Pythagorean generalization of testing the equality of two symmetric positive definite matrices

Jin Seo Cho, Peter C.B. Phillips

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


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 k dimensional symmetric positive-definite matrices A and B: the traces of AB−1 and BA−1 are equal to k if and only if A=B. 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.

Original languageEnglish
Pages (from-to)45-56
Number of pages12
JournalJournal of Econometrics
Issue number1
Publication statusPublished - 2018 Jan

Bibliographical note

Funding Information:
The co-editor, Oliver Linton, associated editor, and two anonymous referees provided helpful comments on several earlier versions of the paper for which we are grateful. The authorsacknowledge helpful discussions with Heejoon Han, Dukpa Kim, Eunyoung Kim, Joocheol Kim, Tae-Hwan Kim, Jin Lee, Nakyeong Lee, Tae Hwy Lee, Timo Teräsvirta, Yoon-Jae Whang, Yohei Yamamoto, Byung Sam Yoo, Jungmo Yoon, and other participants at the KEA conference (Korea University) and the seminar and conference participants at Yonsei University, the Korean Econometric Society, Symposium of the Society for Nonlinear Dynamics and Econometrics (Tuscaloosa, 2016), and the Western Economic Association International (Wellington, 2015). Cho thanks the research grant support by the Ministry of Education of the Republic of Korea and the National Research Foundation of Korea ( NRF-2015S1A5A2A01012140 ), and Phillips acknowledges research support from the National Science Foundation under Grant No. SES 12-58258 .

All Science Journal Classification (ASJC) codes

  • Economics and Econometrics

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