Assignment 2
1. Continuation of question 1, Assignment 1. In answering question 1 of Assignment 1, you showed that the maximum likelihood estimator of the parameter ? of the Poisson distribution is the sample mean, y ¯. Using of Theorem 14.1 of Greene (2012), ?nd the asymptotic distribution of this estimator. (Your answer should include a complete speci?cation of the asymptotic variance of the estimator.) 2. For the Chow test, the unrestricted model is y1 y2 = X1 0 0 X2 ß1 ß2 + e1 e2 . (1)
(a) Show that the OLS estimator of this model is b= b1 b2 = (X1 X1 )-1 X1 y1 (X2 X2 )-1 X2 y2 . (2)
(b) Show that the sum of squared errors of the unrestricted model is e e = e1 e1 + e2 e2 , (3)
where e1 is the vector of residuals from the regression of y1 on X1 and e2 is the vector of residuals from the regression of y2 on X2 . 3. Continuation of question 4, Assignment 1. The ?le slid2010HKontD.dta is a Stata data ?le that that contains data retrieved from the 2010 Survey of Labour and Income Dynamics (SLID). The ?le contains 6,808 observations for individuals living in Ontario. (a) As you did for Assignment 1, construct the variable lhwage, the natural log of hwage. In addition, create the variable exp2, the square of exp. (b) Estimate two human capital earnings models, one for men and one for women. Use lhwage as the dependent variable and include exp, exp2, educ, notgrad, hsgrad somepse, uni1, uni2 and marr as independent variables. Discuss the goodness of ?t of the two equations. (c) Interpret the results of the test of overall signi?cance for each equation. Do not forget to state the decision rule for the test, the level of signi?cance you are using, and the critical value of the test statistic. (d) Interpret the results of individual tests of signi?cance for the coe?cients of exp, exp2, and educ for both men and women. Do not forget to state the decision rule for the test, the level of signi?cance you are using, and the critical value of the test statistic. 1
(e) The equations you estimated in part (b) include two types of human capital measure : (i) years of schooling (educ) and (ii) a number of dummy variables indicating the highest level of education attained by the individual. Does the latter group of variables contribute the explanatory power of the model, or are years of schooling a su?cient measure of human capital ? Carry out a hypothesis test using the test command that will help you answer this question. In your answer, include the null and alternative hypotheses, a formula for the test statistic, the matrix R and the vector q, the distribution of the test statistic, the level of signi?cance that you have chosen, the decision rule for the test and your conclusion. (f) An alternative form for the Chow test F statistic is (e* e* – e1 e1 – e2 e2 ) K (e1 e1 + e2 e2 ) (n – 2K ) where e* e* is the sum of squared residuals of the restricted model, e1 e1 is the sum of squared residuals of for subsample 1, and e2 e2 2. In order to use this formula, one needs to estimate the regression model for three samples : subsample 1 (n1 observations), subsample 2 (n2 observations) and the full sample (n = n1 + n2 observations). Using this formula, carry out a Chow test in order to verify whether the coe?cients of the human capital regression model are the same for men and women. Show all your calculations. In addition, include in your answer the decision rule for the test, the level of signi?cance, and the critical value for the test. (g) Ramsey’s RESET test is a diagnostic test for speci?cation errors, in particular the omission of nonlinear functions of the explanatory variables of a regression 3 2 ,y ˆi model. The test is an F test applied to an augmented model that includes y ˆi 4 et y ˆi as additional explanatory variables. If the original model is y = Xß + e, the augmented model will be
2 3 4 yi = xi ß + ?1 y ˆi + ?2 y ˆi + ?3 y ˆi + ui ,
F =
.
(4)
(5)
where is a random error and xi is the ith row of the matrix X. The null hypothesis for the test is that ?1 = ?2 = ?3 = 0 (i.e., there is no speci?cation error). The command estat ovtest (placed after a regress that estimates the original model) computes the F statistic for testing this hypothesis. Carry out the test for both men and women. What do the results imply with respect to the properties of the OLS estimator of the original model ?
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