CORNELL ECON 3120 - Unbiasedness of Multivariate OLS Estimators (2 pages)

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Unbiasedness of Multivariate OLS Estimators



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Econ 3120 1st Edition Lecture 13 Outline of Current Lecture I Goodness of Fit II Unbiasedness Current Lecture III Unbiasedness of Multivariate OLS Estimators Unbiasedness of Multivariate OLS Estimators In order to show that the OLS estimators described above are unbiased we need to make a number of assumptions along the same lines as SLR 1SLR 4 MLR 1 The model is given by y 0 1x1 2x2 kxk u MLR 2 Random sampling our sample x1 x2 xk y i 1 n is a random sample following the population model in MLR 1 MLR 3 No perfect collinearity In the sample and in the population 1 Each independent variable has sample variation 6 0 2 None of the independent variables can be constructed as a linear combination of the other independent variables MLR 4 Zero conditional mean E u x1 x2 xk 0 Under these assumptions multivariate OLS estimates are unbiased that is E j x1 x2 xk j The proof is somewhat involved algebraically so we will not cover it here See Appendix 3A in Wooldridge for the details 6 Omitted Variable Bias 6 1 Comparison of Bivariate and Multivariate Estimates In order to understand the effects of omitting relevant variables let s return to regression anatomy We ll use this to describe how the estimated s change when the regression is run without one of the x s i e we have an omitted variable Take the example where the true model is of the form y 0 1x1 2x2 u 3 6 but we instead run the regression omitting x2 y 0 1x1 u 4 Let 1 and 2 represent the OLS estimators of 1 and 2 using 3 and 1 be the OLS estimator of 1 using 4 How can we use regression anatomy to figure out what s going on It turns out that 1 1 2 1 where 1 is the estimate from equation 2 above Proof 6 2 Omitted Variable Bias Formula We re now in a position to show what happens to the expectation of our OLS estimators when we exclude a relevant variable in the analysis Suppose the true model is y 0 1x1 2x2 u 5 but we instead estimate y 0 1x1 u 6 We know that estimates from 5 are related to the estimates from 6 by the formula 1 1 2 1 7 Taking expectations with implicit conditioning on the x 0 s E 1 E 1 2 1 E 1 E 2 1 1 E 2 1 1 2 1 This implies that Bias 1 E 1 1 2 1 2 cov dx1 x2 vard x1 7 If we have omitted a relevant variable the bias of our estimator is going to depend on two relationships 1 the the effect of the excluded independent variable x2 on the dependent variable and 2 the correlation between the included independent variable x1 and the excluded variable x2 Example Suppose you know that log wages depend linearly on both These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute education and ability logwagei 0 1educi 2abilityi ui 8 But you only include education as a regressor logwagei 0 1educi ui 9 Formula 7 suggests that your estimation will be biased upwards if 1 ability has a positive effect on wages positive 2 and 2 education and ability are positively correlated positive 1 We can illustrate this using the 1980 National Longitudinal Study which has information on earnings education and IQ score a proxy for ability Estimating Equation 9 yields logdwagei 5 97 0 06educi While estimating Equation 8 yields logdwagei 5 66 0 039educi 0 0059IQi 8 where IQi is the person s IQ score We can see the positive correlation between IQ and education through the regression IQci 53 7 3 53educi We can use these three equations to confirm the omitted variable bias formula 0 06 0 039 0 0059 3 53


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