# Estimating the effect of time-invariant and time-varying regressors using Fixed Effects

I have a question about the Fixed vs Random effects modeling.

It is said that

Fixed effects modeling identifies only parameters for time-varying regressors, not for time-invariant regressors

I can intuitively understand the above statement as follows: If I have a model $$y_{it} = \alpha_i + x^{'}_{it}\beta+ \epsilon_{it}$$, all the effects of time-invariant individual fixed effects are taken care of in the $$\alpha_i$$ term. Hence, we cannot individually estimate the parameters for each time-invariant regressor.

However,

Random effects modeling can identify the parameters for time-varying and time invariant regressors.

I cannot understand how Random effects model is able to estimate the parameters for time-invariant regressors.

Random effects treats individual differences in level as an error term that is not correlated with anything else. This is easiest to understand with an example of just 4 observations. Let x be a time invariant regressor that varies in the crosssection. Rewrite the individual intercepts as deviations from a fixed value: $$\alpha_i = \alpha + u_{i}$$
$$y_{00} = \alpha + \beta x_{0} + \epsilon_{00} + u_{0}$$ $$y_{01} = \alpha + \beta x_{0} + \epsilon_{01} + u_{0}$$ $$y_{10} = \alpha + \beta x_{1} + \epsilon_{10} + u_{1}$$ $$y_{11} = \alpha + \beta x_{1} + \epsilon_{11} + u_{1}$$
$$E[y_{11} - y_{01}|X=x] = \beta (x_{1} - x_{0}) - E[\epsilon_{11} -\epsilon_{01} |X=x] - E[u_{1} - u_0|X=x] = \beta (x_{1} - x_{0})$$ is informative about the coefficient of the time invariant regressor $$\beta$$ and this variation is used, in an optimal way, by the random effects estimator.
The key assumption is $$E[u_{i}|X=x] = 0$$, that the individual level differences are not correlated with $$X$$. That this isn't plausible in many settings is why fixed effects is widely used.