Consistency of averaged impulse response estimators in vector autoregressive models

We show root-T consistency of the smoothed AIC and smoothed BIC model averaging estimators (sAIC, sBIC) of impulse response coefficients in stationary vector autoregressive models of finite lag order. We also show that there is not one unique way to define the sAIC and sBIC estimators, but that instead there is a whole class of each of these defined by a weight scaling factor that allows the averaging estimator to become more similar to either its model selection counterpart or the equal weights averaging estimator. We also show asymptotic validity of a bootstrap method for estimating the averaging estimators' distributions. Simulations illustrate the benefits of using sAIC instead of AIC estimators.