Exogenous variable in both the mean and variance model?

Question

\begin{aligned} y_t &= \beta_0 + \beta_1 y_{t-1} + \beta_2 x_{t-1} + \epsilon_t, \\ \epsilon_t &= \sqrt{h_t}\eta_t, \\ h_t &= \alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \alpha_2 h_{t-1} + \alpha_3 e^{x_{t-1}}, \\ \eta_t &\sim N(0,1) \ \forall \ t. \\ \end{aligned}

1. Does it make sense to include an exogenous variable ($x_t$) in both the mean and the variance model?

2. Do standard inferences about the coefficients still apply here?

3. Given the exogenous variable is in both the mean and the variance, we would see endogeneity and hence biased estimate of the $\beta_2$ parameter?

Answer 1

1. This may make sense. There is nothing contradictory or paradoxical in including $x_t$ in both the conditional mean and the conditional variance equations.

2. It depends on what do you mean by standard. Inference on $\alpha$s is in a way nonstandard even in a simple GARCH model; see Part II of Francq & Zakoian "GARCH Models: Structure, Statistical Inference and Financial Applications" (2010), e.g. Chapter 8 "Tests Based on the Likelihood". Other than that, I do not expect that adding an exogenous variable to the equations should mess up the inference further, at least when $y_t$ and $x_t$ are stationary.

3. No, I do not think there is endogeneity, because $x$ is not a function of $y$. Regarding unbiasedness, even without exogeneous regressors, coefficient estimates may be biased. In a simple AR(1) model, I think both the OLS and the ML estimators are biased. Adding an exogenous regressor and making the conditional variance time varying will not change that.