Diff-in-Diffs Estimation
I’ve been puzzling over the Diff-in-Diffs method since a lunch discussion last week. I’ve clarified my thinking enough to get stymied, so I’ll record my thoughts here, and perhaps someone can explain it further. Non-economists: you will probably be mystified by this, but perhaps amused by the terminology.
Suppose we have 100 counties of data. We want to see if death rates for males depends on whisky consumption in a county. We are reasonably sure that death rates for females do not depend on whisky consumption. One regression we could run is
(1) Maledeath = B1*Femaledeath + B2*Whisky + Epsilon,
where there would be N=100 observations, one for each county, each with its own random disturbance Epsilon.
We would then test to see if B2 is significant, B1 having picked up the effect of most of the omitted relevant variables (e.g., proportion of elderly, income) that vary from county to county.
As I understand it, the standard Diff-in-Diffs approach does something different. It runs this regression:
(2) Deathrate = (County Dummies)+ D1*(Dummy for Maledeath) + D2*(Dummy for Maledeath)*Whisky + Epsilon,
where there would be N=200 observations, two for each county, where “Deathrate” would be Maledeath for 100 observations and Femaledeath for 100 observations, where (Dummy for Maledeath) would be 1 if Deathrate were Maledeath and 0 if it were Femaledeath, and where there would be 100 County Dummies.
We would then test to see if D1 were significant, the County Dummies having picked up most of the omitted relevant variables (e.g., proportion of elderly, income) that vary from county to county.
How do approaches (1) and (2) differ?
(A) How do they control for omitted variables?
Method (1) assumes that in the absence of whisky and Epsilon, the male death rate would be B1 times the female death rate in each county.
Method (2) assumes that in the absence of whisky and Epsilon, the male death rate in a county would equal the Constant plus D1 plus the county’s dummy. I think — but I might be wrong– that the county’s dummy equals its female death rate. D1 would add a constant amount to the female death rate to get the male death rate.
Thus, Method (1) says that to get the male death rate you start with the female death rate and scale it up using D1, whereas Method (2) says you start with the female death rate and add D1 to it.
(B) What do they assume about Epsilon?
Method (1) has 100 observations, with 100 independent county disturbances that add equally to male deaths and to female deaths. Method (2) has 200 observations, with 200 independent county/sex disturbances, but it also has 100 more variables– the county dummies– so it does not have any more degrees of freedom. Is there a difference, then? I think there must be, but I’m not sure what it is. Method (2) really says that there are three independent disturbances, one affecting male deaths, one affecting female deaths, and one common to them both. The County Dummies pick up the common disturbance.