514 Discrete Choice Modeling
11.4.5 Application
Riphahn, Wambach and Million (2003) studied the joint determination of two
counts, doctor visits and hospital visits. One would expect these to be highly cor-
related, so a bivariate probit model should apply to DOCTOR=1 (DocVis>0) and
HOSPITAL= 1 (HospVis> 0 ). The simple product moment correlation coefficient
is inappropriate for binary variables. The tetrachoric correlation is used instead;
this turns out to be the estimate ofρin a bivariate probit model in which both
equations contain only a constant term. The first estimated model in Table 11.4
reports a value of 0.311 with a standard error of only 0.0136, so the results are
consistent with the conjecture. The second set of estimates assumeρ= 0; the esti-
mates for the “Doctor” equation are reproduced from Table 11.3. As noted, there is
evidence thatρis positive. Kiefer’s (1982) LM statistic equals 399.20. The limiting
distribution is chi-squared with one degree of freedom – the 5% critical value is
3.84, so the hypothesis that the outcomes are uncorrelated is rejected. The Wald
and likelihood ratio statistics based on the unrestricted model are 21.496^2 =462.08
and 2[17670.94+8084.465−25534.46]=441.998, respectively, so the hypothesis
is rejected by all three tests. The third model shown in Table 11.4 is the unre-
stricted bivariate probit model, while the fourth is the recursive bivariate probit
model with DOCTOR added to the right-hand side of the HOSPITAL equation. The
results do not support this specification; the log-likelihood is almost unchanged.
It is noteworthy that in this expanded specification, the estimate ofρis no longer
significant, as might have been expected.
Table 11.4 Estimated bivariate probit models (standard errors in parentheses)
(1) (2) (3) (4)
Tetrachoric corr. Uncorrelated Bivariate probit Recursive probitDoctor Hospital Doctor Hospital Doctor Hospital Doctor HospitalConstant 0.329 –1.355 0.155 –1.246 0.155 –1.249 0.155 –1.256
(.0077) (.0107) (.0565) (.0809) (.0565) (.0773) (.0565) (.481)
Age .000 .000 .0128 .00488 .0128 .00489 .0128 .00486
(.000) (.000) (.0008) (.0011) (.0008) (.0011) (.0008) (.0025)
HhNInc .000 .000 –.116 .0421 –.118 .0492 –.118 .0496
(.000) (.000) (.0463) (.0633) (.0462) (.0595) (.0463) (.0652)
HhKids .000 .000 –.141 –.0147 –.141 –.0129 –.141 –.0125
(.000) (.000) (.0182) (.0256) (.0181) (.0257) (.0181) (.0386)
Educ .000 .000 –.0281 –.026 –.028 –.026 –.028 –.026
(.000) (.000) (.0035) (.0052) (.0035) (.0051) (.0035) (.0066)
Married .000 .000 .0522 –.0547 .0519 –.0546 .0519 –.0548
(.000) (.000) (.0205) (.0279) (.0205) (.0277) (.0205) (.0313)
Doctor .00912
(.663)
ρ .311 (.0136) .000 .303 (.0138) .298 (.366)
LnL –25898.27 –1767.94 –8084.47 –25534.46 –25534.46