# ps3 - Problem Set 3 Introduction to Econometrics Fall 2006...

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Problem Set 3 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 0 : β 1 =0 at the 5% signi f cance level. Do you reject the null? Compute the 95% two-sided con f dence interval for β 1 . (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 didn’t the estimated coe cients change? Are the OLS estimators unbiased and consistent in this case? (c) Test the null hypothesis H 0 : β 1 =0 at the 5% signi f cance level using the homoskedasticity- only standard errors, and compute the 95% two-sided con f dence interval for β 1 . Are your results di f erent from those in ( a ) ? (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. 2. When the least squares assumptions hold, the heteroskedasticity-robust variance of b β 0 is: σ 2 b β 0 = Var ( H i u i ) n [ E ( H 2 i )] 2 where: H i =1 µ X E ( X 2 i ) X i In this problem, you will use this formula to derive the variance of the OLS estimator b β 0 in the case the error term is homoskedastic, that is when Var ( u i | X i )= σ 2 u . (a) Write

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ps3 - Problem Set 3 Introduction to Econometrics Fall 2006...

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