This question uses the probit model but requires little knowledge of the model. Let y denote a

Question:

This question uses the probit model but requires little knowledge of the model. Let $y$ denote a binary variable that takes value 0 or 1 according to whether or not an event occurs, let $x$ denote a regressor vector, and assume independent observations.

(a) Suppose $E [y ∣ x] = Φ (x^{'} β)$ , where $Φ (\cdot)$ is the standard normal cdf. Show that $E [(y - Φ (x^{'} β)) x] = 0$ . Hence give the estimating equations for a method of moments estimator for $β$ .

(b) Will this estimator yield the same estimates as the probit MLE? [For just this part you need to read Section 14.3.]

(c) Give a GMM objective function corresponding to the estimator in part (a). That is, give an objective function that yields the same first-order conditions, up to a full-rank matrix transformation, as those obtained in part (a).

(d) Now suppose that because of endogeneity in some of the components $E [y ∣ x] \neq Φ (x^{'} β)$ . Assume there exists a vector $z, \dim [z] > \dim [x]$ , such that $E [y - Φ (x^{'} β) ∣ z] = 0$ . Give the objective function for a consistent estimator of $β$ . The estimator need not be fully efficient.

(e) For your estimator in part

(d) give the asymptotic distribution of the estimator. State clearly any assumptions made on the dgp to obtain this result.

(f) Give the weighting matrix, and a way to calculate it, for the optimal GMM estimator in part (d).

(g) Give a real-world example of part (d). That is, give a meaningful example of a probit model with endogenous regressor(s) and valid instrument(s). State the dependent variable, the endogenous regressor(s), and the instrument(s) used to permit consistent estimation. [This part is surprisingly difficult.]

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