Approximating stationary long memory processes by an AR model with application to foreign exchange rate

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Content APPROXIMATING STATIONARY LONG MEMORY PROCESSES BY AN AR MODEL WITH APPLICATION TO FOREIGN EXCHANGE RATE by Shin-Huei Wang A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ECONOMICS) August 2008 Copyright 2008 Shin-Huei Wang Table of Contents ListofTables iii Abstract iv Introduction 1 Chapter1 4 1.1Introduction 4 1.2 The Model and Main Results 9 1.2.1ApproximatingtheARFIMA(p;d;q)byAnARModel 11 1.2.2 Asymptotic Properties of Haugh's Statistics 13 1.3OrderSelectionforApproximatingLongMemorybyAnARmodel 14 1.4 Monte Carlo Simulation 15 1.4.1FiniteSamplePerformanceoftheARApproximation 17 1.4.2 Finite Sample Properties of Haugh's Statistics 19 1.5ApplicationtoForeignExchangeRatesVolatility 21 1.6Conclusion 22 Chapter2 24 2.1 Introduction 24 2.2TheModelandParameterEstimation 26 2.3 Asymptotic Properties of b A(k) and b § k 31 2.4AsymptoticPropertiesoftheLinearPredictorb y T;k (1) 34 2.5MonteCarloExperiment 36 2.6Conclusion 39 Conclusion 39 References 53 Appendices Appendix A Proof of Chapter 1 59 Appendix B Proof of Chapter 2 71 ii List of Tables Table 1. The Frequencies of Selected Lag Lengths for AIC and C p Respectively When the DGP is the Fractional White Noise Process 41 Table 2. Mean of Residual Variance S 2 e;k When the DGP is the ARFIMA (0;d;0) Process 42 Table 3. Mean of Residual Variance S 2 e;k When the DGP is the ARFIMA (¡0:7;d;0:5) Process 43 Table 4. The Mean of the Estimated ARFIMA (0;d;0) Model Parameter 44 Table 5. The Rejection Percentages of the t When Y 1t and Y 2t , are Fractional White Noise , and ½ 12 (j)=0 for all j 45 Table 6. The Rejection Percentages of the S M Test at 5% Level of Signi¯cance When Y 1t and Y 2t are Fractional White Noise, e 1;t ;e 2;t »N(0;1), and ½ 12 (j)=0 for all j 46 Table 7. The Rejection Percentages of the S ¤ M Test at 5% Level of Signi¯cance When Y 1t and ;Y 2t are Fractional White Noise, e 1;t ;e 2;t »N(0;1), and ½ 12 (j)=0 for all j 47 Table 8. The Rejection Percentages of the S M Test at 5% Level of Signi¯cance When Y 1t and Y 2t are Fractional White Noise, e 1;t ;e 2;t »N(0;1), ½ 12 (0)=0:2, and ½ 12 (j)=0 for all j6=0 48 Table 9. The Rejection Percentages of the S ¤ M Test at 5% Level of Signi¯cance When Y 1t and Y 2t are Fractional White Noise, e 1;t ;e 2;t »N(0;1),½ 12 (0)=0:2, and ½ 12 (j)=0 for all j6=0 49 Table 10. Estimation of ARFIMA (1;d;1) Models for Two Series 50 Table 11. Table 11. s 2 e;k and AIC for AR(k) model ¯tted to Two Series 51 Table 12. Test statistics for correlation in volatility 52 iii Abstract This dissertation focuses on the AR approximation of long memory processes and its applications. The ¯rst chapter proposes an easy test for two stationary au- toregressive fractionally integrated moving average (ARFIMA) processes being un- correlated via AR approximations. We prove that an ARFIMA (p;d;q) process, Á(L)(1¡L) d y t = µ(L)e t , d 2 (0;0:5), where e t is a white noise, can be approxi- mated well by an autoregressive (AR) model and establish the theoretical foundation of Haugh's (1976) statistics to test two ARFIMA processes being uncorrelated. The Haughstatisticisusefulbecauseitcanavoidtheissuesofspuriousregressioninduced by the long memory processes considered by Tsay and Chung (2000). Using AIC or Mallow's C p criterion as a guide, we demonstrate through Monte Carlo studies that a lower order AR(k) model is su±cient to prewhitten an ARFIMA process and the Haugh test statistics after AR pre-whitening perform very well in ¯nite sample. We illustrate the use of our methodology by investigating the independence between the volatility of two daily nominal dollar exchange rates-Euro and Japanese Yen. We ¯nd that there exists " strongly simultaneous correlation " between the volatilities of Euro and Yen within 30 days. The second paper extends the analysis of Lewis and Reinsel (1985) to the r- dimensional I(d) process y t , where d > 0, i.e., we consider the problems of the linear prediction of y t+1 based on y t ;y t¡1 ;:::, using a VAR model of order k ¯tted to a realization of T observations y 1 ;y 2 ;:::;y T . Assuming that k grows with T at a suitable rate, along with other regularity conditions imposed on y t , the consistency of the multivariate least squares (LS) coe±cient estimator and that of the residual covariance matrix estimator b § k are derived, and the one-step ahead prediction error based on the VAR(k) model is shown to converge in probability to its population counterpart. iv Furthermore, a Monte Carlo experiment is carried out to assess the e®ect of es- timating autoregressive parameters on the mean square prediction error. The results reveal that the average observed squared prediction errors from using the VAR(k) model are very close to the ¯nite sample approximation formula § k (h) derived by Lewis and Reinsel (1985) for the weakly dependent processes. v Introduction Autoregressive fractionally integrated moving average process of order p;d;q, de- noted as ARFIMA (p;d;q) or I(d) processes, have been proved as useful tools in the analysis of a time series with long-range dependence, especially for ¯nancial and eco- nomic data. With a fractionally intergrated process analysis, Baillie and Bollerslev (1994, 2000) demonstrated that the forward premia for currencies have long memory. Ding et al. (1993), de Lima and Crato (1993) and Bollerslev and Mikkelsen (1996) proved that the persistent dependence in stock markets' volatility possesses as long memory processes as well. Also, Teyssiere (1997,1998a) revealed that some daily and intraday foreign exchange rate (FX) returns display the same degree of long memory intheircovariancesandconditionalvariances. Yet, estimatingthevalueofthedi®er- encing parameter d remains a di±cult task. While Sowell (1992) outlined the general framework for the maximum likelihood estimation of the long memory processes, the presence of AR parameters greatly complicate the computation of the corresponding autocovariance functions, because they involve hypergeometric functions which need tobeevaluatedwithatruncatedin¯nitesum. Therefore, roundingerroris inevitable in Sowell's (1989) methodology. Due to this heavy computational burden of d, only few papers deal with the estimation of the di®erencing parameters of the VARFIMA process. And also, for the univariate case, when d is close to 0.5 and the sample size is not large enough, the estimation of the di®erencing parameter of the autoregres- sive fractionally intergrated moving average (ARFIMA) process may not be accurate (Sowell (1992)), either. Such poor performances of the existing estimation methods for long memory processes mislead the statistical properties of data. As a result, the main idea of this dissertation is to propose alternative means to avoid inaccurate estimation results and to lessen the computational burden as we deal with the long 1 memory processes. There are two chapters included in this dissertation. The ¯rst chapter of this dissertation is " An easy test for independence between twostationarylongmemoryprocessesviaARapproximations,withanapplicationto the volatility of foreign exchange rates", which is dedicated to propose one possible way to tackle the problems as have been mentioned above, that is, approximating a long memory process by an autoregressive time series (AR) model. Additionally, this chapter establishes the theoretical foundation of Haugh's (1976) statistics to test the independence between two ARFIMA processes. as well as provides the guidance of determination of the order of AR(k) model ¯tted to long memory processes. The simulations con¯rm the desirability of using the Haugh test statistics after AR pre- whitening of ARFIMA processes to test for independence in ¯nite sample. In short, thischaptersuggestsanalternativemethodtoavoidtheinaccurateestimationresults of d, and a test statistic to check the correlation between two stationary I(d) pro- cesses based on AR approximation of long memory processes. It also illustrates the methodology by applying the new tests to the volitility betweeen two daily nominal U.S. dollar exchange rate-Euro and Japanese Yen and ¯nds that there exist strong simultaneous volatilities interaction between them within one month. The chapter 2 of this disseratation concentrates on the approximation of multi- variatelongmemoryprocesses,whichis"Predictionofthemultivariatelongmemory process via a vector autoregressive model". This paper is to extend the analysis car- ried out by Lewis and Reinsel (1985) to the r-dimensional long memory process with the same di®erencing parameter d. Under suitable assumption imposed on the DGP, this chapter proves that the one-step ahead prediction error of the \semiparamet- ric" VAR(k) model converges in probability to its population counterpart when k is chosen appropriately. 2 Forthepurposeofprediction,thischapteralsoestablishesthetheoreticalfoundations ofusingtheVAR(k)model¯ttingforthemultivariatelongmemoryprocess. Aswell, it proves that the multivariate least squares (LS) coe±cient estimator for the \semi- parametric" VAR(k) model is consistent. In this chapter, a Monte Carlo experiment to investigate the ¯nite sample prediction performance of the VAR(k) model is also conducted. Thesimulationsrevealthattheaveragesquaredpredictionerrorsarevery close to their population counterparts multiplied by a factor of (1+kr=T), imply- ing that the ¯nite sample prediction performance of the \semiparametric" VAR(k) model for the long memory process is comparable to the cases where the DGP is the I(0) process. For the purpose of prediction, a great deal of e®ort associated with estimating the di®erencing parameter can be saved, because we do not estimate the values of d any more. 3 Chapter 1 An Easy Test for Independence between Two Stationary Long Memory 1.1 Introduction The extentof ¯nancial integration has received a great deal of attention in recent years. For example, a substantial body of literature have documented the issue of whether there exist period-to-period comovements across di®erent assets or markets (e.g., Baillieand Bollerslev, 1990; Engle et al., 1990; Hamao et al., 1990; Cheungand Ng, 1996 ; Hong Y., 2001; Andersen et al. 2003; Calvet et al. 2006). Accordingly, testing for interrelationships between two time series, Y 1t and Y 2t becomes very im- portant. Recently, we have witnessed increasingly large number of studies using the autoregressivefractionallyintegratedmovingaverageprocessoforder p;d;q, denoted as ARFIMA (p;d;q) or I(d) process, when the integrated process order d, or the dif- ferencingparameterdisafractionalnumbertomodel¯nancialtimeseries. Themain featureofthestationary I(d)processisthatitsautocovariancefunctiondeclinesata slowerhyperbolicrate(insteadofthegeometricratefoundinconventionalstationary ARMA processes). For instances, Baillie (1996) provided some exhaustive presenta- tions for long memory processes. Breidt et al. (1998) modeled the squared stock returns by the long memory stochastic volatility to describe the persistent depen- dence structure. Ray and Tsay (2000) showed that the squares of daily stock return were long range dependent. Many empirical results show that the long range depen- dence in the volatility follows the class of long memory autoregressive conditional heteroskedasticity processes (See Robinson 1991, Ding and Granger 1996 , Bollerslev and Mikkelsen 1996 and Giraitis et al, 2000 ). 4 Teyssiere (1997,1998) revealed that some daily and intraday foreign exchange rate (FX) returns display the same degree of long memory in their covariances and conditional variances. Others, such as Bollerslev and Wright (2000) and Robinson andHenry(2003),providenewestimationmethodsforlongmemoryandextendthem to the empirical applications of volatilites in intra-day foreign exchange returns. One of the advantages of using the ARFIMA process is that it provides a parsi- moniouswaytodescribedataserieswithlong-rangedependence. However,theuseof the ARFIMA process also incurs some problems in some popularly used test statis- tics. TsayandChung(2000)provedthattwostationaryI(d 1 )andI(d 2 processescan yield spurious regression so long as d 1 +d 2 > 0:5, d 1 2 (0;0:5), and d 2 2 (0;0:5). On the other hand, Tsay (Theorem 4, 2000) showed that the insigni¯cant testing problem ¯rst considered by Robinson (1993) could also arise if d 1 d 2 <0, because the t ratio will converge to zero, the actual size of using t-test for the null hypothesis of no relationship between the two unrelated I(d) processes will be signi¯cantly below thenominalsize. Thus,theusualtstatisticcanbeofnouseinempiricalapplications when the DGP are I(d) processes. One possible way to tackle the preceding testing problems is to ¯rst estimate the parametersoftheARFIMAmodelforY 1t andY 2t ,respectively. Undertheassumption that the di®erencing parameters d 1 and d 2 are known, the null hypothesis that Y 1t and Y 2t are independent of each other can be tested with accuracy by Monte Carlo simulation. Nevertheless, it is well known that the estimation of the di®erencing parameter of the ARFIMA process may not be very accurate in ¯nite samples when d is close to 0.5. Therefore, the simulation method may not be a good way to meet our purpose. This chapter proposes a test for correlation between two I(d) processes. Essen- tially, our tests follow the suggestions of Haugh (1976) and Hong (1996) to construct the residual cross-correlation functions from prewhitened weakly dependent time se- ries, say, Y 1t and Y 2t .For I(0) process, Haugh (1976) proposed to prewhiten Y 1t and 5 Y 2t with autoregressive moving average (ARMA) functions, while Hong (1996) em- ployed the autoregression approximation based on the analysis of Berk (1974). We shall¯rstjustifytheuseofHaugh'sstatisticswhentheDGParetheARFIMA(p;d;q) processes. We then show that y t can be approximated well by an AR(k) model, be- cause the OLS residuals from an AR(k) ¯lter mimic the properties of the true error term e t asymptotically. This ¯nding itself contains some interesting empirical im- plications. First, the applicability of Haugh's statistics to the ARFIMA processes followsimmediately. Actually, Haugh'sstatisticsbasedontheARapproximationare theeasiestwaystochecktheindependencebetweentwoARFIMAprocesses, because wedonotneedtoestimatethedi®erencingparametersofthedataseries. Allwehave to do is to ¯t AR models to Y 1t and Y 2t , respectively. These features of the charac- teristics are extremely valuable, because the estimation of the di®erencing parameter d is computationally complicated, and may not be very accurate, in particular, when d is close to 0.5, and the sample size is not very large. Second, through a simulaiton study, Ray (1993) showed that an AR approximation could be useful for long-range forecasting of a long-memory process. Her simulation results are explained by our theoretical analysis. Third, the good performance of the indirect estimation method for the I(d) process proposed by Martin and Wilkins (1999) and that of the e±- cient method of moments (EMM) employed by Gallant, Hsieh and Tauchen (1997) are easily understood, because they both employed an AR model as the auxiliary model. Fourth, theissuesofspuriousregressionbetweentwostationarylongmemory processes considered by Tsay and Chung (2000) will not arise. We also provide criteria for the determination of the order of an AR(k) model ¯ttedtoalongmemoryprocess. In manyapplications, autoregressivemodels oftime series are useful for prediction and statistical inferences. 6 Thus, over the last three decades, order selection methods for autoregressive models are of considerable interest. The methods for determining the order of an autoregression, k, are either with reference to the structure of the model or from the forecasting point of view. One of the popular order selection criteria is the Akaike imformation criterion, AIC (Akaike, 1973). Other model selection criretia, such as corrected Akaike information criterion AICC (Hurvich and Tsai, 1989), Bayesian information criterion BIC (Schwarz, 1978), ¯nal prediction error FPE (Akaile, 1969) and C p (Mallows, 1973), have been proposed and studied for regression and times seiresmodelselectionaswell. Generallyspeaking,allthesecriteriacouldbeclassi¯ed into one of two categories: e±cient (e.g. AIC, AICC, FPE, and C p ) or consistent (e.g. BIC).Whenthetruemodelisofin¯nitedimension, thee±cientcriteriaprovide the best ¯nite dimensional approximating model. However, if the true model is ¯nite dimension, the e±cient criteria do not obtain consistent order selection model. On the other hand, consistency can be obtained only at the cost of asymptotic e±ciency. Shibata (1980) veri¯ed that if Akaike's ¯nal prediction error (FPE) and AIC methods are applied to an in¯nite order autoregressive process, they are asymp- totically e±cient. For the case of an asymptotically optimal selection of regression variables, Shibata (1981) showed that Mallows's C p , Akaike's FPE, and AIC meth- ods are all equivalent as well. Furthermore, in selecting the true generating model in the particular case of ARFIMA (0;d;0) processes, Schmidt and Tschernig (1993) found that the BIC criterion performed poorly compared to the other criteria, such as AIC-type criteria. For issues related to the order of an autoregression selected us- ing information criteria, Ng and Perron (2005) show the formulation of information criteria can a®ect the precision and variability of the selected lag order, especailly for the case that the true model is in¯nite dimensional, the penalty functions de¯ned in textbooks also can be easily misinterpreted. 7 Thus, following the framework of Ng and Perron (2005), we justify the use of the AIC and the Mallow's C p on the basis of approximating for the long memory process byanAR(k)model. OurMonteCarloexperimentsshowthattheresidualsfrom¯nite k, AR approximation can yield the mean of the residual variance estimate very close to the true variance of e t and the lag length needed to approximate the ARFIMA process is quite small. Moreover, our experiment also clearly demonstrates that the good¯nite sample performance of Haugh's statistics eventhe DGP are the ARFIMA processes. The size distortion of the Haugh's statistics is well controlled for various combinationsofd 1 andd 2 andthepowerperformanceisverypromising. Accordingly, all these ¯ndings illustrate that Haugh's statistics based on the AR approximation is very useful in testing the correlation between the two ARFIMA processes, as well as in saving a great deal of e®orts associated with estimationg parameters of long memory processes. In short, this paper suggests an alternative method to avoid the inaccurateestimationresultsofd,andateststatistictocheckthecorrelationbetween twostationary I(d) processes basedon AR approximation oflong memory processes. We illustrate our methodology by applying our tests to the volitility betweeen two daily nominal U.S. dollar exchange rate-Euro and Japanese Yen and ¯nd that there exist strong simultaneous volatilities interaction between them within one month. This Chapter is organized as follows: section 2 provides the theoretical justi- ¯cation of the test statistics, including the proof of approximating a long memory processe by an AR (k) model. In section 3, we provide a criterion for the deter- mination of the order of an AR (k) model to long memory processes. Section 4 reports simulation results. Further justi¯cation of AR (k) approximation is proposed in section 5. Section 6 presents the empirical analysis of the correlation in volatil- ity between Euro and Japanese Yen. The last section summarizes this chapter. All proofs are in the Appendix A. 8 1.2 The Model and The Main Results Haugh(1976)proposedthefollowingteststatisticsforindependencebetweentwo weakly dependent processes: S M =T M X l=¡M b ½ 12 (l) 2 and S ¤ M =T 2 M X l=¡M (T ¡jlj) ¡1 b ½ 12 (l) 2 ; where b ½ 12 (l)= T¡l X t=k+1 b e t;k;1 b e t+l;k;2 Ã T X t=k+1 b e 2 t;k;1 ! 1=2 Ã T X t=k+1 b e 2 t;k;2 ! 1=2 ; and b e t;k;1 and b e t;k;2 are residuals from ¯tting Y 1t and Y 2t with an ARMA model, respectively. Haugh (1976) proved that S M and S ¤ M are both distributed as Â 2 2M+1 when Y 1t and Y 2t are both weakly dependent and independent of each other. Hong (1996) suggested constructing the Haugh's statistics with residuals from an AR(k) ¯lter. BasedontheworkofBerk(1974),HongshowedthatHaugh'sstatisticsperform very well when the DGP are weakly stationary. Thsobjectiveofthispaperistofurtherextendthetwoteststatistics,S M andS ¤ M , to the long memory case, i.e, the ARFIMA(p;d;q) processes. We suppose the DGP, y t ,anARFIMA(p;d;q)process,satis¯esthefollowingAssumption1throughoutthis paper. 9 Assumption 1. y t is generated as: Á(L)(1¡L) d y t =µ(L)e t ; where (i) d 2 (0;0:5); (ii) Á(L), and µ(L) are ¯nite degree polynomials, and the zeroes of Á(L), and µ(L) all lie outside the unit circle; (iii) Á(L) and µ(L) have no common zeroes; (iv) e t is an independently and identically distributed process, with E(e t )=0, E(e 2 t )=¾ 2 , and E ¡ e 4 t ¢ <1. Hosking (1996) shows that a stationary and invertible ARFIMA process with d6=0 has an autocovariance function that satis¯es ° j » ¾ 2 f y (0)¡(1¡2d) ¡(d)¡(1¡d) j 2d¡1 ; as j!1; where ¡(:) is the Gamma function, and f y (0)= (1+µ 1 +:::+µ q ) 2 (1¡Á 1 ¡:::¡Á p ) 2 : Assumption 1 guarantees that the conditions in Theorem 3 of Hosking (1996) hold, and allows us to represent an ARFIMA process y t as: y t = 1 X j=0 Ã j e t¡j ; where Ã j =O ³ j d¡1 ´ as j!1; or y t = 1 X j=1 ¯ j y t¡j +e t ; where ¯ j =O ³ j ¡d¡1 ´ as j!1: Accordingly,weresorttoanalyzingtheasymptoticpropertiesofHaugh'sstatistics when Y 1t and Y 2t are generated as Á(L)(1¡L) d 1 Y 1t =µ(L)e 1;t ;and Á(L)(1¡L) d 2 Y 2t =µ(L)e 2;t ; (1) where d 1 2 (0;0:5) and d 2 2 (0;0:5), respectively, and e 1;t and e 2;t are both zero mean white noise processes with variances ¾ 2 1 and ¾ 2 2 , respectively. Just like Hong (1996), we ¯rst show that an ARFIMA (p;d;q) process can be approximated by an AR model. This theoretical investigation is the building block of our paper and presented in the following section. 10 1.2.1. Approximating an ARFIMA (p;d;q) by an AR Model Based on a binomial expansion, an ARFIMA (p;d;q), y t , de¯ned in Assumption 1 can be represented by an in¯nite order AR process (Brockwell and Davis (1991)), i.e., y t = P 1 j=1 ¯ j y t¡j +e t , where ¯ j = ¡¡(j¡d) ¡(j+1)¡(¡d) ; ¡(:) is the Gamma function, y m = 0 as m · 0, and e t is a zero mean white noise process with variance ¾ 2 . Our analysis centers on ¯guring out the conditions needed for the growth rate of lag length k such that y t can be approximated by an AR(k) model, i.e., y t =e t + k X j=1 ¯ j y t¡j + 1 X j=k+1 ¯ j y t¡j =e t;k + k X j=1 ¯ j y t¡j ; and 1 X j=k+1 ¯ j y t¡j =o p (1): Nevertheless, inempiricalapplications, theOLSestimatoriswidelyusedtocalculate the coe±cient of an AR(k) model. Therefore, the ultimate criteria for deciding whether y t can be well approximated by an AR model is to verify whether the OLS residualsb e t;k from an AR(k) ¯lter are able to mimic the statistical properties of e t asymptotically. Theorem 1-4 establish this is indeed the case. For ease of reading, we put all the proofs in the appendix. Since our job is to check whether the following condition holds: b e t;k =e t +o p (1); (2) weneedtoinvestigatetheasymptoticpropertiesof b ¯ 0 =( b ¯ 1 ;:::; b ¯ k )and P 1 j=k+1 ¯ j y t¡j . We note that the OLS residualsb e t;k can be rewritten as: b e t;k =e t + 1 X j=k+1 ¯ j y t¡j ¡ k X j=1 ³ b ¯ j ¡¯ j ´ y t¡j : 11 In Theorem 1, we discuss the asymptotic properties of the OLS estimator b ¯, which is obtained from regressing y t on y t¡1 ,:::, y t¡k . The asymptotic properties of the OLS residuals b e t;k and those of the residual variance s 2 e;k de¯ned as s 2 e;k = (T ¡ k) ¡1 P T t=k+1 b e 2 t;k are also contained in Theorem 1. Following Berk (1974), we let the norm of a matrix D be de¯ned as kDk=supkDxk; where kxk·1: The Euclidean norm of the column vector x is used, and kxk 2 = x 0 x. Berk (1974) shows thatkDk 2 · P i;j d 2 i;j , where d i;j is the(i;j)th element of matrix D, andkDk is dominated by the largest modulus of D's eigenvalues. THEOREM1. Ifthedatageneratingprocesssatis¯esAssumption1,k=o ³ T 0:5¡d 1¡d ´ then as T !1 and we have: 1. ° ° ° b ¯¡¯ ° ° °=O ¡ k 0:5¡d T d¡0:5 ¢ , when d2(0;0:5). 2. b e t;k =e t +o p (1), when d2(0;0:5). 3. s 2 e;k p ¡!¾ 2 , when d2(0;0:5). Item 1 of Theorem 1 shows that the OLS estimator b ¯ is a consistent estimator for the population parameter vector ¯ where k ¡!1 as T ¡!1. However, item 1 of Theorem 1 also says that the convergence rate of b ¯ j to ¯ j could be slow if d is close to 0.5. Nevertheless,regardlessoftheconvergencerateof b ¯ j ,item2ofTheorem1clearly indicates that a stationary and invertible fractional white noise process can be well approximated by an AR(k) model when the lag length k is chosen appropriately. We note that in Theorem 1, we impose some restrictions on the growth rate of k under various ranges of d. 12 However, this growth rate should be viewed as the su±cient condition to ensure that a stationary and invertible fractional white noise process can be approximated well by an AR model. 1.2.2. Asymptotic Properties of Haugh's Statistics In this subsection we derive the limiting distribution of the Haugh's statistics when the DGP are the ARFIMA (p;d;q). THEOREM2. IfthedatageneratingprocessessatisfyAssumption(1), andallthe conditions in Theorem 1 hold, then as T !1, both S M and S ¤ M are distributed as Â 2 2M+1 when e 1;t is independent of e 2;s for all t and s. Theorem2setsupthetheoreticalfoundationofHaugh'sstatisticswhentheDGP aretheARFIMA(p;d;q)processes. Italsorelievesusfromthecomputationalburden of estimating the di®erencing parameters of Y 1t and Y 2t . This ¯nding is especially useful for empirical applications, because it is well known that the estimation of the di®erencing parameter is problematic when the sample size is small and d is close to 0.5, since many economics and ¯nancial data series contain only 100 to 200 observations. While Theorems 1 and 2 provide the su±cient condition for the order of k to ensure that the Haugh's statisitcs is asymptotically Chi-Square distributed. As a matteroffact, weonlyneeda¯niteordercriterionforchoosingasuitablektoensure the Chi-Square distribution of the Haugh's statistics. THEOREM 3. If the data generating processes satisfy Assumption 1, and the shocks of the two processes e 1;t and e 2;s are independent for all t and s, there exists a positive ¯nite constant c such that for k¸ c, when T !1, both S M and S ¤ M are distributed as Â 2 2M+1 . 13 UsinganAR(k)modelandkwhichisapositive¯niteconstant,suchasp+1,orp+2, Theorem3rebuildsthetheoreticalfoundationofapplicabilityofHaugh'sstatisticsto theARFIMA(p;d;q)processes. Thesigni¯cantdi®erencebetweentheassumptionof Theorem1and2andthatofTheorem3isontheconditionforlaglengthk. Theorem 3 implies the appropriate lag length k needed to approximate ARFIMA (p;d;q) is ¯nite and could be very small. This could be useful for empirical analysis. By Theorem 2 and 3, we can show that the asymptotic power of the Haugh test under the alternative hypothesis of two processes being correlated approaches to 1 as T ¡!1. THEOREM4. IfthedatageneratingprocessessatisfyAssumption(1), andallthe conditions in Theorem 1 hold, then as T !1, the power of both S M and S ¤ M are equal to 1 when e 1;t is correlated with e 2;s . 1.3. Order Selection for Approximating Long Memory by an AR Model The adequacy of an approximate model for the DGP depends on the choice of order, this section considers two commonly used criteria for choosing a suitable k of an AR (k) model for approximating ARFIMA processes, the Akaike's (1973) AIC and the Mallows's C p criterion. The Akaike Informaiton Criterion The Akaike information criterion (AIC) was designed to be an approximately unbiased estimator of the expected Kullback-Leibler information , a useful measure of the discrepancy between the operating and approximating models. Beran (1995) derived a version of the AIC for determining an appropriate autoregressive order for a class of ¯nite order fractional autoregressive processes as AIC(k)=logb ¾ 2 ² + 2(k+2) T ; (3) 14 where b ¾ 2 ² is the estimate of the variance ¾ 2 ² , when the true model is y t = P k j=1 ¯ j y t¡j +e t . However, our true model, y t = P 1 j=1 ¯ j y t¡j +e t , is of in¯nite dimension. When an autoregressive model of order k: y t =¯ 1 y t¡1 +¯ 2 y t¡2 +¢¢¢+¯ k y t¡k +e tk (4) is estimated instead, equation (3) is written as AIC(k)=logb s 2 e;k + 2(k+2) N : where N = T ¡2k, becasue we cannot estimate an in¯nite number of parameters from a ¯nite sample of T observations and we only can use T ¡k observations to estimate the AR(k) model for approximating the true model. Based on the framework of Beran's (1995) procedure, we can similarly estab- lish the Mallow's (1973) criterion for choosing a suitable order of AR(k) models to approximate a long memory process. The C p criterion Let Y 1 =(y t¡1 ;y t¡2 ;¢¢¢;y t¡k ); Y 2 =(y t¡k¡1 ;y t¡k¡2 ;¢¢¢;y t¡1 ) and ¯ =(¯ 1 ;¯ 2 ). Following Mallows (1973) and Ng & Perron (2005), we rewrite our true model as Y =Y 1 ¯ 1 +Y 2 ¯ 2 +e: We estimate the model Y =Y 1 ¯ 1 +e tk and obtain b e k =M 1 Y 2 ¯ 2 +M 1 e 15 where M 1 = £ I¡Y 1 (Y 0 1 Y 1 ) ¡1 Y 0 1 ¤ : Thus, due to E £ e 0 k e k ]=E[¿b ¾ 2 k ]=¯ 0 2 X 0 2 M 1 X 2 ¯ 2 +¾ 2 tr(M 1 )=¯ 0 2 X 0 2 M 1 X 2 ¯ 2 +(N¡k); the mean-squared prediction error of a model with k regressors is E h mse ³ Y 1 b ¯ 1 ;Y¯ ´i =E · ³ Y 1 b ¯ 1 ¡Y¯ ´ 0 ³ Y 1 b ¯ 1 ¡Y¯ ´ ¸ =¾ 2 k+¯ 0 2 Y 0 2 M 1 Y 2 ¯ 2 =¾ 2 k+E £ ¿¾ 2 k ¤ ¡(N¡k)¾ 2 E h mse ³ Y 1 b ¯ 1 ;Y¯ ´i ¾ 2 =k+ E £ ¿b ¾ 2 k ¤ ¾ 2 ¡(T ¡k)= E £ ¿b ¾ 2 k ¤ ¾ 2 +2k¡T where ¿ =T ¡2k as de¯ned in Ng & Perron (2005). Hence, the C p criterion can be given as C p = £ ¿b ¾ 2 k ¤ ¾ 2 +2k¡T: In addition, if N is the same across models, the C p yields the same minimizer as C ¤ p = £ ¿b ¾ 2 k ¤ ¾ 2 +2k; or SC p =lnb ¾ 2 k + 2k ¿ : 16 1.4 Monte Carlo Simulation 1.4.1. Finite Sample Performance of the AR Approximation for ARFIMA model Monte Carlo experiments are conducted in this section to examine the ¯nite sample properties of our analytical results. The Monte Carlo experiment for each model is based on 1,000 replications with di®erent sample size T. To construct T values of a stationary I(d) process, we ¯rst generate T independent values from the standardnormaldistributionandformaT£1columnvectore. Wethencalculatethe T autocovariances of the I(d) process, from which we construct the T£T variance- covariance matrix § and compute its Cholesky decomposition C (i.e., § = CC 0 ). Finally, thevectorpoftheT realizedvaluesoftheI(d)processisde¯nedbyp=Ce. This algorithm was suggested by McLeod and Hipel (1978) and Hosking (1984). We discard the ¯rst 200 values to obtain random starting values. Forthesakeofexaminingthe¯nitesamplepropertiesofvariousselectioncriteria, AIC and C p , for choosing an appropriate AR (k) model to ¯t to the long memory processe, we denote the following set of short run parameters (Á;µ) and di®erenc- ing parameters d = (0:1;0:2;0:3;0:4;0:45) to simulate several ARFIMA (p;d;q) processes of the form as follows: Modle (a). (1+0:7L)(1¡L) d y t =(1+0:5)e t ; Modle (b). (1+0:7L+0:3L 2 )(1¡L) d y t =e t ; Modle (c). (1+0:7L+0:3L 2 )(1¡L) d y t =(1+0:5)e t ; Modle (d). (1+0:7L+0:3L 2 +0:3L 3 )(1¡L) d y t =e t : Special case of ARFIMA (0;d;0) is also considered. The choice of the models was based on the empirical evidence of the ARFIMA processes in foreign exchange rates found by Cheung (1993) and in the forward premia for currencies investigated by Baillie and Bollerslev (1994). All simulations are performed for T =200;500 with 17 k ranging from 1 to some maximal order k max . As a matter of fact, according to Theorem 1, k max increases with sample size T and we can set k max = o(T 0:5¡d 1¡d ). Thus, with respect to the di®erent combinations of d and T, the values of k max can be calculated. We set k max = 9 and k max = 14 for all cases of T = 200 and T =500, respectively, regardless of d. Table 1.1 displays the frequency of the order selected by AIC and C p for model selection of AR (k) ¯tted to the ARFIMA (0;d;0) process. All models were esti- mated by the least squares method. The results reveal that the lag length needed to approximate a fractional white noise processes indeed is very small. Particularly, as d=0:1, no matter what the sample size and model selection criteria used, the value of k which minimizes these four selection criteria is 1. The largest value of k selected by C p for the combination of T = 500 and d = 0:45 is also only 3. In particular, for T =200;d=0:4, the order chosen by the AIC is very similar to those reported in Beran (1995) as well. The further evidence of how well the performance of the chosen k is also presented in Table 1.2. In Table 2.2, we report the mean of residual varianceb s 2 e;k , while the mean ofb s 2 e;k is very close to the simulated ¾ 2 which is set to be ¾ 2 = 1 in our experiment. It means the performance ofb e t;k asymptotically approaches to that of the white noise error e t . Likewise, to compare the small-sample performacne of various selection criteria inthegeneralARFIMA(p;d;q)processescase,Table2.1,3.1,4.1and5.1alsolistthe frequency of orders of AR (k) models selected by the criteria AIC and C p while the DGP are Models (a)-(d), respectively. These results show that the lag length needed to approximate the more general ARFIMA (1;d;1)process is close to that found in Table 2.1. For instance, in Table 2.1, the lag length k needed to approximate long memoryprocesseswellis2whenT =200nomatterwhatmodelselectioncriterionis chosen, while the value of k is only 1 in the case of the fractional white noise process 18 (Table 1.1). Further, as T =500, the largest lag length k to approximate the Model (b) is only 2 for d · 0:3 and 4 for d = 0:4 and d = 0:45. In spite of a relatively higher autogressive order in Model (b), Model (c) and Model (d), the appropriate lag length k to approximate those three models are still very small. Apparently, the results reveal that for T = 200 and T = 500;d = 0:1, k = 2 is large enough to approximate model (b)-(d) well; for the case of T =200;d2(0;0:5) and the case of T =500;d·0:2,k=3andk=4arewhatweneedforeachcase,respectively. And also, Table 2.2, 3.2, 4.2 and 5.2 show the further evidence of how well a appropriate lag length k for approximating Model (b)-Model (d). No doubt, except for k = 1 and k=2, the mean of the residual variance is still close to 1.0 as that in Table 2.2, especially as k=k ¤ which is the suitable lag length chosen by AIC and C p . As a result, according to the above simualtion evidence , we conclude that the sutible lag length k will rise slightly with the increase of the AR order in ARFIMA (p;d;q)processes. Thatistosay, k=p+1ork=p+2istheappropriatelaglength for approximating a ARFIMA (p;d;q) procees with the small sample size T = 200 or 500. 1.4.2. Finite Sample Properties of Haugh's Statistics Before reporting the ¯nite sample performance of Haugh's statistics, we present the conventional procedures of examining the independence between two stationary long memory processes. We ¯rst estimate the di®erencing parameter, d with the maximum likelihood method and then use t-ratio to be the test statistic. Those results are shown in Table 6.1 and Table 6.2. It appears that with the increase of thedi®erencingparameter, d, thebiasofmaximumlikelihoodestimatesofdbecomes larger. In addition, at 5% signi¯cant level, there are signi¯cant size distortions of t ratio tesst. Table 6.2 shows that there appears to have the signi¯cant spurious regression present, when T =200 and choices of M(5;9;15). Table 7.1 and 7.2 19 performance of Haugh's S M and S M ¤ at5% signi¯cant level, respectively, where Y 1t and Y 2t are the ARFIMA (0;d;0) processes and are independent of each other. Due to the similar order choicefrom AIC and C p found in last subsection, our stuidies are all based on k chosen by AIC. Obviously, They show that the rejection percentage of these two tests are very close to 5% for every pair of d 1 and d 2 with AIC model selection criterion. In other words, the size control of Haugh's two tests is extremely welleventhoughtheDGParerelaxtedtobethefractionalwhitenoiseprocesses. We also consider the power of Haugh's statistic by considering processes whose pattern of short cross correlation is as same as those of many ¯nancial time series, is also consideredtoexplorethepowerpropertiesofthesetwostatisticsasdescribedinHong (1996). Weassumethattheerrorterme 1;t ande 2;t aregeneratedas: ½ 12 (j)=0:2for j =0and½ 12 (j)=0forj6=0where½ 12 (j)denotesthecrosscorrelationfunctionof e 1t ande 2t atlagj. TheresultsinTable5.3and5.4showthattherejectionpercentage of 1000 replications at 5% are similar to those of Table 2 in Hong (1996), where two time series he considered are short memory processes. That is to say, not only the size but also the power performacne of Haugh's statistics does not depond on the long memory characteristics of the data series, as long as the data series themselves are stationary. The predicitons made in Theorem 2 are clearly borne out in our simulation studies. Moreover, Table 9.1-12.2 also examine the sampling properties of Haugh's test based on AR(k) approximation while the DGP are Models (a)-(d). All these results suggest that the size is always well controlled and the power of Haugh's statistics are good. Our experiment clearly demonstrates that the Haugh's statistics derived from prewhitening ARFIMA(p;d;q) processes by AR(k) works well in ¯nite sample. 20 1.5. Application to Foreign Exchange Rates Volatility Inthissection,weillustratetheapplicationofourproceduretoaforeignexchange rate study. The emergence of the Euro (EUR) and the Japanese Yen (JPY) as the two major currencies of the world has raised questions about how the two currencies will interact and what it will mean for foreign exchange rate markets. Furthermore, it is appealing to use the data measured at fairly ¯ne intervals of time to examine the possible relationship between di®erent currencies across di®erent markets. The data series under this study are daily spot exchange rates from January 3 , 2000 to August 31, 2005, with totally 1426 observations. De¯ne the volatilities of the EUR (Y Et ) and the JPY(Y Yt ) as squared returns of those curriencies, those are Y Et =R 2 Et = ¡ ln(P Et;t =P Et;t¡1 ) ¢ 2 Y Yt =R 2 Yt = ¡ ln(P Yt;t =P Yt;t¡1 ) ¢ 2 whereP Et andP Yt arethedailypricesoftheEuroandtheJapaneseYen,respectively. As a preliminary data analysis, our ¯st task is to check the statistics properties of Y Et and Y Yt . We use the Conditional Sum of Squares (CSS) estimation method discussed in Chung and Baillie (1993). The results of estimating ARFIMA (1;d;1) models for Y Et and Y Yt are reported in Table 8. The volatility of the Euro and the Japanese Yen are really I(0:3224) and I(0:1209) respectively. Also, the AR and MA parameter estimates (Á;µ) are highly signi¯cant. It appears that the volatility of Euro and Japanese Yen posseses stationary long memoryproperties. WethereforeexaminetheperformanceofAR(k)approximations of stationary long memory processes. Table 9 exhibits the order of AR (k) models ¯tted to Y Et and Y Yt by AIC. The `k' refer to the order of AR (k) approximation of two stationary long memory processes, Y Et and Y Yt . Hence, we set " k" from 16 to 25, because of sample size T =1426 and k Max =25. Apparently, as k=22, for 21 both series, their corresponding values of AIC , -0.141 and -0.382 are smaller than those of AIC as k are other numbers. On the other hand, based on a suitable model selection method to choose a k, such as AIC criterion, a stationary long memory process can be approximated well by an AR (k) model as real evidence shown in Table 13. In spite of noting that the lag lenghth k=22 is more suitable order of an AR(k) for approximating Y Et and Y Yt and in order to obtain robust empirical evidence, Table 14 also reports two test statistics with di®erent order k;k = 16¢¢¢25: The values of S M , for M = 5;10;15;20 and 25, are 39.75, 53.82, 58.32, 67.83, 72.17, respectively, as k=22. These values are signi¯cant at5% level. Furthermore, other than k=22, no matter what values of k chosen are , as shown in Table 14, all values ofS M are signi¯cant at5% level as well for M =5;10;15;20;25. Similarly, S ¤ M has the same performance as that of S M . It follows that our testing procedures o®er the robust empirical evidence, that is, there exists " strong contemporaneous correlation " between the volatility of Euro and Yen within 30 days, even though both of these two series are long memory processes. 1.6 Conclusion This chapter have proved that a stationary long memory ARFIMA (p;d;q) pro- cesscanbeaprroximatedwellbyanAR(k)modelwhenk ischosenappropriatelyand showtheapplicabilityofHaugh's(1976)stasticsbasedonthesamplecross-correlation function between two OLS residuals series to the ARFIMA (p;d;q) processes. The new tests can lessen the computational burden and avoid issues arising from inac- curate estimation of d. Moreover, the issues of spurious regression induced by the longmemoryprocessesconsideredinTsayandChung(2000)canbeavoidedbyusing Haugh'sstatistics. Furthermore,thischapterdemonstratedthedesirabilityofAkaike 22 information criterion for selecting the order of an autoregression for approximating long memory processes. Monte Carlo experiments conducted in this chapter con¯rm the theoretical pre- diction. We¯ndthatHaugh'sstatisticsbasedontheARapproximationisanaccurate and powerful method to detect the independence between two ARFIMA (p;d;q) pro- cesses. We also applied our methodology to investigate the independence between volatilities of two daily nominal dollar exchange rates-Euro and Japanese Yen. We found that there existed " strong contemporaneous correlation " between the volatil- ities of Euro and Yen within 30 days. 23 Chapter 2 Prediction of a Multivariate Long Memory Process via a Vector Sutoregressive Model 2.1 Introduction Approximating an unknown time series with an autoregressive (AR) model has been widely employed in statistics and econometrics. The rationale behind these studies hinges on the fact that the exact AR order of the data generating process (DGP)isusuallyunknownormaynotbe¯niteinpractice. Forexample,Berk(1974) derived the asymptotic distribution of the spectral density estimator constructed by ¯tting an AR model of order k to a univariate data series of sample size T, where k is allowed to grow with T at a suitable rate. Bhansali (1978) employed Berk's methodology to the problems of the prediction of future values of an univariate data series. Hong (1996) applied Berk's approach in testing the independence of two stationary processes. For the multivariate data series, Lewis and Reinsel (1985) extended the analysis of Berk (1974) and that of Bhansali (1978) to the problems of prediction based on a vector autoregressive (VAR) model of order k. LÄ utkepohl (1988) and LÄ utkepohl and Poskitt (1991, 1996), on the other hand, focused on the impulse responses and Granger causality analysis. Overall, the preceding research studies all concentrated on the data series which are stationary and weakly dependent, or I(0) processes. We recently have witnessed fast-growing studies on the I(d) process, that is, the integrated process of order d, where the di®erencing parameter d is a fractional num- ber. Manyeconomicand¯nancialdataarefoundtopossessthiskindofphenomenon. For example, Ding et al. (1993) and Breidt et al. (1998) showed that the absolute value or square of stock returns possesses long memory. Please also refer to Baillie 24 (1996) for more references. Nevertheless, it is well known that estimating the value of d is not an easy task, especially when the DGP is not one-dimensional, i.e., the vectorautoregressivefractionallyintegratedmovingaverage(VARFIMA)process. In theliteratureonlyfewpapersdealwith theestimation ofthe di®erencingparameters of the VARFIMA process, including Hosoya (1996) and Martin and Wilkins (1999). To estimate the di®erencing parameters of the VARFIMA process, Martin and Wilkins (1999) quite interestingly proposed an indirect estimation method which utilizes a VAR model as its auxiliary model. They found that the ¯nite sample per- formanceoftheindirectestimationmethodispromisinginestimatingthedi®erencing parameters of the VARFIMA process. However, Martin and Wilkins (1999) did not provide the theoretical support for their estimation method. The objective of this paper is to extend the analysis carried out by Lewis and Reinsel(1985)tothe r¡dimensionallongmemoryprocesswiththesamedi®erencing parameter d. Under suitable assumption imposed on the DGP, we prove that the one-step ahead prediction error of the \semiparametric" VAR(k) model converges in probability to its population counterpart when k is chosen appropriately. For the purpose of prediction, we establish the theoretical foundations of using the VAR(k) model ¯tting for the multivariate long memory process. Lewis and Reinsel (1985) also showed that the asymptotic e®ect of estimating the autoregressive coe±cients will in°ate the minimum mean square prediction error by a factor of(1+kr=T) when the DGP is an I(0) process. Although we prove that the multivariate least squares (LS) coe±cient estimator for the \semiparametric" VAR(k) model is consistent, we are unable to derive such an approximation factor (1+kr=T)forthelongmemoryprocess. WethusconductaMonteCarloexperiment to investigate the ¯nite sample prediction performance of the VAR(k) model. The simulations reveal that the average squared prediction errors are very close to their 25 population counterparts multiplied by a factor of (1+ kr=T), implying that the ¯nite sample prediction performance of the \semiparametric" VAR(k) model for the long memory process is comparable to the cases where the DGP is the I(0) process. For the purpose of prediction, a great deal of e®ort associated with estimating the di®erencing parameter can be saved, because we do not estimate the values of d any more. Economists are always interested in analyzing the relationship between data se- ries. This explains the popularity of the VAR model advanced by Sims (1980) in theoretical studies and empirical applications. Because the consistency of the mul- tivariate LS coe±cient estimator for the \semiparametric" VAR(k) model and that of the residual covariance matrix estimator are both established in this paper, our study can serve as the basis for future investigations about impulse responses and causality analysis concerning the multivariate long memory process. This chapter is organized as follows: In Section 2 we illustrate the model and the estimation method. The asymptotic properties of the multivariate LS estimator are discussed in Section 3. The asymptotic properties of the one-step ahead forecast error are considered in Section 4. Monte Carlo experiments are conducted in Section 5tostudythe¯nitesampleforecastingperformanceofthe\semiparametric"VAR(k) model. Section 6 concludes. 2.2. The Model and Parameter Estimation Supposey t =(y 1;t ;:::;y r;t ) 0 ,t=0;§1;§2;:::;isanr¡dimensional, stationary, stochastic process with MA representation y t = 1 X j=0 ª j e t¡j ; ª 0 =I r ; where the (r£r) coe±cient matrices ª j are often referred to as impulse response functions,I r isan(r£r)identitymatrix,ande t =(e 1;t ;:::;e r;t ) 0 isanr¡dimensional 26 independent and identically distributed (i.i.d.) white noise process with nonsingular covariance matrix §. It is well-known that the minimum mean square error (MSE) h-step (h¸ 1) forecast error of y t at origin t is P h¡1 j=0 ª j e t+h¡j with corresponding covariance matrix §(h)= P h¡1 j=0 ª j §ª 0 j . Extending Berk's (1974) and Bhansali's (1978) approaches, Lewis and Reinsel (1985) considered the problems of the linear prediction of y t+h based on y t ;y t¡1 ;:::, using a VAR model of order k ¯tted to a realization of T observations y 1 ;y 2 ;:::;y T . They established the consistency and asymptotic normality of the estimated au- toregressive coe±cients when k ! 1 as T ! 1, and P 1 j=0 kª j k < 1 with kª j k 2 = tr(ª 0 j ª j ), among other regularity conditions. Moreover, an asymptotic approximation to the mean square prediction error based on a VAR(k) model was also provided by Lewis and Reinsel (1985). The objective of this paper is to further extend the analysis of Lewis and Reinsel (1985) to the cases whereª j decays at a slow hyperbolic rate, i.e.,kª j k=O ¡ j d¡1 ¢ as j ! 1. In particular, we are concerned with the cases d > 0, because many economicand¯nancialtimeseriesexhibitthiskindofcharacteristic. Wethusneedto reevaluatethetheoreticalfoundationsinLewisandReinsel(1985)sincethepreceding condition P 1 j=0 kª j k<1 fails to hold when d>0. Concerning the U.S. equity market, Bollerslev and Jubinski (1999) showed that the persistence of volume and that of volatility are very close to each other. Ray and Tsay (1998) also considered the problems of testing for common long-range depen- dence in a vector time series. We thus assume that y t is generated as the following multivariate long memory process: h (1¡L) d I r i y t =e t ; (1) where d >0, and denotes the Kronecker product, i.e., we assume the di®erencing parameterof eachelementof y t isidentical toeachother. The case d=0 is excluded 27 from our analysis, because it has been addressed by Lewis and Reinsel (1985). When r = 1, y t is degenerated to be y t = y 1;t = (1¡L) ¡d e 1;t , and y t is the well-known fractional white noise process. This process was ¯rst introduced by Granger (1980, 1981), Granger and Joyeux (1980), and Hosking (1981), and they showed that y t is stationary when d < 0:5. As d > 0, y t is said to have a long memory, because the autocovariance functions of y t are not absolutely summable. Forexpositionalpurposes,weemploythenotationandmethodologyofLewisand Reinsel (1985) frequently. By (1), we rewrite y t as an in¯nite order VAR process, y t = 1 X j=1 A j y t¡j +e t ; A j = ¡(j¡d¡1)! j! (¡d¡1)! I r =¯ j I r ; (2) where ¯ j = O(j ¡d¡1 ) as j ! 1. Denote the autocovariance function of y t by (j)= E (y t y 0 t+j )forj =0;§1;§2;:::. Thisimpliesthat(¡j)=(j) 0 . By(2.3) of Lewis and Reinsel (1985), we note that the minimum mean square error linear predictor of y t+1 based on y t , y t¡1 ,:::, y t¡k+1 is given by y ¤ t;k (1)=A 1k y t +A 2k y t¡1 +:::+A kk y t¡k+1 ; (3) where A jk , j =1;:::;k, satisfy the \Yule-Walker" equations (A 1k ;A 2k ;:::;A kk )=¡ 0 1;k ¡ ¡1 k ; as ¡ 0 1;k = ((1) 0 ;(2) 0 ;:::;(k) 0 ) and ¡ k is a (kr£kr) matrix whose (m;n)th (r£r) block of elements is (m¡n), m;n=1;:::;k. Moreover, we denote § k = E f[y t+1 ¡y ¤ t;k (1)][y t+1 ¡y ¤ t;k (1)] 0 g as the mean square error of the predictor y ¤ t;k (1). Based on a realization of observations y t ;t = 1;:::;T, we estimate A jk , j = 1;:::;k, by the multivariate LS coe±cient estimator b A(k): b A(k)= ³ b A 1k ; b A 2k ;:::; b A kk ´ = b ¡ 0 1;k b ¡ ¡1 k ; (4) 28 where b ¡ 1;k =(T ¡k) ¡1 T¡1 X t=k Y t;k y 0 t+1 ; b ¡ k =(T ¡k) ¡1 T¡1 X t=k Y t;k Y 0 t;k ; and Y t;k = ¡ y 0 t ;y 0 t¡1 ;:::;y 0 t¡k+1 ¢ 0 : Moreover,§ k is estimated by the multivariate LS residualsb e t;k =y t+1 ¡b y t;k (1), i.e., b § k =(T ¡k) ¡1 T¡1 X t=k £ y t+1 ¡b y t;k (1) ¤£ y t+1 ¡b y t;k (1) ¤ 0 ; (5) where b y t;k (1)= k X j=1 b A jk y t¡j+1 (6) is an estimate of y ¤ t;k (1). To derive the asymptotic properties of the LS coe±cient estimator and the cor- responding one-step ahead prediction error based on the VAR(k) model, we impose the following Assumption 1 on e t throughout this paper. Assumption 1. (i) e t =(e 1;t ;:::;e r;t ) 0 is an r¡dimensional disturbance vector for the t-th observation; (ii) e t is an i.i.d. process with E (e t )=0, and E (e 2 i;t )=§ i;i < 1 for all i = 1;2;:::;r; (iii) each element of e t has a ¯nite fourth moment, i.e., E (e 4 i;t )=· i <1 for all i=1;2;:::;r. Assumption1requiresthate t isani.i.dvectorwith¯nitefourthmomentasLewis and Reinsel (1985) assume for the study of short memory process. To allow y t to be a VARFIMA(p;d;q) process, one anonymous referee suggested to relax e t to be a weakly dependent ARMA(p;q) process. His point is that the estimation of d in (1) by Whittle's methods is extremely easy. His suggestion is interesting and we do agree that the extension from (1) to the general VARFIMA(p;d;q) process is useful in empirical applications. However, as we mentioned previously, the objective of this 29 is to consider the forecasting performance of our \semiparametric" VAR(k) model \without" estimating the value of d in the VARFIMA(0;d;0) process. Therefore, we consider our paper as a ¯rst attempt toward the \semiparametric" estimation of the VARFIMA(p;d;q) process where d could be di®erent across di®erent univariate data series. Given y t ;t = 1;:::;T, and the estimators b A(k) and b § k derived from these T observations, we consider the problem of predicting the (T +1)th value of y t , i.e., y T+1 . By (3), the predictor is of the form b y T;k (1)= k X j=1 b A jk y T¡j+1 : (7) Weareinterestedintheasymptoticproperties ofthe predictionerror y T+1 ¡b y T;k (1). In the literature the asymptotic properties of the linear predictorb y T;k (1) de¯ned in (7),anditscorrespondingpredictionerrory T+1 ¡b y T;k (1)havebeenconsideredunder the assumption that the series used for the estimation of parameters and the series used for the prediction are independent of each other, but have the same stochas- tic structure. In particular, b A(k) is estimated from a realization of T observations x 1 ;x 2 ;:::;x T which have the same stochastic structure of y t , but are independent of y t . This simplifying assumption was adopted by Yamamoto (1976), Shibata (1977), Bhansali(1978),andLewisandReinsel(1985). Ontheotherhand,Phillips(1979)in- vestigatedtheconditionaldistributionoftheforecasterrorsgiventhe¯nalperiodob- servation,becausetheprecedingassumptionthatx t andy t areindependentwitheach other is not realistic. However, the simplifying assumption adopted by Yamamoto (1976), Shibata (1977), Bhansali (1978), and Lewis and Reinsel (1985) is used in this paper to derive the asymptotic properties of the prediction error y T+1 ¡b y T;k (1) due to the slow convergence rate of b A(k). 30 2.3. Asymptotic Properties of b A(k) and b § k In this section we ¯rst investigate the asymptotic properties of the coe±cient estimator b A(k), which are important in determining the asymptotic properties of b § k in (5). As b A(k)¡A(k)= b ¡ 0 1;k b ¡ ¡1 k ¡A(k) b ¡ k b ¡ ¡1 k = h b ¡ 0 1;k ¡A(k) b ¡ k i b ¡ ¡1 k ; and b ¡ 0 1;k ¡A(k) b ¡ k = 1 T ¡k T¡1 X t=k 0 @ y t+1 ¡ k X j=1 A j y t¡j+1 1 A Y 0 t;k ; we have b A(k)¡A(k)= 2 4 1 T ¡k T¡1 X t=k 0 @ y t+1 ¡ k X j=1 A j y t¡j+1 1 A Y 0 t;k 3 5b ¡ ¡1 k = 2 4 1 T ¡k T¡1 X t=k 0 @ 1 X j=k+1 A j y t¡j+1 1 A Y 0 t;k 3 5b ¡ ¡1 k + Ã 1 T ¡k T¡1 X t=k e t+1 Y 0 t;k ! b ¡ ¡1 k =U 1T b ¡ ¡1 k +U 2T b ¡ ¡1 k : (8) As shown in Wiener and Masani (1958) and later used by Lewis and Reinsel (1985), kABk 2 ·kAk 2 1 kBk 2 and kABk 2 ·kAk 2 kBk 2 1 ; wherek:k 1 denotes the usual operator norm, namelykBk 2 1 =sup kzk=1 kBzk 2 . Con- sequently, from (8) we have ° ° ° b A(k)¡A(k) ° ° °· ° ° ° b ¡ ¡1 k ° ° ° 1 [kU 1T k+kU 2T k]: (9) As a result, the three terms, kU 1T k, kU 2T k, and ° ° ° b ¡ ¡1 k ° ° ° 1 , are crucial in determining the asymptotic properties of ° ° ° b A(k)¡A(k) ° ° ° and the results are presented in Lemma A.1. Given the results in Lemma A.1, the consistency of b A(k) follows immediately. 31 We note that b § k (de¯ned in (5)) is an estimate of the mean square error of the predictor y ¤ t;k (1) (please refer to (3) of this paper). However, we also can view b § k as the residual covariance matrix estimator for§ when we try to approximate y t with a \misspeci¯ed" VAR(k) model. In fact, b § k is important in carrying out the impulse response and causality analysis concerning the multivariate long memory process. Thus, the consistency of b § k is also considered in Theorem 1. THEOREM 1. Given that y t is generated by (1) and Assumption 1 holds, 1=k+ 1=T !1,suchthatk=o ¡ T 1¡2d ¢ when0:25<d<0:5,andk=o ¡ T 0:5 (log T) ¡0:5 ¢ when 0<d·0:25, then we have the following results: 1. ° ° ° b A(k)¡A(k) ° ° °=O p ¡ k d¡0:5 ¢ when d2(0:25;0:5). ° ° ° b A(k)¡A(k) ° ° °=O p ¡ k ¡d (log k) 0:5 ¢ when d=0:25. ° ° ° b A(k)¡A(k) ° ° °=O p ¡ k ¡d ¢ when d2(0;0:25). 2. b § k p ¡!§. p ¡! denotes convergence in probability. The proofs of Theorem 1 and the following Theorem 2 are in the Appendix. Theorem 1 indicates that b A(k) and b § k are both consistent estimators for their pop- ulation counterparts even though y t is a multivariate long memory process. This implies that, provided k !1 with T at a suitable rate, the omission of the terms, y t¡k¡1 ;y t¡k¡2 ;:::; in the regressors does not incur any trouble in the consistency of b A(k) and b § k asymptotically. There are some remarks to be addressed here. First, we discuss the rationale of imposing the restrictions on the growth rate of k and the moment conditions of e t . 32 By (9), we note that ° ° ° b ¡ ¡1 k ° ° ° 1 is important in determining the asymptotic properties of the LS coe±cient estimator. From (A.1) of the Appendix, we also observe that ° ° ° b ¡ ¡1 k ° ° ° 1 · ° ° ¡ ¡1 k ° ° 1 + ° ° ° b ¡ ¡1 k ¡¡ ¡1 k ° ° ° 1 . In the proof for item 1 of Lemma A.1 (in the Appendix) we ¯rst prove that ° ° ¡ ¡1 k ° ° 1 can be uniformly bounded by a ¯nite positive constant F for all k. If we can show that ° ° ° b ¡ ¡1 k ¡¡ ¡1 k ° ° ° 1 p ¡! 0, then we prove that ° ° ° b ¡ ¡1 k ° ° ° 1 =O p (1). By (A.2), we ¯nd that the necessary condition for ° ° ° b ¡ ¡1 k ¡¡ ¡1 k ° ° ° 1 p ¡! 0 to hold is that ° ° ° b ¡ k ¡¡ k ° ° ° 1 must converge in probability to zero. From (A.3) and (A.4), we note that the asymptotic properties of ° ° ° b ¡ k ¡¡ k ° ° ° 1 actually depend on those of the terms: q mn;ij = (T ¡ k) ¡1 T¡1 P t=k y i;t¡m+1 y j;t¡n+1 ¡ ° jm¡nj § i;j , where (T ¡k) ¡1 T¡1 P t=k y i;t¡m+1 y j;t¡n+1 denotes the sample autocovariance function between y i;t¡m+1 and y j;t¡n+1 , i;j =1;:::;r; m;n=1;:::;k, and ° jm¡nj represents the au- tocovariance function of a univariate fractional white noise process z t at lagjm¡nj, i.e., (1¡L) d z t = b t , such that E(b t ) = 0 and E (b 2 t ) = 1, and § i;j is the (i;j)th element of §. To calculate the probability order of q mn;ij , we extend the methodology in the proof of Theorem 3 of Hosking (1996). Because both y i;t¡m+1 and y j;t¡n+1 are fractional white noise processes, the conditions in (1), (2), and (3) of Hosking (1996, p.262)areautomaticallyful¯lled. Givenitem(iii)ofAssumption1,wehave E (e 4 i;t )< 1and E (e 4 j;t )<1. Thus,alltheconditionsimposedinTheorem3ofHosking(1996) are satis¯ed. In fact, the ¯nite fourth moment condition is not stringent at all, since Lewis and Reinsel (1985) have employed the same moment condition for the study of I(0) process. Given that Assumption 1 holds, in (A.6) of the Appendix we show that max 1·m;n·k 1·i;j·r E (q 2 mn;ij )= 8 < : O ¡ T 4d¡2 ¢ ; if 0:25<d<0:5; O ¡ T ¡1 (log T) ¢ ; if 0<d·0:25. 33 where the \ max 1·m;n·k 1·i;j·r " is over all considered m;n;i, and j, where both m and n might increase with the sample size T. The preceding reasoning in turn leads to (A.7) in the Appendix, and we realize that a proper choice of lag length k makes ° ° ° b ¡ k ¡¡ k ° ° ° 1 p ¡!0. For example, if we choose k=o(T 1¡2d ), then ° ° ° b ¡ k ¡¡ k ° ° ° 1 p ¡!0 as d > 0:25. On the other hand, k = o ¡ T 0:5 (log T) ¡0:5 ) ¢ is what we need when 0 < d · 0:25. Therefore, the restrictions imposed on the growth rate of k under various ranges of d should be viewed as the su±cient condition to ensure the results in our Theorem 1 hold. Second, as mentioned previously, the assumption that the di®erencing parameter of each element of y t is identical to each other is motivated by the ¯ndings in Boller- slev and Jubinski (1999) and the studies in Ray and Tsay (1998). Extending our research to the case where each component of y t has di®erent di®erencing parameter is interesting and important. However, the results in Theorem 1 cannot be easily ex- tended to the case where the di®erencing parameter is not identical to each other. In particular, the exact form of¡ ¡1 k can be complicated, and the asymptotic properties of ° ° ° b ¡ k ¡¡ k ° ° ° 1 need to be carefully reconsidered. If the DGP is assumed to be the VARFIMA(p;d;q) process where d can be di®erent across di®erent univariate data series, then our approach has to be revised signi¯cantly to establish the consistency of b A(k) and b § k . 2.4. Asymptotic Properties of the Linear Predictorb y T;k (1) In this section we consider the asymptotic properties of the linear predictor b y T;k (1) de¯ned in (7). Note that y T+1 ¡b y T;k (1)=[y T+1 ¡y ¤ T (1)]¡ £ b y T;k (1)¡y ¤ T (1) ¤ =e T+1 ¡ £ b y T;k (1)¡y ¤ T (1) ¤ ; (10) 34 where y ¤ T (1) denotes the minimum mean square error linear predictor of the future value y T+1 , based on y T ;y T¡1 ;:::, when the coe±cients A j are known. That is, y ¤ T (1)= 1 X j=1 A j y T¡j+1 : (11) Moreover, e T+1 is independent of b A(k) and y ¤ T (1), because e t is assumed to be an independent white noise process. Undertheassumptionthat b A(k)isestimatedfromarealizationofT observations x 1 ;x 2 ;:::;x T which have the same stochastic structure of y t (but are independent of y t ), Lewis and Reinsel (1985) showed that the asymptotic e®ect of estimating A jk , for j = 1;2;:::;k, in°ates the minimum mean square prediction error by a factor of (1+ kr=T) when y t is an I(0) process. This (1+ kr=T) rate was ob- tained by Lewis and Reinsel (1985) with some speci¯c assumptions. Among them, T 0:5 P 1 j=k+1 kA j k!0 is used to ensure that the multivariate LS coe±cient estima- tor is asymptotically normally distributed (please refer to Theorem 4 of Lewis and Reinsel, 1985). Nevertheless, some of the assumptions used in Theorem 4 of Lewis and Reinsel (1985) cannot be satis¯ed trivially if y t is a long memory process. For example, P 1 j=k+1 kA j k = O ¡ k ¡d ¢ when y t is a long memory I(d) process. This explains why we are unable to obtain the same approximation factor (1+ kr=T) when the DGP is a long memory process by extending the methodology of Lewis and Reinsel (1985). Furthermore, the inability to establish the asymptotic normality of the estimated autoregressive coe±cient b A(k) and that of b § k is also due to the preceding explanations. AlthoughweareunabletoextendsomeoftheresultsinLewisandReinsel(1985) tothelongmemoryprocess, usingthesamesimplifyingassumptionadoptedinLewis and Reinsel (1985), we can prove that the one-step ahead prediction error, y T+1 ¡ b y T;k (1), does converge in probability to its population counterpart, e T+1 , in the following Theorem 2. 35 THEOREM 2. Given that all the conditions in Theorem 1 hold, and b A(k) is esti- mated from the T observations of x t which have the same stochastic structure of y t , but are independent of y t , then as T !1, y T+1 ¡b y T;k (1)=e T+1 +o p (1). There is one more technical issue to be considered, i.e., the way to select the lag length k of the approximating VAR model. For the choice of the \optimal in- sample goodness of ¯t" lag length, the modi¯ed multivariate portmanteau statistic considered in Hosking (1980), Li and McLeod (1981), and Poskitt and Tremayne (1981) has been used to test the white noiseness of the LS residuals when the order of the VAR model is ¯nite and known. This test statistic is de¯ned as: P M =T 2 M X i=1 (T ¡i) ¡1 tr ³ b C 0 i b C ¡1 0 b C i b C ¡1 0 ´ ; where b C i denotes the autocovariance functions of the LS residuals. However, the theoretical foundations of using the modi¯ed multivariate portmanteau statistic is not established when the DGP is a long memory process and the order of the VAR model increases with the sample size. On the other hand, when the choice of lag length is designed for the purpose of \out-of-sample prediction", Lewis and Reinsel (1985, p.408) suggested that Akaike's (1971) FPE criterion is useful if the DGP is an I(0) process. Thus, we might apply Akaike's (1971) FPE criterion to the multivariate long memory process. However, this might prove a nontrivial task. 2.5. Monte Carlo Experiment In this section a Monte Carlo experiment is conducted to investigate the ¯nite sample properties of our analytic studies. The Monte Carlo experiment for each model is based on 2,000 replications with di®erent sample sizes of T. Without loss of 36 generality, we only consider the bivariate case, i.e., r=2. To construct T£2 values of a stationary bivariate I(d) process, we ¯rst generate T £2 independent values from the standard bivariate normal distribution e, such that §= " 1:0 0:5 0:5 1:0 # : (12) We then calculate the T analytic autocovariances of the I(d) process, from which we construct the T £ T variance-covariance matrix ¨ and compute its Cholesky decomposition C (i.e., ¨= CC 0 ). Finally, the vector p£2 of the T realized values of the I(d) process is de¯ned by p = Ce, where four values of d are chosen, d = 0:15;0:25;0:35,and0.45. ThisalgorithmwassuggestedbyMcLeodandHipel(1978) and Hosking (1984). Moreover, two di®erent sample sizes are used: T = 100 and T = 200. Two hundred additional values are generated in order to obtain random starting values. All the programs are written in GAUSS language. The focus of our experiment is on the prediction performance of the \semipara- metric" VAR(k) model ¯tting. Our design basically follows the procedure used in Lewis and Reinsel (1985). Thus, for every sample size T, we generate T +4 real- izations. Four VAR(k) models (k = 2, 3, 4, and 5) were ¯tted to the ¯rst T =100 observationsusing(4)asLewisandReinsel(1985)did. When T =200, anotherfour VAR(k) models (k =5, 6, 7, and 8) were ¯tted to the ¯rst T =200 observations. We admit that the values of the chosen k are arbitrary. However, as noted in the preced- ing section, we do not have a solid theorem to guide us in choosing an \optimal" lag length for the purpose of prediction. The last 5 observations were compared to their corresponding predicted values, which were calculated by the formula given in Lewis and Reinsel (1985, p.403). The \observed" squared prediction error and the average of these \observed" squared prediction errors (based on 2000 replications) are given in Tables 1, 2, 3, and 4. 37 Moreover, to investigate the impact of demeaning on the prediction performance of the \semiparametric" VAR(k) model, we demean the data ¯rst and then recalculate theprecedingaverage\observed"squaredpredictionerrors. Forclarityofcomparison, weputtheresultsfromthedemeaneddataintotheparenthesis. Forthedetailsabout the experiment's design, please also refer to Lewis and Reinsel (1985). One of the important contributions of Lewis and Reinsel (1985) is that they can approximate the mean square error of the h ¸ 1 step ahead predictorb y t;k (h) with § k (h) ´ (1+kr=T)§(h), where §(h) is the mean square error of the \optimal" predictor y ¤ t (h) (Lewis and Reinsel, 1985, p.402). For purpose of comparison, we record the diagonal elements of§(h) and those of§ k (h)´(1+kr=T)§(h) in these tables, even though we are unable to derive such a beautiful approximation factor (1+kr=T) in this paper. TheresultsinTables16,17,18,and19revealthattheaveragesquaredprediction errors are very close to their corresponding counterparts in the diagonal elements of § k (h), no matter if we demean the original data or not. For ease of comparison, we also calculate the ratios of the average squared prediction errors relative to their corresponding values in the diagonal elements of § k (h) from these four tables, and the results are in Tables 20 and 21. Tables 20 and 21 clearly indicate that the approximation formula § k (h) also performs well for the long memory process, because we ¯nd that the ratios are very close to 1. In many situations the ratios are less than 1. This implies that the ¯nite sample prediction performance of the \semiparametric" VAR(k) model for the long memoryprocessisatleastcomparabletothecaseswheretheDGPisanI(0)process. In terms of prediction performance, the usefulness of \semiparametric" VAR(k) model ¯tting is supported by our experimental results. 38 2.6. Conclusion Thischapterprovesthetheoreticaljusti¯cationofthe`semiparametric"approach of predicting future values of a multivariate long memory process by autoregressive model ¯tting under suitable regularity conditions imposed on the DGP. For the pur- poseofprediction, agreatdealofe®ortassociatedwiththeestimationofdi®erencing parameters can be saved. Moreover, the consistency of the LS coe±cient estimator and that of the residual covariance matrix estimator are also established in this pa- per. Ourinvestigationswillserveastheplatformforfuturestudiesabouttheimpulse responses and causality analysis concerning the multivariate long memory process. Conclusion This dissertation have proved that a stationary long memory ARFIMA (p;d;q) process can be aprroximated well by an AR(k) model when k is chosen appropri- ately and show the applicability of Haugh's (1976) stastics based on the sample cross-correlation function between two OLS residuals series to the ARFIMA (p;d;q) processes. The new tests can lessen the computational burden and avoid issues aris- ingfrominaccurateestimationofd. Furthermore,thisdissertationalsodemonstrated the desirability of Akaike information criterion for selecting the order of an autore- gressionforapproximatinglongmemoryprocesses. Inaddition, we¯ndthatHaugh's statistics based on the AR approximation is an accurate and powerful method to detect the independence between two ARFIMA (p;d;q) processes by our simulation. Wealsoappliedourmethodologytoinvestigatetheindependencebetweenvolatilities of two daily nominal dollar exchange rates-Euro and Japanese Yen. Then, it found that there existed " strong contemporaneous correlation " between the volatilities of Euro and Yen within 30 days. As well, this dissertation proves the theoretical validity of the `semiparametric" approach of predicting future values of a multivariate long memory process by au- toregressive model ¯tting under suitable regularity conditions imposed on the DGP. 39 For the purpose of prediction, a great deal of e®ort associated with the estimation of di®erencing parameters can be saved. Furthermore, the consistency of the LS coe±cient estimator and that of the residual covariance matrix estimator are also established in this dissertation. Our investigations will serve as the platform for future studies about the impulse responses and causality analysis concerning the multivariate long memory process. 40 Table 1. The Frequencies of Selected Lag Lengths for AIC and C p repectively When the DGP is the Fractional White Noise Process d k(Lag) 1 2 3 4 5 6 7 8 9 T 0.1 200 AIC 758 ¤ 134 57 25 12 8 1 1 1 C p 758 ¤ 132 55 25 13 10 1 2 2 500 AIC 639 ¤ 192 81 45 24 6 5 6 1 C p 651 ¤ 185 79 43 23 6 5 6 1 0.2 200 AIC 585 ¤ 229 101 41 22 10 5 3 1 C p 583 ¤ 223 100 42 23 15 5 4 1 500 AIC 226 322 ¤ 202 106 72 29 15 15 3 C p 226 321 ¤ 201 105 73 29 16 15 3 0.3 200 AIC 391 ¤ 308 148 78 39 21 4 5 1 C p 380 ¤ 301 154 79 42 23 6 5 2 500 AIC 52 244 275 ¤ 173 107 66 30 27 8 C p 49 241 271 ¤ 172 109 68 30 29 12 0.4 200 AIC 238 361 ¤ 201 101 53 23 10 7 2 C p 231 344 ¤ 201 101 55 31 13 7 6 500 AIC 7 154 262 ¤ 214 143 95 47 27 21 C p 7 151 255 ¤ 209 147 97 46 29 25 0.45 200 AIC 198 365 ¤ 222 110 53 23 16 7 2 C p 190 348 ¤ 218 115 54 32 16 8 7 500 AIC 2 129 259 ¤ 223 146 99 53 32 23 C p 2 123 251 ¤ 220 144 104 54 34 25 Notes: The results are all based on 1,000 replications. ¤ denotes the lag length k ¤ at which simulated numbers chosen by the AIC and C p are the maximum 41 Table 2. Mean of Residual Variance S 2 e;k When the DGP is the ARFIMA (0;d;0) Process k 1 2 3 4 5 6 7 8 d T 0:1 200 0:99 0:99 0:99 0:99 0:99 0:99 0:994 0:994 500 1:00 1:00 1:00 0:99 0:99 0:99 0:999 1:000 0:2 200 1:006 0:998 0:998 0:994 0:994 0:994 0:994 0:994 500 1:017 1:007 1:003 1:001 1:000 1:000 1:000 0:099 0:3 200 1:020 1:003 0:997 0:995 0:994 0:993 0:993 0:993 500 1:040 1:017 1:008 1:005 1:002 1:001 1:000 1:000 0:4 200 1:035 1:007 0:998 0:995 0:994 0:993 0:992 0:993 500 1:063 1:025 1:010 1:007 1:004 1:002 1:001 1:000 0:45 200 1:020 1:003 0:997 0:995 0:994 0:993 0:993 0:994 500 1:072 1:028 1:011 1:008 1:004 1:002 1:001 1:000 42 Table 3. Mean of Residual Variance S 2 e;k When the DGP is the ARFIMA (¡0:7;d;0:5) Process k 1 2 3 4 5 6 7 8 d T 0:1 200 1:026 0:997 0:996 0:995 0:995 0:995 0:995 0:996 500 1:034 1:002 1:002 0:999 1:037 1:044 1:050 1:056 0:2 200 1:053 0:998 0:998 0:994 0:994 0:994 0:994 0:994 500 1:069 1:007 1:003 1:001 1:001 1:000 1:000 0:999 0:3 200 1:020 1:003 0:997 0:995 0:994 0:993 0:993 0:993 500 1:119 1:015 1:009 1:004 1:003 1:001 1:000 1:000 0:4 200 1:035 1:007 0:998 0:995 0:994 0:993 0:992 0:993 500 1:176 1:022 1:011 1:006 1:004 1:002 1:001 1:001 0:45 200 1:020 1:003 0:997 0:995 0:994 0:993 0:993 0:994 500 1:200 1:025 1:012 1:007 1:005 1:001 1:000 0:999 43 Table 4. The Mean of the estimated ARFIMA (0;d;0) Model Parameter d 0:1 0:2 0:3 0:4 0:45 T MLE 100 0:093 0:194 0:311 0:428 0:495 200 0:094 0:195 0:303 0:415 0:482 500 0:096 0:196 0:302 0:411 0:477 44 Table 5. The Rejection Percentages of the t When Y 1t and Y 2t are Fractional White Noise, and ½ 12 (j)=0 for all j Y 1t d 1 0:1 0:2 0:3 0:4 0:45 Y 2t d 2 0:1 10:4 14:5 15:3 18:6 24:1 0:2 14:3 15:0 15:5 19:7 27:3 0:3 15:7 16:0 16:8 24:7 30:2 0:4 18:0 20:2 24:2 28:9 32:7 0:45 24:2 27:8 30:8 33:1 33:4 45 Table 6. The Rejection Percentages of the S M Test at 5% Level of Signi¯cance When Y 1t and Y 2t are Fractional White Noise, e 1;t ;e 2;t »N(0;1), and ½ 12 (j)=0 for all j Y 1t d 1 k M 0:1 0:2 0:3 0:4 0:45 Y 2t d 2 0:1 k AIC 5 4:5 4:3 4:5 4:6 4:8 9 4:3 3:9 3:8 4:2 4:1 15 3:6 3:7 3:7 4:1 3:9 0:2 k AIC 5 4:3 4:5 4:7 4:6 4:7 9 4:4 4:3 4:2 4:3 4:3 15 4:2 3:6 5:1 5:3 4:7 0:3 k AIC 5 4:6 4:7 4:6 4:8 4:8 9 4:3 4:0 4:4 4:6 4:8 15 3:8 3:8 4:1 4:1 4:2 0:4 k AIC 5 4:6 4:8 4:7 4:4 4:5 9 4:2 4:2 4:3 4:8 4:6 15 3:8 3:8 4:1 4:1 4:2 0:45 k AIC 5 4:5 4:6 4:6 4:4 4:5 9 4:6 4:3 4:6 4:5 4:6 15 4:0 4:0 4:1 4:2 4:3 46 Table 7. The Rejection Percentages of the S ¤ M Test at 5% Level of Signi¯cance When Y 1t and Y 2t are Fractioanl White Noise, e 1;t ;e 2;t »N(0;1), and ½ 12 (j)=0 for all j Y 1t d 1 k M 0:1 0:2 0:3 0:4 0:45 Y 2t d 2 0:1 k AIC 5 5:0 4:9 5:0 5:0 5:0 9 5:2 5:3 5:3 5:0 5:0 15 5:7 5:7 5:7 5:4 5:3 0:2 k AIC 5 5:3 4:8 4:9 5:0 5:1 9 5:1 5:2 5:1 5:2 5:1 15 5:6 5:4 5:6 5:8 5:7 0:3 k AIC 5 5:3 5:3 5:1 5:1 5:1 9 5:2 5:3 5:5 5:8 5:3 15 5:6 6:0 5:7 5:7 5:8 0:4 k AIC 5 5:1 5:2 5:4 5:2 5:0 9 5:5 5:4 5:5 5:7 5:6 15 5:6 5:7 5:5 5:8 5:8 0:45 k AIC 5 5:0 5:0 5:2 5:1 5:1 9 5:6 5:4 5:3 5:6 5:6 15 5:3 5:7 5:6 5:9 5:9 . 47 Table 8. The Rejection Percentages of the S M Test at 5% Level of Signi¯cance When Y 1t and Y 2t are Fractional White Noise, e 1;t ;e 2;t »N(0;1), ½ 12 (0)=0:2, and ½ 12 (j)=0 for all j6=0 Y 1t d 1 k M 0:1 0:2 0:3 0:4 0:45 Y 2t d 2 0:1 k AIC 5 40:8 41:0 40:5 40:3 40:0 9 30:2 30:4 30:4 30:1 30:1 15 23:0 23:3 23:6 23:2 23:8 0:2 k AIC 5 40:7 41:4 41:0 40:9 40:6 9 29:4 28:9 29:1 30:3 30:0 15 21:6 21:8 24:4 23:7 23:4 0:3 k AIC 5 40:0 40:2 40:3 40:5 40:7 9 30:2 30:2 29:7 30:5 30:3 15 21:3 21:2 21:6 21:8 21:9 0:4 k AIC 5 39:1 39:6 40:2 39:9 40:2 9 29:3 28:6 30:3 30:3 30:1 15 21:2 21:3 21:7 21:9 21:9 0:45 k AIC 5 38:9 39:5 39:6 40:0 40:1 9 29:9 29:8 28:8 29:3 30:0 15 20:0 20:4 21:9 22:1 22:0 . 48 Table 9. The Rejection Percentages of the S ¤ M Test at 5% Level of Signi¯cance When Y 1t and Y 2t are Fractional White Noise, e 1;t ;e 2;t »N(0;1), ½ 12 (0)=0:2, and ½ 12 (j)=0 for all j6=0 Y 1t d 1 k M 0:1 0:2 0:3 0:4 0:45 Y 2t d 2 0:1 k AIC 5 41:7 41:7 41:4 41:2 41:3 9 32:2 33:1 30:9 30:9 32:2 15 23:5 25:6 26:2 25:2 25:8 0:2 k AIC 5 41:6 41:9 41:7 41:7 41:4 9 33:8 32:3 33:3 32:3 31:4 15 25:9 26:1 25:6 25:7 25:6 0:3 k AIC 5 40:8 41:1 41:7 40:9 41:5 9 32:7 33:0 32:7 32:3 32:3 15 25:8 26:6 26:2 26:0 25:7 0:4 k AIC 5 39:7 40:5 41:0 40:7 40:9 9 31:5 32:3 32:7 32:3 32:4 15 25:9 26:4 26:2 25:9 25:9 0:45 k AIC 5 39:6 40:1 40:9 40:7 40:5 9 32:3 32:4 32:7 32:1 32:2 15 25:6 26:1 26:6 25:8 26:3 49 Table 10. Estimation of ARFIMA (1;d;1) Models for Two Series Estimates Japanese Yen Euro d 0:1209 0:3224 Á 0:5948 0:3703 µ ¡0:6824 ¡0:6982 t d 1:6699 4:1900 t Á 4:0956 6:2149 t µ ¡4:8412 ¡10:2539 50 Table 11. s 2 e;k and AIC for AR(k) model ¯tted to Two Series k 16 17 18 19 20 Japanese Yen s 2 e;k 0:888 0:877 0:870 0:851 0:849 AIC ¡0:104 ¡0:103 ¡0:126 ¡0:130 ¡0:130 Euro s 2 e;k 0:703 0:793 0:792 0:792 0:790 AIC ¡0:361 ¡0:379 ¡0:376 ¡0:379 ¡0:378 51 Table 12. Test statistics for correlation in volatility k 16 17 18 19 20 21 22 23 M S M 5 37:27 ¤ 38:17 ¤ 37:10 ¤ 37:15 ¤ 39:77 ¤ 40:07 ¤ 39:75 ¤ 39:27 ¤ 10 50:33 ¤ 51:46 ¤ 50:33 ¤ 50:25 ¤ 53:28 ¤ 53:82 ¤ 53:82 ¤ 53:62 ¤ 15 55:47 ¤ 51:61 ¤ 55:35 ¤ 55:31 ¤ 57:76 ¤ 58:33 ¤ 58:32 ¤ 58:20 ¤ 20 64:55 ¤ 65:74 ¤ 64:61 ¤ 64:60 ¤ 66:99 ¤ 67:50 ¤ 67:83 ¤ 67:77 ¤ 25 69:31 ¤ 69:36 ¤ 68:26 ¤ 68:59 ¤ 71:23 ¤ 71:85 ¤ 72:17 ¤ 72:17 ¤ 30 72:68 72:93 71:78 72:11 74:60 75:27 75:57 75:68 35 76:31 76:77 75:71 76:35 79:38 79:93 80:20 80:53 40 86:02 86:53 85:77 86:23 88:01 88:81 89:21 89:62 S ¤ M 5 37:30 ¤ 38:21 ¤ 37:14 ¤ 37:18 ¤ 39:81 ¤ 40:10 ¤ 39:78 ¤ 39:31 ¤ 10 50:44 ¤ 51:57 ¤ 50:43 ¤ 50:35 ¤ 53:38 ¤ 53:92 ¤ 53:93 ¤ 53:73 ¤ 15 55:61 ¤ 56:76 ¤ 55:50 ¤ 55:46 ¤ 57:90 ¤ 58:47 ¤ 58:46 ¤ 58:35 ¤ 20 64:81 ¤ 66:00 ¤ 64:87 ¤ 64:87 ¤ 67:25 ¤ 67:76 ¤ 68:10 ¤ 68:04 ¤ 25 69:65 ¤ 69:28 ¤ 68:58 ¤ 68:92 ¤ 71:56 ¤ 72:18 ¤ 72:50 ¤ 72:51 ¤ 30 73:08 73:32 72:17 72:51 74:99 75:67 75:97 76:09 35 76:80 77:25 76:20 76:84 79:89 80:44 80:72 81:06 40 86:77 87:28 86:53 86:99 88:76 89:56 89:97 90:39cr 52 References 1. 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(1991): \Asymptotic Properties of the LSE in a Regression Model with Long-Memory Stationary Errors," The Annals of Statistics, 19, 158-177. 61. Yamamoto, T.(1976): \ Asymptotic Mean Square Prediction Error for An Au- toregressive Model with Estimated Coe±cients," Applied Statistics 25, 123-127. 58 Appendix A: Proof of the Chapter 1 ToproveTheorem1,weneedaseriesoflemmas. Beforepresentingtheselemmas, we note here that the DGP is generated to be the ARFIMA(p;d;q) process de¯ned in Assumption 1 with d2 (0;0:5). For clarity of exposition, let us de¯ne the OLS estimator b ¯ k;T as: b ¯ k;T = Ã T X t=k+1 Y t Y 0 t ! ¡1 T X t=k+1 Y t y t ; where b ¯ 0 k;T =( b ¯ 1;k ;:::; b ¯ k;k ), and Y 0 t =(y t¡1 ;:::;y t¡k ). Assume b R(k)= T X t=k+1 Y t Y 0 t ; and b °(k)= T X t=k+1 Y t y t ; then b ¯ k;T ¡¯ = b R(k) ¡1 T X t=k+1 Y t e t;k ; where e t;k =e t + 1 X j=k+1 ¯ j y t¡j : Suppose R= 2 6 6 6 4 ° 0 ° 1 ::: ° k¡1 ° 1 ° 0 ::: ° k¡2 . . . . . . . . . . . . ° k¡1 ° k¡2 ::: ° 0 3 7 7 7 5 ; where ° s is the autocovariance function of y t at lag s. We then have b ¯ k;T ¡¯ = µ 1 T ¡k b R(k) ¶ ¡1 1 T ¡k Ã T X t=k+1 Y t e t;k ! : Moreover, if we further de¯ne T X t=k+1 Y t e t;k =M and 1 T ¡k b R(k)=R ¤ ; 59 then b ¯ k;T ¡¯ can be rewritten as b ¯ k;T ¡¯ = 1 T ¡k ³ R ¤ ¡1 ¡R ¡1 ´ M + 1 T ¡k R ¡1 Ã T X t=k+1 Y t e t ! : (A:1) Moreover, Let C denote an arbitrary ¯nite positive constant. LEMMAA.1. IfthedatageneratingprocessistheARFIMA(p;d;q)process,then as k!1, T !1 and k=T !0, we have the following results: 1. kR ¡1 k is bounded for d2(0;0:5). 2. ° ° °R ¤ ¡1 ¡R ¡1 ° ° °=O p ¡ kT 2d¡1 ¢ , when d2(0:25;0:5) and k=o ¡ T 1¡2d ¢ ; ° ° °R ¤ ¡1 ¡R ¡1 ° ° °=O p ¡ kT ¡0:5 (log T) 0:5 ¢ whend2(0;0:25]andk=o ¡ T 0:5 (log T) ¡0:5 ¢ . A.1. Proof of Lemma A.1 The spectral density of the the ARFIMA(p;d;q) process y t is f(¸)= ¾ 2 ¯ ¯ ¯µ ³ e ¡i¸ ´¯ ¯ ¯ 2 2¼ ¯ ¯ ¯Á ³ e ¡i¸ ´¯ ¯ ¯ 2 ¯ ¯ ¯1¡e ¡i¸ ¯ ¯ ¯ ¡2d : If we factor Á(L) and µ(L) as follows: Á(L)=1¡Á 1 L¡:::¡Á p L p =(1¡´ 1 L)(1¡´ 2 L):::(1¡´ p L); µ(L)=1+µ 1 L+:::+µ q L q =(1¡» 1 L)(1¡» 2 L):::(1¡» q L); then f(¸) can be rewritten as: f(¸)= ¾ 2 q Y j=1 £ 1¡2» j cos(¸)+» 2 j ¤ 2¼ p Y j=1 £ 1¡2´ j cos(¸)+´ 2 j ¤ ¯ ¯ ¯1¡e ¡i¸ ¯ ¯ ¯ ¡2d 60 BecausetherootsofÁ(L)andµ(L)areassumedtobeoutsidetheunitcircle, f(¸)> 0 for all ¸ in our model setting. Moreover, f(¸)= ¾ 2 [µ(1)] 2 2¼ [Á(1)] 2 ¸ ¡2d ; as ¸!0: We thus have min ¸ f(¸)>0 and ByProposition4.5.3ofBrockwellandDavis(1991), theeigenvaluesofthecovariance matrix R are all greater than 0, and the eigenvalues of R ¡1 is uniformly bounded for all k. This proves item 1 of Lemma A.1. To prove item 2 of Lemma A.1, let Q = R ¤ ¡ R, q = kR ¤ ¡1 ¡ R ¡1 k and p=kR ¡1 k. Observe that q= ° ° R ¤¡1 ¡R ¡1 ° ° = ° ° R ¤¡1 (R¡R ¤ )R ¡1 ° ° · ° ° R ¤¡1 ° ° kR¡R ¤ k ° ° R ¡1 ° ° ·(p+q)kQkp; and q·p 2 kQk(1¡pkQk) ¡1 : (A:2) Equation (A.2) indicates that kQk p ¡! 0 is the necessary condition for q p ¡! 0 to hold. Thus, we need to ¯rst consider the asymptotic properties of Q. Letq i;j bethe(i;j)thelementofQ,wherei;j =1;:::;k,i.e.,Qisak£k matrix with q i;j =(T ¡k) ¡1 T X t=k+1 y t¡i y t¡j ¡° i¡j =b ° i¡j ¡° i¡j ; where b ° s denotes the sample autocovariance function of y t at lag s. We ¯rst note that E(q i;j )=E(b ° i¡j )¡° i¡j =0, it follows that E(q 2 i;j )=E © (b ° i¡j ¡° i¡j ) 2 ª = E [b ° i¡j ¡ E (b ° i¡j )+ E (b ° mn;ij )¡° i¡j ] 2 = E [b ° i¡j ¡ E (b ° i¡j )] 2 =Var(b ° i¡j ): 61 To derive the asymptotic properties of E (q 2 i¡j ) or those of Var(b ° i¡j ), we ¯rst rewrite y t¡i and y t¡j into moving average representations: y t¡i = 1 X s=0 Ã s e t¡i¡s ; y t¡j = 1 X s=0 Ã s e t¡j¡s : (A:3) Both y t¡i and y t¡j satisfy the conditions in equations (1), (2), and (3) of Hosking (1996, p.262) automatically. Given that item (iii) of Assumption 1 is satis¯ed, i.e., E (e 4 t ) < 1, then the conditions imposed in Theorem 3 of Hosking (1996) are all ful¯lled. Following the arguments in the proof of Theorem 3 of Hosking (1996), we show that E (q 2 i;j ) is uniformly bounded by max 1·i;j·k E (q 2 i;j )= 8 < : O ¡ T 4d¡2 ¢ ; if 0:25<d<0:5; O ¡ T ¡1 (log T) ¢ ; if d=0:25. Note that both i and j might increase with the sample size T. For the case 0 < d < 0:25, E (q 2 i;j ) can be uniformly bounded by max 1·i;j·k E (q 2 i;j ) = O ¡ T ¡1 (log T) ¢ ; together with the preceding results, we thus have max 1·i;j·k E (q 2 i;j )= 8 < : O ¡ T 4d¡2 ¢ ; if 0:25<d<0:5; O ¡ T ¡1 (log T) ¢ ; if 0<d·0:25. (A:4) Given the results in (A.4) andkDk 2 · P i;j d 2 i;j , where d i;j is the(i;j)th element of matrix D, it follows that E kQk 2 · X 1·i;j·k max 1·i;j ·k E ° ° ° ° ° (T ¡k) ¡1 T X t=k+1 y t¡i y 0 t¡j ¡°(i¡j) ° ° ° ° ° 2 = 8 < : O ¡ k 2 T 4d¡2 ¢ ; if 0:25<d<0:5; O ¡ k 2 T ¡1 (log T) ¢ ; if 0<d·0:25. 62 Asd2(0:25;0:5),followingtheargumentsofLÄ utkepohlandPoskitt(1996,p.69) to say that ° ° °k ¡1 T 1¡2d ³ b ¡ k ¡¡ k ´° ° ° is bounded in probability, i.e., ° ° ° b ¡ k ¡¡ k ° ° ° = O p ¡ kT 2d¡1 ¢ . The preceding arguments can be similarly applied to the cases d 2 (0;0:25] to show that kQk= 8 < : O p ¡ kT 2d¡1 ¢ ; if 0:25<d<0:5; O p ¡ kT ¡0:5 (log T) 0:5 ¢ ; if 0<d·0:25. (A:7) Given the results in (A.7), k = o(T 1¡2d ) must be imposed to make kQk 2 p ¡!0 as d > 0:25; when 0 < d· 0:25, k = o(T 0:5 (log T) ¡0:5 ) is needed to ensure that kQk 2 p ¡!0. Conditional on these restrictions being imposed, by (A.2) kQk p ¡!0 when ( k=o(T 1¡2d ); if 0:25<d<0:5, k=o(T 0:5 (log T) ¡0:5 ); if 0<d·0:25. (A:8) Thus, when these conditions are all ful¯lled, kQk p ¡! 0, and item 2 of Lemma 1 is established. LEMMAA.2. GiventhattheconditionsinLemmaA.1hold,andforsomepositive constant C and K p as T !1, we have the following results: 1. ° ° ° 1 T¡k P T t=k+1 Y t e t ° ° °=C¾ 2 k 1=2 T 1=2 =O p ³ k 1=2 T ¡1=2 ´ , if d2(0;0:5). 2. ° ° ° 1 T¡k P T t=k+1 Y t (e t;k ¡e t ) ° ° °= CK p ¾ 2 k 0:5¡d T d¡0:5 = O p (k 0:5¡d T d¡0:5 ), if d2 (0;0:5). 3. ° ° ° 1 T¡k P T t=k+1 Y t e t;k ° ° °=O p ¡ k ¡d T 0:5+d ¢ , if d2(0;0:5). 63 A.2. Proof of Lemma A.2 We note that E ° ° ° ° ° 1 T ¡k T X t=k+1 Y t e t ° ° ° ° ° 2 =(T ¡k) ¡2 k X i=1 E Ã T X t=k+1 y t¡i e t ! 2 =(T ¡k) ¡1 k X i=1 ° 0 ¾ 2 =C 1 ¾ 2 kT ¡1 =O(kT ¡1 ); where C is a positive ¯nite constant. Additionally, because y t¡i is independent of e t when i>0, and item 1 of Lemma A.2 is obtained. To prove item 2 of Lemma A.2, we observe that ° ° ° ° ° 1 T ¡k T X t=k+1 Y t (e t;k ¡e t ) ° ° ° ° ° 2 =(T ¡k) ¡2 k X i=1 ( T X t=k+1 y t¡i (e t;k ¡e t ) ) 2 ; where e t;k ¡e t = P 1 j=k+1 ¯ j y t¡j . Observe that the 2nd moment norm of Z = P T t=k+1 P 1 j=k+1 ¯ j y t¡j y t¡i satis¯es © E jZj 2 ª 1=2 ·jEZj+ © E jZ¡EZj 2 ª 1=2 ´A ¤ +B ¤ (A:9) To calculate the asymptotic properties of A ¤ and B ¤ in (A.9), let us ¯rst de¯ne ° s¡t = E (y t¡i y s¡i ) and ° ¤ s¡t = E n³ P 1 j=k+1 ¯ j y t¡j ´³ P 1 j=k+1 ¯ j y s¡j ´o : Using First Moment Bound Theorem in Findley and Wei (1993), we ¯nd that B ¤ ·K p Ã T X s=k+1 T X t=k+1 ° s¡t ° ¤ s¡t ! 1=2 (A:10) where K p denotes some positve ¯nite constant. Furthermore, we note that T X s=k+1 T X t=k+1 ¯ ¯ ° s¡t ° ¤ s¡t ¯ ¯ ·° ¤ 0 T X s=k+1 T X t=k+1 j° s¡t j·° ¤ 0 (T ¡k) T¡k¡1 X i=¡(T¡k¡1) j° i j: (A:11) 64 The de¯nition of the spectral density offy t g is f y (¸)= jµ(e ¡i¸ )j 2 jÁ(e ¡i¸ )j 2 j1¡e ¡i¸ j ¡2d ¾ 2 =(2¼)= jµ(e ¡i¸ )j 2 jÁ(e ¡i¸ )j 2 j2sin(¸=2)j ¡2d ¾ 2 =(2¼) and the autocovariances are °(h)= Z ¼ ¡¼ e ih¸ f(¸)d¸= ¾ 2 ¼ Z ¼ 0 jµ(e ¡i¸ )j 2 jÁ(e ¡i¸ )j 2 cos(h¸)(2sin(¸=2)) ¡2d d¸: h=0;1;2;¢¢¢ Thus, ° ¤ 0 = Z ¼ ¡¼ j 1 X j=k+1 ¯ j e ¡ij¸ j 2 f(¸)d¸ · Z ¼ ¡¼ 1 X j=k+1 1 X h=k+1 ¯ j ¯ h e ¡i(j¡h)¸ f(¸)d¸ ·j 1 X j=k+1 1 X h=k+1 ¯ j ¯ h j Z ¼ ¡¼ e i(h¡j)¸ f(¸)d¸ =j 1 X j=k+1 1 X h=k+1 ¯ j ¯ h j° h¡j ·° 0 0 @ 1 X j=k+1 ¯ j 1 A 2 +j 1 X j=k+1 1 X h=k+1;h6=j ¯ j ¯ h ° h¡j j ·° 0 0 @ 1 X j=k+1 ¯ j 1 A 2 +C 1 X j=k+1 1 X h=k+1;h6=j j ¡d¡1 h ¡d¡1 (h¡j) 2d¡1 ·° 0 0 @ 1 X j=k+1 ¯ j 1 A 2 +C¾ 2 k ¡2d =Ck ¡2d ¾ 2 (A:12) By (A.12), we can get the order of the second term on the right side of (A.11) being O(T 1+2d k ¡2d ). Thus, B ¤ ·K p Ã T X s=k+1 T X t=k+1 ° s¡t ° ¤ s¡t ! 1=2 =C¾ 2 K p T 0:5+d k ¡d (A:13) . 65 To calculate the asymptotic properties of A ¤ in (A.9), we introduce the following notations: y t¡j = 1 X l 1 =0 Á l 1 e t¡j¡l 1 ; y t¡i = 1 X l 2 =0 Á l 2 e t¡i¡l 2 : Using some algebra computation, we ¯nd that A ¤ =jEZj=j E 0 @ T X t=k+1 1 X j=k+1 ¯ j y t¡j y t¡i 1 A j =j 8 < : T X t=k+1 1 X j=k+1 ¯ j 0 @ 1 X l 1 =0 Á l 1 Á (j¡i)+l 1 1 A ¾ 2 9 = ; j »j 8 < : T X t=k+1 1 X j=k+1 ¯ j µZ 1 l 1 =0 l d¡1 1 (l 1 +(j¡i)) d¡1 dl 1 ¶ ¾ 2 9 = ; j (A:14) Thus, we can deduce that the order magnitude of R 1 l 1 =0 l d¡1 1 (l 1 +(j¡i)) d¡1 dl 1 is less than or equal to that of 1 d l d 1 (l 1 +(j¡i)) d¡1 j 1 0 , i.e O µZ 1 l 1 =0 l d¡1 1 (l 1 +(j¡i)) d¡1 dl 1 ¶ ·O µ 1 d l d 1 (l 1 +(j¡i)) d¡1 j 1 0 ¶ or µZ 1 l 1 =0 l d¡1 1 (l 1 +(j¡i)) d¡1 dl 1 ¶ =C ¤ µ 1 d l d 1 (l 1 +(j¡i)) d¡2 j 1 0 ¶ ; where C ¤ denotes some ¯nite positive constant. 66 Hence, following the argument of Robinson (1993, p.693), we note that 1 X j=k+1 j ¡d¡1 » Z 1 k+1 t ¡d¡1 dt=O ³ k ¡d ´ : Then, we can observe (A.14) and have A ¤ = 8 < : ¾ 2 T X t=k+1 1 X j=k+1 ¯ j µZ 1 l 1 =0 l d¡1 1 (l 1 +(j¡i)) d¡1 dl 1 ¶ 9 = ; = 8 < : ¾ 2 C T X t=k+1 1 X j=k+1 ¯ j µ 1 d l d 1 (l 1 +(j¡i)) d¡1 j 1 0 ¶ 9 = ; ·¾ 2 C 8 < : T X t=k+1 1 X j=k+1 ¯ j µ 1 d T d (T +(j¡i)) d¡1 ¶ 9 = ; »¾ 2 C 8 < : T X t=k+1 1 X j=k+1 j ¡d¡1 µ 1 d T d (T +(j¡i)) d¡1 ¶ 9 = ; ·C¾ 2 ( T X t=k+1 k ¡d µ 1 d T d (T +((k+1)¡i)) d¡1 ¶ ) ·C¾ 2 ½ Tk ¡d µ 1 d T d (T +((k+1)¡i)) d¡1 ¶¾ (A:15) Combining the results from (A.9),(A.13) and (A.15), we prove 67 Therefore, ° ° ° ° ° 1 T ¡k T X t=k+1 Y t (e t;k ¡e t ) ° ° ° ° ° 2 =O(k 1¡2d T 2d¡1 ): Thus, item 2 of Lemma 3 is established. Because the order of magnitude of item 3 of Lemma A.2 cannot be greater than that of the sum of the items 1 and 2 of Lemma A.2, we can obtain the asymptotic properties of item 3 of Lemma 2 without di±culty. The details are omitted and the proof of Lemma A.2 is now complete. A.4. Proof of Theorem 1 With the help of Lemmas A.1 and A.2, we are ready to prove Theorem 1. For the item 1 of Theorem 1, we note that ° ° ° b ¯¡¯ ° ° °· ° ° ° ° ³ R ¤ ¡1 ¡R ¡1 ´ 1 T ¡k M ° ° ° ° + ° ° ° ° ° R ¡1 1 T ¡k Ã T X t=k+1 X t e t !° ° ° ° ° =Z 1T +Z 2T +Z 3T =O(k 0:5¡d T d¡0:5 ); 68 because Z 1T · ° ° °R ¤ ¡1 ° ° ° ° ° ° ° 1 T ¡k M ° ° ° ° = ( O(k 1:5¡d T 3d¡1:5 ) if d2(0:25;0:5); O(k 1:5¡d T d¡1 (log T) 0:5 ) if d2(0;0:25], Z 2T · ° ° R ¡1 ° ° ° ° ° ° ° 1 T ¡k T X t=k+1 X t e t ° ° ° ° ° =O(k 1=2 T ¡1=2 ); Z 3T · ° ° R ¡1 ° ° ° ° ° ° ° 1 T ¡k ( T X t=k+1 X t ¡ e t;k ¡e t ¢ )° ° ° ° ° =O(k 0:5¡d T d¡0:5 ) Item 1 of Theorem 1 follows immediately. To prove item 2 of Theorem 1, we note that the OLS residuals are de¯ned as: b e t;k =e t + 1 X j=k+1 ¯ j y t¡j ¡ k X j=1 ³ b ¯ j ¡¯ j ´ y t¡j : (A:17) Thus,theterms P 1 j=k+1 ¯ j y t¡j and P k j=1 ³ b ¯ j ¡¯ j ´ y t¡j mustbothconvergeinprob- ability to zero, otherwise equation (2) cannot be sustained. First, for the term P 1 j=k+1 ¯ j y t¡j , we have 1 X j=k+1 ¯ j y t¡j · 1 X j=k+1 j¯ j jjy t¡j j=O p (1) 0 @ 1 X j=k+1 j¯ j j 1 A =O p (k ¡d ): Second, by Cauchy-Schwarz's Inequality, we obtain k X j=1 ³ b ¯ j ¡¯ j ´ y t¡j · 2 4 k X j=1 ³ b ¯ j ¡¯ j ´ 2 3 5 1=2 2 4 k X j=1 y 2 t¡j 3 5 1=2 =O(k 1¡d T d¡0:5 ) (A:18) Because P 1 j=k+1 ¯ j y t¡j = o p (1) and P k j=1 ³ b ¯ j ¡¯ j ´ y t¡j = o p (1); item 2 of Theorem 1 follows immediately. Item 3 of Theorem 1 can be easily proved with the help of item 2 of Theorem 1, and the details are omitted. 69 A.5. Proof of Theorem 3 To prove Theorem 3, let k be a positive ¯nite constant, such as k = 1;2;¢¢¢;c and c is a positive ¯nite constant. Obviously, it follows that ° ° ° ° ° ° C 2 T ¡k T¡l X t=k+1 e t;1 8 < : k X j=1 ³ b ¯(k)¡¯(k) ´ u t+l¡j 9 = ; ° ° ° ° ° ° =O p (T d¡1 ); (A:32) ° ° ° ° ° ° C T ¡k T¡l X t=k+1 e t+l;2 8 < : k X j=1 ³ b ¯ j ¡¯ j ´ u t¡j 9 = ; ° ° ° ° ° ° =O p (T d¡1 ); (A:33) ° ° ° ° ° ° C T ¡k 8 < : T¡l X t=k+1 e t;1 0 @ 1 X j=k+1 ¯ j u t+l¡j 1 A 9 = ; ° ° ° ° ° ° =O p (T ¡0:5 ); (A:34) ° ° ° ° ° ° C T ¡k 8 < : T¡l X t=k+1 e t+l;2 0 @ 1 X j=k+1 ¯ j u t¡j 1 A 9 = ; ° ° ° ° ° ° =O p ((T ¡0:5 ); (A:35) C T ¡k 8 < : T¡l X t=k+1 0 @ 1 X j=k+1 ¯ j u t¡j 1 A 0 @ 1 X j=k+1 ¯ j u t+l¡j 1 A 9 = ; =O(T ¡1=2 ); (A:36) ° ° ° ° ° ° C T ¡k T¡l X t=k+1 0 @ 1 X j=k+1 ¯ j u t¡j 1 A 8 < : k X j=1 ³ b ¯ j ¡¯ j ´ u t+l¡j 9 = ; ° ° ° ° ° ° =O(k 1¡2d T 2d¡1 ) (A:37) and ° ° ° ° ° ° C 2 T ¡k T¡l X t=k+1 8 < : k X j=1 ³ b ¯ j ¡¯ j ´ u t¡j 9 = ; 8 < : k X j=1 ³ b ¯ j ¡¯ j ´ u t+l¡j 9 = ; ° ° ° ° ° ° =O(T 2d¡1:5 ): (A:38) Thus, together with (A.32), (A.33), (A.34), (A.35), (A.36), (A.37) and (A.38) Theorem 3 can be established. 70 A.6. Proof of Theorem 4 By Theorem 2, and we note that, under the null hypothesis of independence, b ½ 12 (l)= T¡l X t=k+1 e t;1 e t+l;2 Ã T X t=k+1 e 2 t;1 ! 1=2 Ã T X t=k+1 e 2 t;2 ! 1=2 +o p (T ¡1=2 )=½ 12 (l)+o p (T ¡1=2 ); the asymptotic power of the two Haugh tests based on the AR approximation of long memory is lim T!1 Pr(S M >Â 2 2M+1 )=1¡©S M Â 2 2M+1 =1 where © is the cumulative distribution of Â 2 2M+1 . 71 Appendix B: Proof of Chapter 2 B.1. Proof of Theorem 2. By (10), we have y T+1 ¡b y T;k (1)=e T+1 ¡[b y T;k (1)¡y ¤ T (1)]; where b y T;k (1)¡y ¤ T (1)= h b A(k)¡A(k) i Y T;k ¡ 1 X j=k+1 A j y T¡j+1 ; and Y T;k = ¡ y 0 T ;y 0 T¡1 ;:::;y 0 T¡k+1 ¢ 0 : We also can prove that E ° ° ° ° ° ° 1 X j=k+1 A j y T¡j+1 ° ° ° ° ° ° 2 =O(k ¡2d ): (B:1) For the term h b A(k)¡A(k) i Y T;k , by (8), we note ¯rst that h b A(k)¡A(k) i =(U 1T +U 2T ) b ¡ ¡1 k =(U 1T +U 2T )¡ ¡1 k +(U 1T +U 2T ) ³ b ¡ ¡1 k ¡¡ ¡1 k ´ : Thus, by items 2 and 3 of Lemma A.1, and (A.1) together with the conditions imposed in Theorem 1, we show that ° ° ° h b A(k)¡A(k) i ¡(U 1T +U 2T )¡ ¡1 k ° ° °·k(U 1T +U 2T )k ° ° ° b ¡ ¡1 k ¡¡ ¡1 k ° ° ° 1 =o p (1): Furthermore, from the Proposition 6.1.2 and item (iii) of the De¯nition 6.1.4 of Brockwell and Davis (1991), we note that h b A(k)¡A(k) i p ¡!(U 1T +U 2T )¡ ¡1 k : (A:2) Using the Theorem 2.7 of van der Vaart (2000), we have h b A(k)¡A(k) i Y T;k p ¡!(U 1T +U 2T )¡ ¡1 k Y T;k : (B:3) 72 Wenowconsidertheasymptoticpropertiesofthe(r£1)matrix(U 1T +U 2T )¡ ¡1 k Y T;k with its the mean squared error matrix E h (U 1T +U 2T )¡ ¡1 k Y T;k Y 0 T;k ¡ ¡1 k 0 (U 1T +U 2T ) 0 i : Conditionalontheassumptionthat(U 1T +U 2T )isindependentofY T;k (becauseU 1T and U 2T compose of the realizations of x t which are independent of y t as assumed in Theorem 2, and E (Y T;k Y 0 T;k )=¡ k , we obtain E h (U 1T +U 2T )¡ ¡1 k Y T;k Y 0 T;k ¡ ¡1 k 0 (U 1T +U 2T ) 0 i = E h (U 1T +U 2T )¡ ¡1 k 0 (U 1T +U 2T ) 0 i : (B:4) As ¡ ¡1 k is symmetric, we then observe that ° ° °(U 1T +U 2T )¡ ¡1 k 0 (U 1T +U 2T ) 0 ° ° °·k(U 1T +U 2T )k ° ° ¡ ¡1 k ° ° 1 k(U 1T +U 2T )k It follows that E ° ° °(U 1T +U 2T )¡ ¡1 k 0 (U 1T +U 2T ) 0 ° ° °· ° ° ¡ ¡1 k ° ° 1 E k(U 1T +U 2T )k 2 : (B:5) Note that E k(U 1T +U 2T )k 2 · E kU 1T k 2 + E kU 2T k 2 +2 E kU 1T kkU 2T k · E kU 1T k 2 + E kU 2T k 2 +2 ³ E kU 1T k 2 E kU 2T k 2 ´ 1=2 ; we can prove that E k(U 1T +U 2T )k 2 =o(1); (B:6); by item 2 and 3 of Lemma A.1. Using (B.2) and the result that ° ° ¡ ¡1 k ° ° 1 is uniformly bounded by a ¯nite positive constant F, 73 We show that E ° ° °(U 1T +U 2T )¡ ¡1 k 0 (U 1T +U 2T ) 0 ° ° °=o(1): Consequently, we have (U 1T +U 2T )¡ ¡1 k Y T;k =o p (1); (B:7) together with (A.50), we establish that h b A(k)¡A(k) i Y T;k =o p (1): (B:8) Combining the results in (B.7) and (B.9), we show that y T+1 ¡b y T;k (1)=e T+1 +o p (1); (B:9) and Theorem 2 is proved. 74

Abstract (if available)

Abstract This dissertation focuses on the AR approximation of long memory processes and its applications. The first chapter proposes an easy test for two stationary autoregressive fractionally integrated moving average (ARFIMA) processes being uncorrelated via AR approximations. We prove that an ARFIMA (p,d,q) process, \phi(L)(1-L)^{d}y_{t} = \theta(L)e_{t}, d\in (0,0.5), where e_{t} is a white noise, can be approximated well by an autoregressive (AR) model and establish the theoretical foundation of Haugh's (1976) statistics to test two ARFIMA processes being uncorrelated. The Haugh statistic is useful because it can avoid the issues of spurious regression induced by the long memory processes considered by Tsay and Chung (2000). Using AIC or Mallow's C_{p} criterion as a guide, we demonstrate through Monte Carlo studies that a lower order AR(k) model is sufficient to prewhitten an ARFIMA process and the Haugh test statistics after AR pre-whitening perform very well in finite sample. We illustrate the use of our methodology by investigating the independence between the volatility of two daily nominal dollar exchange rates-Euro and Japanese Yen. We find that there exists "strongly simultaneous correlation " between the volatilities of Euro and Yen within 30 days.

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University of Southern California Dissertations and Theses

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A general equilibrium model for exchange rates and asset prices in an economy subject to jump-diffusion uncertainty

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Creator Wang, Shin-Huei (author)

Core Title Approximating stationary long memory processes by an AR model with application to foreign exchange rate

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Economics

Publication Date 08/08/2008

Defense Date 06/15/2007

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

Tag approximation,long memory,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Hsiao, Cheng (committee chair), Hyungsik Roger Moon (committee member), Jakasa Cvitatic (committee member)

Creator Email modelhuei@hotmail.com,shin-huei.wang@uclouvain.be

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1566

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Rights Wang, Shin-Huei

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Repository Location Los Angeles, California

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