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# Testing for Instability in Covariance Structures

## Author

Listed:
• Chihwa Kao
• Lorenzo Trapani
• Giovanni Urga

## Abstract

We propose a test for the stability over time of the covariance matrix of multivariate time series. The analysis is extended to the eigensystem to ascertain changes due to instability in the eigenvalues and/or eigenvectors. Using strong Invariance Principle and Law of Large Numbers, we normalize the CUSUM-type statistics to calculate their supremum over the whole sample. The power properties of the test versus local alternatives and alternatives close to the beginning/end of sample are investigated theoretically and via simulation. The testing procedure is validated through an application to 18 US interest rates over 1997-2011, finding instability at the end-2007/beginning-2008. Key Words: Covariance Matrix, Eigensystem, Changepoint, Term Structure of Interest Rates, CUSUM statistic JEL No. C1, C22, C5

## Suggested Citation

• Chihwa Kao & Lorenzo Trapani & Giovanni Urga, 2012. "Testing for Instability in Covariance Structures," Center for Policy Research Working Papers 131, Center for Policy Research, Maxwell School, Syracuse University.
• Handle: RePEc:max:cprwps:131
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## References listed on IDEAS

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1. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
2. Carriero, A. & Kapetanios, G. & Marcellino, M., 2009. "Forecasting exchange rates with a large Bayesian VAR," International Journal of Forecasting, Elsevier, vol. 25(2), pages 400-417.
3. Aue, Alexander & Horváth, Lajos & Hušková, Marie, 2012. "Segmenting mean-nonstationary time series via trending regressions," Journal of Econometrics, Elsevier, vol. 168(2), pages 367-381.
4. Castle, Jennifer L. & Fawcett, Nicholas W.P. & Hendry, David F., 2010. "Forecasting with equilibrium-correction models during structural breaks," Journal of Econometrics, Elsevier, vol. 158(1), pages 25-36, September.
5. Jushan Bai, 2000. "Vector Autoregressive Models with Structural Changes in Regression Coefficients and in Variance-Covariance Matrices," Annals of Economics and Finance, Society for AEF, vol. 1(2), pages 303-339, November.
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## Citations

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Cited by:

1. Lorenzo Trapani, 2014. "Comments on: Extensions of some classical methods in change point analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 283-286, June.
2. Castagnetti, Carolina & Rossi, Eduardo & Trapani, Lorenzo, 2015. "Inference on factor structures in heterogeneous panels," Journal of Econometrics, Elsevier, vol. 184(1), pages 145-157.
3. Barigozzi, Matteo & Trapani, Lorenzo, 2020. "Sequential testing for structural stability in approximate factor models," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5149-5187.
4. Marco R. Barassi & Nicola Spagnolo & Yuqian Zhao, 2018. "Fractional Integration Versus Structural Change: Testing the Convergence of $$\hbox {CO}_{2}$$ CO 2 Emissions," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 71(4), pages 923-968, December.

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### JEL classification:

• C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
• C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
• C5 - Mathematical and Quantitative Methods - - Econometric Modeling

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