What is the impact of the wage on labor supply? Give the two characteristics that a valid instrument should possess.

Question 1.

Consider the following linear regression model:yi = β0 + β1xi + β2wi + ui  where E(ui) = 0, E (xiui) /= 0, and E (wiui) = 0so that the regressor xi is endogenous. Assume on top of that that we do not observe wi. As we still need to estimate the effect of x on the conditional expectation of y, we consider the regressionyi = β1 + β2xi + vi

1.    Why is it reasonable to believe that regression (2) as endogenous? Let zi be an instrumental variable exogenous in regression (1).

2.    Why the candidate set of instruments that we shall use for an IV es- timation  of  (2)  is  Zi  = (1, zi)j  (before  we  check  whether  zi  is  a  valid instrument or not)?

3.    Give the analytical expression of βˆ2,iv,  the IV estimator of β2.  (Hint: Use the IV estimator formula for just-identified models)

Calculate the probability limit of βˆ2,iv   in the following cases:

i)    wi and zi are not correlated.

ii)    wi and zi are correlated.

4.  From 3.i) and 3.2) why is each of the conditions Cov (zi, xi) = 0 and  Cov (ziwi) = 0 important? 

 Question 2.

In this exercise, we want to determine the effect of the wage on the labor supply. To address this question, we use a sample composed of 4165 US workers. We consider a linear model in which the dependent variable weeks is the number of weeks worked in the year? Explanatory variables are the log of the monthly wage, denoted log(wage), the number of years of schooling (educ), an indicator variable that equals 1 if the wage is set by a union contract (union), an indicator variable that equals 1 if the person is a female (female). The model has been first estimated by OLS. Results are reported in Column (1) of Table 1.

1.    What is the impact of the wage on labor supply?

2.    We suspect the variable log(wage) to be endogenous. Give one possible reason for the potential endogeneity of log(wage). Explain briefly.

3.    Give (in words) the two characteristics that a valid instrument should possess.

4.    A two-stages least squares estimation is performed, using the following instruments: an indicator variable that equals 1 if the individual works in a manufacturing industry (industry) and an indicator variable equal to 1 if the individual lives in an urban area (urban). Discuss briefly whether the two potential instruments are likely to be valid and why.

5.    Estimation results are reported in Column (2) of Table 1. What is the impact of the wage on labor supply according to TSLS (2SLS) results?

6.    To test overidentifying restrictions, a Sargan (or J) test is performed. The value of the test statistic is 1.051. Give its behavior under the null hypothesis, the formal decision rule, and your conclusion based on the above results.

7.    Obtain the Hausman t-test statistic based. Does it suggest you should use the OLS results or the IV results?