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Figure 7.2. Range coefficients under the SB distribution (solid line) and log-normal (dashed line).

Figure 7.2. Range coefficients under the SB distribution (solid line) and log-normal (dashed line).

utility that is used to calculate the logit formulas in L( ) depends on the bounds of the SB distributions. A M-H algorithm is used to take draws from this conditional posterior, similar to that used by Train (2001) for fixed coefficients.

We estimated a model with the upper bound of the SB distribution for the range coefficient treated as a parameter. Using a flat prior, the estimated value was 2.86 with a standard error of 0.42. The log-likelihood of the model dropped slightly from -6,159.7 with the upper bound set at 2.0 to -6,160.56 with the estimated bound. Run time approximately doubled, since the M-H algorithm for the bounds of the SB distribution requires about the same amount of calculation as the M-H algorithm for 3n Vn. As noted above, run times are fairly short with the procedure such that doubling them is not a burden. However, identification becomes an issue when the bounds are treated as parameters, since the difference between the upper and lower bounds, u — I, is closely related to the variance w of the latent normal term. An important area for further work is whether the SB distributions can be re-parameterized in a way that improves identification of each parameter when the researcher does not specify the bounds.

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