# Forecast error variance decomposition

In practice forecast error variance decompositions are also popular tools for interpreting VAR models. Expressing the h-step forecast error from (32.16) in terms of the orthogonalized impulse responses et = (e1t,…, eKt)’ = P 1ut from (32.25), where P is a lower triangular matrix such that PP = X u, gives

уТ+к — yT+hT = ^0ЄТ+к + ^1ЄТ+к-1 + … + ^к-1ЄТ+1.

Denoting the (i, j)th element of ¥n by y^, the kth element of the forecast error vector becomes

h-1

yk, T+h — yk, T+h T = X (¥ k1,n£1,T+h-n + + ¥kK, n £K, T+h-n ).

n=0

Using the fact that the еы are contemporaneously and serially uncorrelated and have unit variances by construction, it follows that the corresponding forecast error variance is

к -1 K

°k(h) = X (Vk1,n + " + ¥kK, n) = X (Vk/,0 + " + ¥2/,h-l).

n=0 j=1

The term (y2j 0 + … + ¥І, к-1) is interpreted as the contribution of variable j to the к-step forecast error variance of variable k. This interpretation makes sense if the eit can be interpreted as shocks in variable i. Dividing the above terms by аі(к) gives the percentage contribution of variable j to the к-step forecast error variance of variable k,

ЩЩ = (y |,0 + … + ¥ |,к-і)/° kW.

These quantities, computed from estimated parameters, are often reported for various forecast horizons. Clearly, their interpretation as forecast error variance components may be criticized on the same grounds as orthogonalized impulse responses because they are based on the latter quantities.

If there are exogenous variables in the system (32.7), the model may also be used directly for policy analysis. In other words, if a policy maker affects the values or properties of zt the effect on the endogenous variables may be investigated within the conditional model (32.7). If the policy maker sets the values of zt the effect of such an action can be analyzed by considering the resulting dynamic effects on the endogenous variables similar to an impulse response analysis. In general, if zt represents stochastic variables, it is more natural to think of policy actions as changes in the distribution of zt. For instance, a policy maker may shift the mean of zt. Again, such changes can be analyzed in the context of our extended VAR models. For details see, for example, Hendry and Mizon (1998).

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