Sorawoot Srisuma

Semiparametric estimation of Markov decision processes with continuous state space

We propose a general two-step estimation method for the structural parameters of popular semiparametric Markovian discrete choice models that include a class of Markovian Games and allow for continuous observable state space. The estimation procedure is simple as it directly generalizes the computationally attractive methodology of Pesendorfer and Schmidt-Dengler (2008) that assumed finite observable states. This extension is non-trivial as the value functions, to be estimated nonparametrically in the first stage, are defined recursively in a non-linear functional equation. Utilizing structural assumptions, we show how to consistently estimate the infinite dimensional parameters as the solution to some type II integral equations, the solving of which is a well-posed problem. We provide sufficient set of primitives to obtain root-T consistent estimators for the finite dimensional structural parameters and the distribution theory for the value functions in a time series framework.

An Alternative Asymptotic Least Squares Estimator for Dynamic Games

The estimation of dynamic games is known to be a numerically challenging task. In this paper we propose an alternative class of asymptotic least squares estimators to Pesendorfer and Schmidt-Denglers (2008), which includes several well known estimators in the literature as special cases. Our estimator can be substantially easier to compute. In the leading case with linear payoffs specification our estimator has a familiar OLS/GLS closed-form that does not require any optimization. When payoffs have partially linear form, we propose a sequential estimator where the parameters in the nonlinear term can be estimated independently of the linear components, the latter can then be obtained in closed-form. We show the class of estimators we propose and Pesendorfer and Schmidt-Denglers are in fact asymptotically equivalent. Hence there is no theoretical cost in reducing the computational burden. Our estimator appears to perform well in a simple Monte Carlo experiment.

Estimation of Structural Optimization Models: A Note on Identification

Bajari, Benkard and Levin (2007) propose an estimation methodology for a broad class of dynamic optimization problems. To carry out their procedure, one needs to select a set of alternative policy functions and compare the implied expected payoffs with that from the data. We show that this can generally lead to objective functions that are not capable of consistently estimating an identified model.

Minimum Distance Estimation for a Class of Markov Decision Processes

We develop a two-step estimator for a class of Markov decision processes with continuous control that is intuitive and simple to implement. Making use of the monotonicity assumption we estimate the expected continuation value functions nonparametrically in the first stage. In the second stage our estimator minimizes a minimum distance criterion that measures the divergence between the nonparametric conditional distribution function and a model implied simulated semiparametric counterpart. We show that our minimum distance estimator is asymptotically normal and converges at the parametric rate under some regularity conditions. We estimate the expected value function by kernel smoothing and derive its pointwise distribution theory. We illustrate how our estimation methodology forms a basis for the estimation of dynamic models with different class of control variable(s) as well as a class of Markovian games.

Ordinary Least Squares Estimation of a Dynamic Game Model

Estimation of dynamic games is known to be a numerically challenging task. A common form of the payoff functions employed in practice takes the linear-in-parameter specification. We show a least squares estimator taking a familiar OLS/GLS expression is available in such a case. Our proposed estimator has a closed form. It can be computed without any numerical optimization and always minimizes the least squares objective function. We specify the optimally weighted GLS estimator that is efficient in the class of estimators under consideration. Our estimator appears to perform well in a simple Monte Carlo experiment.

IDENTIFICATION IN DISCRETE MARKOV DECISION MODELS

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Minimum Distance Estimation of Search Costs using Price Distribution

ABSTRACT It has been shown that equilibrium restrictions in a search model can be used to identify quantiles of the search cost distribution from observedprices alone. These quantiles can be difficult to estimate in practice. This article uses a minimum distance approach to estimate them that is easy to compute. A version of our estimator is a solution to a nonlinear least-square problem that can be straightforwardly programmed on softwares such as STATA. We show our estimator is consistent and has an asymptotic normal distribution. Its distribution can be consistently estimated by a bootstrap. Our estimator can be used to estimate the cost distribution nonparametrically on a larger support when prices from heterogenous markets are available. We propose a two-step sieve estimator for that case. The first step estimates quantiles from each market. They are used in the second step as generated variables to perform nonparametric sieve estimation. We derive the uniform rate of convergence of the sieve estimator that can be used to quantify the errors incurred from interpolating data across markets. To illustrate we use online bookmaking odds for English football leagues’ matches (as prices) and find evidence that suggests search costs for consumers have fallen following a change in the British law that allows gambling operators to advertise more widely. Supplementary materials for this article are available online.

Banking Privatization and Market Structure in Brazil: A Dynamic Structural Analysis

This paper examines the effects of bank privatization on the number of bank branches operating in small isolated markets in Brazil. We estimate a dynamic game played between Brazilian public and private banks. We find private banks compete with each other as expected. We also find public banks generate positive spillovers for private banks. The latter can at least partly be explained by complementarities between credit products offered by different types of banks in Brazil. Our counterfactual study shows that privatization substantially reduces the number of banks. More than half of the markets in our sample would end up without any bank branch if banks were privatized. The government can mitigate the effects of privatization by providing subsidies to private banks. Our model predicts subsidy policies that reduce operating costs are always more cost-effective than entry costs for isolated markets in Brazil.

Joint Analysis of the Discount Factor and Payoff Parameters in Dynamic Discrete Choice Models

Most empirical and theoretical econometric studies of dynamic discrete choice models assume the discount factor to be known. We show the knowledge of the discount factor is not necessary to identify parts, or all, of the payoff function. We show the discount factor can be generically identified jointly with the payoff parameters. It is known the payoff function cannot non-parametrically identified without any a priori restrictions. Our identification of the discount factor is robust to any normalization choice on the payoff parameters. In IO applications normalizations are usually made on switching costs, such as entry costs and scrap values. We also show that switching costs can be non-parametrically identified, in closed-form, independently of the discount factor and other parts of the payoff function. Our identification strategies are constructive. They lead to easy to compute estimands that are global solutions. We illustrate with a Monte Carlo study and the dataset from Ryan (2012).

Local Identification in Markov Decision Models

We provide necessary and sufficient conditions for the local identification of the finite dimensional parameters in a semiparametric dynamic discrete choice model under additive separability and conditional independence assumption (Rust (1987)). We show that the policy value approach commonly used in the two-step estimation methodologies has convenient features so that the conditional version of Rothenbergs (1971) parametric identification results can be readily applied. We provide results for both the single agent problems and a class of games of incomplete information. These conditions are easy to check under the extreme value distributional assumption and when the payoff function has a linear-in-parameter specification. Our approach does not depend on the discreteness of the control variable and can be used to derive analogous conditions in other Markov decision models. Our approach can also be used when the value of the discounting factor not known.