Asymptotic properties of autoregressive integrated moving average processes

AbstractIn this paper we study the asymptotic behavior of so-called autoregressive integrated moving average processes. These processes constitute a large class of stochastic difference equations which includes among many other well-known processes the simple one-dimensional random walk. They were dubbed by G.E.P. Box and G.M. Jenkins who found them to provide useful models for studying and controlling the behavior of certain economic variables and various chemical processes. We show that autoregressive integrated moving average processes are asymptotically normally distributed, and that the sample paths of such processes satisfy a law of the iterated logarithm. We also establish a law which determines the time spent by a sample path on one or the other side of the “trend line” of the process

Rationality in Econometrics

The idea of rationality enters an econometrician's work in many ways; e.g., in his presuppositions about sample populations, in his model selections and data analyses, and in his choice of projects. I shall consider some of these ways and their ramifications for the econometrician's own life and for the development of econometrics. I begin with a discussion of rationality that I have found in the writings of Aristotle and other leading philosophers. My aim here is to establish the characteristics that we in good faith can expect rational members of a sample population to possess. The characteristics with which I end up have no definite meaning. Instead they are like undefined terms in mathematics that an econometrician can interpret in ways that suit the purposes of his research and seem appropriate for the population he is studying. When interpreted, the pertinent characteristics of the rational members of a given population become hypotheses whose empirical relevance must be tested. In rationally designed econometric studies the interpretation of 'rationality' that seems appropriate for a given study is usually an interpretation that the pertinent econometrician extracts from various economic theories. I look at some of these interpretations and discuss their empirical relevance. The interpretations of particular interest concern consumer choice under certainty, choice under risky and uncertain conditions, and choice in game-theoretic situations. These interpretations appear in various representations in the ways econometricians model rationality. I single out for discussion microeconometric models of consumer choice and macroeconometric rational expectations models. In the last section of the paper I consider two lacunas in Kuhn's and Lakatos' theories, and see how econometricians go about solving puzzles and extending positive heuristics. I begin by discussing the considerations that guide an econometrician in his choice of research projects. Then, I argue about the determinants of rational choice in model selection. Finally, I consider the politics of writing research reports. The contents of these sections concern aspects of an econometrician's rational choice that are relevant for the orderly development of econometrics.

Limiting distributions for explosive PAR(1) time series with strongly mixing innovation

This work deals with the limiting distribution of the least squares estimators of the coefficients a r of an explosive periodic autoregressive of order 1 (PAR(1)) time series X r = a r X r--1 +u r when the innovation {u k } is strongly mixing. More precisely {a r } is a periodic sequence of real numbers with period P \textgreater{} 0 and such that P r=1 |a r | \textgreater{} 1. The time series {u r } is periodically distributed with the same period P and satisfies the strong mixing property, so the random variables u r can be correlated

The Status of Bridge Principles in Applied Econometrics

The paper begins with a figurative representation of the contrast between present-day and formal applied econometrics. An explication of the status of bridge principles in applied econometrics follows. To illustrate the concepts used in the explication, the paper presents a simultaneous-equation model of the equilibrium configurations of a perfectly competitive commodity market. With artificially generated data I carry out two empirical analyses of such a market that contrast the prescriptions of formal econometrics in the tradition of Ragnar Frisch with the commands of present-day econometrics in the tradition of Trygve Haavelmo. At the end I demonstrate that the bridge principles I use in the formal-econometric analysis are valid in the Real World—that is in the world in which my data reside

Asymptotic properties of dynamic stochastic parameter estimates (III)

In this paper we establish three theorems concerning the asymptotic distributions of ordinary least-squares estimates of the parameters of a stochastic difference equation. We show that, if there is at least one root of the associated characteristic equation with modulus less than one and if all the roots have moduli different from one, the vector of least-squares estimates converges in distribution to a normally distributed vector. The distribution of the limiting vector is degenerate if there is at least one root with modulus greater than one. The results we obtain represent extensions of results proviously obtained by H. B. Mann and A. Wald, H. Rubin, J. S. White, T. W. Anderson, M. M. Rao, T. J. Muench, and the author.Asymptotic properties stochastic difference equation least square estimates