# Incidental parameter problem

I always struggle to get the true essence of the incidental parameter problem. I read in several occasions that the fixed effects estimators of nonlinear panel data models can be severely biased because of the "well-known" incidental parameter problem.

When I ask for a clear explanation of this problem the typical answer is: Assume that the panel data has N individuals over T time periods. If T is fixed, as N grows the covariate estimates become biased. This occurs because the number of nuisance parameters grow quickly as N increases.

I would greatly appreciate

• a more precise but still simple explanation (if possible)
• and/or a concrete example that I can work out with R or Stata.
• This does not suffice for an answer. The incidental parameters problem can occur in non-linear models which, unlike linear regression, do not have the property of being unbiased estimators. A popular example is probit/logit. These models are consistent estimators, meaning that as the ratio of the number of observations to number of parameters increases, the parameter estimates will converge onto their true values as standard errors become arbitrarily small. The problem with fixed effects is that the number of parameters grows with the number of observations. – Zachary Blumenfeld Dec 10 '15 at 8:08
• Therefore, the parameter estimates can never converge to their true value as the sample size increases. Thus the parameter estimates are severely unreliable. – Zachary Blumenfeld Dec 10 '15 at 8:08
• Thanks for this clarification. I guess I now better understand the problem. So, e.g., if my panel is T=8 and N=2000, I can add T-fixed effects in a probit/logit estimation and get reliable estimates. Otherwise, with N-fixed effects, I would get unreliable ones. Is this correct? – emeryville Dec 10 '15 at 19:29
• Here is a Blog entries illustrates the incidental parameter problem for logit and probit with an example in R: econometricsbysimulation.com/2013/12/… – Arne Jonas Warnke Dec 11 '15 at 14:12
• I just answered a related question on Stack Overflow where I give concrete examples in R and Stata code. – jay.sf Dec 19 '20 at 8:37

In FE models of the type $$y_{it} = \alpha_i + \beta X_{it} + u_{it}$$ $\alpha$ is the incidental parameter, because theoretically speaking, it is of a secondary importance. Usually, $\beta$ is the important parameter, statistically speaking. But in essence, $\alpha$ is important because it provides useful information on the individual intercept.

Most of the panels are short, i.e., T is relatively small. In order to illustrate the incidental parameter problem I will disregard $\beta$ for simplicity. So the model is now: $$y_{it} = \alpha_i + u_{it} \quad \quad u_{it}\sim iiN(0,\sigma^2)$$ So by using deviations from means method we have $\hat{u}_{it} = y_{it}-\bar{y}_i$ - and that's how we can get $\alpha$. Lets have a look on the estimate for $\sigma^2$: $$\hat{\sigma}^2 = \frac{1}{NT}\sum_i\sum_t (y_{it}-\bar{y}_i)^2 = \sigma^2\frac{\chi_{N(T-1)}^2}{NT} = \sigma^2\frac{N(T-1)}{NT} = \sigma^2\frac{T-1}{T}$$

You can see that if T is "large" then the term $\frac{T-1}{T}$ disappears, BUT, if T is small (which is the case in most of the panels) then the estimate of $\sigma^2$ will be inconsistent. This makes the FE estimator to be inconsistent.

The reason $\beta$ is usually consistent because usually N is indeed sufficiently large and therefore has the desired asymptotic requirements.

Note that in spatial panels for example, the situation is opposite - T is usually considered large enough, but N is fixed. So the asymptotics comes from T. Therefore in spatial panels you need a large T!

Hope it helps somehow.

• Could you elaborate a bit more, how $\frac{1}{NT}\sum_i\sum_t (y_{it}-\bar{y}_i)^2$, turned into $\sigma^2\frac{\chi_{N(T-1)}^2}{NT}$? – Mario GS Nov 29 '18 at 15:45
• @Mario GS: The sum of squared normal random variables is chi square distributed – Corel Nov 29 '18 at 19:32
• FYI, the third equality has a random variable on one side and its expectation on the other. I do like this because it's less messy to look at than the corresponding correct statement, but readers must understand that you're ultimately computing the expected value. – eric_kernfeld May 1 '20 at 22:52
• What happens if you include a single global mean parameter $\beta$ in this model? (As if in your initial statement $y_{it} = \beta X_{it} + u_{it}$ $X_it=1$ for all $i$, $t$.) You would probably estimate it as the mean of the $\alpha_i$'s, and the standard error would be estimated as $\frac{\hat \sigma}{\sqrt {NT}}$, right? – eric_kernfeld May 1 '20 at 23:07