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 {pt} that is observed only at a subset of times {t1,…,tn} that depend on the outcome of a probabilistic sampling rule that depends on the history of the process as well as other observed covariates xt. We focus on a particular example where pt denotes the daily wholesale price of a standardized steel product. However there are no formal exchanges or centralized markets where steel is traded and pt 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 pt 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 {pt} process. The PIML estimator also yields estimates of the structural parameters that determine the optimal trading rule. We also introduce an alternative consistent, less efficient, but computationally simpler simulated minimum distance (SMD) estimator that avoids high dimensional numerical integrations required by the PIML estimator. Using the SMD estimator, we provide estimates of a truncated lognormal AR(1) model of the wholesale price processes for particular types of steel plate. We use this to infer the share of the middleman’s discounted profits that are due to markups paid by its retail customers, and the share due to price speculation. The latter measures the firm’s success in forecasting steel prices and in timing its purchases in order to “buy low and sell high’.” The more successful the firm is in speculation (i.e. in strategically timing its purchases), the more serious are the potential biases that would result from failing to account for the endogeneity of the sampling process.
Keywords: Endogenous sampling, Markov processes, Maximum likelihood, Simulation estimation
This paper introduces a new detailed data set of high-frequency observations on inventory investment by a U.S. steel wholesaler. Our analysis of these data leads to six main conclusions: orders and sales are made infrequently; orders are more volatile than sales; order sizes vary considerably; there is substantial high-frequency variation in the firm’s sales prices; inventory/sales ratios are unstable; and there are occasional stockouts. We model the firm generically as a durable commodity intermediary that engages in commodity price speculation. We demonstrate that the firm’s inventory investment behavior at the product level is well approximated by an optimal trading strategy from the solution to a nonlinear dynamic programming problem with two continuous state variables and one continuous control variable that is subject to frequently binding inequality constraints. We show that the optimal trading strategy is a generalized (S,s) rule. That is, whenever the firm’s inventory level q falls below the order threshold s(p) the firm places an order of size S(p) - q in order to attain a target inventory level S(p) satisfying S(p) > s(p), where p is the current spot price at which the firm can purchase unlimited amounts of the commodity after incurring a fixed order cost K. We show that the (S,s) bands are decreasing functions of p, capturing the basic intuition of commodity price speculation, namely, that it is optimal for the firm to hold higher inventories when the spot price is low than when it is high in order to profit from “buying low and selling high.” We simulate a calibrated version of this model and show that the simulated data exhibit the key features of inventory investment we observe in the data.