This paper establishes the asymptotic properties of the quasi-maximum likelihood estimator (QMLE) of a GARCH(1,1) process with time-varying cofficients driven by an exogenous variable, when some true coefficients may be null. The QMLE is shown to be consistent. Its asymptotic distribution is a projection of a normal vector distribution onto a convex cone. Furthermore, the QMLE is shown to converge to its asymptotic distribution locally uniformly. We then consider the problem of testing that one or several coefficients are equal to zero. The null distribution and the local asymptotic powers of the Wald, Rao-Score and Quasi-Likelihood Ratio tests are derived. The results are derived under mild conditions, that do not require the existence of moments of the observed process. The results are illustrated by numerical simulations. The framework developed here allows some intercepts to be null for certain regimes.