Please see attachment for the fully formatted solution.
1) An economics department at a large state university keeps tract of its majors' starting salaries. Does taking econometrics effect starting salary? Let Sal = starting salary in dollars, GPA = grade point average on a 4.0 scale, Metrics = 1 if the student took econometrics, and 0 otherwise. Using a sample size of 50, we obtain
Sal(hat) = 24200 + 1643GPA + 5033METRICS, R-squared=0.74
(se) (1078) (352) (456)
The variance-covariance matrix of the estimated coefficients is
Intercept GPA Metrics
Intercept 116299737 -370463 -124114
GPA -370463 124108 22428
Metrics -124114 22428 208216
a) Show how you can obtain the standard error of the coefficient on GPA, Metrics from the variance-covariance matrix.
b) At the 95% level, testing the testing the overall significance of joint hypotheses.
(Hint: H0: B1 = 0, B2 = 0)
c) At the 95% level, test the hypothesis that the marginal effect of GPA on starting salary is equal to the marginal effect of Metrics on starting salary. (Hint: H0: B1 = B2)© BrainMass Inc. brainmass.com March 21, 2019, 10:15 pm ad1c9bdddf
a) Recall that the i,j-th element on the variance covariance matrix represents cov(xi, xj). Also, cov(xi, xi) = var(xi).
Thus the estimated variances are var(GPA) = 124108 and var(METRIC) = 208216. Standard error (which is the estimated standard deviation) are se(GPA) = 352.29 and var(METRIC) = 456.31. These are the same as the ones calculated by the computer.
b) To test β1 = β2 = 0, we need to use the Bonferroni adjusted t test. To begin, we compute 2 t test statistics
let t1 = (β1_hat - 0)/se(β1_hat) and t2 = (β2_hat - 0)/se(β2_hat)
thus, simplifying gives t1 = 1643/352 = ...
This solution explains how to solve various problems related to economics.