# Difference-in-Differences derivation

#### JohnMayer

##### New Member
Hi everyone,

I have a problem with a derivation in the Diff-in-Diff setting, based on the paper "Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania" by David Card and Alan Krueger.

Card and Krueger estimated their model according to the following equation:
(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:
(2) Y_i = α + γ_t*T_i + γ_d*D_i + τ_DiD*T_i*D_i + ε_i

Now I have to show that the Diff-in-Diff specification (1) is equivalent to the specification in (2), both without covariates but honestly I have no idea how to start.

Can anyone help out? Thanks in advance!