Who Invented Regression? 1. Who invented regression? 2. Omitted Variables and Multivariate Regression3. Omitted Variable Bias (OVB)4. Experiments vs. OVB5. R2Copyright © 2003 by Pearson Education, Inc. 4-21. Who invented regression? • Francis Galton, - climatologist, - gentleman explorer- social scientistCopyright © 2003 by Pearson Education, Inc. 4-3Heredity and Height“regression” to the meanCopyright © 2003 by Pearson Education, Inc. 4-42. Omitted Variables and Omitted Variable Bias• What if you left out an important variable? • Many interesting relationships have more than 2 dimensionsGRE prep course exampleCoffee exampleProblem set and exam example• We need more variables.. “multivariate” regressionCopyright © 2003 by Pearson Education, Inc. 4-52. OLS Multivariate regressionLook familiar? Same criterion with more variables.Copyright © 2003 by Pearson Education, Inc. 4-62. Properties of OLS estimators in Multivariate Regression•Consistent• Unbiased• Approximately N(.) in large samples• Same first order conditions(for 2 or more X’s)00012111===∑∑∑===NiiiNiiiNiieXeXeCopyright © 2003 by Pearson Education, Inc. 4-7First order conditions for multivariate regressionCopyright © 2003 by Pearson Education, Inc. 4-83. Omitted Variable “Bias”• Short regressiony = b0s+ b1sx1+ eS(SR) • Long regressiony = b0L+ b1Lx1+ b2Lx2 + eL(LR) • Claim:b1s= b1L+ b2Lb21 , b21is slope of a regression of x2on x1Copyright © 2003 by Pearson Education, Inc. 4-9Omitted variable bias formula - derivationCopyright © 2003 by Pearson Education, Inc. 4-104. Why experiments eliminate OVBb1s= b1L+ b2Lb21 , • So there’s no OVB if b21 =0i.e., b21 =0 implies b1s= b1L.. Which you can guarantee if you design an experiment in which X1 is uncorrelated with other X’s (omitted variables).Random assignment of X1is sure to do that... Back to examples to demonstrateCopyright © 2003 by Pearson Education, Inc. 4-115. R2– How much Variation Explained?• How much of the variation in Y did we explain with the regression line?∑∑∑∑=−==−=−−−=−−=NiNiNiNiyyeyyyyR12121212^2)(1)()(Copyright © 2003 by Pearson Education, Inc. 4-12Coffee Demand – High R2cupspricequantity demanded qhat0.5 1 1.5 205101520Copyright © 2003 by Pearson Education, Inc. 4-13E.g. Coffee Demand – high R2• p is the price of coffee, • q is the quantity (in cups). reg q p, robustRegression with robust standard errors Number of obs = 23F( 1, 21) = 28.20Prob > F = 0.0000R-squared = 0.7349Root MSE = 4.0549------------------------------------------------------------------------------| Robustq | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------p | -6.246766 1.176301 -5.31 0.000 -8.693018 -3.800513_cons | 17.5064 1.822797 9.60 0.000 13.71568 21.29711------------------------------------------------------------------------------Copyright © 2003 by Pearson Education, Inc. 4-14Eg. Wage Regression - Low R2* Lhwage is log(hourly wage), ed is years of educationregress lhwage ed, robustRegression with robust standard errors Number of obs = 13743F( 1, 13741) = 1795.40Prob > F = 0.0000R-squared = 0.1185Root MSE = .5083------------------------------------------------------------------------------| Robustlhwage | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------ed | .0704563 .0016628 42.37 0.000 .0671969 .0737156_cons | .9852746 .0238393 41.33 0.000 .9385464