Testing Models with Multiple Equilibria by Quantile Methods

We derive an econometric test for the presence of monotone comparative statics in models with multiple equilibria. The test applies to many economic models, such as single-person decision, industrial organization, macroeconomic and game-theory models. These models have complementarities between exogenous and endogenous variables. We show that, as a result, extreme (large and small) conditional quantiles of the endogenous variable are increasing in the exogenous variable. We develop a likelihood-ratio test based on estimates for the conditional quantiles of the endogenous variable, which is an asymptotic extension of Bartholomew's (1959a,b) "chi-bar squared" test. Our assumptions are weak; we remain agnostic about the cardinality, location and probabilities of the equilibrium set, and make no restrictions on the equilibrium-selection procedure.