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final - ECON 139 Duke University Spring 2008 Final Exam...

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ECON 139 Duke University Spring 2008 Final Exam Instructions (PLEASE READ CAREFULLY before starting): ° This test has a total of 4 problems and 100 points . ° You have 150 minute s to °nish it. ° Remember to Show Your Work and please be sure that your handwriting is legible ! ° Be precise but brief. If a correct reply is hidden among wrong, or irrelevant, arguments, you will not get full credit. ° We will grade only what is written on your exam sheet. There should be plenty of space for all your calculations. ° There are two extra pages included at the end of the exam to be used as scratch paper. Do not turn in anything besides your exam sheet . ° If time is running short, you should try to set up the problem without doing the °nal calculations. ° Standard errors are in parentheses (unless otherwise noted). Name: 1
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1. (20 points total) Consider the basic panel data set-up in which the population regression model is given by Y it = ° 0 + ° 1 X it + ± i + u it (1) for i = 1 ; :::; n; t = 1 ; :::; T and where we can assume E ( ± i ) = 0 (because we have included an intercept) and E ( u it j X it ) = 0 . a. (3 points) Given (1), show that the following relation also holds Y i = ° 0 + ° 1 X i + ± i + u i (2) where the overbar represents the average over time for a given individual. b. (6 points) Suppose that u i is uncorrelated with X i ; but cov ( X it ; ± i ) = ² for all t and i: Show that cov ° X i ; ± i + u i ± = ² 2
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c. (5 points) Let e ° 1 be the OLS estimate of ° 1 obtained from estimating equation (2). This is sometimes called the ±between²estimator. Show that plim e ° 1 = ° 1 + ² V ar ° X i ± where the probability limit is de°ned as N ! 1 : d. (3 points) Assume further that the X it ; for all t = 1 ; :::; T are uncorrelated with constant variance ² 2 X : Show that plim e ° 1 = ° 1 + T ² ² 2 X 3
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e. (3 points) If the explanatory variables are indeed uncorrelated across time, what does the result in part d) suggest about whether the inconsistency in the ±between²estimator is smaller when there are more time periods? 4
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2. (34 points total) Continuing a theme from the last exam, we are now going to use the data from the airline airfare question on Midterm 2 to estimate airline demand. Recall from before that we have data from 1,149 U.S. markets (i.e. routes) where a market is a particular city pair (e.g. ³ights connecting Raleigh-Durham (RDU) and Washington National (DCA) airports). We have information on
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