The exact stochastic character of observed data from a Markov process is derived for the case where only aggregate stocks, as opposed to individual transitions, are observed. Particular attention is devoted to the distinction between data generated by a panel study, where a single group of individuals is followed over time, and that generated by random sampling, where the observed groups are not identical over time. Several alternative estimators are developed which take into account the particular stochastic structure of the data.

Zellner [11], and Chow [3], is to deal with the uncertainty by treating the model parameters as independent, identically distributed random variables in each period,' yielding what Zellner calls rules. Although the sequential updating rules do capture the uncertainty, they ignore the possibility of ongoing estimation in the formulation of decision rules. The purpose of this paper is to develop an adaptive learning decision rule for the multiperiod problem. This rule incorporates the effect of policy variables on the learning which is expected to occur throughout the remainder of the planning period. It is a generalization of the rules described above in the sense that both of them can be derived as special cases of this rule. Finally, this decision rule yields to economic analysis in terms of the stock and value of information, a feature which is not found in previous work. Section 2 of the paper describes the multiperiod decision problem with unknown parameters, discusses the assumptions associated with the certainty equivalence and sequential updating rules, and then presents the assumptions employed in this paper. Adaptive learning decision rules are derived in Section 3 and a mathe