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# Testing for Instability in Covariance Structures

## Author

Listed:
• Chihwa Kao

(University of Connecticut)

• Lorenzo Trapani

• Giovanni Urga

(Cass Business School and Universitá di Bergamo)

## 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 Principles and Law of Large Numbers, we normalise the CUSUM-type statistics to calculate their supremum over the whole sample. The power properties of the test versus alternative hypotheses, including also the case of breaks close to the beginning/end of sample are investigated theoretically and via simulation. We extend our theory to test for the stability of the covariance matrix of a multivariate regression model. The testing procedures are illustrated by studying the stability of the principal components of the term structure of 18 US interest rates. JEL Classification: Key words: Covariance Matrix, Eigensystem, Changepoint, CUSUM Statistic.

## Suggested Citation

• Chihwa Kao & Lorenzo Trapani & Giovanni Urga, 2016. "Testing for Instability in Covariance Structures," Working papers 2016-33, University of Connecticut, Department of Economics.
• Handle: RePEc:uct:uconnp:2016-33
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File URL: https://media.economics.uconn.edu/working/2016-33.pdf
File Function: Full text
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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. 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.
4. 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.
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.
Full references (including those not matched with items on IDEAS)

## Citations

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Cited by:

1. 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.
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. 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.
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

### NEP fields

This paper has been announced in the following NEP Reports:

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