Difference in Difference in the non-linear case

[Neeraj Kumar]

Hello,

I read the paper by Patrick A. Puhani "The Treatment Effect, The Cross Difference, and the Interaction Term in Non-Linear Difference in Difference Models". They mentioned that, " in a non-linear model such as probit is correct, the cross difference (or derivative) does not represent the treatment effect and thus not an interesting parameter in a nonlinear "difference-in-difference" model. Instead, it is correct to focus on the coefficient of the interaction term".

I am using difference in difference with the help of logit model

logit i.treatment##i.pre_post other_covariates

margins treatment, dydx(pre_post) pwcompare

Is treatment effect # pure-intervention effect shown by coefficient of interaction term or coefficient of marginal table ?

[John Mullahy]

Neeraj: This is a topic that has generated lots of interest. A key issue here is how one defines "treatment effect." A seminal paper is Athey & Imbens, Econometrica, Vol. 74, No. 2 (March, 2006), 431–497. In a nonlinear model, as you probably know, standard D-I-D doesn't eliminate subject-specific effects like it does in linear models.

Moreover the interpretation of the parameter associated with the "interaction term" also has a different meaning in a nonlinear model. Importantly, in nonlinear models the sign of that parameter doesn't even necessarily coincide with the sign of the cross-difference (second partial derivative or difference) since that cross difference in general depends on all the parameters and covariates.

In addition to Puhani's paper, the following article from Stata Journal is quite helpful.
http://www.stata-journal.com/article...article=st0063