In this paper, we derive asymptotic distribution theory for a general class of models where the identification strength of one parameter is determined by another parameter and where the latter is allowed to be at the boundary of the parameter space—extending the results in Andrews and Cheng (2012). This allows us to derive the asymptotic distribution, under different identification strengths, of the two test statistics that are used in the two-step (testing) procedure proposed in Pedersen and Rahbek (2019). The latter aims at testing the null hypothesis that a GARCH-X type model, with an exogenous covariate (X), reduces to a standard GARCH model, while allowing the “GARCH parameter” to be unidentified. We find that using the second step test statistic together with plug-in least favorable configura- tion (PI-LF) critical values offers (asymptotic) power gains over a wide range of alternatives (for realistic choices of the data generating process) compared to the two-step procedure. Furthermore, we find that the two-step procedure fails to control asymptotic size. Together, our findings provide arguments against the use of the two-step procedure in practice; other tests, such as the aforementioned test using PI-LF critical values, ought to be preferred.