Vecorauoregressive Model and Coinegraion Analysis Par V Time Series Analysis Dr. Sevap Kesel 1
Vecorauoregression Vecor auoregression (VAR) is an economeric model used o capure he evoluion and he inerdependencies beween muliple ime series, generalizing he univariae AR Models. All he variables in a VAR are reaed symmerically by including for each variable an equaion explaining is evoluion based on is own lags and he lags of all he oher variables in he model. A VAR model describes he evoluion of a se of k variables measured over he same sampleperiod (єt) as a linear funcion of only heir pas evoluion. The variables are colleced in a k x 1 vecor y, which has as he ih elemen y i,, he ime observaion of variable yi. For example, if he ih variable is GDP, hen y i, is he value of GDP a. A (reduced) p-h order VAR, VAR(p), is
y c A y A y A y ε = + 1 1 + 2 2 +... + p p + where c is a k x 1 vecor of consans (inercep) Ai is a k x k marix (for every i = 1,..., p) and ε is a k x 1 vecor of error erms saisfying he condiions. Properies: E[ ε ] = 0 every error erm has mean zero E[ εε ] = Ω he conemporaneous covariance marix of errors E[ εε k ] = 0 Order of inegraion of he variables Noe ha all he variables used have o be of he same order of inegraion. We have he following cases: All he variables are I(0) (saionary): one is in he sandard case, ie. a VAR in level
All he variables are I(d) (non-saionary) wih d>1: The variables are coninegraed: he error correcion erm has o be included in he VAR. The model becomes a Vecor error correcion model (VECM) which can be seen as a resriced VAR. The variables are no coinegraed: he variables have firs o be differenced d imes and one has a VAR in difference
Example: VAR(1) Suppose {y 1 } єt denoe real GDP growh, {y 2 } єt denoe inflaion y1 c1 A11 A12 y1, 1 ε1 y = 2 c + 2 A21 A + 22 y 2, 1 ε 2 y = c + A y + A y + ε 1 1 11 1, 1 12 2, 1 1 y = c + A y + A y + ε 2 2 21 1, 1 22 2, 1 2 One equaion for each variable in he model. One equaion for each variable in he model. The curren (ime ) observaion of each variable depends on is own lags as well as on he lags of each oher variable in he VAR.
Srucural VAR (SVAR) wih p lags B y c B y B y B y e 0 = 0 + 1 1 + 2 2 +... + p p + where c 0 is a k x 1 vecor of consans, B i is a k x k marix, i = 0,..., p, and e is a k x 1 vecor of error erms. The main diagonal erms of he B 0 marix (he coefficiens on he ih variable in he ih equaion) are scaled o 1. The error erms e (srucural shocks) saisfy he condiions and pariculariy ha all he elemens off he main diagonal of he covariance marix E(e e ') = Σ are zero. Tha is, he srucural shocks are uncorrelaed.
IMPULSE RESPONSE FUNCTION The key ool o race shor run effecs wih an SVAR is he impulse response funcion. y c A y A y A y ε = + 1 1 + 2 2 +... + p p + can be expressed as MA( ) y = c + ε + ψ 1ε 1 + ψ 2ε 2 +... = Ψ( B) ε y + l ' ε = ψ l he row i, column j elemen of ψ ψ l idenifies he consequences of a one-uni increase in he jh variable s innovaion a dae (εj) for he value of he ih variable a ime +l, holding all oher innovaions a all daes consan. A plo of he row i, column j elemen of as a funcion of lag l is called he non-orhogonalized impulse response funcion. ψ l
Coinegraion If wo or more series are hemselves non-saionary, bu a linear combinaion of hem is saionary, hen he series are said o be coinegraed. Example: A sock marke index and he price of is associaed follow a random walk by ime. Tesing he hypohesis ha here is a saisically significan connecion beween he fuures price and he spo price could now be done by esing for a coinegraing vecor. The usual procedure for esing hypoheses concerning he relaionship The usual procedure for esing hypoheses concerning he relaionship beween non-saionary variables was o run Ordinary Leas Squares (OLS) regressions on daa which had iniially been differenced. Alhough his mehod is correc in large samples, coinegraion provides more powerful ools when he daa ses are of limied lengh, as mos economic imeseries are. The wo main mehods for esing for coinegraion are: The Engle-Granger hree-sep mehod. The Johansen procedure.
Engle-Granger Approach Esimaion of parameers can be done by OLS esimaion of linear regression equaion: Y γ γ Y γ Y ε = 0 + 1 2 +.. + M M + Dickey-Fuller es is applied o he OLS residuals ε ˆ Rejecing he null hypohesis of non-saionariy concludes coinegraion relaionship does exis.
Three-sep approach Deermine he I(d) for every variable Dickey Fuller, Perron ess H 0 : series is non-saionary Esimae he coinegraion relaion by OLS regression Tes he residuals for saionariy y = β + β y + ε ε = y β β y 1 0 1 2 1 0 1 2 ˆ ε = y ˆ β ˆ β y 1 0 1 2 H 0 : series are no coinegraed ADF Tes does no give correc criical values because of he OLS residuals we use MacKinnon Table o deermine he criical values Mulicoinegraion exends he coinegraion echnique beyond wo variables, and occasionally o variables inegraed a differen orders.
Error Correcion Model Granger Represenaion Theorem Deerminaion of he dynamic relaionship beween coinegraed variables in erms of heir saionary error erms.for bivariae case: Two inegraed I(1) variables y 1 and y2 yielding one coinegraed combinaion p 1 y = λ ε + ( a y + a y ) + ε 1 1 1 11i 1 i 12i 2 i 1 i = 1 p 1 y = λ ε + ( a y + a y ) + ε 2 2 1 21i 1 i 22i 2 i 1 i= 1 ε I(0) Esimae parameers by OLS. Regression wih only saionary variables on boh sides.
Mulivariae Coinegraion Analysis - Johansen Tes VAR(1) having M I(1) variables can be expressed as: Y = µ + Γ Y + ε 1 where: Y, ì and å are (Mx1) vecors and à is a (MxM) marix Johansen Tes The approach of Johansen is based on he maximum likelihood esimaion of he marix (Γ - I) under he assumpion of normal disribued error variables. Following he esimaion he hypoheses H0: r = 0, H0: r = 1,, H0:r = M-1 are esed using likelihood raio (LR) ess.
Ex:Exchange rae, ineres raes, S&P 500(GLOBAL) index, ISE index
Series: GLOBAL EXCHANGE_RATE INTEREST_RATE ISE Lags inerval (in firs differences): 1 o 4 Hypohesized Trace No. of CE(s) Eigenvalue Saisic None * 0.065285 156.7717 A mos 1 * 0.017048 38.96042 A mos 2 0.005118 8.955273 A mos 3 1.20E-06 0.002096 Trace es indicaes 2 coinegraing eqn(s) a he 0.05 level * denoes rejecion of he hypohesis a he 0.05 level Unresriced Coinegraion Rank Tes (Maximum Eigenvalue) 0.05 Criical Value 47.85613 29.79707 15.49471 3.841466 Prob.** 0.0000 0.0034 0.3695 0.9599 Hypohesized Max-Eigen 0.05 No. of CE(s) Eigenvalue Saisic Criical Value None * 0.065285 117.8113 27.58434 A mos 1 * 0.017048 30.00514 21.13162 A mos 2 0.005118 8.953177 14.26460 A mos 3 1.20E-06 0.002096 3.841466 Max-eigenvalue es indicaes 2 coinegraing eqn(s) a he 0.05 level Prob.** 0.0000 0.0022 0.2901 0.9599 * denoes rejecion of he hypohesis a he 0.05 level **MacKinnon-Haug-Michelis (1999) p-values
1 Coinegraing Equaion(s): Log likelihood -46009.85 Normalized coinegraing coefficiens (sandard error in parenheses) GLOBAL EXCHANGE _RATE INTEREST_ RATE ISE 1.000000-0.000120-16.55210-0.026394 2 Coinegraing Equaion(s): (0.00014) (1.44457) Log likelihood (0.00218) -45994.85 Normalized coinegraing coefficiens (sandard error in parenheses) GLOBAL EXCHANGE _RATE INTEREST_ RATE ISE Therefore, we can conclude ha in he long erm hese hree variables are coinegraed and here are 2 coinegraion equaions 1.000000 0.000000-15.66567-0.024910 (1.30729) (0.00194) 0.000000 1.000000 7382.376 12.36210 (1649.84) (2.44689)
Pairwise Granger Causaliy Tess Granger Causaliy Tes: In order o compare pairwise variables Granger Causaliy Tess is used Dae: 07/20/08 Time: 10:40 Sample: 1/02/2001 12/31/2007 Lags: 5 Null Hypohesis: INTEREST_RATE does no Granger Cause EXCHANGE_RATE Obs 174 5 F-Saisic 28.3482 Probabiliy 1.1E-27 EXCHANGE_RATE does no Granger Cause INTEREST_RATE 32.1459 2.0E-31 ISE does no Granger Cause EXCHANGE_RATE 174 5 58.2545 3.6E-56 EXCHANGE_RATE does no Granger Cause ISE 2.31559 0.04151 GLOBAL does no Granger Cause EXCHANGE_RATE 174 5 21.4690 6.7E-21 EXCHANGE_RATE does no Granger Cause GLOBAL 1.16105 0.32611 ISE does no Granger Cause INTEREST_RATE 174 5 12.2991 9.7E-12 INTEREST_RATE does no Granger Cause ISE 0.10286 0.99158 GLOBAL does no Granger Cause INTEREST_RATE 174 5 2.98831 0.01084 INTEREST_RATE does no Granger Cause GLOBAL 1.48645 0.19105 GLOBAL does no Granger Cause ISE 174 5 20.7727 3.3E-20 ISE does no Granger Cause GLOBAL 1.91422 0.08894