Constructing smooth tests without estimating the eigenpairs of the limiting process |
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| Institution: | 1. Department of Economics, National Chengchi University, Taipei 116, Taiwan;2. Department of Finance and CRETA, National Taiwan University, Taipei 106, Taiwan;1. Department of Mathematics, National University of Singapore, 0511, Singapore;2. Department of Economics, Ryerson University, Toronto, ON M5B2K3, Canada;1. MIT, United States;2. University of Arizona, United States;1. Department of Economics, Stanford University, 579 Serra Mall, Stanford, CA 94305, United States;2. Department of Economics, University of Kentucky, 335A College of Business and Economics, KY 40506, United States;1. Department of Finance, Kellogg School of Management, Northwestern University, Evanston, IL 60208, United States;2. Department of Economics, Duke University, Durham, NC 27708, United States;1. University of Cambridge and St John''s College, Cambridge, UK;2. Faculty of Economics, University of Cambridge, Sidgwick Avenue, Cambridge, CB3 9DD, UK;3. Toulouse School of Economics (CNRS, GREMAQ), France;1. Department of Economics, University of Rochester, Rochester, NY, 14627, United States;2. Department of Economics, Cornell University, Ithaca, NY 14850, United States;3. Department of Statistical Science, Cornell University, Ithaca, NY 14850, United States;4. Wang Yanan Institute for Studies in Economics (WISE), Xiamen University, Xiamen 361005, China;5. MOE Key Laboratory in Econometrics, Xiamen University, Xiamen 361005, China |
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| Abstract: | Based on the well known Karhunen–Loève expansion, it can be shown that many omnibus tests lack power against “high frequency” alternatives. The smooth tests of Neyman (1937) may be employed to circumvent this power deficiency problem. Yet, such tests may be difficult to compute in many applications. In this paper, we propose a more operational approach to constructing smooth tests. This approach hinges on a Fourier representation of the postulated empirical process with known Fourier coefficients, and the proposed test is based on the normalized principal components associated with the covariance matrix of finitely many Fourier coefficients. The proposed test thus needs only standard principal component analysis that can be carried out using most econometric packages. We establish the asymptotic properties of the proposed test and consider two data-driven methods for determining the number of Fourier coefficients in the test statistic. Our simulations show that the proposed tests compare favorably with the conventional smooth tests in finite samples. |
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| Keywords: | Data-driven method Eigenpairs Fourier representation Karhunen–Loève expansion Smooth test |
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