Fixing Boundary Violations

1.4 Research Purpose

In this article, I propose a revised TRM model (hereafter the TRMCO model) within the framework of nonlinear programming and demonstrates how to improve the current model by eliminating boundary violations. By specifying proper boundary restrictions, the estimation of the TRMCO model will be transformed into a constrained optimization problem (COP). (Bertsekas, 1996) While the ideas involved in solving a COP problem are not dramatically different from those that solve an unconstrained optimization problem, few techniques in numerical analysis are applied to the statistical methods familiar to political scientists.⁴ In the following sections, I first explain the problem of boundary violations in the TRM model. Next, I present a modified procedure of maximum likelihood estimation with the sequential quadratic programming (SQP) algorithm (Nocedal and Wright, 1999, 529) that solves constrained optimization problems. Third, I compare the inferential validity of the modified model with the current one through three simulation tests. Finally, an empirical case is presented to illustrate the remarkable difference when the TRMCO model is applied.

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Footnote

⁴ Previous applications of constrained optimization in political science tend to focus on the formal theory instead of numerical analysis. See Moe (1980) and Sorokin (1994). Recently, political scientists started working on numerical problems with constrained optimization. See Mebane and Sekhon (1998) and Mebane and Sekhon (2011).