I have a question regarding three way interactions.
In traditional three way interactions, the hierarchy principle tells us that you must include all the main effects, the first order effects, and then finally the three way effects. As I understand, this only hold if the all possible permutations are allowed, and the observations are independent of one another.
However, I have a case where the observations are NOT independent of one another. In my regression I interact Time, Treatment, and Firm_size. In my dataset, each firm has two entries; one in Year 1 and one in Year two (it's a panel). Firm_size is not allowed to evolve. It is fixed to what it was in Year_1. The treatment was received between Year 1 and Year 2, so if it turns out that a firm received treatment by the time the survey was done in Year 2, I label it as treated for both years.
Clearly, there is some higher level conditionality in the data (e.g. one firm has two observations and cannot take on any value for treatment or firm size). Now the question is, how do I setup my regression? I think I can argue that I don't need all the first order effects, but in my case which ones can I drop? I lean towards the Time*Treatment one.
Alternatively, I could collapse the firm data into one line per observation by specifying separate variables for the ones in 2006 and 2010 where it applies... Or take differences...
Thanks for any help. Responses as detailed as possible welcome!
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