Two-way fixed effects and the F-test

Question

I have two questions regarding the reporting of scientific results. I was reading the following paper by Grinols and Mustard (2006) which can be downloaded here for free. Their main specification is a two-way fixed effects estimator which I reproduced below:

$$ C_{it} = \alpha + \beta_{i}X_{i} + \gamma_{t}T_{t} + \delta L_{it} + \theta A_{it} + \epsilon_{it} $$

where $C_{it}$ is the crime rate (offenses per 100,000 people) of county $i$ in year $t$, $\beta_{i}$ is the vector of estimated coefficients on the county-level fixed effects that control for unobserved characteristics across counties. The time fixed effect, $T_{t}$, controls for national crime rate trends. $L_{it}$ is a vector of the casino-opening dummy variables.

I included a screen shot of their tabular results in case anyone cannot access the article.

Two questions:

• Why is it necessary to indicate that the constant was removed (or not removed)? Reviewers have asked me to do this and I am not sure why. Most models will estimate a global intercept if you included dummies for all units and time periods.

• I assume the $F$-test is reported to show that the fixed effects are non-zero (i.e., pooled OLS is biased). Thus, is the $F$-test for the model without the fixed effects the normal $F$-test of "overall" significance?

I do not wish to critique the work of this author. Rather, I am using it to facilitate my own understanding about reporting scientific results and to appease several reviewers of my own work.

Answer 1

This is all conjectural, since the paper lacks detail.

My guess is that they used a panel FE model with time dummies since the county FE are nuisance parameters and there are many of them. The time dummies might have a bit more interest as national trends and there are fewer of them, so that makes their inclusion more interesting, computationally easier, and doesn't require scrolling. This was written before there were canned HDFE packages available, so this is probably just a standard panel FE estimator like xtreg, which has a pseudo-intercept. That is my guess as to what that No means. This is also what areg does, which is another popular estimator for such models.

On your second question, I believe that is just the standard F statistic that tests that the coefficients on the regressors are jointly zero. If it was a test of the FE, you should not also have it in the odd rows. This is just the standard Stata set of GoF measures. If they were different tests, I would expect there to be a note.

I would write the corresponding author and ask them to make sure.