Inference in VARs with conditional heteroskedasticity of unknown form

We consider a framework for asymptotically valid inference in stable vector autoregressive (VAR) models with conditional heteroskedasticity of unknown form. A joint central limit theorem for the LS estimators of both the VAR slope parameters as well as the unconditional innovation variance parameters is obtained from a weak vector autoregressive moving average model set-up recently proposed in the literature. Our results are important for correct inference on VAR statistics that depend both on the VAR slope and the variance parameters as e.g. in structural impulse responses. We also show that wild and pairwise bootstrap schemes fail in the presence of conditional heteroskedasticity if inference on (functions) of the unconditional variance parameters is of interest because they do not correctly replicate the relevant fourth moments' structure of the innovations. In contrast, the residual-based moving block bootstrap results in asymptotically valid inference. We illustrate the practical implications of our theoretical results by providing simulation evidence on the finite sample properties of different inference methods for impulse response coefficients. Our results point out that estimation uncertainty may increase dramatically in the presence of conditional heteroskedasticity. Moreover, most inference methods are likely to understate the true estimation uncertainty substantially in finite samples.