Knowledge that Transforms

OVER THE LAST DECADE the problem of measurement errors in the independent variables of a regression equation has attracted renewed interest among econometricians. In the fifties and sixties, the problem was considered to be more or less hopeless due to its inherent underidentification (e.g., Theil (1971)). Apart from instrumental variables, the most frequently cited textbook solution was Wald's method of grouping (Wald (1940)). Recent insight into the properties of the method of grouping can be interpreted as making this method worthless in most practical cases (Pakes (1982)). Since about 1970, new approaches to the problem have been explored, basically along three lines, viz. embedding the error-ridden equation into a set of multiple equations (e.g., Zellner (1970), Goldberger (1972)), into a set of simultaneous equations (e.g., Hsiao (1976), Geraci (1976)), and using the dynamics of the equation, if present (e.g., Maravall and Aigner (1977)). In view of the underidentification of the basic model, it is clear that all these methods invoke additional information of some kind. If this information takes the form of exact or stochastic knowledge of certain parameters in the model, the construction of consistent estimators is fairly straightforward (e.g. Fuller (1980), Kapteyn and Wansbeek (1984)). For an overview of the state of the art, see Aigner et al. (1984). An approach somewhat orthogonal to the ones described above has been to take the model as it is and to use prior ideas about the size of the measurement errors to diagnose how serious the probem is. Examples are Blomqvist (1972), Hodges and Moore (1972), and Davies and Hutton (1975). Leamer (1983) starts from the opposite direction by asking how serious the measurement error problem has to be in order to render the data useless for inference, that is to say, when measurement error is large enough to make it impossible to put bounds on regression parameters. In an empirical example, he shows that even very small measurement errors in some explanatory variables would open up the possibility of perfectly collinear explanatory variables and hence make the data useless for statistical inference (at least without additional prior information). The most systematic analysis of the information loss caused by measurement error is due to Klepper and Leamer (1984). They start out by invoking a minimal amount of prior information and then ask the question under what conditions it is still possible to make some inferences regarding the vector of unknown regression parameters p. In the special case where the measurement errors are assumed uncorrelated and the k + 1 estimates of ,3, obtained by regressing each of the k +1 variables involved (i.e. the one dependent variable and the k independent variables) on the remaining k variables, are all in the same orthant, one can bound the ML estimates of p. In that case, the convex hull of the k + 1 regressions contains all possible ML estimates and any point in the hull is a possible ML estimate. If the k + 1 regressions are not all in the same orthant then the set of ML estimates is unbounded. In that case Klepper and Leamer (1984) introduce extra prior information which allows them to bound the set of maximum likelihood estimates. The prior information comes in two forms. Firstly, a researcher is supposed to be able to specify a maximum value of R2 if all exogenous variables were measured accurately. It is shown that if this maximum is low enough, one can again bound the set of ML estimates by a convex hull. Secondly, if

On obtient un test exact, similaire a celui de Chow (1960), qui comporte une simple integration unidimensionnelle

If it is common knowledge that the players in a game are Bayesian utility maximizers who treat uncertainty about other players' actions like any other uncertainty, then the outcome is necessarily a correlated equilibrium. Random strategies appear as an expression of each player's uncertainty about what the others will do, not as the result of willful randomization. Use is made of the common prior assumption, according to which differences in probability assessments by different individuals are due to the different information that they have (where "information" may be interpreted broadly, to include experience, upbringing, and genetic makeup). Copyright 1987 by The Econometric Society.

The permanent income hypothesis implies that people save because they rationally expect their labor income to decline; they save a rainy day. It follows that saving should be at least as good a predictor of declines in labor income as any other forecast that can be constructed from publicly available information.The paper tests this hitherto ignored implication of the permanent income hypothesis, using quarterly aggregate data for the period 1953-84 in the U.S. A vector autoregression for saving and changes in labor income is used to generate an unrestricted forecast of declines in labor income. In the VAR, saving Granger causes labor income changes as one would expect if the PIH is true. The mean of the unrestricted forecast is far from the mean of saving, but the dynamics of the two series are quite similar.The paper presents both formal test statistics and an informal evaluation of the fit of the permanent income hypothesis. By contrast with most of the recent literature, the results here are valid when income is nonstationary.

The expectati on of the excess holding yield on a long bond is postulated to depend upon its conditional variance. Engle's ARCH model is extended to allow the conditional variance to be a determinant of the mean and is called ARCH-M. Estimation and infer ence procedures are proposed, and the model is applied to three interest rate data sets. In most cases the ARCH process and the time varying risk premium are highly significant. A collection of LM diagnostic tests reveals the robustness of the model to various specification changes such as alternative volatility or ARCH measures, regime changes, and interest rate formulations. The model explains and interprets the recent econometric failures of the expectations hypothesis of the term structure. Copyright 1987 by The Econometric Society.

A general equilibrium model of markets for commodities and assets is considered. The author develops the notion of a price system that admits no arbitrage opportunity and demonstrates the fundamenta l role of this concept for the theory of the existence of a competiti ve equilibrium. A price system admits no arbitrage opportunity for a consumer if every bundle of desired commodities has a positive market value. The author proves that the existence of a price system that a dmits no arbitrage opportunity for all consumers is sufficient for th e existence of an equilibrium. Copyright 1987 by The Econometric Society.

Time series variables that stochastically trend together form a cointegrated system. OLS and NLS estimators of the parameters of a cointegrating vector are shown to converge in probability to the true parameter value at the rate T11d for any positive d. These estim mators can be written asymptotically in terms of relatively simple nonnormal random matrices which do not depend on the parameters of th e system. These asymptotic representations form the basis for simple and fast Monte Carlo calculations of the limiting distributions of th ese estimators. Asymptotic distributions thus computed are tabulated for several cointegrated processes. Copyright 1987 by The Econometric Society.

This paper investigates the consequences of the following modification of Expected Utility theory: instead of requiring independence with respect to probability mixtures of risky prospects, require independence with respect to direct mixing of payments o f risky prospects. A new theory of choice under risk- a so-called Dual theory-is obtained. Within this new theory, the following questions are considered: (1) numerical representation of preferences; (2) properties of the utility function; ( 3) the possibility for resolving the "paradoxes" of Expected Utilit y theory; ( 4) the characterization of risk aversion; and (5) comparative statics. The paper ends with a discussion of other non-Expected Utility theories proposed recently. Copyright 1987 by The Econometric Society.