David M. Grether

Price, Quality and Timing of Moves in Markets with Incomplete Information: An Experimental Analysis

This paper reports the results of theoretical and experimental analyses of search models of shopping behavior. Earlier work which dealt with markets for homogeneous goods in which sellers first determined their prices and buyers then decided upon their information acquisition (shopping) strategies conditional upon the distribution of prices showed that Nash equilibria for these markets were good predictions of outcomes of laboratory experiments. The authors extend the analysis to markets with heterogeneous goods and allow buyers and sellers to move simultaneously. The results of a series of laboratory experiments indicate that the Nash equilibria continue to be good predictions in these more realistic settings. Copyright 1992 by Royal Economic Society.

Analysis of Economic Time Series.

Analysis of Economic Time Series: A Synthesis integrates several topics in economic time-series analysis, including the formulation and estimation of distributed-lag models of dynamic economic behavior; the application of spectral analysis in the study of the behavior of economic time series; and unobserved-components models for economic time series and the closely related problem of seasonal adjustment. Comprised of 14 chapters, this volume begins with a historical background on the use of unobserved components in the analysis of economic time series, followed by an Introduction to the theory of stationary time series. Subsequent chapters focus on the spectral representation and its estimation; formulation of distributed-lag models; elements of the theory of prediction and extraction; and formulation of unobserved-components models and canonical forms. Seasonal adjustment techniques and multivariate mixed moving-average autoregressive time-series models are also considered. Finally, a time-series model of the U.S. cattle industry is presented. This monograph will be of value to mathematicians, economists, and those interested in economic theory, econometrics, and mathematical economics.

THE SPECTRAL REPRESENTATION AND ITS ESTIMATION

This chapter presents two ways of introducing the spectral density function for stationary stochastic processes. First is a development based upon the Wold decomposition theorem and the autocorrelations of a time series and second shows how the spectral distribution function of a stationary time series can be derived from the representation of such a process in terms of random variables defined in the frequency domain rather than in the time domain. The chapter discusses the canonical factorization of the covariance generating function. The autocovariance generating function converges on the unit circle and in an annulus surrounding it. Stationary processes that have both a moving-average and an autoregressive representation are called invertible. One way to view the requirement of invertibility is as a condition for identifiability in the usual econometric sense. The chapter also discusses the spectral representation of a stationary time series. It is not necessary to assume stationarity for the development of the spectral representation but only covariance or weak stationarity. It is convenient to work with complex-valued time series until the last moment and then to make the necessary specialization.

FORMULATION AND ANALYSIS OF UNOBSERVED-COMPONENTS MODELS

This chapter discusses the formulation and analysis of unobserved-components models. It discusses how unobserved-components models, which capture much of the flavor of those used by economic statisticians of the 19th and early 20th centuries, may be formulated by superimposing simple mixed moving-average autoregressive models with independent white noise inputs. If a process is invertible, that is, if the process has a purely autoregressive representation, the autocovariance generating function has no zeros on the unit circle. If the generating function of a sequence is an analytic function in some region, then the infinite series form represents the Laurent expansion of that function in the region. Thus, the autocovariances of a zero-mean stationary purely linearly nondeterministic time series are the coefficients in a Laurent series expansion of the autocovariance generating function in the annulus about the unit circle in which the series converges. Therefore, Cauchys integral formula and the residue theorem may be used to evaluate these autocovariances in a simple fashion for time series having a rational spectral density.

Chapter XI – FORMULATION AND ESTIMATION OF MULTIVARIATE MIXED MOVING-AVERAGE AUTOREGRESSIVE TIME-SERIES MODELS

Publisher Summary This chapter outlines a method for formulating and estimating multiple time-series models for time series that are known from other considerations to be related to one another. It presents a single-equation approach to multiple time-series model formulation and estimation, in which the past and current values of other series are supposed to affect the behavior of the series to be forecast. In general, all tests used for autoregressive integrated moving-average (ARIMA) models can be used to check the models. Some of the ways to check the quality of the fitted model are (1) the correlograms for the innovations, (2) a χ2 test of the hypothesis that these correlations are jointly equal to zero, (3) the covariance generating function implied by the parameters compared with that implied by the sample, and (4) the spectral density of the theoretical model compared with the actual spectral density. To obtain forecasts based on a simultaneous-equations model, one must first specify the nature of the relationships among the variables and the innovations involved. Therefore, it is necessary to discuss the problems of specifying the orders of the polynomials.

Chapter II – INTRODUCTION TO THE THEORY OF STATIONARY TIME SERIES

Publisher Summary This chapter discusses the theory of stationary time series. Time-series analysis is based on two fundamental notions: the idea of unobserved components and a more probabilistic theory based on parametric models. The idea of unobserved components not only lies behind the traditional decomposition of an economic time series into three or four components but is also the central idea in the harmonic analysis of time series. In this type of analysis, the time series, or some simple transformation of it, is assumed to be the result of the superposition of sine and cosine waves of different frequencies. The chapter discusses the Wold decomposition theorem. The object of Wolds theorem is to show how an arbitrary stationary time series can be decomposed into two parts. The Wold decomposition divides the process into an ergodic and a linearly deterministic part. It follows from Wolds theorem that any purely nondeterministic process can be written as a one-sided moving average with a white noise input. This representation is extremely important in spectral theory and in the theory of optimal prediction and extraction.

Chapter VIII – APPRAISAL OF SEASONAL ADJUSTMENT TECHNIQUES

Publisher Summary This chapter discusses the problem of seasonal adjustment of economic time series. It presents several criteria for optimal seasonal adjustment. The use of these criteria lead to relationships between the spectral densities of adjusted and unadjusted series similar to those found between estimated spectra using time series adjusted by official census and the bureau of labor statistics (BLS) methods. The chapter discusses the implications for the parameter estimates of using seasonally adjusted data when fitting econometric models. The problem of seasonal adjustment arises in three different contexts: in historical studies of business cycles, in appraising current economic conditions, and in estimating structural parameters in relationships among economic time series. The decision to use adjusted or unadjusted data in the estimation of economic models can have substantial effects upon the estimates of the structural parameters. There has been some discussion in the literature of such consequences of seasonal adjustment. One of the difficulties in appraising methods for eliminating seasonal variation from economic times series is the lack of precise criteria by which to rank various alternative procedures.

Chapter VI – FORMULATION OF UNOBSERVED-COMPONENTS MODELS AND CANONICAL FORMS1

Publisher Summary This chapter discusses the formulation of unobserved-components (UC) models, their canonical forms, and the ordinary mixed autoregressive moving-average (ARMA) models for both single and multiple time series. The first step in the estimation of either an ARMA model or an UC model is the determination of the orders of the moving averages and autoregressions as well as the number of components in a UC model, which is itself made up of superimposed ARMA models. This process is called model identification. Identification problems in the ordinary econometric sense may arise in the estimation of both ARMA and UC models. The chapter discusses the form of ARMA models using the estimated autocorrelation and partial autocorrelation functions, as suggested by Box and Jenkins. It also describes the formulation of time-series models relating two or more interdependent time series. The formulation of a time-series model means the determination of the appropriate degrees of the polynomials in the lag operators that appear in both the autoregressive part and the moving-average part of an ARMA model.

Chapter XII – FORMULATION AND ESTIMATION OF UNOBSERVED-COMPONENTS MODELS: EXAMPLES

Publisher Summary This chapter discusses the formulation and estimation of unobserved-components models. All models are estimated in both time and frequency domains; however, there had always been convergence problems in the time domain and only a two-component model lacking an irregular component was estimated. The differencing employed in fitting an autoregressive integrated moving-average (ARIMA) model to the data is equivalent to removing a polynomial of appropriate degree corresponding to the order of the difference so that removal of a polynomial trend in fitting an unobserved-components model does not result in an unfair comparison with the former. The theoretical spectral densities of the estimated models are compared with the spectral densities estimated for the actual detrended series.

Chapter V – ELEMENTS OF THE THEORY OF PREDICTION AND EXTRACTION

Publisher Summary This chapter discusses the elements of minimum-mean-square-error prediction and extraction when the true model generating a time series or several such series is known. The white noise inputs in a moving average representation of a purely nondeterministic covariance stationary process are the one-step prediction errors. The chapter discusses the problem of signal extraction. It discusses the theory of least-squares forecasting for nondeterministic covariance stationary processes. The chapter presents a number of examples to illustrate the basic theory and to show convenient methods for calculating and updating the forecasts. For finite autoregressive processes, there is a simple recursive procedure for calculating forecasts for any period in the future. In addition to the coefficients of the autoregression, the only data that are required to forecast a pth-order autoregression are the p most recently observed values of the time series. Moreover, for moving-average processes in general, the forecasts cannot be expressed in terms of a finite number of the observed values of the process.