University of Southern California Dissertations and Theses

Econometric analysis of nonlinear dynamics in a behavioral institutional model of stock price fluctuation

(USC Thesis Other)

Econometric analysis of nonlinear dynamics in a behavioral institutional model of stock price fluctuation

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content E C O N O M E T R IC A N A L A Y S IS O F N O N L IN E A R D Y N A M IC S IN A B E H A V IO R A L -IN S T IT U T IO N A L M O D E L O F S T O C K P R IC E F L U C T U A T IO N by Rajesh Srinivasan A D issertation Presented to the F A C U L T Y O F T H E G R A D U A T E SC H O O L U N IV E R S IT Y O F S O U T H E R N C A L IF O R N IA In P a rtia l F u lfillm e n t of the i Requirem ents for the Degree j D O C T O R O F P H IL O S O P H Y ! (Economics) I I I A u g u s t 1993 i C opyright 199-3 Rajesh Srinivasan UMI Number: DP23402 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. Disssrtation PubJishing UMI DP23402 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 OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90007 This dissertation, written by S r h n W A S A n J under the direction of h Dissertation Committee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillm ent of re quirements fo r the degree of DOCTOR OF PHILOSOPHY Dean of Graduate Studies Date .. ... A ugust 6A.1 9 ? 3 , D IS S E R T A T IO N C O M M IT T E E Chairperson D e d ic a tio n This dissertation is dedicated to my parents. A ck n o w led g em en ts I I would like to thank my com m ittee members, Prof. Richard H. Day, whose pioneer- i _ . . . . | ing work in the area of Nonlinear Dynamics was the inspiration for this research, i Prof. Sankarasubramaniam whose co n trib u tio n to my knowledge of Financial M ar- ^ kets is close to in fin ity and Prof. Chou for introducing me to the exciting w orld of encompassing. I would also like to thank Prof. Andrew Weiss for helpful comments f on the S E TA R methodology, my friends and colleagues for their support, through out my graduate school years and fin a lly my dear friend Jody, for her constant ; ; encouragement which enabled me to overcome m y dissertation blues. C O N T E N T S D e d ic a tio n ii A cknow ledgem ents iii A b s tra c t vii 1 In tro d u c tio n 1 1.1 Stochastic processes and the stock m a r k e t ................................................ 4 1.2 D eterm inistic chaos and the stock m a rk e t................................................. 5 2 N a tu re of S tock P ric e D a ta 8 2.1 Conventional tests of determ inistic structure in d a ta ............................... 8 2.2 A lte rna tive tests for nonlinearity ......................................................... 10 2.2.1 Brock, Dechert and Scheinkman t e s t ........................................... 10 2.2.2 H inich Bispectral te s t........................................................................... 12 2.2.3 Tsay’s F and T A R -F te s t.................................................................... 13 2.2.4 Topological t e s t .................................... 16 2.3 R e s u lts .................................................................................................................... 17 2.3.1 D a t a .......................................................................................................... 17 2.3.2 Sum m ary ............................................................................................ 24 3 In s titu tio n a l and B eh avio ral T h e o ry o f S tock P ric e F lu c tu a tio n 26 3.1 B a c k g ro u n d .......................................................................................................... 26 3.2 Basic Tw o Investor M odel . . . ................................................................ 27 3.2.1 o-Investors ............................................................................................ 27 3.2.2 /3-investors............................................................................................... 28 3.2.3 M arket M e d ia tio n .................................................................................. 29 3.3 Theoretical properties of the basic m o d e l................................................. 31 3.4 M o tiva tio n for tra n s fo rm a tio n .............................................. 32 3.5 Transformed m o d e l........................................................................................... 35 3.6 Sum m ary .................................................................................... 37 4 M e th o d o lo g y 38 IV 4.1 B a c k g ro u n d ......................................................................................................... 38 1.2 Self exciting TA R m o d e l................................................................................ 41 4.2.1 A general model . 41 4.3 SETAR m e th o d o lo g y ....................................................................................... 42 4.4 Our m e th o d o lo g y .............................................................................................. 45 4.4.1 Bootstrap m ethod to test the encompassing principle . . . . 49 4.5 Summary ............................................................................................................ 50 5 E stim atio n and In fe re n ce 51 5.1 D a ta ............................................................................... 51 •5.2 Properties of the transform ed s e rie s ........................................................... 52 5.3 R esults................................................................................................................... 56 5.3.1 E s tim a tio n .............................................................................................. 56 5.3.2 M onte-carlo study of the SETAR m eth o d o lo g y........................ 63 5.4 Extensions of the basic m o d e l...................................................................... 68 5.4.1 Framework for a m ultiple investor m o d e l.................................... 70 5.4.1.1 M u ltip le a -in v e s to rs ......................................................... 70 5.4.1.2 M u ltip le a and .^-investors............................................... 72 5.5 C o n c lu s io n s ........................................................................................................ 75 v L IST O F T A B L E S 2.1 Summ ary Statistics of Raw R e tu rn s ............................................................ 18 2.2 A utocorrelation Coefficients of Absolute R e tu rn s ................................... 20 2.3 A utocorrelation Coefficients of Squared R e tu r n s ................................... 20 2.4 BDS Test for I.I.D of the Returns S e r ie s ................................................. 21 2.5 H inich Bispectral Test for N o rm ality and L in e a rity ............................ 22 2.6 Tsay’s F and T A R -F Tests for N onlinearity in Returns......................... 22 5.1 Bispectral test, Tsay’s-F and T A R -F test for nonlinearity of the qt s e rie s...................................................................................................................... 53 5.2 BDS test for i.i.d. of the qt s e r ie s ............................................................... 53 5.3 Unconstrained TAR(7;2; 1)“ model identified by p lo ttin g A R coeffi cients vs threshold variable, obtained from a recursive autoregression of the qt series...................................................................................................... 58 5.4 Dickey fuller and Augm ented Dickey Fuller test for unit-root nonsta- tio n a rity of the residual series obtained from the estimated T A R ( 7:2:1) m odel ..................................................................................................................' 60 5.5 Unconstrained TA R (5;2;1) model after refining the p a rtitio n spanned by the qt series .................... 61 5.6 Roots of the characteristic polynom ial of the unconstrained TAR(5;2;1) model .................................................................................................................. 61 5.7 Constrained TA R (5;2;1) model for the qt series .................................... 62 5.8 M onte-carlo study of the SE TAR methodology:Benchmark Case . . 65 5.9 M onte-carlo study of the S E TAR methodology for variable window le n g t h .................................................................................................................. 65 5.10 M onte-carlo study of the S E TAR methodology for variable window length w ith 5% noise . .................................. 66 5.11 M onte-carlo study of the S E TAR methodology for variable window length w ith 25% noise ................................................................................... 66 5.12 Bootstrapped BDS test of the Constrained TAR(5;2;1) model . . . 67 5.13 A m ulti-investor model of stock returns for the S&P 500 .................. 77 L IST O F F IG U R E S 2.1 Closereturns histogram for daily S&P 500 r e tu r n s ............................... 23 2.2 Closereturns histogram for weekly S&P 500 r e tu r n s ........................... 23 2.3 Closereturns histogram for m onthly S&P 500 r e t u r n s ........................ 23 3.1 Phase diagram of prices generated from the 2-investor model . . . . 31 3.2 Price trajectory generated by the 2-investor m odel................................ 31 3.3 Standard and P oor’s Q uarterly real prices, 1935-1990............................. 33 3.4 Standard and Poor’s Q uarterly real dividends, 1935-1990..................... 33 3.5 Standard and Poor’s Q uarterly real earnings, 1935-1990........................ 33 5.1 Em pirical d is trib u tio n of m onthly qt series.................................................. 52 5.2 Closereturns histogram of m onthly qt series.............................................. 54 5.3 Plot of recursive estimates o f slope coefficient vs threshold variable. 54 5.4 Autocorrelation & p a rtia l autocorrelation function of qt series. . . . 55 5.5 A utocorrelation & p a rtia l autocorrelation function of slope coefficient 56 5.6 Plot of recursive estimates of the constant coefficient vs threshold variable for the TA R (7;2;1) m o d e l.............................................................. 57 5.7 Plot of recursive estimates of the A R (1) coefficient vs threshold vari able for the TA R (7;2;1) m o d e l..................................................................... 57 5.8 Plot of recursive estimates of the A R (2) coefficient vs threshold vari able for the TA R (7;2;1) m o d e l..................................................................... 57 5.9 Tim e series of sim ulated series, < ? ’ : The first 100 observations were dropped................................................................................................................... 62 5.10 Tim e series o f m onthly qt series....................... 63 5.11 P lot of recursive estimates of the constant coefficient vs threshold variable for the TAR(13;2;1) m o d e l........................................................... 76 5.12 Plot of recursive estimates of A R (1) coefficient vs threshold variable for the TAR(13;2;1) m o d e l............................................................................ 76 5.13 Plot of recursive estimates o f the AR (2) coefficient vs threshold vari able for the TAR (13;2;1) m o d e l.................................................................. 76 A b stra ct One notion of stock price dynamics, which forms the basis of the efficient m arket hypothesis a ttrib u te s the flu ctu a tio n in stock prices to random news. An alterna tive view is th a t, the interaction o f different investor types w ith varying levels of sophistication could cause the stock price to resemble the outcome of a stochastic process. The la tte r view provides the m otivation for this m anuscript. In the form er theory, statistica l models have been used as a descriptive tool to explain stylized facts of the stock m arket such as excess v o la tility relative to fun damental values and mean reversion. In this paper, we have derived a nonlinear time-series m odel called the Self E xcitin g Threshold Autoregression from the alter native theory. In contrast to popular models of excess v o la tility th a t are nonlinear in the variance, the proposed model is nonlinear in the mean. Some of its features such as, lim it cycles, and ju m p phenomena, cannot be captured by linear tim e se ries models. Further, it encompasses a nonhomogeneous linear autoregressive model w ith different variances in different regimes and a mean shift model. Prelim inary investigation of a series of m onthly returns on a composite index such as the Standard and Poor’s 500, suggests the presence of significant systematic nonlinearities. Simple linear and nonlinear statistical models discussed in the lite ra ture are inadequate for m odelling this structure. The suggested tim e series model is found to approxim ate this series well. However, certain a-priori restrictions are not supported by data. This suggests th a t a generalization of the basic model may be required to describe the m o nthly returns process. In conclusion, we have suggested a few extensions to the basic model. C h a p te r 1 i I (In tr o d u c tio n Econom etricians and applied economists often believe th a t random shocks and in- 1 novations co n tin u a lly affect the economy and hence it is essentially stochastic in , nature. B y contrast, some models G randm ont (1985) in the economic theory lit- < erature have suggested th a t the essential n o n lin e a rity o f real econom ic forces is sufficient to cause d e term inistic tim e series to appear random to the naked eye1. In ■ fact, conventional sta tistica l tools have failed to distinguish between the tw o tim e ; < I series. T his has resulted in the developm ent o f a whole new theory o f s ta tistica l j inference. j Results fro m algorithm s designed to distinguish between x'eal data and th a t gen- ‘ ( erated by stochastic difference equations suggest th a t at least in the stock m arket, ; i nonlinearities m ay play an im p o rta n t role in explaining asset returns. T h is im plies ! th a t a relationship as follows: I r t = . .. , r t_fc) + et , ; 4 ; < : . . . . I ' where r t is the re tu rn at tim e t and et an independent and id e n tica lly (i.i.d .) dis- j ! £ 1 trib u te d random variable, could be m odelled w ith m uch sm aller error term s, if is j : allowed to be nonlinear, than if lin e a rity is imposed. ■ For m any years a random w a lk m odel was considered by a substantial num ber . of researchers in the Finance area, to describe stock returns over periods at least a week long, G ranger (1970) and Fama (1970). T he efficient m arket theory o f Samuel- son (1965) and Fama (1970) im p lie d th a t returns are i.i.d . over tim e. Since this i ------------------------------------------------------ ; 1A good overview o f chaos models in economics is covered in B aum ol and Benhabib (1989). hypothesis has been strongly rejected by some recent studies, [Hsieh (1990), H inich ■ & Patterson (1985)], the v a lid ity o f the und e rlyin g linear ra tio n a l expectations asset p ricin g theory has come under close scrutiny. ! The e q u ilib riu m asset p ricing models th a t follow ed the w ork o f R ubin stein (1976), ■Lucas (1978) and others, provided an avenue fo r nonlinearities to enter these models by lin k in g stock returns to consum ption va ria b ility. However, in im plem enting these m odels, ease o f p a ram etrization and estim ation resulted in models where flu c tu a tio n s were nonexistent in the absence of exogenous shocks. j R ejection o f the i.i.d assum ption leads one to suspect th a t the process generating jasset returns could be o f low dim ension. In other words, it may be possible to 'approxim ate the unknow n data generating mechanism w ith re la tive ly few economic I jvariables. T his prem ise has resulted in efforts to m odel the stock price series as 'the outcom e of a determ inistic process, where flu cta tio n s are p rim a rily the result of jintrinsic dynam ics and the economy is b uffetted by occasional random shocks, [see D ay and Huang (1990) and Gu (1992)]. In em pirical research, it is easy to ju s tify in clu d in g a stochastic shock th a t accounts fo r “ real external events,” measurement jerror and the like. [ The m odelling philosophy adopted by tim e-series analysts am ounts to in fe rrin g !a process th a t could have generated the observed series by e xp lo itin g its sta tistica l I 1 {properties. The resulting models m ay not have any econom ic reasoning or easy j [interpre tatio n. In contrast, the in trin s ic th e o ry of Day, Huang and Gu, and the | •focus of m y own research is b u ilt on econom ic specification of investor behavior and i [assumptions about how the m arket operates. j Specifically, the question th a t I address is, “ Do e m p irica l data support the in- ! trin s ic th e o ry whose explanation for the observed flu c tu a tio n in the stock m arket is based on investor behavior and the in s titu tio n a l setup ? ” In contrast to conven tio n a l tim e-series approach, our m ethodology tests the im p lica tio n s o f the proposed : j theory, such as signs and m agnitudes o f coefficients, presence of higher order nonlin- • e a rity etc. using conventional and unconventional tests. T h is involves fo rm a lizing | ia stochastic fram ew ork w ith in w hich estim ation and inference can be carried o u t. | 1 F irs t a stru c tu ra l m odel is derived fro m the theory. Second, th is m odel is trans form ed in to the Self E x c itin g Threshold Autoregressive (S E T A R ) fo rm , the con sistency properties of w hich have been established. T h ird , the S E T A R m odel is e s tim a te d and its long-run properties are used to evaluate the in trin s ic theory. In the next section, we provide a b rie f review of the lite ra tu re th a t has come to dom inate the finance profession for the last 40 years. T he debate of stochastic versus determ inistic explanation o f stock price flu c tu a tio n is discussed. E m p irica l j t studies of system atic nonlinearities in finan cial data relevant to th is study is also ■ highlighted. 3 1.1 S to c h a s tic p r o c e s s e s a n d th e s to c k m a rk et T he developm ent of a theory o f price fo rm a tio n in the stock m arket was w ell pre- jceeded by observation in the m id 1950’s and early 1960’s th a t the behavior o f com m on stock and other speculative prices could be w ell approxim ated by a random w alk. In try in g to rationalize th is e m p irica l evidence, economists came up w ith the th e o ry o f efficient m arkets stated as a random w alk m odel b u t generally im p ly in g a j“fa ir game” 2 m odel. | Bachelier, whose contribu tions were largely ignored u n til the m id 1960’s had stated and proved the random w alk m odel. In his m odel, the random behavior of Iprices was a ttrib u te d to speculation by investors; in p a rticu la r, th e expected pro fits jto the speculator is shown to be zero, w hich essentially im plies a m a rtin g a le sequence fo r the price process. The role o f “ fa ir game” expected re tu rn models in the theory 'of efficient m arkets and the relatio nship of the same to random w a lk models was 'first studied rigorously by Samuelson (1965) and M ande lbrot (1966)3. ; Samuelson (1965) proved th a t when relative prices are co rre ctly adjusted for expected dividends, they are m ore or less indistin guish able fro m ‘white noise\ T his was shown under the assum ption th a t curre nt price is set at the expected discounted value o f its fu tu re dividends, where the fu tu re dividends are assumed to be random jvariables, th a t follow a know n stochastic process. In the efficient m arket m odel of iFama (1970), current price is fu rth e r assumed to “ fu lly reflect” available in fo rm a tio n . T his im plies th a t successive price changes (or one period retu rns) are independent. jF urther, he assumes these returns to be id e n tic a lly d is trib u te d 4. j 2A fa ir game m odel can be w ritte n as follow s: ! E(pt + 1 \It) = {l + E(rt + 1 \It) }p t | where the te rm on the le ft hand side is the expected price at tim e t + 1; rt is the one-period percentage re tu rn (pt+i —Pt)/Pt and It is the in fo rm a tio n set at tim e t. 3M an de lbrot and Samuelson show th a t i f the price o f futures contract a t tim e i is the expected value at t ( given in fo rm a tio n It) o f te rm in a l spot price, then the futures price w ill fo llo w a m artingale w ith respect to the in fo rm a tio n set It \ th a t is, the expected one-period price change ,w ill be zero, and the price changes w ill be a “fa ir game.” ! 4S tric tly speaking prices w ill fo llo w a ‘random w alk w ith d rift ’ o nly i f price changes are i.i.d ., i since expected price changes can be non-zero. I f one-period returns are independent and ide n tica lly |d is trib u te d , then prices w ill n ot follow a random w alk since the d is trib u tio n o f price changes w ill .depend on the price level. However tests o f efficient m arket hypotheses overlook th is d istin ctio n . : E a rly em pirical tests o f the efficient m arke t hypothesis under various in fo rm a tio n sets have led some theorists to conclude th a t for returns longer th a n a day, the m arket is efficient, [see Fama (1970) fo r an extensive review o f the various tests]. In other words investors cannot use p u b lic ly available in fo rm a tio n to earn “ p ro fits ” in excess o f w hat is considered “ n o rm a l” fo r the am ount o f risk undertaken. A lth o u g h , some studies reported evidence o f positive dependence in day-to-day price changes and returns on com m on stocks, the m agnitude of dependence was insufficient to con sistently p ro fit fro m any m echanical tra d in g rules th a t exploited this dependence in returns. T h is was in terpreted as evidence in favor o f the efficient m arket hypothesis, j The conventional n o tio n o f stock m arket dynam ics in the above fram ew ork in volves a random flow o f news (in fo rm a tio n w hich cannot be predicted) th a t causes prices to flu c tu a te ra ndo m ly and result in e q u ilib riu m zero expected returns. Stochas- j tic models have been constructed to describe th is process due to th e ir tra c ta b ility and a b ility to approxim ate the observed pa tte rn s in actual price m ovements over I tim e. However, th e need fo r b e tte r models to characterize stock m arket dynam ics is emphasized by the fa ilu re o f pure ly stochastic m odels to account fo r certain s ty l ized facts such as excess v o la tility re la tive to fundam entals [see Shiller (1990) fo r an ; extensive review of this lite ra tu re ], mean reversion, Summers & P oterba (1987) and - ra n d o m ly sw itching b u ll and bear m arket regimes. In recent tim es, investigators have trie d to provide alte rn a tive explanations fo r these stylized facts. T he next section deals w ith one such theory th a t has challenged the conventional n o tio n of stock m arket dynam ics. J1.2 D e te r m in is tic ch a o s a n d th e sto c k m a rk et In prin cip le , nonlinearities in the tra d in g process, [Day & Huang (1990); M u G u (1992)], in fo rm a tio n dissem ination or even certa in kinds o f news, Shaffer (1991) 'could lead to chaotic price tra je cto rie s5. Such behavior can easily be generated by non-linear dynam ic systems w ith feedback. A good in tro d u c tio n to th e m eaning and properties o f chaos is provided in B aum ol and B enhabib (1989). 1 _ , . . ^ ^ ^ ^ ... I | 5Chaos here is defined as behavior over tim e th a t is non-periodic and sensitive to in itia l 1 conditions. \ The p o ssib ility o f chaotic dynam ics in economics was firs t suggested by M ay and B eddington (1975). Since then chaos in economics has been shown to arise in, sim ple m acroeconom ic models, [Stutzer (1980), D ay and Shafer (1983)]; dynam ic models of choice w ith endogenous tastes, B enhabib and D ay (1980); in models of p ro d u c tiv ity grow th, Baum ol and W o lf (1983); in dynam ic models o f advertising expenditures, B aum ol and Q uandt (1985); and in the stock m arket, Shaffer (1991). In the em pirical area, Chen [(1984), (1987)], Sayers (1985) and B rock (1986) have characterized and tested the presence o f chaos in data. E m p irica l tests of non linear ity and n o n n o rm a lity in economic and fin a n cia l data were advanced among others !b y Ashley, H inich and P atterson (1986) and tests o f dependence by Brock, Dechert |and Scheinkman (1987), and Hsieh (1990). j I I E m p irica l evidence of possible chaotic behavior in economics and finan cial data | has been p u t fo rth by B rock and Sayers (1988) fo r U.S. business cycle data,Scheinkm an1 I 1 j and LeB aron (1989) for weekly stock returns, H in ich and Patterson (1985) fo r daily I stock retu rns, B rock & M a llia ris (1989) for treasury b ill retu rns, Hsieh (1989) and * j K ugler & Lenz (1990) for foreign exchange data. j M ost relevant to the present w ork, Scheinkman and LeB aron (1989), based on ; estim ates of phase-space correlation (to be defined la te r), found evidence o f stock re- j turn s trajectories, consistent w ith nonlinear d e te rm in istic models rath e r th a n purely I stochastic behavior. However, th e y do not suggest a theoretical m odel th a t could i i | help explain the observed phenomenon. The above w ork and th a t o f Sayers, Bar- j n e tt and Chen (1989) was re-exam ined b y Ramsey, Sayers and R othm an (1990). , A fte r correcting for n o n s ta tio n a rity and the sm all sample bias present in dim ension | calculations th e y found less evidence o f chaos in these series, except possibly in the w ork stoppage data. B rock argues th a t short te rm flu ctu a tio n s in stock prices cannot be a ttrib u te d I j to low dim ensional chaos as long as m arkets are frictionless and prices are observed w ith o u t error. His arguem ent is based on the hypothesis th a t investors can resort to sophisticated m odelling o f the und e rlyin g stru ctu re . E fficient forecasts generated I f fro m these models w ould result in profits. T his p o te n tia l w ould drive investors in to i the m arket causing stock prices to change u n til p ro fits are elim inated. Hence, if at a ll stock prices are generated by some low dim ensional d e te rm in istic process, it 6 'must be so subtle th a t the crude tools o f m easurem ent are in sufficient to detect th e ir presence. j The above statem ent cru cia lly depends on the u n d e rlyin g assum ptions. In gen eral stock price movements are discrete i.e., there are lower bounds on the m in im u m jobserved nonzero price m ovem ent as w ell as in d iv is ib ilitie s in the observed price level ( l/ 8 t h ’s). Jensen (1987) has shown th a t such in d iv is ib ilitie s , and m easurem ent er ror, can su b sta n tia lly a lte r any u n d e rlyin g chaotic process. S im ila rly, tran saction ’ costs make it u n p ro fita b le to trad e on the basis o f price movements th a t are too sm all. These m arket frictio n s weaken B ro c k ’s argum ent. However, B rock finds e m pirica l evidence o f nonlinear stru ctu re in w eekly and m o n th ly returns on aggregate stock price indices. F urther, he was able to ru le out a j ijanu ary effect, autoregressive con d itio n a l heteroscedasticity, and n e a r-u n it roots as j possible explanations of nonlinear stru ctu re . S im ilarly, A shley and P atterson (1989), j > ' fin d strong evidence o f nonlinear stru ctu re in d a ily com m on stock retu rns, w hich , m ig h t be consistent w ith chaos. In a subsequent article, H in ich and P atterson (1992) rule out weekends, holidays and day o f the week as possible causes of n o n lin e a rity in in tra -d a y returns. * T he debate about e m pirical fin d in g o f d e te rm in istic s tru ctu re leading to chaos in 'economic and financial data is ongoing as m ore sophisticated and pow erful tests are [constructed. In the m eantim e, evidence in favor o f sig nificant non linearitie s in stock re tu rn s is m ounting. B u t w h at is lacking is a careful e xplana tion of the observed nonlinearity. T he purpose of this study is to continue th e effort to fill th is void. ( 1 | The m anuscript is organized as follows. C hapter 2 discusses the nature o f stock j price data by applying various tests designed to id e n tify non linear stru ctu re in . tim e-series data. C hapter 3 presents the basic m odel and the m o tiv a tio n fo r the I j “ in trin s ic theory.” In C hapter 4 we develop the m ethodology fo r estim a tio n and j .statistical inference by tra n sfo rm in g the basic m odel in to th e Self E x c itin g T hreshold j Autoregression m odel. C hapter 5 presents the results fro m the analysis, suggests some generalizations o f the basic m odel and concludes. 7 |C h a p ter 2 N a tu r e o f S to c k P r ic e D a ta In this chapter we present evidence in favor o f significant nonlinear stru ctu re in stock prices, in p a rtic u la r fo r a com posite index like the S&P 500. Four different test procedures are described and applied to sample data, to analyze the stru ctu re t u n d erlyin g aggregate stock returns. W here appropriate, the deficiences o f these tests are pointed out. R eturns are calculated as the lo g a rith m ic price relatives. I t is shown th a t d a ily and m o n th ly returns e x h ib it significant nonlinear stru ctu re. M ost of the n o n lin e a rity seems to be w ell accom odated by Autoregressive C o ndi tio n a l H eteroscedasticity (A R C H ) class o f m odels, th a t are nonlinear in the variance. However, in out o f sample forecasts, these models seem to do worse th a n a constant variance assum ption. T his is suggestive o f n o n lin e a rity perhaps in the m ean as w ell as th e variance. D a ta also supports the presence of second and higher order nonlinearities and in p a rtic u la r threshold non linearity. T he ore tical m o tiv a tio n for a regim e sw itching (nonlinear in the m ean) m odel is provided in the next chapter and it is form alized in the subsequent chapter. i j i 2.1 C o n v e n tio n a l t e s t s o f d e te r m in is tic s tr u c tu r e in d a ta Tests o f nonlinear dynam ics and other nonlinear structures in fin a n cia l m arkets ! 'analyze w hether asset prices have a certa in degree o f dependency on th e ir own past levels or changes. A conventional test fo r low dim ensional chaos w ould begin t w ith the estim ation o f a certain n o tio n o f a ttra c to r dim ension such as the corre- i la tio n dim ension Grassberger and Procaccia (1983), to see if it is low. N e xt one w ould estim ate a measure o f sensitive dependence on in itia l conditions such as the largest Lyapunov exponent and see if it is positive. T h ird one w ould estim ate the K o lm ogorov entropy and see if the estim ate is consistent w ith chaos. F in a lly the jdeterm inistic m ap generating the series is checked using devices such as th e phase .diagram for the presence o f strange a ttra cto rs. | Scheinkman and LeBaron (1989) used the G rassberger-Procaccia (G P ) plots and [calculated the correlation dim ension o f weekly and d a ily stock returns o f the value jweighted CRSP index. For weekly data, the G P p lo t resulted in a fra c ta l dim ension i of 5.7 w ith an em bedding dim ension o f 13 and fo r d a ily data the estim ated slope was 5.9 w ith an em bedding dim ension o f 20. T his was in te rp re te d as m eaning th a t to m odel weekly or d a ily stock returns one w ould need at the very least, 6 ^variables. T he y then tested for n o n lin e a rity using a novel approach. F irs t returns •were regressed on past returns. T he n they sam pled (w ith replacem ent) fro m the \ residuals and re b u ilt a new data set using the same in itia l values as in the real data. T his scram bled data is in fact d e te rm in istic, since it is com puter generated. For each new data set so constructed, the corre latio n dim ension was reestim ated. T he m o tiv a tio n being th a t, if the o rig in a l series was chaotic, then the scram bled series should be m ore random , and the dim ension estim ate for the la tte r should be higher th a n for the o rig in a l series. O n the other hand, if the o rig in a l series was random , the correlation dim ension estim ate should not change as a result o f the scram bling. Based on th is procedure they were able to provide evidence supp o rting 1 ‘the hypothesis o f significant dete rm in istic structure. F u rth e r, they were able to 'show th a t th is s tru ctu re is inconsistent w ith a nonlinear stochastic a lte rn a tive such I ■ a s the A R C H m odel o f Engle (1982). Peters (1990) using the correlation dim ension approach estim ated a chaotic a t tra c to r fo r a sample o f S&P500 m o n th ly retu rns covering 1950-1989. His estim ate of the fra c ta l dim ension was 2.33 w hich w ould im p ly th a t a m in im u m o f 3 variables w ould be needed to m odel the system th a t generated m o n th ly S&P 500 returns over the relevant period. However, it is not very clear, w h a t those three variables should be. I t is interestin g to note th a t one needs fewer variables to describe m o n th ly re tu rn s process com pared to weekly returns as suggested b y Scheinkm an and LeBaron. J The correlation dim ension test, developed b y Gi'assberger and Procaccia has jbeen w idely used in the n a tu ra l sciences where there is a large am ount o f clean data available and hence, its a p p licatio n to sm all and noisy data sets in economics and finance has been h ig h ly criticised. F u rth e r, standard techniques such as spectral [ analysis or the autocorrela tion fu n ctio n , have not been very useful in distinguishing jwhether a tim e series was generated by a d e te rm in istic or a stochastic m echanism 6 Some o f the deficiencies o f the above approach are: I , 1. T he estim ate o f the correlatio n dim ension converges to the dim ension of the | a ttra c to r as the num ber o f observations goes to in fin ity . In econom ic and finan cial data, convergence m ay be very slow, so over short periods, it im plies I i th a t chaotic behavior m ay no t be detected. j 2. Noise due to exogenous factors in the tim e series could d is to rt dim ension ' calculations. In general, dim ension estim ates are higher w ith a d d itive noise. 3. E stim a tio n of the scaling region, e w hich measures the distance between any i tw o ‘m ’-tu p le , is subjective and requires careful judgem ent. In practice, e is I chosen to be about 2 to 5% of the ‘range’ o f the data. : ! 4. D im ension calculations provide little in fo rm a tio n on how to “ m odel the dy- j nam ics” o f the u n d e rlyin g system. These deficiencies im p ly the need fo r m ore robust tests. W e look at fo u r such tests. T hey are: B rock, Dechert and Scheinkman test fo r n o n lin e a rity, B ispectru m test for lin e a rity and n o rm a lity, T say’s-F test fo r general no n lin e a rity, and Topological test [for the presence o f unstable periodic orbits. j '2.2 A lte r n a tiv e t e s t s for n o n lin e a r ity | 2.2.1 B rock , D ech ert an d S ch ein k m an te st I The problem s w ith using the G P p lo t, and the sm all sample bias in the estim ate of correlation dim ension led B rock, Dechert and Scheinkm an (1987) (hereafter, “ B D S ” ) : i ( ------------------------------------------------------- t j 6(Sakai & Tokam aru, 1980) have shown th a t the autocorrelations o f a sim ple nonlinear deter- [ m in is tic difference equation (N D D E ) coincides w ith those o f the first-o rd e r autoregressive (A R 1 ) process although th e ir s ta tis tic a l properties are quite different. j jto derive a test th a t is app ro priate fo r detecting general stochastic non lin e a rity. To jderive this test, we firs t have to define the corre latio n in tegral. The tim e series con sidered (x(f),t = 1,... , T ) is used to fo rm m histories, x™ — { x t , £<+i ... x t+m^ i}. I These to dim ensional (em bedding dim ension) vectors are used to calculate the cor- f re la tio n in te g ra l7. EE C m , T ( e) ~ Tm (Tm - 1) t < s ) x s ) jwhere Tm = T — to + 1 and I t is the in d ic a to r fu n ctio n o f the event 1 | i ^ G ^ d , 1 , • . • , flT 1 i I !Cm,r(c) can be in terpreted as an estim ate o f the p ro b a b ility th a t x™ and x™ are w ith in a distance e. As e increases, the fra ctio n o f such pairs increases to a m a x i m u m o f u n ity. B y counting the num ber o f a d d itio n a l pairs captured as e increases, jone obtains a measure o f spatia l correlation. For exam ple, the percentage rate of increase o f new pairs captured by a one percent increase in e gives a measure o f the '“ dim ension.” | G iven th is in te rp re ta tio n , under th e n u ll hypothesis th a t the series is inde- I pendently and id e n tic a lly d is trib u te d , B ro ck and Dechert (1988) have shown th a t p m)r(e ) — > C i;r(e )m as T —> oo. In p a rtic u la r, when { x t} is p u re ly random , B rock, jDechert and Scheinkm an have shown th a t the B D S sta tistic: B m ,T{z) = V/T [C 'm!T(e) — C lyT (e)m} j ;converges in d is trib u tio n to a N (0 ,V ) as T — » oo. Here \ / T , and Ar(0, V ) denote ■the square root o f T and th e no rm a l d is trib u tio n w ith m ean 0 and variance V •respectively. F urtherm ore, th e variance V can be consistently estim ated fro m the ,data. Therefore, under the n u lll hypothesis th a t {aq} is p u re ly random , the s ta tis tic I W mtT(e) = 5 m)y (e )/ I 7T his short description o f the test follow s th a t o utline d by Hsieh (1989). P ublished references 'o f the BD S test are Scheinkm an (1990) and B rock (1991) Converges in d is trib u tio n to N ( 0 , 1) as T — > oo. To test fo r randomness o f a given series a ll th a t one needs to do is calculate W. I f W is large in absolute value, the i n u ll of randomness is rejected at the significance level chosen fo r the test. Hsieh (1990) using m onte-carlo sim ulations has shown th a t the BDS test avoids the biases of the correlation dim ension estim ates obtained using the GP p lo t. A lth o u g h th is test is constructed based on properties exh ib ite d by d e te rm in istic processes it is im p o rta n t to rem em ber th a t the rejection o f n u ll hypothesis does no t im p ly accepting the a lterna tive o f chaos. T h is is because the test has good i power against general alternatives, such as stochastic nonlinear processes th a t do not i necessarily lead to chaos. Nevertheless, if the linea r com ponent is p ro p e rly removed, rejection o f the n u ll does correspond to presence o f “ nonlinear dependence” in data. B rock suggested applying the BDS test to the residuals of a fitte d linear m odel. U nder the n u ll hypothesis, the residuals fro m the linear fit are assumed to be i.i.d . ■ I f th e n u ll m odel is correct then th is norm alized s ta tis tic (B D S ) converges to the i norm al d is trib u tio n w ith mean zero and variance one. I f we fa il to reject i.i.d ., th is i is evidence consistent w ith the hypothesis th a t the data is generated by a linear m echanism . O n the other hand if we reject the n u ll hypothesis then we conclude th a t the data exhibits nonlinear dependency. Since the BDS test has strong power against general alternatives, it is app ro priate to use it when we don’t have a strong 1 . p rio r as to the possible directio n o f departure fro m i.i.d .. 1 • I I 2 .2 .2 H inich B isp e ctra l te st Subba Rao and G abr (1980) were the firs t to im plem ent B rillin g e r’s (1979) m ethod ; ■for m easuring the departure o f a process fro m lin e a rity (and n o rm a lity ) b y using j jan estim ate o f the bispectrum o f the observed tim e series. H inich (1982) presented { a p ra ctica l bispectral procedure fo r testin g w hether a given tim e series data are consistent w ith a lin e a rity (and also a n o rm a lity ) hypothesis. T he proposed test is ja non param etric test. | ' I f yt is a sta tio n a ry tim e series, then th e spectrum of yt , w hich is the fo u rie r ! 1 . . . . 1 ■transform o f the autocovariance fu n ctio n , can be w ritte n as: o o j j S ( f ) = J 2 C y ( t ) e x p ( - 2 i r i f t ) , (2.1) j n —G I I 12 ! where Cy(t) is the autocovariance fu n ctio n . In general, yt is serially uncorrelated (w h ite noise), if S(f) is constant. The b isp e ctru m o f yt is the tw o dim ensional fo u rie r tran sform o f the th ird -m o m e n t fu n c tio n given as: ; # ( / i , /2 ) = S 5 3 Cyy(n> ^ )e a ;p (-2 7 r* 7 1n - 27rz/2m ), (2.2) j m n jwhere Cyy( n ,m ) = E[y(t + n)y{t + m)y(t)]. In general if yt is a linear tim e series, then its spectrum is S (f) — a ^ \A (f)\2, and the bispectru m is j = A ( f 1) A ( f 2)A * (fi + f 2)y 3, (2.3) ^ h e re y 3 — E[e3(t)], and A (f) is the tra n sfo rm of the coefficient series : A ( f) — Y^Lo a[t]exp(—2Trift)> and A* denotes the com plex conjugate o f A. From 2.2 and 2.3 it follow s th a t: i \ B { fu f2 ) \2 = v l ( . I W a W i + h ) ^ K j jwhich under the n u ll hypothesis is constant over a ll frequency pairs (fi,/? ) if yt is linear. Sample estim ates, B o f the bispectrum and S o f the spectrum S(f) 'are obtained. T he ra tio in 2.4 is then estim ated at different frequency pairs. | , I f these ratios differ sig n ifica n tly over different frequency pairs, the constancy jof the ra tio , and hence lin e a rity o f the tim e series yt is rejected. However, if the 'estim ated ra tio differs sig n ifica n tly fro m 0, then th e hypothesis th a t yt exhibits n o rm a lity is rejected. N ote th a t the constant /X 3 2/cre 6 is the square o f F isher’s I skewness measure for the e series. The test s ta tis tic th a t H in ich (1982) derives fo r te stin g lin e a rity is based on the in te r q u a rtile range o f the estim ated ra tio over the set o f p e rtin e n t frequency pairs. 'If the ra tio is constant, then the in te r q u a rtile range is sm all and vice versa. The test s ta tis tic for lin e a rity is a s y m p to tic a lly norm al, so significance is easily determ ined fro m standard norm al tables. T he test fo r n o rm a lity likew ise is an a sym p to tica lly fo rm a l test based on the estim ated ra tio . I 2 .2.3 T sa y ’s F an d T A R -F te st jTsay’s F -test is analagous to T uke y’s one degree of freedom test for n o n a d d itiv ity in a regression setting. The m o tiv a tio n behind b o th these tests is w hether forecasts can be im proved upon by in clu d in g nonlinear term s. In p a rtic u la r, if a series {y f } can be expressed as an in fin ite order m oving average (M A ) process where the M A process has nonlinear term s, then th is w ill be reflected in the diagnostics of a fitte d m odel, if the residuals o f the linear m odel are correlated w ith say yt- \ • a nonlinear term . In general, higher order coefficients are unknow n. In w hich case th e forecast squares yf are used to o b ta in qua dratic term s upon w hich the residuals can be correlated. Keenan (1985) in trodu ced a test based on this corre latio n of et w ith y% . T he test of Tsay (1986) is sim ila r to the Keenan procedure, b u t tests the p o s sib ility o f fu rth e r im p ro vin g forecasts by in clu d in g p • (p + l ) / 2 cross-product jterms of the com ponents of yt , o f the fo rm yt~j ■ yt-k, k > j, j, k — 1 ,... , p. This test is d ire c tly designed to test for departures fro m lin e a rity in mean. T he test s ta tis tic is a p p ro xim ate ly F ( k — 1 ,n — k) under the n u ll hypothesis. T he T sa y’s F test procedure can be o u tlin e d as follows. 1. Regress yt on {1 , yt-\, • • •, Ut-p] and calculate the fitte d values yt, the residuals i t and the residual sum o f squares fo r t = p + 1 , ..., n. j 2. Regress vech (V^Vt) on {1 , y t-i, • ■ •, Vt-p} and get the residual vector ^ for j i = p + l , . . . , n . W here “ vech” denotes the stacked vector o f unique lower diagonal elements o f the m a trix (V)14) an(l Vt = {y t-i, ■ • • ? Vt-p}- N ote the j stacking vector has dim ension k = p • (p -\- 1) / 2 . 1 3. Regress residuals fro m step 1, i.e., et on the m a trix o f residual vectors fro m i step 2. F u rth e r get the sum of squared residuals fro m th is step, J2 ef and ; calculate the F -s ta tis tic as follows. i \ {Y ,yt^){T ,y[yt)~ 1{ ' L y A ) l k \ j I £ 3 / ( « - p - * - i ) J 1 } 1 i where yt is the m a trix of residual vectors fro m step 2 and has dim ension j (n — p — 1) x k. The F -s ta tis tic has k and (n — p — k — 1) d.f. T h is test has considerably higher power in detecting nonlinear models like the non- ' linear m oving average and Threshold autoregression (hereafter “ T A R ” ) com pared 1 I 'to the Keenan F-test. i . i I The T A R -F test of Tsay (1989), specifically tests w hether the concerned series i exh ib its any endogenous regim e changes w hich can be approxim ated by local linear ' I _________ L 4 _ i m odels. I f there are at least tw o regimes, then th e overall process is nonlinear in the m ean, where linear models can be used to describe each of those regimes. T he test exploits the properties o f arranged autoregression (explained in d e ta il in C hapter 4) and recursive predictive residuals w hich are i.i.d in th e absence of any m odel change. r ,The n o ta tio n th a t w ill be used to characterize a T A R m odel w ill be T A R (& ; p; d), jwhere k is the num ber o f regimes, p the num ber o f lags (or order o f autoregression) and d is the delay param eter w hich determ ines the threshold variable. T he test procedure can then be b rie fly sum m arized as follows. 1 . Specify the autoregression order p and a set o f possible threshold variables yt-d, ' where, d < p. For a given T A R (2 ;p ,d ) m odel w ith n observations the threshold variable yt-d m ay assume values {yh, ■ ■ ■, yn-d } i where h = m a x { l, p + 1 — d}. Let hi be the tim e index o f the i th sm allest observation o f {yh, * • •, yn-d}- j 2. R e w rite the m odel as j i < s i > s, (2 .6 ) ; where s satisfies yh s < rq < yhs + 1 ■ T h is is an arranged autoregression w ith the | firs t s cases in the firs t regim e and the rest in the second regime. | 3. For fixed p and d, the effective num ber o f observations in arranged autoregres- 1 i sions is n — d — h + 1, w ith h as defined in step 1 above. I ! 4. P erform recursive autoregressions beginning w ith say 6 = n / 10 + p, observa- j tions so th a t there are n — d — b — h + 1 pre d ictive recursive residuals available, j T he predictive residuals are denoted by e. ! 5. Do the least squares regression, I l P | &hi+d = ^0 + ^ WyVhi + d -v + eh,+d, (2-7) | vz= 1 j fo r i = 6 + 1,..., n — d — h + 1, and com pute the F s ta tistic yhi+d 4 + E ^lyhi+d-v + ^hi+d 7f P ^ g + E ^IVh.+d-v + €hi+d ^ V = 1 15 - d - b - p - h ) J where for large n the s ta tis tic F (p , d) follow s app ro xim a te ly an F d is trib u tio n w ith p + 1 and n — d — b — p — h d.f. In p rin cip le since the threshold value is unknow n, one has to perform a sequential estim ation. I f there are enough observations in the firs t regim e then the least squares estim ate fo r the coefficients are consistent in w hich case the p redictive residuals I are w h ite noise a sym p to tica lly and orthogonal to th e regressors. O n the other hand, when the tim e index i, exceeds the threshold s, the p re d ictive residual fo r the observation w ith tim e index hi -f- d is biased because o f the m odel change at tim e '■ h i -f d. In this case the predictive residual is a fu n c tio n o f the regressors and hence the o rth o g o n a lity condition is violated. T h is is picked up by the F sta tistic. 2.2 .4 T o p o lo g ica l te st I W h ile the m e tric test (B D S ) exploits the geom etric corre latio n in data, the to p o logical test is a q u a lita tiv e m ethod. It detects w hether a tim e series exhibits chaotic behavior by searching for evidence of unstable perio dic o rb its. F urther, it differs fro m the m e tric approach in th a t it preserves th e dyna m ic correlation in data by reta in in g the tim e ordering. I t ’s advantage over the m e tric approach is th a t, once chaos has been detected, it facilitate s the characterization o f the und e rlyin g pro cess in a q u a n tita tiv e way although the exact equation sytem th a t generated the jdata is not id entified. However, com peting models th a t are incom p atible w ith the reconstructed strange a ttra c to r can be rejected. ! T he idea behind the topological m ethod is th a t if one observation pi occurs near a periodic o rb it, then subsequent observations w ill evolve near th a t o rb it fo r a w hile i jbefore being repelled away fro m it. S im p ly p u t, fo r a chaotic process the value ;| yi — pi+t |, * = 1,..., N and t = 1,..., TV — i, w hich represents th e close returns for the process, w ould be less th a n a c ritic a l value say e where the tim e series begins ;to follow an unstable perio dic o rb it. O n a color coded graph w ith the h o rizo n ta l axis representing the observation num ber i and the ve rtica l axis the tim e delay t, ;the differences, | pi — pi+t |< e, are coded in black and ones greater th a n e in w hite. I f the data is generated by a de te rm in istic process, a num ber o f h o rizo n ta l line segments w ill be seen and if the data is stochastic, a u n ifo rm array o f dots w ill I appear. G ilm ore (1992)8 has a detailed exposition o f the topological m ethod w ith several em pirical applications o f th is test. A n o th e r way to look at the close returns w ould be to com pute a histogram of the differences where H ( t ) represents the incidence o f a close re tu rn , defined as follows; H it) = where 9 is the Heaviside fu n ctio n w h ich takes the value 1, if | yi — yi+t |< G and 0 otherwise. T he histogram fo r a stochastic tim e series I .will e x h ib it a scattering around a u n ifo rm d is trib u tio n , and the histogram fo r a ^chaotic process w ill have periodic peaks. G ilm ore has shown th a t this test is quite robust to a d d itive noise and th a t tim e averaging is able to re tu rn the signal fro m a (determ inistic process corrupted w ith a d d itive noise. I I 2 .3 R e s u lts 2.3.1 D a ta i Raw data consists o f daily, weekly and m o n th ly re tu rn s fo r th e S & P 500 com posite p o rtfo lio obtained fro m the Center fo r Research in Security Prices (C R S P ). Fob i i I low ing the em pirical w ork in the Finance lite ra tu re , a ll re tu rn s were com puted as I I lo g a rith m ic price relatives, i.e., R t = iog(Pt/P t-i)- D a ily returns are based on end of day prices, weekly returns are com puted fro m Wednesday closing price and m o n th ly I 'returns are based on m o n th ly averages of d a ily closing prices. D a ily and weekly (data cover the period J u ly 1962 - December 1990 whereas m o n th ly d ata is for Jan. 1926 - Dec. 1991. There are a to ta l of 7168 daily, 1370 weekly and 792 m o n th ly observations. Table 2.1 presents some sum m ary statistics fo r the three series. It is clear th a t (d istrib u tio n o f d a ily and m o n th ly retu rns e x h ib it very fa t ta ils; w hile d a ily returns i are negatively skewed, m o n th ly returns show positive skewness. D is trib u tio n of weekly returns does n o t provide any strong evidence o f excess skewness or kurtosis i ;compared to a norm al d is trib u tio n . Fama (1965) p rovid ed sim ila r evidence for d a ily ; l I 1 8See G ilm ore (1992) fo r details on how to choose the correct value fo r c ' • returns o f the Dow Jones In d u s tria l stocks and concluded th a t the d is trib u tio n of price changes conforms m ore closely to the stable P aretian d is trib u tio n w ith char acteristic exponent less th a n 2 th a n the sym m etric student d is trib u tio n suggested by B la ttb e rg and Gonedes (1974). S ignificant excess kurtosis in d a ily retu rns is com m only a ttrib u te d to the persis tence in variance, w hich is related to the rate o f in fo rm a tio n arrivals, level of tra d in g a c tiv ity , and corporate finan cial and ope ra ting leverage decisions w h ich tend to affect the level o f stock price. Seasonal announcements such as the disclosure o f a firm ’s q u a rte rly earnings have been know n to result in re tu rn d is trib u tio n s w ith higher .variance d u rin g the disclosure period than d u rin g the nonannouncem ent periods, see [Beaver (1968); P a te ll and W olfson (1981)]. C h ristie (1983) generalized the idea o f in fo rm a tio n signals causing param eter shifts to encompass a ll firm -specific events by fo rm u la tin g a discrete m ix tu re of tw o norm al d is trib u tio n s m odel where returns ! i I draw n fro m the d is trib u tio n w ith the higher variance represent in fo rm a tio n events w hile the other d is trib u tio n generates n o n in fo rm a tio n random variables. C h ristie (1982) dem onstrates th a t the standard deviation o f a stock’s re tu rn is an increasing fu n c tio n o f financial and operating leverage and e m p iric a lly verifies the finan cial i leverage effect. Table 2.1: S um m ary S tatistics o f Raw R eturns Stock Mean Std. Dev Skew K u rt M in Max N. Obs D a ily W eekly M o n th ly 0.00085 0 . 0 0 1 2 1 0.00420 0.0321 0.0206 0.0430 -2.6016 -0.5576 0.7075 76.126 4.0654 17.903 -0.8433 -0.1666 -0.2647 0.3775 0.0818 0.5030 7167 1447 740 Notes: Raw returns are defined as ln(pt/pt_i) . Std. Dev, Skew and K u rt stand for the standard deviation, measure of skewness and kurtosis respectively. M in, M ax and N. Obs denote the m inim um , the m aximum and the number of observations for each of the series. i ! In order to fu rth e r understand the n ature o f the dependence in the retu rns series, | ; jthe sample autocorrelations are analyzed. Tables 2.2 and 2.3 e x h ib it the estim ated ; lautocorrelations for the series {|72i |} and { R ^}. To test w hether a p a rtic u la r value i ,of the autocorrelation fu n c tio n fo r these series is equal to zero, we use a result ob tained by B a rtle tt9. Using th is ap p ro xim ation, we observe th a t autocorrelations of , ! 9He showed th a t if a tim e series has been generated by a white noise process, i.e., consists jo f independently d is trib u te d random variables, the sample a uto correlatio n coefficients fo r lags i b o th series are greater th a n 6 / y / T , w hich is a sufficie n tly large confidence bound. The autocorrelations o f the absolute and squared re tu rn series are consistently sig- n ific a n tly p o sitive and as the lag increases, autoco rre la tio n fu n c tio n o f b o th series decay slow ly b u t never below 2/ y / T at any lag up to 15. T his fin d in g agrees w ith those reported in the classic w ork o f Fam a (1965), th a t large price changes (re tu rns) are followed by large changes, and sm all, by sm all, o f either sign. M ore generally, the ^distribution o f the next absolute or squared re tu rn depends not o n ly on the current re tu rn b u t also on several previous returns. T h is is a rejection o f the hypothesis jthat re tu rn series are stric t w hite-noise processes. ! T he presence o f linear dependence in re tu rn series o f m arket indexes is often a ttrib u te d to various m arket phenom ena. T he presence of a com m on m arke t factor, i the problem o f th in tra d in g in some stocks, the speed o f in fo rm a tio n processing by m arket p a rticip a n ts, and day-of-the-w eek effects could co n trib u te p a rtia lly to the observed first-o rd e r autocorrelations. j | Table 2.4 shows results o f the BDS test applied to the three series. A t conven- ; jtional significance levels the n u ll hypothesis of i.i.d . is stron gly rejected in a ll cases, j 'P re-filterin g using a linear A R (3 ) m odel leads to the same conclusion, a ltho ugh the 1 j t m agnitude o f the statistics are sm aller. T h is im plies nonlinear dependence in the returns series. In the lite ra tu re , th is non linear dependence is often explained by the w ell-docum ented fact o f changing variances [see, fo r exam ple, Hsu, M ille r, and W ic h e rn (1974) and Tauchen and P itts (1983)]. Changing variance is also used to explain th e high levels o f kurtosis in re tu rn d is trib u tio n s. P re -filte rin g using a non- I lin e a r filte r such as the A R C H (3 ), lends support to the i.i.d . hypothesis in the case of d a ily returns, b u t is strongly rejected fo r the m o n th ly series. T h is suggests th a t (at least in the case o f m o n th ly retu rns, the dependence is not of the fo rm captured by the A R C H class o f models and the dependence in d a ily returns can perhaps be approxim ated by an A R C H (3 ) m odel. However, the perform ance o f these fitte d [ A R C H models to d a ily returns as measured by th e ir sample forecasts fa ll short o f j ;those obtained fro m a constant variance assum ption, [see French et.a l (1989)]. J i ^greater than zero are a pp ro xim a tely d is trib u te d according to a norm al d is trib u tio n w ith m ean , 0 and standard d eviation l / V T (where T is num ber o f observations in the series). Thus, if a .p a rticu la r series consists of, say, 100 d a ta points, we can attach a standard e rro r o f 0.1 to each a uto correlatio n coefficient. Therefore, i f a p a rtic u la r coefficient was greater in m agnitude th an 0.2, we could be 95 percent sure th a t the true auto correlatio n coefficient is not zero. 1 ____________ ,_____________________________________ : __________________ 1 9 - Table 2.2: Autocorrelation Coefficients of Absolute Returns Lag Daily Weekly Monthly P\x\ 1 ) 0.2307 0.2255 0.1900 P\x\ 2 ) 0 . 2 1 0 0 0.1660 0.1391 P\x\ 3) 0.2432 0.2136 0.1263 P\x\ 4) 0.1824 0.1917 0.1829 P\x\ 5) 0.2304 0.1946 0.1060 P\x\ 6 ) 0.1932 0.2168 0.1316 P\x\ 7) 0.1798 0.2223 0.1025 P\x\ 8) 0.1869 0.1902 0.1642 P\x\ 9) 0 . 2 0 2 1 0.1449 0.1633 P\x\ 1 0 ) 0.1652 0.1519 0.1754 Notes: Returns here are computed as log of price changes. The autocorrelation coefficients for the first 10 lags are positive and significantly larger than 3 jy /T for all three series, where T is the to ta l number of observations. Table 2.3: A u to co rre la tio n Coefficients o f Squared R eturns LA G Daily Weekly Monthly P xx (1) 0.2527 0.2194 0 . 2 1 0 0 P x x (2) 0.2300 0.1914 0.1669 P x x (3) 0.2353 0.1241 0.1346 P xx (4) 0.1944 0.1048 0.2099 P x x (5) 0.2440 0.1091 0 . 1 1 0 0 P xx ( 6 ) 0.1782 0.1675 0.1469 P x x (7) 0.1698 0.1533 0.1145 P x x ( 8 ) 0.1969 0.1419 0.1574 P xx (9) 0.1921 0.1226 0.1709 P xx ( 1 0 ) 0.1552 0.1252 0.1775 : ! •Note: Returns were computed as log of price changes. The autocorrelation coefficients , jfor the first 1 0 lags are all positive and significantly larger than 3/y /T for all three series, jwhere T is the tota l number of observations. I Table 2.4: BDS Test for I.I.D of the Returns Series Stock m e/a 0.50 1.00 1.50 2.00 D a ily 2 3.825 2.188 1.135 0.572 3 6.057 3.149 1.570 0.775 4 9.027 4.136 1.972 0.951 5 13.75 5.322 2.396 1.129 W eekly 2 9.238 8.004 7.838 0.136 3 12.92 10.46 9.679 0 . 2 0 1 4 15.60 11.94 10.76 0.256 5 18.99 13.71 11.67 0.303 M o n th ly 2 8.515 8.908 9.833 10.17 3 9.315 9.628 10.81 11.40 4 10.28 10.47 11.27 11.64 5 10.76 10.93 11.75 12.23 ■Notes: The BDS U statistic is distributed N(0,1) under the null hypothesis of i.i.d. Based |on the values of the statistics, the null is rejected for e/a <=1.5 for all series. B o th the raw series as w ell as residuals fro m a linea r fit were used in the a p p lication of the B ispectral test. Table 2.5 presents the results fro m the la tte r. Q u a lita tiv e ly jthe outcomes were sim ilar. T he H s ta tis tic w hich measures the d e viation fro m n or m a lity is large enough to reject the n u ll o f n o rm a lity at any level o f significance. Likew ise, the Z s ta tis tic rejects the n u ll of lin e a rity at conventional levels o f signif icance. T he results o f b o th tests in dica te a greater degree of non linear dependence In the d a ily and m o n th ly returns series. T h is is consistent w ith evidence obtained j ■from the a utocorrela tion o f the absolute and squared returns series above, i Results o f T say-F and T A R -F tests are presented in Table 2.6. Once again, daily returns e x h ib it significant second order non linearity, whereas w eekly and m o n th ly ;series provide very weak evidence fo r higher order non linearities. T he T A R -F test 'was applied to each o f those series w ith a preset p and d w hich was chosen according I [ | jto a c rite ria described in C hapter 4. Results fo r d a ily re tu rn s suggest significant j | t ;model changes over the sample period, w hile weekly returns provide little or no | i . . . . 1 evidence for m odel changes, m o n th ly re tu rn s in d ica te possible regim e changes. ! Table 2.5: H inich Bispectral Test for N orm ality and Linearity Data H Z D a ily W eekly M o n th ly 43.27 21.89 4.28 61.89 8.99 11.06 H = (2 5 )1/ 2 — (2n — l ) 1/ 2, where S is distributed as a chi-squared w ith n d.f. Note 1 that H is approxim ately distributed as N(0,1) under the null hypothesis of norm ality, j Likewise, Z is approxim ately distributed as N(0,1) under the null hypothesis of a linear data generating process. Table 2.6: T say’s F and T A R -F Tests fo r N o n lin e a rity in R eturns Data Tsay’s F TAR-F D a ily W eekly M o n th ly 74.44 ( 0 .0 0 0 ) 1 .2 0 (0.304) 3.432 (0.0023) 6.78 ( 0 .0 0 0 1 ) 0.89 (0.469) 3.35 (0.0099) Tsay’s F statistic for p = 3, is asym ptotically distributed TA R -F is Tsay’s T A R (p,d), F test which is asym ptotically distributed as F p + l , n - d - b - p - h , where p = 3, is [the order of autoregression, d = 1 is the delay parameter, b = re / 1 0 + p is the starting j'number o f observations for recursive autoregressions and h — m ax(l,p + 1 — d). Numbers jw ithin parentheses are p values. I U pon closer exam ination, outliers in d a ily returns data had a significant im pact on the m agnitude of th is s ta tis tic . However, rem oval o f outliers d id not change the q u a lita tiv e results o f the test. ! I Figures 2.1, 2.2 and 2.3 show the closereturns histogram described in section ! '2.2.4, for daily, weekly and m o n th ly returns respectively. D a ily and m o n th ly retu rns , e x h ib it some nonlinear structure. T he evidence fo r nonlinear s tru ctu re fro m weekly ; data is weak at best. T his is consistent w ith evidence obtained fro m the three [previous tests. i 22— I 190 170 150 130 110 o Lag(days) Figure 2.1: Closereturns histogram for d a ily S&P 500 returns 160 120 100 Lag(weeks) Figure 2.2: Closereturns histogram for weekly S&P 500 returns 150 140 130 1 2 0 f 110 | 100 - o o o o Lag(months) Figure 2.3: Closereturns histogram for m o n th ly S&P 500 returns o f contingent claim s. For exam ple, understanding the behavior o f the variance is essential to o p tio n p ricing models. E m p iric a l tests o f asset p ricin g models and the ^efficient m arkets hypothesis draw sta tis tic a l inferences th a t are also cond ition al on I d is trib u tio n a l assum ptions. T he m ost convenient assum ption fo r fin a n cia l th e ory and em pirical analysis is th a t the d is trib u tio n o f security returns be m u ltiv a ria te n o rm a l w ith param eters th a t sta tio n a ry over tim e . Since the no rm a l d is trib u tio n is stable under a d d itio n , any a rb itra ry p o rtfo lio of stocks w ill also be n o rm a lly d istrib u te d . W ith the a d d itio n a l assum ption o f risk aversion, m ean-variance th e ory follow s. F urtherm ore, the assum ptions o f n o rm a lity and param eter s ta tio n a rity are •required fo r m ost o f th e econom etric techniques ty p ic a lly used in em p irica l research. | E m p iric a l evidence on the d is trib u tio n o f d a ily and m o n th ly p o rtfo lio (S & P 500) returns clearly rejects the stationary normal d is trib u tio n m odel. It is, however, the t n o rm a lity hypothesis th a t is crucial to models of fin a n cia l theory. T he hypothesis o f I lin e a rity and i.i.d . are also stron gly rejected. T h is by its e lf does n o t co n tra d ict m ar- jket efficiency. Since m arket efficiency only im plies th a t retu rns are not predicta ble ■consistently, the rejection o f i.i.d . does not provide any direct evidence o f (nor pre clu de) th e presence of a determ instic expla n a tio n fo r the observed behavior. One j lean th in k of at least three alternatives th a t could have lead to the above conclusion: j I I taonstation arity, nonlinear stochastic process and d e te rm in istic process. , I j S ta tio n a rity, is m erely a sam pling assum ption. In fa ct, theory predicts th a t in the jcase of in d iv id u a l stocks, changes in the investm ent and fin a n cia l decision variables 1 iof firm managers w ill result in adjustm ents o f the expected re tu rn and the standard j deviation param eters o f the returns d is trib u tio n . Boness et al. (1974) found th a t w ith weekly re tu rn data, the mean and standard d e viation o f th e returns process ■ ■changes su b sta n tia lly before and after a ca p ita l stru ctu re change. T he y also present evidence th a t the overall series contain su b sta n tia lly m ore departures fro m n o rm a lity :th a n the series of either the pre or post c a p ita l s tru c tu ra l change subperiods, j However, we w ould expect the s tru c tu ra l changes to be infrequent, especially jif the p o rtfo lio is w ell diversified. I f th is were true , then by going to higher and ■higher frequency data, we should be able to remove the effects of s tru c tu ra l changes. ; However, Hsieh (1990), based on the results o f th e BD S test on S&P500 weekly raw {returns fro m 1962 to 1989, d a ily returns fro m 1983 to 1989, and 15 m inute returns d u rin g 1988 concludes th a t a ll three series reject th e i.i.d hypothesis and hence 24 argues, it is u n lik e ly th a t n o n s ta tio n a rity could be a possible cause. However, the rejection o f i.i.d . could very w ell be consistent w ith non independent b u t id e n tica lly d is trib u te d random variables, w hich is w hat one w ould expect in higher frequency data. j Rejection of i.i.d. is also consistent with returns being generated by nonlinear stochastic system s. W e have show n that nonlinear stochastic m odels of the A R C H Jc la s s do not account for all of the dependence, at least in the case of m onthly returns. Further, the type of nonlinear dependence observed seem s to vary with the frequency of observations. This leads us to believe that a deterministic explanation for the observed behavior should be plausible. T he lite ra tu re on “ m arke t efficiency” and its u n d e rlyin g linear ra tio n a l expecta tions models assume th a t the m arket price tru e ly reflects a ll available in fo rm a tio n , and th a t investors have perfect foresight and so fo rth . In rea lity, in fo rm a tio n is 'costly and hence even if investors liked to in co rp o ra te a ll available in fo rm a tio n , j 'they cannot afford to. A m odel w hich exploits th is observation along w ith some j salient features o f the m arke t environm ent is developed and used in the subsequent 'chapters to propose an “ in trin s ic th e o ry” of stock prices w hich is consistent w ith some of the observed em pirical regularities. T he m o tiv a tio n fo r such a m odel is | ! provided in the next C hapter. j I I I .25 I 1 I I I \ jC hap ter 3 i I n s titu tio n a l a n d B e h a v io r a l T h e o r y o f S to c k I P r ic e F lu c tu a tio n l I i jln th is chapter we present th e m o tiv a tio n behind an a lterna te th e o ry o f stock price flu c tu a tio n firs t proposed by D ay and Huang and la te r by D ay and G u (1993). A n illu s tra tiv e m odel is used to form alize the th e o ry and basic properties o f the m odel ( 'are reviewed. Testing the theory involves estim ating th e coefficients o f the m odel. I iTo fa c ilita te this, we suggest a tra n sfo rm a tio n o f the basic m odel and provide a 'm otivation fo r the tran sform ation. i i 3.1 B a c k g r o u n d A behavioral and in s titu tio n a l the o ry o f stock price flu c tu a tio n is m o tiva te d on the prem ise th a t, observed randomness in stock prices can be a ttrib u te d in p a rt to the in te ra ctio n o f investors w ith different investm ent strategies, in a p a rtic u la r m arket fram ew ork, f t is behavioral in th a t, it is designed to approxim ate rules th a t investors and other m arket p a rticip a n ts follow and in s titu tio n a l in th a t, it represents one type |of m arket m echanism o f w h ich the New Y ork Stock Exchange is an example. ! Stock m arkets are organized around buyers and sellers who do not trad e d ire c tly Jwith one another. Instead, they carry o u t th e ir transactions th ro u g h a broker or a ! .financial consultant w ho in tu rn places orders w ith the flo o r tra d e r. In some m arkets, !such as the New Y o rk Stock Exchange, m arke t orders are executed b y a “ specialist” whose social re sp o n sib ility is to m a in ta in an ord e rly m arke t in a p a rtic u la r stock. i i i \ ; — . -26 In other words he adjusts prices in response to dem and and supply conditions so as I to prevent excessive changes in price. The dynam ics o f dem and and supp ly and price setting m echanism in these m ar- cets often lead to flu ctu a tio n s th a t seem irre g u la r on a broader scale b u t e x h ib it pattern s upon closer observation. T he d iffic u lty in discerning these pattern s mo- jtivates the idea th a t investors fo rm u la te a va rie ty o f strategies based on signals ranging fro m sim ple forecasting models to elaborate technical tra d in g rules. i i | T he com ple xity involved in b u ild in g a m odel th a t accounts fo r a ll the different ^strategies and m arket mechanisms led, D ay and H uang (1990) and G u (1992) to fo rm u la te a m odel o f stock price behavior th a t is based on tw o stylized investm ent strategies w hich are associated w ith tw o different types of investors called a and (3. i * i : 3 .2 B a s ic T w o In v e sto r M o d e l 3 .2 .1 cr-In vesto rs H ollow ing D ay and Huang, alpha-investors believe th a t the average price reflects ! th e long ru n value u o f the stock in lig h t o f “ fu n d a m e n ta l” in fo rm a tio n , where u j represents the estim ated investm ent value o f the stock assuming conditions over the ■relevant horizon a ctu a lly occur. Since conditions can change over the investm ent horizon, the estim ate of u in general is lik e ly to be m ore or less v o la tile depending on «the investm ent horizon. Furtherm ore, e stim a tio n o f u requires substantia l am ount o f resources w hich can be very expensive. These investors w ould correspond to Fischer B la c k ’s “ in fo rm a tio n trad e rs” . > a-investors investm ent strategy is based on the com parison between th e invest- jm ent value u and th e current price. T h e y expect the price of a c u rre n tly undervalued istock even tua lly to go up w h ile th a t o f an overvalued stock eventually to go down. |ln other words they have contrarian expectations. T he investm ent strategy consis- j I ' !tent w ith these expectations is to b u y when the stock is su fficien tly undervalued ’(current price is below estim ated value) and to sell when it is su fficie n tly overvalued (current price is above estim ated value). ; Assum ing g to be a tra d in g threshold and h to be a tra d in g bound, (^-investors are I 1 . . . I a ctively b uying or selling if g < | u — pt |< h\ holding th e ir positions if | u — pt |< g) ’ buying or selling if | u — pt |> h. T he corresponding b u y /s e ll threshold prices are p — u — g and p = u -f g. The b o tto m in g and to p p in g prices are pB — u — h, pT = u + h, where a indicates the “ stre n g th ” o f G-investors demand. In general a could vary w ith in a-investor types. In p a rtic u la r, (^-investors w ith sufficien tly large thresholds are lik e ly to express a stronger dem and fo r the stock im p ly in g a larger yalue for a so as to cash in on the discrepancy between the m arke t value and the estim ated investm ent value. Here, fo r convenience we assume one type of g:-investor whose investm ent strategy is given a piecewise linear form , 1 i j j a[h — g] , p < u — h a \ ( u ~ 9 ) - Pt] , u - h < p t < u - g a(pt) = < 0 , u - g < p t < u + g ( 3 . 1 ) i 1 a{{u + g) - pt] , u + g < pt < u + h —a[h — g] , u + h < pt )efine the fo llow ing intervals: i I 1 = 0 , u — h] | 1 2 = u — hjU — g] i I 3 = u - g , u + g\ (3.2) j 1 4 = u + g,u + h] ; I 5 = u + f i , oo] , |3.2.2 /3-investors j ■In general fo rm u la tio n of a type strategies is q u ite expensive. I t requires an investor t to spend a considerable am ount o f tim e and fin a n cia l resources to be succesful in im p le m e n tin g this strategy. O n the other hand, a m a jo rity o f investors use re la tive ly sim ple rules. T he y could estim ate a value fo r u based on current earnings inform a- I jtion w hich is usually provided in a com pany’s q u a rte rly re p o rt. T h e y could then update th is estim ate in an adaptive m anner by gathering relevant m arket inform a- ! I . . . . 1 ition over the investm ent horizon. As a firs t a p p ro xim a tio n , the curre nt price could ; 1 , j ibe used as an estim ate o f the investm ent value u. In w hich case th e ir investm ent .strategy is based on a com parison of the curre nt price to current fundam ental v. v * by d e fin itio n m ay not be very v o la tile b u t nevertheless changing over tim e, i I f the current price is above the curre nt fundam ental, /3-investors in te rp re t the m arket to be b u llish and expect the tre n d to continue. I f however, current price is 1 _________ 2 8 _ j j below this value, they regard the m arke t as bearish and expect a bearish m arket ; likewise. In other words th e y fo rm e xtra p o la tive expectations based on current | in fo rm a tio n . Consequently th e ir investm ent strategy is to b u y in to a risin g m arket ;when it is b u llish and sell o u t o f a fa llin g m arket when it is bearish, j Technical analysts often fo llo w sim ple technical rules such as “ m oving average crossover” strategy, w hich involves com paring a m oving average o f past prices to | current price. I f the current price is higher th a n the m oving average, then a b u ll | m arket is perceived and they become net buyers. I f current price is below the m oving 'average, then a bear m arket is signalled and th e y become net sellers. Hence, f3- j ^ t I in ve sto rs are assumed to be like th e chartists or technical analysts, in so fa r as th e ir | 'estim ate of current fundam entals can be approxim ated by a “ m oving average.” A key difference fro m the o-investo rs’ strategy is th e ir lack o f a w e ll defined tra d in g ; threshold. G iven these assum ptions, /9-investors strategy is sim ply, ! i 1 | P(Pt) := KPt - v) (3.3) j 1 3 .2 .3 M arket M ed ia tio n , In the stylized m arket investors do not trad e w ith each other, instead th e y trade : ' th ro u g h th e ir brokers. T he brokers in tu rn route orders to a m arke t such as the New j Y o rk Stock Exchange where the specialist, who is a m arket m aker, executes orders. T he p rim a ry fu n c tio n of the specialist is to m ediate transactions out o f e q u ilib riu m , i His w illingness to make the m arket w hich is driven by p ro fit o p p o rtu n itie s brings ; about c o n tin u ity o f the m arket. T he m arket m aker announces a price at discrete tim e intervals (called the u n it : tra d in g period) and stands ready to absorb any excess dem and by reducing his in ventory and accum ulating fro m excess supply. In the m eantim e he is exposed to enorm ous risks in quoting prices p rio r to seeing the flow o f p u b lic orders. Conse- i quently, price is adjusted every period so as to balance the in ve n to ry and keep price j flu ctu a tio n s at m oderate levels. T h is n a tu ra lly requires h im to have substantial J i | finan cial resources, and in ad d itio n , access to large am ounts o f credit in tim es of ;need. T he assumed price adjustm e nt fu n c tio n is P t + i P t + A(a(p,) + I. (3.4) (1 + A b )p t — A[6u — a(h ~9)\ , Pt e I 1 ( 1 + A ( b - a))pt - X[bv -- a(h ~9)~ - a(u - gr)] , Pt G 1 2 pt+l = < (1 + A b )p t — Xbv > Pt G I 3 ( 1 + A ( 6 - a ) ) P t + *-< 1 a(h — 9) + a(u + g)] , P t G 14 ( 1 + A b )p t — X[bv + a(h ~9)] > P t G J 5 where A is the price adjustm ent coefficient. From (3.1)-(3.4), the basic m odel can be re w ritte n as, (3.5) D ay and H uang have shown th a t A should be n e ith e r too sm all n or too big in order fo r a feasible m arket to exist. For exam ple, if A is too sm all, then price trajectories w ould converge to one o f tw o te m p o ra ry e q u ilib ria , w hich the m arket cannot sustain. T his is because, fo r specialist in ve n to ry to rem ain constant, the a- investors should have an inexhaustible dem and fo r stocks w h ile /9-investors should have an in exhaustible supply or vice-versa, a and /9-investors are in fu ll-e q u ilib riu m o n ly if current fundam ental value v equals investm ent value u equals current m arket price p. If, however, A is too big, price tra je cto rie s w ill flu ctu a te v io le n tly w ith o u t bound. T he m arket w ill collapse due to the in a b ility o f the specialist to absorb the excess demand. In th e ir basic m odel, they fu rth e r assumed fo r purposes o f the o re tica l analysis, the s ta b ility of the aggregate excess dem and fu n c tio n 10, no exogenous rando m shocks and param eters o f the m odel such as u, v, A, a, b, h and g as constant. F igure 3.111 shows the phase diagram o f th is m odel for an a rb itra rily chosen in itia l price and A value. T he m ap is re la tive ly sim ple and regim e switches are m ore frequent com pared to e m pirical data. For details on how the various param eter values affect the resultin g price d is trib u tio n , see, G u. F igure 3.2 e xh ib its th e tim e- series o f prices generated by the above m odel. 10T h is was obtained by assum ing th a t the flow o f a-investors e xitin g the m arket and inflo w of , new /3-investors, some o f w hom , replace investors w ho exited the m arket and some w ho learned to ! become cr-investors, is ju s t sufficient to keep aggregate excess dem and fixed. li T h is m odel was firs t presented in G u (1992). Figure 3.1: Phase diagram of prices generated from the 2 -investor model w ith p[0] = 1.53, A = 0 .S2 /v Figure 3.2: Price tra je cto ry generated by the 2-investor model. j 3.3 T h e o r e tic a l p r o p e r tie s o f th e b a sic m o d el By construction the dynam ic behavior of price tra je cto ry depends on a and 3 in- ; vestors m arket shares and the specialist’s price adjustm ent rule. Day and Huang ; have shown the follow ing results: a) there exists a unique fu ll e q u ilib riu m at the ; 3 1 jprice where a-inve sto rs’ estim ated fundam en tal value u coincides w ith the current jvalue v assessed by /3-investors, b) if a-investors dom inate, price tra je cto rie s con verge to the fu ll e q u ilib riu m price, c) if /3-investors dom in ate the m arket, price trajectories are explosive, d) if /3-investors o n ly dom inate the m arke t locally, when j prices are around the fundam en tal value b u t not beyond, then th e fu ll e q u ilib riu m is ! not stable. Price tra je c to ry in th is case m ay be cyclic, chaotic and stro n g ly ergodic converging to stable density functions. 3 .4 M o tiv a tio n for tr a n sfo r m a tio n i I In deriving a ll the th e o re tica l properties, a num ber o f variables have been treated as param eters. In p a rtic u la r, u , v, p B , pT , p, p and A should be considered variable. In a d d itio n , the investor stren gth param eters a and b could also be m odelled as changing over tim e depending on past experience and m arke t conditions. ^ u, sim ilar to Shillers’ ex-post value o f an asset, in o ur th e o ry is the a-inve stors’ jbest estim ate o f the price, ta k in g in to account the actual fu tu re dividends the in- |vestors w ill receive over the investm ent horizon, u w ill be high today, if the in ve s tm e n t w ill yie ld high dividends in th e fu tu re . C le a rly investors do not generally ! f # 1 I know the ex-post value today o f an investm ent today, a ltho ugh th e y have im perfect | ■. . i indicators o f it. W e use data to m o tiva te a proxy fo r u. \ j I t is w ell know n th a t the grow th in the stock m arke t reflects general econom ic I grow th. M acro variables such as G N P and price level are know n to e x h ib it a long 'te rm trend. T he real price o f corporate stocks, as measured by a deflated 12 Standard and P oor’s com posite stock price index (F igure 3.3), shows a d is tin c t upw ard trend, ; p a rtic u la rly between th e la te 1940s and the late 1960s. F igure 3.4, presents the same tre n d for real dividends fo r the S&P index. F igure 3.5 however e xh ib its a i i I dow n-w ard tre n d in real earnings, p a rtic u la rly in the la te 60’s. Since u is com puted | I ' as the present value o f th e curre nt and fu tu re dividends, it is a long m oving average , o f dividends. Therefore it is lik e ly to be sm ooth and tre n d in g upw ard. Hence, an I ' adequate test o f our th e o ry using em p irica l data w ould require at the very least th a t : 12A11 nom inal variables have been deflated by annual average o f the producer price index w ith j ■ the base fixed at 1967 = 100. 1.2 1 .8 0.8 0.6 ■ 0.2 0 / V / V • ■ v y . 4 A A 'V \ :'V o o o o o o o o o o c ? CP l H CP u ~) CP up CP uP O l O CP ■ ^ r u o u ~ ) c o u p p — c c c o cr> Figure 3.3: Standard and Poor’s Q u a rte rly real prices, 1935-1990. 0.04 , 0.035 j 0.03 i 0.025 1 0.02 I 0.015 i o.oi y 0.005 j 0 *- •p V U /-vu. o o o o o up CP uP CP Up ■*cr u p U P cjO u p o o o CP Up CP p - r - C O Figure 3.4: Standard and Poor’s Q ua rte rly real dividends, 1935-1990. 0.6 : £ , 0.5 i •5 : : lS ° - 3 A A 0-2- ' c S 0.1 : 0 4 — \a/ , w Figure 3.5: Standard and Poor’s Q u a rte rly real earnings. 1935-1990. 33 ' u grows at a certa in rate. Here, we shall assume th a t: ut+i — (1 -f- S)ut (3-6) i where, 6 is the rate at w hich long te rm fundam entals are assumed to grow. T h is is in contrast to th e constant u assum ption o f D ay and Huang. T he accuracy o f our | a p p ro xim a tio n is like ly to decrease w ith the le n g th o f the investm ent horizon. In 'p a rtic u la r, for shorter horizons the role of exogenous shocks (in the fo rm o f random : news) could be considered q u ite sig nificant resultin g in 3.6 being a poor appro xim a tio n . However, over a longer horizon the a p p ro xim a tio n assumes, th a t p o sitive and negative shocks cancel each other out. In so fa r as this assum ption is reasonable, i our in tre p re ta tio n o f the basic results are valid. For purposes o f s im p lify in g th is analysis, we have assumed a m onth to be a longer investm ent horizon. ’ T he current fundamental value, v in our m odel, is also like ly to be changing 'over tim e . In general, it is based on the m ost recent com pany q u a rte rly reports I on earnings, dividends and deb t-e q u ity ra tio . A lte rn a te ly , it could be the present i value o f current earnings discounted at the curre nt rate o f interest or m u ltip lie d by a suitable price-earnings ra tio , B lack (1986). As a firs t a p p ro xim a tio n , we assume The current fundam ental value to have tw o com ponents, one a price com ponent (depending on current earnings co n d itio n ) and second, a random te rm (th a t captures j \ current exogenous shocks in the fo rm o f random news, m easurem ent error and the jlik e ). W e can fo rm a lly w rite , i i ! vt = pt - 1 + r}t (3.7) i j where, rjt ~ N(0, a 2) captures the effect of any residual in fo rm a tio n th a t could affect I the curre nt fundam ental, b u t not in a system atic way and p t-i is assumed to be | the best estim ate o f the price com ponent. T h is is based on evidence fro m section ! : 2.4 th a t price levels and returns e x h ib it strong first-o rd e r correlation. As a general j ap p ro xim a tio n , the price com ponent can be approxim ated by a m oving average of i | past prices w hich w ould be consistent w ith curre nt em p irica l evidence as w ell as t ! I ! m oving average crossover strategy, discussed earlier. I Param eters g and h w ill in general vary over tim e as a-investors receive new j in fo rm a tio n about com pany’s earnings p o te n tia l. F u rth e r, they m ay vary across in- 'd ivid u a ls. In general, short te rm investors (whose investm ent horizon is say daily, or l 'perhaps even h o u rly ) or traders are m ore interested in curre nt incom e ra th e r than c a p ita l appreciation. Therefore, th e y are lik e ly to have sm aller thresholds com pared to lo ng-te rm investors (whose investm ent horizon is say, m onthly, q u a rte rly or even ye a rly), whose investm ent objectives revolve around ca p ita l appreciation Jand grow th. One can th in k o f m edium te rm investors as well, whose investm ent ! objectives include some am ount o f curre nt incom e and the p o te n tia l for ca p tia l ap- ! p re d a tio n . These investors are like ly to have thresholds w hich on the average w ould I be larger th a n those for traders (sh o rt-te rm investors) b u t sm aller than fo r long- j | te rm investors. Since a-investors investm ent strategy is based on the com parison o f | current price to lo ng-te rm fundam entals, it is o n ly app ro priate th a t th e ir thresholds be a fu n c tio n o f lo ng-te rm fundam entals. In p a rtic u la r, we assume the follow ing. ht = ( 1 + 7 )ut (3.8) 9t = ( 1 + 9)ut (3.9) ; where, 0 < 8 < 7 < 1 . i j For purposes o f th is section, we have assumed th a t a and b, the strength param e te rs of a and /^-investors are not tim e -va ryin g coefficients. However, in general they ' are like ly to be different across investors w ith in each type. In p a rtic u la r, if a and b can be m odelled as evolving over tim e as new in fo rm a tio n flows in and investors accum ulate experience over a period o f tim e , state-space models and tim e -va ryin g param eter models can be used to app ro xim ate the above m odel. T his com pletes the necessary tran sform ations fo r e stim a tio n to be carried out ; and now I w ill show how th is m odel can be reduced to the “ Self E x c itin g Threshold Autoregression” (S E T A R ) form . 1 I 3 .5 T r a n sfo r m e d m o d e l ' For the sim ple one a and one (3 type investor m odel, the price adjustm ent equation I 3.5 allow ing fo r variables u, v, h and g to be tim e varying can be re w ritte n as P t+ l ( 1 + Xb)pt - X[bvt - a(ht - gt)] ( 1 + X(b - a))pt - X[bvt - a(ht - gt) - a(ut - gt )] (1 + A b)pt — Xbvt (1 + X(b - a))pt - X[bvt + a(ht - gt ) + a(ut + gt)] (1 + Ab)pt - X[bvt + a(ht - gt)\ P t € I 1* P t e I 2* Pt e 13* P t € I 4* P t e / 5 * (3.10) N ow , d iv id in g bo th sides o f (3.10) by (3.6) a fte r s u b s titu tin g fo r vt, ht and gt fro m (3.7)-(3.9), we get, Qt+i — + 4> \ \ t + + e {For j — 1 ,..., 5, where, ( i), Ad t 5 P -1 ' <qt < P (3.09) < h = P tlu t 4 1 ’ - = A a ( 7 - 0 ) / ( l + £ ) ■rf2’ = ^ = A c r y / ( T - M ) = 0 = d 3) = 4 5 ) = 1 + A 6 / ( l + 6 ) d 2 ) = d 4 ) = 1 + A ( b — a ) / ( l + S ) 4 ‘ > 4 5 ) = - X b /( 1 + S ) 2 (3.12) | and I : M = 6 (5) = —(A6)?7f/( 1 + S^uo i and the new thresholds are, (3.13) I 1' j 2' I 3' I 4' I 5' [o, ( 1 - 7)] [ ( 1 - 7 ) , ( 1 - 0 ) ] [(1 - 0), (1 + 0)] [(1 + 0), (1 + 7)] [(I+ 7), 00] T h is we observe is a “ Self E x c itin g Threshold A utoregression” in the variable qt, where the thresholds have to be estim ated before e stim a tin g th e coefficients. Equa tio n (3.12) denotes the a -p rio ri cross-equation constraints suggested b y a sim ple i version o f the “ in trin s ic theory.” E q uation (3.13) suggests th a t fo r fin ite L t \ the variance o f the error te rm in each of the regimes is equal and as f lim o o , th is va ri ance approaches zero. A lth o u g h re strictive , th is result sim plifies the analysis. 36 j T he tra n sfo rm a tio n fro m pt to qt is m o tiva te d by the fo llo w in g observation. The basic m odel describes a price process. However, in e m p irica l studies in vo lvin g esti m ation, to avoid the problem of n o n sta tio n a rity, the price series is ofte n transform ed ! to produce returns, where returns are often defined as the n a tu ra l lo g a rith m o f price ] i relatives, i.e., r t — ln (p t/p t^ i). T h is tra n sfo rm a tio n not o n ly takes care o f th e trend in prices b u t also makes the transform ed series u n it free. A n a tu ra l tra n sfo rm a tio n j th a t th e “ in trin s ic th e o ry” suggests is to detrend prices using th e long te rm funda- j m entals ut w hich has an exponential tre n d b u ilt in to it. T his tra n sfo rm a tio n fu rth e r avoids the need fo r deflating, using a price index. Hence for a test o f our theory, jth e variable o f interest is qt- , not pt , where qt can be rough ly in te rp re te d as price in ; excess o f w hat is suggested by lo n g -te rm fundam entals. ! ' l | j 3.6 S u m m a ry j In this chapter, we m o tiva te d the behavioral and In s titu tio n a l th e o ry o f stock price , behavior and presented a d e te rm in istic version o f the m odel. W e reviewed some of its theoretical properties. To fa c ilita te e stim a tio n and hypothesis testing, we ! transform ed the m odel and provid ed a m o tiv a tio n fo r the tra n sfo rm a tio n . | C h a p ter 4 i i M e th o d o lo g y I i In this chapter, I w ill present the m ethodology adopted to test the “ in trin s ic th e ory.” II w ill discuss the conventional m ethodology to describing stock returns process. A i general procedure for m odelling and te stin g the class o f S E T A R models is described. . M y m ethodology also describes how the “ encompassing principle ” can be used to test , the proposed theory. Results fro m e stim a tio n and hypothesis testing are presented ! in the next chapter. 4.1 B a ck g ro u n d I , A large am ount o f em pirical in vestigatio n in the fin a n cia l lite ra tu re is concerned w ith the s ta tis tic a l process th a t governs security re tu rn s and vo la tilitie s . In p a rtic u la r, linear tim e series models have been w id e ly used to describe stock returns. Fama (1965) and G ranger and M orgenstern (1970) observed th a t da ily and m o n th ly stock returns exh ib ite d significant autocorrelations fo r lags fro m 1 to 10. However, they concluded th a t the sm all m agnitude o f serial correlations cannot be used to o b ta in significant excess returns. Consequently, th e rando m -w alk m odel has been regarded as an adequate description o f asset re tu rn s since the 1970s. R ecently French, Schwert and Staum baugh (1987) m odelled the lo g a rith m o f the m o n th ly v o la tility fo r S&P retu rns as an A R IM A (0 ,1 ,3 ) and found the fit to be adequate in describing the I v o la tility process. Schwert (1990) and Schwert and Seguin (1990) used an A R (12) to m odel the m o n th ly v o la tility process. T h e success o f these sim ple linea r models I I is m a in ly because ( 1 ) they provide a good firs t a p p ro xim a tio n to m any higher order i processes; ( 2 ) theory o f sta tis tic a l inference in the case o f linea r Gaussian models is I Jwell developed; (3) a w ell defined m odel b u ild in g k it, [see B ox and Jenkins (1976)] ! is readily available. ] Recently, several authors began stu d yin g nonlinear tim e series models to de- I i scribe the asset returns generating process. Some of the exam ples are G ranger and Anderson (1978), Engle (1982), B ollerslev (1986), H in ich (1985), Hsieh (1989) and Scheinkm an and LeBaron (1989). G ranger and Anderson em ployed a b ilin e a r pro- Icess to m odel d a ily IB M stock-price movements. Com pared w ith a linear m oving , * . . 1 (average m odel, th e y showed th a t a b ilin e a r m odel leads to a sm all b u t significant reduction in mean squared out-of-sam ple forecast error. The m ost com m only used nonlinear tim e series m odel in the recent lite ra tu re l | is the autoregressive co n d itio n a l heteroscedastic (A R C H ) m odel of Engle (1982) j land its generalization, the G A R C H m odel o f B ollerslev (1986). A retu rns process 1 generated by the follow ing equations, j l i 1 fit — a o + £t, (4-1) ! v < t | vt = a o + ^ a i ct - i + X / P jvt - 3 i (4-2) i »=i j= i j i where p > 0 and q > 0 is called a G A R C H (p ,q ) process. T he param eters satisfy ] i the conditions a 0 > 0, cq-, f3j > 0 , i = 1 , . . . , p, j = 1 , . . . , q. X t | u>t-i F{VuVt)- W here F ( /q ,i/t) is the co n d itio n a l d is trib u tio n , w ith co n d itio n a l mean fit and vari- jance vt , cot-i is th e set o f a ll in fo rm a tio n available at tim e t. i The em pirical d is trib u tio n o f such processes are heavy ta ile d com pared to the norm al d is trib u tio n . T he u n co n d itio n a l mean and variance are constant, b u t the j co n d itio n a l mean and variance are tim e dependent. C o n d itio n a l variance here is a ' measure o f the v o la tility o f returns as a fu n c tio n o f past in fo rm a tio n . Hence, past j I patterns o f the stock re tu rn m ovements m ay have p re d ictive power fo r forecasting j | I fu tu re v o la tility o f m ovem ents in returns, even though the past pattern s have no pre d ictive power for assessing w hether returns its e lf w ill go up or down. T h is depen dency o f co n d itio n a l variances on past realized variances is consistent w ith actual ■ v o la tility p a tte rn o f the stock m arket. In p a rtic u la r, a large disturbance Cf_i tends to , be followed by another large disturbance et , re su ltin g in a cluster o f high v o la tilitie s. I A R C H , 13 and G A R C H models have been successfully applied by A k g ira y (1989) to jstock returns data, and by D om ow itz (1985) and M ilh o j (1987) to foreign exchange data. French, Schwert and Staum baugh (1987) have used th e various versions of | G A R C H to exam ine the re la tio n in general, between stock returns and stock m arket v o la tility and in p a rtic u la r the re la tio n between risk p re m iu m and v o la tility . M ore recently, B rock, Lakonishok and LeB aron (1991) used technical tra d in g strategies as tests o f m isspecification where the various versions of G A R C H were used as n u ll ! models o f re tu rn generating process. A n o th e r area where these m odels have found widespread use, is in the op tio n s-p ricin g lite ra tu re . A lth o u g h G A R C H models have been successful in characterizing the n o n lin e a rity in variance or v o la tility , it is only ’ recently th a t a system atic search for the causes o f th is serial co rre la tio n has begun. ! One possible explanation for the prom inence o f G A R C H effects is the presence of ! serially correlated news a rriv a l process, as discussed by D iebo ld and Nerlove (1989) and G allant, Hsieh, and Tauchen (1989). E m p iric a l support fo r th is hypothesis has ibeen produced by Engle, Ito and L in (1990). In a related context B ollerslev and .D o m o w itz (1991) have shown how the actual m arket m echanisms m ay themselves ' I j result in very different tem p ora l dependence in the v o la tility o f transactions prices, j J f ! such as a p a rtic u la r autom ated trad e execution system in d u cin g a very high degree i | o f persistence in the variance process. j The “ in trin s ic th e o ry” has a n a tu ra l e xplana tion fo r th is v o la tility clustering. Let i u s say, a investors dom inate the m arket. Supposing, th e y are led to believe th a t j : in the fu tu re m arket conditions are lik e ly to worsen, th e y w ill revise th e ir estim ate i ; o f ut w hich w ill affect th e ir tra d in g bound and thresholds. So when m arket price i j reaches the tra d in g threshold, they are lik e ly to unload a large q u a n tity o f stocks in the m arket w hich w ill b rin g down the price. Since /? investors are very sensitive j to current price, they are lik e ly to fo llo w the tre n d w hich w ill cause a fu rth e r drop i in price. T h is w ill lead to a clusterin g o f negative retu rns. T he same analysis can be carried out, when a investors expect th e m arket to do b e tte r in th e fu tu re . The I I idea o f “ feedback” is the d riv in g m echanism behind th is phenom ena as explained | , by our theory. , _________________________________ l ' 13N ote: W hen q — 0 in equations 4.1 and 4.2, we have a sim ple A R C H (l) process. T he basic problem w ith the tim e-series approach discussed above is th a t stock retu rns are assumed to be determ ined endogenously in general e q u ilib riu m . H ow ever, one cannot explain the behavior o f stock retu rns in econom ic term s by app lying 'a s ta tis tic a l m odel d ire c tly to retu rns. A n econom ic expla n a tio n requires th a t the I s ta tis tic a l m odel be applied to exogenous variables, w ith the behavior o f stock re tu rn s em erging fro m the solution of an econom ic m odel. We believe our basic m odel 'provides th is o p p o rtu n ity to understand the behavior o f stock retu rns by looking at th e w ay th e m arket behaves. T h e tran sform ed m odel presented in the previous chapter, is a piecewise linear m odel o f the self e xcitin g thre shold autoregression type, Tong (1977). 1 4.2 S elf e x c itin g T A R m o d e l i ! I j T he idea of using piecewise linear models system atically, to m odel discrete tim e ; [series data was firs t m entioned in Tong (1977) and la te r in Tong (1983). A com pre- ’ hensive tre a tm e n t o f various threshold autoregressive (T A R ) models and num erous | a p p lica tio n o f these models in diverse fields is available in Tong and L im (1980). A p a rtic u la r k in d o f T A R m odel called th e self-exciting T A R (S E T A R ), [Tong (1980); Tsay (1987)] has gained p o p u la rity in recent years. j * ; i i ! 4 .2 .1 A gen eral m o d el i i i A general tim e series Yt is a self-exciting threshold autoregressive process, if it follows 1 the m odel j y, = * .« ’ + r,-_, < Y , - i < Tj , (4.3) I j i— 1 where j — 1 , . . . , k and d is a p ositive integer. T he thresholds are - oo = r 0 < rq . . . < rjfc = oo; for each j , is a sequence o f in depe nden tly and id e n tic a lly d is trib u te d ! norm al random variable w ith mean zero and variance <r| such th a t { 4 ^ } and { 4 ^ } I are independent, if i ^ j. I t j Com pared to regim e-sw itching linea r regression m odels, [see Q uandt (1978)], | where the regimes are controlled by the tim e index t, the S E T A R m odel in 4.3 is a piecewise linear m odel in the space Yt-d , where the regimes are controlled by the 41 j threshold variable Yt-d■ The overall process is non linear when there are at least Awo regimes w ith different linear models in each regim e. T his nonlinear tim e series m odel was firs t proposed by Tong (1977) and Tong and L im (1980) as an a lte rn a tive | m odel for describing periodic tim e series. Some o f the features o f S E T A R models j include tim e irre v e rs ib ility , asym m etric lim it cycle and ju m p phenomena. Nevertheless, these models have not been w id e ly used in applications, p rim a rily I because (a) in practice, it is hard to id e n tify the threshold variable and estim ate the associated threshold values, and (b) the m odelling procedure is usua lly q uite : com plicated and ad-hoc. R ecently T say (1989) has proposed a procedure fo r testing I threshold n o n lin e a rity in data and b u ild in g , if necessary, a S E T A R m odel. O ur I (m ethodology combines the S E T A R m odelling procedure and bo o tstra p m ethod (to I be described la te r) to make inferences about the v a lid ity o f our theory. In the next section we describe the m odelling procedure fo r a general S E T A R m odel. I 4.3 S E T A R m e th o d o lo g y i In app lication, there are different m ethods to b u ild T A R m odels. Here we shall use the m odelling procedure o f Tsay (1989). T he central idea o f the procedure is | to tra n sfo rm a regim e sw itching T A R m odel in to a regular change p o in t problem in linear regression analysis. T his is achieved by using the concepts o f arranged ( i autoregression and local estim ation . i i I N o rm ally, in fittin g an A R m odel we p erform a sequential estim a tio n w ith a , fixed w indow th a t controls the num ber of observations used. However, here instead j of using t, we use the m agnitude o f the threshold variable to control the data flow I in the w indow . B y so doing, we tra n sfo rm the threshold problem in to a sw itching * . . . 1 | regression problem . For illu s tra tio n , the fo llo w in g ta b le shows the usual setup o f 1 I fittin g an A R ( 1 ) m odel to 10 consecutive observations o f the T A R (2 ;1 ;1 ) m odel j ! i y f + 4 X ) if < 0.0 \ 0.51^-1 + e|2^ otherwise, where j — 1,2, are tw o independent sequences o f i.i.d N (0,1) random variates ■ and regim e L 1 = ( —oo,0.0) and L 2 = [0 .0 , oo). tim e Yt Y t-i regim e 2 -0.05 0.500 L 2 3 0.320 -0.05 Lx 4 0.660 0.320 l 2 5 1.030 0.660 l 2 6 -0 . 6 8 1.030 l 2 7 -0.79 -0.680 L i 8 0.105 -0.790 L t 9 0.503 0.105 l 2 1 0 -0 . 2 2 0.503 l 2 Such a setup cannot yie ld consistent param eter estim ates. However an arranged autoregression rearranges the setup in to tim e Yt Yt- i regim e 8 0.105 -0.79 Lx 7 -0.79 - 0 . 6 8 Lx 3 0.320 -0.05 Lx 9 0.503 0.105 d > 2 4 0.660 0.320 L 2 2 -0.05 0.500 l 2 1 0 -0 . 2 2 0.503 l 2 5 1.030 0.660 l 2 6 -0 . 6 8 1.030 l 2 ■ based on the m agnitude of the threshold variable Yt- \ . In th is case, the firs t 3 rows 'belong to regim e one o f the T A R (2 ;1 ;1 ) m odel and the rest belong to the second i _ _ | I regim e. A local estim a tio n thus can provide useful estim ates of the coefficients. In | ■ practice, the local estim a tio n can be done e fficie n tly by a recursive least squares j alg o rith m . A m a jo r d iffic u lty in b u ild in g S E T A R models is the specification o f the threshold | variable, w hich plays a crucial role in the nonlinear nature o f the m odel. For the : m odel in equation 4.3 it am ounts to the selection o f the delay param eter d w hich determ ines th e threshold variable, Yt-d■ In contrast to Tong and L im (1980) who used the A kaike in fo rm a tio n c rite rio n (A IC ) 14 to select d a fte r choosing a ll of the 14T he A IC is used as a s ta tis tic to d iscrim in ate between com peting m odels. T he decision rule is ■ to select the m odel fo r w hich the s ta tis tic a tta in s a m in im u m , A IC = —2Iog(x[)) + 2n. T he emphasis | here is on com paring the ‘goodness o f f i t ’ o f various m odels w ith an appro priate allowance made fo r parsim ony. ! i i I 43_l other param eters, we select d before lo ca tin g the threshold values. T he A R order is firs t chosen based on some m odel selection c rite rio n such as the A IC or the p a rtia l I a u tocorrela tion fu n ctio n . For a given S E T A R m odel and an A R order p, one selects an estim ate o f the delay param eter, say dp, such th a t we get th e m ost significant result in testing for threshold non lin e a rity. T h a t is, : F(p>dp) ^ v m e xs { f (p , v) } , (4.4) where F (p ,v ) is the F s ta tis tic described in C hapter 2, the subscript p denotes th a t ' the estim ate o f d m ay depend on p, and s is a set o f prespecified positive integers, th a t is a collection o f possible values o f d. N ote th a t in conventional models the choice o f dp is som ewhat heu ristic. ; Once p and d are determ ined, special care is needed in e stim a tin g th e threshold values, r j ’s. In general, one m ay provide ad-hoc, an in te rva l estim ate fo r each of the , threshold values or use sam ple percentiles as p o in t estim ates. Here we consider the 1 , e m pirical percentiles as candidates fo r the threshold values. Instead of prespecifying ja set o f fin ite num bers o f sample percentiles, we search th ro u g h the percentiles to i | locate the threshold values. ! T he graphical m ethod proposed to locate these thresholds involve looking at the ; scatterplots o f various statistics versus the specified threshold variable. T he plots i j used are (a) the scatte rplot of the p re d ictive residuals versus Yt_dp, and (b) the scatte rplot of t ratios o f recursive estim ates o f an A R coefficient versus Yt_dp. A S E T A R m odel consists of various m odel changes th a t occur at each threshold ' value Tj. Therefore, the pred ictive residuals are biased at the thre shold values. A sca tte rp lo t o f predictive residuals versus the threshold variable is lik e ly to reveal the 1 locations o f the threshold values of a S E T A R m odel. On the otherhand fo r a linear tim e series, th is p lo t is random . W e then consider the sca tte rp lo t o f the recursive A R coefficients or T statis- I tics versus the threshold variable iYt^ ( ip> • In p rin cip le , th is should look like a step | fu n c tio n w ith ju m ps in d ic a tin g the values o f th e threshold variable at w hich the ' regim e changes. In practice, however since the windows used overlap sequentially, I the plots tend to show certain sm ooth tra n s itio n fro m one regim e to another. Once the threshold values are located, we can p a rtitio n the space in to several regimes and estim ate an A R m odel w ith an app ro p ria te order in each regim e. F u rth e r, the estim ated S E T A R m odel can be refined according to some in fo rm a tio n crite rio n fu n c tio n , say the A IC . The general m odelling procedure described above can be sum m arized as follows: t Select a te n ta tiv e A R order ‘p ’ and a set of possible threshold variables. For each threshold variable Yt-dp considered, p erform the T A R -F test for threshold n o n lin e a rity as discussed in C h apter 2. Select the threshold variable based on the test results in step 2 and c rite rio n 4.4. i P erform an arranged autoregression and locate the possible threshold values J using scatter plots o f p re d ictive residuals or recursive A R coefficients versus the specified threshold variable. E stim a te th e specified S E T A R m odel by co n d itio n a l least squares. Check the estim ated S E T A R m odel (lo o kin g at residuals or squared residuals) and refine it if necessary using the p a rtia l a u to co rre la tio n fu n c tio n or A IC . j j N ote th a t the procedure requires o n ly a sortin g ro u tin e and a recursive least-squares | a lg o rith m . I f data is m issing then one can make use o f kalm an -filte rs instead. i I I 4.4 O ur m e th o d o lo g y | i « O u r m ethodology is b u ilt around the idea o f a s tru c tu ra l m odel based on funda- ! m ental econom ic assum ptions w hich, under some reasonable assum ptions can be I transform ed in to a sim ple nonlinear tim e-series m odel, whose properties are w ell , know n. A -p rio ri th eoretical restrictio ns on the fo rm e r impose constraints on th e co- | efficients o f the transform ed m odel, in the fo rm o f sign reversals w h ich are testable. I W e firs t hypothesize the behavior th a t characterizes tw o possible types o f in- , vestors in the m arket. A ra th e r sim ple a p p ro xim a tio n o f the in s titu tio n a l setup is j presented in th e fo rm of a m arket m aker w ith a price adjustm ent rule. T he in te r- ! action o f th e different investor types and the m arket m aker completes the s tru ctu re ! of the m odel. U nder app ro priate assum ptions about the way investors choose th e ir | i i thresholds, the fundam entals generating process and the m arket m aker behavior, we are able to cast the p u re ly d e te rm in istic m odel in a stochastic setting. F urther, ; we are able to show th a t it is equivalent to a nonlinear tim e-series m odel called ith e “ Self E x c itin g Threshold Autoregression” m odel. In contrast to A R C H class o f models, the S E T A R m odel is non linear in the mean. To test the adequacy of th is m odel in characterizing data, we firs t lo ok at the properties o f the transform ed i m odel. ! P roperties o f the transform ed m odel based on a -p rio ri restrictio ns o f the proposed ; theoretical m odel can be sum m arized as follows. | 1. The transform ed variable qt , w hich is a fu n c tio n o f pt and u t e xh ib its significant sh o rt-te rm p ositive correlation. j 2. The sign o f the constant coefficient o f the S E T A R m odel switches between , ' positive and negative, although not perio dica lly. j I I j 3. T he constant coefficient is inversely correlated w ith the slope coefficient, i.e., | j when th e constant coefficient is positive, the associated slope is negative and ! vice versa. 4. R estrictions (m agnitude) on the constant and slope coefficients in each regim e of the th e oretical m odel imposes re strictio n s on th e ir counterparts in the trans- ! form ed m odel, see equation 3.12. 5. As t —> ■ oo, /?-type investors re ly less on vt to determ ine th e ir dem and for i equity. W ith o u t the a d d itio n a l assum ption th a t fresh /3 typ e investors are , I * 1 entering the m arket at the beginning o f every tim e period, the above result j translates in to a purely d e te rm in istic m odel. T his can be observed by lo oking i at the erro r te rm t t in equation 3.13 whose denom inator goes to oo as t — » oo. I ■ One im p lic a tio n is th a t randomness in qt w ill have to be generated in trin sica lly. ; ; . . . . ! 1 O ur approach to testing the theory firs t involves e stim a tin g the transform ed m odel ! i using e m pirical data. E stim a tio n m ethods are as described in the previous section. ’ N ext, properties o u tlin e d above are tested. P roperties 1 , 2 and 3 are tested by ! I I graphical m ethods. P ro p e rty 4 is tested using a w ald typ e s ta tistic. In p a rtic u la r ! f r 1 coefficients o f the estim ated m odel are used to test the cross equation constraints presented in equation 3.12, using a standard % 2 test as w ell as a like lih o o d ra tio test. P ro p e rty 5 is an outcom e o f the tra n sfo rm a tio n . W h a t it says is th a t as tim e progresses, the errors in /? investors estim ate o f cu rre n t fundam entals r]t as a p ro p o rtio n o f long te rm fundam entals ut w ill become negligible. N ote, th is is not a testable proposition. However, the in flu x o f new {3 investors, is lik e ly to make the effect of random errors non-negligble. N e xt, we use a b o o tstra p m ethodology to test the adequacy o f our estim ated m odel in characterizing certa in sta tis tic a l properties th a t data exh ib its. T h is takes advantage o f th e p rin c ip le o f “ e n c o m p a s s in g [H endry and R ichard (1982); M izon j (1984)]. T he encompassing p rin c ip le provides a fo rm a l basis fo r a m odel b u ild e r j \ to analyze w hether his m odel can account fo r the salient features o f riv a l models, i J However, its role extends beyond the co m p u ta tio n o f nested or non-nested test j ! statistics; e.g., H endry and R ichard (1982, 1983) discuss its c ritic a l role in the , evaluation o f a m odel as an adequate representation o f th e data. ! The idea of encompassing as elucidated in M izon (1984, p. 135-140) is para phrased in the fo llo w in g paragraph. The encompassing p rin cip le is concerned w ith the a b ility o f a m odel to account for the behavior of others, or less am bitio usly, to explain the behavior o f relevant characteristics o f other m odels... A general m odel can always explain the behavior o f models th a t are re stricte d versions of it, and so a requirem ent th a t models encompass th e ir rivals could lead to the adoption o f the m ost general m odel feasible, given the data lim ita tions. T he p rin c ip le o f parsim ony though m ilita te s against such action. j 1 Indeed, a m ore interestin g and valuable concept is th a t o f re stricted ; m odel being able to p erform as w ell as the m odel o f w hich it is a special j ! case, once account is taken o f the difference in the degrees o f freedom . If ‘perform as w ell as’ is in te rp re te d in a hypothesis testin g sense then th is : concept o f encompassing leads d ire c tly to nested hypothesis testing. O n s the other hand if ‘perform as w ell as’ is in te rp re te d in a m odel selection sense then the encompassing p rin cip le leads to selection c rite ria such as the ad hoc m axim um Ft2 or the m in im u m Akaike In fo rm a tio n C rite ria ... One o f the a ttra c tiv e features o f the encompassing p rin c ip le is th a t it ! provides a sim ple fram ew ork fo r com paring nested and non-nested (or ' separate) m odel. ' In the context of m odel b u ild in g th e y argue th a t “ an essential characteristic of 1 e m pirical m odelling (and in fact the developm ent of th e o ry m odels) is th a t it is not a once-for-all event, b u t a process in w hich new in fo rm a tio n fro m th e o ry a n d /o r i , data leads to the m o d ifica tio n o f existing m odels.” In other words, the encompassing ; p rin c ip le is to be used as a to o l fo r progressive ra th e r th a n regressive m odelling. I f the m odeller knows the tru e data generating process (D G P ), he could concep tu a lly at least, derive the “ tru e ” co n d itio n a l densities by app ro p ria te co n d itio n in g and m arg ina liza tions and by com paring th e m w ith the m arginal densities fro m a com peting m odel analyze, w hether or not the la tte r are “ useful” app ro xim ations of I the form er for any given purpose. However, the n ature o f econom etric m o d e llin g is | such th a t the tru e process is not know n, and fu rth e rm o re , the lim ite d sam ple evi- ' dence ty p ic a lly means th a t the D G P is u n lik e ly to be discovered. Em phasis is then j on searching for useful approxim ations o f the D G P. Am ong other c rite ria , it seems t : n a tu ra l to ask w hether a specific m odel, say M 1 ? can m im ic the D G P , in th a t statis- ! tics w hich are relevant w ith in g the context of another m odel, M 2 say, behave as they 1 should were M i the D G P. T he statistics used as a basis fo r com parison need not . : be lim ite d to the “ param eters” o f M 2 as determ ined by the m odeller, fo r exam ple, ! ' the fin d in g th a t M v can p redict problem s such as residual a u to co rre la tio n , param - i eter nonconstancy or predictive fa ilu re w ith in M 2, w ould strengthen the c re d ib ility ' o f M i . T he above description fits under the category o f p aram etric encompassing (since b o th com peting models are param etrized). O u r m ethodology exploits the idea o f no n -p a ra m e tric encompassing w h ich is I explained below. W e define a m odel as encompassing a p ro p e rty o f data, if the j » ; I hypothesized em pirical m odel is capable of generating data th a t explains the cho- | sen p ro p e rty of e m pirical data as m easured by a s ta tis tic . In o th e r words, if the | n u ll m odel captures a certain sta tis tic a l p ro p e rty (as measured by a d is c rim in a tin g ! 1 I s ta tis tic ), such as peakedness or fla t ta ils , th a t w hich em pirical data e xh ib its, then I ( we say th a t the form er encompasses data, so fa r as th a t sta tis tic a l p ro p e rty is con- j cerned. A sim ple exam ple w ould help illu s tra te th is idea. I f the e m p irica l returns j d is trib u tio n is observed to be le p to k u rtic , then any econom ically m eaningful com- ; p e tin g e m p irica l m odel th a t is capable o f produ cing le p to k u rtic d is trib u tio n fo r the retu rns series is said to encompass th is p ro p e rty in data. T he p rin c ip le as applied to our m odel is described in the next section. i 1 4.4.1 B o o tstr a p m e th o d to te s t th e en co m p a ssin g p rin cip le A b o o tstra p m ethodology is used to im plem ent the n o n -p ara m etric encompassing test. T he m ethodology consists of setting up a n u ll hypothesis, w hich in o u r case w ould be th a t qt is generated according to th e m odel defined in equation 3.15 and a d is c rim in a tin g s ta tis tic such as the BD S s ta tis tic fo r i.i.d . O u r objective then is to show th a t, the qt derived fro m e m p irica l d ata differs sig n ifica n tly fro m any realization th a t w ould be generated under the n u ll hypothesis as measured by the 'B D S test. We w ant to show th a t the value of this s ta tis tic fo r the o rig in a l (e m p irica l) tim e ! , series is s ta tis tic a lly different fro m th a t obtained fo r the realizations consistent w ith (our m aintained n u ll hypothesis. W h ich in th is case is the m odel defined in equation ] 3.15. Since the value o f the d is c rim in a tin g s ta tis tic w ill vary fo r different realizations 1 under the n u ll hypothesis, we m ust re s tric t ourselves w ith the d is trib u tio n o f the j BD S s ta tis tic under the n u ll hypothesis. T h is is not know n in general. However, I | if m any re a liza tio n o f the o rig in a l d ata were available, we can compare the tw o j em pirical d is trib u tio n s , using fo r instance the K o lm ogorov-S m irnov sta tistic. Since j o n ly one re a liza tio n o f the o rig in a l d ata is available, we have adopted the fo llo w in g ; boo tstra p procedure. j ; U nder the n u ll hypothesis, we generate m any realizations of the tim e series. T he i t ^ I [ BD S test is then applied to each realization. T he mean and standard d e viation o f i the d is trib u tio n o f the BD S s ta tis tic under the n u ll is app ro xim ate d b y the sample ; mean and sam ple standard deviation . Then we define a s ta tis tic called “ significance” J w hich is the absolute value o f the difference between the BD S s ta tis tic fo r the o rig in a l series and the mean BD S value o f the generated series, divid e d by the standard j deviation fo r th e generated series. ■ s = \ B o ~ B g \ ^ ^ : aa j where, B a and B g are the BDS s ta tis tic fo r th e o rig in a l series and the mean value ! of the s ta tis tic for generated series respectively. crg is th e standard deviation o f the statistics fo r generated series. I f ‘S’ is large then we conclude th a t, the o rig in a l series was not generated by the n u ll hypothesis and vice versa. T he “ error b a r” on 5 is ! given by A S and can be com puted b y standard propo gation o f errors m ethodology. I j Here we state th a t th e absolute error bar on S can be w ritte n as, ; A S = ^ / ( l + 2S2) /N g (4.6) where, N g is the num ber of realizations for w hich th e mean and standard deviation have been calculated. A sm all value fo r the significance w ould im p ly th a t the n u ll I m odel cannot be rejected. T h is means th a t the test is inconclusive b u t the h y p o th esized m odel encompasses the data (in term s of e xp la in in g the chosen q u a lita tive p ro p e rty o f data). On the other hand a large value o f significance w ould reject the n u ll hypothesis w hich im plies th a t the proposed m odel does not encompass this f p ro p e rty o f data. In general one can have several d is c rim in a tin g statistics each of I w hich could lead to a different result. However, together th e y are like ly to present j ' a consistent p ictu re o f th e u n d e rlyin g stru ctu re in data, if the statistics are cho- I I sen carefully and if th e n u ll m odel is a reasonable a p p ro xim a tio n o f the tru e b u t I unknow n D G P. Choice o f the d is crim in a tin g s ta tis tic is ofte n m o tiva te d by the as- ■ I ■ sumed n u ll hypothesis. Note, the above m ethod is a n o n -p a ra m e tric m ethod w hich j I is a sym p to tica lly m ore pow erful th a n , a pa ra m e tric test w ith a m isspecified null. 1 4.5 S u m m a ry j j In th is chapter, we contrasted the conventional tim e-series m ethodology w ith our f \ m ethodology of deriving a tim e-series m odel fro m a th e o re tica l m odel. W e discussed a general m odelling procedure fo r b u ild in g nonlinear tim e-series models such as the S E TA R . As p a rt o f the s ta tis tic a l inference m ethodology, in a d d itio n to conventional ! 1 like lih o o d ra tio tests, we proposed a b o o tstra p m ethodology, th a t is a s y m p to tica lly j ! m ore pow erful. T he next chapter presents the data used in the analysis, discusses i i I ! the results and concludes. ■ C h ap ter 5 E stim a tio n an d In feren ce * In this chapter we present the results fro m e stim a tio n of o ur tran sform ed m odel. ! T he m ethodology described in the previous chapter is used to derive inferences ' about the estim ated m odel. Long-run properties o f the m odel are discussed, and ' conclusions o f the study are sum m arized. i ; i ,5.1 D a ta I | Raw data of m o n th ly price series fo r the value-weighted S&P com posite index was | obtained fro m th e CRSP files. C orresponding m o n th ly d ivid e n d series was gener- ; I ated fro m data on q u a rte rly dividends obtained fro m the S & P S e c u rity P ric e R e c o rd . D a ta covers th e period s ta rtin g Jan. 1926 - Dec. 1991, a to ta l of 792 i observations. Follow ing S hiller (1990), u t the investm ent value at t was calculated \ j using the constant g row th m odel. T he te rm in a l price was set equal to the value of j the index in Dec. 1991, and the variable discount rate was set equal to a one m onth T -b ill ra te .15 E stim ate o f m o n th ly fundam entals was obtained using the recurrence re latio n, 1 i ] j Ut = Ht{ut+i + A ) , t = 1 , . . . , T - 1, (5.1) ! where, = 1 / ( 1 + r t ) is the discount ra te and D t is the d ivid e n d at tim e period ! ■ t. One im p lic a tio n o f the above recurrence is th a t, as we move fu rth e r away fro m ; th e te rm in a l date, the im portance o f the te rm in a l value chosen declines. N ote th a t j i 15D a ta on m o n th ly T -b ill rates were obtained fro m U.S. governm ent files on the CRSP tapes. . Figure 5.1: E m pirical d istrib u tio n of m o n th ly qt series. all o f the series used in the analysis are in nom inal term s. T he reason, we have not deflated the data using a consumer price index as is the practice in m ost em pirical research in Finance, is because our transform ation from pt to qt not only takes care of in flation but also detrends the price series. An exponential trend was fit to the estim ated fundam entals and the value o f 6 was obtained. 5.2 P rop erties o f th e transform ed series To have a good understanding o f the nature of this transform ed variable, we applied the various tests described in C hapter 2 . Figure 5.1 shows the em pirical d istrib u tio n of the qt series. The d is trib u tio n has fat tails and is skewed to the right compared to a norm al d istrib u tio n . T his is consistent w ith our observation about the returns series in C hapter 2. It fu rth e r im plies th a t the price is below the long-term fundam entals more often than not. Table 5.1 presents the results of the Bispectra] test. Tsay's F and T A R -F test, for the period 1926-1991 as well as the sub-periods 1926-1960 and 1961-1991. The bispectral statistics H and Z for the whole period, reject the null of norm ality and lin e a rity of the qt series at conventional levels of significance, however for the subperiods, this evidence is very weak. S im ilarly. Tsay's F and T A R -F statistics for the whole period indicate the presence of significant nonlinear structure and in particu la r, threshold type nonlinearity. T he values of p and d that m axim ized the F sta tistic in 4.4 are 2 and 2 respectively. Table 5.2 shows the outcom e of the BDS test for i.i.d. The results suppori the conclusions drawn from C hapter 2 . which used r, instea.d of qt. Once again, for a scaling region equal to the standard deviation of data, the standardized statistic is 84 Table 5.1: Bispectral test, Tsay’s-F and T A R -F test for nonlinearity of the qt series T E S T 1926-1991 1926-1960 1961-1991 H Z Tsay-F T A R -F 3.61 3.15 4.45 (0.004) 6.04 (0.0005) 2.41 1.81 3.25 (0 .0 2 2 ) 2.97 (0.032) 1.72 1.27 0 . 0 1 (0.998) 1.07 (0.362) • H and Z statistics test for norm ality and linearity respectively. Asym ptotically, they are ! both distributed N(0,1). Tsay’s-F statistic was computed for p = 2 and is distributed • F(3,786). TA R -F statistic was computed for p — 2, d — 1 and is distributed as F(3,705). Numbers w ith in the parentheses denote the p-values. 1 i 's ig n ific a n tly larger th a n the c ritic a l values at 5% and 1% significance levels. L inear ! ■ p re -filte rin g using an A R (2 ) m odel does not a lte r th e q u a lita tiv e nature o f the j ' results. F in a lly F igure 5.2 presents the closereturns h istog ra m fo r the qt series. I t i j is suggestive o f significant nonlinear s tru ctu re as observed by perio d ic peaks in the j I histogram . 1 i 1 Table 5.2: BD S test fo r i.i.d . o f the qt series e/ a m, 0.50 1.00 1.50 2.00 2 4.92 5.74 7.18 8 . 0 1 3 5.85 6 . 1 0 7.56 8.62 4 7.08 6 . 8 8 7.90 8.67 5 8.57 7.60 8.62 9.30 Notes: m is the embedding dimension, e measures the size of the sphere around each point in the embedding vector and a is the standard deviation of data. 53 120 100 60 40 ia ' C V I C -vi Lag(months) Figure 5.‘ 2: Closereturns histogram of m onthly qt series. Threshold variab le Figure 5.3: P lo t o f recursive estimates of slope coefficient vs threshold variable. 1 I Figure 5.3 exhibits the property th a t we had described in C hapter 4. The graph ' shows the recursive coefficient of the first lag A R ( 1 ), plo tte d against the threshold j variable, which using the T A R -F test was determ ined to be equal to q t- 1 - The ! signs of the coefficient, switch from positive to negative b u t not in any periodic 1 I manner. However, they are significantly affected by the choice of window length. In the section on SE TAR m ethodology we had described the window length to be t the num ber of observations entering the recursive autoregression. A -p rio ri, a longer ; window results in fewer sign reversals. This imposes a certain bias on the estimates j of the coefficients in the SETAR model. In particular, when the window length is large, the slope coefficient is biased upwards and the constant coefficient is biased downwards. On the otherhand a very short window length, results in spurious ! 5 4 | regime changes. To better understand the bias imposed by choice of window length, we perform a monte-carlo study of the basic model as it appears in 3.5, for fixed parameter values. The role of random noise is also studied w ithin this context in the next section. Here, the optim al window length was chosen to reflect the requirements of our model. The window lengt h chosen was 150 m onths which roughly split up the sample into five regimes as suggested by the “ in trinsic theory.” It should be noted that, when using higher frequency data (w’eekly and d a ily), the window length should be smaller to accomodate for more frequent regime changes. We observe that frequent regime changes in terms of the intrinsic theory calls for m u ltip le investor types. This is explored in greater detail in the next section. Figure 5.4 illustrates the significant short-term autocorrelation and dim inish ing long-term autocorrelation of the qt series. A -p rio ri, the recursive coefficients exh ib it positive short-term autocorrelation and negative long-term autocorrelation, see Figure 5.5. Notice th a t even after 25 lags, the autocorrelation function does not approach the asymptote, suggesting very high correlation in the recursive estimates. However, the partial autocorrelation function is indicative of an AR(2) process for the recursive coefficient. This evidence is consistent w ith the predictions of the “ in trinsic theory.” In the next section, we estim ate the transformed model and infer the properties of the estimated model using our m ethodology described in Chapter 4. C y~ ~ ' r-~ ; r -— u T • - - - - r---- O " ,; r\: 1 I j --------------------------------ACp pACF | Figure 5.4: A u tocorrelatio n partial a u to co rre la tio n function of q: series. 5 5 A C F Figure 5.5: A utocorrelation partial autocorrelation function of slope coefficient I 5.3 R esu lts 5.3.1 E stim ation ; In the previous section we found strong evidence for the presence of threshold non- ' lin e a rity in qt. A lthough, the p value o f the threshold nonlinearity test, described in C hapter 2 was m inim ized for an A R order o f p — 2 and delay parameter d = 2, a-priori our theory suggests a value of p — 2 (num ber of lags in the transform ed model) and d = 1 (as the threshold variable). E stim a tio n proceeds w ith the values for p and d as suggested by our theory. The arranged autoregression methodology ; described in C hapter 4 is used to id e n tify the threshold values. The chosen window length is 150. As mentioned earlier, if the m odel changes, then both the recursive coefficient estimates and their t ratios are likely to change. Plots of recursive AR co efficients versus the threshold variable are shown in Figures 5.6. 5.7 and 5.S. Based ; on these and sim ilar plots of the recursive T ratios versus the threshold variable, i we tentatively identified the following 7 regimes. ( 1 ) : (0. 0.9609]: (2) : (0.9609, 0.970S]; (3) : (0.9708, 0.9792]; (4) : (0.9792, 0.9922]; (5) : (0.9922, 1.006]: (6 ) : (1.006, 1.035], and (7) : (1.035. oo]. C onditional least squares was then used to es tim ate each of these regimes. Table 5.3 presents the estim ated coefficients for each ■ of these regimes along w ith their t-ratios. The a -p rio ri cross equation restrictions from 3.12 have not been imposed for this estim ation. 5 6 Threshold variable Figure 5.6: P lot of recursive estimates o f the constant coefficient vs threshold vari able for the TA R (7;2;1) model Threshold v a ria b le Figure 5.7: P lot o f recursive estimates of the A R (1 ) coefficient vs threshold variable for the TA R (7;2;1) model 0 .5 0 i rv < - 0 .5 i r i : r < . - i v A . V ‘w — - v 'n J ' . J O u o cr. cr; I C -02 0 ^ . 0 ^ ' — o c> c r> o Threshold v a ria b le Figure 5 .S: P lot of recursive estimates of the A R (2) coefficient vs threshold variable for the T A R (7;2;1) model 1 Table 5.3: U nconstrained T A R (7 ; 2; 1)° m odel id entified by p lo ttin g A R coefficients !vs threshold variable, obtained fro m a recursive autoregression o f the qt series. Regime °3 N X2( W 1 0.774 0.459 -0.237 0.0069 165 1 . 8 6 (7.18*) (4.26*) (-3.94*) 2 2.693 -1.812 0.039 0.0028 99 9.21 (2.23*) (-1.45) (0.35) 3 2.678 -2.132 0.397 0.0037 83 7.93 (1.49) (-1.14) (3.23*) 4 0.369 0.684 -0.045 0.0023 115 1.78 (0.31) (0.58) (-0.40) 5 2.574 -1.546 -0.032 0 . 0 0 2 1 1 1 1 14.61 (2 .2 1 *) (-1.32) (-0.27) 6 0.504 0.618 -0.141 0.0016 137 5.77 (1.17) (1.49) (-1.73) 7 0.685 0.292 0.013 0.0041 89 6 . 2 2 (4.39*) (2.31*) ( 0 . 1 2 ) Notes: * denotes t statistics are significant at the 5where p is the AR order, d is the delay | variable and k denotes the number of regimes. | Com pared to th e ir standard errors, regimes 2 , 4 and 6 are m ore or less w hite j noise. T his suggests th a t the lagged variables are not significant in d eterm ining the mean value in these regimes. However, regimes 1 and 7 e x h ib it strong firs t order I autocorrelation. In regim e 3, the second lag is very significant suggesting a delayed I influence on the mean. One possible explana tion w ould be th a t, the fu rth e r away price is fro m lo ng-te rm fundam entals (since qt measures price as a p ro p o rtio n of lo ng-te rm fundam entals), stronger is the effect of past qt and when price is closer to fundam entals, qt behaves m ore or less like w h ite noise. To ensure th a t th is was not a result o f n o n sta tio n a rity, we tested fo r u n it roots in each o f these regimes and were able to reject the n u ll o f u n it roo t at conventional levels o f significance. Table 5.4 j presents the results o f the D ickey F uller (1980) test for u n it ro o t n o n -sta tio n a rity. j In a d d itio n , the a u tocorrela tion fu n c tio n of squared residuals in each regim e was j I . . . I ' used to check fo r m odel adequacy. R esidual heteroskedasticity and hence m odel | ! inadequcy can also be measured by the M cLeod and L i (1980) test, w hich is a y 2 i i ' s ta tis tic , often p rin te d out by standard tim e-series software. T his x 2 s ta tis tic is ! [shown in Table 5.3. Low values fo r the x 2 s ta tis tic suggest th a t the fitte d residuals ! jdo not e x h ib it any significant heteroskedasticity. Note, however th a t the variance ,in each o f these regimes is different. In p a rtic u la r, in extrem e regimes (1 and 7) the Variance is m uch higher th a n in the m id d le regimes. One possible in te rp re ta tio n 'w ould be th a t when price is su fficie n tly fa r away fro m lo n g -te rm fundam entals, m arket correction w ould require price to vary su fficie n tly tow ards fundam entals. T h is is also accom panied by a higher volum e o f a c tiv ity . T h is observation raises an i interestin g question, about the role o f A R C H type models th a t are used to capture i [the persistence and change in v o la tility . j O n the one hand, we have A R C H or G A R C H typ e s ta tis tic a l models th a t are s till |inadequate (as measured by the BD S test) in characterizing all o f the n o n lin e a rity |present in data, on the other hand, an a -p rio ri m odel o f stock price, one th a t is | |nonlinear in the mean, suggests a changing variance. Ideally, one w o u ld w ant a j m odel th a t is not only nonlinear in the mean b u t also nonlinear in the variance to | [capture th e persistence and change in v o la tility . In the next section we discuss some ! ! extensions of our basic m odel th a t can help explain the changing v o la tility . 1 To adhere m ore closely w ith our hypothesized m odel, we decided to refine the 'p a rtitio n s of the threshold variable. T he c rite ria used to refine the p a rtitio n was th e value of the log-likelihood. Table 5.5 shows th e results o f e stim a tio n given the : 'fo llo w in g p a rtitio n s , (1) : (0,0.9609]; (2) : (0.9609,0.9792]; (3) : (0.9792,1.0060]; (4) i (1.0060,1.035], and (5) : (1.035,oo]. T he difference here is in regimes 2 and 3. ! Intervals 2 and 3, and 3 and 4 fro m the previous analysis were m erged to o b ta in ! regimes 2 and 3 respectively. i As seen in Table 5.5, the coefficients in the extrem e regimes are once again sig n ific a n t, whereas those in the m idd le regimes are m ore or less w h ite noise. Note, we s till haven’t im posed the cross equation restrictio ns o f 3.12. Coefficients o f the ex- ' trem e regimes are suggestive o f the m odel being asym m etric, i.e., we w o uld have tw o ■q’s and tw o 0 ’s describing th e tra d in g bounds and tra d in g thresholds respectively. I For exam ple, ~ 0.04, and 01 ~ 0.02, translates in to a 4% tra d in g bound and 2% | tra d in g threshold when current price is below lo n g -te rm fundam entals. Likewise, 7 2 ~ 0.006, and 02 — 0.035, translates in to a 0.6% tra d in g threshold and 3.5% tra d in g bound when current price is above lo n g -te rm fundam entals. T h is asym m etry : can be explained by how a investors vary th e ir tra d in g thresholds based on w hether I m arket is b u llis h or bearish, a investors perceive th a t if th e m arke t has undervalued , Table 5.4: D ickey fu lle r and Augm ented D ickey F u lle r test fo r u n it-ro o t nonstation- a rity o f the residual series obtained fro m the estim ated T A R (7 ;2 ;1 ) m odel Regime N D.F. A .D .F 1 164 -6.78 -7.43 [2] (0.004) ( 0 .0 0 2 ) 2 98 -7.73 -7.92 [3] (0 .0 0 1 ) (0.0008) 3 82 -7.34 -8.39 [3] (0.003) ( 0 .0 0 0 2 ) 4 114 -7.02 -7.86 [4] (0.007) (0.0007) 5 1 1 0 -8.30 -8.33 [2] (0.0003) ( 0 .0 0 0 2 ) 6 136 -7.63 -8.10 [3J ( 0 .0 0 2 ) (0.0009) 7 8 8 -7.40 -7.68 [3] ( 0 .0 0 2 ) (0.006) 'The Dickey-Fuller test is based on the t statistic of the coefficient of lagged residual in a regression of, it — it - 1 on it - i- Where, i t is the estimated residual from fittin g a |TAR(7;2;1). The null hypothesis of unit root implies th a t the coefficient of lagged residual ! equals zero. The Augmented Dickey-Fuller test is based on the t statistic of the coefficient jof lagged residual, where the basic regression is augmented by including lagged differences !of residuals on the right hand side. The number of lagged differences is denoted by the number in square brackets. Numbers in parentheses are the p values of these coefficients obtained from M acKinnon’s (1991) tables. i I The stock, then the discrepancy m ust be sufficie n tly large fo r it to enter the m arket (sufficiently b u llis h ). On th e other hand, the perception th a t the m arket is not lik e ly to overvalue a stock fo r very long, results in a sm aller tra d in g threshold. T he tra d in g bound on th e other hand is ro u g h ly p ro p o rtio n a l on e ithe r end of the m ar ket. T his suggests th a t assum ing sym m e try in our basic m odel m ay be appealing in term s of de rivin g th e oretical properties b u t is inconsistent w ith data. Adequacy o f the estim ated m odel was checked b y lo okin g at the a u tocorrela tion fu n ctio n o f the squared residuals and the re su ltin g \ 2 s ta tis tic . F urther, to check ' the properties o f each o f these regimes, we looked at the roots o f the characteristic polyn o m ia l. Table 5.6 presents the roots in each o f these regimes. C le a rly a ll roots jlie outside the u n it circle suggesting the s ta tio n a rity o f these regimes. F igure 5.9 ; r [Table jthe qt 5.5: U nconstrained T A R (5;2 ;1 ) m odel after re fin in g the p a rtitio n spanned by series Regime <U N x 2( W 1 0.774 0.459 -0.237 0.0069 165 1 . 8 6 (7.18*) (4.26*) (-3.94*) 2 0.511 0.283 0 . 2 0 2 0.0014 172 9.54 (0.94) (0.51) (2.46*) 3 1.16 - 0 . 1 1 2 -0.050 0 . 0 0 2 0 225 2.94 (2 .8 8 *) (-0.28) (-0.63) 4 0.505 0.619 -0.141 0.0016 137 5.77 (1.17) (1.49) (-1.73) 5 0.685 0.292 0.013 0.0040 89 6 . 2 2 (4.39*) (2.31*) ( 0 . 1 2 ) [Notes: * denotes t statistics are significant at the 5% level. -shows one stochastic sim ulation o f the q* series based on th e estim ated coefficients .w hile F igure 5.10 shows the actual qt series. Table 5.6: Roots o f the characteristic p o lyn o m ia l o f the unconstrained T A R (5;2;1) m odel Regime Roots Modulus Period 1 0.229±0.723i 0.236 5.83 2 -0.33 0.33 0.61 0.61 3 -0.05±0.22i 0.05 4.67 4 0.31±0.21i 0.14 10.46 5 0.055 0.055 0.237 0.237 . In Table 5.7, the cross equation constraints o f 3.12 are im posed. I t is im p o rta n t [to note th a t sym m etry o f constant coefficients has not been im posed, since results i 'fro m th e previous e stim a tio n are n o t supp o rtive o f th is hypothesis. A lth o u g h all coefficients are significant a t the 5% level, the n u ll hypothesis th a t our a-p riori restrictio ns are correct is rejected using the likelihoo d ra tio s ta tis tic and the con ventional F -s ta tis tic at any level o f significance. T he like lih o o d ra tio test yields a Table 5.7: Constrained TAR(5;2;1) model for the q t series Regime A j - ~ ) < ? o o\j) 4 ]) 1 0.35 1.700 -0.721 (2 .8 8 *) (91.61*) (-47.69*) 2 0.065 1.716 -0.721 (2 .8 8 *) (108.55*) (-47.69*) 3 0 1.700 -0.721 (91.61*) (-47.69*) 4 -0.563 1.716 -0.721 (-2 .8 8 *) (108.55*) (-47.69*) • 5 -0.468 1.700 -0.721 (-2 .8 S*) (108.55*) (-47.69*) N otes: * denotes t s ta tis tic s are s ig n ific a n t a t th e 5% leve l. * * denotes x l 0 ~ 3. T h e e xtre m e regim es were n o t co n stra in e d fo r s y m m e try , since th e re was no in d ic a tio n o f such fro m th e p re vio u s e s tim a tio n . value of 877.57, and the F test results in a value of 147.70. It im plies th a t the con strained model is very different from the one estim ated from data. In other words, even after allow ing for asym metry, the resulting cross equation restrictions are not valid. As m entioned in C hapter 3, not all of the parameters are estim able. In particular A the price adjustm ent coefficient is not estim able because it is not identified in 3.10. However, we can get an estim ate of a jb which provides a measure of the C3 -c > 0.9 Figure 5.9: T im e series of sim ulated series. q“ \ T h e first 100 observations were dropped. 6 2 1 1.2 0.9 0.6 t Figure 5.10: T im e series of m onthly qt series. i ‘ I ; | 1 relative strength of a over /3-investors demand. T his value equals 0.0171. W hich j im plies that for m onthly returns {qt's), /? investors on the average dom inate the m arket. A ll of the above analysis assumed th a t the SETAR. m ethodology used to I identify the thresholds and estim ate the different regimes was robust to the choice I : of window length, or the num ber of observations entering the recursion. To verify this assumption a m onte-carlo study was undertaken and is the subject of the next J section. I I 5.3.2 M onte-carlo stu d y o f th e S E T A R m eth od ology j i i The crucial components of the SE TAR m ethodology are the order of autoregression ; ! ‘p ’, the delay param eter td'. Once d is determ ined, then the threshold variable yt-d is « I ■ identified. E stim ation of threshold values then proceeds w ith the help of recursive I autoregression. A lthough in general the window length chosen for this recursion j I does not affect the estimates of the threshold values, in certain models, it is likely to j bias the coefficients estim ated using these identified thresholds. The above premise is the m otivation behind this m onte-carlo study. ; 63 : ... J We chose the follow ing m odel fro m Gu (1992), where the param eters in clu d ing the thresholds were fixed at know n values. The m odel has the fo llo w in g three I I regimes P t + i 7.31649 - 4.696593p, , pt < 1.3 -0.8 0495 3 + 0.536635pf , 1.3 < pt < 1.7 (5.2) 9.773289 - 4.696593p, , Pt > 1.7 T he study consists o f three different experim ents. In the firs t one, d e te rm in istic jd ata is generated fro m the above m odel. T he A R order and the delay param eter are ad-hoc fixed at 1. N ext we choose an a rb itra y w indow length to id e n tify the j threshold values. E s tim a tio n is then carried o u t using co n d itio n a l least squares. T his : w ould be referred to as E xp erim e n t 1 . T he results o f w hich are com pared w ith the ; benchm ark case, w hich is the o rig in a l m odel. W e then vary the w indo w length and I ; reestim ate the threshold values and the coefficients in each regim e. T he m agnitudes o f the coefficients are com pared w ith those obtained fro m E xp e rim e n t 1 and the benchm ark case, to id e n tify any inherent bias in the estim ates o f th e coefficients due to the choice o f a certa in w indo w length. T his is id e n tifie d as E xp e rim e n t 2. F in a lly, Gaussian noise equal to 5% and 25% o f the standard d e via tio n in d ata is added to the d e te rm in istic data to generate a noisy signal. T his is referred to as E xp erim e nt 3. The effect o f changing w indow length is once again studied w ith this noisy data. A n in itia l price po = 1.53 was chosen. F irs t 1100 observations were generated using th is seed and the firs t 1 0 0 observations were dropped to remove s ta rtin g value , bias. T he rem aining 1000 observations were used in the analysis. In the case of ; E xp erim e nt three, Gaussian noise was added to th is data. Results o f th is stu d y are 'presented in Tables 5.8-5.11. j Table 5.8 shows the benchm ark case, w hich is the o rig in a l m odel. Table 5.9 j e xhibits the results fo r E xp erim e n t 1 w ith a fixed w indow length o f 50. For w indow ! lengths 100 and 150, the coefficients of the extrem e regimes are consistent w ith the j benchm ark m odel and E xp erim e n t 1. However, the coefficients o f the m id d le regime j are biased upwards com pared to the benchm ark m odel. T h is is tru e , even when the 'e stim ated thresholds are very close to the benchm ark values. One reason could be 1 Table 5.8: Monte-carlo study of the SETAR methodology:Benchmark Case j threshold 1 7.31649 -4.696593 [0, 1.3) 2 -0.80495 0.536635 [1.3, 1.7) 3 9.773289 -4.696593 [1.7,oo) 'N o te s : 4 > q ^ denotes th e c o n s ta n t c o e ffic ie n t a n d (j>^ s ig n ifie s th e slope c o e ffic ie n t o f th e l b e n c h m a rk m o d e l. Table 5.9: M onte-carlo study of the S E T A R m ethodology for variable w indow length W A 2) 4 3 ) 4 3 ) thr. 50 1 0 0 150 2 0 0 7.3159 7.3161 7.3162 0.1145 -4.6969 -4.6972 -4.6970 0.9361 -0.6681 -0 . 6 6 8 8 -0.6682 * 1.4454 1.4462 1.4457 * 9.7729 9.7742 9.7730 9.7729 -4.6963 -4.6951 -4.6966 -4.6963 1.301, 1.701 1.302, 1.701 1.300, 1.701 *, 1.701 N o te s : D e te rm in is tic d a ta w as g e n e ra te d fro m th e b e n c h m a rk m o d e l. W denotes th e : w in d o w le n g th a n d th r , th e e s tim a te d th re s h o ld s u s in g th e S E T A R m e th o d o lo g y . * de n o te s, w e c o u ld n o t e s tim a te these c o e ffic ie n ts because th e th re s h o ld va lu e c o u ld n o t be j id e n tifie d . j I i . th a t, because tw o th ird s o f the observations fa ll in the m id d le regim e th e dispersion i ■ w ith in th is in te rva l is very narrow , w h ich leads to an upw ard bias. W hen the w indow length exceeds 150, we are unable to id e n tify the firs t regim e, since there are only 140 observations in th is regime. T h is results in regim e one and tw o being com bined and consequently the constant coefficient is severly biased downwards w h ile the 'slope coefficient is biased upw ard. Table 5.10 presents the results fo r E xp e rim e n t 3 w ith 5% a d d itiv e noise. Once again, coefficients of the extrem e regimes are fa irly close to the benchm ark case fo r w indo w lengths 50, 100 and 150. However, in the m iddle regim e bo th coefficients are sig n ifica n tly biased upwards. N ote, th e estim ated thresholds vary quite su b sta n tia lly fro m the benchm ark case for w indow le ngth 1 0 0 . i I i W hen w indow length was increased to 200, we c o u ld n ’t id e n tify the firs t regim e and j hence th e results cannot be com pared w ith the benchm ark m odel. However, it i (Table 5.10: M onte-carlo study o f the S E T A R m ethodology fo r variable w indow le n g th w ith 5% noise * I w <t>i] 4 2) 4 > { ? 4 3) thr. 50 1 0 0 150 2 0 0 7.0497 7.0971 6.9613 0.1074 -4.4882 -4.5273 -4.4171 0.9416 -0.6481 -0.5706 -0.6193 * 1.4324 1.3811 1.4117 * 9.0972 8.9238 8.7812 8.6587 -4.3071 -4.2089 -4.1288 -4.0593 1.295, 1.700 1.280, 1.712 1.301, 1.720 *, 1.724 Notes: 5% of Gaussian noise was added to data generated from the benchmark model. W denotes the window length and th r , the estimated thresholds using the SETAR m ethodol ogy. * denotes, we could not estimate these coefficients because the threshold value could not be identified. i I I I Table 5.11: M onte-carlo study o f the S E T A R m ethodology fo r variable w indow | length w ith 25% noise W 4 3) 4 > { ? thr. 50 1 0 0 150 2 0 0 4.0734 4.1471 4.1335 0.3068 -2.1487 -2.2073 -2.1967 0.8041 -0.4831 -0.5096 -0.5359 * 1.3214 1.3401 1.3587 * 4.9332 4.7438 4.5438 4.5387 -1.9027 -1.7948 -1.6828 -1.6778 1.312, 1.693 1.305, 1.680 1.308, 1.661 *, 1.659 ; Notes: 25% of Gaussian noise was added to data generated from the benchmark model. W i .denotes the window length and thr, the estimated thresholds using the SETAR methodol- |ogy. * denotes, we could not estimate these coefficients because the threshold value could | jnot be identified. j I signifies the im portance o f choosing an o p tim a l w indow length, if one is interested in ca pturin g a ll th e regim e changes th a t data exhibits. In Table 5.11 the am ount o f noise added was set at 25% o f the standard devia- j tio n in data. I t resulted in estim ates o f thresholds th a t were considerably different fro m the previous tw o Experim ents. T he associated coefficients are also very differ- 1 ent. For instance, in the extrem e regimes, the constant coefficient is severly biased j j downwards w h ile the slope coefficients are biased upwards. In the m id d le regim e ; however, the results are consistent w ith those fro m the previous tw o E xperim ents for w indow lengths 50, 100 and 150. Once again, having a large w indow leads to jregim e id e n tific a tio n problem s. In sum m ary, th is study illu stra te s the inherent bias in the S E T A R m ethodology to noise as w ell as th e w indow length chosen. W e were i able to show th a t when the noise level is low, there is a range o f w indo w lengths fo r w hich, estim ates o f coefficients are robust. However, when a single regim e has too m any observations, the fitte d m odel fo r th a t regim e o n ly describes an average relatio nship and hence the coefficients are sig n ifica n tly biased. In general, when the noise level is high a n d /o r when the w indow length is sm all, it becomes very d iffic u lt to id e n tify the thresholds and hence the regimes. T h is is m anifested in the in s ta b ility of the estim ates or bias in the coefficients. ' F in a lly , we applied the b o o tstra p m ethodology to verify, if the estim ated m odel (in Table 5.7) is capable o f q u a lita tiv e ly characterizing observed data. M ore pre cisely, estim ated coefficients are used to generate several sets o f data, upon w hich ,the BD S test was applied. T he significance test described in C h apter 4 was em- j ployed to make inferences about the adequacy o f the n u ll m odel in characterizing 1 j q u a lita tiv e ly the observed dependence in the m o n th ly qt series. U sing a sam ple size o f 1000 observations and 500 ite ra tio n s, fo r em bedding dim ensions m — 2, 3 ,4 , 5 and e/a = 1.0, the significance values are as shown in Table 5.12. I t is clear th a t the q u a lita tiv e properties o f the re stricte d m odel as measured b y the BDS s ta tis tic are s ig n ifica n tly different fro m th a t observed in data. I 1 Table 5.12: B o otstrapped B D S test o f the C onstrained T A R (5 ;2 ;1 ) m odel m A c tu a l Sim. M ean Sim. S.D. S 6S 2 5.74 1.34 0.89 4.94 0.159 3 6 . 1 0 1.48 0.97 4.76 0.154 4 6 . 8 8 1.62 1.08 4.87 0.157 5 7.60 1.85 1.17 4.91 0.158 [Notes: m is the embedding dimension. A ctual stands for the BDS statistics of the em- 'p irica l qt series. Sim. Mean and Sim. S.D. stand for the simulated mean and standard deviation of the BDS statistics for bootstrapped data. S is the significance level discussed in Chapter 4 and 6S is the error bar of significance. i i T h is is consistent w ith our previous tests, i.e., the like lih o o d ra tio and the F test. I Results fro m the bootstrap m ethodology and conventional p a ra m e tric tests are not j supp ortive o f the a -p rio ri re strictio n s o f o ur theoretical m odel. However, certain j q u a lita tiv e features (such as sign reversals and s h o rt-te rm vs lo n g -te rm corre lation) ;of the estim ated m odel stron gly support the predictions o f the “ in trin s ic theory.” I The next section develops some m o tiv a tio n fo r extending the basic tw o investor ;m odel. Specifically, the idea th a t a m ore general m odel w ould be required to validate the “ in trin s ic theory,” is tested in the next section. F irs t, we derive a general m u lti-in ve sto r m odel. W e then estim ate the m odel using d a ily d ata and draw some p re lim in a ry conclusions about the usefulness of a m u lti-in ve sto r m odel. Conclusions I jo f this stu dy are sum m arized in the last section. j •5.4 E x te n s io n s o f th e b a sic m o d e l I i j Suggestions fo r extensions o f the basic tw o-inve sto r m odel are based on the q u a lita - | I 1 tiv e perform ance and realism o f the u n d e rlyin g assum ptions of the proposed m odel. | |In p a rtic u la r, we can id e n tify at least three different areas in w hich the m odel can ;be extended. ! , One, the basic m odel assumes th a t the spectrum o f investor types can be b ro ad ly j classified in to tw o categories, a and f3. In re a lity however there are a coun tably in fin ite num ber o f investor types (based on strategies) and hence, to be m ore re a listic |we w ould have to generalize it to a m u ltip le investor setting. T his w ill change the flavor o f the results considerably. E a rlie r, we had observed th a t the estim ated m odel ;T A R (5;2;1) was alm ost w h ite noise in th e m iddle regim e. We believe th a t in th a t (range, there are different investor types who are entering and e x itin g the m arket, 1 . . * w hich is not being captured by our basic m odel. A d d in g m ore investor types, w ill | ■ present a clearer p ic tu re o f the price dynam ics in th a t range. A lth o u g h , it could j ! com plicate the analysis in the fo rm of increased num ber of tra d in g thresholds and { tra d in g bounds (m ore param eters to estim ate). In th e next section we shall lo ok at a | general m odel w ith m u ltip le investors and provid e a m o tiv a tio n for th e ir strategies. ; ! P re lim in a ry w ork using sim ulations o f a three investor (2 a -typ e and 1 /3-type) I m odel provided support for the observed s tru ctu re in d a ily data. In p a rtic u la r, we I | were able to generate the k in d of n o n lin e a rity observed in d a ily data as measured I by the BD S s ta tis tic . T his means, the degrees o f freedom problem observed earlier j in a m u lti-in ve sto r m odel can be overcome by using higher frequency data. T w o, we had assumed th a t on ly vt is buffeted b y random shocks and hence only ^ -ty p e investors introduce certa in randomness in the m odel. In reality, u t is also affected by random news flow , although not on a d a ily basis. T his however has im p lica tio n s fo r our transform ed m odel. In p a rtic u la r, allow ing o n ly vt to flu ctu a te .resulted in a very unique b u t u n re a listic s itu a tio n where in the absence of fresh 1/5-types entering the m arket on a regular basis, as t — > ■ oo, the /5 investors strategies ;became determ inistic. Since a investors strategy had already been assumed to be d e te rm in istic (because ut was d e te rm in istic), a ll random flu ctu a tio n s in this m odel had to be generated in trin s ic a lly . Now, if we allow u t to be affected by random shocks as well, th e thresholds w ill become stochastic. Standard estim ation j jprocedures w ill no longer be applicable to estim ate S E T A R models w ith stochastic jthresholds. However, some pseudo-m axim im um like lih o o d typ e e stim a tio n m ethod 'can be used to jo in tly estim ate th e thresholds as well. i F in a lly, we had assumed u t to be the ex-post fundam entals and vt was chosen as i some ad-hoc linear fu n c tio n o f past prices. It w ould be m ore in fo rm a tiv e to a ctually estim ate ut based on exogenous in fo rm a tio n . In p a rtic u la r if ut is th e investm ent value, as in terpreted by D ay and Huang, then a investors w o uld be em ploying enor- j jm ous quantities o f data and q u a lita tiv e in fo rm a tio n to generate estim ates o f u t. For 1 |practical applications however, the in fo rm a tio n set th a t is p u b lic ly available could j ;be constrained to in fo rm a tio n on trends o f aggregate econom ic variables, in d u s try j [aggregates, in d iv id u a l com pany perform ances and the like. Neverthless, it w ould ; be interesting to test the in trin s ic theory under the assum ption th a t u t is estim ated j based on current in fo rm a tio n on key fin a n cia l and economic variables in contrast to ra tio n a l expectations based ex-post fundam entals, used in the previous analysis. M o difica tions can also be m ade to the assum ption regarding th e fu n c tio n a l form o f vt w hich could accom odate longer lags in the S E T A R m odel ( if a m oving average !of prices is used) or sm oother versions o f the same, if the fu n c tio n involves q u a d ratic j i _ , term s. T his can be m otiva te d based on curre nt evidence on some m acroeconom ic j data th a t is shown to e x h ib it cycles o f grow th and recession where the tra n s itio n is ■ sm ooth ra th e r th a n discrete (as in the S E T A R case). T he above suggestions are by no means com prehensive b u t in d ica tive o f the . | p o te n tia l for fu rth e r research in th is area and specifically the fle x ib ility o f the basic ; m odel. In the next section, we address the issue o f generalizing the basic m odel in 1 |term s o f increasing the num ber o f investor types and te stin g its va lid ity. I 1 ;5.4.1 F ram ew ork for a m u ltip le in vesto r m o d el In a ll o f the previous analysis, we had assumed one a and one /3 type investor fo r sim plicity. In re a lity there are several different types o f investors and m ore im p o r ta n tly w ith in each type (based on strategy), investors have different tra d in g bounds and tra d in g thresholds. To capture th is general m odel in com pact m ath em a tical no ta tio n is fa r m ore easier th a n e m p irica lly estim a tin g and va lid a tin g the u n d e rlyin g “ in trin s ic theory.” In the next section we m o tiva te a general m odel o f our theory. '5 .4 .1 .1 M u lt ip le a -in v e s to rs { ; F irs t we w ill consider increasing o n ly the num ber o f a-types. Suppose, there are '<na classes o f cc-investors, each w ith different tra d in g threshold. Define the fo llo w ing 'intervals: ] i I" := [0, u t - h™ } J° := [ut,u t + gi] J P := [ut - h3 t ,u t - g 3 t ] Jj : = [u t + g3 t ,u t + hJ t ] (5.3) I I 1° := [ a t - 9 t , u t ] Jn := [ut + fc?,oo] I where, 0 < g3 t < h3 t , j = 1,..., IV — 1. F u rth e r assume u t and vt are as given in 3.6 | and 3.7. In a d d itio n , the tra d in g thre shold g\ and tra d in g bound h\ are obtained by j indexing 3.8 and 3.9. G iven 3.1-3.4, 3.6-3.9 and 5.3, th e price adjustm ent equation j j becomes = < i ( 1 + Xb)pt - X[bvt - A J t ] (1 + A( 6 - a3))pt - X[bvt - A \ - a,j(ut - gJ t )} ( 1 + Xb)pt - Xbvt ( 1 + X(b - dj))pt - A[bvt + A J t + a.j(ut + g3 t )] . (1 + Xb)pt — X[bvt + A{] , P t e l n , Pt € P , P t £ l ° U J 0 , pt £ J 1 , P t G J " (5 .4 ) i where i I M ~ 9 \), J = 2 = 1 i n [ 1 ... T o ! land A$ denotes the a investors aggregate dem and or supply in period t, i | h\ := ( 1 + 7 3)u t i i and gJ t := ( 1 + 0r )ut denote the j th a investors tra d in g bound and tra d in g threshold respectively. Note, we s till have o n ly one j3 type investor and so the strength param eter b is n o t indexed. To tran sform 5.3 in to a S E T A R m odel, d iv id in g b o th sides o f 5.3 by 3.6 and 's u b s titu tin g for h°t and g%, we get, I | Q t+ 1 = 4>o^ + 4 ^ Qt + 4 ^ Q t - i + ^ 1 ^ Qt S. T (5-5) fo r j = 1,..., N a , where i | Qt = P t / u t ; 4 1 } = - 4 $ = A E L i a A i 3 - Q3) l l + s ; < 4 m) = 4 > t i] = A £L* «i(y - ^)/i + 6 - A « i( 2 + P ) - 7J ) ' 4 > ( 0 j/2+1) - 0 (5.6) 1 4>?J + l) = 1 + X b / l + 6 4>\^ — 1 + A (6 — o,j)/l 4- 8 = . . . , = # = - \ b / ( i + s)2 ■ and I 4 1} =>•••, = 4 J) = - ( A % * / ( 1 + Sfuo (5.7) i I 5.5 sum marizes the cross equation restrictio ns, if sym m etry is to be im posed. How- 1 ever, e m p irical evidence fro m the basic 2 investor m odel stron gly rejects th is assump tio n . I f we assume th a t each of the ‘ f a -ty p e investors is unique (non overlapping j intervals) and we have one /9-type investor, then our m odel w o uld consist o f 2j + 1 ! regimes. Assum ing sym m etry, the to ta l num ber o f param eters th a t need to be es tim a te d equals 3 ( j -hi)- I f th e co n d itio n on sym m etry o f thresholds is n o t im posed | then the num ber of param eters th a t need to be estim ated increases to 5j + 3 and fu rth e r, if the strength param eter, a:i o f a types are allowed to vary, then a to ta l of 6 j + 3 param eters need to be estim ated. Thus fo r a m odel w ith three a types and 71 jOne j3 type investor w ith no sym m e try assumed and a / s changing over the different thresholds a to ta l o f 21 param eters have to be estim ated. T h is imposes a severe constraint on data. A n a lte rn a tive strategy w ould be to fit as m any regimes as possible to the m odel in 5.3 using th e m ethodology described in C hapter 4 and allow the data to determ ine the num ber o f investor types. B a rrin g problem s regarding the id e n tifica tio n o f the param eters o f the various investor types, th is analysis w ould m ost lik e ly indica te the m in im u m num ber of investor types, to include in a m odel o f the “ in trin s ic theory.” j5 .4 .1 .2 M u lt ip le a an d /3-investors I iSupposing, in a d d itio n to m u ltip le a-investors, if there are m u ltip le /3-type investors, the analysis w ould be even m ore interesting and com plicated. For instance le t us 1 assume th a t /3-investors enter and exit the m arke t w ith respect to buy and sell .thresholds sim ila r to those o f the a-investors. In th is case, however, a buy signal is triggered when the current price moves above the b u y threshold and likewise the sell signal is triggered when current price falls below the sell threshold. Suppose, there are n@ classes o f /3-investors, each w ith different trig g e rin g thresh o ld s . As suggested by D ay and G u, when pt £ [t> * — l\,v t + /J], the /3,-investor does not enter the m arket. W hen l\ < | pt — vt |< k\, he buys or sells in the am ount bi(pt — vt) and when | pt — vt |> k\, he buys or sells the am ount bik\ sign {pt — vt }. G iven these rules the ^ -s tra te g y is (U p) -U K - id 0 U p t - 1\) bi{k\ - / * ) 0 < P t < v t — k\ V i - k \ < p t < v t - l\ vt - l\ < P t < vt + l\ V t + l\ < P tV t + k\ vt + K < p t (5.8) and i = 1,... where bi is the stren gth o f the ^ -in v e s to rs ’ dem and. Define the follow ing intervals: M " := [0 ,vt — fc"] N ° := [vt ,v t + /j] M 8 [v t + l\t vt + k\] (5.9) j M ° := [ v t - l ^ v t ] N n := [vt + K , oo] jG iven 5.8 and 5.9, the aggregate dem and or supp ly o f stocks fro m /^-investors’ strategy is ~ B 8 - B i - b i[(v t - I t ) - p t] 0 { p ) : = 0 B i + bi\pt - ( v t + l\)] B i P t € M n P t G M l P t e M ° U N ° P t G N l P t G AT" ■ From 5.3 and 5.9, the price adjustm e nt equation can be w ritte n as: P t A p, + A[M - Bi] P t - X B i (1 - Xa3)pt + \ [ A 3 t + a3(ut - g3 t )] (1 - $cij + bi))pt + \ [ A J t - B\ + aj(ut - gi) - bi(vt - l$] (1 - A bi)pt - A [BI + bi(vt - l\)] \ P t + i = 0 (1 + Abi)pt + A[B\ — bi(vt -f l\)] (1 + A (6,- — cij))pt + A [B\ — A \ — aj(ut + g{) + bi(vt + /])] (1 + A ctj)pt — \ [ A J t + a3(ut + g3 t )] P t + pt - A [A} - Bi] pt - A A\ where the intervals are defined j * i j * 2 j*3 J * 4 J * 5 J * 6 J * 7 j * 8 j * 9 J * 10 J * l l 1*12 T*13 j n _ j^jn p M n j I n n M n M n - ( / " n M n) r - (P n a p ) P n M i M i - (P n Af*) (1° U J°) U ( M ° U N °) N 8 - (j* n n {) P n n 7 > - ( J 8 n A ") N n - ( j 7 1 n jvn ) N n n J n j n _ ( N n n J n j (5.10) P t G I*1 P t £ l * 2 P t G /*3 P t e I * 4 P t G /*5 P t G / * 6 P t G / * 7 P t G / * 8 P t G / * 9 P t G / * 10 P t G / * U P t G / * 12 P t G i * 13 (5.11) w here i = 1,... , A(g and j = 1 , ..., A ^ . O bviously th is m odel is fa r m ore co m pli cated th a n the tw o investor m odel o f C hapter 3. However, when transform ed in to 73 th e S E T A R fo rm , it can be w ritte n in the same com pact fo rm as in 5.4 and esti m a tio n and te sting can be carried o u t using the m ethodology described in C hapter 4. Table 5.13 presents the results fro m fittin g a m u lti-re g im e m odel to d a ily data. The data chosen fo r the analysis uses d a ily closing prices fo r th e S&P 500 index, covering the period, June 1962 - M ay 1972, a to ta l o f 2362 observations. T his period | was chosen as it covers about 6 b u ll and 7 bear m arkets . 16 T he qt series was derived jby detrending the d a ily price series using m o n th ly fundam entals th a t was obtained !in the previous section. In otherw ords, the fundam entals changed o n ly once every m onth. W e fu rth e r im posed on data the p rio r-re s tric tio n o f p = 2 and d = 1 , so I th a t i becomes the threshold variable. \ j The graphical m ethod o f p lo ttin g the constant and slope coefficients versus the threshold variable helped us determ ine about 13 d is tin c t regimes. T he fa ct th a t .there are considerable num ber o f observations in each regim e gives us confidence | in in te rp re tin g th e results. Three different observations can be made fro m looking 'a t Table 5.13. One, unlike the results based on m o n th ly data, no regim e can be .characterized as w h ite noise. In otherw ords, at least one coefficient in each regim e is /significant. T w o, the constant coefficient switches signs, although n o t consistently 'w ith the sw itch in the signs o f the slope coefficient. T he reason being, th a t in our •m odel we have tw o lags and the coefficient o f the second lag is always negative, save I fo r the firs t regim e, w h ile the sign o f the firs t coefficient is always positive. T h ird , ! prices ro u g h ly ranged about 4.5% below and above the fundam entals fo r the entire I i sample. W h ich translates in to a w ider tra d in g bound com pared to th a t obtained j using m o n th ly data. Figures 5.11, 5.12 and 5.13 e x h ib it the plots o f recursive coefficients against the | threshold variables w hich is used to id e n tify the threshold values and subsequently j the various regimes. For purposes o f th is analysis, the sym m etry co n d itio n was not assumed. However fro m our basic m odel, a negative sign fo r the constant ; coefficient and a positive slope suggests a /7-type investor, w hile, a positive sign fo r the constant w ith a negative slope suggests an ct-type investor. As m entioned ' .earlier, the slope coefficients are biased by the introduction of a second lag term. i _________ ____________ : 16B u ll and bear m arkets have been determ ined based on w hether the d a ily high (low ) was atleast > i 10% over (under) the previous high (lo w ) , see M cln ish and W ood (1992). J Each o f the regimes however e x h ib it s ta b ility , w hich was observed by lo okin g at the roots o f th e ir respective characte risitic polynom ials. T his com bined w ith the fact Ithat the estim ated residuals e x h ib it no significant a u to co rre la tio n suggests a good t !fit. F urtherm ore, if we assume th a t each regim e indicates a unique investor class, then we have id entified at least 13 different classes o f investors. However, the in te rp re ta tio n o f th is m odel becomes com plicated. In p a rticu la r, one is unable to disentangle th e effects o f a and /3 types. Nevertheless, there is l strong support for the presence o f m u ltip le investors in the fa ct th a t a m u lti-re gim e m odel can be fitte d to this data. F urtherm ore, coefficients in each o f these regimes are s ta tis tic a lly significant adding to the grow ing evidence th a t non linearitie s in the I jinvestor behavior could lik e ly cause stock prices to flu ctu a te random ly. In p a rtic u la r ‘th is n o n lin e a rity can be captured by sim ple piece-wise linear tim e series models, such as the S E TA R . The next section concludes th is study. I 5.5 C o n c lu sio n s I I (Casual observation o f the stock m arket led early researchers to fo rm u la te sim ple ; linear tim e-series models th a t were adequate in describing the observed flu c tu a tio n . \ \ [As the economy grew, there was a fundam en tal need fo r b e tte r models th a t could not j o n ly describe the observed flu c tu a tio n b u t could also provide an econom ic ra tio na le i I l fo r the same. Furtherm ore, new tests fo r detecting s tru ctu re in data were able to j garner sufficient evidence towards the presence of significant nonlinear stru ctu re in j stock returns. N onlinear stochastic models o f the A R C H v a rie ty have been used j |to explain the observed dependence, p a rtic u la rly the changing variances o f stock ■returns. However, research in the area o f nonlinear dynam ics has suggested th a t even sim ple d e te rm in istic processes are capable o f producing com plex pattern s w hich j are hard to distinguish fro m th e ir stochastic counterparts using conventional tests, j (This idea was the fundam ental m o tiv a tio n behind this m anuscript. j l A -2 -4 cn o Threshold v a ria b le Figure 5.11: Plot o f recursive estimates of the constant coefficient vs threshold variable for the TA R (13;2;1) model Threshold v a ria b le Figure 5.12: P lot of recursive estimates o f AR(1) coefficient vs threshold variable for the TAR.(13;2;1) model o J^\ C D Threshold v a ria b le Figure 5.13: P lot of recursive estimates of the A R (2) coefficient vs threshold variable ■ for the TA R (13;2;1) model 7 6 Table 5.13: A m ulti-investor model of stock returns for the S&P 500 Threshold <f>2} N [0.0,0.9591] 0.070 0.922 0.006 265 (1.89) (18.80) (0.17) (0.9591,0.9649] 0.496 0.756 -0.269 106 (1.35) (1.99) (-4.08) (0.9649,0.9692] 0.052 0.926 0 . 0 2 0 105 (0 .1 1 ) ( 1 .8 8 ) (0.27) (0.9692,0.9746] -0.281 1.514 -0.223 153 (-0 .6 8 ) (3.50) (-2.56) (0.9746,0.9812] -0.084 1.411 -0.325 206 (-0.34) (5.38) (-4.45) (0.9812,0.9835] 0.634 0.630 -0.275 76 (0.58) (0.56) (-2.81) (0.9835,0.9893] 0.076 1.081 -0.159 238 (0 .2 0 ) (2.61) (-1.39) (0.9893,0.9966] -0.363 1.689 -0.323 198 (-1.03) (4.45) (-2.92) (0.9966,1.0012] -0.284 1.443 -0.159 172 (-0.61) (3.00) (- 1 .2 2 ) (1.0012,1.0093] -0.565 1.813 -0.251 2 1 0 (-2.05) (6.24) (-2.43) (1.0093,1.0230] 0.543 1.781 -0.469 241 (0.58) (3.52) (-4.37) (1.0230,1.0436] -0.244 1.472 -0.237 185 (-1.61) (8.28) (-2.31) (1.0436, oo) 0.053 1.032 -0.084 207 (2.90) (31.7) (-2.85) Notes: D aily data was used to identify the different regimes. The window length used was 100. Numbers w ithin parentheses denote the ‘^’statistics. Based on results obtained by researchers such as Scheinkm an, LeBaron and Shaf- ; fer, Day, Huang and G u had hypothesized th a t a d e te rm in istic m odel o f the inter- j action o f different investor types in a given in s titu tio n a l fram ew ork is capable of , explaining certain stylized facts in the m arket such as excess v o la tility , and ran- j i dom ly sw itching b u ll and bear m arkets. O ur objective was to test th is theory. j I To fa c ilita te th is process, we transform ed the basic m odel in to a fram ew ork w hich preserved the essence o f the th e o ry b u t enabled us to estim ate and test certain a- p rio ri restrictio ns th a t were suggested b y th e proposed theory. j We have draw n fo u r d is tin c t conclusions fro m th is study. F irs t, based on con ventional and unconventional tests, we concluded th a t m o n th ly S&P 500 returns exhibited significant nonlinear dependence, in p a rtic u la r o f the threshold type. T his is in d ic a tiv e o f different linear models at different thresholds fo r the retu rns process, j Therefore, a S E T A R could be used to describe the m o n th ly returns process. Second, upon suitable tra n sfo rm a tio n , our hypothesized m odel can be cast in the fram ew ork of a S E T A R m odel. T he general T A R m ethodology was shown j to approxim ate the “ in trin s ic theory,” in a q u a lita tiv e sense. In p a rtic u la r, we fin d I 1 considerable support fo r certain a -p rio ri re strictio n s such as coefficient sign reversals and sh o rt-te rm vs long-term a u to co rre la tio n in the qt series. T h ird , a -p rio ri cross equation re strictio n s im p ly in g sym m etry o f investor behav- | io r around the fundam ental value, is not supported by data. W e em ployed the | conventional like lih o o d -ra tio test as w ell as a new b oo tstra p m ethodology w hich j ! exploits th e p rin cip le o f encompassing and concluded th a t the constrained m odel j ; does not encompass the observed dependence in m o n th ly data as measured by the : BDS d is c rim in a tin g sta tistic. T h is however, does not rule out the p o s s ib ility th a t ' our m odel encompasses other linear or non linear com peting models. T h is is an area j where fu rth e r research is needed. ‘ F in a lly, the hypothesis th a t a m ore general m odel th a t accounts fo r different i j investor types was entertained. E s tim a tio n provided strong support fo r the idea of ' m u ltip le regimes. However fro m the unconstrained m odel, it is not very clear how ' m any different classes o f investors we can id entify. To sharpen the analysis, and draw m ore concrete conclusions, we suggest a few areas where fu rth e r research is ; ( needed. | i B ibliography 1. Akgiray, V. (19S9), Conditional heleroscedaslicily in tim e series of stock re turns: evidence and forecasts, Journal of Business, 62, 55-81. 2. Ashley, R., Hinich, M. J. and D. M. Patterson (1986), A diagnostic test for nonlinear serial dependence in tim e series fittin g errors, Journal of Time Series Analysis, 7, 165-178. 3. Ashley, R., and D. M. Patterson (1989), Linear versus nonlinear macroeco nomics: a statistical test, International Economic Review, 30, 685-794. j 4. Barnett, W. A. and P. Chen (1989), The aggregation-theoretic monetary ag gregates are chaotic and have strange attractors: an econometric application I of mathematical chaos, in W. Barnett, E. Berndt, and H. W hite (eds.), D y- j n a m ic E c o n o m e tric M o d e lin g , Cambridge University Press, Cambridge, 199-246. 5. Baumol, W. J. and J. Benhabib (1989), Chaos : significance, mechanism and economics applications, Journal of Economic Perspectives, 3, 77-105. i 6 . Baumol, W . J. and R. E. Quandt (1985), Chaos models and their implications for forecasting, Eastern Economic Journal, 11, 3-15. 7. Baumol, W. J. and E. N. W olf (1983), Feedback from productivity growth to ; R &D , Scandinavian Journal of Economics, 85, 147-157. j 8 . Beaver, W . (1968), The inform ation content of annual earnings announce- j ments, Empirical Research in Accounting: Selected Studies, Supplement to J Journal of Accounting Research, 6 , 67-92. \ 9. Benhabib, J. and R. H. Day (1980), Erratic accumulation, Economics Letters, 6 113-117. 1 I 10. Black, F. (1986), Noise, Journal of Finance, 41, 529-543. j I 1 1 . Blattberg, R. and N. Gonedes (1974), A comparison of the Stable and Student j distributions as statistical models for stock prices, Journal of Business, 47, J 244-80. j 12. Bollerslev, T. (1986), Generalized autoregressive conditional beteroscedastic- j ity, Journal of Econometrics, 31, 307-27. j 13. Bollerslev, T. and I. Domowitz (1991), Price, volatility, spread variability and ■ the role of alternative market mechanisms, Review of Futures Markets, 14, j 12-34. 7 9 14. Boness, A. J., Chen, A., and S. Jatusipitak (1974), Investigations of nonsta- tio n a rity in prices, Journal of Business, 47, 518-37. j 15. B rillinger, D. (1979), T im e Series, D a ta A nalysis and T h e o ry , H olt, Reinhart and W inston, New York. 1 16. Brock, W . A. (1986), D istinguishing random and determ inistic systems: abridged I version, Journal of Economic Theory , 40, 168-195. ' • 17. Brock, W . A., Dechert, W . D. and J. A. Scheinkman (1987), A test for independence based on correlation dimension, working paper, U niversity of ; W isconsin-Madison. ' 18. Brock, W . A ., LeBaron, B. and A. Lakonishok (1991), Simple technical trading rules and the stochastic properties of stock returns, W orking paper, U niversity of W isconsin-Madison. I 19. Brock, W . A., and A. G. M alliaris (1989), D iffe re n tia l equations, s ta b ility and chaos in d yn am ic econom ics, Elsevier-Am sterdam. 20. Brock, W . A., and C. L. Sayers (1988), Is the business cycle characterized by determ inistic chaos ?, Journal of Monetary Economics, 22, 71-90. 1 21. Chen, P. (1984), A possible case of economic attractor, Prigogine center for studies in statistical mechanics, U niversity of Texas at Austin. 22. Chen, P. (1987), Nonlinear dynamics and business cycles, Ph.D. Dissertation, ■ U niversity of Texas, Austin. j 23. Chen, P. (1988), E m pirical and theoretical evidence of economic chaos, System | Dynamics Review, 4, 81-108. 24. Christie, A. (1982), The stochastic behavior of common stock variances: value, leverage, and interest rate effects, Journal of Financial Economics, 10, 407-32. ! 25. Christie, A. (1983), On inform ation arrival and hypothesis testing in event j studies, working paper, U niversity of Rochester. ! 26. Day, R. H. and W . Huang (1990), Bears, bulls and m arket sheep, Journal of Economic Behavior and Organization, 14, 1-15. 27. Day, R. H. and W . Shafer (1983), Keynesian chaos, Departm ent of Economics, M R G working paper, USC. 28. Day, R. H. and Gu, M. (1993), W hy stock markets churn: P art I. A behavioral, ! in s titu tio n a l theory of a stock exchange w ith specialist m ediation, Unpublished m anuscript, U.S.C., Los Angeles. 29. Dickey, D. A. and W . A. Fuller (1979), D istrib ution of the estim ators for autoregressive tim e series w ith a u n it root, Journal of the American Statistical Association, 74, 427-431. 30. Diebold, F. X. and M . Nerlove (1989), The dynamics of exchange rate v o la til ity : a m ultivariate latent factor AR C H model, Journal of Applied Eoconomet- rics, 4, 1-21. 31. Dom owitz, I. and C. Hakkio (1985), C onditional variance and the risk pre m ium in the foreign exchange m arket, Journal of International Economics, 50, 987-1008. I 32. Engle, R. F. (1982), Autoregressive conditional heteroscedasticitj' w ith esti mates of the variance of U .K . in flation, Econometrica, 50, 987-1008. \ 33. Engle, R. F,, Ito, T . and W . Lin (1990), M eteor showers or heat waves ? ! Heteroskedastic in tra daily v o la tility in the foreign exchange m arket, Econo- \ metrica, 58, 525-542. i | 34. Fama, E. F. (1970), Efficient capital m arket : review of theory and em pirical ! work, Journal of Finance, 10, 383-417. 35. French, K. R., Schwert, G. and R. Staumbaugh (1987), Expected stock returns and vo la tility, Journal of Financial Economics, 19, 3-29. i [ 36. G ilm ore, E. (1992), A new test for Chaos, working paper, K in g ’s College. : 37. G randm ont, J. M . (1985), On endogenous com petitive business cycles, Econo- 1 metrica, 53, 995-1045. i 38. Granger, C. W . J. and A. P. Andersen (1978), A n In tro d u c tio n to B ilin e a r i T im e Series M o d els, Vandenhoek and Ruprecht, G ottingen. 39. Granger, C. W . J. and O. Morgenstern (1970), P re d ic ta b ility o f stock price, Heath, Lexington, Massachusetts. 40. Grassberger, P. and I. Procaccia (1983), Measuring the strangeness of strange ! attractors, Physica, 90, 189-208. I 41. Gu, M. (1992), A theory of stock price behavior, Ph.d thesis, U niversity of i Southern C alifornia, Departm ent of Economics, Los Angeles. i | 42. Hendry, D. F. and J. F. Richard (1982), On the form ulation of em pirical models in dynam ic econometrics, Journal of Econometrics, 20, 3-33. 43. Hendry, D. F. and J. F. Richard (1983), The econometric analysis o f economic tim e series, International Statistical Review, 51, 111-163. 44. Hinich, M. J. (1982), Testing for gaussianity and linearity of a stationary tim e series, Journal of Time Series Analysis, 3, 169-176. 45. Hinich, M . J. and D. M. Patterson (1985), Evidence of nonlinearity in stock returns, Journal of Business and Economic Statistics, 69-77. 46. Hinich, M . J. and D. M. Patterson (1992), Intra-day Nonlinear Behavior of Stock Prices, U niversity of Texas at A ustin, V P I & SU, working paper. 47. Hsieh, D. (1989), Testing for nonlinear dependence in daily foreign exchange rates, Journal of Business, 62, 339-364. 48. Hsieh, D. (1990), Chaos and nonlinear dynamics : A pplication to financial markets, Unpublished m anuscript, Duke U niversity, Durham , NC. 49. Hsu, D., M ille r, R. and D. W ichern (1974), On the stable Paretian behavior of stock m arket prices, Journal of American Statistical Association, 69, 108-113. 50. Jensen, R. V. (1987), Classical Chaos, American Scientist, 75, 168-181. 51. Keenan, D. M . (1985), A tukey non additivity-type test for tim e series nonlin earity, Biometrika, 72, 39-44. 52. Kugler, P. and C. Lenz (1990), Chaos, AR C H and the foreign exchange m arket: em pirical results from weekly data, Unpublished m anuscript, Zurich. 53. LeBaron, B. (1990), Some relations between v o la tility and serial correlations in stock m arket returns, working paper, U niversity of W isconsin-Madison. 54. Lucas, R. E. (1978), Asset prices in an exchange economy, Econometrica, 46, 1429-1445. 55. M andelbrot, B. (1966), Forecasts of future prices, unbiased markets, and m ar tingale models, Journal o f Business, 39, 242-245. 56. May, R. M . and J. R. Beddington (1975), Nonlinear difference equation: stable points, stable cycles; chaos, Unpublished m anuscript. 57. M ilh o j, A. (1987), A conditional variance model for daily deviations of an exchange rate, Journal of Business and Economic Statistics, 5, 99-103. 58. M izon, G. E. (1984), The encompassing approach in econometrics in Econo m e tric and Q u a n tita tiv e Econom ics, ed. D. F. Hendry and K. F. W allis, O xford: Basil Blackwell. 59. Patell, J., and M . Wolfson (1981), The ex-ante and ex-post price effects of quarterly earnings announcements reflected in O ption and Stock prices, Jour nal of Accounting Research, 19, 434-458. 60. Peters, E. E. (1990), A Chaotic a ttra cto r for the S&P 500, Financial Analysts Journal. 61. Q uandt, R. E. and J. Ramsey (1978), E stim ating m ixtures of norm al d istrib u tions and switching regressions, Journal of American Statistical Association, 730-738. 62. Ramsey, J. B., Sayers, C. L. and P. Rothm an (1990), The statistical properties of dimension calculations using small data sets : some economic applications, International Economic Review, 31, 991-1020. 63. Rubinstein, M . (1976), The valuation of uncertain income streams and the pricing of options, Bell Journal of Economics, 7, 407-425. 64. Sakai, H. and H. Tokum aru (1980), Autocorrelations of certain chaos, IE E E Transactions on Acoustics, 28, 588-590. 65. Samuelson, P. A. (1965), Proof th a t properly anticipated prices fluctuate ran domly, Industrial Management Review, 6, 41-49. 66. Sayers, C. L. (1985), W ork stoppages: exploring the nonlinear dynamics, work ing paper, U niversity of W isconsin-Madison. 67. Scheinkman, J. A. and B. LeBaron (1989), Nonlinear dynamics and stock returns, Journal of Business, 3, 311-337. 68. Schwert, G. W . (1990), Stock v o la tility and the crash of ’87, Review of Finan cial Studies, 3, 77-102. 69. Schwert, G. W . and P. J. Seguin (1990), Heteroscedasticity in stock returns, Journal of Finance, 4, 1129-1155. 70. Shaffer, S. (1991), S tructural shifts and the v o la tility of chaotic markets, Jour nal of Economic Behavior and Organization, 15, 201-214. 71. Shiller, R. J. (1990), Market Volatility, M IT Press, Cambridge, Massachusetts. 72. Stutzer, M. (1980), Chaotic dynamics and bifurcation in a macro model, Jour nal of Economic Dynamics and Control, 2, 353-376. 73. Subba, Rao T . and Ga.br, A4. (1980), A test for lin e a rity of stationary tim e series, Journal o f Time Series Analysis, 1, 145-158. 74. Summers, L. and J. Poterba (1986), The persistence of v o la tility and stock market fluctuations, American Economic Review, 76, 1142-1151. 75. Tauchen, G. and M . P itts (1983), The price variability-volum e relationship in speculative markets, Econometrica, 51, 485-505. 83 76. Tong, H. (1977), Some comments on the Canadian Lynx data - w ith discussion, Journal of Royal Statistical Society, A 140, 432-435, 448-468. 77. Tong, H., and K . S. L im (1980), Threshold autoregression, lim it cycles and cyclical data, Journal o f Royal Statistical Society, B 42, 245-292. 78. Tong, H. (1983), T h resh o ld M o d e ls in N o n lin e a r T im e Series Analysis, Springer Verlag, New York. 79. Tong, H. (1990), N o n lin e a r T im e Series: A D y n a m ic a l S y te m A p proach, O xford University Press, London. 80. Tsay, R. S. (1986), N onlinearity tests for tim e series, Biometrika, 73, 461-466. 81. Tsay, R. S. (1987), C onditional heteroscedastic tim e series models, Journal of the American Statistical Association, 82, 590-604. 82. Tsay, R. S. (1989), Testing and m odeling threshold autoregressive processes, Journal of the American Statistical Association, 84, 231-240. 84

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

A theory of stock price behavior

PDF

Model selection in dynamic models

PDF

Two essays on the valuation of options: A martingale analysis

PDF

Adaptive economizing in disequilibrium: Essays on economic dynamics

PDF

Econometric modelling of United States regional electricity demand under peak-load pricing

PDF

The price of oil and OPEC behavior: A utility maximization model

PDF

Futures markets and production decisions

PDF

A nonlinear disequilibrium simulation model of an open economy

PDF

An econometric approach to insurance ratemaking

PDF

An economic analysis of cooperative group behavior: Theory and evidence

PDF

Studies of behavioral axioms for comparative statics under under uncertainty

PDF

A theory of the rational pricing of mortgage contracts: An application of contingent claim analysis

PDF

Food policy choices in an uncertain world: Policy analysis with a probabilistic world food model

PDF

The estimation of mortgage prepayment probabilities and the valuation of mortgage-backed securities

PDF

The impact of time-of-use electricity pricing: A firm level econometric approach

PDF

A new econometric approach to modelling peak load pricing policies: The case of electricity demand by large industrial customers

PDF

Interaction between the public and private sectors in regional I-O models

PDF

The relationship between industrial and legal jurisprudence

PDF

The drivers of economic change: Technology, globalization, and demography

PDF

An analysis of the effects of investment expenditures on the balance of payments of Iran, 1955-1962

Asset Metadata

Creator Srinivasan, Rajesh (author)

Core Title Econometric analysis of nonlinear dynamics in a behavioral institutional model of stock price fluctuation

Degree Doctor of Philosophy

Degree Program Economics

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

Tag Economics, Finance,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Day, Richard H. (committee chair), [illegible] (committee member), Sankarasubramanian, L. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c20-280880

Unique identifier UC11259319

Identifier DP23402.pdf (filename),usctheses-c20-280880 (legacy record id)

Legacy Identifier DP23402.pdf

Dmrecord 280880

Document Type Dissertation

Rights Srinivasan, Rajesh

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA