A Vector Auto-Regressıve (VAR) Model for the Turkish Financial Markets

Transcription

1 MPRA Munich Personal RePEc Archive A Vecor Auo-Regressıve (VAR) Model for he Turkish Financial Markes Selcuk Bayraci and Yakup Ari and Yavuz Yildirim Yediepe Universiy 24. April 2011 Online a hp://mpra.ub.uni-muenchen.de/30475/ MPRA Paper No , posed 27. April :15 UTC

2 A VECTOR AUTO-REGRESSIVE (VAR) MODEL FOR THE TURKISH FİNANCIAL MARKETS Selcuk Bayraci, PhD Candidae Yediepe Universiy / Financial Economics selcuk.bayraci@alumni.exeer.ac.uk Yakup Ari, PhD Candidae Yediepe Universiy / Financial Economics yakupari@gmail.com Yavuz Yildirim, PhD Candidae Yediepe Universiy / Financial Economics yildirimyz@gmail.com Absrac In his paper, we develop a vecor auoregressive (VAR) model of he Turkish financial markes for he period of June June and forecass ISE100 index, TRY/USD exchange rae, and shor-erm ineres raes. The ou-ofsample forecas performance of he VAR model is compared wih he resuls from he univariae models. Moreover, he dynamics of he financial markes are analyzed hrough Granger causaliy and impulse response analysis. Keywords: mulivariae financial ime series, vecor auo-regressive (VAR) model, impulse response analysis, Granger causaliy JEL Classificaion: C01, C51

3 1. Inroducion Modeling he dynamics of financial markes is gaining populariy among researchers because of heoreical and echnical reasons. Economic agens, boh privae and public, have close ineres wih he movemens of he sock marke index, ineres raes, and exchange raes in order o make invesmen and economic policy decisions. Therefore, building efficien forecasing models for hese variables play imporan roles in he decision making processes. Alhough, univariae models, ARMA(p,q) and GARCH(p,q), are widely used in he lieraure by he researchers for modeling and forecasing purposes, i is also imporan o analyse he ineracion beween variables in a mulivariae framework. In his paper, we move forward ino his area by applying a vecor auoregressive (VAR) model in modeling he financial variables of Turkish marke. For his purpose, daily observaions of IMKB100 index, TCMB benchmark bond raes, and USD/TRY exchange raes beween he four years period of and are used. The res of he paper is; par wo deals wih he lieraure review and relaed work, par 2 and 3 give deailed descripion of mehodology and daa analysis, empirical resuls are discussed in par 5 and par 6 concludes. 2. Lieraure Review The vecor auoregression (VAR) model is one of he mos successful, flexible,and easy o use models for he analysis of mulivariae ime series. VAR models in economics were made popular by Sims [8]. I is a naural exension of he univariae auoregressive model. The VAR model is useful for describing he dynamic behavior of financial ime series and for forecasing. The superior forecass o hose from univariae ime series models and elaborae heory-based simulaneous equaions models can be provided by using VAR models.. Forecasing is quie flexible since hey can be made condiional on he poenial fuure pahs of specified variables in he model. There are many sudies abou modeling financial ime series wih VAR models.the mos imporan one is he book of Culberson[3] ha is abou socks,

4 bonds and foreign exchange.bu here are a few sudy abou Turkish Financial Marke especially in he period which includes 2008 financial crisis. In addiion o daa descripion and forecasing, he VAR model is also used for srucural inference and policy analysis. In srucural analysis, cerain assumpions abou he causal srucure of he daa under invesigaion are imposed, and he resuling causal impacs of unexpeced shocks or innovaions o specified variables on he variables in he model are summarized.these causal impacs are usually summarized wih impulse response funcions and forecas error variance decomposiions. The definiive echnical reference for VAR models is Lukepohl [5], and updaed surveys of VAR echniques are given in Wason [11] and Lukepohl [6] and Waggoner and Zha [10]. Applicaions of VAR models o financial daa are given in Hamilon [4], Campbell, Lo and MacKinlay [2],Culberson [3], Mills [7] and Tsay [9]. 3. Mehodology When building a VAR model, he following seps can be used. Firs we can use he es saisic M(i ) or he Akaike informaion crierion o idenify he order, hen esimae he specified model by using he leas squares mehod (if here are saisically insignifican parameers, by removing hese parameers he model shoul be reesimae), and finally use he Qk (m) saisic of he residuals o check he adequacy of a fied model. Oher characerisics of he residual series, such as condiional heeroscedasiciy and ouliers, can also be checked. 3.1 Vecor AR(p) Models The ime series Y follows a VAR(p) model if i saisfies Y 0 1Y 1... py p a, p > 0, (1) where 0 is a k-dimensional vecor, and a is a sequence of serially uncorrelaed random vecors wih mean zero and covariance marix Σ. In applicaion, he

5 covariance marix Σ mus be posiive definie; oherwise,he dimension of Y can be reduced. The error erm a is mulivariae normal and j are k k marixes. Using he back-shif operaor B, he VAR(p) model can be wrien as (I 1B p p B ) Y = 0 + a, where I is he k k ideniy marix. In a compac form as follows Φ(B) Y = 0 + a, p where Φ(B) = I- 1B- - p B is a marix polynomial. If Y is weakly saionary, hen we have μ = E( Y ) = (I p ) 1 0 = [Φ(1)] 1 0 provided ha he inverse exiss since deerminan of Φ(1)] is differen from zero. Le Y ~ = Y - μ. Then he VAR(p) model becomes Y ~ ~ = 1 Y p 1 ~ + a. (2) Y p Using he equaion(2) below resuls can be obained Cov( Y, a ) = Σ, he covariance marix of a ; Cov( Y l, a ) = 0 for l > 0; l = 1 l p 1 l p for l > 0. (3) The equaion (3) is mulivariae version of Yule Walker equaion and i is called he momen equaions of a VAR(p) model 3.2 Building a VAR(p) Model The concep of parial auocorrelaion funcion of a univariae series can be generalized o specify he order p of a vecor series. Consider he following consecuive VAR models:

6 Y Y Y a 0 1Y 1 2Y2 a... =... Y...= Y 1... iy i a (4) The ordinary leas squares (OLS) mehod is used for esimaing parameers of hese models.this is called he mulivariae linear regression esimaion in mulivariae saisical analysis.[9] ( ) For he i h equaion in Eq. (3), le_ j be he OLS esimae of j and ˆ i he esimae of 0, where he superscrip (i ) is used o denoe ha he esimaes are for a VAR(i ) model. Then he residual is aˆ ( i) i i Y ˆ ( ) ( ) 1 Y 1... ˆ i Y i For i = 0, he residual is defined as_ Y ˆ (0) of Y.The residual covariance marix is defined as ˆ 1 ˆ ˆ (5) / i a T i T 2i ( i) ( ) a 1 i1 ˆ( i ) j be Y Y, where Y is he sample mean To specify he order p, we can use he i h and (i 1)h equaions in Eq. (4) o, esing a VAR(i ) model versus a VAR(i 1) model and es he hypohesis H 0 : l 0 versus he alernaive hypohesis H a : l 0 sequenially for l = 1, 2,.. [1].The es saisic is ˆ 3 i M i T k i ln 2 ˆ i1 The disribuion of M(i ) is a chi-squared disribuion wih 2 k degrees of freedom.

7 Alernaively, he Akaike informaion crierion (AIC) can be used o selec he order p. Assume ha a is mulivariae normal and consider he i h equaion in Eq. (4). One can esimae he model by he maximum likelihood (ML) mehod. For AR models, he OLS esimaes 0 and j are equivalen o he (condiional) ML esimaes. However, here are differences beween he esimaes of Σ.The ML esimae of Σ is [9] ˆ 1 ˆ ˆ (6) / i a T ( i) ( ) i a T i 1 The AIC of a VAR(i ) model under he normaliy assumpion is defined as 2 ~ 2k i AIC( i) ln( i ) (7) T For a given vecor ime series, one selecs he AR order p such ha AIC(p) = min1 i p AIC(i ), where p is posiive ineger. 3.3 Esimaion and Model Checking Boh of he ordinary leas squares mehod or he maximum likelihood mehod can be used o esimae yhe parameers of VAR model sine he wo mehods are asympoically equivalen. The esimaes are asympoically normal under some regulariy condiions, Afer consrucing he model, adequacy of he model should hen be checked. The Qk (m) saisic can be applied o he residual series o check he assumpion ha here are no serial or cross-correlaions in he residuals. For a fied VAR(p) model, he Qk (m) saisic of he residuals is asympoically a chi-squared 2 disribuion wih k m g degrees of freedom, where g is he number of esimaed parameers in he AR coefficien marixes.[9] 3.4 Srucural Analysis by Impulse Response Funcions

8 The general form of he VAR(p) model is shown in eq.(1). VAR(p) model also has a Wold represenaion as follows Y a 1a 1 2a2... (8) Where s ij s are moving average nxn marices. To inerpre he (i,j)-h elemen,, elemen of he marix s as he dynamic muliplier or impulse response y a i, s j, y a i, j, s s ij i,j=1,2,..,n (9) The condiion for he eq.(9) is var( a ) = Σ is a diogonal marix. If Σ is diogonal, i shows he elemen of Σ, a, are uncorrelaed. One way o make he errors uncorrelaed is o esimae he riangular srucural VAR(p) model ' ' y1 c1 11Y pY p 1 ' ' y2 c1 21 y1 21Y py p 2 : y n c ' ' 1 n1 y1... n, n1 yn 1, n1y 1... npy p n (10) he esimaed covariance marix of he error vecor uncorrelaed/orhogonal errors is diagonal. The are referred o as srucural errors. The Wold represenaion of Y based on he orhogonal errors is given by Y Where 0 B (B is he lower riangular marix of i, j in eq. (10). The diagonal elemens of he B is 1.) The The impulse responses o he orhogonal shocks j are

9 y i, s y j, i, j, s s ij s, where ij is he (i,j) h elemen of s. The plo of agains s is called he orhogonal impulse response funcion of yi wih respec o j. s ij 3.5 Srucural Analysis by Granger Causaliy In order o invesigae he causal relaionship beween he variables of he sysem, he linear Granger causaliy ess should be applied by using following sraegy. Compare he unresriced models; (11) (12) wih he resriced models (13) (14) where m1 m2 y a y x e 1 1i i 1 j ji 1 i1 j1 x and y m1 m2 x a x y e 2 2i i 2 j ji 2 i1 j1 m1 y a y e 1 1i i 1 i1 m1 x a x e 2 2i i 2 i1 are he firs order forward differences of he variables, a,, are he parameers o be esimaed and, e1, e 2 are sandard random errors. The lag

10 order m are he opimal lag orders chosen by informaion crieria.the equaions described above, are convenien ools for analyzing linear causaliy relaionship beween he variables. If 1 is saisically significan, and 2 is no, i can be said ha changes in variable y Granger cause changes in variable x or vice versa. If boh of hem are saisically significan here is a bivariae causal relaionship beween he variables, if boh of hem are saisically insignifican neiher he changes in variable y nor he changes in variable x have any effec over oher variable. 3.6 Forecasing If he fied model is adequae, hen i can be used o obain forecass.for forecasing, same echniques in he univariae analysis can be applied. To produce forecass and sandard deviaions of he associaed forecas errors can be done as following. For a VAR(p) model, he 1-sep ahead forecas a he ime origin h is p Y h ( 1) 0 iyh1i, and he associaed forecas error is e h a h 1. The i01 covariance marix of he forecas error is Σ. If Y is weakly saionary, hen he l- sep ahead forecas Y h (l) converges o is mean vecor μ as he forecas horizon increases. 4. Daa Analysis For his paper, daily observaions of TCMB benchmark bond rae, USD/TRY foreign exchange rae, and IMKB100 index values for he four year period beween and are used. Daa beween and (980) are used in-sample esimaion and daa beween and

11 are used for he ou-of-sample forecasing purposes. Figure 1 below shows he ime series plos of he hree variables during he sample period. Figure 1: Time Series Plos of he Variables Ineres Rae USD/TRY Exchange Rae IMKB100 Index Source: TCMB Daabase hp://evds.cmb.gov.r In order o build an appropriae model, all series ha are used in analysis mus be saionary herefore we should check he uni-roo srucure of he daa. Alhough above graph gives us a rough idea abou he saionariy srucure of he series we need more formal ess o check he saionary. We have applied Augmened Dickey-Fuller es o series in order o es uni-roos. Table 1 exhibis he resuls from ADF es applied o boh levels and firs differences of he series

12 Table 1: ADF Uni-Roo Tes Resuls Variable Deerminisic erms Lags Tes value Criical values 1% 5% 10% ineres consan,rend Δineres consan fx consan,rend Δfx consan xu100 consan,rend log(δxu100) consan Source: Own Sudy The ADF es resuls indicae ha all variables are non-saionary by no rejecing he null hypohesis of uni-roo a all levels of criical values, bu hey are all saionary afer firs differencing. Therefore, we use differenced series in our analysis, figure 2 below ime series plos of he differenced series. Figure 2: Time Series Plo of he Differenced Variables Change in Ineres Rae Change in USD/TRY Exchange Rae Log Change in IMKB100 Index Source: Own Sudy

13 5. Empirical Resuls In his par, our firs aim is o deermine he rue lag order for he model as Lukepohl [5] poins ou ha selecing a higher order lag lengh han he rue lag lenghs increases he mean square forecas errors of he VAR, and selecing a lower order lag lengh han he rue lag lenghs usually causes auocorrelaed errors. Therefore, accuracy of forecass from VAR models highly depends on selecing he rue lag lenghs. There are several saisical crierion for selecing a lag lengh. We have idenified a VAR(p) model for he analysis by using penaly selecion crieria such as Akaike Informaion Crierion (AIC), Bayesian Informaion Crierion (BIC), and Hannan-Quinn Informaion Crierion (HQC). The able 2 below shows he resuls of he selecion crierion. Table 2: VAR(p) Model Order Selecion Crierion Crieria Lag Order AIC HQC BIC Minimum Lag Order Source: Own Sudy The resuls from he able 2 sugges ha he appropriae model for our daa is VAR(2) because all hree mehods gave lag order 2 as minimum lag order. Afer we have idenified a VAR(2) model, we move forward o model esimaion process. The model esimaion resuls from he VAR(2) model are given in he following ables.

14 Table 3.1 Coefficien Esimaes for Ineres Rae Equaion Coefficien Sd.Error -value -prob Dineres_ Dineres_ Dfx_ Dfx_ DLxu100_ DLxu100_ Consan sigma RSS Table 3.2 Coefficien Esimaes for Exchange Rae Equaion Coefficien Sd.Error -value -prob Dineres_ Dineres_ Dfx_ Dfx_ DLxu100_ DLxu100_ Consan sigma RSS Table 3.3 Coefficien Esimaes for IMKB100 Equaion Coefficien Sd.Error -value -prob Dineres_ Dineres_ Dfx_ Dfx_ DLxu100_ DLxu100_ Consan sigma RSS Source: Own Sudy

15 When we look a he coeffiens for he ineres rae equaions apar from consan erm and firs lag of he ineres rae, are all saisically significan in erms of - value. All coeffiens are significan in foreign exchange equaions, whereas only is firs lag has saisically significan effec on sock index variable. Afer we have esimaed a suiable VAR(2) model for he variables, his sage of he analysis deals wih he diagnosic checking process. There are several mehods ha conrol he robusness of he model, we have used graphical analysis ools and saisical ess for he residuals for he diagnosic checks. The able 4 below exhibis he resuls of he serial correlaion, normaliy and heeroskedasiciy ess of he residuals. And he figure 3 and 4 shows he ACF and densiy plos of he model residuals. The diagnosic resuls imply ha VAR(2) model should be exended by making heavy ailed disrubiuonal assumpions of he residuals as he disribuional properies of he residuals are no normal. Also, heeroskedasiciy esing resuls sugges he applicaion of a mulivariae GARCH model for he series. We can say ha for he ineres rae and IMKB100 index VAR(2) could be a good model as i eliminaes he serial correlaion, bu for he foreign exchange series here is sill correlaed residuals as he es saisics sugges he rejecion of he null hypohesis of no serial correlaion unil lag 12. Perhaps, insead of symmeric lag order, we can use assymmeric lag order model for he variables. Table 4: Residual Diagnosic Tess Dineres Serial Correlaion Tes F(12,958) = [0.1231] Dfx Serial Correlaion Tes F(12,958) = [0.0006]** DLxu100 Serial Correlaion Tes F(12,958) = [0.0833] Dineres Normaliy Tes Chi^2(2) = [0.0000]** Dfx Normaliy Tes Chi^2(2) = [0.0000]** DLxu100 Normaliy Tes Chi^2(2) = [0.0000]** Dineres Heeroskedasiciy Tes F(12,957) = [0.0000]** Dfx Heeroskedasiciy Tes F(12,957) = [0.0000]** DLxu100 Heeroskedasiciy Tes F(12,957) = [0.0000]** Source: Own Sudy

16 Figure 4: Correlaions of Residuals Auocorrelaions wih 2 Sd.Err. Bounds Cor(DI,DI(-i)) Cor(DI,DFX(-i)) Cor(DI,DLX(-i)) Cor(DFX,DI(-i)) Cor(DFX,DFX(-i)) Cor(DFX,DLX(-i)) Cor(DLX,DI(-i)) Cor(DLX,DFX(-i)) Cor(DLX,DLX(-i)) Source: Own Sudy Figure 5: Residuals Densiy Plos 0.75 Densiy r:dineres N(0,1) Densiy r:dfx N(0,1) Densiy 0.6 r:dlxu100 N(0,1) Source: Own Sudy

17 In order o see he dynamics of he variables we have applied impulse response analysis and Granger causaliy ess. Figure 6: Impulse Response Funcion.004 Response of DI o Cholesky One S.D. Innovaions.012 Response of DFX o Cholesky One S.D. Innovaions DI DFX DLX Response of DLX o Cholesky One S.D. Innovaions DI DFX DLX DI DFX DLX Source: Own Sudy The figure 6 shows he combined graph of he impulse responses of each variable. As we can see from he graph ha one exogenous shocks o ineres raes and IMKB100 index have immediae effec on foreign exchange rae, whereas ineres rae responses are no significan. And, IMKB100 index has very lile response o exogenous shocks o oher variables.

18 Table 5: Granger Causaliy Tes Pairwise Granger Causaliy Tess Sample: Lags: 2 Null Hypohesis: Obs F-Saisic Probabiliy DFX does no Granger Cause DINTEREST E-05 DINTEREST does no Granger Cause DFX DLXU100 does no Granger Cause DINTEREST E-37 DINTEREST does no Granger Cause DLXU DLXU100 does no Granger Cause DFX E-83 DFX does no Granger Cause DLXU Source: Own Sudy Table 5 shows he Granger causaliy es resuls. The es resuls indicae ha here is a bivariae causal relaionship beween ineres raes and foreign exchange raes by rejecing he null hypohesis of no Granger causaliy. Whereas, here is one way causal relaionships beween ineres raes and IMKB100 index, and beween foreign exchange raes and IMKB100 index. While changes in IMKB100 index have direc effec over oher variables, he changes in ineres raes and foreign exchange raes do no cause changes in he IMKB100 index. Afer we have esimaed and checked our model for in-sample analysis, his sage deals wih he ou-of-sample forecasing performance analysis. We have used 21 observaion for he forecas purposes and compare he resuls of he VAR(2) model wih he univariae models which are chosen for each variable by penaly selecion crieria. ARMA(1,1) model is chosen as bes suiable model for ineres rae and foreign exchange rae series, and ARMA(1,3) for he IMKB100 index

19 series. Roo mean squared error (RMSE) saisics are used for he performance evaluaion ool. The able 6 shows he es resuls. Table 6: RMSE Saisics for Forecas Performance VAR(2) Model Univariae Model Dineres Dfx DLxu Source: Own Sudy According o RMSE saisic, univariae model gives beer ou-of-sample performance for ineres rae series, whereas VAR(2) model ouperforms univariae models in forecasing he foreign exchange raes and IMKB100 index. The saisic also sugges ha VAR(2) models ou-of-sample forecasing performance for ineres raes are beer han for he oher variables. 6. Conclusion In his paper, we have aemped o build a mulivariae ime series model for he Turkish financial markes. We applied vecor auoregressive (VAR) model in modeling and forecasing he Turkish ineres raes, USD/TRY exchange raes, and IMKB100 index for he four year period beween VAR(2) model has been choosen as bes candidae model for he varibles in sample period.model esimaion resuls, impulse response analysis and Granger causaliy ess indicae ha while VAR(2) model is a saisfacory model for ineres raes and exchange raes, i is no a suiable for he sock marke dynamics. A furher sudy on coninuous-ime sochasic models should be beer for modeling he dynamics of Isanbul Sock Exchange. Also, heeroskedasicy ess show ha volailiy of he series are no consan, an exended sudy on mulivariae GARCH models would be beer for modeling he series for he sample period.

20 References [1] Box, G.E.P. and G.C Tiao (1977), A canonical analysis of muliple ime series, Biomerica, 64, [2] Campbell, J.A. Lo and C. MacKinlay (1997). The Economerics of Financial Markes. Princeon Universiy Press, New Jersey [3] Culberson, K (1996). Quanaive Financial Economics: Socks, Bonds and Foreign Exchange, 1996, John Wiley & Sons [4] Hamilon, J.D. (1994). Time Series Analysis. Princeon Universiy Press, Princeon [5] Lukepohl, H. (1991). Inroducion o Muliple Time Series Analysis, Sprinder- Verlag: Berlin [6] Lukepohl., H (1999). Vecor Auoregressions, unpublished manuscrip, Insiu für Saisik und Ökonomerie, Humbold-Universia zu Berlin. [7] Mills, T.C. (1999). The Economeric Modeling of Financial Time Series, Second Ediion. Cambridge Universiy Press, Cambridge. [8] Sims, C.A. (1980) Macroeconomics and Realiy, Economerica, 48, 1-48 [9] Tsay, R.S. (2001). Analysis of Financial Time Series, Springer, p.p [10] Waggoner, D.F., and T. Zha. (1999). Condiional Forecass in Dynamic Mulivariae Models, Review of Economics and Saisics, 81 (4), [11] Wason, M. (1994). Vecor Auoregressions and Coinegraion, in Handbook of Economerics, Volume IV. R:F. Engle and D. MacFadden (eds.). Elsevier Science Ld., Amserdam [12] Zivo, E. and J.Wang (2006). Modeling Financial Time Series wih S- Plus,2nd Ediion. Springer, p.p ,

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