Estimating the Variance of a Combined Forecast: Bootstrap-Based Approach

This paper considers bootstrap inference in model averaging for predictive regressions. We first consider two different types of bootstrap methods in predictive regressions: standard pairwise bootstrap and standard fixed-design residual-based bootstrap. We show that these procedures are not valid in the context of model averaging. These common bootstrap approaches induce a bias-related term in the bootstrap variance of averaging estimators. We then propose and justify a fixed-design residual-based bootstrap resampling approach for model averaging. In a local asymptotic framework, we show the validity of the bootstrap in estimating the variance of a combined forecast and the asymptotic covariance matrix of a combined parameter vector with fixed weights. Our proposed method preserves non-parametrically the cross-sectional dependence between different models and the time series dependence in the errors simultaneously. The finite sample performance of these methods are assessed via Monte Carlo simulations. We illustrate our approach using an empirical study of the Taylor rule equation with 24 alternative specifications.