Inverse Mills ratio in system of equations

External question — Cross Validated Stack Exchange Author: research111 Original post: https://stats.stackexchange.com/questions/228868 License: CC BY-SA 3.0 — https://creativecommons.org/licenses/by-sa/3.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I have been stuck on the following selection/endogeneity problem for a while: I have a panel dataset with 1.000 products becoming fair trade products, all at very different times between 2010 and 2014. For each product, I have weekly sales data for 6 months before they become fair trade up to 1 year after they become fair trade. In step 1, I estimated the effect of becoming fair trade using a (step) dummy variable that becomes 1 from the week onwards they become fair trade using OLS for each product individually . In step 2, I regress the coefficients from the step dummy on about 12 variables (moderaters). I use a 2 step procedure, because 12 interaction effects are not possible to estimate in the same equation due to multicollinearity. The problem, however, is that brands self-select into joining fair trade or not. I do have data on all products, including those who did not become fair trade. In short, my question is: Can I estimate 1 (panel?) selection equation across time across products, and include the same single inverse Mills ratio in each product equation seperately? Or are there other, better approaches to my problem?

External answer — Cross Validated Stack Exchange Author: asdir Original post: https://stats.stackexchange.com/a/231321 License: CC BY-SA 3.0 — https://creativecommons.org/licenses/by-sa/3.0/ Adaptation: HTML converted to plain text; contact email addresses removed. To answer your question: No, you cannot use the IMR from a pooled selection equation in the regression in an outcome equation with just a subset of the pooled observations. The outcome equation has to contain all observations, except for those cases that do not fulfill your selection criterion. However, I think might be a much simpler resolution to your situation: If the multicollinearity you are concerned about his multicollinearity between the interaction term and its components, you could try standardizing your variables, as suggested here . Hope that helps.