THE LINEAR PROBABILITY MODEL: USING LEAST SQUARES TO ESTIMATE A REGRESSION EQUATION WITH A DICHOTOMOUS DEPENDENT VARIABLE
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1 PLS 802 Spring 2018 Professor Jacoby THE LINEAR PROBABILITY MODEL: USING LEAST SQUARES TO ESTIMATE A REGRESSION EQUATION WITH A DICHOTOMOUS DEPENDENT VARIABLE This handout shows the log of a Stata session that performs least-squares regression analysis on some data about state Electoral College votes in the 1992 U.S. presidential election. The model is estimated two ways: First, using ordinary least squares; second, using OLS with robust standard errors. - (FIRST FEW LINES OMITTED) > * Retrieve the Clinton state voting data. use clint92; > * Describe the contents of the dataset, > * and calculate summary statistics. describe; Contains data from clint92.dta obs: 48 vars: 5 14 Apr :09 size: storage display value variable name type format label variable label - state str2 %9s State name ideol float %9.0g Ideology of state electorate party float %9.0g Partisanship of state electorate black float %9.0g Pct African American vote92 float %9.0g Clinton won state Electoral College vote - Sorted by:. summarize; Variable Obs Mean Std. Dev. Min Max state 0 ideol party black vote > * Use OLS to estimate the coefficients of > * a regression equation with clinton vote as the > * dependent variable, electorate ideology, electorate > * partisanship, and pct African American > * as independent variables
2 Page 2. regress vote92 ideol party black; Source SS df MS Number of obs = F(3, 44) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = vote92 Coef. Std. Err. t P> t [95% Conf. Interval] ideol party black _cons > * Standardize independent variables and > * re-estimate model. egen ideol2 = std(ideol);. egen party2 = std(party);. egen black2 = std(black);. regress vote92 ideol2 party2 black2; Source SS df MS Number of obs = F(3, 44) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = vote92 Coef. Std. Err. t P> t [95% Conf. Interval] ideol party black _cons > * Obtain predicted values, sort dataset, and print data. predict yhat, xb;. sort yhat;. list state ideol party black > vote92 yhat; state ideol party black vote92 yhat 1. WY UT MS ID SC
3 Page 3 6. SD VA AL AZ NH NE NC TX OK OR FL IN VT KS MI AR CT NJ MO TN ND IL WI OH LA GA PA MT NV WV CO MN NM CA NY WA MD IA KY DE ME MA RI > * Graph dependent variable against fitted > * values, and add OLS line. Save graph > * to file, creating Figure 1. twoway (scatter vote92 yhat, > scheme(s1color) > jitter(3) > msymbol(oh) > mcolor(black) > ysize(4.5) > xsize(4.5) > xaxis(1 2) yaxis (1 2)
4 Page 4 > xtitle(" *Party *Ideol *Black", axis(1)) > ytitle("clinton received state EC vote in 1992", axis(1)) > ylabel(, axis(1) nogrid) > ylabel(, axis(2) nolabel) > xlabel(, axis(2) nolabel) > legend(off) > ) > (lfit vote92 yhat) > ;. graph export "lpm fit.pdf", replace; (file lpm fit.pdf written in PDF format) > * Obtain residuals, construct > * residual plot, and save. > * This creates Figure 2.. predict esubi, residuals;. graph twoway > scatter esubi yhat, > ; (GRAPH OPTIONS OMITTED). graph export resid1.pdf, replace; (file resid1.pdf written in PDF format) > * Create graph of dependent variable against > * fitted values, with nonparametric regression > * line superimposed over points. Save graph > * to external file, creating Figure 3.. twoway (scatter vote92 yhat, > scheme(s1color) > jitter(3) > msymbol(oh) > mcolor(black) > ysize(4.5) > xsize(4.5) > xaxis(1 2) yaxis (1 2) > xtitle(" *Party *Ideol *Black", axis(1)) > ytitle("clinton received state EC vote in 1992", axis(1)) > ylabel(, axis(1) nogrid) > ylabel(, axis(2) nolabel) > xlabel(, axis(2) nolabel) > legend(off) > ) > (lowess vote92 yhat) > ;. graph export "lowess curve.pdf", replace; (file lowess curve.pdf written in PDF format) > * Estimate the same model, but > * use robust standard errors > * which correct for the inherent > * heteroskedasticity of the linear > * probability model
5 Page 5. regress vote92 ideol party black, vce(robust); Linear regression Number of obs = 48 F(3, 44) = Prob > F = R-squared = Root MSE = Robust vote92 Coef. Std. Err. t P> t [95% Conf. Interval] ideol party black _cons > * Calculate percent correctly predicted and > * correlation between actual and predicted > * values of dep variable as > * alternative goodness of fit measures. generate dichot_yhat = yhat;. replace dichot_yhat = 0 if yhat <= 0.5; (12 real changes made). replace dichot_yhat = 1 if yhat > 0.5; (36 real changes made). tabulate vote92 dichot_yhat, > cell chi2 ; Key frequency cell percentage Clinton won state Electoral College dichot_yhat vote 0 1 Total Total Pearson chi2(1) = Pr = 0.000
6 Page 6. correlate dichot_yhat vote92; (obs=48) dichot~t vote dichot_yhat vote display ^2; log close; - Figure 1: Scatterplot of dichotomous Clinton won state variable versus the best-fitting linear combination of state partisanship, ideology, and race (obtained from the OLS estimates). The data points are jittered in the vertical direction to reduce overplotting problems, and the bivariate OLS regression line is shown. Linear prediction Clinton received state EC vote in *Party *Ideol *Black
7 Page 7 Figure 2: Residual plot from linear probability model of Clinton voting in 1992 Electoral College. A loess curve is fitted to the data. Residuals Predicted values Figure 3: Plot of the dichotomous dependent variable (jittered) versus the predicted values from the linear probability model (OLS estimates) with a nonparametric loess curve superimposed over data. Linear prediction Clinton received state EC vote in *Party *Ideol *Black
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