ON GENERATING MECHANISMS AND DETECTION OF OUTLIERS IN MULTIVARIATE TIME SERIES

Abstract

Outliers are aberrant observations that adversely affect parameter estimation and predictive capability of a given model. The problem of outlier detection in time series has gained much attention in the literature and various methods of detection have been developed, but are limited to univariate time series with its attendant swamping effect. This work is focused on developing outlier Generating Mechanisms (GMs) for the detection of outliers in the Multivariate Time Series (MTS) setting that is capable of ameliorating the swamping effect. Two-variable Vector Autoregressive (VAR) models were considered, where Xit and Xjt-1, i,j=1, 2 were the current and lagged values of the response and explanatory variables respectively, , i,j=1, 2, were coefficients, t is the time and and were distributed as N(0,?). Each series was assumed to have been generated by the model f(Z_t,?_t (?),??_t^T) where Zt is an outlier free time series, is a time indicator where for all otherwise, = 1- ?1B-�� � ?pBp were polynomials of order p and ?=(?_1,�,?_k )^' were the magnitude of outliers. The nature of effect of outlier on uncontaminated series determines the model which could be Innovative (IO), Additive (AO), Multiplicative (MO), and Convolution (CO) which is the combination of IO and AO effects. These models were used to develop four GMs for detection of outliers in multivariate time series. The magnitudes of outliers and their variances with the test statistics were derived for the four generating mechanisms. Simulation data of sample sizes of 10, 50, and100 were used to establish the validity of the developed models.Data on Nigerian Gross Domestic Product (GDP) and Consumer Price Index (CPI), commercial bank deposits and loans were also used. Estimates of the magnitude and residual variance of outliers were obtained using method of least squares. The percentages of outliers detected for simulated data and the number of detected outliers in data sets were observed. The relative efficiency of the models was evaluated in determining the best outlier generating mechanism. The developed generating mechanisms were: ?? ?(?)(1+?), ?? (?+?(?)),X ??? ? (?)and ?? [2?(?)+?(1+?(?))] for IO, AO, MO and CO respectively. The performance of the generating mechanisms based on simulations showed that the percentages of outliers detected using IO, AO, MO, and CO were 21%, 71%, 86%, and 100% respectively. For GDP and CPI, 30 outliers were detected by CO; 29 each by IO and AO while MO was unable to detect any outlier because it did not exhibit any multiplicative effect on the data. For deposit and loan, 6 outliers each were detected by all the GMs except MO. The CO gave a high precision with low percentage of variation compared with other generating mechanisms. It was observed that whenever the explanatory variable was infested with outlier, the response variable is also contaminated. The derived outlier generating mechanisms were able to detect potential outlier independently in multivariate time series with the swamping effect ameliorated. The pairwise relative efficiency of the variances indicated that convolution model was the best. It is therefore recommended for outlier detection in multivariate time series setting.