1. Suppose the model of interest is Yi = 0 + 1X1i + 2X2i + ui , where E(u|X)=0 and E(u 2 |X)= and X1 and X2 are uncorrelated in your sample. Will the bivariate regression of Y on Xi have the same coefficient estimate and standard error for 1 as the multivariate regression of Y on X1 and X2?

2. Express the price elasticity of demand in terms of the coefficients in (1)

3. ln(Yt).

4. Commercial rental price (dollars/square foot/month) in Cambridge in that year. [

5. The average length of hair of individuals appearing in People magazine in that year.

able IV from Lemieux, MacLeod, and Parent (Quarterly Journal of Economics, 2009; see the following page) shows results from a regression of log wages on a dummy for whether a job has pay linked to performance (e.g. salespeople paid on commission) and other variables. The data are panel data on workers. In addition to the reported coefficients, the regressions include industry, occupation, and year dummies; county unemployment; and marital status, race dummies, and union status. Standard errors are in parentheses. The model also includes quadratic functions of experience (number of years in the workforce) and tenure (number of years at this specific job). The row labeled "Experience x performancepay" is the effect of experience at 20 years interacted with performance pay. Similarly, the row labeled "Tenure x performance pay" is the effect of tenure (evaluated at ten years) interacted with performance pay. 1. Based on column (3), is the return to education higher at performance pay jobs or nonperformance pay jobs? What is the difference and is it statistically significant?

2. Again using column (3), what is the return to having a performance pay job for somebody with a college degree (16 years of education), 20 years of experience, and 10 years of tenure?

3. Regression (4) includes worker-level fixed effects. The coefficient on years of education falls from .0637 in (3) to .0167 in (4). Is this a large change in economic terms? Explain

4. Provide an explanation for the difference in the coefficients discussed in question 3 (.0637 vs. .0167). Be concrete

5. Consider three possible ways to compute standard errors for the regressions in Table IV: homoskedasticity-only; heteroskedasticity-robust; and clustered at the individual-job level. Which is the most appropriate method, and why?