# Incidental parameter problem for fixed effects OLS regression with logarithmic dependent variable on a short panel

I have panel data over 7 years and 6000 observations in total, on which I am running an OLS regression with around 600 fixed effects dummies. The dependent variable is logarithmic. I have heard about the incidental parameter problem, which biases the regression in short non-linear panels. Is an OLS regression with a logarithmic dependent variable a non-linear panel for which the incidental parameter problem applies, or are only models like logit and probit counted as non-linear panels?

• I think of “non-linear” as referring to the functional form of the model predicting the dependent variable. Probit and logit have a non-linear functional form—a linear model with a logarithmic DV still has a linear function form. – paqmo Aug 5 '20 at 15:16

Do you care about the fixed effects $$\alpha_i$$ for some reason? Usually they are a kind of a nuisance parameter that you either get rid of (or control for) to get the other common parameters right (the $$K$$ coefficients on time-varying Xs of interest).
In a short panel, the $$N$$ FEs will be inconsistent if you estimate them with dummies because with short panels the asymptotics involve adding more panels (growing $$N$$). The problem is that each time you do that you get a new $$\alpha_i$$ to estimate, so you never get to learn more about the FEs. You only have T (or fewer) observations to estimate them and that never changes as your data grows. You find yourself in this boat now, with largish N = 600 and small T = 7 (so not entirely sure how you get $$600 \cdot 7 < 6K$$ observations). In a linear model, you can still estimate the $$K$$ common parameters consistently, unless $$T\rightarrow \infty$$ as well as $$N$$. This was a surprising result for me, and one of the many charms of linear models.
Your model is linear in parameters (which is what really matters) and not growing $$T$$, so you are safe here. You can use all kinds of variable transformations, include polynomial terms, add interactions and still maintain the linearity.
The problem with nonlinear panel models is that the inconsistency spreads from the FEs to the $$K$$ coefficients that are common to all $$N$$, even though those are estimated using $$NT \rightarrow \infty$$. This is just the standard MLE small-sample bias. There are no general solutions here, though there are special cases that allow you to get consistent $$K$$ coefficients after getting rid of the nuisance parameters.
To be a bit clearer, this whole idea of adding $$N$$ is mostly a thought experiment. You are stuck with the data you have, though sometimes you can get more by sampling some new panels and/or waiting longer. And sometimes you can't: if you want to put in US state effects, there is only ~51. But the mathematical recipes for calculating your parameters and their uncertainty try to guarantee how they behave if you were to add more data. And those guarantees make assumptions about how you get more data (more $$N$$, more $$T$$, or both). With real, unsimulated data you never know just where on the road to Asymptopia you are for the guarantee of good behavior to be meaningful. But it is still useful to keep in mind which of the three Asymptopias you are heading to when you apply a statistical method.