Difference-In-Difference with Multiple Time Periods

#1
Hi! This is an equation I'm using for a difference-in-differences model. I'm appending four periods of survey data and observing around a dozen policy shifts.
Screenshot 2018-05-23 13.52.56.png
Where:
  • alpha is the intercept term
  • Y is the wages made in one week by one person in state s in time t
  • STATE accounts for fixed effects in each state (10 states and perhaps a control if I need one? I don’t know how this will work exactly)
  • YEAR is fixed effects in each year (1983-2001 is the scope as of right now although if time permits I’m hoping to extend, although I may ultimately end up working in 3 month blocks in which case it would be fixed effects for each three month block I am thinking)
  • LEGAL is an interaction term that multiplies a dummy variable that is one for if the state has ever had a prohibition law and 0 if it’s a control (if I end up having controls) and another dummy variable that’s 1 if the state currently has an active prohibition law and 0 if not. My issue here- is that the second dummy is always just equivalent to total variable, so why even have the interaction term? But I do know I need to have an interaction term in order for the equation to be a DiD and it needs to be a DiD for me to get any sort of causal results. Although, I am open to using another model.
  • LEGAL2 is an interaction term that multiplies a dummy variable that is one for if the state has ever had a partial-prohibition law and 0 if it’s a control (if I end up having controls) and another dummy variable that’s 1 if the state currently has an active partial-prohibition law and 0 if not.
Overall:
  • I think I ideally I am looking to perform the following dif-in-difs, although I do wonder if it would be productive to see what a move from prohibition to partial prohibition is or vice-versa in addition to looking at no prohibition to prohibition and no prohibition to partial-prohibition.

Screenshot 2018-05-23 15.18.27.png
  • I am flexible to changing both the mathematics and the concept of my model. I'm really struggling to understand how to write this so any help would be so appreicated!!!