Comparing coefficients in IV regression with period interactions and Year FEs

[Hakan Gunduz]

Hi everyone,

I am working with a panel data IV regression using ivregress. My objective is to split the sample into two periods (pre-2008 and post-2008) to examine the structural break and compare the coefficients between these sub-samples.

Initially, I ran the regressions separately for each sub-sample. Then, to formally test the differences in coefficients, I attempted to replicate these results using a fully interacted model (interacting the regressors and instruments with a post2008 dummy).

However, my specification requires the inclusion of year fixed effects. Here is the simplified logic of my code:

ivregress 2sls y x i.year (z=a b), cluster(id) // Full sample
ivregress 2sls y x i.year if year > 2008 (z=a b), cluster(id) // Post-2008
ivregress 2sls y x i.year if year < 2008 (z=a b), cluster(id) // Pre-2008

gen zpost2008 = z * post2008
gen apost2008 = a * post2008
gen bpost2008 = b * post2008

ivregress 2sls y c.x##i.post2008 i.year (z zpost2008 = a b apost2008 bpost2008), cluster(id)

As expected, in the fully interacted model, Stata drops one of the year fixed effects (in addition to the base year) to resolve perfect collinearity with the post2008 dummy. However, I observed that the coefficients for the variables of interest match those obtained from the separate split-sample regressions. Given that my primary goal is to compare the slopes between the two periods, is this interaction approach considered methodologically correct despite the collinearity issue with the year fixed effects?

Thank you!

[Clyde Schechter]

The fully-interacted model is entirely equivalent to the separate subset regressions approach with regard to the coefficients it produces. The colinearity is a non-issue: the only difference is that when you do the interaction approach the omission of a second time indicator is visibly called to your attention, whereas in the separate subset regressions it is there but you probably won't notice it unless you look for it. That is, in the separate subset regressions, you have one time indicator dropped in each subset, a total of two for the two regressions considered together. In the interaction model, you have two time indicators dropped--exactly the same number.