Suppose I have a longitudinal dataset in which each country $i$ is observed at different points in time $t$. Suppose that my dependent variable is a dummy variable, and I want to estimate the probability of success, say $y_{it}=1$, conditional to a set of predictors. To do this I estimate a pooled probit model with time fixed-effects: \begin{equation} P(y_{it}=1|x) = \Phi(x_{it} \beta + \delta_t + \delta_r) \end{equation} Here, I include a full set of time and regional dummies (e.g. Europe, Asia...), $\delta_t$ and $\delta_r$, respectively. However, in some countries, the dependent variable ALWAYS takes the value zero. Consequently, it is preferable not to include country dummies ($\delta_i$) in the regression to control for cross-country unobserved heterogeneity. According to my notes, it is more efficient to control for regional unobserved heterogeneity rather than country-specific heterogeneity. This is because, for some countries in the sample, the dependent variable is ALWAYS zero, leading to their exclusion from the regression analysis. Why are these countries dropped?
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1$\begingroup$ Why is country a fixed effect? $\endgroup$– Peter FlomCommented Mar 24 at 15:26
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$\begingroup$ I have just made the question more clear $\endgroup$– MaximilianCommented Mar 24 at 15:48
1 Answer
I assume with regional dummies you are referring to groups of countries that have the same dummy, so e.g. a dummy for Europa, a dummy from Asia etc.
It is indeed more efficient to control for regional unobserved heterogeneity. Let's say you have 196 countries and 6 regions. Generally, one would expect that the country-specific heterogeneity is small / non-existent within a region. In such case, estimating 196 parameters vs. estimating 6 parameters makes a huge difference in terms of consistency. For the 6 parameters, you have a lot more the data so the variance of the parameter estimates will be smaller. However, in case there is actually strong country-specific variation within a region, you will be missing out on this and this could lead to biased estimates. When this variation is small or negligible (depends on the application), controlling for regions likely gives more accurate results.
However, there is another problem with including dummies for all countries and/or time periods. This is called the incidental parameter problem (see this paper http://www.econ.brown.edu/Faculty/Tony_Lancaster/papers/IncidentalParameters1948.pdf for a review). When you have a large number of countries (N) and a small number of time periods (T), including dummies for all countries will lead to inconsistent estimates. This could also be the other way around when you have a large T compared to N and include dummies for all time periods. For linear models, you can apply within transformations to small obtain consistent parameter estimates for $\beta$. For non-linear models such as probit, this is not possible and you estimate parameters by MLE generally. However, for the logit model, there is an approach that can be interpreted as the "within transformation" from linear models. However, for this approach you need to condition on all units that have both zeros and ones. Thus, you need to drop observations that only have one of the two. I think this could be the reason for excluding them. See Section 15.8 of the book by Woolridge for an in-depth discussion of such methods: https://ipcig.org/evaluation/apoio/Wooldridge%20-%20Cross-section%20and%20Panel%20Data.pdf .
