Richard E. Quandt

The Estimation of the Parameters of a Linear Regression System Obeying Two Separate Regimes

Abstract In attempting to estimate the parameters of a linear regression system obeying two separate regimes, it is necessary first to estimate the position of the point in time at which the switch from one regime to the other occurred. The suggested maximum likelihood estimating procedure is based upon a direct examination of the likelihood function. An asymptotic and a small-sample test are suggested for testing the hypothesis that no switch occurred against the single alternative that one switch took place. The procedure is illustrated with a sampling experiment in which the true switching point is correctly estimated. * I am indebted to Professors F. Anscombe, R. Dorfman and F. Stephan and the referees of this paper for criticism and helpful suggestions. I am also indebted to Professor John S. Chipman for originally suggesting the problem. The responsibility for errors is, of course, mine.

TESTS OF THE HYPOTHESIS THAT A LINEAR REGRESSION SYSTEM OBEYS TWO SEPARATE REGIMES

Abstract Several approaches are explored for testing the hypothesis that no switch has occurred in the true values of the parameters of a linear regression system. The distribution of the relevant likelihood ratio λ is analyzed on the basis of the empirical distribution resulting from some sampling experiments. The hypothesis that −2 log λ has the χ2 distribution with the appropriate degrees of freedom is rejected and an empirical table of percentage points is obtained. Finally some small sample tests are suggested.

Some Tests for Homoscedasticity

Abstract Two exact tests are presented for testing the hypothesis that the residuals from a least squares regression are homoscedastic. The results can be used to test the hypothesis that a linear [ratio] model explains the relationship between variables as opposed to the alternative that the ratio [linear] specification is correct. The first test is parametric and uses the F-statistic. The second test is nonparametric and uses the number of peaks in the ordered sequence of unsigned residuals. In conclusion, the results of some experimental calculations of the powers of the tests are discussed.

A New Approach to Estimating Switching Regressions

Abstract In recent years much attention has been focussed on the problem of discontinuous shifts in regression regimes at unknown points in the data series. This article approaches this problem by assuming that nature chooses between regimes with probabilities λ and 1 — λ. This allows formulation of the appropriate likelihood function maximized with respect to the parameters in the regression equations and λ. The method is compared to another recent procedure in some sampling experiments and in a realistic economic problem and is found satisfactory.

Estimating Mixtures of Normal Distributions and Switching Regressions

Abstract Since the likelihood function corresponding to finite mixtures of normal distributions is unbounded, maximum likelihood estimation may break down in practice. The article introduces the “moment generating function estimator” defined as the estimator which minimizes the sum of squares of differences between the theoretical and sample moment generating functions. The consistency and asymptotic normality of the estimator are proved and its finite sample behavior is compared to that of the standard method of moments estimator by Monte Carlo experiments. The estimator is applied to the Hamermesh model of wage bargain determination.

Racetrack betting and informed behavior

Abstract Horse racing data permit interesting tests of attitudes toward risk. The present paper studies a new sample of racetrack results from Atlantic City, New Jersey. The questions examined are: (1) Are the market odds the best data for predicting the order of finish? (2) Do horses go off at odds that reflect their true probability of winning? (3) Is there any evidence that late bettors have better information than early bettors? It is found that market odds predict the order of finish well, but that ‘favorites’ are good bets and ‘long shots’ are poor ones. The data suggest that there does exist an ‘informed’ class of bettors and that bettors are on the whole neither risk neutral nor risk averse.

Estimation of Disequilibrium Aggregate Labor Market

AN important question in contemporary 1AILeconomics is whether or not the real wage clears the labor market. Its answer has bearing on issues as diverse as the nature of unemployment, the efficacy of fiscal and monetary policies, and the incidence of income taxes. Unfortunately, consensus as to the correct answer seems to be lacking. While much of modern macroeconomic theory allows for the possibility that the real wage fails to equate the supply and demand of labor (Barro and Grossman, 1971; Korliras, 1975), much analysis is based on the assumption of equilibrium in the labor market (Patinkin, 1965). The purpose of the present paper is to carry out an econometric test for which view of the labor market is more appropriate. Although the model we build is very aggregative and much too crude to be used as a basis for policy, we believe that it provides a first step in making operational the theoretical literature on disequilibrium macro models. Our tentative conclusion is that the hypothesis of a labor market in continuous equilibrium must be rejected. In section II we describe briefly some earlier work on modelling the aggregate supply and demand for labor. It is shown that prior studies either assume equilibrium in the labor market, or deal with disequilibrium inadequately. In section III we specify the disequilibrium model. Section IV contains a discussion of estimation problems, an interpretation of the results, and a comparison with an equilibrium version of the model. A concluding section has a summary and an agenda for future research. II. Antecedents

Econometric disequilibrium models

Four basic strands in the disequilibrium literature are identified. Some examples are discussed and the canonical econometric disequilibrium model and its estimation are dealt with in detail. Specific criticisms of the canonical model,dealing with price and wage rigidity, with the nature of the min condition and the price-adjustment equation, are considered and a variety of modifications is entertained. Tests of the “equilibrium vs. disequilibrium” hypothesis are discussed, as well as several classes of models that may switch between equilibrium and disequilibrium modes. Finally, consideration is given to multimarket disequilibrium models with particular emphasis on the problems of coherence and estimation.

A Probabilistic Theory of Consumer Behavior

I. The state of affairs, 507. — II. The deficiency of the Complete Ordering Axiom, 508. — III. Redefinition of preference and the choice mechanism, 510. — IV. Intransitivity, 518. — V. Expected behavior under price and income changes, 525. — VI. Conclusion, 536.

COMPUTATIONAL PROBLEMS AND METHODS

Publisher Summary This chapter discusses some of the important computational methods and problems. The emphasis is on algorithms and general procedures for solving problems. The chapter discusses matrix methods involved in estimating the parameters of single and simultaneous equation models. It discusses various aspects of numerical optimization. These methods become relevant whenever the first-order conditions for a maximum are not linear in the parameters to be estimated. The chapter presents a survey of the typical functions that are optimized and discusses the basic theory of optimization. Special purpose algorithms and simplifications useful in econometrics; and some further aspects of algorithms are discussed. The chapter discusses particular difficulties encountered only in problems of certain types and focuses on numerical integration and random number generation.