Data-Snooping Biases We quantify these effects in the following sections by appealing to asymptotic results for induced order statistics,and show that even mild forms of data snooping can change inferences substantially.In Section 1.1,a brief summary of the asymptotic properties of induced order statistics is provided.In Section 1.2,results for tests based on individual securities are presented,and in Section 1.3,corresponding results for portfolios are reported.We provide a more positive inter- pretation of data-snooping biases as power against deviations from the null hypothesis in Section 1.4. 1.1.Asymptotic properties of induced order statistics Since the particular form of data snooping we are investigating is most common in empirical tests of financial asset pricing models,our exposition will lie in that context.Suppose for each of N securities we have some consistent estimator &of a parameter a,which is to be used in the construction of an aggregate test statistic.For example, in the Sharpe-Lintner CAPM,@would be the estimated intercept from the following regression: Rn-Rn=:+(Rm-Rn)B:+en (1) where RR,and R are the period-t returns on security i,the market portfolio,and a risk-free asset,respectively.A test of the null hypothesis that a,=0 would then be a proper test of the Sharpe- Lintner version of the CAPM;thus,@may serve as a test statistic itself. However,more powerful tests may be obtained by combining the a,'s for many securities.But how should we combine them? Suppose for each security i we observe some characteristic X,,such as its out-of-sample market value of equity or average annual earnings, and we learn that X,is correlated empirically with a,.By this we mean that the relation between X,and @is an empirical fact uncovered by "searching"through the data,and not motivated by any a priori the. oretical considerations.This search need not be a systematic sifting of the data,but may be interpreted as any one of Leamer's (1978)six specification searches,which even the most meticulous of classical statisticians has conducted at some point.The key feature is that our interest in characteristic X:is derived from a look at the data,the same data to be used in performing our test.Common intuition sug- gests that using information contained in the X,'s can yield a more powerful test of economic restrictions on the a,'s.But if this char- acteristic is not a part of the original null hypothesis,and only catches our attention after a look at the data (or after a look at another's look at the data),using it to form our test statistics may lead us to reject those economic restrictions even when they obtain.More formally, 435