Why are my constraints getting dropped?

Question

I'm currently applying the Roy Zelner test of poolability as shown in the excellent article of Andrea Vaona, in fact I'm working with panel N=17 T=5, and my model looks like this : $$Y_{it}= a_0+B_1X_1+B_2X_2+B_3X_3+B_4X_4+e_{it}$$

My question is the following. When I'm testing for coefficient equality of the unpooled data (the last stage), many of my constraints are getting dropped. This impacts the degrees of freedom of $\chi^2$, and I would like to understand the reason? Is this because the time dimension of my panel is too small? or because the number of my constraints is too high?

Ama

Answer 1

You have a panel data regression

$$y_{it}=x_{it}'\beta+u_{it},$$

where $x_{it}$ in your case is $(1,X_1,X_2,X_3,X_4)$. Poolability tests test whether alternative model is actually correct:

$$y_{it}=x_{it}'\beta_i+u_{it}.$$

So the null hypothesis is that $\beta_i=\beta$. To test this hypothesis we need to estimate $\hat{\beta_i}$. In your case, you need to estimate 17 $\beta_i$. Since $T=5$, your are estimating regressions with 5 parameters having 5 observations. This of course gives you a lot of problems, since the usual practice for statistical packages in this case is to drop some of the variables from the regression.

In general if $T$ is small do not test whether you can pool the data. Simply use panel data regression and check whether the resulting model is appropriate.