Auditing, Stratified Sampling, Substantive test, Stochastic modeling

In this note we introduce a weak optimality condition for tests, called complete consistency. We argue that complete consistency is a more appropriate weak optimality condition for tests than is test consistency. Complete consistency is a testing analogue of estimator consistency. It is shown that a sequence of estimators is consistent, if and only if certain tests based on the estimators (such as Wald or likelihood ratio tests) are completely consistent, for all simple null hypotheses. The above notwithstanding, the relationship between consistent and completely consistent tests shows that test consistency is a relevant concept. Consistent tests can be used to show the existence of, and to construct, completely consistent tests. Further, completely consistent tests cannot be generated from nested families of inconsistent tests.

Donald W. K. Andrews, A Note on the Unbiasedness of Feasible GLS, Quasi-Maximum Likelihood, Robust, Adaptive, and Spectral Estimators of the Linear Model, Econometrica, Vol. 54, No. 3 (May, 1986), pp. 687-698

THIS PAPER INVESTIGATES a property of estimators called stability. The stability exponent of an estimator is a measure of the magnitude of the effect of any single observation in the sample on the realized value of the estimator. A number of reasons related to robustness suggest that often it is desirable for an estimator to be relatively insensitive to any particular observation in the sample, i.e., to have high stability. In addition, it is useful for diagnostic purposes to have knowledge of the stability exponents of different estimators, in order to know which estimators are likely to rely more heavily on some single observation. The paper is organized as follows: Section 1 introduces the basic idea contained in the paper, motivates it, and summarizes the results in an informal manner. Section 2 presents definitions, assumptions, and the general results. For purposes of illustration, the linear regression model with the least squares estimator is used as a running example throughout this section. Section 3 discusses numerous additional applications of the general results. An Appendix contains proofs of

Several predetermined variables that reflect levels of bond and stock prices appear to predict returns on common stocks of firms of various sizes, long-term bonds of various default risks, and default-free bonds of various maturities. The returns on small-firm stocks and low-grade bonds are more highly correlated in January than in the rest of the year with previous levels of asset prices, especially prices of small-firm stocks. Seasonality is found in several conditional risk measures, but such seasonality is unlikely to explain, and in some cases is opposite to, the seasonal found in mean returns.