Should a covariate be lagged in a GARCH-X model? I am modelling Dow Jones returns using a GARCH(1,1) model but I also want to estimate a GARCH(1,1) by inserting a covariate to check if this covariate affects the volatility in some ways. The covariate I want to insert is an index called EnvP which measures the salience of US environmental policy over the 1981-2019 period.
What I want to ask is how can I do it. I estimated a first GARCH(1,1) to give an initial interpretation of the parameters and to use it as a sort of benchmark. The results are:

How can I interpret those results?
Now, if I want to add the covariate, do I have to add a lag to it? To give you a better understanding:

The model on the left contains the Envp Index at the time t, the one on the right contains the EnvP Index lagged(1). Is one better than the other? Is it methodologically correct? What's the interpretation?


How can I interpret those results?

Interpretation of a GARCH(1,1) is textbook material; it should not be hard to find. At the same time, there is not much to it. The model specifies that the current conditional variance $\sigma_t^2$ of the variable of interest $y_t$ is a weighted sum of the previous conditional variance $\sigma_{t-1}^2$ and the previous squared error $\varepsilon_{t-1}^2$, plus a constant $\omega$. Since this applies to the conditional variance at all time points, we can iterate and express the conditional variance as a weighted sum of all past errors $\varepsilon_{t-1},\dots,\varepsilon_0$ (with exponentially decreasing weights) and a (different) constant.

Is it one better than the other? Is it methodologically correct? What's the interpretation?

No, one is not necessarily better than the other. From a descriptive perspective, both can be equally valid. A model with a contemporaneous external variable $x_t$ specifies the conditional variance given that value. That may be relevant if you want to know whether there is correlation between the size of the error (as specified by the conditional variance) and $x_t$. A model with lagged external variable $x_{t-1}$ specifies the conditional variance as a function of that value, anf the interpretation is analogous. From a predictive perspective, only the latter model may be useful, since when predicting the series of interest $y_t$ at time $t-1$ you only know $x_{t-1}$ but not $x_t$. (This assumes $y_t$ and $x_t$ become available at exactly the same time. If $x_t$ becomes known a while before $y_t$, you could use it to predict $y_t$.)
