Implicit Profiling Estimation for Semiparametric Models with Bundled Parameters

TL;DR

This paper introduces an implicit profiling algorithm for semiparametric models that combines the efficiency of iterative methods with the optimality of Newton's method, improving computational speed and estimation accuracy.

Contribution

The paper proposes a novel implicit profiling algorithm that efficiently handles infinite-dimensional parameters in semiparametric models, enhancing convergence speed and estimation precision.

Findings

Demonstrated computational efficiency in semiparametric transformation models.

Achieved high statistical precision in GARCH-M models.

Solved local quadratic programming problems in two steps.

Abstract

Solving semiparametric models can be computationally challenging because the dimension of parameter space may grow large with increasing sample size. Classical Newton's method becomes quite slow and unstable with intensive calculation of the large Hessian matrix and its inverse. Iterative methods separately update parameters for finite dimensional component and infinite dimensional component have been developed to speed up single iteration, but they often take more steps until convergence or even sometimes sacrifice estimation precision due to sub-optimal update direction. We propose a computationally efficient implicit profiling algorithm that achieves simultaneously the fast iteration step in iterative methods and the optimal update direction in the Newton's method by profiling out the infinite dimensional component as the function of the finite dimensional component. We devise a first order approximation when the profiling function has no explicit analytical form. We show that our implicit profiling method always solve any local quadratic programming problem in two steps. In two numerical experiments under semiparametric transformation models and GARCH-M models, we demonstrated the computational efficiency and statistical precision of our implicit profiling method.