* Tobit Model with Multiplicative Heteroskedasticity * References: * The data set and application of the Tobit model is presented in: * Ernst R. Berndt, The Practice of Econometrics, Addison-Wesley, 1991. * Chapter 11, Exercise 4, pp. 657-658. * For Multiplicative Heteroskedasticity in the Tobit Model see: * W.H. Greene, Econometric Analysis, Fourth Edition, 2000. * Chapter 20, Example 20.11, p. 913 SAMPLE 1 753 READ (mroz.txt) / NAMES * Analyze wife's property income GENR PRIN=(FAMINC-WW*WHRS)/1000 GENR WA2=WA*WA DIM LWW 753 * Restrict the sample to those who work SAMPLE 1 428 GENR LWW=LOG(WW) * Estimate a wage determination equation OLS LWW WA WA2 WE CIT AX * Estimate a predicted wage for non-workers FC / PREDICT=LWW BEG=429 END=753 * TOBIT estimation SAMPLE 1 753 TOBIT WHRS LWW PRIN KL6 K618 WA WE / COEF=ALPHA * Get the regression coefficients GEN1 SIGMA=SQRT(\$SIG2) SAMPLE 1 7 GENR BHAT=ALPHA*SIGMA * Now use the NL command for maximum likelihood estimation to * replicate the results of the TOBIT command. * First, use OLS to set some starting values for the estimation. * Restrict the sample to those who work SAMPLE 1 428 DIM BCON 8 OLS WHRS LWW PRIN KL6 K618 WA WE / HETCOV COEF=BETA GEN1 BCON:8=SQRT(\$SIG2) * Calculate the Goldberger-Greene adjusted estimates as * suggested in Berndt, Exercise 4 (b), p. 657. * Proportion working GEN1 P=428/753 PRINT P SAMPLE 1 7 GENR BCON=BETA/P PRINT BHAT BETA BCON * Define an equation in a SHAZAM character string. * Description of character strings is in the chapter SHAZAM PROCEDURES * in the SHAZAM User's Reference Manual. * Note that the expression is enclosed by brackets -- this is to * ensure correct evaluation of the expression when it is used later on * the EQ command. XB:(B1*LWW+B2*PRIN+B3*KL6+B4*K618+B5*WA+B6*WE+B0) SAMPLE 1 753 GENR const=-LOG(2*\$PI) * Set LIMIT=1 when WHRS > 0 and LIMIT=0 otherwise GENR LIMIT=DUM(WHRS) * On the NL command, the LOGDEN option specifies that the log-density * of a single observation is entered on the EQ command. * See Greene, Equation (20-13), p. 911. * Note that the use of this option requires careful checking and * testing of both the specification of the log-likelihood function * and the SHAZAM commands that program the formula. * NCDF(z) is the standard normal cumulative distribution function. * Starting values are specified with the START= option on the NL command. DIM BTOBIT 14 NL 1 / NCOEF=8 LOGDEN START=BCON COEF=BTOBIT EQ (1-LIMIT)*LOG(1-NCDF([XB]/sig))+ & LIMIT*(const-LOG(sig**2)-((WHRS-[XB])/sig)**2)/2 END * The value of the log-likelihood function is available in the * temporary variable \$LLF. GEN1 LLF0=\$LLF * Now specify the form of the variance equation for a model * with Multiplicative Heteroskedasticity - see Greene, p. 913. SIG:(A0*(EXP(A1*LWW+A2*PRIN+A3*KL6+A4*K618+A5*WA+A6*WE)**(1/2))) * Estimation of the Tobit Model with Multiplicative Heteroskedasticity. * The starting values are the coefficient estimates from the * previous Tobit estimation - note that the value of the * log-likelihood function at the first iteration is identical * to the Tobit log-likelihood function value. NL 1 / NCOEF=14 LOGDEN START=BTOBIT EQ (1-LIMIT)*LOG(1-NCDF([XB]/[SIG]))+ & LIMIT*(const-LOG([SIG]**2)-((WHRS-[XB])/[SIG])**2)/2 END GEN1 LLF1=\$LLF * Likelihood ratio test statistic for heteroskedasticity GEN1 LR=2*(LLF1-LLF0) * Calculate p-value for the test SAMPLE 1 1 DISTRIB LR / TYPE=CHI DF=6 CDF=cdf GEN1 p_value=1-cdf PRINT LR p_value STOP