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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.

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    $\begingroup$ 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. $\endgroup$
    – tdm
    Nov 18, 2021 at 6:30
  • $\begingroup$ @tdm please consider expanding it into an answer, comments should not be used for answers because they cant be accepted $\endgroup$
    – 1muflon1
    Nov 18, 2021 at 12:43
  • $\begingroup$ "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. $\endgroup$
    – Papayapap
    Nov 19, 2021 at 8:28

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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.
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    $\begingroup$ Interestingly, this is called an "incidental parameters problem." Although, check out this link: ocw.mit.edu/courses/economics/… . It seems as though the estimates of these FE are problematic only in non-linear models estimated by MLE.. $\endgroup$
    – ChinG
    Nov 18, 2021 at 13:49
  • $\begingroup$ And the term “incidental parameters” was first introduced by Neyman and Scott (1948, Econometrica). Lancaster (2000, JoE) writes “It was, I believe, the only occasion on which Neyman, arguably the second most influential statistician of the twentieth century, published in an economics journal.” $\endgroup$
    – chan1142
    Nov 18, 2021 at 17:02
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    $\begingroup$ BTW, the estimators of the “fixed” effects are still unbiased (under regularity, of course, for linear models), so people do not hesitate to find useful information from the estimates. $\endgroup$
    – chan1142
    Nov 18, 2021 at 17:05
  • $\begingroup$ Thanks (+1), but I am not sure if the answer captures everything. Usually fixed effects panel regression is done in large N (not too large T) setting, so consistency is almost surely done on N. @chan1142 could you elaborate on the unbiasedness? $\endgroup$
    – Papayapap
    Nov 19, 2021 at 8:24
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    $\begingroup$ 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. $\endgroup$
    – chan1142
    Nov 19, 2021 at 8:32

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