Question 3 (36 marks) We have the model with one fixed regressor yi…
Question Answered step-by-step Question 3 (36 marks) We have the model with one fixed regressor yi… Question 3 (36 marks) We have the model with one fixed regressor yi = ?xi + ui (1) for individuals i = 1, . . . , n with uncorrelated homoskedastic error terms ui ? N(0, ?2 ). Suppose that the value for ? 2 is known. 3(a) Show that the OLS estimator ? = (X0X) ?1X0y = P Pi xiyi i x 2 i for ? is unbiased. (2 marks) 3(b) Derive an expression for the standard error of the OLS estimator for ? in terms of xi and ?. (5 marks) Suppose that the individuals are divided into groups j = 1, . . . , J each with nj observations respectively, and we only observe the reported group means ¯yj and ¯xj . The model becomes y¯j = ?x¯j + ¯uj , (2) with error terms ¯uj = 1 nj Pnj i=1 uij , where uij indicates error term ui of individual i belonging to group j. 3(c) Show that the error terms ¯uj are heteroskedastic. (4 marks) 3(d) Show that the OLS estimator for ? is unbiased. (3 marks) 3(e) Derive an expression for the standard error of the OLS estimator for ? in terms of xij and ?, where xij indicates xi of individual i belonging to group j. (6 marks) 3(f) What are the consequences of heteroskedasticity in the errors for the OLS estimator of the parameters, their usual OLS standard errors reported by statistical packages, and the standard t-test and F-test for these parameters? (4 marks) 3(g) Test the null-hypothesis that H0 : E[¯u 2 j |x¯j ] = ? 2 for j = 1, . . . , J, against the alternative that the variance is a smooth unknown function of ¯xj . Explicitly state which regression(s) you use, the null and the alternative, and the test statistic with its distribution under the null. (5 marks) 3(h) Suppose that we observe the group size nj for j = 1, . . . , J. Regress ¯yj ? nj on ¯xj ? nj . Show that the error terms of this regression are homoskedastic. (4 marks) 3(i) Apart from the problem of heteroskedasticity in the errors, what would be another reason to prefer the regression with individual data over the regression with grouped data? (3 marks) Business Economics Econometrics ECON 2206 Share QuestionEmailCopy link Comments (0)
