## Random Effects

My last post was on the fixed effects model. I established that the fixed effects model assumes that variables not included in the regression are correlated with the variables included in the regression, and thus the results of the regression cannot be used to assess the effects of unobserved variables. The random effects model, on the other hand, assumes that unobserved variables are not correlated with observed variables, and allows the regression to be used to investigate the effects of variables not included in the regression.

In my past posts on the panel data model and its specific variations, I explained that the general form of the panel data model is $y_{it} = \alpha_i + \beta'x_{it} + \epsilon_{it}$, and the general form of the fixed effects model is $y_{it} = D \alpha_i + \beta'x_{it} + \epsilon_{it}$. With the random effects model, the general form is $y_{it} = \alpha + \beta'x_{it} + u_i + \epsilon_{it}$. In this model, $\alpha$ is taken to be constant, and $u_i$ is a measurement of random disturbance for each cross-sectional unit.

In choosing whether to use a fixed effects model or a random effects model, one must first test to see if individual effects exist. This is done using a Langrange Multiplier (LM) test. If they do indeed exist, then a Hausman test can be used. The Hausman test uses a hypothesis test to determine whether or not the fixed effects model and the random effects model have the same variance. If their variances are the same, then a random effects model may be used. If not, the more restricting fixed effects model must be used.

Recall the general form of a panel data model: $y_{it} = \alpha_i + \beta'x_{it} + \epsilon_{it}$. The fixed effects model assumes that the individual effects coefficients ($\alpha_i$) vary across each cross-sectional unit, while the $\beta'$ coefficient is held constant.  As a result of the variability in the individual effects coefficients, it becomes necessary to use dummy variables representing each cross-sectional unit in order to properly estimate the regression. The resulting equation is $y_{it} = D \alpha_i + \beta'x_{it} + \epsilon_{it}$ where $D$ is the set of dummy variables.