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There is one problem that arises when using a fixed-effects model. Assume that you have a sample of observations for a large number of individuals over a period of years. If you use a fixed-effects model, you will not be able to find parameter estimates for any variable like race or sex that do not change over the time period of the sample. The reason for this limitation is that the time-constant variables are perfectly correlated with the dummy variables used for the fixed-effects. A similar problem arises if the fixed-effects are for years (rather than individuals). You cannot include a variable is constant for all individuals in any given year. Quite often the individual-constant (or time-constant) variable is not of interest and nothing is lost by not having the parameter estimate. On the other hand, the random-effects model does not have this problem because the estimation makes use of differences amongst the individuals to estimate a parameter for the individual-constant variable. Another way to think about this point is to remember that, unlike the fixed-effects model, the random-effects does not use dummy variables to summarized the unknown characteristics; thus, there is no problem with multicollinearity. We discuss in the next section an example in which this “problem” arises.
What would be nice is if there were a statistical test that allows us to decide if the random-effects model is the appropriate model? The Hausman test offers such a statistical test. The Hausman (specification) test exploits the fact that the parameters for the random-effects model should be not be statistically different from those found using a fixed-effects specification. If one observes a chi-squared value greater than the critical value you can conclude that the parameter estimates for the random-effects model are statistically different from the parameter estimates for a model using an assumption of fixed-effects, then you can conclude that the random-effects model is misspecified. Unfortunately, the misspecification could be due to the fact that the fixed-effects model is appropriate or it could be due to the unobserved error terms being correlated with the included explanatory variables. If the latter is the case, then one might consider augmenting the model with an appropriate measure of the part of the unobserved effect that is correlated with the error term. What we are describing is that same thing that happens when omitted variables are correlated with the error term—the parameter estimates are biased. We include an example of how to use Stata to perform the Housman specification test.
There are three commands that matter in setting up the panel data. The first two commands precede the regression command because they establish which variable denotes the time period and which variable denotes the cross-sectional unit. These commands are:
.iis [variable name]
.tis [variable name]
The command for estimating the fixed-effects model is:
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