Identification and Statistical Decision Theory

TL;DR

This paper explores how identification analysis can inform statistical decision theory, especially under partial identification and ambiguity, by providing bounds on decision performance and suggesting randomized strategies.

Contribution

It demonstrates that identification analysis offers valuable bounds for decision-making performance and introduces randomized decision rules to improve outcomes under ambiguity.

Findings

Identification bounds inform decision performance limits.

Randomized decision rules can enhance outcomes under ambiguity.

Sample data can be used to optimize decision strategies.

Abstract

Econometricians have usefully separated study of estimation into identification and statistical components. Identification analysis, which assumes knowledge of the probability distribution generating observable data, places an upper bound on what may be learned about population parameters of interest with finite sample data. Yet Wald's statistical decision theory studies decision making with sample data without reference to identification, indeed without reference to estimation. This paper asks if identification analysis is useful to statistical decision theory. The answer is positive, as it can yield an informative and tractable upper bound on the achievable finite sample performance of decision criteria. The reasoning is simple when the decision relevant parameter is point identified. It is more delicate when the true state is partially identified and a decision must be made under ambiguity. Then the performance of some criteria, such as minimax regret, is enhanced by randomizing choice of an action. This may be accomplished by making choice a function of sample data. I find it useful to recast choice of a statistical decision function as selection of choice probabilities for the elements of the choice set. Using sample data to randomize choice conceptually differs from and is complementary to its traditional use to estimate population parameters.