Department of Economics Prof. Derek DeLia Econometrics 220:322:B6 Summer 2009 DICHOTOMOUS DEPENDENT VARIABLES Use of dummy variables (0-1) as dependent variables Part of a larger class of models called limited dependent variables OLS estimation w/dichotomous dependent variables Simple model: EMBED Equation.3 How should we interpret EMBED Equation.3 ? Notice that EMBED Equation.3 Also, EMBED Equation.3 2 issues Issue #1: Predicted probability may fall outside of 0-1 range. Issue #2: Variance of the error term is different for each observation (i.e., heteroskedastcity) Notice that EMBED Equation.3 There are potential ways to ?save? the OLS model. But we will use another approach entirely. Limited dependent variable approach Underlying propensity for Y to equal 1 Limited/latent variable Z measures this propensity Z is unobservable but can be modeled Threshold value Z* determines value of Y If EMBED Equation.3 , Y=1 If EMBED Equation.3 , Y=0 Z* is a random variable Model Z as a function of explanatory variables and link to observable data EMBED Equation.3 Prob(Y=1) = Prob(Z>Z*) = Prob(Z* F = 0.0000 Residual | 550.409286 5537 .099405686 R-squared = 0.1307 -------------+------------------------------ Adj R-squared = 0.1293 Total | 633.193077 5546 .114171128 Root MSE = .31529 ------------------------------------------------------------------------------ uninsured | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- lowinc | .1270746 .0112499 11.30 0.000 .1050205 .1491288 hisp | .1129134 .0171238 6.59 0.000 .079344 .1464828 lowhisp | .1060818 .023797 4.46 0.000 .0594305 .1527332 black | .014766 .0127011 1.16 0.245 -.0101331 .0396651 othrace | .0494592 .0201576 2.45 0.014 .0099424 .088976 age0_12 | -.1149144 .0115954 -9.91 0.000 -.1376459 -.092183 age13_18 | -.0621357 .0145587 -4.27 0.000 -.0906766 -.0335949 age19_25 | .1104876 .0150686 7.33 0.000 .0809471 .140028 age46_64 | -.0360741 .0118088 -3.05 0.002 -.059224 -.0129243 _cons | .0893935 .0085939 10.40 0.000 .072546 .106241 ------------------------------------------------------------------------------ Probit regression Probit regression Number of obs = 5547 LR chi2(9) = 650.86 Prob > chi2 = 0.0000 Log likelihood = -1832.8043 Pseudo R2 = 0.1508 ------------------------------------------------------------------------------ uninsured | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- lowinc | .6745101 .0583733 11.56 0.000 .5601006 .7889197 hisp | .63502 .0854861 7.43 0.000 .4674704 .8025697 lowhisp | .0888297 .1111701 0.80 0.424 -.1290597 .3067192 black | .1115549 .0673409 1.66 0.098 -.0204309 .2435407 othrace | .3068138 .1046316 2.93 0.003 .1017396 .511888 age0_12 | -.6523329 .0684125 -9.54 0.000 -.7864189 -.5182469 age13_18 | -.3380097 .0813446 -4.16 0.000 -.4974422 -.1785772 age19_25 | .4096575 .0689405 5.94 0.000 .2745366 .5447784 age46_64 | -.1862669 .065177 -2.86 0.004 -.3140114 -.0585224 _cons | -1.440218 .0482102 -29.87 0.000 -1.534708 -1.345727 ------------------------------------------------------------------------------ Probit is a non-linear model ==> interaction effect is not straightforward More detailed calculations are needed Interaction effect could be different for different individuals (w/different values for independent variables) Calculations in Stata Page PAGE 1 of NUMPAGES 8