A Generalized Argmax Theorem with Applications

TL;DR

This paper generalizes the argmax theorem to allow maximization over converging subsets, simplifying proofs and enabling new results in structural break estimation, boundary parameter estimation, and weak identification.

Contribution

It introduces a generalized argmax theorem applicable to sequences of subsets, broadening its utility in econometric and statistical inference.

Findings

Simplifies proofs for existing estimation results.

Enables new theoretical results in structural break analysis.

Applicable to boundary and weakly identified parameters.

Abstract

The argmax theorem is a useful result for deriving the limiting distribution of estimators in many applications. The conclusion of the argmax theorem states that the argmax of a sequence of stochastic processes converges in distribution to the argmax of a limiting stochastic process. This paper generalizes the argmax theorem to allow the maximization to take place over a sequence of subsets of the domain. If the sequence of subsets converges to a limiting subset, then the conclusion of the argmax theorem continues to hold. We demonstrate the usefulness of this generalization in three applications: estimating a structural break, estimating a parameter on the boundary of the parameter space, and estimating a weakly identified parameter. The generalized argmax theorem simplifies the proofs for existing results and can be used to prove new results in these literatures.