COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1376 Econometric Methods for Endogenously Sampled Time Series:
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. |
