Generalized Impulse Responses - TeXlips

1 Generalized Impulse ResponsesAnders WarneFebruary 27, 2008 Abstract:This note discusses how to compute Generalized Impulse Responses and their asymp-totic distribution. The results I present are essentially vector versions of what has already beenshown by, , Pesaran and Shin (1998). The value added is therefore measurable in terms ofproviding simpler algorithms for writing the computer code needed to make use of generalizedimpulse Responses in :Asymptotics, Impulse response Classification contrast with Impulse response functions for structural models, Generalized Impulse re-sponses do not require that we identify any structural shocks. Accordingly, Generalized impulseresponses cannot explain how, say, inflation reacts to a monetary policy shock.

2 Instead, gener-alized Impulse Responses provides a tool for describing the dynamics in a time series model bymapping out the reaction in, say, inflation to a one standard deviation shock to the residual inthe interest rate general setup we shall consider is a VAR process for somepdimensional time seriesxtgiven byxt Dt k i 1 ixt i t,t 1,..,T,(1)whereDtis a vector with deterministic variables. The processxtmay be covariance stationary,integrated of orderd(and possibly cointegrated), while tispdimensional and assumed to with zero mean and positive definite covariance matrix .Theh-steps ahead forecast error forxtis given by:xt h E xt h|It h 1 j 0Cj t h j,(2)whereItis an information set which includes the history ofxsup to and including periodtaswell as the entire time path forDt.

3 Thep pmatricesCjare given byC0 IpandCj mink,j i 1 iCj i,j 1,so that allCjmatrices can be determined recursively from the , Pesaran and Potter (1996) defined the Generalized Impulse response function by:GIx h, ,It 1 E xt h| t ,It 1 E xt h|It 1 ,(3)where is some known vector. For the VAR process this means that:GIx h, ,It 1 Ch .The choice of is therefore central to determining the time profile for any Generalized im-pulse response function. As an alternative to shocking all elements of tone may consider justRemarks:Copyrightc 2004 2008 Anders one element such that jt j. We may now define the Generalized Impulse responsesas:GIx h, j,It 1 E xt h| jt j,It 1 E xt h|It 1 .(4)Letting j jj, the standard deviation of jt, and assuming that tis Gaussian, it followsthat:E t| jt jj ej 1/2jj,(5)whereejis thej:th column h, jj,It 1 Ch ej 1 measures the response inxt hfrom a one standard deviation shock to jt,wherethecorrelation between jtand itis taken into account.

4 Defining the diagonalp pmatrix as: diag e 1 e1 1/2 e 2 e2 1 e p ep 1/2 ,(6)we may express the Generalized Impulse Responses in matrix form as:GIx h, 11,.., pp,It 1 Ch ChB Ah,(7)where columnjis given byGIx h, jj,It 1 .When is diagonal, thenB 1/2 1,adiagonal matrix with standard deviations along the order to determine to asymptotic covariance matrix for an estimate ofChBwe need to makea few assumptions. Suppose thatChdepends on aKdimensional vector RKand thatChis differentiable with respect to . Relative to the VAR model, includes the elements of iorsome transformations thereof, but they do not include any element from or . IncasetheVAR model includes cointegration rank restrictions, then does not include the cointegrationvectors but only the parameters on stationary transformations cointegrated of order (1,1), this means that only includes parameters on lagged firstdifference ofxtand on the 0<r<pcointegration relations xt , assume that we have an estimator of , denoted by , based on a sample ofTobservations, which satisfies: T d NK 0, ,(8)withNKbeing aK-dimensional Gaussian distribution,d denoting convergence in distribution,and being positive semidefinite.

5 Furthermore, let vech , with vech being the columnstacking operator which only takes the elements on and below the diagonal. The estimator of , denoted by , is assumed to satisfy: T d Np p 1 /2 0, ,(9)while and are asymptotically independent. In case tis Gaussian and, ,xtis cointegratedof order (1,1) these assumptions are all satisfied as long as there are no restrictions whichinvolve both and . Furthermore, for such a model 2D p D p,where is the Kronecker product,Dpis the duplication matrix (cf. Magnus and Neudecker,1988), andD p D pDp 1D pis the Moore-Penrose inverse our assumptions it follows that the asymptotic distribution of the matrix form of thegeneralized Impulse Responses in equation (7) can be expressed as: T vec Ah vec Ah d Np2 0, Ah ,(10) 2 where Ah B Ip vec Ch B Ip vec Ch Ip Ch vec B Ip Ch vec B.

6 The partial derivatives vec Ch / are readily available from several sources (see, , Warne,1993, or Vlaar, 2004). Hence, what remains to be shown is what the matrix with partial deriva-tives vec B / looks can be shown that: vec vec 12Lp 3L p,whereLpis ap2 p0-1 matrix defined byLp e1e 1e2e p .It then follows that the differential of vec B satisfies:dvec B Ip dvec 12 Ip Lp 3L pdvec .Sincedvec Dpd we have that: vec B Ip 12 Ip Lp 3L p Dp.(11)Ifxtis cointegrated of order (1,1) withrcointegration vectors, denoted by the full rankp rmatrix , we may also define Generalized Impulse Responses for the cointegration that we have an estimator of , denoted by ,suchthat T p 0,wherep denotes convergence in probability.

7 Estimators of , such as the ML estimator sug-gested by Johansen (1996), typically satisfy this assumption. Let the cointegrating relations bedefined byzt xt. The Generalized Impulse response function forzt hfrom one standarddeviation shocks to tis then given byGIz h, 11,.., pp,It 1 Ah,It can now be established that an estimator of Ahsatisfies: T vec Ah vec Ah d Np2 0, Ip Ah Ip .(12)The reason for this result is, of course, that is T-consistent whereas Ahis cointegrated of order (1,1), we may rewrite the VAR in VEC form such that xt Dt k 1 i 1 i xt i xt 1 t,where and are full rankp rmatrices (0<r<p); see, , Johansen (1996) for this case we may define vec 1 k 1 and thep pmatrix:C 1 ,with Ip k 1i 1 Ah CB, 3 whilelimh Ah , the long-run Generalized Impulse Responses in levels depend on the long-run impactmatrixCand converge to finite matrix , while the long-run Generalized Responses for the coin-tegration relations converge to zero.

8 The asymptotic distribution ofCBis readily determinedfrom the above results and those in, , Paruolo (1997) regarding the asymptotic distributionfortheMLestimatorofC; see also Johansen (1996).Specifically, lettingA CBthen T vec A vec A d Np2 0, A ,where A B Ip vec C B Ip vec C Ip C vec B Ip C vec B .The matrix with partial derivatives vec B / is given in equation (11). Furthermore, it isreadily shown that vec C C ,(13)where is anp k 1 pmatrix given by 1 C Ip .The Generalized Impulse Responses forzprovides us with a tool to measure how quicklythe long-run relations converge to their steady state values. Since thepshocks may result in Ahej 0fordifferenth, we may, for example, choose a convergence horizonh based on theslowest Generalized Impulse Responses are equal to Impulse Responses from a structural VARwhen the structural shocks are identified from a recursive structure and is diagonal.

9 In allother circumstances will the Generalized Impulse Responses differ from the Impulse Responses ofa structural , S. (1996),Likelihood-Based Inference in Cointegrated Vector Autoregressive Models,2nded., Oxford: Oxford University , G., Pesaran, M. H. and S. M. Potter (1996), Impulse response Analysis in NonlinearMultivariate Models ,Journal of Econometrics, 74, 119 , J. R. and H. Neudecker (1988), matrix Differential Calculus: With Applications in Sta-tistics and Econometrics, New York: John Wiley & , P. (1997), Asymptotic Inference on the Moving Average Impact matrix in CointegratedI(1) Systems ,Econometric Theory, 13, 79 , M. H. and Y. Shin (1998), Generalized Impulse response Analysis in Linear Multivari-ate Models ,Economics Letters, 58, 17 , A.

10 (1993), A Common Trends Model: Identification, Estimation and Inference , Semi-nar Paper No. 555, Institute for International Economic Studies, Stockholm , P. J. G. (2004), On the Asymptotic Distribution of Impulse response Functions with LongRun Restrictions ,Econometric Theory, 20, 891 903. 4