Financial markets are highly correlated systems that reveal both the inter-market dependencies and the corrletations among their different components. Standard analyzing techniques include correlation coefficients and auto- or cross-correlation functions used in a case of one, a pair or at most a few signals under study, as well as correlation matrices useful if one deals with rich multivariate data. In the latter case, if the data consists of N signals, one constructs an N×N real symmetric matrix with N real non-negative eigenvalues describing the correlation structure of the data. Properties of this matrix can then be compared with predictions of Random Matrix Theory and the related ensemble of Wishart matrices in order to reveal any genuine nonuniversal properties of the system under study.

However, a serious limitation of this approach is that such a formalism is a good choice only in a case of signals recorded simultaneously in time orof signals without this time-synchronism but still in absence of any variable time delays. In the opposite case, if one performs a correlation-function-like analysis of multivariate data, when a stress is put on investigation of delayed dependencies among different types of signals, one has to calculate the correlation matrix in a slightly different way which results in its different properties. For example, if there are two sets, each of them comprising N/2 signals, and one is to describe solely the cross-correlations between these sets without looking at the intra-set dependencies, it is recommended to calculate an (N/2)×(N/2) matrix with elements being the τ-delayed correlation coefficients for pairs of signals such that each signal in a pair comes from a different set. Now the matrix is no longer Hermitean and has the spectrum consisting of pairs of complex conjugate eigenvalues with some real eigenvalues also possible. From the RMT point of view this kind of matrices is closely, though not exactly, related to the so-called Ginibre Orthogonal Ensemble (GinOE) and the RMT predictions for this ensemble can be treated as a null hypothesis in this case.

In our talk we present a few examples of practical application of real non-Hermitean matrices in correlation analyses of empirical data. Treating the time lag τ as a variable, we are able to identify temporal structure of the inter-market correlations.

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