Finite Sample Hypothesis Tests for Stacked Estimating Equations

TL;DR

This paper develops a bootstrap-based hypothesis test for stacked estimating equations that avoids reliance on asymptotic approximations by using sample splitting and multiple resampling, providing finite sample validity.

Contribution

It introduces a novel bootstrap hypothesis testing method for stacked estimating equations that is valid in finite samples, unlike traditional asymptotic approaches.

Findings

Bootstrap test performs well in finite samples

Sample splitting reduces variability in estimates

Limiting distribution matches that of stacked estimating equations

Abstract

Suppose there are two unknown parameters, each parameter is the solution to an estimating equation, and the estimating equation of one parameter depends on the other parameter. The parameters can be jointly estimated by "stacking" their estimating equations and solving for both parameters simultaneously. Asymptotic confidence intervals are readily available for stacked estimating equations. We introduce a bootstrap-based hypothesis test for stacked estimating equations which does not rely on asymptotic approximations. Test statistics are constructed by splitting the sample in two, estimating the first parameter on a portion of the sample then plugging the result into the second estimating equation to solve for the next parameter using the remaining sample. To reduce simulation variability from a single split, we repeatedly split the sample and take the sample mean of all the estimates. For parametric models, we derive the limiting distribution of sample splitting estimator and show they are equivalent to stacked estimating equations.