Skip to main content
Log in

Testing for covariance matrices in time-varying coefficient panel data models with fixed effects

  • Research Article
  • Published:
Journal of the Korean Statistical Society Aims and scope Submit manuscript

Abstract

In this paper, we study the tests for sphericity and identity of covariance matrices in time-varying coefficient high-dimensional panel data models with fixed effects. In order to construct the effective test statistics and avoid the influence of the unknown fixed effects, we apply the difference method to eliminate the dependence of the residual sample, and further construct test statistics using the trace estimators of the covariance matrices. For the estimators of the coefficient functions, we use the local linear dummy variable method. Under some regularity conditions, we study the asymptotic property of the estimators and establish the asymptotic distributions of our proposed test statistics without specifying an explicit relationship between the cross-sectional and the time series dimensions. We further show that the test statistics are asymptotic distribution-free. Subsequently simulation studies are carried out to evaluate our proposed methods. In order to assess the performance of our proposed test method, we compare with the existing test methods in panel data linear models with fixed effects.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Anderson, T. W. (2003). An introduction to multivariate statistical analysis. Hoboken: Wiley.

    MATH  Google Scholar 

  • Bai, Z. D., Jiang, D. D., Yao, J. F., & Zheng, S. R. (2009). Corrections to LRT on large-dimensional covariancematrix by RMT. The Annals of Statistics, 37, 3822–3840.

    Article  MathSciNet  Google Scholar 

  • Baltagi, B. H., Feng, Q., & Kao, C. (2011). Testing for sphericity in a fixed effects panel data model. The Econometrics Journal, 14, 25–47.

    Article  MathSciNet  Google Scholar 

  • Baltagi, B. H., Kao, C., & Peng, B. (2015). On testing for sphericity with non-normality in a fixed effects panel data model. Statistics and Probability Letters, 98, 123–130.

    Article  MathSciNet  Google Scholar 

  • Birke, M., & Dette, H. (2005). A note on testing the covariance matrix for large dimension. Statistics and Probability Letters, 74, 281–289.

    Article  MathSciNet  Google Scholar 

  • Cai, T. T., & Liu, W. D. (2016). Large-scale multiple testing of correlation. Journal of the American Statistical Association, 111, 229–240.

    Article  MathSciNet  Google Scholar 

  • Cai, T. T., Liu, W. D., & Xia, Y. (2013). Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings. Journal of the American Statistical Association, 108, 265–277.

    Article  MathSciNet  Google Scholar 

  • Chen, F., Li, Z. X., Shi, L., & Zhu, L. X. (2015). Inference for mixed models of ANOVA type with high-dimensional data. Journal of Multivariate Analysis, 133, 382–401.

    Article  MathSciNet  Google Scholar 

  • Chen, S. X., Zhang, L. X., & Zhong, P. S. (2010). Tests for high dimensional covariance matrices. Journal of the American Statistical Association, 105, 810–819.

    Article  MathSciNet  Google Scholar 

  • Cho, J. S., & Phillips, P. C. B. (2018). Pythagorean generalization of testing the equality of two symmetric positive definite matrices. Journal of Econometrics, 202, 45–56.

    Article  MathSciNet  Google Scholar 

  • Feng, S. Y., Li, G. R., Peng, H., & Tong, T. J. (2019). Panel data varying coefficient models with interactive fixed effects. Statistica Sinica. https://doi.org/10.5705/ss.202018.0248.

  • Hu, X. M. (2014). Estimation in a semi-varying coefficient model for panel data with fixed effects. Journal of Systems Science and Complexity, 27, 594–604.

    Article  MathSciNet  Google Scholar 

  • Jiang, D. D., Jiang, T. F., & Yang, F. (2012). Likelihood ratio tests for covariance matrices of high-dimensional normal distributions. Journal of Statistical Planning and Inference, 142, 2241–2256.

    Article  MathSciNet  Google Scholar 

  • John, S. (1971). Some optimal multivariate tests. Biometrika, 58, 123–127.

    MathSciNet  MATH  Google Scholar 

  • John, S. (1972). The distribution of a statistic used for testing sphericity of normal distributions. Biometrika, 59, 169–173.

    Article  MathSciNet  Google Scholar 

  • Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. The Annals of Statistics, 29, 295–327.

    Article  MathSciNet  Google Scholar 

  • Ledoit, O., & Wolf, M. (2002). Some hypothesis tests for the covariance matrix when the dimension is large compared to rhe sample size. The Annals of Statistics, 30, 1081–1102.

    Article  MathSciNet  Google Scholar 

  • Li, D. G., Chen, J., & Gao, J. T. (2011). Non-parametric time-varying coefficient panel data models with fixed effects. The Econometrics Journal, 14, 387–408.

    Article  MathSciNet  Google Scholar 

  • Li, G. R., Lian, H., Lai, P., & Peng, H. (2015). Variable selection for fixed effects varying coefficient models. Acta Mathematica Sinica, English Series, 31, 91–110.

    Article  MathSciNet  Google Scholar 

  • Li, W. M., & Qin, Y. L. (2014). Hypothesis testing for high-dimensional covariance matrices. Journal of Multivariate Analysis, 128, 108–119.

    Article  MathSciNet  Google Scholar 

  • Lin, R. T., Liu, Z. Y., Zhang, S. R., & Yin, G. S. (2016). Power computation for hypothesis testing with high-dimensional covariance matrices. Computational Statistics & Data Analysis, 104, 10–23.

    Article  MathSciNet  Google Scholar 

  • Mao, G. Y. (2016). Testing for error cross-sectional independence using pairwise augmented regressions. The Econometrics Journal, 19, 237–260.

    Article  MathSciNet  Google Scholar 

  • Nagao, H. (1973). On some test criteria for covariance matrix. The Annals of Statistics, 1, 700–709.

    Article  MathSciNet  Google Scholar 

  • Pei, Y. Q., Huang, T., & You, J. H. (2018). Nonparametric fixed effects model for panel data with locally stationary regressors. Journal of Econometrics, 202, 286–305.

    Article  MathSciNet  Google Scholar 

  • Peng, L. H., Chen, S. X., & Zhou, W. (2016). More powerful tests for sparse high-dimensional covariances matrices. Journal of Multivariate Analysis, 49, 124–143.

    Article  MathSciNet  Google Scholar 

  • Rodriguez-Poo, J. M., & Soberon, A. (2015a). Nonparametric estimation of fixed effects panel data varying coefficient models. Journal of Multivariate Analysis, 133, 95–122.

    Article  MathSciNet  Google Scholar 

  • Rodriguez-Poo, J. M., & Soberon, A. (2015b). Differencing techniques in semi-parametric panel data varying coefficient models with fixed effects: A Monte Carlo study. Computational Statistics, 30, 885–906.

    Article  MathSciNet  Google Scholar 

  • Roy, S. N. (1957). Some aspects of multivariate analysis. New York: Wiley.

    Google Scholar 

  • Srivastava, M. S. (2005). Some tests concerning the covariance matrix in high dimensional data. Journal of the Japan Statistical Society, 35, 251–272.

    Article  MathSciNet  Google Scholar 

  • Srivastava, M. S., & Yanagihara, H. (2010). Testing the equality of several covariance matrices with fewer observations than the dimension. Journal of Multivariate Analysis, 101, 1319–1329.

    Article  MathSciNet  Google Scholar 

  • Sun, Y. G., Carroll, R. J., & Li, D. D. (2009). Semiparametric estimation of fixed effects panel data varying coefficient models. Advances in Econometrics, 25, 101–129.

    Article  MathSciNet  Google Scholar 

  • Wu, J. H., & Li, G. D. (2014). Moment-based tests for individual and time effects in panel data models. Journal of Econometrics, 178, 569–581.

    Article  MathSciNet  Google Scholar 

  • Xia, Y., Cai, T. X., & Cai, T. T. (2015). Testing differential networks with applications to the detection of gene-gene interactions. Biometrika, 102, 247–266.

    Article  MathSciNet  Google Scholar 

  • Xia, Y., Cai, T. X., & Cai, T. T. (2018). Multiple testing of submatrices of a precision matrix with applications to identification of between pathway interactions. Journal of the American Statistical Association, 131, 328–339.

    Article  MathSciNet  Google Scholar 

  • Xu, W. L., Li, Y. W., & Song, D. W. (2013). Testing normality in mixed models using a transformation method. Statistical Papers, 54, 71–84.

    Article  MathSciNet  Google Scholar 

  • Zhang, R. M., Peng, L., & Wang, R. D. (2013). Tests for covariance matrix with fixed or divergent dimension. The Annals of Statistics, 41, 2075–2096.

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the Editor, Associate Editor and two anonymous reviewers for their insightful comments and suggestions that have dramatically improved an earlier version of this paper. Gaorong Li and Ranran Chen’s research was supported by the National Natural Science Foundation of China (11871001 and 11971001), the Beijing Natural Science Foundation (1182003) and the Fundamental Research Funds for the Central Universities (2019NTSS18). Sanying Feng’s research was supported by the National Natural Science Foundation of China (11501522) and the Excellent Youth Foundation of Zhengzhou University (32210452).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gaorong Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, R., Li, G. & Feng, S. Testing for covariance matrices in time-varying coefficient panel data models with fixed effects. J. Korean Stat. Soc. 49, 82–116 (2020). https://doi.org/10.1007/s42952-019-00007-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42952-019-00007-x

Keywords

Mathematics Subject Classification

Navigation