Zero-lag white noise vector bilinear autoregressive time series models

Abstract

The non linear part of a mixed bilinear time series structure seems to pose difficulty if we are to extract the pure autoregressive (AR) bilinear form from the mixed process with the condition that the outcome of such extraction clearly defines itself as an extension from its parent linear AR model. It is therefore of immense interest to address a ‘ bilinear’ situation where the same “order” identified in the linear AR processes are extended to cover the linear and non linear components of a bilinear process with an exception that the lagged white noise process is allowed to remain in its present state. This research focused on these two innovations where the white noise is lagged zero to isolate a pure vector AR bilinear model from a mixed process based on the distribution of autocorrelation and partial autocorrelation function of the different series involved in a vector process, and the extension of the linear ‘orders’ to bilinear ‘orders’. To achieve the aforementioned, we formulated a matrix for a general case of n-dimensional vector for an AR process and then considered a special case of zero lag of white noise. With given conditions, and introduction of diagonal matrix of lagged vector elements, special bilinear expressions reflecting the same ‘orders’ of the corresponding linear forms emerged. The zero lagged white noise denoted by it-0 clearly defined our models as pure AR bilinear models since the lag l = 0 and is equivalent to the current state white noise t of the linear AR process. These gave a brilliant meaning to vector bilinear AR processes in terms of linear AR ‘orders’. The workability of these special bilinear models was assessed by applying them to revenue series and the result showed that the models gave a good fit, in support of our idea.