External regressors in mean or variance equation of AR(1)-GARCH(1,1)? What is the difference between entering my external regressors in the mean equation and entering them in the variance equation in an AR(1)-GARCH(1,1) model? I get more explanatory results with the external regressors in the variance equation than in the mean, but am not sure as to what the actual difference is. I am modelling returns (the dependent variable) and the external regressors are variables (in time $t$) that should affect the return. 
 A: An AR(1)-GARCH(1,1) model without external regressors is 
\begin{aligned}
y_t &= \mu_t + u_t, \\
\mu_t &= \varphi_1 y_{t-1}, \\
u_t &= \sigma_t \varepsilon_t, \\
\sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\
\varepsilon_t &\sim i.i.d(0,1). \\
\end{aligned}
It specifies the entire conditional distribution of the variable: its conditional mean, conditional variance, and density (its location given by the mean and its scale determined by the variance): $y_t\sim d(\mu_t,\sigma_t^2)$.
If you include regressors in the conditional mean model only (the equation for $\mu_t$), for different values of regressors you get different fitted and predicted values of $\mu_t$. Hence, you get different point forecasts (location shifts) but the same variance (and density, adjusted for the location) forecasts around the different points.
If you include regressors in the conditional variance model only (the equation for $\sigma_t^2$), for different values of regressors you get different fitted and predicted values of $\sigma_t^2$. Hence, you get different variance (and thus density) forecasts (scale shifts) around the same point forecasts.
If you include regressors in both the conditional mean and the conditional variance models, for different values of regressors you get different fitted and predicted values of $\mu_t$ and $\sigma_t^2$. Hence, you get different point forecasts (location shifts) and different variance forecasts (scale shifts) around the different point forecasts. So the density both shifts and scales due to the effects of regressors.
