On bootstrapping tests of equal forecast accuracy for nested models

The asymptotic distributions of the recursive out‐of‐sample forecast accuracy test statistics depend on stochastic integrals of Brownian motion when the models under comparison are nested. This often complicates their implementation in practice because the computation of their asymptotic critical values is burdensome. Hansen and Timmermann (2015, Econometrica) propose a Wald approximation of the commonly used recursive ‐statistic and provide a simple characterization of the exact density of its asymptotic distribution. However, this characterization holds only when the larger model has one extra predictor or the forecast errors are homoscedastic. No such closed‐form characterization is readily available when the nesting involves more than one predictor and heteroscedasticity or serial correlation is present. We first show through Monte Carlo experiments that both the recursive ‐test and its Wald approximation have poor finite‐sample properties, especially when the forecast horizon is greater than one and forecast errors exhibit serial correlation. We then propose a hybrid bootstrap method consisting of a moving block bootstrap and a residual‐based bootstrap for both statistics and establish its validity. Simulations show that the hybrid bootstrap has good finite‐sample performance, even in multi‐step ahead forecasts with more than one predictor, and with heteroscedastic or autocorrelated forecast errors. The bootstrap method is illustrated on forecasting core inflation and GDP growth.