Estimation of Nonparametric Functions in Simultaneous Equations Models, with an Application to Consumer Demand

We present a method for consistently estimating nonparametric functions and distributions in simultaneous equations models. This method is used to identify and estimate a random utility model of consumer demand. Our identification conditions for this particular model extend the results of Houthakker (1950), Uzawa (1971) and Mas-Colell (1977), where a deterministic utility function is uniquely recovered from its deterministic demand function.

A Nonparametric Maximum Rank Correlation Estimator

This paper presents a nonparametric and distribution-free estimator for the function h*, of observable exogenous variables, x, in the generalized regression model, y - G(h*(x), mu). The method does not require a parametric specification for either the function h* or for the distribution of the random term mu. The estimation proceeds by maximizing a rank correlation criterion (Han (1987)) over a set of functions that are monotone increasing, concave, and homogeneous degree one; the function h* is assumed to belong to this set of functions. The estimator is shown to be strongly consistent

Estimation of Multinomial Models Using Weak Monotonicity Assumptions

This paper introduces a semiparametric method of estimating multinomial models that imposes extremely weak monotonicity assumptions about a function of observable characteristics. Previous methods have imposed stronger, typically parametric, conditions on this function. The only assumptions made in this paper about the function of characteristics are its monotonicity, upper-semicontinuity, and uniform boundedness. The method is applicable, among others, to polychotomous choice models. The estimation method is shown to be strongly consistent. A technique to calculate the estimator is provided

Nonparametric and Distribution-Free Estimation of the Binary Choice and the Threshold-Crossing Models

This paper studies the problem of nonparametric identification and estimation of binary threshold-crossing and binary choice models. First, conditions are given that guarantee the nonparametric identification of both the function of exogenous observable variables and the distribution of the random terms. Second, the identification results are employed to develop strongly consistent estimation methods that are nonparametric in both the function of observable exogenous variables and the distribution of the unobservable random variables. The estimators are obtained by maximizing a likelihood function over nonparametric sets of functions. A two-step constrained optimization procedure is devised to compute these estimators

Nonparametric Tests of Maximizing Behavior Subject to Nonlinear Sets

This paper extends the axiomatic theory of revealed preference to choices that are generated by the maximization of a strictly concave and strictly monotone function subject to nonlinear constraint sets. I characterize finite sets of observations on choice behavior that are consistent with the maximization of a strictly concave and strictly monotone objective function. Both nonconvex and convex choice sets are considered. The analysis applies, for example, to consumers who face either regressive or progressive taxes and to households that produce commodities according to either a convex or a concave production function. For choice sets that possess convex and monotone complements, my characterization provides a nonparametric test for the maximization hypothesis. For choice sets that can be supported by unique hyperplanes at the chosen elements, the result provides a nonparametric test for the strict concavity and strict monotonicity of the maximized function

Least Concavity and the Distribution-Free Estimation of Non-Parametric Concave Functions

This paper studies the estimation of fully nonparametric models in which we can not identify the values of a symmetric function that we seek to estimate. I develop a method of consistently estimating a representative of a concave and monotone nonparametric systematic function. This representative possesses the same isovalue sets as the systematic function. The method proceeds by characterizing each set of observationally equivalent concave functions by a unique “least concave” representative. The least concave representative of the equivalence class to which the systematic function belongs is estimated by maximizing a criterion function over a compact set of least concave functions. I develop a computational technique to evaluate the values, at the observed points, and the gradients, at every point and up to a constant, of this least concave estimator. The paper includes a detailed description of how the method can be used to estimate three popular microeconometric models

Semiparametric Estimation of Monotonic and Concave Utility Functions: The Discrete Choice Case

This paper develops a semiparametric method for estimating the nonrandom part V ( ) of a random utility function U ( v ,ω) – V ( v ) + e (ω) from data on discrete choice behavior. Here v and ω are, respectively, vectors of observable and unobservable attributes of an alternative, and e(ω) is the random part of the utility for that alternative. The method is semiparametric because it assumes that the distribution of the random parts is know up to a finite-dimensional parameter θ, while not requiring specification of a parametric form for V ( ). The nonstochastic part V ( ) of the utility function U ( ) is assumed to be Lipschitzian and to possess a set of properties, typically assumed for utility functions. The estimator of the pair ( V ,θ) is shown to be strongly consistent

Least Concavity and the Distribution-Free Estimation of Non-Parametric Concave Functions

This paper studies the estimation of fully nonparametric models in which we can not identify the values of a symmetric function that we seek to estimate. I develop a method of consistently estimating a representative of a concave and monotone nonparametric systematic function. This representative possesses the same isovalue sets as the systematic function. The method proceeds by characterizing each set of observationally equivalent concave functions by a unique "least concave" representative. The least concave representative of the equivalence class to which the systematic function belongs is estimated by maximizing a criterion function over a compact set of least concave functions. I develop a computational technique to evaluate the values, at the observed points, and the gradients, at every point and up to a constant, of this least concave estimator. The paper includes a detailed description of how the method can be used to estimate three popular microeconometric models.

Observing trajectories with weak measurements in quantum systems in the semiclassical regime

We propose a scheme allowing to observe the evolution of a quantum system in the semiclassical regime along the paths generated by the propagator. The scheme relies on performing consecutive weak measurements of the position. We show how weak trajectories" can be extracted from the pointers of a series of measurement devices having weakly interacted with the system. The properties of these "weak trajectories" are investigated and illustrated in the case of a time-dependent model system.Comment: v2: Several minor corrections were made. Added Appendix (that will appear as Suppl. Material). To be published in Phys Rev Let