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The multi-dimensional vector autoregressive (VAR) time series is often used to model the impulse-response functions of macroeconomics variables. However, in some economical applications, the variable of main interest is the product of time series describing market variables, like e.g. the cost, being the product of price and volume. In this paper, we analyze the product of the bi-dimensional VAR(1) model components. For the introduced time series, we derive general formulas for the autocovariance function and study its properties for different cases of cross-dependence between the VAR(1) model components. The theoretical results are then illustrated in the simulation study for two types of bivariate distributions of the residual series, namely the Gaussian and Student’s t. The obtained results are applied for the electricity market case study, in which we show that the additional cost of balancing load prediction errors prior to delivery can be well described by time series being the product of the VAR(1) model components with the bivariate normal inverse Gaussian distribution.

Product of bi-dimensional VAR(1) model components. An application to the cost of electricity load prediction errors