# Consistent estimation of fixed effects

Estimating firm fixed effects is very popular in labor economics.

I wonder why this is legit? The estimates shouldn't be consistent, the more firms we have the more parameters we have to estimate.

• That is correct. The coefficients on the fixed effects will usually not be consistent. However, the coefficients on the other covariates will be consistent, and these are usually the ones we are interested in.
– tdm
Nov 18, 2021 at 6:30
– 1muflon1
Nov 18, 2021 at 12:43
• "We" might be a different literature, because the estimation of the worker and especially firm fixed effects is very common in labor economics, see for example the lecture notes I hyperlinked. Nov 19, 2021 at 8:28

Let $$i$$ index firms and $$t$$ time. Consider the following type of regression: $$y_{i,t} = \alpha + \beta_i + X_{i,t}\gamma + \varepsilon_{i,t}$$ where $$\beta_i$$ are the firm fixed effects and $$X_{i,t}$$ is a set of other covariates.
• If the number of time periods and firms goes to $$\infty$$ then both the estimates of $$\beta_i$$ and $$\gamma$$ will be consistent.
• If the number of periods is finite while the number of firms goes to $$\infty$$ then you are correct that the fixed effect estimates will not be consistently estimated. However, the estimates of $$\gamma$$ are still consistent. Usually, these latter are the ones we are interested in.
• Unbiasedness and consistency are different. When $T$ is fixed $\hat\alpha_i$ is certainly not consistent but is still unbiased (in the context of repeated sampling, and under regularity). It is similar to small samples. When the sample size is 7, say, we can still do OLS and inferences although the sampling variability of the estimator is possibly huge. The same thing happens here. When $T$ is small, the sampling variability of $\hat\alpha_i$ can be huge, but they can still do LSDV (i.e., FE) and report coefficient estimates. Nov 19, 2021 at 8:32