Linear Simultaneous Equations Model Research Articles

Estimation with weak instruments: Accuracy of higher‐order bias and MSE approximations

In this paper, we consider parameter estimation in a linear simultaneous equations model. It is well known that two‐stage least squares (2SLS) estimators may perform poorly when the instruments are weak. In this case 2SLS tends to suffer from the substantial small sample biases. It is also known that LIML and Nagar‐type estimators are less biased than 2SLS but suffer from large small sample variability. We construct a bias‐corrected version of 2SLS based on the Jackknife principle. Using higher‐order expansions we show that the MSE of our Jackknife 2SLS estimator is approximately the same as the MSE of the Nagar‐type estimator. We also compare the Jackknife 2SLS with an estimator suggested by Fuller (Econometrica 45, 933–54) that significantly decreases the small sample variability of LIML. Monte Carlo simulations show that even in relatively large samples the MSE of LIML and Nagar can be substantially larger than for Jackknife 2SLS. The Jackknife 2SLS estimator and Fuller's estimator give the best overall performance. Based on our Monte Carlo experiments we conduct informal statistical tests of the accuracy of approximate bias and MSE formulas. We find that higher‐order expansions traditionally used to rank LIML, 2SLS and other IV estimators are unreliable when identification of the model is weak. Overall, our results show that only estimators with well‐defined finite sample moments should be used when identification of the model is weak.

Pre-test estimation in Poisson regression model

Pre-test estimation has been studied extensively for linear regression and simultaneous equation models. Recently attention has turned to pre-test estimation in non-linear models. This article studies pre-test maximum likelihood estimation in Poisson regression model. It presents its risk characteristics and compare them with those of restricted and unrestricted maximum likelihood estimators based on squared error loss function in a Monte Carlo experiment.

The curse of non-investment grade countries

Some of the most influential theories of contagion are those based on liquidity arguments. Nevertheless, very little evidence exists on their importance. This paper shows that liquidity-based stories are responsible for one-fourth of the observed correlation in sovereign debt. Mexico was upgraded from non-investment to investment grade in March of 2000. This paper examines the impact of this event on the properties of the transmission of shocks between Mexico and several Latin American countries. The paper shows that there is a statistically significant change in the propagation of shocks around the time the upgrade was announced. Moreover, the change in the estimated coefficients can explain between one-fourth and a third of the unconditional co-movement that these assets experienced before the upgrade. This is strong evidence in favor of the theories of contagion based on market structure as the source of the propagation mechanism. From the methodological point of view, the paper extends existing identification procedures that solve the problem of estimation in linear simultaneous equations models. The methodology can be used in other Macro and Finance applications where problems of simultaneous equations represent important limitations in the estimation.

OPTIMAL INFERENCE WITH MANY INSTRUMENTS

In this paper, I derive the efficiency bound of the structural parameter in a linear simultaneous equations model with many instruments. The bound is derived by applying a convolution theorem to Bekker's (1994, Econometrica 62, 657–681) asymptotic approximation, where the number of instruments grows to infinity at the same rate as the sample size. Usual instrumental variables estimators with a small number of instruments are heuristically argued to be efficient estimators in the sense that their asymptotic distribution is minimal. Bayesian estimators based on parameter orthogonalization are heuristically argued to be inefficient.

Consistent estimation of the random structural coefficient distribution from the linear simultaneous equations system

For the linear simultaneous equations model (LSEM) with random structural coefficients, it is sometimes of interest to identify the distribution of those structural coefficients. It is shown that the structural parameter distribution can be identified if the distribution of the exogenous variables is reasonably smooth.

Data mining and the selection of instruments

Instrumental variables estimation is widely applied in econometrics. To implement the method, it is necessary to specify a vector of instruments. In this paper, it is argued that there are compelling reasons to use the data for instrument selection, but that it is desirable to ensure the resulting estimator still behaves in the way predicted by standard textbook theory. These arguments lead one to propose three criteria for data based instrument selection. The remainder of the paper assesses the extent to which these criteria are met by two algorithms for data based instrument selection. The first algorithm is the method of structurally ordered instrumental variables proposed in the context of economy-wide linear simultaneous equation models. The second algorithm is proposed in the context of the method of generalized instrumental variables, which is commonly used to estimate the parameters of Euler equation models.

Efficient estimation in the linear simultaneous equations model with vector autoregressive disturbances

This paper obtains the score vector, the information matrix and the asymptotic Cramer-Rao lower bound for the parameters of the linear simultaneous equations model with vector autoregressive disturbances. The Cramer-Rao lower bound for the covariance matrix of a consistent estimator of the parameters of primary interest is obtained for the case where the autoregressive coefficients are known and for the case where these coefficients are unknown, thus enabling us to demonstrate how the lack of knowledge of the autoregressive coefficients increases the bound. These lower bounds are used then to evaluate the asymptotic efficiency of various estimators proposed in the literature.

Robust estimators for simultaneous equations models

This paper presents a class of robust estimators for linear and non-linear simultaneous equations models, which are a direct generalization of the maximum likelihood estimator. The new estimators are obtained as solutions of a generalized likelihood equation. They are resistant to deviations from the model distribution, to outlying observations, and to some model misspecifications. An optimality principle leads to the construction of an optimal robust estimator which is the best trade-off between efficiency at the model and robustness.

Robust estimators for simultaneous equations models

This paper presents a class of robust estimators for linear and non-linear simultaneous equations models, which are a direct generalization of the maximum likelihood estimator. The new estimators are obtained as solutions of a generalized likelihood equation. They are resistant to deviations from the model distribution, to outlying observations, and to some model misspecifications. An optimality principle leads to the construction of an optimal robust estimator which is the best trade-off between efficiency at the model and robustness.

Asymptotically Optimal Tests Using Limited Information and Testing for Exogeneity

By appropriately partitioning the joint hypothesis of weak exogeneity and the maintained overidentifying restrictions in the linear dynamic simultaneous equations model and showing that the component subhypotheses are separable, asymptotically optimal tests for the weak exogeneity hypothesis may be constructed using limited information statistics. A necessary and sufficient condition for the separability of parametric hypotheses of the mixed implicit function and constraint equation type is derived which generalizes conditions previously obtained in the literature. Consequently, limited and full information procedures for testing the weak exogeneity hypothesis are asymptotically equivalent. The impact of these results for testing strong exogeneity in the linear dynamic simultaneous equations model is also explored.

A systems approach to the calibration of deterministic dynamic nonlinear simultaneous equation models with incomplete data

AbstractThis paper develops an interactive three–stage systems approach for the calibration of the structural parameters and missing data within a deterministic, dynamic non–linear simultaneous equations model under arbitrary configurations of incomplete data. In Stage One, we minimize a quadratic loss function in the differences between the actual endogenous variables and the predicted solution values, relative to any feasible choice of the structural parameters. Missing exogenous variables and initial endogenous variables are treated as additional parameters to be calibrated; whereas missing current endogenous variables are treated by the missing data updating condition, in which the current solution values iteratively and sequentially replace those absent. Stage One may or may not lead to unique calibrations of the structural parameters—a fact that can be monitored a posteriori using singular value decompositions of the relevant Jacobian matrix. If not, there is an equivalence class of parameter values, all of which result in the same loss function value. If Stage Two is necessary, we attempt to exploit the non–linearity and simultaneity of the structural system to extract further information about the parameters from the same database, by minimizing the distance between the restricted and unrestricted reduced forms, while constraining the parameters also to lie within the Stage One equivalence class. This requires the use of higher–order numerical derivatives, and probably restricts its use in all but the simplest of cases to the next generation of supercomputers with massive numbers of parallel processors and much larger word–sizes. In Stage Three, various methods by which the original structural model can be simplified, given a non–unique Stage One calibration, are entertained.

Testing for structural change in a linear simultaneous equations model under limited information

The testing of structural change in the linear simultaneous equations model is considered under limited information on a single structural equation. Four testing problems are formulated according to the state of structural change in other structural equations, and appropriate likelihood functions and large sample locally most powerful tests for each problem are presented. The Lagrange multiplier and likelihood ratio (or asymptotically equivalent) tests based on the concentrated likehood function with the well known limited information maximum likehood (or two-stage least-squares) estimator as the maximum likelihood (or its asymptotically equivalent) estimator under null hypotheses are shown invalid for testing structural change of a single structural equation. However, the Wald test is shown effective and convenient for testing structural change of a single structural equation.

A monte carlo study of collinearity in linear simultaneous equation models∗

Although the collinearity issue has been studied in previous simulation studies with a simultaneous system of equations, alternative estimators to circumvent this problem have received little attention. Monte Carlo techniques are used to examine the performance of several estimators under a squared error loss criterion. In particular, this study considers the Vinod–Ullah ridge-type estimators at the first and/or second stage of 2SLS. Ridge regression in the second stage only of 2SLS but not the first stage only, seems to be a practical alternative to 2SLS, especially in situations of strong collinearity. The OLS estimator and the ordinary ridge regression estimator also yield favorable results in situations of moderate to strong collinearity.

Gains by the common structural variance

Consider a linear simultaneous equations model with a structural change. We present gains by the common structural variance in identification and asymptotic efficiency of the limited information maximum likelihood (LIML) estimator of structural coefficients in two regimes. The gains are obtained by comparing two LIML estimators derived with and without the common structural variance restriction.

Classical tests for contemporaneously uncorrelated disturbances in the linear simultaneous equations model

This paper obtains the score vector, the information matrix, and the Cramer-Rao lower bound for the parameters of the linear simultaneous equations model. These concepts are then used to construct the Lagrange multiplier test statistic and the Wald test statistic for the null hypothesis that the disturbances of the model are contemporaneously uncorrelated. A straightforward procedure is given for forming the Lagrange multiplier test statistic.

Testing inequality constraints in linear econometric models

This paper develops three asymptotically equivalent tests for examining the validity of imposing linear inequality restrictions on the parameters of linear econometric models. First we consider the model y = Xβ + ε, where ε is N(0, Σ), and the hypothesis test H: Rβ≥ r versus K: β ϵ R K . Later we generalize this testing framework to the linear simultaneous equations model. We show that the joint asymptotic distribution of these test statistics and the test statistics from the hypothesis test H: Rβ = r versus K: Rβ ≥ r is a weighted sum of two sets of independent χ 2-distributions. We also derive a useful duality relation between the multivariate inequality constraints test developed here and the multivariate one-sided hypothesis test. In small samples, these three test statistics satisfy inequalities similar to those derived by Berndt and Savin (1977) for the case of equality constraints. The paper also contains an illustrative application of this testing technique.

Computational aspects of robust estimators for linear regressions

This paper presents estimation procedures for some robust regression methods: the Bounded-Influence estimator for both a single linear equation (Krasker and Welsch, 1982) and a linear simultaneous equation model (Krasker and Welsch, 1985); the linear version of the Huber estimator for both a single equation (Huber, 1973, 1981) and a simultaneous equations model. The procedures are written in the RATS econometric language, which is widely available on microcomputers and mainframes.

A comparison of structured modeling and GAMS

In recent years a number of integrated modeling systems have been developed which combine modeling languages with database capabilities to greatly increase the productivity of linear programming, nonlinear programming, and simultaneous equation model builders (Palmer, 1984, Meeraus, 1983, Geoffrion, 1986). Among these systems is GAMS which was developed at the World Bank by Alexander Meeraus. Alongside the development of these systems Arthur Geoffrion has created a general framework for model development which he calls ‘Structured Modeling’.

Preliminary Data Errors and Their Impact on the Forecast Error of Simultaneous-Equations Models

Econometric models, especially when designed for forecasting purposes, tend to use updated economic series, the last figures(s) of which embed first-published or preliminary data errors. This article identifies and assesses the contribution of errors in preliminary data to the forecast error and to the forecast error variance of linear dynamic simultaneous equation models. The effect of preliminary data errors is shown to be pervasive, although not necessarily weighty. The suggested decomposition of the forecast error is applied to a small macroeconometric model of the Italian economy.

Reduced form estimation and prediction from uncertain structural models

Abstract This paper proposes a direct hierarchical model for the analysis of the reduced forms (RF) of the classical linear simultaneous equations models (SEM). The mean of the prior p.d.f. on RF coefficients is partially assessed with the aid of the traditionally exact a priori restrictions. Its variance may be partially specified to reflect degrees of model uncertainty and reliability of inferences that are asymptotically valid. Bayes or ‘maximum joint probability’ estimates are derived for both the RF and the ‘structural’ parameters. One variant of the RF estimator is shown to have desirable limiting and finite sample properties. It performs well in an empirical example which incorporates some elements of the two real-life problems of concern to us: simultaneous presence of different types of misspecification and small sample sizes.