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Unformatted text preview: Problem Set 3. Answer Key. Introduction to Econometrics - Fall 2006 Due Date: Thursday, October 26th, BEFORE CLASS. Each student is responsible for handing one handwritten solution. If working in groups, indicate the members of the group (to facilitate grading). PLEASE SHOW YOUR WORK . 1. Consider the following OLS regression line: b Y i = 4 : 52 (1 : 04) + 1 : 53 (0 : 86) X i R 2 = 0 : 25 where the numbers in parentheses are the heteroskedasticity-robust standard errors. Assume that the three least squares assumptions hold. (a) Test the null hypothesis H : & 1 = 0 at the 5% signi&cance level. Do you reject the null? Compute the 95% two-sided con&dence interval for & 1 : Solution : The t-statistic is: t = 1 : 53 & : 86 = 1 : 779 < 1 : 96 , so the null hypothesis is not rejected. The 95% con&dence interval for & 1 is [ & : 1556 ; 3 : 2156] (b) Now you decide to run the regression again and compute the homoskedasticity-only standard errors. You obtain the following estimates: b Y i = 4 : 52 (0 : 96) + 1 : 53 (0 : 75) X i R 2 = 0 : 25 Why didnt the estimated coe cients change? Are the OLS estimators unbiased and consistent in this case? Solution: Homoskedasticity and heteroskedasticity of the error term do not a/ect the properties of the OLS estimators and the value of the coe cients estimated from a given sample. (c) Test the null hypothesis H : & 1 = 0 at the 5% signi&cance level using the homoskedasticity- only standard errors, and compute the 95% two-sided con&dence interval for & 1 : Are your results di/erent from those in ( a ) ? Solution : In this case the t-statistic is: t = 1 : 53 & : 75 = 2 : 04 > 1 : 96 , so the null hypothesis is rejected. The 95% con&dence interval for & 1 is [0 : 06 ; 3] (d) Assume that Y i is yearly earnings in thousand of dollars and X i is years of schooling. Which results are more reliable, those in ( a ) or those in ( c ) ? Explain. Solution : The results in ( a ) are more reliable, because there are no reasons to believe the error term in this model is homoskedastic. In fact, as explained in your book (p.165), there are more reasons to believe the error term is heteroskedastic in this case. 2. When the least squares assumptions hold, the heteroskedasticity-robust variance of b & is: 2 b & = V ar ( H i u i ) n [ E ( H 2 i )] 2 where: H i = 1 & X E ( X 2 i ) X i In this problem, you will use this formula to derive the variance of the OLS estimator b & in the case the error term is homoskedastic, that is when V ar ( u i j X i ) = 2 u : (a) Write V ar ( H i u i ) = E h ( H i u i ) 2 i & [ E ( H i u i )] 2 . Using the &rst least squares assumption (the conditional mean assumption) and the law of iterated expectations, show that E ( H i u i ) = 0 : Solution: E ( H i u i ) = E & u i & X E ( X 2 i ) X i u i = E ( u i ) & X E ( X 2 i ) E ( X i u i ) But E ( u i ) = 0 and E ( X i u i ) = E [ X i E ( u i j X i )] = 0 using the law of iterated expectations and...
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- Fall '08