The problem considered is inference in a simple errors-in-variables model where consistent estimation is impossible without introducing additional exact prior information. The probabilistic prior information required for Bayesian analysis is found to be surprisingly light: despite the model's lack of identification a proper posterior is guaranteed for any bounded prior density, including those representing improper priors. This result is illustrated with the improper uniform prior, which implies marginal posterior densities obtainable by integrating the likelihood function; surprisingly, the posterior mode for the regression slope is the usual least squares estimate. KEYwoRDs: Errors-in-variables, Bayesian inference, identification, improper priors, proper posteriors, finitely additive probabilities, coherence. 1.
In this paper, the author considers how boundedly rational agents learn rational expectations when all equilibrium price functions or forecasts of future equilibrium prices are required to be computable. The paper examines two learning environments. In the first, agents have perfect information about the state of nature. In this case, the theory of machine inference can be applied to show that there is a broad class of computable economies whose rational expectations equilibria can be learned by inductive inference. In the second environment, agents do not have perfect information about the state of nature. In this case, a version of Godel's incompleteness theorem implies that rational expectations equilibria cannot be learned. Copyright 1989 by The Econometric Society.
This paper is concerned with the use of power properties of tests in econometric applications. Inverse power functions are defined. These functions are designed to yield summary measures of power that facilitate the interpretation of test results in practice. Simple approximations are introduced for the inverse power functions of Wald, likelihood ratio, Lagrange multiplier, and Hausman tests. These approximations readily convey the general qualitative features of the power of a test. Examples are provided to illustrate their usefulness in interpreting test results. A COMMON PROBLEM faced in applied econometrics is that of interpreting the results of a hypothesis test when the test fails to reject the null hypothesis. Most practitioners realize that just because a test fails to reject a hypothesis one cannot claim to accept it. Nevertheless, it is common for this to be ignored, since the practitioner is often in a position where he would like the outcome of the test to provide useful inferences whether or not the test rejects. The purpose of this paper is to introduce inverse power (IP) summary measures that enable the practitioner to avoid such errors and make valid inferences when a test fails to reject the null hypothesis. These summary measures are widely applicable, easy to use (especially in the common case of a test concerning a single restriction), and simple to compute. When a test rejects the null hypothesis, the implication is that the data are inconsistent with each parameter point in the null in the sense that the probabil- ity of type I error for each point is small, viz., a or less. Correspondingly, when a test fails to reject the null hypothesis an analogous statement is needed regarding the error probabilities for points in the alternative hypothesis. It is not the case that all points in the alternative are inconsistent with the data in the sense that their probability of type II error is small (a or less). It is possible, however, to determine the region S in the alternative parameter space that is inconsistent with the data in this sense. The IP function introduced below evaluated at
In this note we derive the general form of a utility based system of demand equations that are quadratic in income with coefficients that are functions of prices.Our result turns out to be slightly more general than known till now.
This paper proposes a simple modification of a conventional method of moments estimator for a discrete response model, replacing response probabilities that require numerical integration with estimators obtained by Monte Carlo simulation.This method of simulated moments (MSM) does not require precise estimates of these probabilities for consistency and asymptotic normality, relying instead on the law of large numbers operating across observations to control simulation error, and hence can use simulations of practical size.The method is useful for models such as high-dimensional multinomial probit (MNP), where computation has restricted applications.
T. J. Rothenberg's (1984) Edgeworth test size correction for the linear model with a nonscalar covariance matrix is applied to the special case of AR(1) errors. Simulations show that the correction reduces the overrejection that is commonly encountered in this model, although substantial overrejection remains when the original amount is large. When the regressors are autocorrelated and collinear (for example, two trended regressors) there is not nearly as much overrejection when testing a single restriction as there is when the model contains only one autocorrelated regressor. Copyright 1989 by The Econometric Society.
Properties of t ratios associated with the LIML, TSLS, and OLS estimators in a structural form estimation are studied. The existence of moments of these t ratios including the LIML form is proved first. Second, Monte Carlo simulations are performed to find out real sizes of the t test and the likelihood ratio test. Third, asymptotic expansions of the distributions of t ratios are derived under the null hypothesis to find out deviations of real sizes from nominal sizes theoretically. The asymptotic power functions are also derived
A transferable utility economy in which each agent holds a resource which can be used in combination with the resources of other agents to generate value (according to the characteristic function V) is studied using a dynamic model of bargaining. The main theorem establishes that the payoffs associated with efficient equilibria converge to the agents' Shapley values as the time between periods of the dynamic game goes to zero. In addition it is demonstrated that an efficient equilibrium exists and is unique when an additivity condition is satisfied. To demonstrate the sensitivity of the solution to the institutional detail we modify the model to allow for partnerships and show that the Shapley value is no longer achieved.
This paper gives a solution to the problem of estimating coefficients of index models, through the estimation of the density-weighted average derivative of a general regression function. A normalized version of the density-weighted average derivative can be estimated by certain linear instrumental variables coefficients. The estimators, based on sample analogies of the product moment representation of the average derivative, are constructed using nonparametric kernel estimators of the density of the regressors. Consistent estimators of the asymptotic variance-covariance matrices of the estimators are given, and a limited Monte Carlo simulation is used to study the practical performance of the procedures. Copyright 1989 by The Econometric Society.