Is it considered necessary to include a constant term in the definition of a panel data model which. among other regressors, includes a variable representing fixed effects?

  • $\begingroup$ No. A simple example is one where you have G=2 groups and include group dummies only. In this case without a constant the coefficients are a kind of group average. A "kind of" because each individual enters into the group average with a weight depending on how many time periods the individual is observed. .... When you include a constant you have to set one of the group coefficients (fixed effects) to zero for identification, which is done by leaving one of the group dummy variables out. The same hold for the case where each group are the group of observations for a particular individual. $\endgroup$ – Jesper for President May 12 '20 at 8:44
  • $\begingroup$ Many thanks for your prompt and helpful response. If I understood well, the constant term is set ("forced") to zero when all the individual fixed effects are to be used. $\endgroup$ – JCost May 12 '20 at 11:22
  • $\begingroup$ The model $y_{it} = \beta_0 + \mathbf x_{it}^\top \beta + \mu_i + \epsilon_{it}$ is the same as $y_{it} = \mathbf x_{it}^\top \beta + \lambda_i + \epsilon_{it}$ with $\lambda_i := \mu_i + \beta_0$ so leaving out the constant (forcing it to zero as you say) simply adds the constant value to the values of the fixed effects. When you recover $\hat \lambda_i$ from estimation of the second model and $\hat \mu_i$ from estimation of the first model the difference should be $\hat \beta_0$. $\endgroup$ – Jesper for President May 12 '20 at 12:52
  • $\begingroup$ I am grateful to you for taking the time to provide this detailed description. I was a bit confused because some empirical studies report a constant term in their regression results, although they have stated in previous pages that their model looks like the one you described on the right. They may have forgotten to ask from their software to exclude the constant term. But as I can see, the difference actually lies in the interpretation of the results with respect to the constant term and the fixed effects. Thanks again. $\endgroup$ – JCost May 12 '20 at 16:42
  • $\begingroup$ The constant term can be hidden in x as a column of ones. But yes in the end it is a matter of interpretation and usually the level of the fixed effect/an estimated constant term, will be uninterpretable. $\endgroup$ – Jesper for President May 12 '20 at 20:10

The intercept should almost always be included, that is, if you do not include, you should know why. Specific reasons and discussion can be found in the similar question When is it ok to remove the intercept in a linear regression model?, which is almost a duplicate.

In comments is mentioned the case of a categorical predictor for $k$ groups, with all $k$ dummys included. But that is not really an exception, because in that case the intercept is implicitely included, as the column space of the design matrix have the constant vector.


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