The unconditional volatility of financial return is frequently time-varying. To model this, the most common approach is to decompose the conditional volatility $\sigma_t^2$ multiplicatively into a smoothly varying non-stochastic process $g_t$, and a de-scaled stochastic process $h_t$: $\sigma_t^2 = g_th_t$. We prove the consistency and asymptotic normality of the single-step QMLE for the parameters of $g_t$ for a broad class of specifications $g_t$. Next, we derive a simple but robust and consistent estimator of the coefficient covariance. The exact specification of $h_t$ need not be estimated or known, and $h_t$ can be non-stationary in the distribution. This is important in empirical applications, since financial returns are frequently characterised by a non-stationary zero-process. We compare our single-step estimator with the multi-step iterative estimator of Amado and Terasvirta (2013), and illustrate our results in an empirical application.