On large-sample estimation and testing via quadratic inference functions for correlated data

TL;DR

This paper develops a quadratic inference function (QIF) approach for large-sample estimation and testing in correlated data, extending GMM principles to account for correlation structures with theoretical and algorithmic advancements.

Contribution

It formulates a covariance matrix for extended score functions, develops a unified large-sample theory, and proposes a new test statistic and algorithm for correlated data analysis.

Findings

QIF provides consistent and efficient estimates for correlated data.

The proposed test statistic follows the asymptotic distribution under null and alternative hypotheses.

Monte Carlo experiments show the impact of basis matrices on test size and power.

Abstract

Hansen (1982) proposed a class of "generalized method of moments" (GMMs) for estimating a vector of regression parameters from a set of score functions. Hansen established that, under certain regularity conditions, the estimator based on the GMMs is consistent, asymptotically normal and asymptotically efficient. In the generalized estimating equation framework, extending the principle of the GMMs to implicitly estimate the underlying correlation structure leads to a "quadratic inference function" (QIF) for the analysis of correlated data. The main objectives of this research are to (1) formulate an appropriate estimated covariance matrix for the set of extended score functions defining the inference functions; (2) develop a unified large-sample theoretical framework for the QIF; (3) derive a generalization of the QIF test statistic for a general linear hypothesis problem involving correlated data while establishing the asymptotic distribution of the test statistic under the null and local alternative hypotheses; (4) propose an iteratively reweighted generalized least squares algorithm for inference in the QIF framework; and (5) investigate the effect of basis matrices, defining the set of extended score functions, on the size and power of the QIF test through Monte Carlo simulated experiments.