Abstract
We consider the problem of estimating a Markov process that is endogenously sampled. We observe a discrete-time Markov process {pt} at a set of random times {t1,…,tn} that depend on the outcome of a probabilistic sampling rule that depends on the state of the process and other observed covariates xt. We focus on a particular example where pt is the daily wholesale price of a standardized steel product. The endogenous sampling problem arises from the fact that we only observe pt on the days the firm purchases steel. We show how to solve this problem under two different assumptions about firm behavior: (1) optimality: the timing of steel purchases is governed by an optimal purchasing strategy that maximizes expected discounted profits, and (2) potential suboptimality: we allow the firm to use any randomized, Markovian purchasing strategy. In the latter case, the estimation problem becomes semi-parametric and we use the method of sieves to estimate a flexible parametric approximation to the firm’s purchasing behavior that best fits the data without imposing optimality. We show how estimation of this model becomes tractable under either of these assumptions using the method of simulated moments (MSM). We simulate realizations of wholesale steel prices and sample them in the same way as they are sampled in the actual data, i.e. only on days where purchases occur. We use the MSM estimator to estimate a truncated lognormal AR(1) model of the wholesale price processes for particular types of steel plate and test and reject the assumption that the firm is behaving optimally.