Regression Forced MarchSlide 2Black Elected Officials Example IStop a second: What is the correlation between beo & bpop? .72, .82, .92?The Linear Relationship between Two VariablesThe Linear Relationship between African American Population & Black LegislatorsHow did we get that line? 1. Pick a representative value of YiHow did we get that line? 2. Decompose Yi into two partsHow did we get that line? 3. Label the pointsStop a moment: What is gi?The Method of Least SquaresSolve forSolve forSlide 14About the Functional FormBlack Elected OfficialsLog transformationsHow “good” is the fitted line?Judging resultsDetermining Goodness of Fit IStandard error of the regression pictureDetermining Goodness of FitR2 pictureSlide 24Determining Goodness of FitReturn to Black Elected Officials ExampleResidualsSlide 28One important numerical property of residualsRegression Commands in STATAWhy It’s Called RegressionSome RegressionsTemperature and LatitudeSlide 34Slide 35Slide 36Bush Vote and Southern BaptistsSlide 38Slide 39Weight by State PopulationSlide 41Midterm loss & pres’l popularitySlide 43Slide 44Slide 45Slide 46Regression Forced March17.871Spring 2006Regression quantifies how one variable can bedescribed in terms of anotherBlack Elected Officials Example Ibeobpop1.2 30.8010.8Stop a second:What is the correlation between beo & bpop? .72, .82, .92?beobpop1.2 30.8010.8The Linear Relationship between Two VariablesiiiXY10The Linear Relationship between African American Population & Black Legislatorsbeobpop beo Fitted values0 10 20 300510359.031.110How 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 Yi into two partsbeobpop beo Fitted values0 10 20 300510How did we get that line?3. Label the pointsbeobpop beo Fitted values0 10 20 300510YiYi^εiYi-Yi^iiiXY )(10“residual”Stop a moment: What is i?•Vagueness of theory•Poor proxies (i.e., measurement error)•Wrong functional form•See Utts & Heckard discussion about the difference between deterministic relationships and statistical relationshipsThe Method of Least SquaresniiiniiiXYYY12102110)(or )ˆ(minimize to and Pick beobpop beo Fitted values0 10 20 300510YiYi^εiYi-Yi^Solve for 0)(11210niiiXY)var(),cov(or )())((1211XYXXXXXYYniiniii(Utts & Heckard,p. 164)Solve for0)(01210niiiXY ....rearrange.you if that Note 1010XYXY(Utts & Heckard,p. 164)beobpop beo Fitted values0 10 20 300510Y X 0 1About the Functional Form•Linear in the variables vs. linear in the parameters –Y = a + bX + e (linear in both)–Y = a + bX + cX2 + e (linear in parms.)–Y = a + Xb + e (linear in variables)–Y = a + lnXb/Zc + e (linear in neither)•Utts & Heckard pp. 174-175Black Elected Officials0 5 10 15leg/Fitted values0 10 20 30popleg Fitted valuesFitted valuesLog transformationsY = a + bX + e b = dY/dX, orb = the unit change in Y given a unit change in XTypical caseY = a + b lnX + e b = dY/(dX/X), orb = the unit change in Y given a % change in XCases where there’s a natural limit on growthln Y = a + bX + e b = (dY/Y)/dX, orb = the % change in Y given a unit change in XExponential growthln Y = a + b ln X + e b = (dY/Y)/(dX/X), orb = the % change in Y given a % change in X (elasticity)Economic productionHow “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^εiYi-Yi^Add these up after squaringDetermining Goodness of Fit•Standard error of the regression or standard error of estimate (Root mean square error in STATA)..)ˆ(...12fdYYeesniiid.f. = n-2(Yi-Yi)^10.8-.884722R2 picture Y_(Yi-Y)(Yi-Y)^010beobpop beo Fitted values1.2 30.8beobpop beo Fitted values1.2 30.8-.88472210.8Y_(Yi-Y)(Yi-Yi)^(Yi-Y)^( ) " "() " "() " "Y Y t o t a l s u m o f s q u a r e sY Y r e g r e s s i o n s u m o f s q u a r e sY Y r e s i d u a l s u m o f s q u a r e siiniini iin 212121__010Determining Goodness of Fit •R-squaredexplained"" riancepercent vaor ˆ12122niinii)Y(Y)YY(r“coefficient of determination”Return to Black Elected Officials Example. reg beo bpop Source | SS df MS Number of obs = 41-------------+------------------------------ F( 1, 39) = 202.56 Model | 351.26542 1 351.26542 Prob > F = 0.0000 Residual | 67.6326195 39 1.73416973 R-squared = 0.8385-------------+------------------------------ Adj R-squared = 0.8344 Total | 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------------------------------------------------------------------------------Residualsei = Yi – B0 – B1Xibeobpop beo Fitted values01020300510ALILOne important numerical property of residuals•The sum of the residuals is zero.Regression Commands in STATA•reg depvar indvars•predict newvar•predict newvar, residWhy It’s Called RegressionHeight of FathersHeight of SonsSome RegressionsTemperature and LatitudePortlandORSanFranciscoCALosAngelesCAPhoenixAZNewYorkNYMiamiFLBostonMANorfolkVABaltimoreMDSyracuseNYMobileALWashingtonDCMemphisTNClevelandOHDallasTXHoustonTXKansasCityMOPittsburghPAMinneapolisMNDuluthMN0 20 40 60 80JanTemp25 30 35 40 45latitude. reg jantemp latitude Source | SS df MS Number of obs = 20-------------+------------------------------ F( 1, 18) =
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