Fixing Boundary Violations

4.2 Numerical Estimation

Regarding the numerical setups, the maximum iteration is 100, and the tolerance value is set to $10^{-4}$. The criteria to evaluate the validity of parameter estimates are admissible solution and greater loglikelihood. Admissible solution refers to the non-violation of boundary restrictions from $g_{1}$ to $g_{2m+6}$, where $m=2$. A greater loglikelihood value also indicates a better solution if admissibility is satisfied. Any inadmissible solution is regarded as a failed estimate, and the comparison of the loglikelihood value is only carried out when the TRM and TRMCO models generate admissible estimates.

Both the TRM and TRMCO adopt the centering specification in the regression model. The initial value for the TRMCO model is the parameter estimate of the TRM model. According to the theory of nonlinear programming, when the TRMCO solution satisfies the KKT conditions, it generates the best lower bound of optimality for the Lagrange dual function given the property of weak duality. (Boyd and Vandenberghe, 2004, 225) Therefore, the TRMCO solution could theoretically be inferior to the TRM solution, since the latter only needs to satisfy boundary restrictions, which is only one of the four KKT conditions. Whether the loglikelihood value is larger for the TRM or TRMCO solution is an empirical question. When the TRMCO model reaches the maximum number of iterations, the iteration that generates the largest loglikelihood and satisfies the boundary restrictions is reported, except in the first iteration.¹³

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Footnote

¹³According to the weak duality theorem, violation of the complementary slackness condition (the fourth KKT conditions) sometimes occurs and leads to nonconvergence because of the duality gap. (Murty, 2010, 260-261) When the estimation process reaches the maximum iterations, the best eligible TRMCO estimate is considered a reasonable result to report if the boundary restrictions are satisfied. Otherwise, the result is marked as an ineligible solution. The same criterion is applied to the TRM estimate under the Stata environment.