Believe Least-Squares Assumptions

Posted: March 29th, 2022

 
 
The data set in lawsch85.dta contains information for 1985 cohort of the top 156 law schools in the US. Variables in the dataset include rank, law school ranking, salary, median starting salary, cost, law school cost.
i. Compute the average starting salary across law schools in the sample. Do you think it coincides with the average starting salary across law students?1
ii. Regress starting salaries on the law school’s ranking: salaryi =β0 β1 ×ranki ui compute standard errors and a 95% confidence interval for β1. Report your results.
iii. What is the expected difference in starting salary between the 20th top law school with the 40th top law school? Construct a 95% confidence interval for the difference. Report your results.2
iv. Now regress the cost of attending law school on the school’s ranking: costi =β0 β1 ×ranki ui compute standard errors and a 95% confidence interval for β1. Report your results.
v. What is the expected difference in cost between the 20th top law school with the 40th top law school? Construct a 95% confidence interval for the difference. Report your results. 1Hint: think about the number of student in different schools and the law of iterated expectations 2Hint: the standard error for 2βˆ1 is 2se(βˆ1). More generally, for any number ∆, the standard error for ∆βˆ1 is |∆|se(βˆ1); standard errors cannot be negative. 1
vi. Given the results in ii-iii. and iv-v. discuss the relative benefits and costs of attending a more prestigious program.
vii. Construct a plot with rank on the x-axis and cost on the y-axis. Do you believe Least-Squares Assumptions (LSA) 1-3 are reasonable assumptions in this setting? Plot rank against salary in the same manner and comment on LSA 1-3.
viii. Construct a plot with rank on the x-axis and log(salary) on the y-axis.3 Comment on LSA 1-3.
ix. Repeat ii. but this time regressing log(salary) on rank: log(salaryi) = β0 β1 × ranki ui, compute standard errors and a 95% confidence interval for β1. Remark: This is still a linear model as we saw in class, everything we have seen so far applies to this regression. The only difference is in the interpretation of β1, when x is a continuous regressor: β1 = dlog(yi) = dyi dxi yi because d log(x) = dx/x. This means that 100 × β1 is (roughly) the percentage increase in y when x changes by one unit. Economists often look at log(salary) instead of salary to make statements in terms of percentage increases/decreases. Here x is discrete, so 100 × β1 is just the percent change in log(salary) when we change rank by one unit.