Efficient estimation for longitudinal data by combining large-dimensional moment conditions

Hyunkeun Cho,Annie Qu

doi:10.1214/15-ejs1036

Abstract

The quadratic inference function approach is able to provide a consistent and efficient estimator if valid moment conditions are available. However, the QIF estimator is unstable when the dimension of moment conditions is large compared to the sample size, due to the singularity problem for the estimated weighting matrix. We propose a new estimation procedure which combines all valid moment conditions optimally via the spectral decomposition of the weighting matrix. In theory, we show that the proposed method yields a consistent and efficient estimator which follows an asymptotic normal distribution. In addition, Monte Carlo studies indicate that the proposed method performs well in the sense of reducing bias and improving estimation efficiency. A real data example of Fortune 500 companies is used to compare the performance of the new method with existing methods.

Highlights

Longitudinal data arise frequently in many studies where repeated measurements from a subject are correlated
The generalized estimating equation (GEE) provides a consistent estimator regardless of whether the working correlation is correctly specified or not, the estimator can be inefficient under misspecified correlation structures. [13] developed the quadratic inference function (QIF) based on the generalized method of moments [8] to achieve better estimation efficiency
We show that the proposed criterion is able to select the number of principal components consistently, when the sample size goes to infinity

Summary

Introduction

Longitudinal data arise frequently in many studies where repeated measurements from a subject are correlated. We apply a spectral decomposition of the covariance matrix for the moment conditions and select an optimal number of linear combinations of the moment conditions through a new objective function based on a Bayesian information type of criterion [14]. The proposed method performs well in the sense of reducing bias and improving the efficiency of QIF estimation, and is especially effective when the dimension of moment conditions is high compared to the sample size. It is capable of incorporating a set of preselected moment conditions, in conjunction with selecting the optimal linear combinations of remaining moment conditions. All proofs of the lemmas and theories are provided in the Appendix

Notation and preliminaries

QIF with large-dimensional moment conditions

Methodology

Asymptotic properties

Implementation with unbalanced data

Continuous responses

25 QIFI GEEEX

Binary responses

Fortune 500 data example

Discussion