Including Time Invariant Covariates in a Random Intercept Model

Question

Let us say we have a random intercept model for $n$ individuals

$$y_{i,t} = x_{i,t}'\beta + \alpha_i + \epsilon_{i,t} \hspace{35pt} i = 1,...,n$$

where $x_{i,t}'\beta$ is a set of time variant explanatory variables, $\alpha_i$ is the individual specific heterogeneity and $\epsilon_{i,t}$ is an idiosyncratic errot term.

We could inlcude a time constant fixed effect $x_i$, which differs among individuals (for example the gender of $i$).

$$y_{i,t} = x_{i,t}'\beta + x_i + \alpha_i + \epsilon_{i,t}$$

However, if accurate predictions are our only goal, is there anything we gain by doing so? Should we not expect to obtain the same results whether we include $x_i$ or not?

Answer 1

If the covariate $x_i$ is strongly associated with the outcome, then you would expect to gain in predictive performance irrespectively of the fact that this covariate is not time-varying.

For example, say that $x_i$ is a dummy variable for sex with levels male and female and that there is a big difference in the levels of $y_i$ between the two groups. Then by not including it in your model, you would not appropriately capture this difference. The random effects would try to compensate, but in general, it is always better to try including the covariate.

Answer 2

As long as the variable $x_i$ varies across $i$, meaning there are deviations from the mean that will help in explaining or shifting the regression line, it does not need to vary across $t$ for it to improve your prediction. The easiest way to think of it is, does it improve your $R^2$? If it does, it must also help in improving your prediction.