Efficient Tests Under a Weak Convergence Assumption


  • Ulrich K. Müller

    1. Dept. of Economics, Princeton University, Princeton, NJ 08544, U.S.A.; umueller@princeton.edu
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    • A previous version of this paper was circulated under the title “An Alternative Sense of Asymptotic Efficiency.” The author would like to thank Whitney Newey, three anonymous referees, Mark Watson, seminar participants at Berkeley, Brown, Harvard/MIT, Penn State, and Stanford, and conference participants at the NBER Summer Institute 2008 and the ESEM Summer Meeting 2008 for helpful comments and discussions, as well as Jia Li for excellent research assistance. Financial support by the NSF through Grant SES-0751056 is gratefully acknowledged.


The asymptotic validity of tests is usually established by making appropriate primitive assumptions, which imply the weak convergence of a specific function of the data, and an appeal to the continuous mapping theorem. This paper, instead, takes the weak convergence of some function of the data to a limiting random element as the starting point and studies efficiency in the class of tests that remain asymptotically valid for all models that induce the same weak limit. It is found that efficient tests in this class are simply given by efficient tests in the limiting problem—that is, with the limiting random element assumed observed—evaluated at sample analogues. Efficient tests in the limiting problem are usually straightforward to derive, even in nonstandard testing problems. What is more, their evaluation at sample analogues typically yields tests that coincide with suitably robustified versions of optimal tests in canonical parametric versions of the model. This paper thus establishes an alternative and broader sense of asymptotic efficiency for many previously derived tests in econometrics, such as tests for unit roots, parameter stability tests, and tests about regression coefficients under weak instruments.