(Structural) VAR Models with Ignored Changes in Mean and Volatility

Abstract

We discuss estimation of so-called long vector autoregressions for multivariate series exhibiting possibly time-varying mean and (co)variances. In applied work, such changes often escape undetected, and we ask how standard tools (least squares estimation, point forecasts, and estimated impulse responses) are affected when ignoring the changes altogether. Keeping the order of the autoregression fixed is known to lead to asymptotic bias in autoregressive parameter estimators in the presence of ignored changes in the mean. Yet we show that allowing the complexity of the model to increase with the sample size leads to consistent estimators of the AR coefficient matrices individually. The fitted long VAR models appear to have unit root behavior, in spite of the absence of any stochastic trend in the model, and may even mimic cointegration; but, in spite of the structural change in the data generating process, out-of-sample forecasts based on long VARs are consistent. These findings hold under constant as well as under time-varying covariances. In what concerns estimated impulse responses, their sampling behavior depends primarily on whether the residual covariance matrix is employed for identification of the structural shocks or not. While MA coefficient matrices obtained by inversion of the fitted long VAR model are consistent for the true coefficients under mild additional restrictions even under time-varying error (co)variances, the residual covariance matrix estimator converges to an "average" covariance matrix, such that localized estimators may be more suitable for a precise identification. Monte Carlo simulations and empirical illustration support our theoretical findings. Empirical relevance of the theory is illustrated through two illustrations: (i) international dynamics of inflation and (ii) uncertainty and economics activity.