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The authors study the asymptotic distribution of econometric tests involving nuisance parameters that are not identified under the null hypotheses. In general, the asymptotic distributions depend upon a large number of unknown parameters. The authors show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters and they show that this transformation can be easily approximated via simulation. The theory is applied to threshold models. Monte Carlo methods are used to assess the finite sample distributions. The authors' tests show that S. M. Potter's (1995) finding of a threshold effect in U.S. GNP growth rates can be possibly explained by sampling variation. Copyright 1996 by The Econometric Society.