Alastair Hall ECG 790F: Microeconometrics Spring 2006 Computer Handout # 2 Estimation of binary response models : part II In this handout, we discuss the estimation of binary response models with and without heteroscedasticity using proc qlim. We use the same example as on Computer Handout # 1, and for brevity focus purely on the probit model. Recall that in this example, we wish to model the probability that an individual votes yes in a referendum on whether the local tax rate should be increased to provide additional funding for schools. The probability of voting yes is assumed to be determined as follows, where P (YESVM)=f(PUB 12,PUB 34,PUB 5,PRIV,YEARS,SCHOOL,LOGINC,PTCON) YESVM = I{individual votes yes}, PUB 12 = I{individual has 1 or 2 children in public school}, PUB 34 = I{individual has 3 or 4 children in public school}, PUB 5 = I{individual has 5 or more children in public school}, PRIV = I{individual has 1 or more children in private school}, SCHOOL = I{individual is employed as a teacher (public or private)}, YEARS= # of years living in Troy community, LOGINC = log of annual household income in $, PTCON = log of property taxes paid per year in $, and I{.} denotes an indicator variable that takes the value one if the event in the curly brackets occurs. In Computer Handout 1, we desribed how this model can be estimated using proc logistic. The model can also be estimated in proc qlim as follows: 1 proc qlim data=main; 1 This assumes that the data are read into the data set main using the same commands as in Computer Handout #1. 1
model yesvm = PUB1 2PUB34 PUB5 PRIV YEARS SCHOOL loginc PTCON/ type=bprobit covest=hess optmethod=nr; endogenous discrete=(yesvm 0 1); The role of the commands is as follows: type=bprobit: this implements a binary response probit estimation; for logit use blogit. covest=hess: this causes the covariance matrix of the MLE to be estimated via the hessian of the log likelihood function; the other option is op which would cause the estimate to be based on the outer product gradient estimator. optmethod=nr: this option sets the numerical optimization to the Newton-Raphson; another option is qn which yields the so-called quasi-newton method. endogenous discrete=(yesvm 0 1): this instructs SAS that the dependent variable is discrete and takes values zero or one. It is also possible to control the number of iterations within the routine using maxiter= -the default is 100 - and the starting values using start=. The output is as follows: The QLIM Procedure Binary Probit Estimates Algorithm converged. Model Fit Summary Dependent Variable YESVM Number of Observations 95 Log Likelihood -52.70244 Maximum Absolute Gradient 3.0704E-12 Number of Iterations 6 Optimization Method Newton-Raphson AIC 123.40488 Schwarz Criterion 146.38977 2
Discrete Response Profile Index YESVM Frequency Percent 0 0 36 37.89 1 1 59 62.11 Goodness-of-Fit Measures for Discrete Choice Models Measure Value Formula Likelihood Ratio (R) 20.669 2 * (LogL - LogL0) Upper Bound of R (U) 126.07-2 * LogL0 Aldrich-Nelson 0.1787 R / (R+N) Cragg-Uhler 1 0.1955 1 - exp(-r/n) Cragg-Uhler 2 0.2661 (1-exp(-R/N)) / (1-exp(-U/N)) Estrella 0.2115 1 - (1-R/U)^(U/N) Adjusted Estrella 0.0280 1 - ((LogL-K)/LogL0)^(-2/N*LogL0) McFadden s LRI 0.1639 R / U Veall-Zimmermann 0.3133 (R * (U+N)) / (U * (R+N)) McKelvey-Zavoina 0.3918 N = # of observations, K = # of regressors Parameter Estimates Standard Approx Parameter DF Estimate Error t Value Pr > t Gradient Intercept 1-4.1560 4.5863-0.91 0.3648 2.54E-13 PUB1_2 1 0.1859 0.4352 0.43 0.6693 9.45E-14 PUB3_4 1 0.5324 0.4799 1.11 0.2673 1.16E-13 PUB5 1 0.2039 0.7683 0.27 0.7907-555E-17 PRIV 1-0.3298 0.4649-0.71 0.4780-644E-17 YEARS 1-0.0145 0.0153-0.95 0.3440 3.07E-12 SCHOOL 1 1.6654 0.8616 1.93 0.0532 3E-13 loginc 1 1.4591 0.4800 3.04 0.0024 2.5E-12 PTCON 1-1.4777 0.6489-2.28 0.0228 1.68E-12 3
Notice the following features of this output: The estimates are the same as those obtained in Computer Handout # 1 using proc logistic. The reported statistics are different from those generated by proc logistic. A lot of goodness of fit statistics have been proposed for this model! One advantage of proc qlim is that it can be used to estimate binary response models with hetroscedasticity. Suppose in our example that we wish to estimate the model with heteroscedasticity of the form σ 2 i = exp(θpriv i). This can be implemented as follows. proc qlim data=main; model yesvm = PUB1 2PUB34 PUB5 PRIV YEARS SCHOOL loginc PTCON/ type=bprobit covest=hess optmethod=nr; endogenous discrete=(yesvm 0 1); hetero PRIV; The only difference from our original program is the inclusion of the command hetero PRIV. The output is as follows. The QLIM Procedure Binary Probit Estimates with Heteroscedasticity Algorithm converged. Model Fit Summary Dependent Variable YESVM Number of Observations 95 Log Likelihood -52.27072 Maximum Absolute Gradient 0.0001472 Number of Iterations 14 Optimization Method Newton-Raphson AIC 124.54143 Schwarz Criterion 150.08020 4
Discrete Response Profile Index YESVM Frequency Percent 0 0 36 37.89 1 1 59 62.11 Parameter Estimates Standard Approx Parameter DF Estimate Error t Value Pr > t Gradient Intercept 1-7.2962 5.8095-1.26 0.2092 0.000014 PUB1_2 1 0.001571 0.5628 0.00 0.9978 0.000021 PUB3_4 1 0.3540 0.5928 0.60 0.5504 0.000013 PUB5 1 1.1178 1.0963 1.02 0.3079-0.00001 PRIV 1-0.4496 7.4650-0.06 0.9520 8.574E-8 YEARS 1-0.006539 0.0179-0.37 0.7148-0.00001 SCHOOL 1 1.9683 1.0146 1.94 0.0524 2.307E-6 loginc 1 1.8180 0.5637 3.23 0.0013 0.000147 PTCON 1-1.5335 0.7258-2.11 0.0346 0.0001 HET1 1 5.8697 23.5334 0.25 0.8030-2.68E-7 The estimate of θ is denoted HET1 in the output. It is natural to want to test whether heteroscedasticity is present, that is to test the null hypothesis H 0 : θ = 0 (homoscedasticity) versus H 1 : θ 0 (heteroscedasticity of the form given). This can be done two ways using the output above: using the Wald test which is just the square of the t statistic given in the output, that is W =(0.25) 2 =0.0625 with a p-value of 0.803; using the LR test which is 2{( 52.70244) ( 52.27072)} = 0.8634 with a p-value of 0.6472. Both statistics are insignificant at conventional values and so we fail to reject the null of homoscedasticity in this case. Other forms of heteroscedasticity can be estimated using the following options in the hetero command; here we assume the heteroscedastcity is driven by x θ where x =(x 1,x 2 ) are variables in the data set and θ =(θ 1,θ 2 ) unknown parameters. hetero x1 x2 σi 2 = exp(x iθ); in this case the output contains coefficient estimates 5
HET1andHET2 which are estimates of θ 1 and θ 2 respectively; hetero x1 x2 / link=exp default); σi 2 = exp(x iθ)(i.e. same as previous case and so exp is hetero x1 x2 / link=linear σ 2 i = x i θ; hetero x1 x2 / link=linear square σ 2 i =(x i θ)2 ;thesquare option can also be used with link=exp. 6