , u t i l i t y m axim izatio n re d u c e s to m ax im iz atio n of w e a lth when th e u t i l i t y f u n c t i o n i s l i n e a r . This w i l l , t h e r e f o r e , be th e assumed o b j e c t i v e f u n c t i o n of th e a g e n t. In a d d i t i o n , we assume t h a t a d ju s tm e n t c o s t s a r e q u a d r a t i c . A q u a d r a t i c c o s t f u n c t i o n im p l i e s i n c r e a s i n g m arg in al c o s t of a d ju s tm e n t. Hence, we can not i n t e r p r e t the a d ju s tm e n t c o s t s as being only of the t r a n s a c t i o n ty p e , but a l s o a s some i m p l i c i t i n c r e a s i n g s u b j e c t i v e c o s t t h a t the i n v e s t o r a t t a c h e s t o l a r g e r p o r t f o l i o a d ju s tm e n ts . This 95 may be so due to th e c o s t of g a t h e r i n g in f o r m a tio n about th e economies of th e c o u n t r i e s i s s u i n g th e v a r io u s a s s e t s or the u n c e r t a i n t y about th e e x a c t f u t u r e r e t u r n which depend on f u t u r e r a t e s of exchange. Although th e i n v e s t o r s a r e assumed to form r a t i o n a l e x p e c t a t i o n s , t h a t does not imply knowledge of the a c t u a l f u t u r e v a lu e s . Moreover, l a r g e r p o r t f o l i o a d ju s tm e n ts imply th e p o s s i b i l i t y of a l a r g e r d iv e rg e n c e from the o p tim al in v e s tm e n t plan had th e i n v e s t o r had p e r f e c t f o r e s i g h t . The o b j e c t i v e f u n c t i o n in t h i s case can be w r i t t e n a s : ( 4 .1 ) N E0 lim ^ t j [ r t T£ + r ? T? + • • +r t Tt ] 1/2 (T1, ! 1 ) t t - 1 1 (T1, ! 1 ) t t - 1 <T? - T? -I ) t t -1 ■ • D (T2-T 2 ) t t - 1 • + Yt (Tn -Tn ) t t - 1 , n* n . ' W i t or u sin g m a tr ix n o t a t i o n N ( 4 .1 a ) Eq lim N ->° t=0 - 1 /2 (T t-T t-i),D (VT t-i) + Y t } where D i s an nxn p o s i t i v e d e f i n i t e d ia g o n a l m a trix r e p r e s e n t i n g c o s t s of a d ju s tm e n t of a s s e t h o l d in g s . To s i m p li f y th e a n a l y s i s we can r e w r i t e th e term 96 [ dCUTj. ] 1 [ d(L ) t J , where D(L) ~ Dg + D -j L i s th e l a g o p e r a t o r Dq and D-| n x n d ia g o n a l m a t r ic e s and Dq i s of f u l l rank The o b j e c t i v e f u n c t i o n becomes: -[D (L )T t ] '[ D ( L ) T t ]+ Yt which has a l i n e a r q u a d r a t i c form. The d e r i v a t i o n of th e d e c i s i o n r u l e s i s l e f t to Appendix A. In t h i s s e c t i o n we h i g h l i g h t th e i s s u e s in v o lv e d in t h i s d e r i v a t i o n , and how th e r a t i o n a l e x p e c t a t i o n s assu m p tio n , and th e assu m p tio n s r e g a r d i n g th e s t o c h a s t i c p r o c e s s e s g e n e r a t i n g th e exogenous v a r i a b l e s a re embodied in th e a n a l y s i s . The an alo gu e of e q u a tio n ( 2 . 6 ) , th e s t o c h a s t i c E u ler e q u a tio n , i s in t h i s case a l i n e a r f i r s t - o r d e r d i f f e r e n c e e q u a tio n given by ( 4 .3 ) Tt + D ' 97 The d e c i s i o n r u l e given by ( 4 .3 ) e x p r e s s e s c u r r e n t a s s e t i h o ld in g s as a f u n c t i o n of p a s t h o l d in g s , c u r r e n t and ex p ec te d f u t u r e a s s e t p r i c e s and r a t e s r e t u r n e x p re s s e d in dom estic c u rr e n c y . The r a t i o n a l e x p e c t a t i o n s h y p o t h e s is m a i n ta in s t h a t a g e n ts form e x p e c t a t i o n s about f u t u r e v a r i a b l e s , based on a l l in f o r m a tio n a v a i l a b l e a t th e tim e, t h a t a re c o n s i s t e n t w ith t h e i r a c t u a l p r o c e s s e s . Expanding th e polynom ial in D, 3 , and L, and s u b s t i t u t i n g f o r the ex p ec te d f u t u r e v a r i a b l e s th e o p tim a l p r e d i c t i o n s based th e a ssum ption t h a t e x p e c t a t i o n s a re formed r a t i o n a l l y , we end up w ith the a s s e t demand e q u a t io n s : k - D6"LTt - i +1/2 S y v k J =1 1 / 2 2 Nj ( xj ^2 + (1 “ j =1 q - i q J x | a (A.e r 1 1+ £ ( Z ( ? * J L 1=1 k=i+l where J t = < ( > = ( i n °q -n ) Rt " < * > 2Xt » ^2 " ( °n I n °q -2n > w hile Xt i s th e a u t o r e g r e s s i v e p ro c e s s d e fin e d in Chapter I I by (2 .3 ) q(L )X t = Ut and A. and Ni are f u n c t i o n s of th e e le m e n ts of D (L ). J J 98 The dynamic d e c i s i o n r u l e s given by (4 .4 ) r e p r e s e n t a system of demand e q u a tio n s f o r n a s s e t s t h a t a r e in c lu d e d w i t h i n an i n t e r n a t i o n a l f i n a n c i a l p o r t f o l i o . To c a r r y out th e r e q u i s i t e sim u lta n e o u s e q u a tio n e s t i m a t i o n of exchange r a t e s and t h e u n d e r ly in g s t o c h a s t i c p r o c e s s e s f o r our e co n o m etric problem we r e s t r i c t th e p o r t f o l i o to t h r e e a s s e t s . T h is a llo w s us to study how the exchange r a t e , a r e l a t i v e p r i c e of two a s s e t s , i s a f f e c t e d by a n o th e r a s s e t i n th e p o r t f o l i o , in t h i s case gold, and y e t l e a v e s th e model as sim p le as p o s s i b l e . To s i m p li f y th e n o t a t i o n , l e t th e t h r e e a s s e t s be denoted as f o l lo w s : th e dom estic c u rrency denom inated c o n so l, Tfc, th e f o r e i g n cu rre n cy denominated c o n so l, Tt , and gold, Gt . I f we l e t th e f o r e i g n currency be th e U.S. d o l l a r , th e v e c t o r of a s s e t p r i c e s becomes 4 = < 1 e t Pgt > and Rj- = ( r t r* 0 ) where e^. i s th e r e a l exchange r a t e e x p re s s e d in term s of th e p r i c e of a U.S. d o l l a r , Pg^ i s th e r e a l p r i c e of gold, r^. and r ^ a r e th e r e a l r e t u r n s on th e dom estic and f o r e i g n a s s e t s r e s p e c t i v e l y . In Appendix A, we show t h a t th e a g e n t ' s d e c i s i o n r u l e s can be s i m p l i f i e d to xt 9 9 ( 4 .5 ) T = T - 1/2 D"1 (J ) t t - 1 t r q-1 1 + 1 / 2 D “ % 2 a ( 3 ) ” ^ 1 + ^ C ^ 3 a k ^ i = l k = i+ l E qu atio n (4 .5 ) d e f i n e s a system of t h r e e demand e q u a t io n s f o r T^, Tj., and G^, where th e a ' s a re p a ra m e te rs of the s t o c h a s t i c p ro c e s s Xt which i n c l u d e s th e ele m e n ts of R^, J t and o t h e r v a r i a b l e s t h a t h e lp p r e d i c t Rt d e fin e d by e q u a tio n ( 2 . 3 ) . For t e c h n i c a l convenience we wish to l i m i t th e number of v a r i a b l e s and l a g s in Xt and, t h e r e f o r e , in c lu d e only the most r e l e v a n t i n f o r m a ti o n . R e s u lts from th e p re c e d in g c h a p t e r , where s t r u c t u r e - f r e e v e c to r a u t o r e g r e s s i o n s were e s t im a t e d , a re used t o r e s t r i c t th e o rd e r of th e s t o c h a s t i c p r o c e s s e s g o v ern in g i n t e r e s t r a t e s and t o i n d i c a t e which v a r i a b l e s h e lp f o r e c a s t th e s e exogenous v a r i a b l e s . As i n d i c a t e d in Chapter I I I , a l a g o rd e r of two i s a p p r o p r i a t e f o r c h a r a c t e r i z i n g th e dynamics of th e s e p r o c e s s e s . Moreover, th e f i n d i n g s s u g g e s t t h a t U.S. money i s th e main f a c t o r t h a t i n f l u e n c e s i n t e r e s t and exchange r a t e s . Keeping in mind th e b e n e f i t s from u sin g a l a r g e r number of i n f o r m a tio n v a r i a b l e s vs. th e c o s t of l o s i n g d e g re e s of freedom in our a lr e a d y n o t - t o o - l a r g e sample, we chose to use only U.S. money fo r p r e d i c t i n g f o r e i g n i n t e r e s t r a t e s . In th e e q u a tio n f o r dom estic r a t e s , only U.S. i n t e r e s t r a t e s a re in c lu d e d s i n c e the i n f l u e n c e of U.S. money on 1 0 0 d om estic i n t e r e s t i s b a s i c a l l y due to i t s e f f e c t on U.S. i n t e r e s t r a t e s . P a st exchange r a t e s a re s p e c i f i c a l l y exclud ed from th e s e e q u a t io n s a s s u g g e s te d by th e t e s t s of c a u s a l i t y t h a t were r e p o r t e d in Chapter I I I . In a d d i t i o n t o a l l of th e above v a r i a b l e s , th e c u r r e n t acco u n t w i l l be in c lu d e d i n th e e q u a tio n f o r f o r e c a s t i n g f u t u r e exchange r a t e s . The e x c l u s i o n r e s t r i c t i o n s on a ( L ) which in c lu d e th o se r e s t r i c t i o n s c o n c e rn in g th e U.S. money and p r i c e of gold e q u a t io n s allow us to w r i t e th e s t o c h a s t i c p r o c e s s e s f o r th e s e v a r i a b l e s as f o l lo w s : (2.3a) rt = a2r*_2+ a3rt _1+ a4rt _2+ udt (2.3b) r* = b lr*_1+ b2r ‘ _2+ b3r t _1+ b4rt _2+ + Vt-2+ U f t (2 .3 0 ) M* = o i r *_1+ c 2r *_2+ °3Mt - l + C4Mt - 2 + umt (2 .3 d ) Pgt = 62Pgb-2't* g3Mt - l + e 4Mt - 2 + ug t $ $ ^ ^ ( 2 .3 e ) e t = h ] / t _3+ h 2r t~2+ h 3r t ~ l + h 4r t - 2 + h 5Mt - l + h 6Mt - 2 + h_.P ,+ h„P ^ „+ h .e ,. ,+ h e +hCA 7 g t - 1 8 g t - 2 9 t - 1 1 0 t _2 1]L t - 1 + h CA + u 12 t - 2 x t where i s U.S. money. C o nsequently, th e demand e q u a t io n s f o r th e f o r e i g n a s s e t and gold can be w r i t t e n a s : l o i C 4 .5 .a) T* = t - 1 - V t + ^2r t - l + V t + V t - ! + V \ + 7T 6Mt - l + Tr7e t ( 4 . 5 . b) Gt = 1 / 2 d 3Pg t where th e tt* s a r e d e riv e d e x p l i c i t l y in Appendix A i n terras of th e p a ra m e te rs of ot(L), 6, and D. Our s p e c i f i c a t i o n of the s t o c h a s t i c a u t o r e g r e s s i v e p r o c e s s e s o f the exogenous v a r i a b l e s to have a l a g o r d e r of two im p l i e s t h a t th e r e l e v a n t v a lu e s of th e s e v a r i a b l e s in d e te rm in in g th e demand f o r th e v a r i o u s a s s e t s a re th o se of th e c u r r e n t and th e p a s t p e r i o d s . Whereas, th e absence of i n t e r r e l a t e d c o s t s of a d ju s tm e n ts l e a d s to i n d i v i d u a l a s s e t demand f u n c t i o n s t h a t depend on th e p a s t h o l d in g s of th e a s s e t i t s e l f , but not on th o s e of o th e r a s s e t s . Indeed, one may loo k a t th e s e e q u a t io n s as though they r e p r e s e n t a d ju s tm e n ts t o th e d e s i r e d a s s e t h o l d in g s . The s i m p l i f y i n g a ssu m p tio n s t h a t were in tr o d u c e d to allow us t o s o lv e th e model a n a l y t i c a l l y account f o r th e absence of any v a r i a b l e b e s id e s th e p r i c e of gold in th e e q u a tio n f o r gold. The i m p l i c a t i o n from t h i s i s d i s t u r b i n g s in c e i t s t a t e s t h a t th e d e s i r e d change in gold h o ld in g s w i l l always be in t h e same d i r e c t i o n depending on th e sig n of d-g. Had we assumed th e e x i s t e n c e of i n t e r r e l a t e d c o s t s of a d ju s tm e n ts or a convex from f o r th e u t i l i t y f u n c t i o n , 1 0 2 th e gold demand e q u a tio n would have in c lu d e d o t h e r v a r i a b l e s such as th e ex p ec te d r e t u r n s on o t h e r a s s e t s . Then, the d e s i r e d change in gold h o ld in g s could be e i t h e r p o s i t i v e or n e g a t iv e depending on a l l th e s e f a c t o r s . Note t h a t d^ r e p r e s e n t s th e c o n t r i b u t i o n of changing gold h o ld in g s to th e o v e r a l l c o s t of managing th e p o r t f o l i o which, among o t h e r t h i n g s , i n c l u d e s i n f o r m a ti o n c o s t s and a r i s k premium. In r e a l i t y , in f o r m a tio n c o s t s and r i s k premiums change depending on economic and w orld e v e n ts and sho uld in d ee d have been m od elled as tim e v a ry in g . In o rd e r to u n d e rs ta n d how t h i s would have a f f e c t e d th e r e s u l t s we can, fo r example, assume t h a t th e c o s t of a d j u s t i n g gold h o ld in g s i s d^ x 1^. where 1^ i s an i n d i c a t o r f u n c t i o n t h a t r e f l e c t s i n v e s t o r s ' c o n fid e n c e in i n t e r n a t i o n a l economic and p o l i t i c a l c o n d i t i o n s and t h e i r e x p e c t a t i o n s r e g a r d i n g th e f u t u r e p r i c e of g o ld . Under s t a b l e economic c o n d i t i o n s one would e x p e c t th e demand f o r gold to be downward s l o p i n g and Ij. would, t h e r e f o r e , be p o s i t i v e . T his would a ls o imply a n et d i s i n v e s tm e n t i n gold and a sw itch to i n t e r e s t b e a r in g a s s e t s . C on versely , in u n s t a b l e s i t u a t i o n s , i n v e s t o r s would sw itc h from nominal to r e a l a s s e t s , and th e demand f o r gold may be p o s i t i v e l y slo p e d . That i s , a n e g a t iv e v a lu e f o r I t would r e f l e c t a l a c k of c o n fid e n c e by i n v e s t o r s and an i n c r e a s e i n th e p r i c e of gold may le a d to 1 0 3 an i n c r e a s e d demand f o r gold due to e x p e c t a t i o n s t h a t th e p r i c e of gold may i n c r e a s e f u r t h e r . Since our model i s r e s t r i c t e d to s t a b l e en v iro n m en ts, we ig n o re t h i s problem in th e s p e c i f i c a t i o n of th e gold e q u a t io n . One l a s t remark i s t h a t th e s t o c h a s t i c n a t u r e of th e exogenous v a r i a b l e s t h a t ap pear on th e r i g h t hand s id e of ( 4 .5 ) im p lie s c o n tin u o u s ( y e t not i n s t a n t a n e o u s ) p o r t f o l i o a d ju s tm e n ts where no a s s e t n e c e s s a r i l y commands a c o n s ta n t sh a re i n th e p o r t f o l i o . To com plete th e model we need to d e s c r i b e e q u il i b r i u m in th e f o r e i g n exchange m arket. I t i s assumed t h a t th e c u r r e n t a c c o u n t, CA, which i s e x p re s s e d in u n i t s of the f o r e i g n c u rre n cy p lu s i n t e r e s t e a r n in g s on d o m e s ti c a l ly held f o r e i g n a s s e t s , must be equal to th e change in f o r e i g n c u rre n cy h o ld in g s , F , t h a t i s , CAt = Ft “ Ft-1 + v t where v i s a v a r i a b l e r e p r e s e n t i n g e r r o r s and ommisions which need not be w h ite n o ise or s e r i a l l y u n c o r r e l a t e d . The change in f o r e i g n c u rre n cy h o ld in g s a r i s e s from changes i n both f o r e i g n a s s e t s and gold h o ld in g s s in c e th e l a t t e r * s p r i c e i s denominated in th e f o r e i g n c u rr e n c y , t h a t i s , th e U.S. d o l l a r . T h e re fo re , CAt = CT* - T*_i> ♦ Pgt(Ct- Ct.i) + »t 104 S u b s t i t u t i n g th e demand e q u a t io n s f o r th e f o r e i g n a s s e t and gold, ( 2 . 1 1 ) , in th e above e q u a tio n : CAt = 7 T 2 r t - l + 7 1 3rt + ^4r t - l +7T5 Mt + ¥7e t - 1 /2 d 3P| t + vt D ivide th rou gh by -Tr^, we g e t th e f i n a l r e g r e s s i o n e q u a tio n f o r th e exchange r a t e (4.6) e t = Ai r *+ X 2r * . 1+ x3rt+ X 4r t . 1+ x5M *+ + *7pg t+ ^8CAt + wt The p a ra m e te rs of the exchange r a t e e q u a tio n a re f u n c t i o n s of th e p a ra m e te rs of th e exogenous s t o c h a s t i c p r o c e s s e s and can be w r i t t e n a s: = ( - a 33- a 4S2 ) / I A | A3 = ( b3 3+b4 32 ) / i A ! X5 = Xj_( b5 3 + b6 B2 ) / ( c3 B + c4 32 ) (4.7) ^2 = § ^ ^1^2+ ^2a2 + ^ 3 °2 ^ ^ 4 = B ( A ^ b 4 + A 2 a 4 ) X 6 = B ( A ] _ b 6 + A 3 c 4 ) Ay — — d 2/ d ^ Ag = - 2 d 2 105 where i A 1 = ( l - b ^ B - ^ B 2) + ( b^B+b^B 2) ( 1-c-^3-C 2^) x (anB+a^B^J/Cc^B+c^B2) + ( 1 - a l B-a 2 B2) ( b 3 B+b4 B2) The r a t i o n a l e x p e c t a t i o n s a ssu m p tio n i m p l i e s t h a t e f f i c i e n c y w i l l be l o s t u n le s s we e s t im a t e th e f i v e e q u a tio n system s i m u lt a n e o u s ly . In Chapter I I , we noted t h a t th e e sse n ce of r a t i o n a l e x p e c t a t i o n s models i s not to d e s c r i b e th e b e h a v io r of a g e n ts , b ut th e outcome of t h e i r b e h a v io r . Thus, in such models d e c i s i o n r u l e s based on o p t im iz in g b e h a v io r a re re a c h e d a s though a g e n ts know the t r u e p r o b a b i l i t y d i s t r i b u t i o n of f u t u r e v a r i a b l e s . As a r e s u l t , j o i n t e s t i m a t i o n of th e p a ra m e te rs of th e exchange r a t e e q u a tio n and of th o se d e s c r i b i n g th e s t o c h a s t i c p o c e ss e s g e n e r a t i n g t h e exogenous v a r i a b l e s t h a t a g e n ts fa c e i n our model i s a n e c e s s a r y f e a t u r e of th e r a t i o n a l e x p e c t a t i o n s a ssu m p tio n . In t h i s model we do not have enough exogenous v a r i a b l e s to a c h ie v e i d e n t i f i c a t i o n in th e u su a l sense in term s of th e o rd e r c o n d i t i o n s of sim u lta n e o u s e q u a t io n s m odels. However, due to th e r a t i o n a l e x p e c t a t i o n s a ssu m p tio n we a re a b le to s u b s t i t u t e f o r th e ex p ec te d f u t u r e v a lu e s of the v a r i a b l e s t h a t a r e c o n s id e r e d to be exogenous t o th e 1 0 6 p o r t f o l i o s e l e c t i o n problem t h e i r p a s t v a lu e s . N o tice , though, t h a t th e s e v a r i a b l e s a r e endogeneous i n th e e co n o m etric sen se s in c e we w i l l be e s t i m a t i n g th e a u t o r e g r e s s i v e p r o c e s s e s t h a t a r e g e n e r a t i n g them. Hence, th e l a c k of h aving a s u f f i c i e n t number of s t r i c t l y exogenous v a r i a b l e s i s b ein g compensated f o r by h avin g a s u f f i c i e n t number of la g g e d endogeneous v a r i a b l e s in c lu d e d as p r e d e te rm in e d v a r i a b l e s w ith th e r e g r e s s o r s . The e x o g e n e ity t e s t s r e p o r t e d in th e p re v io u s c h a p te r p e rm it us to t r e a t t h e s e v a r i a b l e s in t h i s manner. Moreover, e s t i m a t i n g th e model e q u a t io n s s i m u lta n e o u s ly g iv e s r i s e to c r o s s - e q u a t i o n r e s t r i c t i o n s summarized in ( 4 .7 ) which p r o v id e s us w ith i d e n t i f y i n g c r i t e r i a f o r e s t i m a t i n g th e f r e e s t r u c t u r a l p a ra m e te rs . N otice t h a t th e model i s o v e r - i d e n t i f i e d s in c e t h e r e a re only t h r e e p a ra m e te rs namely /?, d2 > and d^, t h a t appear in e q u a tio n ( 4 .6 ) but do not ap pear i n th e o t h e r e q u a tio n s and t h a t need to be i d e n t i f i e d , w h ile t h e r e a r e seven c o e f f i c i e n t s i n a d d i t i o n to th e p a ra m e te rs of th e exogenous p r o c e s s e s to be e s t i m a t e d . Indeed, d^ and dg, th e per u n i t c o s t s of a d j u s t i n g d o l l a r and gold h o ld in g s i n a p o r t f o l i o , can be e x a c t l y i d e n t i f i e d s i n c e they appear only in X and 1X . 7 8 Hence, th e o v e r - i d e n t i f y i n g r e s t r i c t i o n s in v o lv e only 3 , th e d i s c o u n t f a c t o r . 107 Another im p o r ta n t p r o p e r ty of r a t i o n a l e x p e c t a t i o n s models i s t h a t th e e q u il i b r i u m exchange r a t e t h a t i s im p lie d by th e model i s c o n s i s t e n t w ith th e e x p e c t a t i o n s t h a t i n v e s t o r s form r e g a r d i n g th e f u t u r e path of exchange r a t e s . Unless th e a c t u a l s t o c h a s t i c p r o c e s s e s g o v e rn in g exchange r a t e s and o t h e r a s s e t p r i c e s and r e t u r n s a re e x p l i c i t l y s p e c i f i e d an i n f i n i t e number of r a t i o n a l e x p e c t a t i o n s e q u i l i b r i a w i l l r e s u l t . R e ca ll t h a t i n v e s t o r s were assumed to view exchange r a t e s , th e p r i c e of gold and i n t e r e s t r a t e s as e le m e n ts in a v e c t o r - a u t o r e g r e s s i v e p ro c e s s , X ^. , where th e exchange r a t e i s d e s c r i b e d by e q u a tio n ( 2 , 3 e ) . Our p re v io u s assu m p tio n s and s p e c i f i c a t i o n of th e p r o c e s s t h a t d r i v e s exchange r a t e s w i l l i n s u r e t h a t a s t a t i o n a r y unique e q u il i b r i u m c o n d i t i o n a l on w i l l e x i s t . Comparing e q u a tio n (4 .6 ) to ( 2 .3 e ) we can see t h a t a reduced form e q u a tio n f o r the exchange r a t e d e riv e d from ( 4 .6 ) and (2 .3 ) i n c l u d e s th e same r i g h t hand s i d e v a r i a b l e s as ( 2 .3 e ) e x ce p t t h a t t h e r e a re two l a g s of th e p r i c e of gold t h a t ap pear in n o n - l i n e a r form and t h e p a s t v a lu e s of th e exchange r a t e a re ex clu d ed . Note t h a t th e e r r o r term v t , and t h e r e f o r e wt , c a p t u r e in p r i n c i p l e th e e f f e c t of th o se v a r i a b l e s t h a t do not app ear in ( 4 . 6 ) . As w i l l be r e p o r t e d l a t e r , wt t u r n s ou t to be s e r i a l l y c o r r e l a t e d , which r e q u i r e s us to a p p r o p r i a t e l y 103 d i f f e r e n c e th e v a r i a b l e s i n t h a t e q u a t io n . This w i l l be i d e n t i c a l to i n c l u d i n g lag g e d v a lu e s of th e exchange r a t e w h ile r e s t r i c t i n g t h e i r c o e f f i c i e n t s t o be equal to th e c o rre s p o n d in g c o r r e l a t i o n c o e f f i c i e n t s . T h e r e fo r e , we a re a b le to a s c e r t a i n t h a t th e p ro c e ss d e te rm in in g th e exchange r a t e i n our model i s in d ee d th e same as t h a t on which i n v e s t o r s base t h e i r e x p e c t a t i o n s and t h a t we do in f a c t have a unique s t a t i o n a r y r a t i o n a l e x p e c t a t i o n s e q u i l i b r i u m p a th , c o n d i t i o n a l on X^, fo r exchange r a t e s . There rem ain s f o r us to v e r i f y e m p i r i c a l l y how w ell ( 4 .6 ) p e rfo rm s in p r e d i c t i n g f u t u r e exchange r a t e s i n com parison t o ( 2 . 3 e ) . The f o l lo w in g p r o p e r t i e s r e g a r d i n g th e e r r o r term s a re a ls o assumed to h o ld : E (Ut ) = E (wt ) = 0 where U». = ( ud t uf t umt ug t u xt ) a su b v e c to r of Ufc e cutup * n t and E (w w|_) = where 0 t i s a d ia g o n a l m a trix w ith p o s i t i v e d ia g o n a l e le m e n ts, i . e . , E(Ut Ut ^j )=0, and A t i s a p o s i t i v e s e m i - d e f i n i t e m a tr ix f o r a l l t , i . e . , a llo w in g f o r s e r i a l a u t o c o r r e l a t i o n between th e w 's . 109 Here we should note t h a t th e a s s e t demand e q u a t io n s from which th e exchange r a t e e q u a tio n was d e riv e d a r e e x a c t e q u a t io n s , t h a t i s , no e r r o r term e x i s t s . T his would pose a problem were we to e s t i m a t e th e a s s e t demand e q u a t io n s s e p a r a t e l y s in c e we would no t be a b le to e x p l a i n th e r e s i d u a l s . G e n e ra lly , s t u d i e s t h a t do e s t im a t e such demand e q u a t io n s j u s t i f y th e e x i s t e n c e of a r e g r e s s i o n r e s i d u a l u sin g an o m itte d v a r i a b l e or a measurement e r r o r app roach . In t h i s study s in c e our b a s i c i n t e r e s t i s in e s t i m a t i n g a model f o r exchange r a t e s , and no t th e a s s e t demand e q u a t io n s i n d i v i d u a l l y , th e i n c l u s i o n of an e r r o r s and o m issio n s v a r i a b l e t h a t i s p a r t of th e b a la n c e of payments a c c o u n ts seems t o be th e most p l a u s i b l e economic i n t e r p r e t a t i o n f o r th e e x i s t e n c e of an e r r o r term in th e exchange r a t e e q u a t io n . That i s , th e a c t u a l c u r r e n t acco u n t i s assumed to c o n s i s t of th e measured c u r r e n t accou nt p lu s v , where v i s contem poraneously u n c o r r e l a t e d w i th CA . To r e p e a t , the o b j e c t i v e i s t o e s t i m a t e a l l the p a ra m e te rs i n th e system of e q u a tio n s given by ( 4 . 6 ) , ( 2 . 3 a ) , ( 2 . 3 b ) , ( 2 .3 c ) and ( 2 .3 a ) s im u lta n e o u s ly imposing a l l th e c r o s s - e q u a t i o n r e s t r i c t i o n s d i s p l a y e d by ( 4 . 7 ) . In th e p ro c e s s we t e s t s e v e r a l assu m p tio n s t h a t were made in c o n s t r u c t i n g th e model. B a s i c a l l y , we would l i k e to 1 1 0 i n v e s t i g a t e how the exogenous p r o c e s s e s , alo n g w ith changes i n th e s e p r o c e s s e s r e f l e c t i n g a l t e r a t i o n s in th e f o r e i g n a s s e t m a rk e ts , a f f e c t a c o u n t r y 's exchange r a t e . The f i n d i n g s undoubtedly depend on how w e ll th e fo rm u la te d model based on th e assum ption of r i s k n e u t r a l i t y a p p ro x im ate s i n v e s t o r s ' t r u e b e h a v io r. Although th e m o ti v a ti o n f o r t h i s stu dy was d i s c u s s e d in Chapter I, b e fo r e going on t o th e model e s t i m a t i o n we wish to r e i t e r a t e th e main aims. F i r s t , t h i s stu d y , f o llo w in g a r e c e n t t r e n d i n i t i a t e d by Lucas in h i s c r i t i q u e of e co n o m etric p o l ic y e v a l u a t i o n , aims a t i l l u s t r a t i n g how s t r u c t u r a l r e s t r i c t i o n s on b e h a v io r a l models can be f o r m u la te d and t e s t e d a s an i n i t i a l s t e p in u sin g e co n o m e tric models f o r a n a l y s i s and p o lic y e v a l u a t i o n . Here, we a re p a r t i c u l a r l y i n t e r e s t e d in a p p ly in g t h i s approach to e x p l a i n t h e dynamics of exchange r a t e s , in th e same way t h a t i t was a p p l i e d by S a r g e n t ( 6 2 ) , M eese(52), and E c k s te in ( 2 1 ) in e s t i m a t i n g dynamic f a c t o r demand and supply f u n c t i o n s . Secopd, we aim to answer an im p o r ta n t q u e s t i o n in p o r t f o l i o models namely as t o how a s s e t demand f u n c tio n s ! i and, t h e r e f o r e , p r i c e s a r e a f f e c t e d by e x p e c t a t i o n s of th e r e t u r n s on o t h e r a s s e t s . P o r t f o l i o th e o ry p r e d i c t s t h a t an un expected i n c r e a s e i n th e r e t u r n on one a s s e t r e l a t i v e to I l l th e o th e r a s s e t s w i l l r e s u l t i n a p o r t f o l i o s h i f t tow ards t h a t a s s e t , and v ic e v e r s a . T his p a r t i c u l a r p r e d i c t i o n has m o tiv a te d s e v e r a l e co n o m ists t o t e s t th e s t a t i s t i c a l s i g n i f i c a n c e of th e e f f e c t of changes in i n t e r e s t d i f f e r e n t i a l s on d i f f e r e n t c u r r e n c i e s on t h e i r r e s p e c t i v e exchange r a t e s . E m p irica l s t u d i e s c i t e d e a r l i e r have p ro v id e d mixed r e s u l t s i n models w ith two m onies. As in th e s t r u c t u r e d model, we a re se e k in g t o e v a l u a t e th e r o b u s t n e s s of th e s e f i n d i n g s in our r e s t r i c t e d s t r u c t u r a l model w ith gold and to a s c e r t a i n th e r o l e p layed by gold in exchange r a t e d e t e r m i n a t i o n . The d a ta used i n e s t i m a t i n g th e s t r u c t u r a l model in c lu d e th o se used in Chapter I I I p lu s d a ta on th e c u r r e n t a cc o u n t v a r i a b l e s . In g e n e r a l , only a g g re g a te f i g u r e s on f o r e i g n a s s e t h o l d in g s and t r a d e flo w s a r e a v a i l a b l e . These were used as a proxy fo r th e b i l a t e r a l f i g u r e s w ith th e U nited S t a t e s . The f i n d i n g s sh o u ld n o t be d r a s t i c a l l y a f f e c t e d s i n c e b i l a t e r a l c a p i t a l and t r a d e flo w s w ith th e U nited S t a t e s form a l a r g e p e rc e n ta g e of the a g g re g a te f i g u r e s . Data on e x p o r t s and im p o rts were o b ta in e d from I n t e r n a t i o n a l F i n a n c i a l S t a t i s t i c s t a p e s (a s w e ll as d ata on f o r e i g n a s s e t s ) , e x p re s s e d in term s of d o m estic c u rr e n c y . Net e x p o r t s d iv id e d by th e exchange r a t e f o r t h a t p e r io d t r a n s f o r m s th e measure i n t o U.S. d o l l a r s . 1 1 2 Since th e p a r t i c u l a r tim e s e r i e s on exchange r a t e s t h a t we use i s a c t u a l l y used i n IFS as th e t r a d e c o n v e rsio n f a c t o r , no i n c o n s i s t e n c i e s a r i s e from such a t r a n s f o r m a t i o n . A ggregate f o r e i g n a s s e t h o l d in g s i n p r i v a t e banks were used as a m easure f o r a s s e t h o ld in g s f o r Canada, France and Germany. For S w itz e r la n d no f i g u r e s were a v a i l a b l e on h o ld in g s i n p r i v a t e banks which l e d us to use d a ta on f o r e i g n a s s e t h o l d in g s i n th e N a tio n a l Bank which a c t s as th e c e n t r a l bank. Hence th e f o r e i g n c u rre n c y h o ld in g s f o r t h i s c o u n try a re th e o f f i c i a l r e s e r v e s and n ot p r i v a t e h o l d in g s . T h e r e fo r e , th e r e s u l t s f o r S w itz e r la n d sho uld r e f l e c t th e r o l e of o f f i c i a l r e s e r v e s i n exchange r a t e d e te r m i n a ti o n . 4 .2 METHODOLOGY E x i s t i n g s methods f o r th e e s t i m a t i o n of n o n - l i n e a r eco n o m etric m odels a r e th e f o l l o w i n g ^ : 1. E s t im a ti n g th e b e s t l i n e a r a p p ro x im atio n of th e model g e n e r a l l y given by th e f i r s t two term s of th e T ay lo r s e r i e s e x p an sio n of th e model e q u a tio n s around th e s t r u c t u r a l p a ra m e te rs ( M a lin vaud ’ s l i n e a r pseudomodel ). ^ Bard(5) and G o ld fe ld and Q u a n d t(2 9 ), p r e s e n t an e x c e l l e n t d e s c r i p t i o n of methods used in n o n - l i n e a r e s t i m a t i o n . 113 2. N o n - lin e a r l e a s t s q u a r e s methods. 3. L im ite d i n f o r m a ti o n and f u l l in f o r m a tio n maximum l i k e l i h o o d e s t i m a t i o n . 4. H a n se n * s(37) g e n e r a l i z e d method of moments e s t i m a t o r ( GM M ). The f i r s t t h r e e e s t i m a t o r s a r e w id ely used and t h e s e methods a r e s i m i l a r i n p r i n c i p l e to t h e i r c o u n t e r p a r t s in l i n e a r r e g r e s s i o n m odels. The f i r s t t e c h n iq u e re d u c e s th e model to a l i n e a r one which a llo w s th e use of any method f o r l i n e a r sim u lta n e o u s e q u a t io n s system s where the a s y m p to tic p r o p e r t i e s of th e v a r io u s e s t i m a t o r s a r e w e ll e s t a b l i s h e d . N o n - lin e a r l e a s t s q u a re s e s t i m a t o r s in c l u d e Amemiya’ sC D tw o - s ta g e l e a s t s q u a r e s and G a l l a n t ‘ s( 27) t h r e e - s t a g e l e a s t s q u a r e s . These e s t i m a t o r s , as w e ll as th e maximum l i k e l i h o o d e s t i m a t o r s , a re based on th e assum ption t h a t th e e r r o r term s a r e a d d i t i v e . Again t h e i r a sy m p to tic p r o p e r t i e s have been d e r iv e d and may, th u s , be used i n s e l e c t i n g a method. The l a s t e s t i m a t o r has been only r e c e n t l y developed and f o r t h i s re a s o n i s n o t w id ely a p p l i e d . The GM M e s t i m a t o r b a s i c a l l y e x p l o i t s th e I o r t h o g o n a l i t y c o n d i t i o n s t h a t a r i s e in dynamic economic m odels. Hansen proved th e a s y m p to tic e q u iv a le n c e of th e GM M and th e maximum l i k e l i h o o d e s t i m a t o r s . 114 Judge, G r i f f i t h , H i l l and Lee(43) argue t h a t s i n c e , in g e n e r a l , a sy m p to tic p r o p e r t i e s of e s t i m a t o r s f o r n o n - l i n e a r models have been d e r iv e d , whereas, t h e i r sm a ll-sa m p le p r o p e r t i e s have n o t, one sho uld fo cu s on f i n d i n g c o n s i s t e n t e s t i m a t o r s . However, s in c e our model embodies complex n o n - l i n e a r l i t i e s , c o n s i s t e n c y alone sho uld no t be th e c r i t e r i o n of e s t i m a t i o n . The p resen c e of lag g e d dependent v a r i a b l e s i n a l l of our r e g r e s s i o n e q u a t i o n s makes t h i s even more t r u e . Moreover, s in c e we do not presume a p r i o r i t h a t th e m a tr ix a u t o r e g r e s s i v e component of our system of e q u a t io n s i s i n v e r t i b l e , th e c h o ice of an e s t i m a t i o n te c h n iq u e i s somewhat p r o b le m a tic . That i s , t h e r e e x i s t s no n a t u r a l t r a n s f o r m a t i o n of th e model y i e l d i n g exogenous r e g r e s s o r s , u n c o r r e l a t e d e r r o r s and s t r o n g n o n - c o l l i n e a r i t y of r e g r e s s o r s . T h e r e fo r e , i t i s a p p r o p r i a t e to t r a d e - o f f some d e s i r a b l e a s y m p to tic p r o p e r t i e s in one e s t i m a t i o n method f o r a n o th e r and to opt f o r th e s i m p le s t method of e s t i m a t i o n s a t i s f y i n g our a sy m p to tic p r o p e r t y c r i t e r i a . For th e same re a so n t h a t th e i n d i r e c t l e a s t s q u a re s e s t i m a t o r i s g e n e r a l l y i n c o n s i s t e n t i n l i n e a r sim u lta n e o u s e q u a t io n s m odels, th e n o n - l i n e a r l e a s t s q u a re s e s t i m a t o r w i l l a ls o be i n c o n s i s t e n t in n o n - l i n e a r m odels. Moreover, t h i s e s t i m a t o r w i l l be i n e f f i c i e n t i n th e p re se n c e of lag g e d dependent v a r i a b l e s and a u t o c o r r e l a t e d e r r o r s . 115 Amemiya(2) shows t h a t both th e n o n - l i n e a r f u l l i n f o r m a tio n maximum l i k e l i h o o d ( NLFI ) and n o n - l i n e a r t h r e e - s t a g e l e a s t s q u a r e s ( NL3S ) e s t i m a t o r s a r e c o n s i s t e n t when th e e r r o r s a r e norm ally d i s t r i b u t e d , w h ile th e NLFI i s g e n e r a l l y a s y m p t o t i c a l l y more e f f i c i e n t th an t h e NL3S e s t i m a t o r e x ce p t when t h e r e g r e s s i o n e q u a tio n s a t i s f i e s a p a r t i c u l a r form. However, i f th e d i s t r i b u t i o n of the e r r o r term i s n o t normal, th e NLFI e s t i m a t o r w i l l be i n c o n s i s t e n t , w hereas, th e NL3S e s t i m a t o r r e t a i n s i t s c o n s i s t e n c y . T his makes t h e l a t t e r a more a t t r a c t i v e e s t i m a t o r due to i t s r o b u s t n e s s when t h e n o rm a lity of th e r e s i d u a l s cann ot be e s t a b l i s h e d . T his f i n d i n g i s c o n tr a r y to th e e q u iv a le n c e of th e l i n e a r t h r e e - s t a g e l e a s t s q u a re s and f u l l i n f o r m a tio n maximum l i k e l i h o o d e s t i m a t o r s i r r e s p e c t i v e of th e d i s t r i b u t i o n of th e e r r o r s . Given th e f a c t t h a t th e a s y m p to tic p r o p e r t i e s of th e NL3S and NLFI a re dependent on th e n o rm a lity of th e e r r o r s , and t h a t they a re c o m p u ta tio n a lly i m p r a c t i c a l , th e f i r s t method which i n v o lv e s an i t e r a t i v e e s t i m a t i o n scheme of th e b e s t l i n e a r a p p ro x im a tio n of th e model a p p e a rs to be th e most a t t r a c t i v e in e s t im a i n g our model. M a lin v a u d (50) shows t h a t , g r a n te d one chooses i n i t i a l v a lu e s f o r th e c o e f f i c i e n t s t h a t a re c lo s e to th e t r u e v a lu e s , t h i s method w i l l alm ost s u r e l y converge. B ard (5, pp. 99-100) and 1 1 6 Amemiya(3) show t h a t t h i s method i s s i m i l a r to th e Gauss-Newton i t e r a t i v e r o u t i n e f o r o b t a i n i n g t h e s i n g l e e q u a tio n n o n - l i n e a r l e a s t s q u a re s e s t i m a t o r . The e q u iv a le n c e can be e a s i l y extend ed t o th e sim u lta n e o u s e q u a t io n s c a se . Moreover, t h i s method o f f e r s t h e l i b e r t y of c ho osing an a p p r o p r i a t e l i n e a r e s t i m a t o r w ith d e s i r a b l e a sy m p to tic p r o p e r t i e s depending on th e p r o p e r t i e s of th e r e g r e s s i o n model. The i t e r a t i v e p ro ce d u re we have chosen t o use i n v o lv e s l i n e a r i z i n g th e model around some i n i t i a l c o n s i s t e n t e s t i m a t e s o f th e f r e e p a ra m e te rs u sin g th e f i r s t two term s of a T ay lo r s e r i e s e x p a n sio n . Subsequent i t e r a t i o n s use th e new e s t i m a t e s to d e r i v e new v a lu e s of th e p a ra m e te rs around which th e T aylor s e r i e s ex p an sio n i s r e c a l c u l a t e d . The i t e r a t i v e scheme i s te r m in a t e d when convergence i s a c h ie v e d . For our pu rp o se s we d e f in e convergence i n th e norm of th e p aram eter v e c t o r . In what rem ains i n t h i s s e c t i o n we w i l l d e s c r i b e th e s t r a t e g y used in o b t a i n i n g i n i t i a l c o n s i s t e n t e s t i m a t e s , and th e n d e s c r i b e th e i t e r a t i v e scheme. There i s only one e q u a tio n in th e s t r u c t u r a l model, namely th e exchange r a t e e q u a tio n , which d i s p l a y s n o n - l i n e a r i t y . The o t h e r e q u a tio n s a re l i n e a r in both th e p a ra m e te rs and th e r e g r e s s o r s . Moreover, we have over 117 i d e n t i f y i n g r e s t r i c t i o n s on th o se c o e f f i c i e n t s t h a t a re n o n - l i n e a r so t h a t i f we can o b t a i n some i n i t i a l e s t i m a t e s f o r th e p a ra m e te rs of th e exchange r a t e e q u a tio n and th e s t o c h a s t i c p r o c e s s e s , we can s o lv e f o r th e o th e r f r e e p a ra m e te r s . Hence, our f i r s t aim i s to o b t a i n c o n s i s t e n t e s t i m a t e s f o r th e c o e f f i c i e n t s of th e exogenous p r o c e s s e s and use th e s e a lo n g w ith the c r o s s - e q u a t i o n r e s t r i c t i o n s to o b t a i n i n i t i a l e s t i m a t e s f o r th e o t h e r f r e e p a ra m e te rs ( 3 , d 2 > d ^ )♦ These r e s t r i c t i o n s u n iq u e ly i d e n t i f y the a d ju s tm e n t c o s t s dg and d^ bu t not g. However, i t i s p o s s i b l e to use a convex com bin atio n of th e r e s t r i c t i o n s to make th e model e x a c tl y i d e n t i f i e d , and t h e r e f o r e , o b t a i n a s o l u t i o n f o r . The p a r t i c u l a r com bin atio n used i s : w h ile 62 = - 1 / 2 o and d3 = 1/2 Xg/X7 The method used f o r o b t a i n i n g i n i t i a l e s t i m a t e s i s H a ta n a k a 's (39) e f f i c i e n t t w o - s te p e s t i m a t o r which he I p ro p o se s f o r dynamic s im u lta n e o u s e q u a tio n models w ith a u t o r e g r e s s i v e r e s i d u a l s . Hatanaka proves th e a s y m p to tic e q u iv a le n c e of t h i s e s t i m a t o r to th e f u l l in f o r m a tio n maximum l i k e l i h o o d e s t i m a t o r which makes h i s e s t i m a t o r both 118 c o n s i s t e n t and e f f i c i e n t . The f i r s t s t e p of t h i s p ro ce d u re i n v o lv e s o b t a i n i n g t h e r e s i d u a l s from a tw o -s ta g e l e a s t s q u a r e s e s t i m a t i o n and e s t i m a t i n g an a u t o r e g r e s s i o n of th e s e r e s i d u a l s . The new r e s i d u a l s of th e s e a u t o r e g r e s s i o n s w i l l be s p h e r i c a l . The e s t im a t e d c o e f f i c i e n t s from th e r e s i d u a l a u t o r e g r e s s i o n s a r e used f o r d i f f e r e n c i n g th e dependent and in d e p e n d e n t v a r i a b l e s t h a t ap pear in th e c o rr e s p o n d in g s t r u c t u r a l e q u a tio n . I n s t r u m e n t s f o r th e in c l u d e d c u r r e n t endogeneous v a r i a b l e s a r e th e p r e d i c t e d v a l u e s o b ta in e d from r e g r e s s i n g each of th e s e endogeneous v a r i a b l e s on a l l c u r r e n t and lag g e d exogenous v a r i a b l e s , and lag g e d in c lu d e d endogeneous v a r i a b l e s . A d d i t i o n a l l a g s , up to th e o rd e r of th e s e r i a l c o r r e l a t i o n in th e r e s i d u a l s , a re a ls o in c lu d e d i n th e l i s t of i n s t r u m e n t a l v a r i a b l e s . That i s , i f th e r e s i d u a l s were found t o fo llo w a f i r s t o rd er a u t o r e g r e s s i v e p r o c e s s , then th e number of la g g e d exogenous and endogeneous v a r i a b l e s used as i n s t r u m e n t s would be one more than th e c o rre s p o n d in g number of l a g s t h a t ap pear i n th e s t r u c t u r a l model. In th e second s t e p of H a ta n a k a 's p ro c e d u re , the e s t im a t e d v a r i a n c e - c o v a r i a n c e m a trix i s used to d e r i v e th e g e n e r a l i z e d l e a s t s q u a r e s e s t i m a t o r which y i e l d s e f f i c i e n t i n i t i a l e s t i m a t e s f o r th e s t r u c t u r a l p a ra m e te rs . 119 In th e i t e r a t i v e p a r t of our e s t i m a t i o n scheme th e exchange r a t e e q u a tio n i s r e p l a c e d by a l i n e a r a p p ro x im atio n around th e i n i t i a l e s t i m a t e s of th e f r e e p a ra m e te r s . The l i n e a r pseudo-model i s e s t im a t e d u s in g A i t k e n 's g e n e r a l i z e d l e a s t s q u a re s te c h n iq u e w ith th e same v a r i a n c e - c o v a r i a n c e m a tr ix used in th e f i r s t s t e p , and s u b j e c t t o th e c r o s s - e q u a t i o n r e s t r i c t i o n s given by ( 4 . 7 ) . These e s t i m a t e s a r e th en used t o o b t a i n a new l i n e a r ap p ro x im a tio n of th e n o n - l i n e a r exchange r a t e e q u a tio n and th e p ro ce d u re i s r e p e a t e d u n t i l convergence i s o b t a i n e d . Bard (5 , pp. 99-100) does not recommend th e use of s t a n d a r d sim u lta n e o u s e q u a t io n s te c h n iq u e s in e s t i m a t i n g th e sequence of m u l t i p l e l i n e a r r e g r e s s i o n problem s s in c e s i n g u l a r i t y may o f t e n a r i s e . Indeed, t h i s problem came up when we t r i e d to e s t i m a t e th e u n r e s t r i c t e d model. M arquadt(51) s u g g e s te d u s in g a r i d g e r e g r e s s i o n p ro c e d u re f o r d e a l i n g w ith t h i s problem. Ridge r e g r e s s i o n i s an o f t used method f o r d e a l in g w ith m u l t i c o l l i n e a r i t y y e t i t i s c r i t i c i z e d because i t cam ouflages th e problem r a t h e r than s o l v e s i t . Moreover, in ch o o sin g th e r i d g e c o e f f i c i e n t s , one o f t e n f a c e s a problem because r i d g e r e g r e s s i o n i s in p r i n c i p l e e q u i v a l e n t to u s in g B ayesian s h r i n k i n g p r i o r s . Hence, an ad hoc c h o ic e of th e r i d g e c o e f f i c i e n t s i s e q u i v a l e n t to im posing ad hoc p r i o r r e s t r i c t i o n s on t h e 1 2 0 p a ra m e te r s . For t h i s re a s o n i t i s more d e s i r a b l e to lo o k f o r a l t e r n a t i v e ways of e s t i m a t i n g th e r e s t r i c t e d p a r a m e t e r s . Since our r e s t r i c t i o n s a r e i d e n t i f y i n g a ssu m p tio n s and n ot t e s t a b l e h y p o th e s e s , a p ro ce d u re t h a t would allow us to e s t i m a t e only th e r e s t r i c t e d p a ra m e te r s would be s a t i s f a c t o r y . Lawson and Hanson(45) prove t h a t im posing a s e t of l i n e a r r e s t r i c t i o n s i s e q u i v a l e n t to a l i n e a r t r a n s f o r m a t i o n of th e r e g r e s s o r s . Based on t h i s p r i n c i p l e H i r s c h b e r g ( 4 0 , Appendix A) developed a te c h n iq u e which allo w s one to o b t a i n e f f i c i e n t p aram eter e s t i m a t e s in c a se s where u n r e s t r i c t e d p a ra m e te r e s t i m a t e s can not be o b ta in e d . T his p a r t i c u l a r te c h n iq u e in v o lv e s s o l v i n g th e l i n e a r system composed of th e l i n e a r r e s t r i c t i o n s and th e s t r u c t u r a l e q u a t io n s to o b t a i n a r e s t r i c t e d red uced form m odel. T h e r e fo r e , given c o n s i s t e n t and e f f i c i e n t i n i t i a l e s t i m a t e s , and assum ing a unique g lo b a l minimum e x i s t s , our i t e r a t i v e p ro ce d u re c on verges a s y m p t o t i c a l l y to th e t r u e v a lu e of th e p a ra m e te r s . Moreover, by u s in g an e f f i c i e n t e s t i m a t i o n method f o r d e r i v i n g su b se q u e n t e s t i m a t o r s , we can a s c e r t a i n t h a t our f i n a l param eter e s t i m a t e s w i l l be a s y m p t o t i c a l l y e f f i c i e n t . 1 2 1 4 .3 RESULTS In t h i s s e c t i o n , th e r e s u l t s from e s t i m a t i n g th e model fo r exchange r a t e s a re examined w ith two i s s u e s in mind. F i r s t , th e s i g n s and v a lu e s o f th e p a ra m e te rs a r e examined t o see w heth er the r e s u l t s s u p p o r t th e i m p l i c a t i o n s of the p o r t f o l i o th e o r y of f o r e i g n exchange r a t e s . Second, th e q u e s t i o n of w hether t h i s p a r t i c u l a r s p e c i f i c a t i o n of th e model can e x p l a i n th e ob serv ed dynamics of exchange r a t e s w i l l be e v a l u a t e d . In a d d i t i o n , th e im pact of s t r u c t u r a l changes in th e exogenous p r o c e s s e s on exchange r a t e s i s a n a ly z e d by i n s p e c t i n g th e s ig n s of th e d e r i v a t i v e s of the c o e f f i c i e n t s i n th e exchange r a t e e q u a tio n w ith r e s p e c t to th e p a ra m e te rs of th e s t o c h a s t i c p r o c e s s e s . In f i n d i n g i n i t i a l e s t i m a t e s f o r th e r e s t r i c t e d model of s e c t i o n 4 . 1 , a f i r s t o rd e r s e r i a l c o r r e l a t i o n in th e r e s i d u a l s o f th e exchange r a t e e q u a tio n was d e t e c t e d . A f te r c o r r e c t i n g f o r t h i s u sin g H a t a n a k a 's t w o - s te p method, th e model was e s t im a t e d u s in g th e i t e r a t i v e te c h n iq u e o u t l i n e d i n th e p re v io u s s e c t i o n f o r Canada, F ran ce, Germany and S w it z e r l a n d . Convergence was a c h ie v e d w ith f o u r , e i g h t , tw elve and two i t e r a t i o n s f o r th e above c o u n t r i e s r e s p e c t i v e l y . For a l l th e s e c o u n t r i e s th e e s t im a t e d d is c o u n t f a c t o r was between 0 and 1 a t th e tim e of convergence which s u p p o r ts th e t o l e r a n c e l e v e l chosen in our convergence c r i t e r i a . 122 TABLE 6 R e g ressio n E s tim a te s of the Exchange Rate E quation Canada et = 1,006 r*+ .009 r*_i - .055 rt + .100 rt _]_ + .004 M*+ .0001 .120 P2 t - .257 CAt R2 = .91 , D-W = 1.76 3 = .28 (1 .06) , d2 = .13 , do =-1.07 ( 2 .2 2 ) France e t = 1.508 r*+ .203 r* .494 r«_- .296 r. 1 t t -1 t t - i + .021 M*+ .035 .202 p | t - .892 CAt R2 = .78 , D-W =1.33 e = .41 (1 .5 6 ) , dp = .46 , do =2.20 (1 .6 0 ) 3 Germanv e t = 1.033 r * - .041 r*_x + .01 r t - .012 rt _1 + .003 M*+ .001 M* .124 P2 - 2.795 CA. t t - i gc t R2 = .46 , D-W = 1.66 3 = .18 ( .24) , dp = 1 .40 , do =11 .30 (.1 4 ) 3 S w it z e r e t = .618 r - .0005 *+ .110 .144 r t - .087 r t - l M*- .002 M*_1+ .141 P2t - 2.915 CAt la n d R2 = 0 .6 2 , D-W =1.19 8 = - .5 4 (1 . 2 2 ) , dp = .146 , do =-10.36 (4 .8 2 ) 5 No t - s t a t i s t i c s a re r e p o r t e d f o r th e c o e f f i c i e n t s of the exchange r a t e e q u a tio n s in c e th e s e were not d i r e c t l y e s t im a t e d and t h e i r v a r i a n c e s a r e not known. Only those f o r 3 and d2 a re a v a i l a b l e . 123 The c o e f f i c i e n t of d e t e r m i n a ti o n f o r the models f o r Canada, F rance, Germany and S w it z e r la n d , were c a l c u l a t e d t o be .9 1 , .7 8 , .46 and .62 r e s p e c t i v e l y . T his i n d i c a t e s t h a t a t l e a s t h a l f of th e v a r i a n c e i n exchange r a t e s may be e x p la in e d by th e model. The Durbin-W atson and Q s t a t i s t i c s i n d i c a t e d no r e s i d u a l s e r i a l c o r r e l a t i o n in th e e s t im a t e d models e x c e p t f o r S w itz e r la n d where th e D-W s t a t i s t i c was s i g n i f i c a n t l y s m a lle r than 2. However, one has to be c a r e f u l in i n t e r p r e t i n g th e D-W s t a t i s t i c s i n c e i t w i l l be b ia s e d in th e p re se n c e of lag g e d dependent v a r i a b l e s . Using t h e s e e s t im a t e d f r e e p a ra m e te rs , we c a l c u l a t e the o r i g i n a l n o n - l i n e a r c o e f f i c i e n t s of th e exchange r a t e e q u a t io n . I n s p e c t i n g th e s ig n s of th e c a l c u l a t e d n o n - l i n e a r c o e f f i c i e n t s r e v e a l s t h e f o l lo w in g . The n e g a t iv e sig n of th e c o e f f i c i e n t s of th e c u r r e n t acco u n t ( n e t e x p o r t s and r e t u r n on f o r e i g n a s s e t s ) f o r Canada, F rance, Germany and S w itz e r la n d conform s w ith t h e o r e t i c a l p r e d i c t i o n s . An i n c r e a s e i n th e b a la n c e of t r a d e would be e x p e c te d to r a i s e th e v alu e of th e dom estic c u rre n cy r e l a t i v e to th e d o l l a r and, t h e r e f o r e , lower the exchange r a t e . The t - s t a t i s t i c c a l c u l a t e d f o r t h i s c o e f f i c i e n t i n d i c a t e s t h a t i t i s s i g n i f i c a n t l y d i f f e r e n t from 0 fo r Canada and S w itz e r la n d but not f o r France and Germany. T his may be an i n d i c a t i o n t h a t th e c u r r e n t acco u n t has no 124 s h o r t - t e r m i n f l u e n c e on th e Deutschmark and French f r a n c , where p o r t f o l i o e f f e c t s may p o s s ib ly be th e m ajor f a c t o r d e te rm in in g t h e i r r a t e s of exchange w ith r e s p e c t to th e U.S. d o l l a r . U n f o r t u n a l t e l y , t h i s i s th e only c o e f f i c i e n t f o r which a t - s t a t i s t i c can be c a l c u l a t e d from the v a r i a n c e s of the f r e e p a ra m e te rs due to th e n a tu r e of the n o n - l i n e a r r e s t r i c t i o n s . The c o e f f i c i e n t s of the U.S. i n t e r e s t v a r i a b l e s a r e a ls o of th e c o r r e c t s ig n . An i n c r e a s e in i n t e r e s t r a t e s in th e U.S. r e l a t i v e to d om estic r a t e s w i l l induce an ou tflo w of i c a p i t a l to th e U.S. which w i l l cause th e dom estic cu rre n cy to d e p r e c i a t e . These c o e f f i c i e n t s a re p o s i t i v e f o r both c u r r e n t and lag g e d v a r i a b l e s e x c e p t f o r th e la g g e d r a t e in th e model fo r Germany which may r e f l e c t th e c o r r e c t i o n f o r over a d ju s tm e n ts o f t e n c i t e d in e m p i r ic a l s t u d i e s on exchange r a t e s . O v e rsh o o tin g of exchange r a t e s , and su b se q u e n t c o r r e c t i o n s a r e observed t o be common phenomena in i n t e r n a t i o n a l m a rk e ts . An i n i t i a l i n c r e a s e in f o r e i g n r e l a t i v e to do m estic i n t e r e s t r a t e s may cause th e dom estic c u rre n c y to d e p r e c i a t e beyond i t s e q u i l i b r i u m l e v e l due to i n v e s t o r s 1 a n t i c i p a t i o n s of f u t u r e a p p r e c i a t i o n in th e c u rr e n c y . The re a so n t h a t t h i s r e s u l t i s o b se rv ed only fo r Germany may be due to th e l a r g e r r o l e th e Deutschmark p la y s i in i n t e r n a t i o n a l p o r t f o l i o s r e l a t i v e to th e o th e r c u r r e n c i e s . 125 I f i n t e r e s t a r b i t r a g e were to h o ld , one would e x p e c t th e sig n on dom estic i n t e r e s t to be n e g a t iv e s in c e an i n c r e a s e in d om estic r a t e s r e l a t i v e to th o s e on f o r e i g n in v e s tm e n ts sh o u ld i n c r e a s e th e demand fo r th e d o m estic c u rre n c y and, t h e r e f o r e , lower th e exchange r a t e . This n e g a t iv e i n f l u e n c e a p p e a rs w ith a l a g . That i s , e x c e p t f o r Canada, an i n c r e a s e in d o m estic r a t e s does not le a d to a contemporaneous a p p r e c i a t i o n in th e v a lu e of dom estic c u rre n c y but i t w i l l cause i t to a p p r e c i a t e a f t e r a la g of one p e r io d . The p o s i t i v e i n f l u e n c e of c u r r e n t and lag ged U.S. money jD n th e Deutschmark, French f r a n c and Canadian d o l l a r may be a t t r i b u t e d to demand f a c t o r s . That i s , s in c e i t i s not i 1 | p o s s i b l e to i d e n t i f y w hether an i n c r e a s e in th e U.S. money v a r i a b l e r e f l e c t s an i n c r e a s e in demand or su p p ly , th e sig n of th e money c o e f f i c i e n t may be e i t h e r p o s i t i v e or n e g a tiv e . Hence, an i n c r e a s e in th e demand f o r U.S. money W ill put upward p r e s s u r e on U.S. i n t e r e s t r a t e s and le a d to c a p i t a l in flo w i n t o th e U.S. c a u sin g th e d o l l a r to a p p r e c i a t e a g a i n s t o t h e r c u r r e n c i e s . In th e case of the Swiss f r a n c , th e n e g a t iv e i n f l u e n c e of U.S. money may be an i n d i c a t i o n t h a t supply f a c t o r s or c o n t r o l s on demand dom inate demand f a c t o r s . 126 The sig n of th e c o e f f i c i e n t s of th e p r i c e of gold v a r i a b l e i s p o s i t i v e f o r Canada and S w itz e r la n d and n e g a t i v e fo r Germany and F rance. This mixed e v id e n c e i s not c o n s i s t e n t w ith any h y p o t h e s is r e g a r d i n g th e r e l a t i o n s h i p between th e gold and f o r e i g n exchange m a rk e ts . The e a r l i e r d i s c u s s i o n c o n c e rn in g i n t e r p r e t i n g th e demand f o r gold e q u a tio n and th e i m p l i c a t i o n s r e g a r d i n g th e sig n o f d^ s u g g e s ts t h a t th e r o l e of gold in th e d e te r m i n a ti o n of f o r e i g n exchange r a t e s emerges mainly under u n s t a b l e c o n d i t i o n s which p o s s i b l y le a d to s t r u c t u r a l changes in th e way i n v e s t o r s form t h e i r e x p e c t a t i o n s . The f a c t t h a t we nave r e s t r i c t e d o u r s e l v e s to e s t i m a t i n g th e model over a s t r u c t u r a l l y s t a b l e p e rio d of our sample may o f f e r an e x p la n a tio n as to why th e i n f l u e n c e of gold on f o r e i g n exchange r a t e s i s not c l e a r . F i n a l l y , the e s t im a t e d s u b j e c t i v e d is c o u n t f a c t o r i s s t a t i s t i c a l l y i n s i g n i f i c a n t f o r a l l of th e c o u n t r i e s c o n s id e r e d , w h ile i t s v a lu e f a l l s between 0 and 1 e x ce p t f o r S w itz e r la n d where i t i s - . 5 4 . This may be r e l a t e d to jthe f a c t t h a t n o n - r e s i d e n t s earn ed n e g a t i v e e f f e c t i v e i n t e r e s t r a t e s on t h e i r Swiss a c c o u n ts d u rin g p a r t of th e sample p e rio d u sed . Even i f we n e g l e c t th e t - s t a t i s t i c s , the e s t im a t e d v a lu e s f o r ( .28 , .4 1, .1 8 , - . 5 4 ) do not i n d i c a t e any c o n s i s t e n t r a t e of tim e p r e f e r e n c e and p erh ap s r e f l e c t d i f f e r i n g i n s t i t u t i o n a l s t r u c t u r e s . 127 While we cannot examine th e s i g n i f i c a n c e of the r e l a t i o n s h i p s between exchange r a t e s and a s s e t m ark e ts t h a t a re im p lie d by th e e s tim a te d s t r u c t u r a l model, we can o b se rv e how th e s e r e l a t i o n s h i p s a r e in f lu e n c e d by s t r u c t u r a l changes in a s s e t m a rk e ts by means of th e c r o s s - e q u a t i o n r e s t r i c t i o n s on th e model p a ra m e te rs . A s t r o n g e r a s s o c i a t i o n between f o r e i g n and dom estic I n t e r e s t r a t e s i s p r e d i c t e d to le a d to an i n c r e a s e d i n f l u e n c e of do m estic r e l a t i v e to f o r e i g n a s s e t m a rk e ts on che p ro c e s s of f o r e i g n exchange d e te r m i n a t i o n . T his i s o b serv ed by examining how such a s t r o n g e r a s s o c i a t i o n ( r e f l e c t e d by l a r g e r v a lu e s f o r a-j , a £ , b^ and b^ in e q u a t io n s ( 2 . 3 a) and (2 . 3 b) ) a f f e c t s th e c o e f f i c i e n t s of che exchange r a t e e q u a tio n . With S w itz e r la n d as an e x c e p ti o n , a 10% i n c r e a s e in a-| , &2» ^3 anc* ^4 le a d s to an i n c r e a s e d i n f l u e n c e of c u r r e n t dom estic i n t e r e s t r a t e s on exchange r a t e s by an a v e ra g e of 10%, w h ile th e c o e f f i c i e n t s of lag g e d U.S. i n t e r e s t f a l l by about 2%. Since the j a r i a n c e s of th e c o e f f i c i e n t s of th e exchange r a t e e q u a tio n a re not known we a r e un a b le to i n f e r w hether t h i s change i s s t a t i s t i c a l l y s i g n i f i c a n t . Note t h a t a s t r o n g e r a s s o c i a t i o n between d o m estic and f o r e i g n i n t e r e s t r a t e s may ae im p lie d by i n t e r e s t a r b i t r a g e . I f one assumes t h a t o r e ig n exchange m ark e ts a re e f f i c i e n t and t h a t i n t e r e s t 128 a r b i t r a g e h o ld s , th e hig h c o r r e l a t i o n between dom estic and f o r e i g n i n t e r e s t r a t e s may e x p la in th e weak e v id e n c e in ■support of the view t h a t i n t e r e s t d i f f e r e n t i a l s a re an im p otant d e te r m in a n t of f o r e i g n exchange r a t e s . Another i n t e r e s t i n g o b s e r v a t i o n , which i s a g a in more pronounced f o r Germany and F rance, i s t h a t th e more p r e d i c t a b l e th e U.S. money p ro c e s s and U.S. and dom estic i n t e r e s t r a t e s a re from t h e i r own p a s t , th e l a r g e r th e i n f l u e n c e of f o r e i g n a s s e t m ark e ts on th e exchange r a t e . jThis f i n d i n g was based on o b s e rv in g how a g r e a t e r deg ree of a s s o c i a t i o n between c u r r e n t and lag g e d v a lu e s f o r each of the money and i n t e r e s t v a r i a b l e s ( im p lied by l a r g e r v a lu e s f o r ag, ajj, b-j, b2> Cg and c^ ) i n f l u e n c e s th e c o e f f i c i e n t s of th e exchange r a t e e q u a t io n . A 10% in c r e a s e in t h e s e c o e f f i c i e n t s l e a d s to an a v e ra g e of 10% i n c r e a s e in th e i n f l u e n c e of U.S. money and i n t e r e s t on th e Canadian jd o lla r, French f r a n c and Deutschm ark. We su g g e s t th e f o llo w in g i n t e r p r e t a t i o n . When th e b e h a v io r of a s s e t m arkets can be w e ll p r e d i c t e d from t h e i r b e h a v io r in th e p r e v io u s p e r i o d s , u n c e r t a i n t y w i l l be reduced and i n v e s t o r s t I would be u n d e rta k in g b ig g e r and more f r e q u e n t a d ju s tm e n ts in t h e i r p o r t f o l i o s and, th u s , e x e r t i n g more p r e s s u r e on th e f o r e i g n exchange m ark e t. In th e l i m i t i n g c a s e , where i n v e s t o r s have p e r f e c t f o r e s i g h t a p o r t f o l i o problem would 129 y ie ld a c o rn e r s o l u t i o n where in each p e rio d i n v e s t o r s w i l l lo ld th e h i g h e s t y i e l d i n g a s s e t th e re b y l e a d in g to a v o l a t i l e path f o r exchange r a t e s over tim e. We a ls o o b se rv e t h a t an i n c r e a s e d i n f l u e n c e of U.S. money on U.S. i n t e r e s t r a t e s l e a d s to a p r o p o r t i o n a l d e c l i n e in th e i n f l u e n c e of U.S. money on th e Deutschmark, French f r a n c and Canadian d o l l a r . This may be an I n d i c a t i o n t h a t a m onetary p o lic y in th e U.S. t h a t i s based on money t a r g e t s r a t h e r th an on s t a b i l i z i n g i n t e r e s t r a t e s w i l l le a d to demand-induced f l u c t u a t i o n s in U.S. i n t e r e s t r a t e s which in t u r n reduce th e i n f l u e n c e of U.S. money on exchange r a t e s . In deed , t h i s has been th e case s in c e November 1979 w ith th e o n s e t of th e V olcker regime where a sw itc h to a money r u l e was i n i t i a t e d . Perhaps th e r o l e :h a t U.S. money p la y s in exchange r a t e d e te r m i n a ti o n w i l l lot app ear to be as im p o r ta n t i f we were to e s t i m a t e the model over a sample p e rio d c o v e rin g th e V olcker regim e. F u rth e rm o re , t h i s f i n d i n g may e x p la in th e c o n t r a d i c t o r y ev id e n ce o b ta in e d in v a r i o u s s t u d i e s o f f l e x i b l e exchange r a t e e p is o d e s t h a t use d a ta f o r d i f f e r e n t c o u n t r i e s w ith i d i f f e r e n t p o l ic y r u l e s . j F i n a l l y , th e e f f e c t of p o r t f o l i o a d ju s tm e n t c o s t s on exchange r a t e s can be e a s i l y o b se rv e d . R e c a ll t h a t the c o e f f i c i e n t of th e c u r r e n t acco u n t v a r i a b l e i s equal to 130 - 2 d 2 w h ile t h a t of th e p r i c e of gold i s - d ^ d ^ . An i n c r e a s e in th e c o s t of a d j u s t i n g nominal a s s e t s (d 2 ) w i l l i n c r e a s e th e e f f e c t of th e c u r r e n t a cc o u n t and gold on th e exchange r a t e . In e s s e n c e , t h i s im p lie s t h a t th e c o s t l i e r i t i s to g a th e r in f o r m a ti o n and a d j u s t f o r e i g n a s s e t h o l d in g s , the sm a lle r w i l l t h e i r i n f l u e n c e be on exchange r a t e s s i n c e s m a lle r and l e s s f r e q u e n t a d ju s tm e n ts w i l l be made. C o n v ersely , the h ig h e r th e c o s t of a d ju s tm e n t of gold h o l d i n g s , th e s m a lle r th e i n f l u e n c e of gold m a rk e ts on f o r e i g n exchange r a t e s . The combined ev id e n ce from e s t i m a t i n g th e model c o e f f i c i e n t s and t h e i r r e s p o n s e s to s t r u c t u r a l changes in th e u n d e r ly in g economic p r o c e s s e s s u p p o r ts a p o r t f o l i o i n t e r p r e t a t i o n of f o r e i g n exchange r a t e s . F o re ig n money s to c k s and i n t e r e s t r a t e s have th e ex p ec te d i n f l u e n c e on exchange r a t e s where t h i s i n f l u e n c e w i l l be g r e a t e r when th e f u t u r e co u rse of th e exogenous v a r i a b l e s can be b e t t e r p r e d i c t e d from t h e i r own p a s t . Moreover, i t i s p o s s i b l e t h a t weak e v id e n c e c o n c e rn in g th e i n t e r e s t d i f f e r e n t i a l view i s due to th e speed w ith which i n v e s t o r s r e a c t to changes in and news about a s s e t m ark e ts and ta k e ad van tage of any e x i s t i n g a r b i t r a g e due to i n t e r e s t d i f f e r e n t i a l s . In a d d i t i o n , we o b se rv e t h a t th e type of p o l ic y t h a t m onetary a u t h o r i t i e s fo llo w w i l l d e te rm in e which f a c t o r s 131 w i l l be im p o rta n t d e te r m i n a n ts of exchange r a t e s . The i n f l u e n c e of money w i l l be g r e a t e r under an i n t e r e s t r a t e r u l e ( s i n c e i n t e r e s t r a t e s w i l l not be allow ed to vary much ), w h ile i n t e r e s t r a t e s w i l l have an i n c r e a s e d i n f l u e n c e under money t a r g e t r u l e s . T his f i n d i n g i s in c o n c e r t w ith p o r t f o l i o b a la n c e th e o ry s i n c e i t im p l i e s t h a t a s s e t demands w i l l i n f l u e n c e exchange r a t e s e i t h e r by c r e a t i n g an e x c e ss demand or by in d u c in g v a r i a t i o n s in i n t e r e s t r a t e s , both of which w i l l c r e a t e i n c i p i e n t c a p i t a l flow s and e x e r t p r e s s u r e on exchange r a t e s . Ji3 2 Chapter V SUMMARY AND CONCLUSIONS En t h i s c h a p te r we p r e s e n t a s y n t h e s i s o f th e r e s u l t s of the p re v io u s two c h a p t e r s and draw c o n c lu s io n s from th e e v iden ce p ro v id ed by th e two s e p a r a t e e m p i r ic a l a n a l y s e s . R e c a ll t h a t the o v e r a l l aim of t h i s stud y was to e m p i r i c a l l y examine th e p ro c e s s of f o r e i g n exchange d e t e r m i n a t i o n . In p a r t i c u l a r , we aimed to t e s t the i m p l i c a t i o n s of th e p o r t f o l i o b a la n c e view and examine the "ole of gold in a dynamic s e t t i n g . This work i s b e s t j/iewed as a s u g g e s t i o n and a d e m o n s tra tio n of a p a r t i c u l a r approach to th e m o d e llin g and t e s t i n g o f a p o r t f o l i o th e o ry of the d e te r m i n a t i o n o f f o r e i g n exchange r a t e s w i t h i n th e framework o f R a tio n a l E x p e c ta tio n s m odels. By combining th e ev id e n ce gain ed from e s t i m a t i n g our s t r u c t u r a l p o r t f o l i o model f o r exchange r a t e s and t h a t from analy zing th e d a ta w ith s t r u c t u r e - f r e e tim e s e r i e s t e c h n i q u e s , we a tte m p te d to g a in a more com plete p i c t u r e of she dynamics o f exchange r a t e s . This p ro v id e s us w ith more c o n fid e n c e in a d d r e s s in g th e f o llo w in g h y p o th e s e s posed about gold and th e p o r t f o l i o app roach : 133 ( i ) t h a t th e b e h a v i o r a l b a s i s o f p o r t f o l i o b a la n c e th eo ry p ro v id e s an e x p l a n a t i o n of th e observ ed f l u c t u a t i o n s in exchange r a t e s d u rin g th e p e rio d of f l o a t i n g r a t e s which s t a r t e d in th e e a r l y 70s. ( i i ) t h a t f l u c t u a t i o n s in th e p r i c e of gold a r e c l o s e l y r e l a t e d to th o se in th e v a lu e of th e d o l l a r and may, t h u s , be used to shed l i g h t on th e f o r c e s d r i v in g exchange r a t e s . The b a s i c re a so n f o r e s t i m a t i n g th e two s e p a r a t e jeconometric models was t h a t , i f th e e v id e n ce from th e two e m p i r ic a l models were c o n s i s t e n t , then we co uld a s s e r t t h a t jthe r e s u l t s have not been b ia s e d by th e s i m p l i f y i n g a ssu m p tio n s imposed on th e s t r u c t u r a l p o r t f o l i o model t h a t were n e c e s s a r y fo r c a r r y i n g out an e m p i r i c a l a n a l y s i s . S i m i l a r l y , by u sin g d a ta f o r fo u r d i f f e r e n t c o u n t r i e s , the I f a c t o r s t h a t i n f l u e n c e exchange r a t e s in a f l e x i b l e r a t e regim e, and which a re not c o u n t r y - s p e c i f i c , may be •iso la te d . One obvious way of ju d g in g th e r e s u l t s from th e s t r u c t u r e - f r e e model w ith r e s p e c t to th e s t r u c t u r a l p o r t f o l i o model i s to examine th e f o r e c a s t e r r o r v a r i a n c e , a c c e p t in g as more n e a r l y c o r r e c t th e model w ith minimum f o r e c a s t e r r o r v a r i a n c e . I f t h i s i s done u sin g th e T h e il U c r i t e r i o n or q u a d r a t i c l o s s f u n c t i o n , th e s t r u c t u r e f r e e 134 model i s c l e a r l y s u p e r i o r . We would have a c c e p te d t h i s c o n c lu sio n had i t not been f o r th e f a c t t h a t both models are s u f f i c i e n t l y i n c o r r e c t , t h a t i s both a re c h a r a c t e r i z e d cy s p e c i f i c a t i o n e r r o r s a r i s i n g from th e s i m p l i f y i n g " e s t r i c t i o n s in th e s t r u c t u r a l model, and l i m i t i n g th e cumber of v a r i a b l e s in th e s t r u c t u r e - f r e e model. R ather, che c o n s i s t e n c y of r e s u l t s over s p e c i f i c a t i o n e r r o r s i s th e ev id en ce we sought to f i n d . Moreover, by i t s e l f , the v e c t o r a u t o r e g r e s s i v e model, even when c o r r e c t l y s p e c i f i e d , does not p ro v id e us w ith any e x p l a n a t i o n o f th e b e h a v i o r a l s t r u c t u r e t h a t u n d e r l i e s th e p ro c e s s of exchange r a t e d e te r m i n a t i o n . Hence, e x p e rim e n tin g w ith d i f f e r e n t s t r u c t u r a l models i s n e c e s s a r y i f economic th e o r y i s to o ro v id e a c o r r e c t b e h a v i o r a l i n t e r p r e t a t i o n of movements in exchange r a t e s . With r e s p e c t to th e f i r s t h y p o t h e s i s , th e ev id e n ce from she two e m p i r ic a l models m ild ly su p p o rte d t h e p o r t f o l i o view of f o r e i g n exchange r a t e d e te r m i n a t i o n , a lth o u g h a l a r g e f r a c t i o n of exchange r a t e v a r i a n c e C in some c a se s over one h a l f ) was not e x p la in e d by th e p o r t f o l i o f a c t o r s c o n s id e r e d . This i n d i c a t e s t h a t t h e r e a re o t h e r im p o rta n t d e te r m in a n ts of exchange r a t e s t h a t were not in c lu d e d in our a n a l y s i s which in t u r n im p lie s t h a t e i t h e r the s t r u c t u r a l and th e u n s t r u c t u r e d models a r e both 135 i n i s s p e c i f i e d , or t h a t f o r c e s o t h e r than p o r t f o l i o d i v e r s i f i c a t i o n , such as t r a d e or perhaps i n t e r v e n t i o n and c o n t r o l s , c o n t r i b u t e to th e movements in exchange r a t e s . In th e u n s t r u c t u r e d model, U.S. money was im p lie d to be the most s i g n i f i c a n t f a c t o r i n f l u e n c i n g exchange r a t e s . Moreover, U.S. money shocks had an i n s t a n t a n e o u s y e t a e r s i s t e n t e f f e c t on f o r e i g n exchange m a rk e ts . However, in Doth e co n o m etric m odels, i t was not c l e a r w hether demand or supply f a c t o r s dom inate s i n c e th e d i r e c t i o n of th e re s p o n se to money shocks was n ot c o n s i s t e n t f o r a l l th e c o u n t r i e s c o n s i d e r e d . As f o r th e do m estic money ( i . e . t h a t of th e c o u n t r i e s s t u d i e d ), th e ev id e n ce did n ot a t t a c h any s i g n i f i c a n t r o l e to i t . Yet, we could d e t e c t some feed back from exchange r a t e s to do m estic money in th e case of Canada and S w itz e r la n d which i n d i c a t e s t h a t t h e i r c u r r e n c i e s ( as w e ll Ls o t h e r s t h a t were not s t u d i e d ) were not r e a l l y f l e x i b l e cut r a t h e r f o llo w in g a d i r t y f l o a t over th e sample p e rio d . As p r e d i c t e d by p o r t f o l i o t h e o r y , both eco n o m etric models i n d i c a t e d t h a t an i n c r e a s e in i n t e r e s t r a t e s in th e J.S . r e l a t i v e to th o se abroad c a u se s th e d o l l a r to a p p r e c i a t e a g a i n s t o t h e r c u r r e n c i e s . But, th e ev id e n ce a ls o i n d i c a t e d t h a t i n t e r e s t r a t e d i f f e r e n t i a l s m a t t e r only !in th e case of th e Deutschmark where t h e i r i n f l u e n c e i s 136 much more s i g n i f i c a n t th an t h a t of U.S. money. The s t r u c t u r a l r e l a t i o n s h i p between th e p r o c e s s e s d r i v i n g i n t e r e s t and exchange r a t e s t h a t i s im p lie d by th e r a t i o n a l e x p e c t a t i o n s h y p o t h e s i s in our p o r t f o l i o model su g g e ste d t h a t m onetary p o l ic y aimed a t s t a b i l i z i n g i n t e r e s t r a t e s would redu ce th e i n f l u e n c e of i n t e r e s t r a t e s on f o r e i g n exchange m a rk e ts . F l u c t u a t i o n s in i n t e r e s t r a t e s would be e x p e c te d tQ be m inim ized under an i n t e r e s t r a t e r u l e , th u s , r u l i n g out any r o l e f o r i n t e r e s t d i f f e r e n t i a l s in exchange r a t e d e t e r m i n a t i o n . F u rth e rm o re , i n v e s t o r s would e x p ec t a u t h o r i t i e s to i n t e r v e n e whenever i n t e r e s t r a t e s d e v i a t e from th e s e t t a r g e t s , and th e re b y would not r e a c t as q u ic k ly to tem porary changes in i n t e r e s t r a t e s . There a re a ls o o t h e r p o s s i b l e re a s o n s f o r t h i s f i n d i n g . One has to do w ith c o n t r o l s and r e s t r i c t i o n s on f o r e i g n c a p i t a l in v e s tm e n ts and flow s which would m i t i g a t e any i n f lu e n c e t h a t i n t e r e s t d i f f e r e n t i a l s would have on exchange r a t e s . Such c o n t r o l s a re known to be e x e r c i s e d by Dhe c o u n t r i e s s t u d i e d , and were d e s c r i b e d in Chapter I I I . The a n a l y s i s in t h a t c h a p t e r a ls o i n d i c a t e d th e e x i s t e n c e of fee d b ac k from exchange r a t e s to money which e n d o rse s Dhis i n t e r p r e t a t i o n . Another rea so n could be th e f a c t t h a t a s s e t m a rk e ts a r e not p e r f e c t . In such a c a s e , i n t e r e s t ~ate p a r i t y would not n e c e s s a r i l y p r e v a i l . The f a c t t h a t 137 t h e r e a re more than one p o s s i b l e i n t e r p r e t a t i o n f o r t h i s r e s u l t d e m o n s tra te s th e v i r t u e of u sing a l t e r n a t i v e e m p i r ic a l ap p ro ach es to t e s t i n g th e d e s i r e d h y p th e s e s in o rd e r to overcome any s p e c i f i c a t i o n b i a s . With r e s p e c t to th e second h y p o t h e s i s , no c l e a r ev id e n ce t h a t gold p la y s an im p o rta n t r o l e in f o r e i g n exchange m ark ets was found. In th e s t r u c t u r e - f r e e model, i n n o v a t i o n s in th e p r i c e of gold acco u n ted f o r alm ost n o th in g in th e v a r ia n c e in exchange r a t e s , w h ile in th e s t r u c t u r a l p o r t f o l i o model th e d i r e c t i o n of th e i n f l u e n c e of gold on exchange r a t e s was not c o n s i s t e n t among our sam ples. The f i n d i n g t h a t U.S. money and p r i c e s acco u n t f o r most of th e v a r i a t i o n s in th e p r i c e of gold s u g g e s ts t h a t gold i s viewed as a s t o r e of v a lu e t h a t i n v e s t o r s sw itc h to when w orld p r i c e s a re not s t a b l e . In th e v e c t o r a u t o r e g r e s s i v e model, 90% o f th e v a r i a n c e in th e p r i c e of gold co u ld be e x p la in e d by U.S. money and U.S. and German p r i c e s . I f one c o n s i d e r s th e combined e v id e n c e , one i m p l i c a t i o n seems to a r i s e from t h i s stu d y , t h a t p o r t f o l i o s u b s t i t u t i o n t a k e s p la c e between t h e U.S. d o l l a r , th e Deutschmark and g o ld , b ut not w ith o th e r c u r r e n c i e s . In deed, t h i s i s c lo s e to th e f i n d i n g of D o rn b u sch (15) where he p o i n t s a t e v id e n ce i n d i c a t i n g t h a t th e Deutschmark sh ou ld have commanded a 138 b ig g e r s h a r e in i n t e r n a t i o n a l p o r t f o l i o s th a n i t did in 1980. This g iv e s r i s e to a q u e s t io n t h a t may be w othw hile i i n v e s t i g a t i n g , t h a t w ith th e e v o l u t i o n o f th e I n t e r n a t i o n a l Monetary System and th e in c r e a s e d s o p h i s t i c a t i o n in in v e s tm e n t s t a r t e g i e s , i n v e s t o r s 1 n o tio n o f a s a f e c u rre n cy changed from gold to U.S. d o l l a r s . But w ith th e g r e a t e r prominance th e Deutschmark i s a t t a i n i n g in i n t e r n a t i o n a l p o r t f o l i o s , an e m p i r i c a l stu dy of b i l a t e r a l exchange r a t e s v i s - a - v i s th e Deutschmark may r e v e a l some i n t e r e s t i n g i n f o r m a ti o n . To summarize th e e v id e n c e , U.S. money was found t o be th e most im p o r ta n t s i n g l e f a c t o r t h a t e x p l a i n s th e v a r i a n c e in exchange r a t e s and th e p r i c e of g o ld . Ind eed, i t i s money demand f a c t o r s t h a t seemed to dom inate as i n d i c a t e d by a p o s i t i v e c o r r e l a t i o n between U.S. money and th e v a lu e of th e d o l l a r r e l a t i v e to o t h e r c u r r e n c i e s . T his l e a d s us to th e c o n c lu s io n t h a t , when i t comes to c u rre n c y d i v e r s i f i c a t i o n , only U.S. money m a t t e r s . The i n d i c a t i o n t h a t demand f a c t o r s a re dominant i m p lie s t h a t e x p e c t a t i o n s r e g a r d i n g th e U.S. economy, and s u b s e q u e n tly t h e i r i n f l u e n c e on th e d o l l a r , a re th e most im p o r ta n t f a c t o r s t h a t i n f l u e n c e exchange r a t e s . How th e r o l e of th e d o l l a r in i n t e r n a t i o n a l p o r t f o l i o s w i l l change when t h e movements in exchange r a t e s a g a i n s t th e d o l l a r and th e Deutschmark 139 a re s t u d i e d s im u lt a n e o u s ly w i l l be l e f t f o r f u t u r e r e s e a r c h . In c o n c lu s io n , th e e m p i r ic a l e v id e n c e p ro v id ed h e re on th e i m p l i c a t i o n s o f dynamic p o r t f o l i o th e o ry c o n c e rn in g th e b e h a v io r of f o r e i g n exchange r a t e s in a f l e x i b l e exchange system , w h ile not s t r o n g l y in f a v o r of th e view t h a t i n t e r e s t d i f f e r e n t i a l s have a m ajor i n f l u e n c e on f o r e i g n exchange r a t e s , i n d i c a t e s t h a t p o r t f o l i o e f f e c t s may be m a n if e s te d by d i f f e r e n t f a c t o r s depending on th e i n s t i t u t i o n a l framework and p o l i c i e s d i r e c t l y i n f l u e n c i n g I a s s e t m a rk e ts . Thus, i t i s our b e l i e f t h a t f u t u r e r e s e a r c h which a p p r o p r i a t e l y models i n s t i t u t i o n a l f a c t o r s and p o l ic y r u l e s alo n g w ith i n v e s t o r s ' b e h a v io r in d i f f e r e n t s e t t i n g s may shed l i g h t on th e u n d e r ly in g phenomena g o v ern in g exchange r a t e s . i 140 Appendix A DERIVATION OF ASSET DEMAND EQUATIONS T h is appendix c o n t a i n s a d e r i v a t i o n of e q u a tio n (4 .5 ) of C hapter IV. The problem s t a t e d has th e form of a g e n e ra l c l a s s of dynamic l i n e a r r a t i o n a l e x p e c t a t i o n s m odels which a re d i s c u s s e d in Hansen and S a rg e n t( 3 5 ) and ( 3 6 ) . The s o l u t i o n t e c h n iq u e used h e re i s c l o s e l y r e l a t e d to th e one develop ed by Hansen and S a rg e n t u sin g th e Wiener-Kolmogorov p r e d i c t i o n fo rm u la s. Let us b egin w ith th e o b j e c t i v e f u n c t i o n (4 .2 ) N (1 A) E lim £ 3 t T - J» (T -T ) 0 i=0 ' t t t t t-1 - [ D U ) T t ] ' [ D ( L ) T t ] - Yt } T his problem can be so lv e d by f i r s t c o n s i d e r i n g th e c e r t a i n t y case u s in g t h e c e r t a i n t y e q u iv a le n c e or s e p a r a t i o n theorem . In p a r t i c u l a r , th e sequ ences of OO 00 oo exogeneous v a r i a b l e s {Rt }t= 0 , ] J t }t==Q and { Yt } t==0 a re r e g a r d e d a s known. The o p t i m i z a t i o n i s over th e sequencej o o | of a s s e t h o ld in g s |T t }t _Q w ith T_-j g iv e n . Let N >> 1 and j c o n s id e r th e N -p e rio d problem w ith N f i x e d . D i f f e r e n t i a t e (1A) w ith r e s p e c t to Tt to o b t a i n t h e f i r s t o rd e r n e c e s s a ry 141 c o n d i t i o n s . For th e i n f i n i t e time problem, th e E u ler e q u a t io n s a r e th e same but th e t r a n s v e r s a l i t y c o n d i t i o n i s found by t a k i n g l i m i t s as N ap p ro a ch e s i n f i n i t y . The E uler e q u a t io n s a r e : (2A) 3 J / 9 T f e = Bt [ Rt “ J t + 3 J t + l ^ “ 3 - £ /9 T t N where L - E B ^ D C D T J « [ D (L ) T ] t=0 t z N = E Bt [T 'D '+ T t D»] [D T +D T ] t O t - 1 l O t 1 t - 1 E v a lu a tin g th e d e r i v a t i v e of X w ith r e s p e c t t o T t f o r t = 0 , . . . , N-1 a Z / 3 T ^ = 3 t D»D(L)T + B t+1D*D(L)T , z 0 t 1 t+1 + Bt {[D(L)T(;] 'D 0 } ' + B t+1{[D(L)Tt + 1 ]'D 1 = 2 B t rD, D( L) +BL "1D » D ( L ) I t L 0 1 J t = 2Bt D(3L“ 1 ) »D(L)T and 3 ^ 1 /3 T ■ = 23 D'D(L)T N NO N T h e r e f o r e , to maximize (1A) w ith r e s p e c t to Tt w ith N f i x e d , th e E u le r c o n d i t i o n s must be s a t i s f i e d (3A) 3J/3Tt = 6t {[Rt - J t + 3 J t+1 ] -'2DX 3L“ 1 ) ’D(L)Tt ] | = 0 f or t = 0 , 1 , . . . , N-1 s u b j e c t to th e t r a n s v e r s a l i t y c o n d i t i o n 142 (4A) 3J/3T. N - s N [ j n + 2 D6 D<L)TJ - 0 We can r e w r i t e th e E uler e q u a tio n as (5A) D (8L-1)'D (L )T = 1/2 ( R - J + 6 J ) t t t t+1 which can be so lv e d s u b j e c t to th e i n i t i a l c o n d i t i o n T_-^ and t e r m in a l c o n d i t i o n given by (4A) by ta k i n g l i m i t s a s N -*-°°. The s o l u t i o n t o th e E u ler e q u a t io n s which s a t i s f i e s t h e t r a n s v e r s a l i t y c o n d i t i o n i s s in c e th e t r a n s v e r s a l i t y c o n d i t i o n l e a d s us t o s o lv e th e s t a b l e r o o t s backwards and th e u n s t a b l e r o o t s fo rw a rd s. S ince and a re given m a t r i c e s , th e "fee d b ac k " polynom ial i s a l s o known. In f i n d i n g th e " f e e d f o r w a r d ” polynom ial we have to o b t a i n an i n v e r s i o n fo rm ula f o r D(L)T t or J P r e m u lt i p ly i n g both s i d e s by g iv e s (6A) 143 Using th e same te c h n iq u e used in (38) we begin by e v a l u a t i n g d e t M(z), Note t h a t 3det M (z)/3z = t r a c e |[8 d e t M (z)/3M (z)] [ 8M(z ) / 8 z3' = d e t M ( z ) . t r a c e [ M(z)” 1 8 M (z )/8 z ] L et M(z) = j^3M (z)/Sz3 M(z)“ ^. I t s T ay lo r s e r i e s e x p an sio n ab ou t z e ro i s M(z) = Mq + M^z + M^z . . . Then M(z) . M(z) = 3M(z)/az = D*D 1 0 S u b s t i t u t i n g f o r M(z) i t s T ay lor s e r i e s e x p a n sio n , and e q u a t in g c o e f f i c i e n t s on both s i d e s o f the above e q u a tio n we g e t MD * D = D’ D 0 0 0 1 0 VoD o * M j-lD iD 0 * 0 or 144 M = 0 D‘ D '“ 1 1 0 Hi = • -M D* D! "O' 1 0 • M. = 3 - M Df D,_1 j - 1 1 0 From th e above we have r e c u r s i v e fo rm u la s Let k = n, and w r i t e f o r th e M 1 s. j k d e t M(z) = c + c z + . . . + c z 0 1 k D i f f e r e n t i a t i n g w . r . t . z we o b t a i n 9 det M ( z ) / 9z.= c^ + 2CpZ + . . . + kc^z^ -1 \ = d e t M(z) . t r a c e ^ Mq + 1 M + . . . J E q u atin g c o e f f i c i e n t s d l - d g tr a c e Mq 2d 2 • = dQtrace M -^ + d ^ tr a c e Mq •. kdk = dQ trace . . . + dk _-^trace M q w h ile dg . d e t (D-V ______________ ■ — — __ 145 We have d e riv e d r e c u r s i v e fo rm u la s f o r th e polynom ial c o e f f i c i e n t s f o r d e t M(z) Using t h e s e fo rm u la s and a nu m erical f a c t o r i z a t i o n a lg o r ith m , we can e x p r e s s d e t M( z ) a re th e r o o t s of d e t M(z), a l l g r e a t e r where z than i n m ag nitu de. Note a ls o t h a t The T aylor s e r i e s e x p a n sio n about z e r o f o r wri t t e n k-1 adj M(z) k-1' N o tice t h a t We e q u a te c o e f f i c i e n t s to d e r i v e D ’ D k -1 or 146 M 0 ■ V W 1 M i * V%V'1 - M o D iD o'1 M* = d (D’ D ) " 1 - M* D’ D ' - 1 k - 1 k - 1 0 0 k - 2 1 0 which p r o v id e s us w ith r e c u r s i v e fo rm u la s f o r th e Mj t s * F i n a l l y , we have to expand th e m a tr ix f o r M(z)” 1 by m a trix p a r t i a l f r a c t i o n s to o b t a i n M(z)' ■ 1 _ N N ( z -z ^ ) + • • • + k ( z - z ) k where R e w ritin g ( 1 / 0 + M l z j + k - 1 M( z) -1 -Vz i (1 - z /z ^ ) - N k /Zk F i n a l l y , s u b s t i t u t i n g 3 z” 1 f o r z g iv e s M(Bz“ 1 ) ” 1 = (D’ D + D’D 3 z " 1 ) “ 1 0 0 10 -Vz i 1 - Bz“ 1/ z V zk 1 - Bz“ 1/ z 147 Using th e above e x p r e s s io n f o r M( 3 z“ 1 ) ~ 1 th e d e c i s i o n ru le (6 A ) can be w r i t t e n a s : (7A) Tt +EgDl Tt _l = 1 /2 X ;- j = l where X • = 1/ z . 3 3 Using th e e x p an sio n -1-X .gL -1 [V J t+ S J t+ i] [■- (X . )3L' J -1 i *i = Z (x. e ) l 1=0 (7A) can be r e w r i t t e n a s: k (8A) T = -D*D T - 1 / 2 E X N t 0 1 t - 1 . 1 j j OO Z ^ . ^ ( r . - J .+e<J . £_Q 3 \ t + 1 t+1 t+ l-fl / To go back to th e u n c e r t a i n t y case where th e exogeneous v a r i a b l e s fo llo w random s t o c h a s t i c p r o c e s s e s , we r e p l a c e Rt + i and J t + i in th e °Ptim a l r u l e w ith Et |~Rt+ iJ and Et |^Jt + ^j (9A) Tt = - D J D ^ ^ - 1 / 2 1 : X.N. OO S (V )lE t(R t+i-JtHt8Jt+l+l) 1=0 v N o tice t h a t i=0 = 1/X. Z (X P^-E (J ) - 1/X J 3i =o 3 t t + i j t The d e c i s i o n r u l e becomes 143 (10A) Tt = 1 / X ^ t )' Since our o b j e c t i v e i s to o b t a i n a c lo s e d form d e c i s i o n r u l e we need to w r i t e th e g e o m e tric sums in E fe . ] and we w i l l use th e Wiener-Kolmogorov optim al p r e d i c t i o n fo rm u la . R e c a ll t h a t Rt and J t were p e rc e iv e d by th e ag en t to be e le m e n ts o f an a u t o r e g r e s s i v e s t o c h a s t i c p r o c e s s Xt d e s c r i b e d by: ( 2 .3 ) a (L ) X t = Ut where X£ = ( J^. R^. ) J t = ^ i Xt » ^1 = ^ I n ° q -n ^ Rt = 4>2Xt * ^2 = ^®n ^n ®q-2n^ and where th e s u b v e c to r S^. c o n t a i n s a l l o th e r v a r i a b l e s t h a t h e lp p r e d i c t Rt and J^.. Thus, we can o b t a i n th e op tim al p r e d i c t o r s f o r R^. and once we know th e p r e d i c t o r f o r Xt . A s u f f i c i e n t a ssu m p tio n t h a t w i l l allow us to e x p re s s e x p e c te d f u t u r e v a lu e s of X ^. in term s of i t s p a s t v a lu e s i s t h a t th e p ro c e s s a(L ) has a mean e x p o n e n tia l o rd e r l e s s tL t + i j in term s of v a r i a b l e s known a t time t . To do t h i s k •1/2 2 j = l • N- ) J 1 1 CO / Z <Xj B)1/ R t + i - d - l / A O J t + i i = 0 ' 149 th an (v/3")""'*• To f a c i l i t a t e th e d e r i v a t i o n , we w i l l s t a r t w ith a more s t r i n g e n t a ssum p tio n on th e o rd e r of a(L) r e q u i r i n g th e r o o t s of d e t a ( z ) = 0 to l i e o u t s i d e th e u n i t c i r c l e . L a t e r , i t w i l l be c l e a r why, fo r our p u rp o se s, i t i s s u f f i c i e n t to r e q u i r e th e r o o t s to be g r e a t e r than y/W. Given t h i s a ssu m p tio n , th e i n v e r s e of a(L) e x i s t s and (2 .3 ) can be e x p re s s e d as U CO <11 A) X = a(L)U = E fa. 1 j - 0 L J J where a(L) = a ( L ) ” ^ The Wiener-Kolmogorov p r e d i c t i o n fo rm ula i s : k - i [ E akLk - ] U, k -- S t a r t i n g w ith th e term E t( x t+ i > = L ,4 a k L ' k= i E a L L k=i k k - i E C A .3 )XE (X ) = E U . B ) 1 i=0 ~ i=0 J °° 0 3 Tc— i E E ak Li=0 k = i = t J jCDU^ OO 00 ip (L) = E E (A.3 ) . L i=0 k= i I n t e r c h a n g i n g o r d e r s of summation O O O O ■iKL) = E E (Xi 3)1 av Lk . l' 1 k=0 i=0 J r- 0 0 “IP " * 1 ■ £ v k][£ u 1 5 0 Let 0 . = A . 3 3 3 ip (L) = y a L k i k k=0 k+1 -k -1 1 - e . . L J 1 - e^L -1 r k=0 o o 00 k Z k < ! v - 1 -a kL - V - J r o V j 1 - e .l- j 1 oo ^ th e n Et (Xt + i ) = i=0 a(L) —0j-L ^ . a ( 0j ) 1 “ 6 j L -1 U b ut Ufc = a“ 1(L)Xt = a(L)X X (A13)1Et (Xt + i ) = i=0 a(L)-(Aj3)L“ 1. a(Aj B) 1 - (A jB )L -1 a 4 (L)X I-(Aj 3)L“ 1.a(A^B).a- 1(L) 1 - (Aj B)L“ 1 I-( A^B)L-1 .a(AjB) ” 1 . ot( L) 1 - (A j3)L~1 151 By long d i v i s i o n of a(L ) / 1 - (Xj 3 ) L” 1 1 = 0 = a ( Xj 3) ~ q-1 q-1 i * Z < Z . (X j3 )k 1ak )L:i i = l k = i+ l C o n se q u e n tly , Et (Rt + i ) = ^2 2 (Xn3)' Et (Xt + i ) i= 0 i= 0 Z U j B ) Et(Jt+1) = ^ £ ( ^ 6 ) M xt+i> and i=0 J ' i=0 which a llo w s us to w r i t e th e d e c i s i o n r u l e a s : k (12A) Tt = -D5BJ,t _1+ k j-1 - 1 / 2 I Nj j =i Xj $2 + (1 - X x < ot( Xj 8) j ) * i ] q q “ 1 • ‘ I ) X ( X <Xj e > _ i a k )L J x t k = i+ l where ( 2 .3 ) a ( L) Xt = U and Xj and Nj a re g iven above i n term s of e le m e n ts of D(L). ( 12A) and (2 .3 ) and Xj and Nj e x p re s s r e s t r i c t i o n s imposed a c r o s s th e d e c i s i o n r u l e p a ra m e te rs and th o s e of th e s t o c h a s t i c p r o c e s s d r i v i n g J t and Rt , Both c u r r e n t and la g g e d v a lu e s o f J t , Rt and M t app ear i n (12A) w ith th e number of l a g s b eing one l e s s th a n th e o r d e r of th e a u t o r e g r e s s i v e p r o c e s s . 152 A p p l i c a t i o n f o r E s tim a tin g Demand E q u atio n s f o r Two A s s e ts and G old. We s h a l l e s t i m a t e our eco n o m etric model u sin g d a ta from fo u r c o u n t r i e s . Our o b j e c t i v e i s t o examine th e r a t e of exchange of a c o u n t r y ’ s d o m estic c u rre n c y w ith r e s p e c t to th e d o l l a r and th e r o l e of gold i n d e te r m in in g t h a t r a t e . The above p o r t f o l i o model i s used a s th e b e h a v i o r a l framework d e s c r i b i n g how dom estic i n v e s t o r s make t h e i r p o r t f o l i o a l l o c a t i o n d e c i s i o n s . Let c o u n try 1 be th e i s s u e r of th e dom estic c u rre n c y , th e U.S. be co u n try 2 then Rt = . < r t r * 0 > Tf. = ( T t T* Gt ) ■ < 1 e t Pgt> where e £ i s th e r e a l exchange r a t e e x p re s s e d in term s of th e p r i c e of a U.S. d o l l a r , P i s th e r e a l p r i c e of gold, g U and Gt i s th e amount of gold h e ld . I f we assume f u r t h e r t h a t t h e r e a r e no i n t e r r e l a t e d c o s t s of a d ju s tm e n t, th e m a tr ix D i n th e term - 1 / 2 ( T - T . )'D (T -T - ) in e q u a tio n t t - 1 t t - 1 (4 .1 ) w i l l n e c e s s a r i l y be a 3x3 p o s i t i v e d e f i n i t e d ia g o n a l m a tri x 153 D = d 0 0 1 0 d 2 0 0 0 d w ith p o s i t i v e d ia g o n a l e le m e n ts. R e w ritin g th e above term a s : [D(L)Tf.] '[ D ( L ) T t ] where D(L) = Dq + D-^L i m p l i e s Dq = -D-^ = d ia g ( d2 dg ) Under th e s e c o n d i t i o n s th e "fee d b ac k " and " fe e d fo rw ard " p o ly n o m ia ls in e q u a tio n (6A) w i l l red uce to -1 Tt - Tt _1= 1/2[D (1-3 L “ 1)] [(R t - J t+ BJt+1 )] = 1 /2 (1 -B L " 1 ) " 1D-1 (Rfc- J t + 6 Jt+1 ) 0 0 Using th e e x p r e s s io n = X) 31 L”1 i=0 The d e c i s i o n r u l e becomes oo D 1 [ Rt+ i~ J t + i + e J t + i + l ] .=0 Using a change of s u b s c r i p t t r a n s f o r m a t i o n T t = Tt - 1 + 1 / 2 < i=0 Z ^ J t+ j4 i> = Z J t + i - J t J=0 J i=0 We g e t 154 00 Tt = Tt-l + 1 / 2 2 s i D-''[R ] - 1 / Z D - ^ j J i=0 Given th e model assu m p tio n s d e s c r i b e d in th e t e x t , only th e demands f o r th e f o r e i g n a s s e t and gold e n t e r th e e q u i l i b r i u m c o n d i t i o n f o r th e f o r e i g n exchange m arket. T h e r e fo r e , we w i l l need t o w r i t e e x p l i c i t l y th e demand e q u a t i o n s f o r th e s e two a s s e t s o n ly . D e riv in g th e demand e q u a t io n f o r gold i s s t r a i g h t fo rw a rd s i n c e gold i s a n o n - i n t e r e s t b e a r i n g a s s e t and no e x p e c t a t i o n s w i l l be i n v o lv e d . Hence, G = G - 1/2d ~P t t - 1 3 g t Whereas, the demand f o r th e f o r e i g n a s s e t w i l l be: 00 r * t = TL l + 1 / 2 d 2 S b V - 1 / 2 d 2e 1 = 0 R e c a ll t h a t R = ( r r* 0 ) = ^oX t. t t ^ t where X' = ( J R M CA ) , < j> _ = ( Oo I , 0? ) t t t t t 2 5 5 ^ and a ( L) X = U t t In th e t e x t some r e s t r i c t i o n s ( im p lie d by th e a n a l y s i s of Chapter I I I ) were imposed on a ( L ) , namely t h a t i t i s a second o rd e r m a t r ix polynom ial t h a t in v o lv e s c e r t a i n z e ro r e s t r i c t i o n s . The e x c l u s i o n r e s t r i c t i o n s allow us to w r i t e 155 th e p ro c e s s g o v e rn in g X ^. ( a f t e r r e a r r a n g i n g rows to f a c i l i t a t e t a k i n g th e i n v e r s e of a ( L ) ) as f o l lo w s : where a(L )X t = Ut ' b l b 3 b 5 0 0 0 0 0" C M b4 b 6 0 0 0 0 0" 3 1 a 3 0 0 0 0 0 0 a 2 a 4 0 0 0 0 0 0 C 1 0 c 3 0 0 0 0 0 c 2 0 c 4 0 0 0 0 0 a ( L ) = I - 0 0 S i ®3 0 0 0 0 L+ 0 0 g 2 ®4 0 0 0 0 h l h 3 h 5 h 7 h 9 h n O 0 h 2 h 4 h 6 h8 h 1 0 h 1 2 ° 0 p l P3 P5 P7 P9 p l l ° 0 P 2 P4 P6 P8 P 1 0 P 1 2 ° 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0_ D 0 0 0 0 0 0 0_ X. . ( r* r t P g t e t GAt 1 0 ) and CA i s th e c u r r e n t a c c o u n t. p r e d i c t i o n form u la f o r E r = i =0 E r , . t t + i ] ■ c p t + i ] 1 (j) a( 8)~^ The Wiener-Kolmogorov im p l i e s t h a t I + Be^L X, where < p = (1 Oj ) I f we p a r t i t i o n a(L) so t h a t 0 a ( L) = 156 then a(L) = 5 ' l 1 -1 -1 -1 6 6 6 6 3 2 1 3 Given th e form of c |> we need only know th e e le m e n ts i n th e f i r s t row of a ( L ) ” 1 . Now 6^(L) i s t h e 3x3 u p p e r - l e f t su bm atri X of a ( L) , which f w r i t t e n as • • / ^ B11 b34 B56 6X(L) ■ ( a 12 a34 0 \ ‘ C12 0 c 34 Once mor e, we use th e expre i n v e r s e to o b t a i n : . ( L) h(L) f ( L) m(L) 6, (L ) - 1 - 1 - Ahm“ " * -m- f A m” ^+m” ^ f Ahm” ^ where A = (p - hm "^f)” ^ and f o r n o t a t i o n a l s i m p l i c i t y m = m(L ) , e t c . . . L et the f i r s t row of 6 ^ (L) be ( S1 S2 S3 ) . To f i n d S1 , S2 and S3 l e t us s t a r t by s o l v i n g f o r A. A Then where | a s : Ij Also S3 s : F i n a l l y r 157 S, = ^3 4 / ' A| S2 =“ ^3 4/ ! A| A i i s th e d e te r m in a n t of A which may be e x p re s s e d M = [b 12- b5 6 c 12/ c 34J a 34- b3 4 a 12 which i s th e f i r s t elem ent of -Ahm“ ^ - w i l l be 3 = " s 1b56/ c 34 T w i l l be = ( S1 S2 S3 0 4 ) x ( I + S o i L )Xfc = S1rt+ S2rt+ S3Mt+ (S1b2+S2a2+S3c2)r*_x •k + 3( S 3b^+ B2a^)r^._3+ g(S-^bg+ S3 c 4 )M^._3 Appendix B STATISTICAL TABLES TABLE 7 T e s t s 3 on Model Lag Length 159 : ( 1974-1980) E quation Canada France Germany S w itz e r la n d F ( 12,39) F (12,3 9) F( 14,35) F( 14,35) Money-US = 1.07 . = 1.17 = .75 = 1 .26 ( . 4 1 ) b ( .3 3 ) ( .70) ( .27) Domesti c 1 .82 2.42 1 .28 1 .13 Money ( .07) ( .01 ) ( .26) ( .3 6 ) I n t e r e s t 1 .43 1 .39 .95 3 .02 D i f f e r - t i a l ( .1 8 ) ( .20) ( .51 ) ( .0003) U.S. 1 .24 .90 .81 .87 P r i c e s ( .2 8 ) ( .56) ( .64) ( .59) Domestic .81 .49 .46 .42 P r i c e s ( .64) ( .9 2 ) ( .93) ( .95) P r i c e Of 1 .40 .63 .62 .47 Gold ( .20) ( .81) ( .82) ( .92) Exchange .85 1 .26 1 .61 1 .37 Rate ( .61) ( .27) ( .12) ( .21 ) O v e ra ll X2(98) X2 (98) X2(98) X2(98) system =105.46 =107.73 = 89.12 =108.20 ( .28) ( .2 3 ) ( .72) ( .22) 3 The above s t a t i s t i c s c o rre sp o n d t o t e s t i n g a l a g o rd e r of two as a r e s t r i c t i o n of a l a g o rd e r of f o u r . b The f i g u r e s in p a r a n t h e s i s a re the l e v e l s . m arg in al s i g n i f i c a n c e 1 6 0 T A B L E 8 T e s ts f o r S t r u c t u r a l S t a b i l i t y : 1974-1976 vs. 1977-1980 E quation Canada France Germany S w itz e r la n d F ( 13,39) F( 13,39) F ( 13,39) ’ FC13,39) Money U.S. = 1 .08 = .78 = .81 = .67 Domestic .61 2.71 1 .22 2.16 Money ( .008) ( .29) ( .03) I n t e r e s t D i f f e r e n t i a l .67 .28 1 .00 .91 U.S. p r i ces 1.19 1 .07 .89 .75 Domestic P r i c e s .61 1 .37 .96 1.15 P r ic e of gold 1 .65 .91 1 .00 .16 exchange r a t e .95 1 .08 .85 .84 O v e ra ll X 2( 9 1 ) X2( 91) X2C 91 ) x 2( 91) system = 85.85 = 97.26 = 86.09 = 88.40 (.63) ( .30) ( .62) (.55) 161 TABLE 9 T e s ts fo r S t r u c t u r a l S t a b i l i t y : 1974-1977 vs. 1978-1980 e q u a tio n Canada France Germany S w itz e r la n d Money U.S. F ( 13,39) = .69 F ( 13,39) = .68 F ( 13,39) = .94 F (1 3 ,3 9) = .61 Domesti c Money .45 1.11 1 .91 ( .05) 2.43 ( .01 ) I n t e r e s t D i f f e r e n t i a l U.S. P r i c e s .92 .33 1 .02 .81 .93 .64 .72 .57 Domestic P r i c e s .58 1 .40 1 .46 1 .01 P ri ce of Gold 2.00 ( .0 4 ) 1 .50 1 .24 .25 Exchange Rate .65 .83 .62 .99 O v e ra ll system l X2( 91) = 74.51 ( .8 9 ) X2(91 ) = 88.77 ( .54) X2(91) = 96.50 ( .3 2 ) X2 C 91) = 87.13 ( .5 9 ) 162 TABLE 10 T e s ts f o r S t r u c t u r a l S t a b i l i t y : 1974-1979 vs. 1979-1980 E quation Canada France Germany S w itz e r la n d F (1 3 ,4 1) F ( 13 ,41 ) F( 13,41) F( 13 ,41) Money U.S. = .74 = .54 = .31 = .25 Domestic Money • oo 1 .39 1 .26 1.15 I n t e r e s t 16.60 4.38 1 .96 2.29 D i f f e r e n t i a l (.00001 ) ( .0001 ) ( .05) ( .02) U.S. 2.57 1 .79 1 .72 1 .67 P r i c e s ( .01 ) ( .07) ( .09) ( .10) Domesti c Pr i ces .98 3.77 ( .0005) 1.18 .64 P r ic e 9 .80 11 .42 10.0 10.92 of Gold (0) (0) (0) (0) Exchange Rate .44 1 .04 1.14 • O O 00 O v e ra ll X (91) X (91) X (91) X (91) system =255.63 =218,21 =204.06 = l8 9 a3 (0) ( 1 0 " 1T ) (10“ 9 ) (10~8 ) CANADA - TABLE 11 T r i a n g u l a r i z e d I n n o v a t i o n s i n : 163 FORECAST ERROR IN K MUS M D INT PUS PD PGOLD EX 1 1 .0 0 0 0 0 0 0 3 .90 .05 .03 0 0 0 01 MUS 6 .87 .04 .07 .01 0 0 .01 12 .81 .03 .12 .01 .01 0 .01 24 .78 .04 .13 .01 .01 0 .02 1 .02 .98 0 0 0 0 0 3 .08 .86 .01 .01 .02 .01 .01 HD 6 .08 .64 .03 .02 .05 .07 .10 12 .06 .52 .03 .04 .05 .12 .16 24 .07 .50 .03 .06 .05 .13 .16 1 .21 0 .78 0 0 0 0 3 .47 .01 • 49 0 .02 0 0 INT 6 .58 .04 .34 0 .04 0 .0 12 .63 .05 .27 0 .03 0 .01 24 .61 .06 .23 .01 .02 .01 .05 1 .13 .01 .07 .79 0 0 0 3 .17 .01 .08 .67 .01 .03 .02 PUS 6 .10 .03 .10 .63 .02 .05 .06 12 ,10 .02 .18 .50 .05 .03 .12 24 .28 .02 .22 .31 .05 .02 .09 1 .03 .05 .01 .21 .71 0 0 3 .03 .04 .06 .38 .39 0 . 11 PD 6 .02 .03 .07 .34 .19 .01 .34 12 .08 .03 .13 .23 .12 .02 .39 24 .30 .03 .17 .14 .09 .02 .25 1 .18 0 .13 .11 .02 .56 0 3 .49 0 .11 .15 .03 .18 .02 PG OLD 6 .43 0 .15 .24 .03 .12 .03 12 .36 0 .22 .24 .05 .09 .04 24 .44 .01 .24 .17 .05 .06 .03 1 0 0 .02 0 .08 .13 .76 3 .01 0 .03 .01 .05 .23 .67 EX 6 .10 0 .04 .02 .03 .23 .57 12 .26 .01 .07 .04 .03 .18 .42 24 .31 .02 .07 .04 .02 .15 .38 1 6 4 TABLE 12 FRANCE - T r i a n g u l a r i z e d I n n o v a t i o n s i n : FORECAST ERROR IN K MUS M D INT PUS PD PGOLD EX 1 1 .0 0 0 0 0 0 0 3 .95 0 .01 .02 .01 0 0 MUS 6 .91 .02 .01 .04 .01 0 .01 12 .84 .02 .02 .05 .02 0 .05 24 .82 .02 .03 .05 .02 0 .06 1 .13 .87 0 0 0 0 0 3 .12 • 71 .02 .01 .07 .06 .0 M D 6 . 1 4 .58 .10 .01 .10 .07 .0 12 .18 .50 .12 .02 .10 .06 .02 24 .23 .44 .11 .03 .11 .05 .03 1 0 .03 .96 0 0 0 0 3 .13 .03 .81 .01 0 .01 0 INT 6 .34 .07 .44 .02 .06 .01 .05 12 .43 .16 .20 .02 .12 .03 .04 24 .45 .14 .16 .05 .10 .02 .08 1 .13 .02 .03 .82 0 0 0 3 .18 .04 .01 .75 0 .01 0 PUS 6 .18 .07 .06 .67 0 .01 0 12 .29 .07 .14 .45 .02 .01 .01 24 .49 .09 .09 .25 .07 .01 .01 1 .17 .04 .05 .29 .45 0 0 3 .26 .05 .02 .45 .20 .01 0 PD 6 .27 .06 .04 .50 .10 .01 .02 12 .33 .04 .15 .38 .07 0 .03 24 .49 ,06 .11 .23 ,08 ,01 .02 1 .15 0 0 .37 .02 .45 0 3 .42 .06 .01 .34 .03 .13 0 PGOLD 6 .42 .09 .01 .35 .04 .08 .01 12 .44 .06 .08 .31 .03 .04 .04 24 .54 .05 .08 .21 .05 .03 .03 1 .09 .01 .02 0 .01 .04 .83 3 .07 .16 .01 0 .08 .01 .66 EX 6 .10 .31 .01 .04 .08 .03 .43 12 .19 .33 .01 .12 .10 .03 .22 24 .32 .23 .03 .14 .09 .02 .16 GERMANY - TABLE 13 T r i a n g u l a r i z e d I n n o v a ti ons i n : 165 FORECAST ERROR IN K MUS M D INT PUS PD PGOLD EX 1 1 .0 0 0 0 0 0 0 3 .91 .01 .03 .02 .01 .02 0 MUS 6 .84 .01 .05 .02 .05 .01 .01 12 .77 .02 .06 .02 .12 .01 .01 24 .72 .02 .06 .06 .12 .01 .01 1 .07 .93 0 0 0 0 0 3 .12 .77 .05 .01 .03 .01 .01 M D 6 .18 .71 .04 .01 .02 .02 .02 12 .26 .61 .05 .01 .02 .02 .01 24 .26 .61 .06 .01 .02 .02 .01 1 .04 0 .46 0 0 0 0 3 .10 .10 .76 0 .01 0 .01 INT 6 .29 .13 .52 .01 .01 .02 .02 12 .38 .12 .42 .01 .03 .02 .02 24 .38 .12 .41 .02 .03 .02 .02 1 .16 0 0 .83 0 0 0 3 .20 0 0 .78 0 .01 .0 PUS 6 .13 .02 .01 .73 .06 .03 .01 12 .09 .12 .04 .46 .23 .03 .02 24 .26 .13 •04 .30 .21 .03 .02 1 0 .01 0 .01 .97 0 0 3 .07 .02 0 .04 .86 0 .0 PD 6 .19 .04 .01 .11 .65 0 .0 12 .30 .02 .04 .22 .38 .02 .01 24 .26 .08 .07 .21 .32 .03 .03 1 .15 0 .01 .30 0 .54 0 3 .47 .04 .04 .25 0 .20 0 PGOLD 6 .42 .03 .06 .28 .08 .13 .0 12 .30 .06 .05 .23 .26 .10 .0 24 .35 .07 .05 ,21 .23 .08 .01 I 1 .07 0 .03 .01 .05 .22 .62 3 .05 .02 .22 .02 .03 .17 .49 EX 6 .09 .01 .27 .01 .15 .12 .34 12 .10 .01 .22 .03 .27 .10 .26 24 .14 .02 .20 .10 .23 .08 .22 SWITZERLAND - TABLE 14 T r i a n g u l a r i z e d I n n o v a t i ons i n : 166 FORECAST ERROR IN K MUS M D INT PUS PD PGOLD EX 1 1 .0 0 0 0 0 0 0 3 .90 0 .03 .03 0 0 .02 MUS 6 .15 .01 .03 .06 .01 0 .04 12 .81 .01 .03 .07 .04 0 .03 24 .78 .02 .03 .07 .06 .01 .03 1 0 .997 0 0 0 0 0 3 .02 .81 0 .02 .05 .06 .03 M D 6 .08 .38 0 .11 .05 .09 .29 12 .20 .23 0 .15 .03 .05 .34 24 .21 .20 .03 .13 .04 .05 .35 1 .13 .01 .86 0 0 0 0 3 .21 .05 .66 0 .05 .03 0 INT 6 .34 .05 .45 0 .05 .04 .06 12 .42 .04 .31 0 .06 .04 .13 24 .45 .03 .28 0 .07 .03 .13 1 .19 0 .01 .80 0 0 0 3 .28 .01 0 .68 .01 .01 0 PUS 6 .28 .06 .02 .61 .01 .01 .02 12 .31 .08 .04 .34 .01 .04 .17 24 .32 .05 .03 .20 .03 .03 .24 1 0 .03 .06 .04 .86 0 0 3 .22 .03 .06 .10 .56 .02 0 PD 6 .24 .02 .10 .13 .45 .04 .01 12 .20 .04 .15 .15 .38 .03 .05 24 .28 .03 .13 .09 .23 .02 .22 1 .16 0 0 .29 0 .54 0 3 .45 .01 .05 .28 .01 .20 0 PGOLD 6 .49 .01 .04 .32 0 .13 0 12 .50 .04 .03 .26 .03 .10 .04 24 .51 .03 ,02 .19 .04 .08 .11 1 .03 .02 0 .05 .11 .04 .75 3 .05 .09 .01 .05 .10 .01 .69 EX 6 .14 .10 .03 .04 .09 .01 .59 12 .26 .09 .03 .05 .07 .02 .48 24 .27 .09 .03 .06 .08 .02 .45 167 TABLE 15 K-Step Ahead S ta n d a rd E rro r of F o r e c a s t FORECAST ERROR IN K CANADA FRANCE GERMANY SWITZERLAND 1 1 .841 1 .942 1 .755 1 .876 3 3.137 3 .122 2.952 3.118 MUS 6 3.823 3.538 3.662 3.906 12 4.405 4.478 3.977 4.379 24 4.968 4.8'15 4.279 4.559 1 .244 4.924 2.433 .606 3 .359 7.066 2.930 .811 M D 6 .421 8.116 3.314 1 .220 12 .472 8.866 3.625 1 .713 24 .483 9.892 3.643 1 .884 1 .439 .460 .791 .709 3 .840 .728 .908 .875 INT 6 1 .248 1 .070 1.115 1 .064 12 1 .647 1 .613 1 .279 1 .287 24 1 .851 1 .914 1 .309 1 .374 1 .377 .372 .372 .370 3 .783 .785 .711 .753 PUS 6 1 .204 1 .107 .963 1 .022 12 1 .686 1 .555 1 .334 1 .424 24 2.215 2.335 1 .773 1 .929 1 .273 .257 .159 .302 3 .431 .498 .244 .549 PD 6 .631 .705 .338 .733 12 .872 .959 .466 .864 24 1 .121 1 .289 .551 1 .133 1 11 .528 11 .408 11 .530 12.055 3 21 .040 21.118 19.858 22.163 PGOLD 6 28.637 28.510 24.268 28.239 12 37.265 36 .920 29.454 33 .872 24 46.411 45.585 36.407 40.110 1 .009 .061 .031 .044 3 .016 .115 .057 .096 EX 6 .021 .167 .074 .429 12 .025 .236 .084 .145 24 .028 .297 .095 .150 1 6 8 TABLE 16 R e g ressio n E s tim a te s f o r The P o r t f o l i o Model - Canada (1) e t = 1.006 r*+ .009 r * - l - .055 r t + .100 r t - 1 + .004 M*+ .0001 M*_1+ .120 p | t - .257 CAt R2 = .91 , D-W = 1 .76 (2) r t = .177 r* - .187 r* 9 + 1.841 r. , - .775 rr (1 .9 6 ) (1 .9 5 ) (4 .3 7 ) (2 .0 2 ) R2 = .85 , D-W =1.77 (3) # r t = .737 r* + .019 r* 9- .196 r .+ .317 r. 9 (2 .3 5 ) t " i (.0 7 2 ) t ~z (1 .0 9 ) (1 .7 6 ) + .012 mJ _ i - .001 Mt_2 (3 .8 7 ) ( .1 4 ) R2 = .94 , D-W =1.96 (4) mJ = .172 r !L i+ .501 r* 9+ .667 M? i + .190 mJ . ? ( .2 7 ) ( .3 7 ) t ~2 (1 .0 1 ) ( .3 2 ) R2 = .84 , D-W =1 .98 (5) Pe t = *926 Ppf- t + .077 Pcrt - 2 + *027 M t - 1 " -026 Mt - 2 g (2 .4 3 ) S ( .2 1 ) g (2 .6 0 ) (2 .4 3 ) R2 = .89 , D-W =2.07 jThe f i g u r e s in p a r a n t h e s e s a r e t s t a t i s t i c s . None a re a v a i l a b l e f o r th e p a ra m e te rs of th e exchange r a t e e q u a tio n 'since th e s e were not d i r e c t l y e s tim a t e d , and t h e i r v a r i a n c e s 'are not known. 1 6 9 TABLE 17 R e g ressio n E s t im a te s f o r The P o r t f o l i o Model - France (1) e t = .508 r*+ .203 r * _ 1+ .494 r t - .296 r t _i + .021 M*+ .035 *202 p | t- .892 CAt R2 = .78 , D-W =1.33 ( 2 ) r t = .0 3 0 r* .016 r!! 9 + .665 r^ 1+ .247 r+- 9 (1 .2 9 ) ( .6 9 ) (4 .6 6 ) ( 1 . 7 6 ) R = .90 , D-W =1.42 (3) * r t •392 r * „x+ .331 r j „ 2+ -771 r t ^1- .555 r t _2 ( 1 . 8 2 ) ( 1 . 7 6 ) C (5 .9 7 ) (3 .8 5 ) + .025 M^ ,+ .0 3 2 M? p ( 2 . 10) ( 2 . 3 7 )" R2 = .93 , D-W =1.45 (4) M* = .727 r* .043 r* 0- 1.144 M* 1+ 1 .793 M? 9 ■ (1 .4 8 ) ( .1 4 ) (1 .7 9 ) (3 .1 8 ) R2 = .81 , D-W =1.76 (5) Pat- = 1 .026 Pcrt-1 - -179 Pgtf-2+ -282 m J„i- .205 M *-2 ( 8 . 3 0 ) 5 (1 .5 2 ) 5 ( 7 . 9 8 ) (5 .5 3 ) R2 = .8 3 , D-W =1.87 170 TABLE 18 R e g ressio n E s tim a te s f o r The P o r t f o l i o Model - Germany (1) e t = .033 r * - .041 r* t - l + *01 rt " *012 rt - l + .003 M*+ .001 M*^1- .124 P2 - 2.795 CAt R = .46 , D-W =1.66 (2) r t = .153 r j i - .029 r t _ 2+ .342 r t _1+ .197 r t _2 (1 .7 8 ) 1 ( .2 6 ) (1 .3 5 ) (1 .4 9 ) R = .68 , D-W =1.35 (3) r* = 1 .042 r* - .222 r* 9+ .064 r,. - .065 r ^ o C (3 .8 1 ) c 1 ( .8 7 ) (.4 2 ) ( .4 3 ) + .012 M* .005 M* o ( 1 .74) ( .6 8 ) R2 = .93 , D-W =1.80 (4) M* = .630 r j i - .093 rf. 9+ .852 M? i - .016 m!L2 ( .6 6 ) ( .1 2 ) (1 .6 0 ) ( .0 3 ) R = .84 D-W =1.93 (5) g t = 1 .052 P. (3 .1 1 ) (2 .6 9 ) (1 .89) R2 = .85 , D-W = 1 .7 9 1 7 1 T A B L E 1 9 jRegression E s t im a te s f o r The P o r t f o l i o Model - S w itz e r la n d (1) e t = .618 r*+ .110 r * ^ 1+ .144 r t - .087 r fc_1 - .0005 M*- .002 .141 P2t - 2.915 CAt R2 = .62 , D-W =1.19 (2) r t = .703 r* + .217 r* 9 + .315 r t + .153 r t 9 (1. 3 9) t_ (. 52 ) (1 . 0 2 ) ( 1.1 1) W R2 = .58 , D-W =1.92 (3) r t = 1.297 r l i - .379 r* 2 - .364 r t „1+ .227 r t _2 (5 .9 6 ) (1 .8 7 ) (1 .6 5 ) (1 .1 3 ) + .005 Mt_l+ .005 Mt o (.6 5 ) (.7 0 ) R2 = .93 , D-W =1.71 (4) M* = .962 r* - .100 r* _ + .489 M* -,+ .316 M J 9 C ( .8 7 ) ( .0 8 ) 2 (1 .0 3 ) (.7 6 ) R2 = .84 , D-W =1.94 (5) P0.4 ~ = 1 .035 Pe t - 1 - *163 Pert-2+ • 034 M? i - .024 m! 2 g t (3 .8 4 ) g ( .6 5 ) g (2 .1 5 ) (1 .5 0 ) R 2 = . 8 6 , D - W = 1 . . 8 7 172 Appendix C CHARTS FROM SIMULATING SYSTEM RESPONSE TO RANDOM SHOCKS This appendix c o n t a i n s th e re s p o n se c h a r t s o b t a i n e d from a s i m u la t io n of th e moving a v erag e r e p r e s e n t a t i o n of th e s t r u c t u r e - f r e e model fo r exchange r a t e s . Each of th e c h a r t s r e p r e s e n t s th e r e s p o n s e s of a v a r i a b l e or s e t s of v a r i a b l e s t o d i f f e r e n t t y p i c a l shocks t o th e system . 17 3 CHART 6 CANADA RESPONSE OF THE EXCHANGE RATE O .O IO 0 - 0 0 9 0 .0 0 8 0 -0 0 7 0 .0 0 6 0.005- 0 .0 0 4 0 .0 0 3 0.002 0.001 o . 000 0.001 0.002 0 . 0 0 3 0 .0 0 4 0 . 0 0 5 - MONTHS THE FOLLOUINC SYMBOLS REPRESENT THE RESPONSE OF EXCHANGE RATE TO SHOCKS TO* + U S. MONEY DOMESTIC MONEY D INTEREST DIFFERENTIAL A U.S. PRICES 0 DOMESTIC PRICES o PRICE OF COLD X EXCHANGE RATE 174 CHART 7 FRANCE RESPONSE OF THE EXCHANGE RATE O 06- 0 05- 0 001- H 0.0 0 X C H - 0 . 0 1 - A N G E - 0 - 0 2 - R A E - 0 . 0 4 - - 0 . OS- 36 30 32 34 28 26 24 16 18 20 22 2 4 6 8 10 I 2 1 4 MONTHS IH F f o l l o w i n g s y m b o l s r e p r e s e n t the r e s p o n s e OF EXCHANCE RATE TO SHOCKS TO" + 11.S MONEY • DOMESTIC MONEY D INTEREST DIFFERENTIAL A U.S. PRICES 0 DOMESTIC PRICES o PRICE OF GOLD X EXCHANGE RATE ! CHART 8 175 GERMANY RESPONSE OF THE EXCHANGE RATE 0 0 3 0 0 . 0 2 5 - 0 . 0 2 0 - 0.015- 0 0 1 0 0 . 0 0 5 0.000 - 0 . 0 0 5 -0.010 - 0 0 1 5 THE FOLLOWING SYMBOLS REPRESENT THE RESPONSE OF EXCHANGE RATE TO SHOCKS TO* + U S MONEY DOMESTIC MONEY U INTEREST DIFFERENTIAL A U.S. PRICES 0 DOMESTIC PRICES o PRICE OF GOLD X EXCHANGE RATE MONTHS < > x m c z > i o x n ita - no </>mv»aEOTJ v>mx 176 CHART 9 SWITZERLAND RESPONSE OF THE EXCHANGE RATE 0 .0 5 0 .0 4 0 - 0 3 - 0 .0 2 0-00 - 0 . 0 2 - - 0 . 0 3 - - 0 0 4 - 2 e 16 18 4 6 12 20 22 26 28 30 32 34 10 1 4 24 36 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSE OF EXCHANGE RATE TO SHOCKS TO* + U.S. MONEY • DOMESTIC MONEY n INTEREST DIFFERENTIAL A U.S PRICES 0 DOMESTIC PRICES o PRICE OF COLD X EXCHANGE RATE 177 CHART 10 FRANCE RESPONSE OF PRICE OF GOLD i O H 9 - R e s p o N S E S 0 F P R ! C E 0 F G 0 L 0 - 5 - 3 6 34 3 0 32 2 4 2 6 2 6 22 20 THE FOLLOWING STH 80LS REPRESENT THE RESPONSE OF PRICE OF GOLD TO SHOCKS TO- + U.S. MONEY DOMESTIC MONEY □ INTEREST DIFFERENTIAL & U.S. PRICES O DOMESTIC PRICES o PRICE OF COLD X EXCHANGE RATE o r u ; c o a. o » c o u * o u . CHART 11 173 GERMANY RESPONSE OF PRICE OF GOLD 9 - 8- 5 - 4 - 3~ 2- P H i C E 0 F C 0 L 0 - 3 - - 5 - ~6~ 36 34 32 3 0 28 24 26 22 18 20 16 12 10 1 4 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSE OF PR ICE OF GOLD TO SHOCKS T0< + U.S. MONEY . OOMESTIC MONEY D INTEREST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES o PRICE OF COLD X EXCHANGE RATE © r© o “n o m o — z*o m o cnmtnar o ' s w n i 179 CHART 12 SWITZERLAND RESPONSE OF PRICE OF GOLD 10- 8- 6- 2- - 4 - -6 2 4 6 B 10 12 20 I 4 1 6 18 22 24 26 32 3 6 2 8 30 34 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSE OF PRICE OF COLO TO SHOCKS TO* + U.S. MONEY • DOMESTIC MONEY n INTEREST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES o PRICE OF COLD X EXCHANGE RATE w > ??oo2 t « < n i* 0 3 * • c ©~» w m w x D x w m x 1 8 0 CHART 13 CANADA RESPONSES TO U.S. MONEY SHOCKS 2 * 6 a 10 12 14 16 18 2 0 2 2 2 4 2 6 2 8 3 0 3 2 34 3 6 MONTHS THE FOLLOWING STMBOLS REPRESENT THE RESPONSES TO U .S . MONET SHOCKS OF i + U.S. MONEY • DOMESTIC MONEY I a INTEREST DtfTERENTIAL | 6 U.S. PRICES i O DOMESTIC PRICES X EXCHANCE RATE o PRICE OF OOtO 'W^noiw— <mz'03— n ^^Ff2"oD“ 0'^— 1 8 1 C H A R T 1 4 CANADA RESPONSES TO DOMESTIC MONEY SHOCKS 0.£H o . ? l 0 - 6-1 0 -5 -4 0 . 3-1 - 0 . 2-1 - 0 . 3 - - 0 .5 - 1 2 2 20 24 2 8 3 0 2 6 3 2 3 4 36 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSES TO DOMESTIC MONEY SHOCKS OF + U.S. MONEY DOMESTIC MONEY n INTEREST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES X EXCHANCE RATE o PRICE OF GOLD 182 CHART 15 CANADA RESPONSES TO INTEREST DIFFERENTIALS 0 .9 -1 0.8- -0.2- ____________ ......... , ..... ,........I T l . . . I 1 1 [....M. I . , . 2 4 6 0 10 12 14 16 18 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 MONTHS THE FOLLOWING STMBOLS REPRESENT THE RESPONSES TO INTEREST D IFFEREN TIA L SHOCKS OF + U.S. MONEY ♦ DOMESTIC MONEY D INTEREST DIFFERENTIAL. A U.S. PRICES O DOMESTIC PRICES X EXCHANCE RATE o PRICE OF QOLD 183 C H A R T 16 CANADA RESPONSES TO U.S. PRICE SHOCKS 2 X 6 8 10 12 I 4 16 18 2 0 2 2 2 4 2 6 2 6 3 0 3 2 3 4 36 MONTHS THE FOLLOW 1NC SYMBOLS REPRESENT THE RESPONSES TO U .S . PRIC E SHOCKS OF + U.S. MONEY DOMESTIC MONEY Q INTEREST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD CHART 17 184 CANADA RESPONSES TO DOMESTIC PRICE SHOCKS 0 . 9 - 0 • B- R 0 . 7 - E S P o.e- 0 N S 0 . S - , E T 0 . 4 - 0 0 0 3 - 0 M E 0 . 2 - S T I 0 . 1 - C P 0 . 0 - R C -o.l- E s - 0 . 2 - H 0 C - 0 . 3 - K S - 0 . 4 - 1 o MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSES TO OOMEST1C PR IC E SMOCKS OF + U S MONEY • OOMESTIC MONEY Q INTEREST DIFFERENTIAL A U S. PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE OF COLD 185 CHART 18 CANADA RESPONSES TO PRICE OF GOLD SHOCKS R E S P 0 N S E T 0 P R t C E 0 F C 0 L D S H 0 C K S MONTHS THE FOL LOW 1NC SYMBOLS REPRESENT THE RESPONSES TO PR IC E OF COLO SHOCKS O F + U.S. MONEY DOMESTIC MONEY a INTERCST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES X ex c h a n g e rate O PRICE OF COED w x o o i w m -i>x m o a o x o x m o - * 186 CHART 19 CANADA RESPONSES TO EXCHANGE RATE SHOCKS 0.6- 1 0 1 2 1 4 1 6 1 8 2 0 22 24 2 6 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSES TO EXCHANGE RATE SHOCKS OF + U.S. MONEY DOMESTIC MONEY U INTEREST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES X EXCHANGE RATE o PRICE OF COLD w x o o i w -c.*nacc3 • • c o-« corner, a ro n c n m a j 187 CHART 20 FRANCE RESPONSES TO U.S. MONEY SHOCKS 1 -25-1 1 . 0 0 - 0 . 5 0 - 0.00 0 - 2 5 - 0 . 50 7 5 - .00- 24 30 36 2 8 2 2 26 28 3 2 34 6 1 0 18 20 4 16 I 4 MONTHS THE FOLLOWING STMBOLS REPRESENT THE RESPONSES TO U .S . MONET SHOCKS OF + U.S. MONEY DOMESTIC MONEY a INTEREST DIFFERENTIA. A U.S PRICES O DOMESTIC PRICES X EXCHANCE RATE o PRICE OF GOLD I w x o c i w -«!»t203 n — 'w m jo o o — * m w i o n w n j 188 CHART 21 FRANCE RESPONSES TO DOMESTIC MONEY SHOCKS 0 . 8 - 0 6 - 0.0 - 0.6 - 0 - 8 - 2 4 26 30 32 34 2 4 6 8 1 0 12 14 16 18 22 28 20 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSES TO DOMESTIC MONEY SHOCKS OF + ■ U.S. MONEY DOMESTIC MONEY n INTEREST DIFFERENTIAL 0 U.S. PRICES O DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD CHART 22 189 FRANCE RESPONSES TO INTEREST DIFFERENTIALS Z MONTHS - 0 . 5 THE F0LL0U1NC SYMBOLS REPRESENT THE RESPONSES TO INTEREST D IFFER EN TIA L SHOCKS OF + ■ U S MONEY DOMESTIC MONEY a INTEREST DIFFERENTIAL A U S PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE OF COLD 190 CHART 23 FRANCE RESPONSES TO U.S. PRICE SHOCKS R E :S 'P 0 ;N S ■ E 0.8 0 . 6 - T 0 0 . 4 - U [S 0 . 2 - P R £ ,S s - 0.6 2 6 6 1 0 12 1 4 18 16 20 22 26 2 8 3 0 34 2 4 32 3 6 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSES TO U .S . PR IC E SHOCKS OF + U.S. MONEY • DOMESTIC MONEY a INTEREST DIFFERENTIAL A U.S. PRICES 0 DOMESTIC PRICES X EXCHANCE RATE o PRICE OF GOLD 1 9 1 CHART 24 FRANCE RESPONSES TO DOMESTIC PRICE SHOCKS o. r-\ 0 6 - R I E S ;P 0 « S E 0 5 0 4 - r o 03- 0 0 M E S | r i c 0 . 2 - P R £ S M 0 - 0 . 2 - r . k s - 0 . 3 - 2 6 4 8 10 12 14 18 20 16 22 24 26 28 30 32 36 MONTHS 1 HE FOLLOWING STM 80LS REPRESENT THE RESPONSES TO DOMESTIC P R IC E SHOCKS OF f U S . MONEY DOMESTIC MONEY U INTEREST DIFFERENTIAL | A U.S. PRICES O DOMESTIC PRICES X EXCHANCE RATE o PRICE OF COLD t s ^ o c i w o r o t i -no no — a t o — m u i t o t m n i s CHART 25 1 9 2 FRANCE RESPONSES TO PRICE OF GOLD SHOCKS 0.7 0.6 0-5 0 4 0.3 0.2 0 1 - 0.0 -0- I - 0.2 - 0 3 -0-4- MONTHS l £ E „ F ? L !:0W,NG1 SYMBOLS R EPR ESEN T THE R E SP O N S E S TO P R IC E OE COLD SHOCKS OF + U .S. MONEY • DOMESTIC MONEY Q INTEREST DIFFERENTIAL A U.S. PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD 19 3 1 . 0 - 0 . 9 - 0 0 - R S o . T ~ f 0 N 0 . 6 - S E S 0 . 5 - T 0 0 . 4 - E X c 0 3 - H A 0 . 2 - N G £ 0 . 1 - R A 0 . 0 - I t - 0 . 1 - H 0 - 0 . 2 - C S - 0 . 3 - - 0 . 4 - - 0 S - CHART 26 FRANCE RESPONSES T O EX CH A N G E RATE SH O CK S 18 20 MONTHS I HE FOLLOWING STMBOLS REPRESENT THE RESPONSES ro e x c h a n g e r a t e s h o c k s o f + U.S. MONEY DOMESTIC MONEY U INTEREST DIFFERENTIAL A U S PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE OF COLD 194 CHART 27 GERMANY RESPONSES TO U.S. MONEY SHOCKS 2 4 6 8 10 12 14 16 18 20 22 24 26 26 30 32 34 36 MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSES TO U .S . MONET SHOCKS OF + U S MONEY » DOMESTIC MONEY n INTEREST DIFFERENTIAL A U.S. PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE Of GOLD 195 CHART 28 GERMANY RESPONSES TO DIMESTIC MONEY SHOCKS R E S P 0 N S E T 0 0 □ h E S T I C M 0 N E y s H 0 c K S - 0 . 3 MONTHS I HE FOLLOWING SYMBOLS REPRESENT THE R E SPO N SE S TO DOM ESTIC MONEY SHOCKS OF U.S. MONEY • DOMESTIC MONEY Q INTEREST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES X EXCHANCE RATE o PRICE OF COLD 196 CHART 29 GERMANY RESPONSES TO INTEREST DIFFERENTIALS I ■ CM 0 . 9 - 0 . 8 - 2 * 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 MONTHS I HE FOLLOW ING SYMBOLS REPRESENT THE R E SP O N S E S TO IN T ER EST D IF F E R E N T IA L SHOCKS OF + U .S MONEY « DOMESTIC MONEY U INTEREST DIFFERENTIAL A U .S PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD 19.7 CHART 30 GERMANY RESPONSES TO U.S. PRICE SHOCKS i.iH MONTHS THE FOLLOWING SYMBOLS R EPRESEN T THE R E SP O N S E S TO U - S - P R IC E SHOCKS OF + U.S MONEY * DOMESTIC MONEY n INTEREST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD 193 CHART 31 GERMANY RESPONSES TO DOMESTIC PRICE SHOCKS R E S P 0 N S £ T 0 D 0 h t S r I c p r l c E S I H 0 C K S MONTHS - 0.6 THE FOLLOWING STMBOLS REPRESEN T THE R E SP O N S E S TO D O M E S !1 C P R IC E SHOCKS OF + U.S MONEY . DOMESTIC MONEY n INTEREST DIFFERENTIAL. A U .S PRICES O DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD K * n o ; « — c r r - o o — n o — n n - n — o ’--' n m o n t B n s 199 CHART 32 GERMANY RESPONSES TO PRICE OF GOLD SHOCKS 0 6 - 0 . 5 - 0 . 3 - 0.0 - 0 . 2 - - 0 . 5 - 22 20 2 4 2 6 3 0 3 4 3 2 3 6 MONTHS [ n ED» ? M ° ) ( c NL SI M 2 0LS "EPSESENT THE RESPONSES TO PRICE OF COLO SHOCKS OF • DOMESTIC MONEY n in t e r e s t d if f e r e n t ia l A U S PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD 2 0 0 CHART 33 - 0 . 5 GERMANY RESPONSES TO EXCHANGE RATE SHOCKS 0 9 - 0 . 8 - 0 . 6- MONTHS THE FOLLOW ING SYMBOLS REPRESEN T THE R E SPO N SE S TO EXCHANGE RATE SHOCKS OF + U .S. MONEY ♦ DOMESTIC MONEY a INTEREST DIFFERENTIAL 6 U S PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD 2 0 1 CHART 34 SWITZERLAND RESPONSES- TO U.S. MONEY SHOCKS 1 - CM ...............I " ’ ........................... 1......... 1............ .TTTvr p . . ..... x r r , ■ 2 4 6 8 10 12 14 16 18 20 22 24 26 2 8 30 32 34 36 MONTHS IH E F0L LD U 1N C SYMBOLS REPRESEN T THE R E SP O N S E S TO U . S . MONEY SMOCKS OF ♦ U S MONEY * OOMESTIC MONEY O INTEREST DIFFERENTIAL A U.S. PRICES O DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD i n ?(-n-on/)—* « m-x-o-3-— o—— « - t o -*> 3 Oc— O T>t^n x- 2 0 2 CHART 35 SWITZERLAND RESPONSES TO DOMESTIC MONEY SHOCKS 0 . 9 - 0 . 8- 0 . 6- 0 5 - 0 . 2 - 0.0 -02 - - 0 . 5 - 2 4 6 8 12 10 1 4 16 18 20 22 24 26 28 36 30 32 34 MONTHS THE FOLLOWING SYMBOLS REPR ESEN T THE R E SP O N S E S TO DOM ESTIC MONET SHOCKS OF + • U.S MONEY « DOMESTIC MONEY H INTEREST DIFFERENTIAL A U.S. PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD CHART 36 203 SWITZERLAND RESPONSES TO INTEREST DIFFERENTIALS I . 0 - 0 . 9 - 0 8 - 0 7 - H S 0 . 6 - p n N 0 . 5 - S E 0 - 4 - T 0 0 . 3 - N r 0 - 2 - E R E 0 . 1 - S T 0 . 0 - S H 0 - 0 . 1 - C K S i O fw ) , , 1 , - 0 - 3 - “ 0 . 4 - - 0 - 5 - MONTHS THE FOLLOWING SYMBOLS REPRESENT THE RESPONSES 10 INTEREST D IF F E R E N T IA L SHOCKS OF + U S- MONEY • DOMESTIC MONEY INTEREST DIFFERENTIAL U.S. PRICES DOMESTIC PRICES EXCHANGE RATE PRICE OF GOLD 20 4 CHART 37 SWITZERLAND RESPONSES TO U.S. PRICE SHOCKS f* R I C £ I s M 0 P K s 0 . 9 - THE FOLLOW INC SYMBOLS REPRESENT THE RESPONSES TO U .S . PRICE SHOCKS OF + U.S. MONEY DOMESTIC MONEY n INTEREST DIFFERENTIAL A U S PRICES © DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD MONTHS 205 CHART 38 SWITZERLAND RESPONSES TO DOMESTIC PRICE SHOCKS 1 .0- 0 . 9 - 0.8- H E S P 0 7- 0 N 0.6- S E 0 5 - r 0 0 . 4 - u 0 M 0 . 3 - E S T 0.2- C 0.1- P R 1 0 0 - C E -0. 1 S H 0 -0.2- C K S - 0 . 3 - - 0 . 4 - -o.s- MONTHS THE FOLLOW ING SYMBOLS R EPR ESEN T THE R ESPO N SES TO DO M ESTIC P R IC E SHOCKS OF + U S . MONEV DOMESTIC MONEV U INTEREST DIFFERENTIAL A U.S PRICES 0 DOMESTIC PRICES X EXCHANCE RATE o PRICE OF COLD 206 CHART 39 SWITZERLAND RESPONSES TO PRICE OF GOLD SHOCKS 0 . 8 0 . 7 R E S P 0 N S E T 0 P R I C E 0 F 0 0 L 0 1 s 0 I C I * S MONTHS - 0 3 - 0 - 4 THE FOLLOWING SYMBOLS R EPR ESEN T THE R E SP O N S E S TO P R IC E OF GOLD SHOCKS OF + U.S MONEY • DOMESTIC MONEY n INTEREST DIFFERENTIAL A U S PRICES 0 DOMESTIC PRICES X EXCHANGE RATE o PRICE OF GOLD 207 CHART 40 SWITZERLAND RESPONSES TO EXCHANGE RATE SHOCKS 1 • 25-T I .00- R E $ P 0 - 7 5 - 0 f t S : E S 0 . 5 0 - ! 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Creator Azar, Randa (author)

Core Title Examining the portfolio balance theory of foreign exchange rates: A time series analysis

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Economics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Economics, Finance,Economics, theory,OAI-PMH Harvest

Language English

Advisor Nickelsburg, Gerald (committee chair), Laffer, Arthur B. (committee member), Nugent, Jeffrey B. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c20-265741

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