ECON0060 Advanced Microeconometrics
Problem set 6: Due before your tutorial.
1. Consider taking a large random sample of workers at a given point in time. Let sicki = 1 if person i called in sick during the last 90 days and sicki = 0 otherwise. Let zi be a vector of individual and employer characteristics including local cigarette taxes and smoking laws. Let cigsi be the number of cigarettes individual i smokes per day on average.
(a) Explain the underlying experiment of interest when a researcher wants to examine the effect of cigarette smoking on workdays lost.
(b) Why might cigsi be endogenous, that is, correlated with unobservables that affect sicki?
(c) Suppose that
Pr (sick = 1| z, cigs, q) = Φ (z1δ1 + γ1cigs + γ2q)
where Φ(·) is the standard normal cumulative distribution function, z1 is a subset of z, and q is an unobservable variable that is possibly correlated with cigs which could cause omitted variable bias in the probit estimator for γ1. Assume that q | (z1, cigs) ∼ N(µ · cigs, σg2). Suppose you ignore q and estimate a probit model for sick including only the variables (z1, cigs) and obtain the estimate ˆγ1. Derive the formula for the probability limit of ˆγ1 in terms of (γ1, γ2, µ, σg2).
(d) Is it possible that the distribution of cigs conditional on z is normal? Explain.
(e) Explain how to test whether cigs is endogenous. Does this test rely on the assumption that cigs is normally distributed conditional on z?
(f) Explain under what conditions you could use a control function approach to estimate the model parameters. Explain how you would use the control function approach to do so.
2. Consider the probit model with continuous endogenous regressor discussed in Wooldridge (2010)
Section 15.7.2 and in the lecture notes from lecture 5.
with
(a) Show that
(b) Explain how you would construct a consistent estimate for ˆε2.
(c) Explain how you could estimate
(d) Derive formulas for (β1, α, θ) as functions of (β1ρ, αρ, θρ).
(e) Explain how you can estimate (β1, α, θ) and also how you can estimate standard errors for these parameter estimates.