Moving block bootstrapping for a CUSUM test for correlation change
Introduction
In global economy, in empirical finance or in other many areas, interdependence among countries, among financial institutions or among time series variables exist generally. Accordingly, correlation coefficient is widely used as a measure of the extent of the dependence. For example, Campbell et al. (1997) and Engle (2008) presented that the correlation is instrumental in determining the risk as a main factor of the optimal portfolio selection. The correlation can be taken as constant or as time-varying with persistent dynamics in various ways. However, since the correlation tends to increase in times of financial crisis, the constancy of the correlation is not satisfied over time, see Krishnan et al. (2009). Therefore, interest in testing constancy of the correlation is gradually increasing.
Several studies are conducted for testing constancy of correlation. Galeano and Peña (2007) and Aue et al. (2009) provided break tests for the covariance structure. Wied et al. (2012) proposed a CUSUM type correlation constancy test in order to detect changes in correlation rather than changes in covariance. Wied et al. (2012) demonstrated that their test based on the correlation change is generally more powerful than the test based on covariance change for the alternatives of correlation changes. Galeano and Wied (2014) developed a method to estimate both the time and the number of multiple change points by applying the usual binary segmentation procedure to the CUSUM test of Wied et al. (2012). Wied (2017) proposed a nonparametric procedure for the break test using bootstrapping variance. On the other hand, as will be demonstrated, all of these break tests show severe size distortion in the samples having conditional heteroscedasticity and/or serial correlation.
Size performance can be improved using bootstrap methods. For example, see Jouini (2010) for single structural change tests, Hlávka et al. (2015) for autoregressive models, He et al. (2008) for stationary vector autoregressive models against continuous change, Cavaliere and Taylor (2008) for unit root tests with nonstationary volatility and Buesetti and Sanzo (2012) for LR tests of stationarity, common trends and cointegration. For weakly dependent data, the moving block bootstrap of Künsch (1989) is widely used, see Liu and Singh (1992), Lahiri, 1993, Lahiri, 2003, and many others. We apply the moving block bootstrap to the correlation break test of Wied et al. (2012) and develop a bootstrap test.
The bootstrap test is constructed by applying quantiles of bootstrapped CUSUM test statistic as critical values of the CUSUM test of Wied et al. (2012). The critical values will be shown to be valid in that the CUSUM test and bootstrapped CUSUM test have the same limiting null distribution. Consistency of the bootstrap test will also be proved under an alternative hypothesis of a correlation break.
A Monte Carlo experiment reveals that the proposed bootstrap test has good size in conditionally heteroscedastic samples and in serially correlated samples, which resolves the over-size problem of the existing tests under conditional heteroscedasticity and/or under serial correlation. On the other hand, the bootstrap test has some worse power performance than the existing tests in a case of double breaks but not in some cases of a single break.
The proposed bootstrap test can improve the break time detection procedure of Galeano and Wied (2014) which first detects a presence of break by the CUSUM test of Wied et al. (2012) and next detects the time of break if any. In case of conditional heteroscedasticity and serial correlation, since the test of Wied et al. (2012) is over-sized, the procedure of Galeano and Wied (2014) would over-detect breaks and break times. This over-detection problem may be resolved if test of Wied et al. (2012) in the first step of Galeano and Wied (2014) is replaced by our bootstrap test.
The remaining of the paper is organized as follows. Section 2 describes the CUSUM test for correlation break and its asymptotic null distribution. Section 3 develops the bootstrap CUSUM test and establishes its asymptotic validity. Section 4 contains a finite sample Monte Carlo simulation. Section 5 applies the bootstrap CUSUM test to real data sets. Section 6 gives a conclusion.
Section snippets
Correlation break tests
Let be a sequence of bivariate random vectors with finite th absolute moments for some . Assume that a sample is given. We wish to test whether the correlation between and changes during the observation time. The null hypothesis of constant correlation is for a constant and the alternative hypothesis is . Wied et al. (2012) proposed a CUSUM test for correlation break given by
A moving block bootstrap test
We construct a moving block bootstrapping (MBB) test and prove its asymptotic validity. The MBB is proposed by Künsch (1989) and the validity of the MBB is shown by many researchers, see Lahiri (2003) and others. The MBB is suitable for weakly dependent data in that the bootstrap samples have the same serial correlation structures as the original samples. We prove consistency of the null bootstrapping distribution of and propose a bootstrap test which remedy the size distortion problem
Monte Carlo simulation
Finite sample performances of the moving block bootstrap test proposed in Section 3 is compared with the existing tests of Wied et al. (2012) and Wied (2017) based on central limit theorem (CLT). A Monte Carlo experiment will show stabler size of the bootstrap test than the CLT tests. The CLT test by Wied et al. (2012) is in (1). The CLT test of Wied (2017) is, in the bivariate cases, where , , , and
Examples
We revisit the data set of the daily log returns of the Euro Stoxx 50 index and the stock prices of its major three companies (Sanofi, Siemens, BASF) for the period of 01/01/2007–01/06/2012, which was considered by Wied (2017) to see correlation breaks in the return pairs during the period of the global financial crisis. Correlation breaks in the volatility pairs are also investigated by analyzing the pairs of absolute return values. The data set is obtained from the websites (//finance.yahoo.com
Conclusion
We have constructed a correlation break test applying moving block bootstrap. The bootstrap test is constructed by replacing the critical values of the CUSUM test of Wied et al. (2012) with the quantiles of the bootstrapped CUSUM statistic. Consistencies of the bootstrap test are established both under the null and under the alternative hypothesis by establishing the functional central limit theorem for the partial sum process of the moving bootstrap samples. A Monte Carlo simulation reveals
Acknowledgments
The authors are very thankful for the fact that the paper improves a lot by exploiting the constructive comments of two referees. This study was supported by a grant from the National Research Foundation of Korea (2016R1A2B4008780) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827).
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