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·	From Theory To Practice

The final element in the analysis of unemployment durations is to investigate the robustness of these results to model specification. To do this, I have estimated a variety of alternative models of the determinants of duration. These are reported in Tables 35-38.

First, I addressed a set of leading alternatives to the Cox partial likelihood approach, a set that includes a variety of parametric models of duration. That is, the overall hazard is now viewed as

h(t,X(t))=b(t,0)e^x(1)'B

where X(t) is a vector of explanatory variables and B is the associated vector of coefficients, but b(t,0) is now taken to follow a particular parametric form. The exponential model obtains when

b(t,0)=

and the related Weibull model adds a shape parameter p so that

b(t,0)=pt^p-1.

The Gompertz model, still in the proportional hazard framework, has

b(t,0)=e^yt

so that the overall (proportional) hazard is

h(t,X(t))=e^yte^x(t)'B.

In contrast, the three other functional forms I estimated are all in the "accelerated failure time" (AFT) framework. This means that larger values of the control variables translate into an "acceleration" of the failure time, rather than a proportional shift of the entire estimated hazard. Specifically, these models are estimated as

lnt=X(t)'b+z

and the nature of the model depends on the assumed distribution for the error term z. I estimated three possibilities, according to whether this distribution was LogNormal (the natural logarithm of time is assumed to follow a Normal distribution), LogLogistic (the natural logarithm of time follows a Logistic distribution), or Gamma.

The key results of this investigation are reported in Table 35, although other coefficient estimates are not reported for brevity. For the three proportional hazard models, the first three reported in the Table, the results are quite robust. The estimated effects of "ei" and "intro" are always positive and significant, at least for models 3 and 4. This suggests that the Cox partial likelihood results reported above would not be seriously modified if one of these alternative models of duration is used. In contrast, the three AFT models produce mixed and inconsistent results, so we are inclined to put less weight on these specifications.

With the proportional hazard framework, I have also addressed some richer models of the determinants of unemployment duration. In Table 36, I report the results from a Cox partial likelihood model where I explicitly model a phase of uninsured unemployment (a period in which UI/EI coverage has been exhausted). This time-varying covariate allows the hazard to shift up or down when UI/EI expires and is estimated within the context of the pooled data with quarterly dummy variables for the seasonal effects. The results of the full sample for "ei" and "intro" are very similar to those reported in Table 19. Small positive and significant effects on "ei" are reported for models 3 and 4. Significant positive effects for all four model specifications for the phase-in variable "intro" are reported. Interestingly, the "unins" variable that captures the uninsured period of unemployment has a positive coefficient (significantly so for models 1 and 2), so that the hazard is raised in the period when coverage has been exhausted.

Relatedly, I address issues of anticipated benefit exhaustion in Table 37. A set of time-varying covariates are included to account for the possibility of 1-3, 4-6 and 7-9 weeks of UI/EI coverage remaining in the unemployment spell (ben13, ben46, and ben79, respectively). Again, there is no essential change in the coefficients on "ei" and "intro" for any of the four model specifications, relative to the basic Cox model without the benefit exhaustion variables, suggesting that those results are quite robust. The benefit exhaustion effects exert a rising influence on the hazard as UI/EI exhaustion approaches (the coefficient on ben13 exceeds that on ben46, which in turn exceeds that on ben79). This is true for each specification. Of the three exhaustion variables, only ben13 is individually significant, but this effect is strong and fairly constant across models 2-4.

The final variant I have examined is an alternative set of duration models in a PGM framework after Prentice and Gloeckler (1978) and Meyer (1990). These models are estimated for the full sample and the VQ/Dis and SW/Oth groups and incorporate three alternative approaches to duration. First, I incorporated the logarithm of duration in addition to the "ei" and "intro" variables. Second, I used a fourth-order polynomial in duration as a flexible means of capturing non-monotonicity of the hazard with respect to duration. Third, I estimated a fully non-parametric model where each duration in the grid (from 1 week to 54 weeks) has its own dummy variable, thereby permitting any pattern whatsoever to the estimated duration effects. The results from these procedures are reported in Table 38, where the six model specifications correspond to the three duration methods, each estimated with and without the set of demographics and the local labour market conditions variable. The estimated "ei" and "intro" effects are very stable across all six models, with significantly positive point estimates for the "ei" coefficient in the 0.06 to 0.16 range. In each case, the duration controls alone result in an estimate under 0.10 while the addition of the demographics raises the estimate to the 0.13-0.16 range. For the phase-in variable "intro", this pattern of higher UI/EI effects when demographics are controlled for is also present, although the range of variation of the key coefficient is small.

I have also estimated the log duration and the polynomial in duration models (models 1 and 3, respectively) in this framework with allowance for Gamma distributed unobserved heterogeneity. This may control for unobserved factors that might influence the hazard and lead to a bias in the estimated duration effects, together with the potential of bias in the estimated effects of the set of control variables. For model 1, the "ei" coefficient drops to 0.036 (t-statistic of 1.10) and the "intro" variable coefficient drops to 0.090 (2.16), while for model 3, the respective estimates allowing for unobserved heterogeneity are 0.031 (0.80) and 0.094 (1.90). Thus, there is some sign that allowance for unobserved heterogeneity lowers the estimated effects, particularly for the "ei" variable. Estimated models with unobserved heterogeneity for the other specifications in Table 38 failed to converge, however.

Overall, although these final two sets of results led to smaller point estimates than the earlier models without unobserved heterogeneity, I find the broad consistency of these full sample results with those from the Cox partial likelihood model (Table 19) striking. It suggests that these conclusions on the behavioural effects of UI/EI are robust.