handout8_l (1)

handout8_l (1) - Econ 139: Introduction to Econometrics...

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Unformatted text preview: Econ 139: Introduction to Econometrics Andrew Sweeting 1 Department of Economics Duke University Spring 2011 Econ 139 Handout 8 (Duke) Multivariate Regression Spring 2011 1 / 74 Omitted Variable Bias The methodology we&ve covered so far has (at least) one big limitation: there&s only one RHS variable explaining Y Consider the Test Scores regression from Chapter 4 Notice the low R 2 What if STR is picking up something besides just the student-teacher ratio? Econ 139 Handout 8 (Duke) Multivariate Regression Spring 2011 2 / 74 Omitted Variable Bias In other words, what if something else is driving test scores? For example percent of English learners, teacher quality, richer school, richer neighborhood, parent&s education... Why do we care? We&d like to establish a causal e/ect. We don&t want STR getting credit (or blame) for the e/ect of something else Worse, what if STR is signicant only because other variables are correlated with both STR and TESTSCR ? Both problems are examples of omitted variable bias. Denition: If a regressor is correlated with a variable that has been omitted from the analysis but that determines (in part) the dependent variable, then the OLS estimator will have omitted variable bias. Econ 139 Handout 8 (Duke) Multivariate Regression Spring 2011 3 74 Omitted Variable Bias Omitted variable bias (OVB) occurs when two conditions hold: 1 The omitted variable is correlated with the included regressor (OVB 1) 2 The omitted variable is a determinant of the dependent variable (OVB 2) Examples: Education and wages Wage = + 1 Educ + u Omitting ability will cause you to overestimate the importance of schooling. Can you see why? Soft drink sales and holidays Formally, omitted variable bias occurs when we don&t include in our regression all the variables that are correlated with Y and one (or more) of the regressors ( X &s). Econ 139 Handout 8 (Duke) Multivariate Regression Spring 2011 4 / 74 Carbonated Beverages Aggregate Unit Sales and Price Reductions 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 J a n u a r y 4 , 2 4 F e b r u a r y 1 , 2 4 F e b r u a r y 2 9 , 2 4 M a r c h 2 8 , 2 4 A p r il 2 5 , 2 4 M a y 2 3 , 2 4 J u n e 2 , 2 4 J u ly 1 8 , 2 4 A u g u s t 1 5 , 2 4 S e p te m b e r 1 2 , 2 4 O c to b e r 1 , 2 4 N o v e m b e r 7 , 2 4 D e c e m b e r 5 , 2 4 Week Proportion of Price Reductions 7 7.5 8 8.5 9 9.5 10 10.5 11 Millions Unit Sales Proportion of SKUs on Sale Units Econ 139 Handout 8 (Duke) Multivariate Regression Spring 2011 5 / 74 Omitted Variable Bias Let&s see what happens when we omit a relevant variable from our analysis. Suppose the true model is: Y i = + 1 X 1 i + 2 X 2 i + u i (1) with E [ u i j X 1 i , X 2 i ] = So 1 is the true slope of X 1 Notice that, using the LIE, we still have E [ u i j X 1 i ] = E [ E [ u i j X 1 i , X 2 i ] j X 1 i ] = 0 (*) It&s also useful to note that for any variables P and Q : & P i & P & Q i & Q = & P i & P Q i (**) Why? Just expand the sum and cancel.Why?...
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handout8_l (1) - Econ 139: Introduction to Econometrics...

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