  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).
