Practical Problems in Applied Econometrics

This post summarize some practical problems in applied econmetrics

• Incidental parameter problem

Incidental parameter problem

the fixed effects estimators of nonlinear panel data models can be severely biased because of the “well-known” incidental parameter problem.
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:

$$y_{it}=\alpha_i+u_{it},\ \ u_{it} \sim N(0,\sigma^2)$$

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 $\sigma^2$:

$$\sigma^2=\frac{1}{NT} \sum_{i} \sum{t}(y_{it}-\bar{y}_i) \sum_i \sum_t (y_{it}−\bar{y}_i)^2=\sigma^2 \frac{\xi^2_N(T−1)}{NT}=\sigma^2 \frac{N(T−1)}{NT}= \sigma^2 \frac{T−1}{T}$$

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 σ2 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 $T$!

Selection Bias

Selection bias is the bias introduced by the selection of individuals, groups or data for analysis in such a way that proper randomization is not achieved, thereby ensuring that the sample obtained is not representative of the population intended to be analyzed.[1] It is sometimes referred to as the selection effect. The phrase “selection bias” most often refers to the distortion of a statistical analysis, resulting from the method of collecting samples. If the selection bias is not taken into account, then some conclusions of the study may not be accurate.