Solving Problems in the Panel Regression Model for Truncated
Dependent Variables: A Constrained Optimization Method
3.3 Numerical Estimation
We apply the sequential quadratic programming (SQP) algorithm to solve a COP problem and derive parameter estimates. The idea of the SQP algorithm is to break down a complicated COP problem into a series of osculating quadratic problems (OQP) (Gilbert, 2009). Given that quadratic problems are much easier to solve, we can approach the optimal solution by solving the subproblems until we reach convergence throughout a sequence of iterations (Powell, 1978). To execute the SQP algorithm, we need to have information about the constraint vector, the first derivative matrix of the constraint vector, the gradient vector of the objective function, and the hessian matrix of the Lagrangian function. Beginning with giving the initial value,14 we will derive a new set of solutions by updating those four matrices, and then we will evaluate whether the convergence is achieved by checking the Karush-Kuhn-Tucker conditions (Kuhn and Tucker, 1951), which are necessary terms for an optimal solution to exist in nonlinear programming. If the solution reaches convergence, we stop the iteration process and report the optimal solution.15 If the maximum number of iterations runs out, we check the admissibility of the best available solution and report it.16 If no admissible solution is available, then we report the estimation as failed.
Supplementary material A and B explain the details of model specification, mathematical exposition, and simulation findings. According to previous research presented in these documents, applying constrained optimization to the modified truncated regression model exhibits unconditional superiority in terms of eliminating boundary violations and deriving the best admissible likelihood measure. Comparing the OLS method and the current truncated regression model, the regression model that incorporates constrained optimization can always find an admissible solution and effectively solves the out-of-bound violations, regardless of how they are empirically or theoretically defined.
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Footnote
14 The initial parameter values are set by the solution of the current panel regression.
15 We adopt the SQP algorithm suggested by Bonnans et al. (2006: 257).
16 Due to the large sample size, we reduce the maximum number of iterations to 31. The tolerance value applied to check the KKT conditions is set to $10^-4$. The step-size parameter is set to $\tau=50$.
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