The Limits of Mathematics and NP Estimation in ... - Chichilnisky

The Impact of Government-SponsoredTraining Programs on the Labor Market Transitions of Disadvantaged MenThe Impact of Government-Sponsored Training Programs on the Labor Market Transitions of Disadvantaged Men 2571No Het Non-Para Log-Normal Weibull Weibull Weibull2 Factors 2 Factors 3 FactorsProbability 0.614 †ω 1 -1.866 †ω 2 -2.364 †b 2 -0.116 -0.221 -0.365 -0.184 -2.980 †b 3 3.757 † 7.781 † 8.304 † 8.209 † 8.173 †b 4 2.337 † 5.455 † 6.201 † 6.037 † 8.879 †b 5 1.323 0.447 0.244 -1.091 -1.691b 6 2.585 † 5.493 † 6.432 † 6.168 † 7.722 †b 7 -1.180 † -2.501 † -2.938 † -2.876 † -4.902 †b 1 ′ 2.768 †b 2 ′ 3.093 †b 3 ′ -7.131 †b 4 ′ 0.333b 5 ′ -8.563 †b 6 ′ -0.437b 7 ′ 0.738 †μ -1.629 †σ -0.566 †λ 9.493 † 9.057 † 14.685 †γ 0.163 † 0.147 † 0.216 †Log-likelihood -150668.6 -149774.2 -149847.8 -149818.9 -149474.8 149422.6Table 5. Heterogeneity Parameters† Statistically significant at 5%‡ Statistically significant at 10%mentioned earlier, the slope parameters of these specifications are sufficiently similar to omitthem from the tables. 16The first specification of the table does not control for unobserved heterogeneity and is thusa special case of all the other specifications. A simple likelihood-ratio test strongly rejectsthe first specification in favour of any specification that includes unobserved heterogeneity.The second specification is a standard non-parametric two-factor loading model and waspresented in equation (4). Most parameter estimates are statistically significant, except forb 2 and b 5 which concern transitions into welfare training programs and UI training programs,respectively. Accordingly, these estimates suggest there is little, if any, selectivity into thesetwo training programs.The third column of the table reports the parameter estimates of a parametric two-factorloading model. As was mentioned earlier, the weibull distribution function was preferredover all other distribution functions that were investigated. Notice that as in thenon-parametric specification, only b 2 and b 5 are not statistically significant. The last twolines of the table report the parameter estimates of the weibull distribution, λ and γ. 17Finally, the last column of the table presents the parameter estimates of the three-factor16 Bonnal et al. (1997) also found the slope parameters to be relatively insensitive to the distributionalassumptions of the unobserved heterogeneity variables. In their work, they compare a two-factorloading model with a finite number of points of support with a single-factor loading model that drawsheterogeneity terms from an i.i.d. IN (0, 1) distribution. The insensitivity of the slope parameters tothe distributional assumption is consistent with the results of Heckman & Singer (1984) using singledurations data.17 The weibull distribution function of a random variable x is given by F(x) =1 − exp [−λx γ ].