Two-step estimation in linear regressions with adaptive learning

TL;DR

This paper develops a two-step estimation method for linear regressions with adaptive learning, addressing the asymptotic properties and distributional challenges when the gain parameter is estimated in a preliminary step.

Contribution

It introduces a novel two-step estimation approach that accounts for the uncertainty in the first-step gain parameter estimation in adaptive linear regression models.

Findings

The two-step estimator is weakly consistent and asymptotically normal.

The limiting distribution is singular and influenced by the first step's sampling uncertainty.

Under certain conditions, the generated-regressor issue is eliminated.

Abstract

Weak consistency and asymptotic normality of the ordinary least-squares estimator in a linear regression with adaptive learning is derived when the crucial, so-called, `gain' parameter is estimated in a first step by nonlinear least squares from an auxiliary model. The singular limiting distribution of the two-step estimator is normal and in general affected by the sampling uncertainty from the first step. However, this `generated-regressor' issue disappears for certain parameter combinations.