Allowing for weak identification when testing GARCH-X type models

TL;DR

This paper analyzes the asymptotic properties of tests for GARCH-X models with weak identification, revealing size distortions in existing procedures and proposing a new, more reliable test with better finite-sample performance.

Contribution

It extends existing asymptotic theory to models with parameters at the boundary, characterizes size distortions in current tests, and introduces a new test that controls size and improves power.

Findings

The two-step procedure does not control asymptotic size.

Existing tests suffer from overrejection in finite samples.

The new test controls size and is more powerful when the ARCH parameter is small.

Abstract

In this paper, we use the results in Andrews and Cheng (2012), extended to allow for parameters to be near or at the boundary of the parameter space, to derive the asymptotic distributions of the two test statistics that are used in the two-step (testing) procedure proposed by Pedersen and Rahbek (2019). The latter aims at testing the null hypothesis that a GARCH-X type model, with exogenous covariates (X), reduces to a standard GARCH type model, while allowing the "GARCH parameter" to be unidentified. We then provide a characterization result for the asymptotic size of any test for testing this null hypothesis before numerically establishing a lower bound on the asymptotic size of the two-step procedure at the 5% nominal level. This lower bound exceeds the nominal level, revealing that the two-step procedure does not control asymptotic size. In a simulation study, we show that this finding is relevant for finite samples, in that the two-step procedure can suffer from overrejection in finite samples. We also propose a new test that, by construction, controls asymptotic size and is found to be more powerful than the two-step procedure when the "ARCH parameter" is "very small" (in which case the two-step procedure underrejects).