Hi everyone,

I am aware that this forum is in particular for questions regarding Stata but my question arose while trying to re-estimate the paper by David Card and Alan Krueger "Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania".

I have a problem with a derivation in the Diff-in-Diff setting:

Card and Krueger estimated their model according to the following equation:
Code:
(1) ∆ E_i= α+ b*X_i+ c*NJ_i+ε_i
where
∆ E_i is the change in employment from wave 1 to wave 2 at store i
X_i is a set of characteristics of store i (we can cross X_i out because its not relevant in our case)
NJ_i is a dummy variable that equals 1 for stores in New Jersey
ε_i is the error term

In general, the average treatment effect of the treated (ATET) is estimated using linear regression:
Code:
(2) Y_i = α + γ_t*T_i + γ_d*D_i + τ_DiD*T_i*D_i + ε_i
Now I am trying to figure out if the Diff-in-Diff specification (1) is equivalent to the specification in (2), both without covariates (it should be!).

Can anyone help out?
Thanks in advance!