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Suppose I have a GARCH(p,q) model with constant conditional mean, \begin{aligned} y_t &= \mu + u_t, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dotsc + \alpha_s u_{t-q}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-p}^2, \\ \varepsilon_t &\sim i.i.d(0,1). \\ \end{aligned} I want to test whether $\mu=0$. I could estimate the model and examine the significance of the estimated $\hat\mu$. But as an alternative, could I regress $y_t$ on a constant and use HAC standard errors for testing the hypothesis that $\mu=0$? That is, will HAC work as it should, or is it not applicable under GARCH-type conditional variance?

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