Pythagorean generalization of testing the equality of two symmetric positive definite matrices |
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Authors: | Jin Seo Cho Peter CB Phillips |
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Institution: | 1. School of Economics, Yonsei University, Republic of Korea;2. Department of Economics, Hong Kong University of Science and Technology, Hong Kong;3. Yale University, United States;4. University of Auckland, New Zealand;5. Singapore Management University, Singapore;6. University of Southampton, United Kingdom |
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Abstract: | We provide a new test for equality of two symmetric positive-definite matrices that leads to a convenient mechanism for testing specification using the information matrix equality or the sandwich asymptotic covariance matrix of the GMM estimator. The test relies on a new characterization of equality between two dimensional symmetric positive-definite matrices and : the traces of and are equal to if and only if . Using this simple criterion, we introduce a class of omnibus test statistics for equality and examine their null and local alternative approximations under some mild regularity conditions. A preferred test in the class with good omni-directional power is recommended for practical work. Monte Carlo experiments are conducted to explore performance characteristics under the null and local as well as fixed alternatives. The test is applicable in many settings, including GMM estimation, SVAR models and high dimensional variance matrix settings. |
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Keywords: | C01 C12 C52 Matrix equality Trace Determinant Arithmetic mean Geometric mean Harmonic mean Sandwich covariance matrix Eigenvalues |
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