Multiple break detection in the correlation structure of random variables

Abstract

Correlations between random variables play an important role in applications, e.g. in financial analysis. More precisely, accurate estimates of the correlation between financial returns are crucial in portfolio management. In particular, in periods of financial crisis, extreme movements in asset prices are found to be more highly correlated than small movements. It is precisely under these conditions that investors are extremely concerned about changes on correlations. A binary segmentation procedure to detect the number and position of multiple change points in the correlation structure of random variables is proposed. The procedure assumes that expectations and variances are constant and that there are sudden shifts in the correlations. It is shown analytically that the proposed algorithm asymptotically gives the correct number of change points and the change points are consistently estimated.It is also shown by simulation studies and by an empirical application that the algorithm yields reasonable results.

Dating multiple change points in the correlation matrix

“…The ideas of the proofs are similar to those in Galeano and Wied (2013) and Wied (2014), whereas we additionally have to ensure that the bootstrap variance estimatorÊ…”

“…As in Galeano and Wied (2013), we assume that there are dominating change points (if any) in order to obtain analytic results.…”

“…Moreover, as in Galeano and Wied (2013), one could relax Assumption 2 by only assuming the existence of finite q-th moments for a q > 1. Finally, we do not need Assumption 5 on a block length as the bootstrap estimator is excluded from the argmax estimator.…”

“…Applying the proof of Theorem 1 in Galeano and Wied (2013) component by component and making use of the fact that a variance estimator likeÊ −1/2 η(l 1 ),ξ(l 2 ) is excluded, one shows that, uniformly on [l 1 , l 2 ], B η(l 1 ),ξ(l 2 ) (z) → a.s. CP l 1 ,l 2 (z) for a certain constant C. Then, the theorem follows by applying the argmax continuous mapping theorem.…”

“…The proposed procedure is a binary segmentation type procedure with a refinement step to avoid the presence of false change points due to multiple testing. The method is an extension of the method proposed by Galeano and Wied (2013) for detecting multiple change points in the correlation between two random variables. The procedure proceeds as follows: first, we determine the "dominating" change point and decide if this point is statistically significant.…”