1. You have been contracted by the Center for Disease Control (CDC) to measure the impact of mental health on labor force participation.  Over several weeks at a CDC clinic that specializes in treating patients suffering from depression, you collect patient data on labor force participation, levels of depression, and the results of a genetic test for a gene that pre-disposes people to depression.

A. Assuming that there is no missing data and that information was collected error-free, would a regression of labor force participation on levels of depression using this sample yield unbiased estimates?  Explain why or why not.

B. Why might we want to include the results of the genetic test for depression as a proxy variable in the regression from part A? 

C. What would be necessary for this to be a good proxy variableto add if we wish to estimate the effect of depression on labor market participation?  Similarly, can you think of any reason why it would not be a good proxy?

D. Suppose that as you collected the data, only certain people at the clinic would give you information about measured levels of depression, though you still had full information about the results of the genetic test due to a clinic-wide confidentiality agreement.  How would this affect your estimation of levels of depression on labor force participation?

2. Every year in Massachusetts, each town sends out local assessors to determine the value of each town's houses.  The job of an assessor is to approximate what the sale price of each house would be if it were to be put up for sale this year. 

In theory the assessed value of each house (A) and a house's sale price on the market (P) should be very close to each other, but some studies have claimed that there is a "regressive" relationship between A and P.  This means that low value houses are systematically over-assessed (so these owners overpay on their property taxes), and high-value houses are systematically under-assessed (so these owners underpay on their property taxes).

A. Suppose you were considering two models to examine this relationship:

       (1) P_i = β₀ + β₁A_i + μ_i

(2) ln(P_i) = β₀ + β₁ln(A_i) + μ_i

What could help you decide between using models (1) and (2)?  Explain.

B. If you ran a regression of P on A, would this satisfy all of the Gauss-Markov assumptions for unbiasedness?  Explain why or why not.

C. How would issues of assessor accuracy/error impact a regression of P on A?  Would assessor error matter if, on average, assessors could correctly appraise the value of houses? Explain.