VECTOR BILINEAR AUTOREGRESSIVE TIME SERIES MODEL AND ITS SUPERIORITY OVER ITS LINEAR AUTOREGRESSIVE COUNTERPART

In this research, a vector bilinear autoregressive time series model was proposed and used to model three revenue series   t t t X X X 3 2 1 , , . The “orders” of the three series were identified on the basis of the distribution of autocorrelation and partial autocorrelation functions and were used to construct the vector bilinear models. The estimates obtained from the bilinear fits were compared graphically with those obtained from fitting linear (autoregressive) models. Residual variance and Box-Ljung Q statistic comparisons were also made. The result showed that vector bilinear autoregressive (BIVAR) models provide better estimates than the long embraced linear models.


INRODUCTION
In the past seven decades, a time series was usually modeled as a linear function of its own past, using autoregressive (AR) or mixed autoregressive moving average (ARMA) frame work. This was because these models are easy to analyze and they provide fairly good approximations for the true underlying process.
However, the underlying structure of the series may not be linear and what is more, the series may not be Gaussian. In these situations, second order properties, such as co-variances and spectra, can no longer adequately characterize the properties of the series. This called for the emergence of non linear models in which bilinear forms a class. Weiner (1958) considered a nonlinear relationship between an input t u and an output t X (both observable) using Volterra series expansion given by ...
From a given finite realization of a process, one cannot estimate the parameters     ,...

, ,
efficiently. To overcome this difficulty, Granger and Anderson (1978) introduced a class of nonlinear models called "bilinear" in the time series context [assuming  Maravall (1983) used a bilinear model to forecast Spanish monetary data and reported a near 10% improvement in one-step ahead mean square forecast errors over several autoregressive moving average (ARMA) alternatives. There is no doubt that most of the economic or financial data assume fluctuations due to certain factors. James (2014) used a bilinear model to forecast South Africa's gross domestic product (GDP). Comparison was made with the forecasts generated by vector autoregressive (VAR) models. The result showed that bilinear forecasts were better than the VAR forecasts.
The general form of the bilinear model according to Rao (1980) is given by the difference equation: Where   ) (t e is an independent white noise process  l and 1 l . Iwueze (2002) studied the existence and computation of all second order moments of the vector valued time series of the form He found that the vectorial representation leads to an important result on matrix algebra with respect to the spectral radius of Kronecker product of matrices. Boonchai and Eivind (2005) gave the general form of multivariate bilinear time series models as: Here the state ) (t and recorded its advantages over the pure vector autoregressive moving average models. We have noted here that except for Boonchai and Eivind (2005) who gave a theoretical form in population dynamics, other works in bilinear time series were based either on mixed ARMA univariate cases or vector of lagged variables of the same time series.
In this research, however, an AR bilinear process is isolated from the vector framework of Iwok and Etuk (2009). That is, a multivariate bilinear AR case where each element of the vector is being explained by the lag values of itself and other time series variables in both the linear and non linear components of the model. This differs from the work of Iwueze (2002) where only one time series was involved and the vector form referred to lagged variables of the same series. Our objectives extend to comparing the performance of our new vector concept with the long embraced linear model.

METHODS OF ESTIMATION
be a vector of n-dimensional time series.

Linear AR Model (i) Univariate case:
This is a model in which the current value of the process is expressed as a finite, linear aggregate of previous values of the process and a shock it where p    ,..., , 2 1 are constants and   it  are a purely random processes.

(i) Vector Case:
The general vector analogue to the univariate Autoregressive time series models for the n-series is:

Vector Non Linear Models
Given the vector elements , the non linear model for a pure AR process is: where ir kl.
 are the bilinear parameters of the product series and

Bilinear Vector Autoregressive Model (BIVAR)
Combining equations (7) and (8), the BIVAR model emerges: Unlike (6), Equation (9) comprises both the linear and non linear components. This study seeks to compare the performances of the two models (Linear and Bilinear). The parameters of the different models are estimated using linear and intrinsic linear regression techniques.

Estimates for the linear models:
The distribution of autocorrelation and partial autocorrelation functions of the non stationary series suggested pure AR process of order 3 for t X 1 , AR process of order 2 for t X 2 and AR of order 1 for t X 3 . The regression estimates obtained provide the following Autoregressive models for the three series of the vector: (i) VECTOR BILINEAR AUTOREGRESSIVE TIME SERIES MODEL AND ITS SUPERIORITY Table1: Three sources of internal generated revenue (X 1t ,X 2t ,X 3t ) S/N X 1t X 2t X 3t S/N X 1t X 2t X 3t S/N X 1t X 2t

Estimates for the BIVAR models:
The bilinear vector autoregressive model consists of two parts. The first part is the linear vector AR process, while the second part is the product of lagged vector elements and white noise. Estimates of the BIVAR parameters and fits were obtained by treating equation (9) as an intrinsically linear model. The following parameter estimates were obtained: As could be seen above, these models are linear in states k it X  but non-linear jointly with l it   as the name 'bilinear' implies.

A Comparison of the Linear Model and Vector Bilinear Autoregressive Time Series Models (i) Residual Variances
After fitting the models, the calculated residual variances from the estimated linear equations (10)- (12)   Examination of the actual and estimates plots above show that AR model estimates exhibit less interwoven behaviour with the real data than BIVAR estimates plots. This is an indication of a high degree performance of the bilinear models.

DISCUSSION
As noted in the literature review, most works assume that a bilinear model is a function of lag variables of the dependent variables. This is a situation where the same series is modeled using the lag values of itself. This work, however, differs in this approach. In this work, each element of the vector is being explained by the lag values of itself and other time series variables in both the linear and non linear components of the model. It is believe that this research has provided another approach to bilinear time series modeling.

CONCLUSIONS
As mentioned earlier, a linear time series model such as AR expresses itself as a linear combination of its past and has been widely used in diverse fields. Due to non stationarity of most series, however, bilinear models have replaced linear models by offering better analytical tools for analyzing several time series data.
From the minimum variance property, Q statistic and graphical verdict shown in this work, there is no gain saying the fact that some series especially, revenue series assume not only linear component but also non linear part. This is so because of the random nature of observations assume by certain processes. The result of this work confirms that non linear models such as 'bilinear vector AR' are superior to pure linear AR models.