Testing in GARCH-X models: boundary, correlations and bootstrap theory

In this article, we consider the so-called Fixed Shrinkage (FS) bootstrap for the class of GARCH models with explanatory variables (GARCH-X). Under the assumption of stationary covariates, the proposed FS bootstrap does not require modeling the covariates, as these are kept fixed in the bootstrap generating process. Our main focus is on testing whether one or more of the covariates can be excluded in the GARCH-X model. As is well-known the limiting distribution of the likelihood-ratio (LR) statistic in this setting is non-standard and depends in particular on whether nuisance parameters are on the boundary or in the interior of the parameter space. In particular, and as detailed here, the non-standard limiting distribution depends on correlations, or dependence, between the explanatory variables. The FS bootstrap takes the presence of nuisance parameters into account by implementing shrinking as proposed in Cavaliere et al. (2022) for pure ARCH models. We establish asymptotic validity of the FS bootstrap for GARCH-X models, and demonstrate by simulations that the bootstrap-based test performs extremely well even when nuisance parameters lie on the boundary of the parameter space. The empirical illustration amplifies that the presence of nuisance parameters (especially whether or not on the boundary) are vital for interpreting the dynamics of conditional volatility in financial stock market indices.