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Unformatted text preview: Economics 20 Prof. Patricia M. Anderson Study Guide for the Midterm This is not meant to represent an exhaustive list of the knowledge required to do well on the midterm. Rather, it outlines the major areas we’ve covered, helping you to pinpoint areas where you might need to do further work to fully understand the material. Basics of the OLS Estimator The goal of econometrics is to use sample data to obtain estimates of unknown population parameters. For the population model: u x x y k k + + + + = β β β ... 1 1 , we can use ordinary least squares (OLS) to obtain a sample model: i ik ik i i u x x y ˆ ˆ ... ˆ ˆ 1 1 + + + + = β β β . and thus get a sample prediction: ik ik i i x x y β β β ˆ ... ˆ ˆ ˆ 1 1 + + + = . The OLS estimator can be derived based on one crucial assumption (and one incidental assumption). The incidental assumption is that E( u ) = 0. The crucial assumption is that E( u|x ) = E( u ) = 0 (zero conditional mean). These assumptions allow us to derive that ( 29 ( 29 ( 29 ∑ ∑ ∑--- = 2 1 ˆ x x y y x x i i i β for the simple regression. This can also be interpreted as the sample covariance between x and y divided by the sample variance of x . We can think of the OLS estimator as having minimized the sum of squared residuals ( û ), where i i i u y y ˆ ˆ + = and ( 29 2 ∑- y y i = SST ( 29 2 ˆ ∑- y y i = SSE ∑ 2 ˆ i u = SSR so that SST = SSE + SSR. The R 2 is the fraction of the variation in y that is explained by the estimated model, and thus measures goodness of fit. It is defined as: R 2 = SSE/SST = 1 – SSR/SST The R 2 can also be interpreted as the square of the correlation between y and ŷ. 1 Unbiasedness & Omitted Variable Bias OLS is an unbiased estimator, provide the following 4 assumption hold: 1. The population model is linear in parameters 2. We can use a random sample from the population to estimate the model 3. E( u| x ) = E( u ) = 0 4. None of the x ’s is constant, and there are no exact linear relationships Omitting a variable that belongs in the model will often violate assumption 3 and can lead to omitted variable bias. We can sign the bias by deriving that if when the true population model is u x x y + + + = 2 2 1 1 β β β we estimate the simple regression, where 1 ~ β is the OLS estimate, then ( 29 δ β β β ~ ~ 2 1 1 + = E where δ ~ is the sample covariance between x 1 and x 2 divided by the sample variance of x 1 . We just need to use common sense to think about what signs β 2 and δ ~ are likely to have, then we can decide if 1 ~ β is too big (positively biased) or too small (negatively biased) relative to β 1 . Note that asymptotic bias (consistency) can be thought of in exactly the same manner, except that technically we are thinking about δ , the population covariance divided by the population variance, instead of δ ~ ....
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