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 costly. Hansen and Timmermann
(2015, Econometrica) propose a Wald approximation of the commonly used recursive Fstatistic
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 heteroskedasticity is present. We first show both the recursive F-test and its Wald
approximation have poor finite-sample properties, especially when the forecast horizon
is greater than one. We then propose a hybrid bootstrap method consisting of a block
moving bootstrap (which is nonparametric) and a residual based bootstrap for both
statistics, and establish its validity. Simulations show that our hybrid bootstrap has good
finite-sample performance, even in multi-step ahead forecasts with heteroscedastic or
autocorrelated errors, and more than one predictor. The bootstrap method is illustrated
on forecasting core inflation and GDP growth.