Introduction to Econometrics 2022/2023 5 of 5 Assignment Two QUESTION FIVE For this question we assume that the residuals are homoscedastic. Three researchers investigating the determinants of hourly earnings have the following data for a sample of 104 male workers in the United States in 2006: E, hourly earnings in dollars; S, years of schooling; NUM, score on a test of numeracy; and VERB, score on a test of literacy. The MUM and VERB tests are marked out of 100. The correlation between them is 0.81. Defining LGE to be the natural logarithm of E, Researcher 1 fits the following regression (standard errors in parentheses; RSS = residual sum of squares): LGE = 2.02 + 0.063S + 0.0044NUM + 0.0026VERB RSS =2, 000 (1.81) (0.007) (0.0011) (0.0010) Researcher 2 defines a new variable SCORE as the average of MUM and VERB. She fits the regression LGE = 1.72 + 0.050S + 0.0068 SCORE RSS = 2, 045 (1.78) (0.005) (0.0010) Researcher 3 fits the regression LGE = 2.02 + 0.0635 + 0.0088 SCORE - 0.0018VERB RSS = 2, 000 (1.81) (0.007) (0.0022) (0.0012) a. Show that the specification of Researcher 2 is a restricted version of the specification of Researcher 1, stating the restriction. b. Perform an F test of the restriction. What are the degrees of freedom of the F-test? Use either the critical value or the p-value hypothesis testing approach at 1% significance level. c. Show that the specification of Researcher 3 is a reparameterized version of the specification of Researcher 1 and hence perform a t test of the restriction in the specification of Researcher 2, knowing that the critical values for the f-test at the 5% significance level is +1.98 and at the 10% significance level is 1.66 d. Explain whether the F test in b. and the f test in c. could have led to different results