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interval.41,42 In models 2,5, and 6, the coefficients of inclusive values in the two branches are constrained to be the same. This is not required by the TEV model; however, differing values can be accommodated in the sequential estimation procedure only by imposing a nonlinear constraint that the coefficients in one branch be a scalar multiple of the coefficients in the other branch, or else by treating the coefficients in the two branches as independent.

The alternative models 1 through 8 yield coefficients of cost, on-vehicle time, and access time of expected sign.43 There is considerable variation between the models in the magnitude of coefficients, with models 2 and 3 implying a sharper discrimination among costs than the remainder. Estimated values of on-vehicle time range from 27 percent of the wage in model 2 to 92 percent in model 6. Extimated values of access time range from 79 percent of the wage in model 2 to 394 percent in model 5, with most values in the 100 to 200 percent range. Thus the value of time estimates are quite sensitive to model specification.

Full information maximum likelihood estimates of models 2 and 6 have been calculated by Cosslett (1978), and are given in table 5.4. In these models the coefficients of inclusive value in the two branches are allowed to differ. Thus the log likelihood is larger for these estimates both because of full model maximization and because of an additional parameter. Cosslett's estimates of model 2 do not reduce the log likelihood substantially compared to sequential estimation. However, there are substantial changes in coefficient values, implying a sensitivity to estimation method not reflected in the asymptotic standard errors. Thus caution should be exercised in interpreting these coefficients. In the FIML estimates of model 2, the coefficients of inclusive values in the two branches are not significantly different.44 FIML estimation of model 6 results in a sub-

41. The inclusive value coefficients in models 2, 4, and 5 are significantly different from one at the 1 percent confidence level, and in model 3 at the 15 percent confidence level. In all four models the coefficient of inclusive value is significantly different from zero at the 1 percent level. It should be remembered that these tests are not independent.

42. The test statistic T = 2[L(k) - £(!)], where L(k) is the log likelihood for model k, has an asymptotic distribution satisfying Prob[7"g ;] S a for X?(a) = t. Hence T> Xf(a) implies that the null hypothesis model 1 holds can be rejected with significance level at least a. By this criterion model 1 is rejected at least at the 5 percent level in (nonindependent) tests against models 2, 3, 4, and 6.

43. The coefficient of on-vehicle time in model 7 is reversed in sign but insignificant.

44. Under the null hypothesis that these coefficients are the same, the difference in the estimated coefficients is asymptotically normal with mean zero and variance equal to left coefficient variance + right coefficient variance —2 (covariance of coefficients).

Table 5.4

Full information estimates of MNML models

Model 2 6

Preference tree Own auto access Transit distinct

Parameter estimates2 (standard errors)

Model 2 6

Preference tree Own auto access Transit distinct

Parameter estimates2 (standard errors)

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