What are the probability limits of the resulting OLS estimates 𝜃̂0 and 𝜃̂1 consistent?

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Suppose we have data on N people collected at time peroid b. Each of these people began to be unemployed at time ai. When we observe them, some people are now employed; denote the duration of their unemployment Di, For the people that remained unemployment, we know their unemployment spell is at leaset b-ai long. We as reaserchers are interested in the following questions: how does the amount of time someone remains  umemployed Di, impact their wage wi. when they do become emploled?? To answer this question, we specify the following two equations:

for part (a)-(c), assume for simplicity that we actually observe duration Di, and wages Wi,

For part (d), we only observe the full duration Di if the person employed by thime b and we only observe wage Wi if the person is employed at time b.

1. Suppose we observe Di and Wi. Suppose we estimate equation 1 using ordinary least squares. what are the probability limits of the resuling OLS estimates.

2. Suppose we observe D * and W *. Now, we estimate equation (2) using ordinary least squares (OLS). 

What are the probability limits of the resulting OLS estimates 𝜃̂0  and 𝜃̂1 consistent? 

and when are these estimates

3. Suppose we observe D * and W *. Propose an instrumental variables (IV) estimator for equation (2)

and prove that your IV estimator is consistent for true parameters 𝜃0 and 𝜃1.

4. No longer assume that we observe D * and W *. We assume that we know σ . Write down the

maximum likelihood estimator for equation (1) and the asymptotic distribution of 𝛽̂0 using maximum likelihood.

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