Stochastic orders and the anatomy of competitive selection

Noe, Thomas (2015) Stochastic orders and the anatomy of competitive selection. Saïd Business School Working Paper, Oxford.

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Abstract

This paper determines the conditions under which stochastic orderings of random variables, e.g., stochastic dominance and the monotone likelihood ordering, are preserved or reversed by conditioning on competitive selection. A new stochastic order over unconditional distributions is introduced—geometric dominance—which is sufficient for a random variable, conditioned on competitive selection, to be stochastically dominant and both necessary and sufficient for it to be dominant under the monotone likelihood ratio ordering. Using geometric dominance, we provide an “anatomy of selection bias” by identifying the conditions under which competitive selection conditioning preserves and reverses unconditional stochastic order relations. We show that, for all standard error distributions (e.g., Normal, Logistic, Laplace), competitive selection preserves stochastic order relations. One implication of this result is that, even in the presence of self-selection bias, the sign of treatment affects can be identified by standard uncorrected OLS and Logit/Probit models. Another is that, for almost all “textbook” distribution families, MLRP ordering is preserved by competitive selection.

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