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Unformatted text preview: ECON 140, Fall 2008  10/30 Alex Rothenberg Practice Problems: Endogeneity, Measurement Error, Omitted Variables, Diagnostics Problem 1 The following set of questions ask you about whether or not OLS will produce biased estimates of in the following model: Y i = + X i + i (1) You will make heavy use of the following formula for the expected value of OLS : E [ OLS ] = + Cov( X, ) Var( X ) We proved this formula in class, but its worth going through the proof at least once so it makes sense to you. 1. Suppose that, when we record data for our regression, we measure Y i with error. That is, we actually observe e Y i , which is the sum of truth and random noise: e Y i = Y i + i Assuming that everything else in our model satisfies the usual assumptions, when will OLS be an unbiased estimator of ? 2. Now, suppose that, when we record data for our regression, we measure each X i with error. That is, we actually observe f X i , which is the sum of truth and random noise: f X i = X i + i Assuming that everything else in our model satisfies the usual assumptions, when will OLS be an unbiased estimator of ? When will we overestimate ? When will we underestimate ? 1 ECON 140, Fall 2008  10/30 Alex Rothenberg 3. Now, suppose that weve omitted an important variable from our model. That is, while we run (1), the true model is actually the following: Y i = + X i + A i + i Assuming that everything else in our model satisfies the usual assumptions, when will OLS be an unbiased estimator of ? When will we overestimate ? When will we underestimate ? 4. Now, suppose that we want to estimate the returns to schooling, in a model similar to (1): log( W i ) = + S i + i However, were worried that weve BOTH omitted an important variable from our model (ability), AND weve measured schooling, S i , with error. That is, the true model is actually the following, log( W i ) = + S i + A i + i And we really observe: e S i = S i + i where i is random measurement error. Is there any way that we could still get an unbiased estimate of using OLS? 2 ECON 140, Fall 2008  10/30 Alex Rothenberg Problem 2 This problem is about regression diagnostic techniques. Im interested in estimating the returns to schooling in Honduras, and I estimated the following model for a sample of workers: log( W i ) = + S i...
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 Spring '09

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