Knowledge that Transforms

Knowledge of the asymptotic variance of an estimator is important for large sample inference, efficiency, and as a guide to the specification of regularity conditions.The purpose of this paper is the presentation of a general formula for the asymptotic variance of a semiparametric estimator.A particularly important feature of this formula is a way of accounting for the presence of nonparametric estimates of nuisance functions.The general form of an adjustment factor for nonparametric estimates is derived and analyzed.The usefulness of the formula is illustrated by deriving propositions on asymptotic equivalence for different nonparametric estimators of the same function, conditions for estimation of the nuisance functions to have no effect on the asymptotic variance, and the form of a correction term for the presence of linear function of a conditional expectation estimator, or other projection estimator (e.g.partially linear and/or additive nonparametric projections), and for a function of a density.Specific results cover a semiparametric random effects model for binary panel data, nonparametric consumer surplus, nonparametric prediction, and average derivatives.Regularity conditions are given for many of the propositions.These include primitive conditions for v'n-consistency, asymptotic normality, and consistency of an asymptotic variance estimator with series estimators of conditional expectations (or projections), in each of the examples.

The authors provide a convergence theory for adaptive learning algorithms useful for the study of learning by economic agents. Their results extend the framework of L. Ljung previously utilized by A. Marcet-T. J. Sargent and M. Woodford by permitting nonlinear laws of motion driven by stochastic processes that may exhibit moderate dependence, such as mixing and mixingale processes. The authors draw on previous work by H. J. Kushner and D. S. Clark to provide readily verifiable and/or interpretable conditions ensuring algorithm convergence, chosen for their suitability in the context of adaptive learning.

This paper provides a general framework for proving the "square root of" T-consistency and asymptotic normality of a wide variety of semiparametric estimators. The class of estimators considered consists of estimators that can be defined as the solution to a minimization problem based on a criterion function that may depend on a preliminary infinite dimensional nuisance parameter estimator. The method of proof exploits results concerning the stochastic equicontinuity of stochastic processes. The results are applied to the problem of semiparametric weighted least squares estimation of partially parametric regression models. Primitive conditions are given for "square root of" T-consistency and asymptotic normality of this estimator.

A model of trade with m buyers and m sellers is considered in which price is set to equate revealed demand and supply. In a Bayesian Nash equilibrium, each trader acts not as a price-taker, but instead misrepresents his true demand/supply to influence price in his favor. This causes inefficiency. We show that in any equilibrium the amount by which a trader misreports is O(1/m) and the corresponding inefficiency is O(1/m2). The indeterminacy and the inefficiency that is caused by the traders' bargaining behavior in small markets thus rapidly vanishes as the market increases in size.

We study the indeterminacy of equilibria in infinite horizon capital accumulation models with technological externalities. Our investigation encompasses models with bounded and unbounded accumulation paths, and models with one and two sectors of production. Under reasonable assumptions we find that equilibria are locally unique in one-sector economies. In economies with two sectors of production it is instead easy to construct examples where a positive external effect induces a two-dimensional manifold of equilibria converging to the same steady state (in the bounded case) or to the same constant growth rate (in the unbounded case). For the latter we point out that the dynamic behavior of these equilibria is quite complicated and that persistent fluctuations in their growth rates are possible.

This paper is concerned with the refined asymptotic properties of several tests for the admissibility of a subset of (overidentifying) instrumental variables. It derives maximum likelihood and linearized maximum likelihood tests and calculates size corrections to the order 1/T. The local power function of the size-corrected tests is the same to the order 1/T, irrespectively of the form of the test statistic or the limited information estimator used in its computation. Futher, it compares these tests with two previously proposed tests. The size and the power of the original and the size-corrected tests are compared by Monte Carlo experiments.

The Bertrand-Edgeworth model describes competition among price setting sellers with production capacity constraints. The authors report on laboratory experiments that permit evaluation of different theories of Bertrand-Edgeworth competition: competitive pricing, Edgeworth cycles in prices, mixed strategy Nash equilibrium pricing, and tacit collusion. Each of the theories helps to explain some aspects of the data. However, none of these theories are completely consistent with the data. In relative terms, the Edgeworth cycle theory provides better predictions of key aspects of the data than the other theories. Coauthors are Stephen Rassenti, Stanley S. Reynolds, and Vernon L. Smith.

In this paper I develop a practical extension of McFaddens method of simulated moments estimator for limited dependent variable models to the panel data case. The method is based on a factorization of the MSM first order condition into transition probabi

The author derives some exact finite sample disbibutions and characterizes the tail behavior of maximum likelihood estimators of the cointegrating coefficients in error correction models. The reduced rank regression estimator has a distribution with Cauchy-like tails and no finite moments of integer order. The maximum likelihood estimator of the coefficients in a particular triangular system representation has matrix t-distribution tails with finite integer moments to order T - n + r, where T is the sample size, n is the total number of variables, and r is the dimension of cointegration space. This helps explain some recent simulation studies where extreme outliers occur more frequently for the reduced rank regression estimator than for alternative asymptotically efficient procedures based on triangular representation.

Two of the most important refinements of the Nash equilibrium concept for extensive form games are (trembling hand) perfect equilibrium and sequential equilibrium. It is shown here that, for almost all assignments of payoffs to outcomes, the sets of sequential and perfect equilibrium strategy profiles are identical. This result is obtained by exploiting the semialgebraic nature of equilibrium correspondences, following from a deep theorem of mathematical logic.