Regression Forced March17.871Spring 20022/28/02Regression quantifies how one variable can bedescribed in terms of anotherBlack Elected Officials Example Ibeobpop1.2 30.8010.8The Linear Relationship between Two VariablesiiiXY εββ ++=10The Linear Relationship between African American Population & Black Legislatorsbeobpop beo Fitted values0 10 20 300510359.031.110=−=ββHow did we get that line?1. Pick a representative value of Yibeobpop beo Fitted values0 10 20 300510YiHow did we get that line?2. Decompose Yiinto two partsbeobpop beo Fitted values0 10 20 300510How did we get that line?3. Label the pointsbeobpop beo Fitted values0 10 20 300510YiYi^eiYi-Yi^The Method of Least Squares∑∑==−−−niiiniiiXYYY12102110)(or )ˆ(minimize to and Pick ββββSolve for 0)(11210=∂−−∂∑=βββniiiXY)var(),cov(or )())((1211XYXXXXXYYniiniii∑∑==−−−=βSolve for0)(01210=∂−−∂∑=βββniiiXY ....rearrange.you if that Note 1010XYXYββββ+=−=How to Think About Regression Results• Deterministically• ExpectationsHow “good” is the fitted line?beobpop beo Fitted values1.2 30.8-215smallybpop smally Fitted values1.2 30.8-215bigybpop bigy Fitted values1.2 30.8-215Judging results• Substantive interpretation of coefficients• Technical judgment of regression– Judgment of coefficients– Judgment of overall fitDetermining Goodness of Fit I• Coefficients– Standard error of a coefficient– t-statistic: coeff./s.e.Standard error of the regression picturebeobpop beo Fitted values0 10 20 300510YiYi^eiYi-Yi^Add these up after squaringDetermining Goodness of Fit• Standard error of the regression (Root mean square error in STATA and Freedman)..)ˆ(...12fdYYeesniii∑=−=R2picture Ibeobpop1.2 30.8-.88472210.8Y_(Yi-Y)∑=−niinYY12)(R2picture IIbeobpop beo Fitted values1.2 30.8-.88472210.8Y_(Yi-Y)(Yi-Yi)^(Yi-Y)^beobpop beo Fitted values1.2 30.8-.88472210.8Y_(Yi-Y)(Yi-Yi)^(Yi-Y)^( ) " "($) " "($) " "Y Y total sumof squaresY Y regression sumof squaresY Y residual sumof squaresiiniini iin− = − = − = ===∑∑∑212121__Determining Goodness of Fit • R-squaredexplained"" riancepercent vaor ˆ12122∑∑==−−=niinii)Y(Y)YY(rReturn to Black Elected Officials Example. reg beo bpopSource | SS df MS Number of obs = 41-------------+------------------------------ F( 1, 39) = 202.56Model | 351.26542 1 351.26542 Prob > F = 0.0000Residual | 67.6326195 39 1.73416973 R-squared = 0.8385-------------+------------------------------ Adj R-squared = 0.8344Total | 418.898039 40 10.472451 Root MSE = 1.3169------------------------------------------------------------------------------beo | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------bpop | .3584751 .0251876 14.23 0.000 .3075284 .4094219_cons | -1.314892 .3277508 -4.01 0.000 -1.977831 -.6519535------------------------------------------------------------------------------Looking at
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