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140_problems6 - ECON 140 Fall 2008 10/30 Alex Rothenberg...

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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 β in the following model: Y i = α + βX i + ε i (1) You will make heavy use of the following formula for the expected value of ˆ β OLS : E [ ˆ β OLS ] = β + Cov( X, ε ) Var( X ) We proved this formula in class, but it’s 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 + ν i Assuming that everything else in our model satisfies the usual assumptions, when will ˆ β OLS be an unbiased estimator of β ? 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 + η i Assuming that everything else in our model satisfies the usual assumptions, when will ˆ β OLS be an unbiased estimator of β ? When will we overestimate β ? When will we underestimate β ? 1
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ECON 140, Fall 2008 - 10/30 Alex Rothenberg 3. Now, suppose that we’ve omitted an important variable from our model. That is, while we run (1), the true model is actually the following: Y i = α + βX i + γA i + υ i Assuming that everything else in our model satisfies the usual assumptions, when will ˆ β OLS be an unbiased estimator of β ? When will we overestimate β ? When will we underestimate β ? 4. Now, suppose that we want to estimate the returns to schooling, in a model similar to (1): log( W i ) = α + βS i + ε i However, we’re worried that we’ve BOTH omitted an important variable from our model (ability), AND we’ve measured schooling, S i , with error. That is, the true model is actually the following, log( W i ) = α +
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