# continuous difference in difference state policy

I have seen a paper where the authors run the following regression:

$$Y_{st} = \lambda_s+\mu_t + \beta log(Min Wage)_{st} + controls + error$$

where $$\lambda$$ and $$\mu$$ are state and time fixed effects. I am leaving out one parameter on the EITC to make the question simpler.

The min wage variable is the min wage in the state. They also say "we can obtain causal estimates of policy effects by comparing states that have different minimum wages and EITC rates within the same year" I want to make sure I understand what variation is actually being used here.

1. Is it the case that the levels of minimum wage are absorbed out by the states, so they arent comparing states with high minimum wages to states with lower minimum wages, they are comparing states who have higher changes in minimum wages to those with lower changes?

2. Why does this make sense to say they are 'comparing states with different minimum wages.. within the same year' then? Is it because you can think of it as comparing the higher minimum wage minus low minimum wage states before and after?

• Seems surprising that their regression involves taking the log of the minimum wage - since there are states where the minimum wage is effectively zero, and $log(0)$ is undefined. Nov 26, 2020 at 9:15
• Without having read the paper (you have linked a summary of it, and not the paper itself), I have a strong feeling that the reasoning by the authors which you are experiencing doubt over does not relate to the regression models per se, but rather the design of difference-in-differences (DiD) as quasi-experiments (for inferring causality). In particular, the parallel-trend assumption in DiD estimates. So my recommendation would be to do some digging on DiD, then understand how it has been applied to your particular use case. Dec 1, 2020 at 15:52