• Bernoulli
  • Volume 24, Number 1 (2018), 740-771.

Testing for instability in covariance structures

Chihwa Kao, Lorenzo Trapani, and Giovanni Urga

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

Bernoulli, Volume 24, Number 1 (2018), 740-771.

Received: September 2014
Revised: August 2016
First available in Project Euclid: 27 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

changepoint covariance matrix CUSUM statistic eigensystem


Kao, Chihwa; Trapani, Lorenzo; Urga, Giovanni. Testing for instability in covariance structures. Bernoulli 24 (2018), no. 1, 740--771. doi:10.3150/16-BEJ894.

Export citation


  • Andrews, D.W.K. (1987). Asymptotic results for generalised Wald tests. Econometric Theory 3 348–358.
  • Andrews, D.W.K. (1993). Tests for parameter instability and structural change with unknown change point. Econometrica 61 821–856.
  • Andrews, D.W.K. and Ploberger, W. (1994). Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62 1383–1414.
  • Audrino, F., Barone-Adesi, G. and Mira, A. (2005). The stability of factor models of interest rates. J. Financ. Econom. 3 422–441.
  • Aue, A., Hörmann, S., Horváth, L. and Reimherr, M. (2009). Break detection in the covariance structure of multivariate time series models. Ann. Statist. 37 4046–4087.
  • Aue, A. and Horváth, L. (2013). Structural breaks in time series. J. Time Series Anal. 34 1–16.
  • Bliss, R.R. (1997). Movements in the term structure of interest rates. Economic Review, Federal Reserve Bank of Atlanta Q 4 16–33.
  • Bliss, R.R. and Smith, D.C. (1997). The stability of interest rate processes. Federal Reserve Bank of Atlanta, Working Paper 97-13.
  • Castle, J.L., Fawcett, N.W.P. and Hendry, D.F. (2010). Forecasting with equilibrium-correction models during structural breaks. J. Econometrics 158 25–36.
  • Cheng, X., Liao, Z. and Schorfheide, F. (2014). Shrinkage estimation of high-dimensional factor models with structural instabilities. Mimeo.
  • Corradi, V. (1999). Deciding between $I(0)$ and $I(1)$ via FLIL-based bounds. Econometric Theory 15 643–663.
  • Csörgő, M. and Horváth, L. (1997). Limit Theorems in Change-Point Analysis. Wiley Series in Probability and Statistics. Chichester: Wiley.
  • Davidson, J. (1994). Stochastic Limit Theory: An Introduction for Econometricians. Advanced Texts in Econometrics. New York: The Clarendon Press, Oxford Univ. Press.
  • Davidson, J. (2002). Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes. J. Econometrics 106 243–269.
  • Davis, C. and Kahan, W.M. (1970). The rotation of eigenvectors by a perturbation. III. SIAM J. Numer. Anal. 7 1–46.
  • Eberlein, E. (1986). On strong invariance principles under dependence assumptions. Ann. Probab. 14 260–270.
  • Gallant, A.R. and White, H. (1988). A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. Oxford: Basil Blackwell.
  • Han, X. and Inoue, A. (2011). Tests for parameter instability in dynamic factor models. TERG Discussion Paper No. 306.
  • Hill, J.B. (2010). On tail index estimation for dependent, heterogeneous data. Econometric Theory 26 1398–1436.
  • Hill, J.B. (2011). Tail and nontail memory with applications to extreme value and robust statistics. Econometric Theory 27 844–884.
  • Horn, R.A. and Johnson, C.R. (1999). Matrix Analysis. Cambridge: Cambridge Univ. Press.
  • Horváth, L. (1993). The maximum likelihood method for testing changes in the parameters of normal observations. Ann. Statist. 21 671–680.
  • Horváth, L. and Rice, G. (2015). Testing for changes in the means and correlations between panels. Mimeo.
  • Jandhyala, V., Fotopoulos, S., MacNeill, I. and Liu, P. (2013). Inference for single and multiple change-points in time series. J. Time Series Anal. 34 423–446.
  • Kao, C., Trapani, L. and Urga, G. (2016). Supplement to “Testing for instability in covariance structures.” DOI:10.3150/16-BEJ894SUPP.
  • Kollo, T. and Neudecker, H. (1997). Asymptotics of Pearson–Hotelling principal-component vectors of sample variance and correlation matrices. Behaviormetrika 24 51–59.
  • Ling, S. (2007). Testing for change points in time series models and limiting theorems for NED sequences. Ann. Statist. 35 1213–1237.
  • Litterman, R. and Scheinkman, J.A. (1991). Common factors affecting bond returns. J. Fixed Income 1 54–61.
  • Móricz, F. (1983). A general moment inequality for the maximum of the rectangular partial sums of multiple series. Acta Math. Hungar. 41 337–346.
  • Müller, U.K. (2014). HAC corrections for strongly autocorrelated time series. J. Bus. Econom. Statist. 32 311–322.
  • Perignon, C. and Villa, C. (2006). Sources of time variation in the covariance matrix of interest rates. Journal of Business 79 1535–1549.
  • Pesaran, H.H. and Shin, Y. (1998). Generalized impulse response analysis in linear multivariate models. Econom. Lett. 58 17–29.
  • Phillips, P.C.B. and Solo, V. (1992). Asymptotics for linear processes. Ann. Statist. 20 971–1001.
  • Stock, J.H. and Watson, M.W. (1999). Forecasting inflation. J. Monet. Econ. 44 293–335.
  • Stock, J.H. and Watson, M.W. (2002). Forecasting using principal components from a large number of predictors. J. Amer. Statist. Assoc. 97 1167–1179.
  • Stock, J.H. and Watson, M.W. (2005). Implications of dynamic factor models for VAR analysis. Manuscript.
  • Stock, J.H. and Watson, M.W. (2012). Disentangling the channels of the 2007-09 recession. Brookings Pap. Econ. Act. 1 81–135.

Supplemental materials

  • Supplement to “Testing for instability in covariance structures”. We provide technical Lemmas, and further Monte Carlo output.