Difference-in-Differences derivation

[John Mayer]

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:

(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 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!

[Clyde Schechter]

I have heard of the Card and Kreuger study, but have not read the paper.

They are more or less equivalent, though not exactly so.

The α in model (2) has no corresponding term in model (1). (N.B., it is not the same thing as the α in model (1).)

γ_d in model (2) also has no corresponding term in model (1).

γ_t in model (2) corresponds to α in model (1).

τ_DiD in model (2) corresponds to c in model (1).

When I say "corresponds to," it means that if there is complete data both pre- and post- intervention for all observations, the corresponding coefficients, standard errors, etc. will come out the same when you estimate the models. If you have incomplete data, however, then there can be discrepancies in the estimates.

Basically, model (2) separately models the pre- and post intervention outcomes, although they are combined into a single equation by use of the interaction mechanism. Model (1) derives from model (2) by subtracting the pre- intervention version of the equation from the post-intervention equation. The parameters in model (2) that have no corresponding term in model (1) are parameters that refer to specific time periods, and hence are not identifiable when looking only at the change in outcome, not at the separate pre- and post- outcomes.