Cross-codifference for bidimensional VAR(1) time series with infinite variance

In this paper, we consider the problem of a measure that allows us to describe the spatial and temporal dependence structure of multivariate time series with innovations having infinite variance. By using recent results obtained in the problem of temporal dependence structure of univariate stochastic processes, where the auto-codifference was used, we extend its idea and propose a cross-codifference measure for a general bidimensional vector autoregressive time series of order 1 (bidimensional VAR(1)). Next, we derive analytical results for VAR(1) model with Gaussian and stable sub-Gaussian innovations, that are characterized by finite and infinite variance, respectively. We emphasize that obtained expressions perfectly agree with the empirical counterparts. Moreover, we show that for the considered time series the cross-codifference simplifies to the well-established cross-covariance in the case when the innovations of time series are given by Gaussian white noise. The last part of the work is devoted to the statistical estimation of VAR(1) time series parameters based on the empirical cross-codifference. Again, we demonstrate via Monte Carlo simulations that the proposed methodology works correctly.