On the asymptotic optimality of the LIML estimator with possibly many instruments

doi:10.1016/j.jeconom.2009.12.001

Journal of Econometrics

Volume 157, Issue 2, August 2010, Pages 191-204

https://doi.org/10.1016/j.jeconom.2009.12.001 Get rights and content

Abstract

We consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables. We derive some asymptotic properties of the limited information maximum likelihood (LIML) estimator when the number of instruments is large; some of these results are new as well as old, and we relate them to results in some recent studies. We have found that the variance of the limiting distribution of the LIML estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed, that is, for the micro-econometric models of some cases recently called many instruments and many weak instruments.

Introduction

Over the past three decades there has been increasing interest and research on the estimation of a single structural equation in a system of simultaneous equations when the number of instruments (the number of exogenous variables excluded from the structural equation), say $K_{2}$ , is large relative to the sample size, say $n$ . The relevance of such models is due to the collection of large data sets and the development of computational equipment capable of analysis of such data sets. One empirical example of this kind often cited in econometric literature is Angrist and Krueger (1991); there has been some discussion by Bound et al. (1995) since then. Asymptotic distributions of estimators and test criteria are developed on the basis that both $K_{2} \to \infty$ and $n \to \infty$ . These asymptotic distributions are used as approximations to the distributions of the estimators and criteria when $K_{2}$ and $n$ are large.

Bekker (1994) has written “To my knowledge a first mention of such a parameter sequence was made, with respect to the linear functional relationship model, in Anderson (1976, p. 34). This work was extended to simultaneous equations by Kunitomo, 1980, Kunitomo, 1982 and Morimune (1983), who gave asymptotic expansions for the case of a single explanatory endogenous variable”. Following Bekker there have been many studies of the behavior of estimators of the coefficients of a single equation when $K_{2}$ and $n$ are large.

The main purpose of the present paper is to show that one estimator, the Limited Information Maximum Likelihood (LIML) estimator, has some optimum properties when $K_{2}$ and $n$ are large. As background we state and derive some asymptotic distributions of the LIML and Two-Stage Least Squares (TSLS) estimators as $K_{2} \to \infty$ and $n \to \infty$ . Some of these results are improvements on Kunitomo, 1981, Kunitomo, 1982, Morimune (1983) and Bekker (1994), Chao and Swanson (2005), van Hasselt (2006), Hansen et al. (2008), and they are presented in a unified notation.

In addition to the LIML and TSLS estimators there are other instrumental variables (IV) methods. See Anderson et al. (1982) on the earlier studies of the finite sample properties, for instance. Several semiparametric estimation methods have been developed including the generalized method of moments (GMM) estimation and the empirical likelihood (EL) method. (See Hayashi, 2000 for instance.) However, it has been recently recognized that the classical methods have some advantages in microeconometric situations with many instruments.

In this paper we shall give the results on the asymptotic properties of the LIML estimator when the number of instruments is large and we develop the large- $K_{2}$ asymptotic theory or the many instruments asymptotics including the so-called case of many weak instruments. The TSLS and the GMM estimators are badly biased and they lose even consistency in some of these situations. Our results on the asymptotic properties and optimality of the LIML estimator and its variants give new interpretations of the numerical information of the finite sample properties and some guidance on the use of alternative estimation methods in simultaneous equations and micro-econometric models with many weak instruments. There is a growing amount of literature on the problem of many instruments in econometric models. We shall try to relate our results to some recent studies, including Donald and Newey (2001), Hahn (2002), Stock and Yogo (2005), Chao and Swanson, 2005, Chao and Swanson, 2006, van Hasselt (2006), van der Ploeg and Bekker (unpublished), Bekker and van der Ploeg (2005), Chioda and Jansson (unpublished), Hansen et al. (2008), and Anderson et al. (forthcoming).

In Section 2 we state the formulation of a linear structural model and the alternative estimation methods of unknown parameters with possibly many instruments. In Section 3 we develop the large- $K_{2}$ asymptotics (or many instruments asymptotics) and give some results on the asymptotic normality of the LIML estimator when $n$ and $K_{2}$ are large. Then we shall present some results on the asymptotic optimality of the LIML estimator in the sense that it attains the lower bound of the asymptotic variance in a class of consistent estimators with many instruments under reasonable assumptions. Also we discuss a more general formulation of the models and the related problems. In Section 4 we show that the asymptotic results in Section 3 agree with the finite sample properties of estimators. Then brief concluding remarks will be given in Section 5. The proof of our theorems will be given in Section 6.

Section snippets

Alternative estimation methods in structural equation models with possibly many instruments

We first consider the estimation problem of a structural equation in the classical linear simultaneous equations framework.¹ Let a single linear structural equation in an econometric model be $y_{1 i} = β_{2}^{'} y_{2 i} + γ_{1}^{'} z_{1 i} + u_{i} (i = 1, \dots, n),$ where $y_{1 i}$ and $y_{2 i}$ are a scalar and a vector of $G_{2}$

Asymptotic normality of the LIML estimator

We state the limiting distribution of the LIML estimator under a set of alternative assumptions when $K_{2 n}$ and $Π_{2 n}$ can depend on $n$ and $n \to \infty$ . We first consider the case when $(I) \frac{K_{2 n}}{n} ⟶ c (0 \leq c < 1),$ $(II) \frac{1}{n} Π_{22}^{{(n)}^{'}} A_{22.1} Π_{22}^{(n)} \overset{p}{⟶} Φ_{22.1},$ where $Φ_{22.1}$ is a nonsingular constant matrix.

Condition (I) implies that the number of coefficient parameters is proportional to the number of observations. Because we want to estimate the covariance matrix of $v_{i}^{(n)} (i = 1, \dots, n)$ , we want $c < 1$ . Then (I) implies $q_{n} ⟶ \infty$ as $n ⟶ \infty$ . Condition

Discussion of asymptotic properties and finite sample properties

It is important to investigate the finite sample properties of estimators partly because they are not necessarily similar to their asymptotic properties. One simple example would be the fact that the exact moments of some estimators do not necessarily exist. (In that case it is meaningless to compare the exact MSEs of alternative estimators and their Monte Carlo analogues.) Although we discuss the asymptotic properties of the LIML estimator, we need to investigate their relevance for practical

Concluding remarks

In this paper, we have discussed the asymptotic optimality when the number of instruments is large in a structural equation of the simultaneous equations system. Although the limited information maximum likelihood (LIML) estimator and the two stage-least squares (TSLS) estimator are asymptotically equivalent in the standard large sample theory, they are asymptotically quite different in the large- $K_{2}$ asymptotics with many instruments or many weak instruments. In some recent microeconometric

Proof of theorems

In this section we give the proofs of Theorems and the mathematical derivation in Section 3.

Proof of Theorem 1

Substitution of (2.2) into (2.4) yields $G = (Π_{n}^{'} Z^{'} + V^{'}) Z_{2.1} A_{22.1}^{- 1} Z_{2.1}^{'} (Z Π_{n} + V) = Π_{2 n}^{'} A_{22.1} Π_{2 n} + V^{'} Z_{2.1} A_{22.1}^{- 1} Z_{2.1}^{'} V + Π_{2 n}^{'} Z_{2.1}^{'} V + V^{'} Z_{2.1} Π_{2 n} .$ Then $G - [Π_{2 n}^{'} A_{22.1} Π_{2 n} + K_{2 n} Ω] = Π_{2 n}^{'} Z_{2.1}^{'} V + V^{'} Z_{2.1} Π_{2 n} + [V^{'} Z_{2.1} A_{22.1}^{- 1} Z_{2.1}^{'} V - K_{2 n} Ω] .$ Condition (II) implies that as $n ⟶ \infty$ $\frac{1}{n} Π_{2 n}^{'} Z_{2.1}^{'} V \overset{p}{⟶} O,$ and $\frac{1}{n} [V^{'} Z_{2.1} A_{22.1}^{- 1} Z_{2.1}^{'} V - K_{2 n} Ω] \overset{p}{⟶} O .$ Then as $n ⟶ \infty$ , $\frac{1}{n} G \overset{p}{⟶} G_{0} = [\begin{matrix} β_{2}^{'} \\ I_{G_{2}} \end{matrix}] Φ_{22.1} (β_{2}, I_{G_{2}}) + c Ω$ and $\frac{1}{q_{n}} H \overset{p}{⟶} Ω .$ Then ${\hat{β}}_{L I} \overset{p}{\to} β$ and $λ_{n} \overset{p}{\to} c$ as $n \to \infty$ .

Define $G_{1}, H_{1}, λ_{1 n}$ , and $b_{1}$

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For example, the two stage least squares (TSLS) estimator tends to have large bias. Although the limited information maximum likelihood (LIML) estimator is less biased, its distribution is often more dispersed than the limiting distribution based on the conventional asymptotics (see, e.g., Anderson et al., 1982, 2010, 2011). We impose the following assumptions on the reduced form.
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^☆: AKM09-11-23-2. This is a revised first part of Discussion Paper CIRJE-F-321 under the title “A New Light from Old Wisdoms: Alternative Estimation Methods of Simultaneous Equations and Microeconometric Models” (Graduate School of Economics, University of Tokyo, February 2005) which was presented at the Econometric Society World Congress 2005 at London (August 2005). We thank Yoichi Arai, the co-editor and the referees of this journal for some comments to the earlier versions.

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On the asymptotic optimality of the LIML estimator with possibly many instruments☆

Abstract

Introduction

Section snippets

Alternative estimation methods in structural equation models with possibly many instruments

Asymptotic normality of the LIML estimator

Discussion of asymptotic properties and finite sample properties

Concluding remarks

Proof of theorems

Journal of Econometrics

Journal of Multivariate Analysis

Journal of Multivariate Analysis

Estimation of linear functional relationships: Approximate distributions and connections to simultaneous equations in econometrics

Journal of the Royal Statistical Society, B

Estimating linear statistical relationships

Annals of Statistics

An Introduction to Multivariate Statistical Analysis

Evaluation of the distribution function of the limited information maximum likelihood estimator

Econometrica

Estimation of the parameters of a single equation in a complete system of stochastic equations

Annals of Mathematical Statistics

The asymptotic properties of estimates of the parameters of a single equation in a complete system of stochastic equation

Annals of Mathematical Statistics

Does compulsory school attendance affect schooling and earnings

Quarterly Journal of Economics

Alternative approximations to the distributions of instrumental variables estimators

Econometrica

Instrumental variables estimation based on group data

Statistica Neerlandica

Problems with instrumental variables estimation when the correlation between the instruments and the endogenous explanatory variables is weak

Journal of the American Statistical Association

Consistent estimation with a large number of weak instruments

Econometrica

Asymptotic normality of single-equation estimators for the case with a large number of instruments