Simultaneous Equation Estimation Research Articles

On the specification of structure-performance relationships

This paper argues that most estimated structure-profitability equations in industrial economics have not been derived from a single coherent view of the behaviour of firms. The independent variables in such equations have often been derived on the basis of a literature search and by appeal to conflicting approaches (e. g. constrained and unconstrained profit maximisation). The use of a coherent approach indicates that structure-performance equations are likely to be highly non-linear, and that simultaneous equation estimation is neither required nor appropriate. In some cases single-equation estimation is adequate, and in other cases reduced form equations are needed.

Recursive estimation of simultaneous equation models

A recursive instrumental variable estimator is derived. For simultaneous equation estimation, the choice of the instruments is discussed. A computationally simple and asymptotically efficient recursive estimator is proposed in this context.

Local Tax and Expenditure Capitalization: Integrating Evidence From the Market and Political Processes

Numerous empirical studies have attempted to verify the Tiebout hypothesis through analysis of the capitalization of differential tax rate and expenditure levels into property values at the local level. A major weakness in this literature is the neglect of voting, the other principal method of preference revelation for local public services. This article seeks to integrate the analysis of the outcomes of voting and mobility decisions by linking a model of residential property value determination and a public choice model of local public expenditure determination. Capitalization of fiscal differentials is shown to affect voting decisions through the tax price of local services. The interrelationship creates a justification for simultaneous equations estimation of both models. An illustrative empirical analysis is presented in which residential property value and local public expenditure models are estimated with ordinary and two-stage least square techniques. Substantial differences are found between the two techniques supporting the hypothesis that significant interrelationships exist between mobility and voting outcomes in metropolitan areas.

The structure of simultaneous equations estimators: A comment

This comment presents a generalised derivation of the maximum likelihood estimator of the parameters in a simultaneous equation model which incorporates an important class of models not included in Hendry (1976), namely those which have singular error covariance matrices. In so doing it conveniently brings under the scope of Hendry's analysis those models associated with systems of demand equations, models with identities and the joint structural and reduced form formulations proposed by Court (1974).

A Monte Carlo Study of Small-Sample Properties of Simultaneous Equation Estimators with Normal and Nonnormal Disturbances

Abstract In this paper we consider four alternative forms of two-parameter normal and nonnormal error distributions and report on a Monte Carlo study of the small-sample properties of least squares, two-stage least squares, three-stage least squares and full information maximum likelihood estimators. On the basis of 1,000 replications of sample size 20 in two experiments on an overidentified model, we found that the small-sample rankings of econometric estimators of both structural coefficients and forecasts of endogenous variables, according to parametric and nonparametric measures of bias, dispersion, and dispersion including bias, do not change for any of the four error distributions. Further, least squares is the most biased and maintains the Gauss-Markov property of minimum variance. The large bias of least squares, however, more than offsets the small variance, so that least squares exhibits the largest mean squares of the four estimators. Maximum likelihood is least biased and most efficient, excep...

A ridge-like method for simultaneous estimation of simultaneous equations

In the context of full information estimation in a linear simultaneous equations model, this paper considers a ridge-like modification of the 3SLS estimator. The proposed method is particularly desirable where the square matrix of the 3SLS normal equationsis singular (or near-singular) leading to non-existence (or poor performance) of the estimator. Furthermore, the type of solution suggested here does seem to result in the existence of the finite sample moments of the estimator even when the degrees of over identification are as low as zero (just identified models). This paper considers only a simple scalar form of the ‘ridge-matrix” with a relatively simple choice of the modifying scalar that preserves the asymptotic properties of the 3SLS estimator. A value of this scalar is derived which minimizes an appropriatequadratic risk criterion. The approximate quadratic risk function is based upon the asymptotic approximation of the relevant moments in the manner of Nagar (1959). A range of risk reducing values of the ‘ridge-scalar” is also given.

Simultaneous equation estimation based on maximum entropy moments

A simulation study is made of three k-class estimators based alternatively on conventional sample moments or on maximum entropy (ME) moments. The most promising estimator is LIML based on ME moments which uses all predetermined variables of the system as instrumental variables.

Empirical Criminology

Statistical results based on aggregate data are, in part, artifacts of discre tionary research activity. In demonstrating this, the author examines the rela tionships between theoretical constructs and empirical concepts and between empirical concepts and statistical proxies, as well as the sensitivity of statistical results to the selection of control variables, the structure of the explanatory model (single v. multiple equation systems), and the mode of esti mation (ordinary least squares v. a simultaneous equation estimation tech nique). Further, aggregation can produce heteroskedastic disturbances, heter oskedasticity produces biased statistical results, and there is no general pro cedure for eliminating aggregation bias. Improper correction for bias can pro duce misleading results. The author proposes that, since empirical research is far from pure, it behooves empiricists to develop a formal metastatistical methodology.

A model of inflation and expectations in Latin America

This paper attempts to overcome certain shortcomings in the existing tests of various theories of inflation and cyclical income growth. It does so by appending a simple model of expectations formation to a simple macroeconomic simultaneous equation model of aggregate supply and demand. Some of the theoretical building blocs are reformulated so as to be more realistic for developing countries. Simultaneous equation estimation techniques are applied to a pooled sample of time series data for 16 Latin American countries. The authors provide empirical evidence supporting the revised hypotheses and offering new insights into the relationships among the actual and expected rates of inflation, the rate of income growth, and the growth rate of the money supply.

Small-Sample Properties of Simultaneous Equation Estimators with Multicollinearity

Abstract A Monte Carlo simulation is employed to reinvestigate the general disagreement in the literature regarding the small-sample properties of simultaneous equation estimators under widely different levels of multicollinearity. This disagreement has been attributed to the greater use by more complex estimators of information on variables excluded from each equation, producing the advantage in inference of increasing precision at low levels of collinearity, but the disadvantage of greater estimator sensitivity with higher collinearity. Although simulation results indicate improvement of some less complex techniques relative to their more complex counterparts as collinearity rises, extreme collinearity is required to generate substantial reversals in relative performance.

Foundations of Econometrics.

Advanced Textbooks in Economics, Volume 7: Foundations of Econometrics focuses on the principles, processes, methodologies, and approaches involved in the study of econometrics. The publication examines matrix theory and multivariate statistical analysis. Discussions focus on the maximum likelihood estimation of multivariate normal distribution parameters, point estimation theory, multivariate normal distribution, multivariate probability distributions, Euclidean spaces and linear transformations, orthogonal transformations and symmetric matrices, and determinants. The manuscript then ponders on linear expected value models and simultaneous equation estimation. Topics include random exogenous variables, maximum likelihood estimation of a single equation, identification of a single equation, linear stochastic difference equations, and errors-in-variables models. The book takes a look at a prolegomenon to econometric model building, tests of hypotheses in econometric models, multivariate statistical analysis, and simultaneous equation estimation. Concerns include maximum likelihood estimation of a single equation, tests of linear hypotheses, testing for independence, and causality in economic models. The publication is a valuable source of data for economists and researchers interested in the foundations of econometrics.

The Asking Wage-Duration of Unemployment Relation Revisited

The purpose of this paper is to specify and estimate a model of the short-run asking wage behavior of unemployed individuals. The level of the asking wage is presumed to affect the probability of reemployment, and therefore the duration of unemployment. Duration is expected to influence the level of the asking wage through economic and social pressures. Therefore, a two-equation model is presented in which both the level of the asking wage and duration of unemployment are endogenous. The first section provides a brief reference to papers in which others have discussed the asking wage-duration (hereafter AW-D) relation. The simultaneous equation system is introduced in a second section. After a brief description in section three of the source of data, we compare the single equation and simultaneous equation estimates of the AW-D relation.

A note on simultaneous equation estimation techniques applications with a regional econometric model

A large-scale regional econometric model is estimated using six estimation techniques, including Iterated Instrumental Variables and Iterated Two-Stage Least-Squares. Following estimation the model is simulated and a seventh technique called PANGLOSS is derived. The seven techniques are then compared in ex post and ex ante tests.

A Program for Econometric and Spectral Analysis-EAS

and FORTRAN IV which is designed to accommodate most researchers' everyday econometric needs. However, this program is particularly useful when spectral methods are combined (in an ex post sense) with regression and simultaneous equations estimation., Residuals from regression or simultaneous equations estimation are easily saved by EAS and used in later spectral computations by using only two program statements.1 All spectral computations are highly efficient since the fast Fourier transform techniques developed by Cooley and Tukey [2] are used throughout. The program also allows algebraic expressions to be used directly in regression statements to define dependent or independent variables. Hence, special regression equations like the harmonic analysis model can be easily estimated with a single program statement. A partial enumeration of the program's capabilities is as follows: ordinary and weighted least squares regression; multivariate regression and estimation of Zellner's seemingly unrelated regression system; structural estimation of simultaneous equations by two-stage least squares, three-stage least squares, limited information maximum likelihood, k-class, double k-class, h-class, and Nagar's minimum bias k-class methods; power spectrum analysis, cross spectrum analysis, and simple frequency domain regression; random number generation and Monte Carlo methods; principal components analysis; estimation of partially nonlinear models by likelihood search techniques; and estimation of certain distributed lag models by Dhrymes' [3] methods. The program accommodates problems in which the sample size and number of yariables do not exceed 32,767 and 1,823, respectively, but the dynamic core allocation features of PL/1 are used to economize on all smaller problems. The design of the program is especially useful when a large data bank (subject to the abQve limitations) is to be maintained, updated, and periodically accessed for various types of econometric analyses. Most small- to medium-sized

Measurement in the Social Sciences: Theories and Strategies.

Among the frustrations constantly confronting the social scientist are those associated with the general process of measurement. The importance of good measurement has long been recognized in principle, but it has often been neglected in practice in many of the social sciences. Now that the methodological tools of multivariate analysis, simultaneous-equation estimation, and causal modeling are diffused more widely into the social sciences, and now that the very serious implications of random and non-random measurement errors are being systematically investigated, it is all the more important that social scientists give top priority to the quality of their data and the clarity of their theoretical conceptualizations. The book is organized so that, one proceeds from problems of data collection to those of data analysis. It is not intended to be a complete work covering all types of measurement problems that have arisen in the social sciences. Instead, it represents a series of studies that are deemed to be crucial for the advancement of social science research but which have not received sufficient attention in most of the social sciences. The basic purpose is to stimulate further methodological research on measurement and to study the ways in which knowledge that has been accumulated in some fields may be generalized. Part I is concerned with applying scaling approaches developed in psychometrics to problems that arise in other social sciences. The focus is on finding better ways to ask questions of respondents so as to raise the level of measurement above that of simple ordinal scales. Part II focuses on multiple-indicator theory and strategies as applied to relatively complex models and to change data. In this section the emphasis shifts to how one analyzes fallible data through the construction of explicit measurement-error models. Part III deals with the statistical analysis of ordinal data, including the interpretation and empirical behaviors of various ordinal measures of association.

Information and computation in simultaneous equations estimation

The complexity and size of simultaneous equations systems necessitates great care with computations for parameter estimation. In three-stage least-squares (3SLS) large matrix inversions are required, and because of the sensitivity of many economic systems to key parameters, accuracy in estimation is important. There are many numerical techniques available which yield accurate solutions to systems of equations. We make use of Householder transformations and recursive triangulation solutions in presenting numerical algorithms for the computation of 3SLS and k-class estimates. Another numerical technique, the singular value decomposition is valuable in providing additional information in k-class estimation. The values of k for which this estimator does not exist are accurately derived, their use being demonstrated by an example.

EFFICIENT ESTIMATION OF MULTIVARIATE FINANCIAL RELATIONSHIPS

RECENTLY Simkowitz and Jones [14] have noted the potential importance of the tools of multivariate analysis and simultaneous equation estimators of econometrics as tools for finance methodology. They note that empirical research in finance dealing with such variables as the value of financial securities has used, almost exclusively, cross sectional analysis . . where all firm effects other than those described by the independent variables are assumed away.' More extensive estimation efforts require either the use of other multivariate techniques, such as cannonical correlation (for correlating sets of variables) or more explicit modeling of the structure of the relationships between, say, different firms' securities' prices before the simultaneous equations estimators can be successfully employed. The purpose of this paper is to propose an approach that is intermediate between these extremes but which will allow us to gauge the importance of unexplained joint effects as between different securities' prices. Additionally, it provides a means of extending the methods proposed by Fried [3] for estimating single asset return equations as an input to the Markowitz model. The technique we propose is an Aitken estimator for a system of multivariate linear relationships which need not be theoretically linked. It was originally introduced by Zellner [19] and has been successfully employed in the joint estimation of many relationships in economics, including production functions (Moroney and Ferguson [10]), the determinants of state and local expenditures (Smith and Fibiger [17]) and several others. We feel it holds great promise for the many problems in finance. In addition to the estimation of different firms' securities prices equations over time,' we might consider the problem of estimating such functions for each time period across firms, thereby examining the sensitivity of the specified dimensions (i.e., cross section and time series) of a pooled data set.3 Section II of this paper describes the technique with specific reference made to finance problems. It is followed by an application which considers the

DISCRETE VARIATIONS ACROSS SUBSETS OF PARAMETERS IN SIMULTANEOUS EQUATIONS MODELS

AbstractDummy variables have proven to be an analytically convenient and empirically productive means of handling econometric problems relating to seasonality, structural change and time series and cross section data pooling. Despite the advantageous of the approach there are few results regarding its use in simultaneous equation estimation. In this paper the problem of discrete parameter variations in simultaneous equation estimation is systematically investigated. The estimation procedures proposed are based on extensions of existing methods for seemingly unrelated regressions and two and three‐strage least squares. In addition to obtaining the desired estimators and their properties a number of convenient advantages resulting from the use of dummy variables to treat parameter variations across subsets of simultaneous equations are indicated.

The Small Sample Properties of Selected Econometric Estimators in the Context of Alternative Macro-Models

techniques, has in the past ignored the fundamental importance of the frame of reference for that evaluation. While Cragg [3, 4, 5, 6] has noted that the specific model used in a given small sample study has an effect upon the observed properties of the estimators, there has been little systematic testing of this proposition.2 The problem is particularly important in view of the diversity of econometric models.3 A decision rule derived from studies which implicitly assume the neutrality of simultaneous equation estimators across economic structures would appear to be of little value. The purpose of this paper is to commence an inquiry into the effect of econometric model upon estimator evaluation. Three single equation estimators have been chosen. The design of the experiments performed in this study precludes the use of full model estimators.4 The three techniques studied were: ordinary least squares applied to the structural form (DLS = direct least squares), two stage least square (2SLS) and limited information single equation estimates (LISE). The evaluation of these estimators will be undertaken with each of three macro-econometric models. The performance of the techniques will be noted in each frame of reference individually. The overall conclusions of each experiment will then be considered in conjunction with the characteristics of the models used. The three models used are: Model I from Klein's Economic Fluctuations in the United States (KF), a linearized version of the Klein-Goldberger Model of the United States (KG) and the Dutta-Su Model of Puerto Rico (DS).5 The paper is partitioned into four sections after these introductory remarks. The first of these, Section 2, discusses the primary characteristics of each of the three models. Section 3 outlines the design of each of the Monte Carlo experiments. It is followed by a discussion of the results derived from each. Finally, the last section specifies some tentative conclusions which might be postulated on the basis of the results.

Efficient Estimation of Simultaneous Equations with Auto-Regressive Errors by Instrumental Variables

T HE purpose of this paper is to point out how the efficient instrumental-variables technique discussed by Brundy and Jorgenson (1971) can be modified to take into account auto-regressive properties of the error terms. The limited-information and full-information estimators proposed in this paper are consistent and have the same asymptotic distributions as the limited-information and full-information maximum likelihood estimators, respectively. The full-information estimation of simultaneous equations models with auto-regressive errors has been discussed by Sargan (1961), Hendry (1971), Chow and Fair (1973), and Dhrymes (1971). Sargan originally proposed the full-information maximum likelihood estimation of such models, and Hendry and Chow and Fair have recently developed computationally feasible methods for obtaining the maximum likelihood estimates. Hendry considered only the case of completely unrestricted auto-regressive coefficient matrices (i.e., no zero elements), whereas Chow and Fair considered the case of restricted auto-regressive coefficient matrices as well. Dhrymes has recently proposed the three-stage least squares estimator of simultaneous equations models with auto-regressive errors. Dhrymes also considered only the case of completely unrestricted auto-regressive coefficient matrices. The limited-information estimation of simultaneous equations models with auto-regressive errors has been discussed by Sargan (1961), Amemiya (1966), and Fair (1970), among others. Sargan proposed the limited-information maximum likelihood estimation of such models, and Amemiya and Fair considered various two-stage least souares estimators of such models. Most of the work on limited-information estimators has been concerned with the case of diagonal auto-regressive coefficient matrices. Brundy and Jorgenson's criticism of the twoand three-stage least squares estimators, namely, that the first stage involves estimating reduced form equations with a very large number of variables included in them, holds even more so for models with auto-regressive errors. For these models, the reduced form equations include not only all of the predetermined variables in the system but also all of the lagged endogenous and lagged predetermined variables. In fact, one of the main purposes of the work by Fair (1970) was to suggest ways in which the number of variables used in the first stage regressions of two-stage least squares might be decreased with perhaps small loss of asymptotic efficiency. The advantage of the instrumental-variables techniques proposed in the Brundy-Jorgenson paper and in this paper is that the first stage regressions need not be run.