## Econometric Methods for Endogenously Sampled Time Series: The Case of Commodity Price Speculation in the Steel Market (2002)

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@MISC{Hall02econometricmethods,

author = {George Hall and John Rust},

title = {Econometric Methods for Endogenously Sampled Time Series: The Case of Commodity Price Speculation in the Steel Market},

year = {2002}

}

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### Abstract

This paper studies the econometric problems associated with estimation of a stochastic process that is endogenously sampled. Our interest is to infer the law of motion of a discrete-time stochastic process that is observed only at a subset of times 1 , . . . , t n that depend on the outcome of a probabilistic sampling rule that depends on the history of the process as well as other observed covariates x t . We focus on a particular example where p t denotes the daily wholesale price of a standardized steel product. However there are no formal exchanges or centralized markets where steel is traded and p t can be observed. Instead nearly all steel transaction prices are a result of private bilateral negotiations between buyers and sellers, typically intermediated by middlemen known as steel service centers. Even though there is no central record of daily transactions prices in the steel market, we do observe transaction prices for a particular firm --- a steel service center that purchases large quantities of steel in the wholesale market for subsequent resale in the retail market. The endogenous sampling problem arises from the fact that the firm only records p t on the days that it purchases steel. We present a parametric analysis of this problem under the assumption that the timing of steel purchases is part of an optimal trading strategy that maximizes the firm's expected discounted trading profits. We derive a parametric partial information maximum likelihood (PIML) estimator that solves the endogenous sampling problem and efficiently estimates the unknown parameters of a Markov transition probability that determines the law of motion for the underlying process. The PIML estimator also yields estimates of the structural parameters that determine the...