Difference-in-Difference Estimators

Difference-in-difference (DD) estimators assume that in absence of treatment the difference between control (B) and treatment (A) groups would be constant or ‘fixed’ over time. DD estimators are a special type of fixed effects estimator.

(A-B) = Differences in groups pre-treatment represent the ‘normal’ difference between groups.

(A’-B) = total post treatment effect = normal effect (A-B) + treatment effect (A’-A)

DD estimates compare the difference in group averages for ‘y’ pre-treatment to the difference in group averages post treatment. The larger the difference post treatment the larger the treatment effect. 

This can also be represented in the regression context with interactions where t = time indicating pre and post treatment and x is an indicator for treatment and control groups. At t= 1 there are no treatments so those terms --> 0. The parameter b₃ on the interaction term is our difference in difference estimator as shown below.

y = b₀ + b₁ x + b₂ t+b₃ xt + e

References: 

Program Evaluation and the
Difference-in-Difference Estimator
Course Notes
Education Policy and Program Evaluation
Vanderbilt University
October 4, 2008

Difference in Difference Models, Course Notes
ECON 47950: Methods for Inferring Causal Relationships in Economics
William N. Evans
University of Notre Dame
Spring 2008