Estimating multi-way error components models with unbalanced data structures

Abstract

I develop simple matrix algebra techniques that simplify and unify much of the previous literature on estimating error components models (ECMs). In fact, the simple analytic results provided here are useful for analyzing a very broad set of models with complex error structures. To illustrate the techniques, I develop the algebra for three- and four-way ECMs explicitly. In addition, I provide Monte Carlo simulation evidence on the performance of several estimators for the three-way ECM and estimate the model using data from a retail market where the three dimensions of data variation are products selling in many locations over time.

Introduction

The error components model (ECM) is one of the most frequently used econometric models for panel data. Work by Balestra and Nerlove (1973), Wallace and Hussain (1969), Amemiya (1971), Nerlove 1971a, Nerlove 1971b, Mazodier (1972), and Fuller and Batese (1974) among others, provide a comprehensive analysis of the one- and two-way ECM for balanced panels. In practice, researchers often have to deal with missing observations. Consequently, Baltagi (1985) and Wansbeek and Kapteyn (1989) (henceforth WK), respectively, developed the one- and two-way ECM for unbalanced data.

In a related literature, the nested ECM has also been developed. Most recently, Baltagi et al. (2001) and Antweiler (2001), respectively, develop the two- and three-way nested ECM for unbalanced panels, building on earlier work by Fuller and Battesse (1973). The nested ECM has proven particularly useful for datasets with a natural grouping structure. For example, data on firms may be grouped by industry.

The aim of this paper is to extend the literature in a number of directions. First, I provide a series of mathematical results which allow an elegant analysis of the ECM for unbalanced panel data with an arbitrary number of error components.¹ I illustrate the results by developing estimators for the three-way ECM in the text while results for the four-way model are presented in the appendix.² By developing the paper's results as recurrence relations, the extensions to the five- or more-way ECMs are easily developed, while the one- and two-way ECMs are observed as special cases.

Second, I show that the standard and nested ECMs have the same mathematical structure and therefore the results I present can be applied to both. In fact, the results are useful for estimating a broad array of models with complex error structures. In each case, modeling the correlation structure among the errors is important because it allows efficient estimation of the parameters of interest and consistent estimates of the standard errors of the parameters. Naturally, the latter is crucially important for performing inference correctly.

Third, whenever possible, I relax WK's assumption that all regressors are exogenous, thereby allowing estimation of the general ECM using instrumental variables. Applied demand analysis motivated these extensions of the literature and this aspect of the results is particularly important in that and other contexts.³

The paper proceeds as follows: 2 The fixed effects model, 3 The random effects model develop the multi-way fixed- and random-effects models, respectively, for given estimates of the variance of each error component. In Section 4 quadratic unbiased estimators (QUEs), minimum norm quadratic unbiased estimators (MINQUEs) and maximum likelihood estimators (MLEs) of the variance of each error component are developed. Section 5 provides Monte Carlo evidence on estimators of the three-way ECM. In Section 6, I apply the results by estimating a differentiated product demand equation using a new dataset from a retail market. Section 7 concludes.