Arnold Zellner

An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias

Abstract In this paper a method of estimating the parameters of a set of regression equations is reported which involves application of Aitkens generalized least-squares [1] to the whole system of equations. Under conditions generally encountered in practice, it is found that the regression coefficient estimators so obtained are at least asymptotically more efficient than those obtained by an equation-by-equation application of least squares. This gain in efficiency can be quite large if “independent” variables in different equations are not highly correlated and if disturbance terms in different equations are highly correlated. Further, tests of the hypothesis that all regression equation coefficient vectors are equal, based on “micro” and “macro” data, are described. If this hypothesis is accepted, there will be no aggregation bias. Finally, the estimation procedure and the “micro-test” for aggregation bias are applied in the analysis of annual investment data, 1935–1954, for two firms.

An introduction to Bayesian inference in econometrics

Remarks on Inference in Economics. Principles of Bayesian Analysis with Selected Applications. The Univariate Normal Linear Regression Model. Special Problems in Regression Analysis. On Errors in the Variables. Analysis of Single Equation Nonlinear Models. Time Series Models: Some Selected Examples. Multivariate Regression Models. Simultaneous Equation Econometric Models. On Comparing and Testing Hypotheses. Analysis of Some Control Problems. Conclusion. Appendices. Bibliography. Indexes.

Three-Stage Least Squares: Simultaneous Estimation of Simultaneous Equations

In simple though approximate terms, the two-stage least squares method of estimating a structural equation consists of two steps, the first of which serves to estimate the moment matrix of the reduced-form disturbances and the second to estimate the coefficients of one single structural equation after its jointly dependent variables are “purified” by means of the moment matrix just mentioned. The three-stage least squares method, which is developed in this paper, goes one step further by using the two-stage least squares estimated moment matrix of the structural disturbances to estimate all coefficients of the entire system simultaneously. The method has full-information characteristics to the extent that, if the moment matrix of the structural disturbances is not diagonal (that is, if the structural disturbances have nonzero “contemporaneous” covariances), the estimation of the coefficients of any identifiable equation gains in efficiency as soon as there are other equations that are over-identified. Further, the method can take account of restrictions on parameters in different structural equations. And it is very simple computationally, apart from the inversion of one big matrix.

Bayesian Estimation and Prediction Using Asymmetric Loss Functions

Abstract Estimators and predictors that are optimal relative to Varians asymmetric LINEX loss function are derived for a number of well-known models. Their risk functions and Bayes risks are derived and compared with those of usual estimators and predictors. It is shown that some usual estimators, for example, a scalar sample mean or a scalar least squares regression coefficient estimator, are inadmissible relative to asymmetric LINEX loss by providing alternative estimators that dominate them uniformly in terms of risk.

ESTIMATORS FOR SEEMINGLY UNRELATED REGRESSION EQUATIONS: SOME EXACT FINITE SAMPLE RESULTS

Abstract The finite sample properties of an asymptotically efficient technique (JASA, June, 1962) for estimating coefficients in certain generally encountered sets of regression equations are studied in this paper. In particular, exact first and second moments of the asymptotically efficient coefficient estimator are derived and compared with those of the usual least squares estimator. Further, the exact probability density function of the new estimator is derived and studied as a function of sample size. It is found that the approach to asymptotic normality is fairly rapid and that for a wide range of conditions an appreciable part of the asymptotic gain in efficiency is realized in samples of finite size.

Posterior odds ratios for selected regression hypotheses

SummaryBayesian posterior odds ratios for frequently encountered hypotheses about parameters of the normal linear multiple regression model are derived and discussed. For the particular prior distributions utilized, it is found that the posterior odds ratios can be well approximated by functions that are monotonic in usual sampling theoryF statistics. Some implications of this finding and the relation of our work to the pioneering work of Jeffreys and others are considered. Tabulations of odds ratios are provided and discussed.

Bayesian and Non-Bayesian Analysis of the Regression Model with Multivariate Student- t Error Terms

Abstract The linear multiple regression model is analyzed assuming the error vector has a multivariate Student-t distribution with zero location vector and scalar dispersion matrix; the multivariate Cauchy and normal distributions are special cases. It is found that the usual least squares coefficient estimate is the maximum likelihood estimate and the mean of the posterior distribution under a diffuse prior distribution. Inferences based on usual t- and F-statistics are shown valid for a range of error distributional assumptions including the multivariate-t assumption. Inferences about the scale parameter of the multivariate-t distribution can be made using an F-distribution rather than the usual χ2 (or inverted χ2) distribution.

Estimating Gross Labor-Force Flows

We present and apply an adjustment procedure for the Bureau of the Census and Bureau of Labor Statistics gross labor-force flows data that addresses two major defects in the data. First, an adjustment procedure is developed to take account of individuals with missing labor-force classifications who are not missing at random. Second, we provide a procedure for adjustment for individuals with spurious labor-force transitions arising because of classification errors in either the current or the previous Current Population Survey. Our procedures are applied to compute adjusted monthly gross change data for the period January 1977–December 1982. The average adjustment for nonrandom missing classifications ranges from –12% to 15% of the unadjusted gross change data. The average adjustment for spurious labor-force transitions reduces estimated movements by 8%–49%. The classification adjustment also increases estimated consecutive periods of unemployment by 18%. We apply several internal and external consistency ...

Optimal Information Processing and Bayes's Theorem

Abstract In this article statistical inference is viewed as information processing involving input information and output information. After introducing information measures for the input and output information, an information criterion functional is formulated and optimized to obtain an optimal information processing rule (IPR). For the particular information measures and criterion functional adopted, it is shown that Bayess theorem is the optimal IPR. This optimal IPR is shown to be 100% efficient in the sense that its use leads to the output information being exactly equal to the given input information. Also, the analysis links Bayess theorem to maximum-entropy considerations.

Prediction and Decision Problems in Regression Models from the Bayesian Point of View

Abstract In this paper we review the derivation of the predictive density function for the normal multiple regression model, state and prove a general theorem on optimal point prediction, and show how the predictive density can be employed in the analysis of an illustrative investment problem. Then we derive the predictive density function for the multivariate normal regression model and indicate how it can be used in the analysis of several problems.