Knowledge that Transforms

This paper proposes new estimators of the latent regression function in nonparametric censored and truncated regression models. Our estimators are computationally convenient, consisting only of two nonparametric regressions and a univariate integral. We establish consistency and asymptotic normality for an implementation based on local linear kernel estimators. An extension permits estimation in the presence of a general form of heteroscedasticity.

We study the problem of identi cation of the long regression E(y j x � z) when the short conditional distributions P (y j x) and P (z j x) are known but the long conditional distribution P (y j x � z) is not known. This problem often arises when a researcher utilizes data from two separate data sets. (A leading example is the ecological inference problem of political science, where voting behavior across electoral districts is observed from administrative records, the demographic composition of voters within a district is observed from census data, and the researcher wants to infer voting behavior conditional on district and demographic attributes.) We isolate an identi cation region containing feasible values of the long regression, and show that this region forms a sharp bound on the long regression. The identi cation region can be calculated precisely when y has nite support. When y has in nite support we characterize two sets, one that contains the identi cation region, and one that is contained by it. Following this completely nonparametric analysis, we examine the identifying power yielded by exclusion restrictions across distinct covariate values. Such restrictions cause the identi cation region to shrink, in many cases to a single point. To illustrate the theory, we pose and address this hypothetical question: What would be the outcome if the 1996 U.S. presidential election were re-enacted in a population of di erent demographic composition, ceteris paribus? We have bene tted from the opportunity to present this research in seminars at Northwestern

In previous work on cheap talk, uncertainty has almost always been modeled using a single–dimensional state variable. In this paper we prove that the dimensionality of the uncertain variable has an important qualitative impact on results and yields interesting insights into the “mechanics” of information transmission. Contrary to the unidimensional case, if there is more than one sender, full revelation of information in all states of nature is generically possible, even when the conflict of interest is arbitrarily large. What really matters in transmission of information is the local behavior of senders’ indifference curves at the ideal point of the receiver, not the proximity of players’ ideal point.

We present an axiomatic model depicting the choice behavior of a self-interest seeking moral individual over random allocation procedures. Individual preferences are decomposed into a self-interest component and a component representing the individual's moral value judgment. Each component has a distinct utility representation, and the preference relation depicting the choice behavior is representable by a real-valued function defined on the components utilities. The utility representing the self-interest component is linear and the utility representing the individual's moral value judgment is quasi-concave. The addition of a hexagon condition implies that the utility representing the individual's preference is additively separable in the components utilities.

The paper examines within a unified methodology expectational coordination in a series of economic models. The methodology views the predictions associated with the Rational Expectations Hypothesis as reasonable whenever they can be derived from the more basic Common Knowledge Hypothesis. The paper successively considers a simple non-noisy N-dimensional model, standard models with “intrinsic” uncertainty, and reference intertemporal models with infinite horizon. It reviews existing results and suggests new ones. It translates the formal results into looser but economically intuitive statements, whose robustness, in the present state of knowledge, is tentatively ascertained.

This paper formalizes the relation between comparative statics and the out-of-equilibrium explanation for how a system evolves after a change in parameters. The paper has two main results. First, an increase in an exogenous parameter sets o# learning dynamics that involve larger values of the endogenous variables. Second, equilibrium selections that are not monotone increasing in the exogenous variables must be predicting unstable equilibria. Moreover, under some conditions monotone comparative statics and stability are equivalent. JEL Classification: C61, C62, C72, C73 Keywords: Monotone comparative statics, supermodularity, strategic complements, learning, correspondence principle. # Discussions with Ilya Segal and Chris Shannon were very important for this work, I am very grateful for all their help. For comments and advice, I also thank Robert Anderson, Juan Dubra, Nestor Gandelman, Ernesto Lopez Cordova, Marcelo Moreira, Charles Pugh, Matthew Rabin, Tarun Sabarwal and Miguel Villa...

This paper analyzes the complexity of the contraction fixed point problem: compute an e-approximation to the fixed point V * = Γ(V * ) of a contraction mapping r that maps a Banach space B d of continuous functions of d variables into itself. We focus on quasi linear contractions where Γ is a nonlinear functional of a finite number of conditional expectation operators. This class includes contractive Fredholm integral equations that arise in asset pricing applications and the contractive Bellman equation from dynamic programming. In the absence of further restrictions on the domain of Γ, the quasi linear fixed point problem is subject to the curse of dimensionality, i.e., in the worst case the minimal number of function evaluations and arithmetic operations required to compute an e-approximation to a fixed point V * e B d increases exponentially in d. We show that the curse of dimensionality disappears if the domain of Γ has additional special structure. We identify a particular type of special structure for which the problem is strongly tractable even in the worst case, i.e., the number of function evaluations and arithmetic operations needed to compute an e-approximation of V * is bounded by Ce -p where C and p are constants independent of d. We present examples of economic problems that have this type of special structure including a class of rational expectations asset pricing problems for which the optimal exponent p = 1 is nearly achieved.

Band spectral regression with both deterministic and stochastic trends is considered. It is shown that trend removal by regression in the time domain prior to band spectral regression can lead to biased and inconsistent estimates in models with frequency dependent coefficients. Both semiparametric and nonparametric regression formulations are considered, the latter including general systems of two-sided distributed lags such as those arising in lead and lag regressions. The bias problem arises through omitted variables and is avoided by careful specification of the regression equation. Trend removal in the frequency domain is shown to be a convenient option in practice. An asymptotic theory is developed and the two cases of stationary data and cointegrated nonstationary data are compared. In the latter case, a levels and differences regression formulation is shown to be useful in estimating the frequency response function at nonzero as well as zero frequencies.

Important estimation problems in econometrics like estimating the value of a spectral density at frequency zero, which appears in the econometrics literature in the guises of heteroskedasticity and autocorrelation consistent variance estimation and long run variance estimation, are shown to be “ill-posed” estimation problems. A prototypical result obtained in the paper is that the minimax risk for estimating the value of the spectral density at frequency zero is infinite regardless of sample size, and that confidence sets are close to being uninformative. In this result the maximum risk is over commonly used specifications for the set of feasible data generating processes. The consequences for inference on unit roots and cointegration are discussed. Similar results for persistence estimation and estimation of the long memory parameter are given. All these results are obtained as special cases of a more general theory developed for abstract estimation problems, which readily also allows for the treatment of other ill-posed estimation problems such as, e.g., nonparametric regression or density estimation.