Teaching Notes on Impulse Response Function and …

1 Teaching Notes on Impulse Response Functionand structural VARJin-Lung LinInstitute of Economics, Academia SinicaDepartment of Economics, National Chengchi UniversityMay 2, 20061 IntroductionStructural VAR embeds economic theory within time series models, providing aconvenient and powerful framework for policy analysis. Impulse Response func-tion (IRF) tracks the impact of any variable on others in the system. It is an essen-tial tool in empirical causal analysis and policy effectiveness analysis. This notereviews important concepts related to Impulse Response Function and Impulse Response functionLetYtbe ak-dimensional vector series generated byYt=A1Yt 1+ +ApYt p+Ut= (B)Ut= i=0 iUt i(1)I= (I A1B A2B ApBp) (B)(2)wherecov(Ut) = , iis the MA coefficients measuring the Impulse specifically, jk,irepresents the Response of variablejto an unit Impulse invariablekoccurringi-th period ago. IRF are used to evaluate the effectiveness ofa policy change, say increasing rediscount is usually non-diagonal, it is impossible to shock one variable with othervariables fixed.

2 Some kind of transformation is needed. Cholesky decompositionis the most popular one which we shall turn to now. LetPbe a lower triangularmatrix such that =P P . then eq. (1) can be rewritten asYt= i=0 iwt iwhere i= iP, wt=P 1Ut, andE(wtw t) =I. LetDbe a diagonal matrixwith same diagonals withPandW=P D 1, =DD . After some manipula-tions, we obtainYt=B0Yt+B1Yt 1+ +BpYt p+VtwhereB0=Ik W 1, W=P D 1, Bi=W 1Ai. Obviously,B0is a lowertriangular matrix with 0 diagonals. In other words, Cholesky decomposition im-poses a recursive causal structure from the top variables to the bottom variablesbut not the other way remarks1. For a K-dimensional stationary VAR(p) process, jk,i= 0,forj6=k, i= 1,2, is equivalent to jk,i= 0fori= 1, , p(K 1).In other words, if the firstpK presponses of variablejto an Impulse invariablekis zero, then all the following responses are all zero. (LutkepohlProposition ).2. Variablekdoes not cause variablejif and only if jk,i= 0, i= 1,2.

3 Critiques of IRF1. Sensitive to variables Impulse Response Function by Pesaran offers a partial solutionand Granger and Swanson (1997) proposed a different but more Omitting important variables may lead to major distortions in IRF and makethe empirical results worthless. However, its impact on forecasting Generalized Impulse Response functionTo circumstance the problem of ordering dependence of IRF, Pesaran and Shin(1998) proposed the i=1 AiXt i+Ut= i=0 iUt i i=A1 i 1+A2 i 2+ +Ap i p, i= 1,2, 2whereE(UtU t) = Cholesky decomposition of , P P = so thatXt= i=0(AiP)(P 1Ut i)IRF is oj(n) = nP ej, n= 0,1,2, whereejis anm 1selection vector with unity as itsj-th element and is defined as :GIRFx(n, j, t 1) =E(Xt+n|ujt= j, t 1) E(Xt+n| t 1)Assume normal distribution forUtE(Ut|Ujt= j) = ( 1j, 2j, , mj) jj j= Ujej 1jj jUnscaled GIRF is:( n ej jj)( j jj), n= 0,1,2, Scaled GIRF by setting j= jj, gj(n) = 1/2jj n Uj, n= 0,1,2, Forecast error decomposition: oij= nl=0(U l jP Uj)2 nl=0(U i l A lUi); gij= 1ii nl=0(U i l Uj)2 nl=0U i l lUi, i, j= 1, , mNote that mj=1 ojj(n) = 1, mj=1 gjj(n)6= GIRF and IRF1.

4 In stead of controlling the impact of correlation among residuals, GIRF fol-lows the idea of nonlinear Impulse Response Function and compute the meanimpulse Response Function . When one variable is shocked, other variablesalso vary as is implied by the covariance. GIRF computes the mean byintegrating out all other When is diagonal, GIRF is the same as GIRF is unaffected by ordering of variables4. The generalized Impulse Response of the effect of an unit shock toj-th equa-tion is the same as that of an orthogonal Impulse Response but different forother shocks. To be specific, g1(n) = o1(n) gj(n)6= oj(n), j= 2,3, , mThus the GIRF can be easily computed by usual IRF with each variable asleading The formula of GIRF is derived under the assumption of multivariate nor-mality that might not be true for some empirical Unit Root, Cointegration and IRF1. If there exists unit roots and/or cointegration, then estimated IRF is inconsis-tent at long horizons in unrestricted VARs.

5 Error correction model producesconsistent IRF and optimal IRF estimates based upon ECM is Proper procedures for computing IRF for a cointegrated system are:(a) Determine the cointegration rank by LR test;(b) Estimate the ECM model: Yt= Yt 1+ p 1i=1 i Yt i+ Dt+Ut;(c) Converted the ECM back to VAR model;(d) Use the resulting VAR model to perform structural VARThis part is taken from the written by Norman Morin ( ).REDUCED FORMYt=A1 Yt 1+ +ApYt p+W Z(t) +c+d t+Ut4where Y denotes vector of endogenous variables of interest, X vector of exoge-nous variables, U is vector of residuals andEUtU t= . The innovations can bewritten terms of uncorrelated error termsUt=G Ut+EtE(EtE t) =DwhereDis a diagonal matrix whose diagonals are the variances ofEandGhaszeroes on the diagonals. Now, letB Ut=EtorA Et=UtwhereB=I G,andA=B 1, whereBandAhave unit diagonals Thus,B B =D=E(EtE t)A D A = =E(UtU t)This will yield the structural form based on the Yt=B1 Yt 1+ +Bp Yt p+F X(t) +v+k t+EtwithBi=B Ai, i= 1, , p, F=B W, v=B c, andk=B B and D, one can write a structural form vector moving average basedon the reduced form matricesA1, , ApYt=Ut+C1 Ut 1+C2 Ut 2+ Yt=M0 Et+M1 Et 1+M2 Et 2+ The coefficient(i, j)thelement ofMkis the effect on variable i of a shock toj-th structural form innovation k periods various choices of orthogonalizations for Impulse responses place condi-tions on the structural form matricesBandD:1.

6 CHOLESKY:Factors intoP P wherePis lower triangular whose diagonals are thestandard deviations ofE. Thus, the first variable in the VAR is only affectedcontemporaneously by the shock to itself. The second variable in the VARis affected contemporaneously by the shocks to the first variable and theshock to itself, and so 1D1/22. BERNANKE-SIMS:Factors intoB 1DB 1 whereDis diagonal (with the variances ofE),Bhas unit diagonals, but allows for the user to force certainB(i, j) = 0,5(not fori=j) and will test these restrictions. You are asked for the numberof nonzero NONDIAGONAL free coefficients, and is then asked to inputthe row number and column number for each non-diagonal free coefficient(this is done by entering the row number and column number separated bya comma (preferred) or a space).3. HARVEY-SARGAN:Factors intoB 1DB 1 where the user can distribute unit coefficientsand zeros among both B and D, but one or the other will have unit diag-onals. The user chooses which matrix will contain unit diagonals, is thenprompted for the number of NON-DIAGONAL free coefficients in B and inD, and is then asked to input the row number and column number for eachnon-diagonal free coefficient (this is done by entering the row number andcolumn number separated by a comma (preferred) or a space).

7 4. IDENTITY:Assumes is diagonal, , the reduced form innovations are contempora-neously BLANCHARD-QUAH:Factors intoP P whereP=C 11 GwhereGis the LT Cholesky decom-position ofC1 1C 1andC1is the sum of the -order VMA coefficientsfrom the Wold decomposition of the VAR. This yields Impulse responsessuch that the 1st variable may have long run effects on all variables, the 2ndmay have long run effects on all but the 1st, the 3rd on all but the 1st and2nd, etc.. In the BQ article, shocks are assigned as supply and de-mand shocks, without reference to a variable ordering. Here, the shocksare labeled with the ordering of the variables in the VAR, but need not begiven that 1andD= that 2, 3, and 4 will test the restrictions. For 2 and 3, the number of freecoefficients (restrictions) should be less than or equal top(p+ 1)/2, wherepisnumber of variables, and there must be no zeros on the diagonalsSoftwares1. Impulse responses: Reduced form and structural form by Norman Morin by Antonio Lanzarotti and Mario Seghelini6 VAR/View/ Impulse /Eviews FinMetrics/Splus2.

8 Cointegration: CATS/RATS COINT2/GAUSS VAR/Eviews urca/R FinMetrics/Splus3. Impulse Response under cointegration constraint:CATS,CATSIRFS/RATSR eferences1. Lutkepohl, HelmutIntroduction to multiple time series analysis, 2nd , Amisano, Gianni and Carlo Giannini (1997),Topics in structural Var Econo-metrics, 2nd ed. New York: Springer-Verlag3. Bernanke, B. S. (1986), Alternative explanations of the money-income cor-relation, Carnegie-Rochester Conference Series on Public Policy, 25, Blanchard, O. J. and D. Quah (1989) The dynamic effects of aggregatesupply and demand disturbance, American economic Review, 77, Gordon, R (1997), The time varying NAIRU and its implications for eco-nomic policy, Journal of economic Perspectives, 11:1, Granger, CWJ and Jin-Lung Lin, 1994, Causality in the long run, wEconometric Theory, 11, 530-536,7. Johansen, S. (1995)Likelihood-based inference in cointegrated vectorautoregressive models, Oxford: Oxford University Press78.

9 Phillips, (1998) Impulse Response and Forecast Error Variance Asymp-totics in Nonstationary VAR s, Journal of Econometrics, 83 Lin, Jin-Lung, (2003), An investigation of the transmission mechanism ofinterest rate policy in Taiwan, Quarterly Review, Central Bank of China,25, (1), 5-47. (in Chinese).10. Swanson, N. and C. W. J. Granger (1997), Impulse Response functionsbased on the causal approach to residual orthogonalization in vector autore-gressions, Journal of the American Statistical Association, 92.