Can I do panel regression if all my covariates are time-invariant? We have a data set where our outcome of interest varies over 10 years, but the explanatory variable of interest and all of the potential confounders are time-invariant. I am quite certain that a panel regression is not possible with this data set but want to be sure I'm not missing something. With a fixed regression all the covariates drop out and the assumptions of random effects do not hold with these data. And practically speaking, does there not need to be at least some variation in the explanatory variables to model the relationship with a time-varying outcome?
 A: Under some assumptions you can recover the coefficients of time invariant variables in fixed effects regressions. The reference for this would be Abowd et al. (1999). Even though their model is for hierarchical panel data the logic should also apply to the standard panel data setting. If you have a regression of the form
$$y_{it} = x_{it}\beta + \theta_i + \epsilon_{it}$$
where $y_{it}$ is the outcome, $x_{it}$ are the time-varying observed variables, $\theta_i$ are the individual fixed effects, and $\epsilon_{it}$ is a stochastic error term. Let
$$\theta_i \equiv \alpha_i + u_i\eta$$
where $u_i$ are observed time-invariant individual characteristics with the corresponding vector of regression coefficients $\eta$, and $\alpha_i$ is the unobserved time-invariant heterogeneity. Estimate the first equation with the standard fixed effects estimator. Then Abowd et al. (1999) obtain the overall individual heterogeneity component as
$$\widehat{\theta}_i = \overline{y}_i - \overline{x}_i\widehat{\beta}$$
The bar variables have been averaged over time for each individual. Now take the above equivalence and regress
$$\widehat{\theta}_i = \text{constant} + u_i\eta + \text{error}_i$$
In order to identify the vector $\eta$ (the coefficients of your time-invariant observed variables), you must assume that $Cov(\alpha_i, u_i) = 0$. This is important. Given that $\alpha_i$ is unobserved it is included in the error term $\text{error}_i$ and therefore causes omitted variable bias if $Cov(\alpha_i, u_i) \neq 0$. Except for this you also need the strict exogeneity assumption $E[\epsilon_{it}|x_{it},\theta_i] = 0$ which you generally require in panel data methods.
The alternative, as mentioned by CloseToC in the comments, is to find an instrument for your variable of interest. However, for that you need some other sort of variation in the data, e.g. your variable does not vary over time but it varies for different states. Finding good instrumental variables for such cases is difficult in general.
