## Bernoulli

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

### Testing for instability in covariance structures

Chihwa Kao, Lorenzo Trapani, and 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 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

**Source**

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

**Dates**

Received: September 2014

Revised: August 2016

First available in Project Euclid: 27 July 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1501142461

**Digital Object Identifier**

doi:10.3150/16-BEJ894

**Mathematical Reviews number (MathSciNet)**

MR3706775

**Zentralblatt MATH identifier**

06778346

**Keywords**

changepoint covariance matrix CUSUM statistic eigensystem

#### Citation

Kao, Chihwa; Trapani, Lorenzo; Urga, Giovanni. Testing for instability in covariance structures. Bernoulli 24 (2018), no. 1, 740--771. doi:10.3150/16-BEJ894. https://projecteuclid.org/euclid.bj/1501142461

#### Supplemental materials

- Supplement to “Testing for instability in covariance structures”. We provide technical Lemmas, and further Monte Carlo output.Digital Object Identifier: doi:10.3150/16-BEJ894SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.