Consider the program evaluation problem where w is the binary treatment indicator and the potential outcomes are
Question:
Consider the program evaluation problem where w is the binary treatment indicator and the potential outcomes are y(0) and y(1), as we discussed in Sections 3-7e, 4-7, 7-6a, and elsewhere. For a set of control variables x1, x2, . . . , xk, define conditional mean functions for the two counterfactuals:
If y(0) and y(1) are binary, these are the response probabilities. If we wish to move beyond a linear probability model, we would likely use a logit or probit model. If the outcome is a count variable, we would likely use exponential models. We can also use the means from a Tobit if y(0), y(1) are corner solutions.
Recall that the average treatment effect is
(i) Explain why
where both expectations are necessarily over the distribution of x.
(ii) If you knew the functions m0(x) and m1(x), and you have a random sample
would be an unbiased estimator of τate.
(iii) Now consider the case where we have models, m0(x, θ0) and m1(x, θ1), where θ0 and θ1 are parameters and we have estimators θ̂0 and θ̂1. Now how would you estimate τate?
(iv) Define the observed response as usual:
Show that if w is independent of [y(0), y(1)] conditional on x, then
Therefore,
(v) If y(0) and y(1) follow logit or probit models with response probabilities
respectively, how would you estimate all of the parameters? How would you estimate τate?
Step by Step Answer:
Introductory Econometrics A Modern Approach
ISBN: 9781337558860
7th Edition
Authors: Jeffrey Wooldridge