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Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size

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  • Olivier Ledoit
  • Michael Wolf

Abstract

This paper analyzes whether standard covariance matrix tests work when dimensionality is large, and in particular larger than sample size. In the latter case, the singularity of the sample covariance matrix makes likelihood ratio tests degenerate, but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well-defined. We study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together, with their ratio converging to a finite non-zero limit. We find that the existing test for sphericity is robust against high dimensionality, but not the test for equality of the covariance matrix to a given matrix. For the latter test, we develop a new correction to the existing test statistic that makes it robust against high dimensionality.

Suggested Citation

  • Olivier Ledoit & Michael Wolf, 2001. "Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size," Economics Working Papers 575, Department of Economics and Business, Universitat Pompeu Fabra.
  • Handle: RePEc:upf:upfgen:575
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    Cited by:

    1. Anatolyev, Stanislav, 2012. "Inference in regression models with many regressors," Journal of Econometrics, Elsevier, vol. 170(2), pages 368-382.
    2. Arnab Bhattacharjee & Jagjit Chadha & Qi Sun, 2010. "Productivity, Preferences and UIP Deviations in an Open Economy Business Cycle Model," Open Economies Review, Springer, vol. 21(3), pages 365-391, July.
    3. Olivier Ledoit & Sandrine P�ch�, 2009. "Eigenvectors of some large sample covariance matrices ensembles," IEW - Working Papers 407, Institute for Empirical Research in Economics - University of Zurich.

    More about this item

    Keywords

    Concentration asymptotics; equality test; sphericity test;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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