Impulse Response Analysis for Unrestricted VAR Models

Impulse response analysis used to identify dynamic causal relationships among variables (Brandt and Williams 2007: 36). When one variable reacts to an impulse in another variable, the latter may be called causal for the former. This type of causality can be analysed by tracing out the effect of an exogenous shock in one variable on another variable (Lütkepohl 2005: 51). Typically, the magnitude of the innovation is one standard deviation of the residuals in the VAR model. Then, the initial responses are traced out as functions of time (Brandt and Williams 2007: 41).

5 Methodology 5.3 VAR/VEC Analysis

When the process yt in an unrestricted VAR model is I(0), the response to shocks in the

variables of a given system can best be seen in the Wold moving average (MA) representation [5.16]

_=F

_+F

₊

_F

₊L,

u

u

u

y

where

F

=I

[5.17]

_å

A

=

F

=

F

s

...

,

2

,

1

,

can be estimated recursively from the reduced-form coefficients of a VAR in levels.The coefficients of the MA representation then reflect the responses to impulses introduced into the system. The (i, j)th elements of the matrices Фs are viewed as a function of s. They trace

out the expected response of yi, t+s to a unit change in yjt, thereby holding constant the past

values of yt. Because the change in yit given {yt-1, yt-2, …}, is measured by innovation uit, the

elements of Фs represent the impulse responses of the elements of yt with respect to the

innovations ut. For an unrestricted stationary VAR, Фs → 0 as s → ∞. That is, the response

to the impulse is transitory and vanishes over time.

The analysis of impulse responses has been criticised because the underlying shocks are unlikely to occur in isolation when the components of ut are instantaneously correlated or

rather when Ɖu is not diagonal. Hence, orthogonal impulses, which can be calculated by

using a Cholesky decomposition of the covariance matrix Ɖu, are used by default. When Β is

a lower triangular matrix so that Ɖu= ΒΒ’, the orthogonalised shocks are given by εt= Β-1ut.

Therefore, the following can be obtained form [5.16]:

[5.18]

_=Y

_+Y

₊L,

y

e

e

where Ψi = ФiB (i = 0, 1, 2,…). Here, Ψ0 = B is lower triangular. Hence, an ε shock in the

first variable has an instantaneous effect on all other variables in the system while a shock in the second variable cannot have an instantaneous effect on the first variable, the third variable cannot have an instantaneous effect on the first and the second variable, and so on (see 5.2.4). Given that the ε shocks in the Wold causal chain are instantaneously uncorrelated or orthogonal, the impulse responses are often referred to as orthogonalised impulse responses (Breitung et al. 2004: 165f.).

5 Methodology 5.3 VAR/VEC Analysis

When using the Cholesky decomposition, the ordering of the variables in the equations of the VAR model is crucial. According to Brandt and Williams (2007: 91), “changing the ordering alters the normalisation of the Cholesky decomposition and the order in which the equations

are shocked in the computation of the moving average response.” If the correlations among the residuals are rather low, the ordering is not a decisive factor when computing the impulse responses. When there is strong correlation among the series, however, the ordering affects the interpretation of the results.

In case the innovations in the variables in the system are uncorrelated, the choice about how to compute the contemporaneous correlations is not of primary importance. When the contemporaneous correlations are highly correlated, however, different approaches should be taken into consideration:

1. In case not all of the variables in the VAR model are highly contemporaneously correlated, the subset of correlated variables should be placed together in the Cholesky ordering. While not too much can be learned about the impacts of the contemporaneously correlated variables on each other, more can be learned about their impact on the other variables in the system.

2. A sensitivity analysis can be conducted to see how the dynamics of the system’s impulse responses are altered by the different choices regarding the contemporaneous orderings. (Brandt and Williams 2007: 37ff.).

As described above, the IRFs indicate how a VAR/VEC model reacts to a specific impulse (Möller 2004). In this case, the response of steel exports from different countries to the U.S. in quantity metric tons to a shock in real oil prices is estimated, thereby using a Cholesky one-standard deviation as a shock introduced into the system. The responses to the shock, which is introduced into the system in period one, are then calculated for a fifteen month time period. The focus of the analysis is on the immediate reaction to the shock until the line representing the impulse responses crosses the zero line and the shock starts to die down over time. The IRFs are displayed graphically (see appendix, section A4) and in tabular form (see chapter 6).