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

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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 {p_t} that is observed only at a subset of times {t_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 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 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 {p_t} 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.

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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1376.

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Length: 57 pages
Date of creation: Jul 2002
Date of revision:
Handle: RePEc:cwl:cwldpp:1376

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Web page: http://cowles.econ.yale.edu/
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Endogenous sampling; Markov processes; Maximum likelihood; Simulation estimation;

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  1. Williams,Jeffrey C. & Wright,Brian D., 1991. "Storage and Commodity Markets," Cambridge Books, Cambridge University Press, number 9780521326162, October.
  2. Deaton, A. & Laroque, G., 1989. "On The Behavior Of Commodity Prices," Papers 145, Princeton, Woodrow Wilson School - Development Studies.
  3. George Hall & John Rust, 1999. "An Empirical Model of Inventory Investment by Durable Commodity Intermediaries," Macroeconomics 9904005, EconWPA.
  4. Hugo Benitez-Silva & John Rust & Gunter Hitsch & Giorgio Pauletto & George Hall, 2000. "A Comparison Of Discrete And Parametric Methods For Continuous-State Dynamic Programming Problems," Computing in Economics and Finance 2000 24, Society for Computational Economics.
  5. John Rust & George Hall, 2001. "Middle Men Versus Market Makers: A Theory of Competitive Exchange," Cowles Foundation Discussion Papers 1299, Cowles Foundation for Research in Economics, Yale University.
  6. Tülin Erdem & Michael P. Keane, 1996. "Decision-Making Under Uncertainty: Capturing Dynamic Brand Choice Processes in Turbulent Consumer Goods Markets," Marketing Science, INFORMS, vol. 15(1), pages 1-20.
  7. Lee, Bong-Soo & Ingram, Beth Fisher, 1991. "Simulation estimation of time-series models," Journal of Econometrics, Elsevier, vol. 47(2-3), pages 197-205, February.
  8. John Rust & Hui Man Chan & George Hall, 2004. "Price Discrimination in the Steel Market," Econometric Society 2004 North American Summer Meetings 245, Econometric Society.
  9. McFadden, Daniel, 1989. "A Method of Simulated Moments for Estimation of Discrete Response Models without Numerical Integration," Econometrica, Econometric Society, vol. 57(5), pages 995-1026, September.
  10. J. Rust & J. F. Traub & H. Wozniakowski, 2002. "Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?," Econometrica, Econometric Society, vol. 70(1), pages 285-329, January.
  11. Duffie, Darrell & Singleton, Kenneth J, 1993. "Simulated Moments Estimation of Markov Models of Asset Prices," Econometrica, Econometric Society, vol. 61(4), pages 929-52, July.
  12. Alastair R. Hall & Atsushi Inoue, 2005. "The Large Sample Behaviour of the Generalized Method of Moments Estimator in Misspecified Models," Econometrics 0505002, EconWPA.
  13. Yacine Ait--Sahalia & Per A. Mykland, 2003. "The Effects of Random and Discrete Sampling when Estimating Continuous--Time Diffusions," Econometrica, Econometric Society, vol. 71(2), pages 483-549, March.
  14. repec:rus:hseeco:72158 is not listed on IDEAS
  15. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
  16. Aguirregabiria, Victor, 1999. "The Dynamics of Markups and Inventories in Retailing Firms," Review of Economic Studies, Wiley Blackwell, vol. 66(2), pages 275-308, April.
  17. Kamran Moinzadeh, 1997. "Replenishment and Stocking Policies for Inventory Systems with Random Deal Offerings," Management Science, INFORMS, vol. 43(3), pages 334-342, March.
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