# Impulse response analysis

Tracing out the effects of shocks in the variables of a given system may also be regarded as a type of causality analysis. If the process yt is I(0), it has a Wold moving average (MA) representation

yt = Фои + Ф1Иt_1 + Ф2щ_2 + …, (32.23)

where Ф0 = IK and the Ф8 may be computed recursively as in (32.16). The coefficients of this representation may be interpreted as reflecting the responses to impulses hitting the system. The (i, j )th elements of the matrices Ф8, regarded as a function of s, trace out the expected response of yitt+s to a unit change in yjt holding constant all past values of yt. Since the change in yit given {yt-1, yt-2,…} is measured by the innovation uit, the elements of Ф8 represent the impulse responses of the components of yt with respect to the ut innovations. In the presently considered I(0) case, Ф8 ^ 0 as s ^ ^. Hence, the effect of an impulse is transitory as it vanishes over time. These impulse responses are sometimes called forecast error impulse responses because the ut are the one-step ahead forecast errors.

Although the Wold representation does not exist for nonstationary cointegrated processes it is easy to see that the Ф8 impulse response matrices can be computed in the same way as in (32.16) (Lutkepohl, 1991, ch. 11; Lutkepohl and Reimers, 1992). In this case the Ф8 may not converge to zero as s ^ ™ and, consequently, some shocks may have permanent effects. Assuming that all variables are I(1), it is also reasonable to consider the Wold representation of the stationary process Ayt,

Ay t = S оЩ + S 1Mt_1 + S2Ut-2 + …, (32.24)

where S0 = IK and Sj = Фj _ Ф;-1 (j = 1, 2,…). Again, the coefficients of this representation may be interpreted as impulse responses. Because Ф8 = X)=0S j, s = 1, 2,…, the Ф8 may be regarded as accumulated impulse responses of the representation in first differences.

A critique that has been raised against forecast error impulse responses is that the underlying shocks are not likely to occur in isolation if the components of U are not instantaneously uncorrelated, that is, if Xu is not diagonal. Therefore, in many applications the innovations of the VAR are orthogonalized using a Cholesky decomposition of the covariance matrix X u. Denoting by P a lower triangular matrix such that X u = PP’, the orthogonalized shocks are given by et = P-1u t. Hence, in the stationary case we get from (32.23),

y t = 4^ + ^£t_1 + …, (32.25)

where ‘¥i = Ф^ (i = 0, 1, 2,…). Here ¥0 = P is lower triangular so that an e shock in the first variable may have an instantaneous effect on all the other variables as well, whereas a shock in the second variable cannot have an instantaneous impact on y1t but only on the remaining variables and so on.

Since many matrices P exist which satisfy PP’ = Xu, using this approach is to some extent arbitrary. Even if P is found by a lower triangular Choleski decomposition, choosing a different ordering of the variables in the vector yt may produce different shocks. Hence, the effects of a shock may depend on the way the variables are arranged in the vector yt. In view of this difficulty, Sims (1981) recommends trying various triangular orthogonalizations and checking the robustness of the results with respect to the ordering of the variables. He also recommends using a priori hypotheses about the structure if possible. The resulting models are known as structural VARs. They are of the general form (32.7). In addition, the residuals may be represented as vt = Ret, where R is a fixed (K x K) matrix and et is a (K x 1) vector of structural shocks with covariance matrix E(et e’) = Xe. Usually it is assumed that Xe is a diagonal matrix so that the structural shocks are instantaneously uncorrelated. The relation to the reduced form residuals is given by Г *ut = Ret.

In recent years, different types of identifying restrictions were considered (see, e. g. Watson (1994) and Lutkepohl and Breitung (1997) for discussions). The aforementioned triangular system is a special case of such a class of structural models with P = Г *-1R. Obviously, identifying restrictions are required to obtain a unique structural representation. In the early literature, linear restrictions on Г * or R were used to identify the system (e. g. Pagan, 1995). Later Blanchard and Quah (1989), King, Plosser, Stock, and Watson (1991), Gali (1992) and others introduced nonlinear restrictions. To motivate the nonlinear constraints it is useful to consider the moving average representation (32.24) and write it in terms of the structural residuals:

АУ = 00£f + ®1£f-1 + 02£f-2 + . . . , (32.26)

where 0s = Ssr*-1R (s = 0, 1,…). The long run impact of the structural shocks on yt is given by lim^^dyt+n/de’t = limП^„ФПГ*-1R = XS=00s = 0. If the shock j has a transitory effect on yit, then the (i, j)th element of 0 is zero. Hence, the restriction that Zjt does not affect yit in the long run may be written as the nonlinear constraint

the ith (jth) column of the identity matrix. It can be shown that for a cointegrated system with cointegrating rank r, the matrix 0 has rank n – r so that there exist n – r shocks with permanent effects (e. g. Engle and Granger, 1987).

Imposing this kind of nonlinear restrictions in the estimation procedure requires that nonlinear procedures are used. For instance, generalized methods of moments (GMM) estimation may be applied (see Watson, 1994). If an estimator a, say, of the VAR coefficients summarized in the vector a is available, estimators of the impulse responses may be obtained as ф^h = ф^а). Assuming that a has a normal limiting distribution, the фij, h are also asymptotically normally distributed. However, due to the nonlinearity of the functional relationship, the latter limiting distribution may be singular. Moreover, the asymptotic covariance matrix of a may also be singular if there are constraints on the coefficients or, as mentioned earlier, if there are I(1) variables. Therefore, standard asymptotic inference for the impulse response coefficients may fail.

In practice, bootstrap methods are often used to construct confidence intervals (CIs) for impulse responses because these methods occasionally lead to more reliable small sample inference than asymptotic theory (e. g. Kilian, 1998). Moreover, the analytical expressions of the asymptotic variances of the impulse response coefficients are rather complicated. Using the bootstrap for setting up CIs, the precise expressions of the variances are not needed and, hence, deriving the analytical expressions explicitly can be avoided. Unfortunately, the bootstrap does not necessarily overcome the problems due to the aforementioned singularity in the asymptotic distribution. In other words, in these cases bootstrap CIs may not have the desired coverage. For a critical discussion see Benkwitz, Lutkepohl, and Neumann (2000).

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