Andriy Norets

Inference in Dynamic Discrete Choice Models With Serially orrelated Unobserved State Variables

This paper develops a method for inference in dynamic discrete choice models with serially correlated unobserved state variables. Estimation of these models involves computing high-dimensional integrals that are present in the solution to the dynamic program and in the likelihood function. First, the paper proposes a Bayesian Markov chain Monte Carlo estimation procedure that can handle the problem of multidimensional integration in the likelihood function. Second, the paper presents an efficient algorithm for solving the dynamic program suitable for use in conjunction with the proposed estimation procedure. Copyright 2009 The Econometric Society.

APPROXIMATION OF CONDITIONAL DENSITIES BY SMOOTH MIXTURES OF REGRESSIONS

This paper shows that large nonparametric classes of conditional multivariate densities can be approximated in the Kullback‐Leibler distance by dierent specifications of finite mixtures of normal regressions in which normal means and variances and mixing probabilities can depend on variables in the conditioning set (covariates). These models are a special case of models known as mixtures of experts in statistics and computer science literature. Flexible specifications include models in which only mixing probabilities, modeled by multinomial logit, depend on the covariates and, in the univariate case, models in which only means of the mixed normals depend flexibly on the covariates. Modeling the variance of the mixed normals by flexible functions of the covariates can weaken restrictions on the class of the approximable densities. Obtained results can be generalized to mixtures of general location scale densities. Rates of convergence and easy to interpret bounds are also obtained for dierent model specifications. These approximation results can be useful for proving consistency of Bayesian and maximum likelihood density estimators based on these models. The results also have interesting implications for applied researchers.

Semiparametric Inference in Dynamic Binary Choice Models

We introduce an approach for semi-parametric inference in dynamic binary choice models that does not impose distributional assumptions on the state variables unobserved by the econometrician. The proposed framework combines Bayesian inference with partial identification results. The method is applicable to models with finite space of observed states. We demonstrate the method on Rusts model of bus engine replacement. The estimation experiments show that the parametric assumptions about the distribution of the unobserved states can have a considerable effect on the estimates of per-period payoffs. At the same time, the effect of these assumptions on counterfactual conditional choice probabilities can be small for most of the observed states.

Posterior Consistency In Conditional Density Estimation By Covariate Dependent Mixtures

This paper considers Bayesian nonparametric estimation of conditional densities by countable mixtures of location-scale densities with covariate dependent mixing probabilities. The mixing probabilities are modeled in two ways. First, we consider finite covariate dependent mixture models, in which the mixing probabilities are proportional to a product of a constant and a kernel and a prior on the number of mixture components is specified. Second, we consider kernel stick-breaking processes for modeling the mixing probabilities. We show that the posterior in these two models is weakly and strongly consistent for a large class of data-generating processes. A simulation study conducted in the paper demonstrates that the models can perform well in small samples.

On the surjectivity of the mapping between utilities and choice probabilities

This note considers a standard multinomial choice model. It is shown that if the distribution of additive utility shocks has a density, then the mapping from de- terministic components of utilities to choice probabilities is surjective. In other words, any vector of choice probabilities can be obtained by selecting suitable utilities for alternatives. This result has implications for at least three areas of interest to econometricians: the Hotz and Miller (1993) estimator for structural dynamic discrete choice models, nonparametric identification of multinomial choice models, and consistency of conditional density estimators based on co- variate dependent mixtures. Keywords. Multinomial choice models, identification, Hotz and Miller estimator, covariate dependent mixtures. JEL classification. C35, C61.

Estimation of Dynamic Discrete Choice Models Using Artificial Neural Network Approximations

I propose a method for inference in dynamic discrete choice models (DDCM) that utilizes Markov chain Monte Carlo (MCMC) and artificial neural networks (ANNs). MCMC is intended to handle high-dimensional integration in the likelihood function of richly specified DDCMs. ANNs approximate the dynamic-program (DP) solution as a function of the parameters and state variables prior to estimation to avoid having to solve the DP on each iteration. Potential applications of the proposed methodology include inference in DDCMs with random coefficients, serially correlated unobservables, and dependence across individual observations. The article discusses MCMC estimation of DDCMs, provides relevant background on ANNs, and derives a theoretical justification for the method. Experiments suggest this to be a promising approach.

Continuity and differentiability of expected value functions in dynamic discrete choice models

This paper explores the properties of expected value functions in dynamic dis- crete choice models. The continuity with respect to state variables and parame- ters, and the differentiability with respect to state variables are established under fairly general conditions. The differentiability with respect to parameters is proved when some state variables do not affect the state transition probabilities and, thus, the expected value functions. It is shown that such variables are needed so as to apply the implicit function theorem used in the proof. The results are of particular relevance to estimable dynamic discrete choice models. Keywords. Dynamic discrete choice models, continuity, differentiability. JEL classification. C35, C61.

Adaptive Bayesian Estimation Of Conditional Densities

We consider a non-parametric Bayesian model for conditional densities. The model is a finite mixture of normal distributions with covariate dependent multinomial logit mixing probabilities. A prior for the number of mixture components is specified on positive integers. The marginal distribution of covariates is not modeled. We study asymptotic frequentist behavior of the posterior in this model. Specifically, we show that when the true conditional density has a certain smoothness level, then the posterior contraction rate around the truth is equal up to a log factor to the frequentist minimax rate of estimation. An extension to the case when the covariate space is unbounded is also established. As our result holds without a priori knowledge of the smoothness level of the true density, the established posterior contraction rates are adaptive. Moreover, we show that the rate is not affected by inclusion of irrelevant covariates in the model. In Monte Carlo simulations, a version of the model compares favorably to a cross-validated kernel conditional density estimator.

Credibility of Confidence Sets in Nonstandard Econometric Problems

Confidence intervals are commonly used to describe parameter uncertainty. In nonstandard problems, however, their frequentist coverage property does not guarantee that they do so in a reasonable fashion. For instance, confidence intervals may be empty or extremely short with positive probability, even if they are based on inverting powerful tests. We apply a betting framework and a notion of bet‐proofness to formalize the “reasonableness” of confidence intervals as descriptions of parameter uncertainty, and use it for two purposes. First, we quantify the violations of bet‐proofness for previously suggested confidence intervals in nonstandard problems. Second, we derive alternative confidence sets that are bet‐proof by construction. We apply our framework to several nonstandard problems involving weak instruments, near unit roots, and moment inequalities. We find that previously suggested confidence intervals are not bet‐proof, and numerically determine alternative bet‐proof confidence sets.

Coverage Inducing Priors in Nonstandard Inference Problems

ABSTRACT We consider the construction of set estimators that possess both Bayesian credibility and frequentist coverage properties. We show that under mild regularity conditions there exists a prior distribution that induces (1 − α) frequentist coverage of a (1 − α) credible set. In contrast to the previous literature, this result does not rely on asymptotic normality or invariance, so it can be applied in nonstandard inference problems. Supplementary materials for this article are available online.