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.


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.


It uses variation in the regressor across units.

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: $\alpha_i = \alpha + u_{i}$

$$y_{00} = \alpha + \beta x_{0} + \epsilon_{00} + u_{0}$$ $$y_{01} = \alpha + \beta x_{0} + \epsilon_{01} + u_{0}$$ $$y_{10} = \alpha + \beta x_{1} + \epsilon_{10} + u_{1}$$ $$y_{11} = \alpha + \beta x_{1} + \epsilon_{11} + u_{1}$$

Then a simple difference of eg the second observation of each unit,

$$E[y_{11} - y_{01}|X=x] = \beta (x_{1} - x_{0}) - E[\epsilon_{11} -\epsilon_{01} |X=x] - E[u_{1} - u_0|X=x] = \beta (x_{1} - x_{0})$$ 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.


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