"Equivalence" between -eteffects-, -ivregress- and -teffects psmatch-

[Parul Gupta]

I’m working on an empirical project using observational data where I suspect endogeneity between the treatment variable and the outcome. My goal is to estimate the causal effect of a binary treatment variable (D) on an outcome variable (Y). The main concern is that D may be endogenous due to selection bias.

I am trying to use different approaches and want to confirm if the following three specifications are "equivalent". I understand that the estimates from the three approaches will be different because the estimation method is different. I want to check if I am setting up the three commands properly.

D is the treatment variable assumed to be influenced by $p and district fixed effects. This is being used as the exclusion criteria.


*treatment effects model
eteffects (y $l $s $v $stateFE) (D $p $l $s $v $district_code), vce(cluster district_code) atet


*IV
ivregress 2sls y $l $s $v $stateFE (D=$p $district_code), vce(cluster district_code)

*PSM
teffects psmatch (y) (D $p $l $s $v), vce(robust) atet

In my case, PSM doesn't allow district fixed effects to be included, so the third specification doesn't have district dummies.

I might not to be using PSM at all, but I want to make sure that IV and eteffects are the same specifications.

[Parul Gupta]

Edit to #1: I might not end up using PSM (unable to edit original post)

[Jeff Wooldridge]

Parul: PS matching is essentially like using regression, except it is based on the propensity score rather than a parametric model for the conditional mean. It is not like the control function method in eteffects or the IV method in ivregress. Both of these require instrumental variables. I would actually implement the full endogenous switching regression method, as I discuss in my 2015 Journal of Human Resources survey. eteffects almost does that, but not quite.

[Parul Gupta]

Thank you for your response.

I am not too convinced about PSM myself, but I was doing it as a robustness check.

Regarding eteffects, my understanding is that it implements something similar to the Heckman style model. I used this source here: https://us.sagepub.com/sites/default...4_Chapter4.pdf

I am using eteffects instead of treatreg because my outcome variable is binary and treatreg only allows continuous outcome. I tried -cmp- but it didn't converge, and estimation of std errors was taking too long with -eprobit-. The effect sizes were almost the same with eteffects and eprobit so I decided to go ahead with eteffects.

Is the JHR paper similar to Heckman-type model? Is there an in-built command to implement it?

Finally, in case I end up having to use -eteffects- and -ivregress- (or ivprobit), are the specifications shown in #1 correct? My main concern is whether I should use "$p $l $s $v $district_code" as the instruments (in ivreg), or only "$p $district_code" for it to be similar to the eteffects specification?

[Jeff Wooldridge]

This is a bit subtle. The methods in eteffects are generally approximations because they insert first-stage residuals into nonlinear functions. This control function approach is sometimes called "two-stage residual inclusion." The are based on the assumption that the functional form -- such as probit -- is valid once you condition on the CF. I made a case for this in my 2014 Journal of Econometrics paper, but it's not something everyone embraces.

If you want to do a Heckman-type switching regression for probit that is valid when the two models are probit (before conditioning on the CF), you can apply "heckprobit" to the control and treated units separately. Then, you compute the predicted values in each case and average the difference. This gives the average treatment effect. You can put all steps in a bootstrap routine to get a standard error. I should have this somewhere with my graduate econometrics course.