* Probit Model with Heteroskedasticity * W.H. Greene, Econometric Analysis, Fourth Edition, 2000. * Example 19.7, p. 830. SAMPLE 1 32 * Table A19.1 READ (grade.shd) / NAMES * Probit Estimation GEN1 NK=5 DIM BETA NK PROBIT GRADE GPA TUCE PSI / COEF=BETA INDEX=XB GEN1 LLFPROB=$LLF GEN1 NOBS=$N * Construct a Lagrange Multiplier test for heteroskedasticity. * The error assumption in Greene, 2000, is Var(e) = EXP(B5*PSI) * Reference: R. Davidson and J.G. MacKinnon, "Convenient Specification * Tests for Logit and Probit Models", Journal of Econometrics, * Vol 25, 1984, pp. 241-262. * Davidson and MacKinnon propose a variety of tests. * The test reported in Greene, based on the BHHH estimator, is * called the LM1 test in Davidson and MacKinnon. SET NOOUTPUT Y: GRADE * Davidson and MacKinnon - Equation (26), p. 247. GENR ONE=1 GENR Z=-XB*PSI COPY GPA TUCE PSI ONE Z XMATRIX * Davidson and MacKinnon - Equation (5), p. 243 DISTRIB XB / TYPE=NORMAL CDF=CDF PDF=PDF MATRIX G=([Y]/CDF +([Y]-1)/(1-CDF))*PDF*XMATRIX * Artificial regression - Davidson and MacKinnon - Equation (14), p. 245 OLS ONE G / NOCONSTANT * LM test statistic - explained sum of squares GEN1 LM=$ZSSR PRINT LM * Set up the log-likelihood function for the Probit model with * heteroskedasticity (Greene, 2000, Equation 19-33, p. 829). SET OUTPUT * Formula for model with heteroskedasticity - this formula is used to * construct the log-likelihood function (see Greene, 2000, p. 829). MATHEQ: (B1*GPA+B2*TUCE+B3*PSI+B4)/EXP(B5*PSI) GEN1 DELTA=1/(10**5) NL 1 / NCOEF=NK LOGDEN START=BETA COEF=BETAP STDERR=SE STEPSIZE=DELTA EQ GRADE*LOG(NCDF([MATHEQ]))+ & (1-GRADE)*LOG(1-NCDF([MATHEQ])) END * The LR statistic. GEN1 LR=2*($LLF-LLFPROB) * The Wald statistic. GEN1 GAMMA=BETAP(NK) GEN1 GSE=SE(NK) PRINT GAMMA GSE GEN1 WALD=(GAMMA/GSE)**2 PRINT LR LM WALD * * In Greene, 2000, from the numerical results in Table 19.3 the Wald * test statistic should be: (1.09337/0.8806)**2 = 1.542 * The difference betweeen SHAZAM's Wald statistic and that reported by * Greene results from differences in the numerical estimation algorithm. STOP