<p>Let <span class="math-container">$i$</span> index firms and <span class="math-container">$t$</span> time. Consider the following type of regression:
<span class="math-container">$$
y_{i,t} = \alpha + \beta_i + X_{i,t}\gamma + \varepsilon_{i,t}
$$</span>
where <span class="math-container">$\beta_i$</span> are the firm fixed effects and <span class="math-container">$X_{i,t}$</span> is a set of other covariates.</p>
<ul>
<li>If the number of time periods and firms goes to <span class="math-container">$\infty$</span> then both the estimates of <span class="math-container">$\beta_i$</span> and <span class="math-container">$\gamma$</span> will be consistent.</li>
<li>If the number of periods is finite while the number of firms goes to <span class="math-container">$\infty$</span> then you are correct that the fixed effect estimates will not be consistently estimated. However, the estimates of <span class="math-container">$\gamma$</span> are still consistent. Usually, these latter are the ones we are interested in.</li>
</ul>

CC BY-SA 4.0
2021-11-18T13:49:59.687Z
Interestingly, this is called an "incidental parameters problem." Although, check out this link: https://ocw.mit.edu/courses/economics/14-385-nonlinear-econometric-analysis-fall-2007/lecture-notes/lec21_22_nlpan.pdf   .  It seems as though the estimates of these FE are problematic only in non-linear models estimated by MLE..
ChinG
CC BY-SA 4.0
2021-11-18T17:02:06.867Z
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.”
chan1142
CC BY-SA 4.0
2021-11-18T17:05:42.880Z
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.
chan1142
CC BY-SA 4.0
2021-11-19T08:24:32.730Z
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?
Papayapap
CC BY-SA 4.0
2021-11-19T08:25:29.897Z
Another thing I thought about: in this type of data that labor economists use we have multiple workers per firm. So perhaps n (number of workers) increases to guarantee consistency? However, I am not sure this is the whole story because people actually estimate both the worker AND the firm fixed effect.
Papayapap
CC BY-SA 4.0
2021-11-19T08:32:48.710Z
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.
chan1142
CC BY-SA 4.0
2021-11-18T13:34:25.420Z
2021-11-18T13:34:25.420Z
tdm
<p><a href="https://uploads-ssl.webflow.com/5d8e3657fd776a7142924af1/5dea0a068be68880b143ee61_lecture8_akm.pdf" rel="nofollow noreferrer">Estimating firm fixed effects</a> is very popular in labor economics.</p>
<p>I wonder why this is legit? The estimates shouldn't be consistent, the more firms we have the more parameters we have to estimate.</p>

CC BY-SA 4.0
2021-11-18T06:30:02.650Z
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
CC BY-SA 4.0
2021-11-18T12:43:28.320Z
@tdm please consider expanding it into an answer, comments should not be used for answers because they cant be accepted
1muflon1
CC BY-SA 4.0
2021-11-19T08:28:17.523Z
"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.
Papayapap
<econometrics><labor-economics><fixed-effects>
Consistent estimation of fixed effects
Papayapap
# Consistent estimation of fixed effects

(Asked by: username_0 on 2021-11-18)

[Estimating firm fixed effects](https://uploads-ssl.webflow.com/5d8e3657fd776a7142924af1/5dea0a068be68880b143ee61_lecture8_akm.pdf) 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.

**Tags:** econometrics, labor-economics, fixed-effects

> **Comments:**
> **username_1**: 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.  
> **username_2**: @username_1 please consider expanding it into an answer, comments should not be used for answers because they cant be accepted  
> **username_0**: "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.  

---

## 1 answers

### Answer by username_1 on 2021-11-18. Score: 4

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.

> **Comments:**
> **username_3**: Interestingly, this is called an "incidental parameters problem." Although, check out this link: https://ocw.mit.edu/courses/economics/14-385-nonlinear-econometric-analysis-fall-2007/lecture-notes/lec21_22_nlpan.pdf   .  It seems as though the estimates of these FE are problematic only in non-linear models estimated by MLE..  
> **username_4**: 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.”  
> **username_4**: 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.  
> **username_0**: 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. @username_4 could you elaborate on the unbiasedness?  
> **username_0**: Another thing I thought about: in this type of data that labor economists use we have multiple workers per firm. So perhaps n (number of workers) increases to guarantee consistency? However, I am not sure this is the whole story because people actually estimate both the worker AND the firm fixed effect.  
> **username_4**: 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.  

---

