On the Stationarity of Multivariate Time Series

Abstract

The purpose of this work is to investigate the effect of non-stationary process in the multivariate time series.
The investigation was carried out using three vector series X1t, X2t and X3t representing the average, the urban and the rural consumer price indices respectively in Nigeria. The data were collected from CBN Statistical Bulletin from 1995-2018. Of the three vector series, X2t (urban price index) was conditioned to be in the state of disequilibrium (non-stationary state). The cross-autocorrelation matrix of the three vector series up to the second lag produced nine sub-cross-autocorrelation matrices. The sub-matrices are made up of negative and positive cross-autocorrelations. The positive definiteness property was investigated for each sub-autocorrelation matrix. The joint processes with positive cross-correlations met the stationarity condition, while processes with negative cross-correlations violated the condition. The violation of the stationarity condition was as a result of the unstable urban consumer price index. The investigation revealed that when a state is unstable in the multivariate time series, the stability of the multivariate process is affected by way of partial stationarity.