Paul A. Bekker

ALTERNATIVE APPROXIMATIONS TO THE DISTRIBUTIONS OF INSTRUMENTAL VARIABLE ESTIMATORS

This paper considers the instrument variable and method-of-moments estimators, as generalizations of two-stage least squares and limited information maximum likelihood, of the coefficients of a single equation. Motivated by a simple principle, asymptotic distributions are derived based on a parameter sequence where both the number of instruments and the sample size increase. The approximations to the distributions provided by this sequence are more accurate than traditional ones. The instrument variable estimator appears to be inconsistent. The asymptotic covariance matrix of the method-of-moments estimator can be consistently estimated under the alternative parameter sequence. The resulting approximate confidence regions have exact levels very close to their nominal levels. Copyright 1994 by The Econometric Society.

The Rank of Reduced Dispersion Matrices.

Psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements. Psychometricians concentrate on cases in which low rank can be attained, preferably rank one, the Spearman case. Econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one, the Frisch case. In this paper we give an extensive historial discussion of both fields, we prove the two key results in a more satisfactory and uniform way, we point out various small errors and misunderstandings, and we present a methodological comparison of factor analysis and regression on the basis of our results.

Identification of linear stochastic models with covariance restrictions

Abstract The purpose of this paper is to provide a systematic treatment of the problem of identification in systems of linear structural equations where some of the disturbances are uncorrelated.

IDENTIFICATION IN RESTRICTED FACTOR MODELS AND THE EVALUATION OF RANK CONDITIONS

Abstract This paper presents a rank condition for the local identification in restricted factor analysis models and it provides a method for the computation of the exact rank in the condition. The method can be applied to evaluate the rank of rather general Jacobian matrices.

FINITE-SAMPLE INSTRUMENTAL VARIABLES INFERENCE USING AN ASYMPTOTICALLY PIVOTAL STATISTIC

The paper considers the K-statistic, Kleibergen’s (2000) adaptation of the Anderson-Rubin (AR) statistic in instrumental variables regression. Compared to the AR-statistic this K-statistic shows improved asymptotic efficiency in terms of degrees of freedom in overidenti?ed models and yet it shares, asymptotically, the pivotal property of the AR statistic. That is, asymptotically it has a chi-square distribution whether or not the model is identi?ed. This pivotal property is very relevant for size distortions in ?nite-sample tests. Whereas Kleibergen (2000) focuses especially on the asymptotic behavior of the statistic, the present paper concentrates on finite-sample properties in a Gaussian framework. In that case the AR statistic has an F-distribution. However, the K-statistic is not exactly pivotal. Its finite-sample distribution is affected by nuisance parameters. Here we consider the two extreme cases, which provide tight bounds for the exact distribution. The first case amounts to perfect identification —which is similar to the asymptotic case—where the statistic has an F-distribution. In the other extreme case there is total underidentification. For the latter case we show how to compute the exact distribution. Thus we provide tight bounds for exact con?dence sets based on the efficient K-statistic. Asymptotically the two bounds converge, except when there is a large number of redundant instruments.

THE POSITIVE SEMIDEFINITENESS OF PARTITIONED MATRICES

Abstract The positive semidefiniteness of a partitioned matrix is characterized in terms of its submatrices. The result is applied to a variety of problems concerning Lowner ordered matrices, which need not be partitioned themselves.

A note on the identification of restricted factor loading matrices

It is shown that problems of rotational equivalence of restricted factor loading matrices in orthogonal factor analysis are equivalent to problems of identification in simultaneous equations systems with covariance restrictions. A necessary (under a regularity assumption) and sufficient condition for local uniqueness is given and a counterexample is provided to a theorem by J. Algina concerning necessary and sufficient conditions for global uniqueness.

Measurement Error and Endogeneity in Regression: Bounds for ML and 2SLS Estimates

We consider the single equation errors in variables model and assume that a researcher is willing to specify upper bounds on the possible measurement errors in the exogenous variables. We prove that as a result the set of possible ML estimates is bounded by an ellipsoid. The result is generalized to IV estimation of a structural equation of a simultaneous system, which has only endogenous variables on the right hand side.

The APT Model as Reduced-Rank Regression

Integrating the two steps of an arbitrage pricing theory (APT) model leads to a reduced-rank regression (RRR) model. So the results on RRR can be used to estimate APT models, making estimation very simple. We give a succinct derivation of estimation of RRR, derive the asymptotic variance of RRR estimators for a general case, and discuss how undersized samples (more assets than time periods) can be dealt with.

Some clarifications of the TUCKALS2 algorithm applied to the IDIOSCAL problem

Kroonenberg and de Leeuw have suggested fitting the IDIOSCAL model by the TUCKALS2 algorithm for three-way components analysis. In theory, this is problematic because TUCKALS2 produces two possibly different coordinate matrices, that are useless for IDIOSCAL unless they are equal. Kroonenberg has claimed that, when IDIOSCAL is fitted by TUCKALS2, the resulting coordinate matrices will be identical. In the present paper, this claim is proven valid when the data matrices are semidefinite. However, counterexamples for indefinite matrices are also constructed, by examining the global minimum in the case where the data matrices have the same eigenvectors. Similar counterexamples have been considered by ten Berge and Kiers in the related context of CANDECOMP/PARAFAC to fit the INDSCAL model.