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the marginal effects, 8 = dE [y | x]/dx, are shown in Table 22.6 for three models: the ZIP model, Fair's original tobit model, and the tobit model estimated with the doubly censored count. The estimates for the ZIP model are considerably lower than those for Fair's tobit model. When the tobit model is reestimated with the censoring on the right, however, the resulting marginal effects are reasonably close to those from the ZIP model, though uniformly smaller. (This result may be from not building the censoring into the ZIP model, a refinement that would be relatively straightforward.)

We conclude that the original tobit model provided only a fair approximation to the marginal effects produced by (we contend) the more appropriate specification of the Poisson model. But the approximation became much better when the data were recorded and treated as censored. Figure 22.3 also shows the predictions from the ZIP model (narrow bars). As might be expected, it provides a much better prediction of the dependent variable. (The integer values of the conditional mean function for the tobit model were roughly evenly split between zeros and ones, whereas the doubly censored model always predicted y = 0.) Surprisingly, the treatment of the highest observations does greatly affect the outcome. If the ZIP model is fit to the original uncensored data, then the vector of marginal effects is 8 = [-0.0586, 0.2446, -0.692, 0.115, -0.787], which is extremely large. Thus, perhaps more analysis is called for—the ZIP model can be further improved, and one might reconsider the hurdle model—but we have tortured Fair's data enough. Further exploration is left for the reader.

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