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

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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
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Omitted Variable Bias The methodology we°ve covered so far has (at least) one big limitation: there°s only one RHS variable ±explaining² 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
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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 signi³cant only because other variables are correlated with both STR and TESTSCR ? Both problems are examples of omitted variable bias. De³nition: 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
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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 = β 0 + β 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
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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 January 4, 2004 February 1, 2004 February 29, 2004 March 28, 2004 April 25, 2004 May 23, 2004 June 20, 2004 July 18, 2004 August 15, 2004 September 12, 2004 October 10, 2004 November 7, 2004 December 5, 2004 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
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Omitted Variable Bias Let°s see what happens when we omit a relevant variable from our analysis. Suppose the true model is: Y i = β 0 + β 1 X 1 i + β 2 X 2 i + u i (1) with E [ u i j X 1 i , X 2 i ] = 0 So β 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.
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