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Bootstrap Testing of the Rank of a Matrix via Least-Squared Constrained Estimation

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  • François Portier
  • Bernard Delyon

Abstract

To test if an unknown matrix M 0 has a given rank (null hypothesis noted H 0 ), we consider a statistic that is a squared distance between an estimator and the submanifold of fixed-rank matrix. Under H 0 , this statistic converges to a weighted chi-squared distribution. We introduce the constrained bootstrap (CS bootstrap) to estimate the law of this statistic under H 0 . An important point is that even if H 0 fails, the CS bootstrap reproduces the behavior of the statistic under H 0 . As a consequence, the CS bootstrap is employed to estimate the nonasymptotic quantile for testing the rank. We provide the consistency of the procedure and the simulations shed light on the accuracy of the CS bootstrap with respect to the traditional asymptotic comparison. More generally, the results are extended to test whether an unknown parameter belongs to a submanifold of the Euclidean space. Finally, the CS bootstrap is easy to compute, it handles a large family of tests and it works under mild assumptions.

Suggested Citation

  • François Portier & Bernard Delyon, 2014. "Bootstrap Testing of the Rank of a Matrix via Least-Squared Constrained Estimation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(505), pages 160-172, March.
    • Handle: RePEc:taf:jnlasa:v:109:y:2014:i:505:p:160-172
    DOI: 10.1080/01621459.2013.847841
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    Cited by:

    1. François Portier, 2016. "An Empirical Process View of Inverse Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(3), pages 827-844, September.
    2. Anyck Dauphin & Bernard Fortin & Guy Lacroix, 2018. "Is consumption efficiency within households falsifiable?," Review of Economics of the Household, Springer, vol. 16(3), pages 737-766, September.
    3. Alain Guay, 2020. "Identification of Structural Vector Autoregressions Through Higher Unconditional Moments," Working Papers 20-19, Chair in macroeconomics and forecasting, University of Quebec in Montreal's School of Management.
    4. Valentin Jouvanceau & Ieva Mikaliunaite, 2020. "Euro Area Monetary Communications: Excess Sensitivity and Perception Shocks," Bank of Lithuania Working Paper Series 79, Bank of Lithuania.
    5. Guay, Alain, 2021. "Identification of structural vector autoregressions through higher unconditional moments," Journal of Econometrics, Elsevier, vol. 225(1), pages 27-46.
    6. Miguel Cabello, 2022. "Robust Estimation of the non-Gaussian Dimension in Structural Linear Models," Papers 2212.07263, arXiv.org, revised Sep 2023.

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