INTRODUCTION
Most time series behave as though they have no fixed mean. Such series have arisen in forecasting and control problems, and all of them exhibit behavior suggestive of non stationarity.
Evidence of non linearity which is usually found in the dynamic behavior of such data implies that classical linear models are not appropriate for most time series. This called for the emergence of nonlinear models in which bilinear forms a special class.
Maravall (1983) used a bilinear model to forecast Spanish
monetary data and reported a near 10% improvement in onestep 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.
On the contrary, Imeh (2007) observed that in certain
time series applications, univariate linear estimates are comparatively better
than those obtained from bilinear models.
Similar research engaged the attention of John (2008).
He investigated the relative merits of multivariate linear process and univariate
bilinear process using Canadian money and income data. The reports revealed
that the linear models performed better than the bilinear models.
However, it is the objective of this study to establish a bilinear concept from a vector point of view and compare it performance with the vector linear model using a trivariate case of time series. That is, two multivariate cases of three vector elements each.
The general form of the bilinear model according to Granger
and Anderson (1978) is given by:
Similarly, Rao (1981) described bilinear time series
model BL (p, r, m, k) as given by the difference equation:
where, {e(t)} is an independent white noise process and C_{0} = 1. {X(t)} is termed the bilinear process. The autoregressive moving average model ARMA(p, r) is obtained from Eq. 2 by setting d_{ll}1 = 0 ∀ l and l^{1}.
Parameter estimation of bilinear processes has been studied for particular
cases by Bouzaachane et al. (2006) .
Boonchai and Eivind (2005) gave the general form of
multivariate bilinear time series models as:
Here, the state X(t) and noise e(t) are nvectors and the coefficients A_{i}, M_{j} and B_{dij} are n by n matrices. If all B_{dij} = 0, we have the class of wellknown vector ARMAmodels.
Iwok and Akpan (2007) established the matrix form of
vector autoregressive time series as:
and recorded its advantages over the univariate case.
In this study, we are interested in the comparative performance of vector linear models and vector bilinear models. We considered three series of a vector and each was taken as response variable and the remaining two were lagged predictors. The data used for estimation are three sources of monthly generated revenue (for a period of ten years) from Ik. L.G.A. in Nigeria.
MATERIALS AND METHODS
Let X^{l}_{it} be a vector of ndimensional time series.
Linear Model
The general vector (VARMA) analogue to the univariate autoregressive moving
average (ARMA) for the nseries is:
where, γ_{k.ir} and λ_{l}.it are the autoregressive (AR) and Moving Average (MA) parameters. p and q are the AR and MA orders. ε^{l}_{it} [ε_{1i}, ε_{2t}, ..., ε_{nt}] is a vector of white noise. k and l represent the lags of AR and MA models.
is the vector AR part.
is the vector moving average part.
If λ_{l}.it = 0 for all lagged white noise, the linear vector AR (VAR) model can be isolated and written in the form:
Similarly, vector MA (VEMA) part can be obtained by setting γ_{k.ir} = 0 and the resulting expression is:
Vector Non Linear Models
Autoregressive (AR) Process
Given the vector elements X_{1t},X_{2t}, ....., X_{nt},
the non linear model for an AR process is:
where, β_{kl.ir} are the bilinear parameters of the product series and l = 0 ∀ q.
Moving Average (MA) Process
For the moving average process, the non linear model is expressed as:
where, β_{kl.ir} are the bilinear parameters and k = 0 ∀ q.
Bilinear Vector AutoregressiveMoving Average Model (BIVARMA)
Combining Eq. 69, the BIVARMA model
emerges:
Unlike Eq. 5, Eq. 10 comprises both the vector linear and vector non linear components. This study seeks to compare the performances of the two vector models (Linear and Bilinear).
RESULTS
Estimates for the VARMA Model
The distribution of partial autocorrelation function of the non stationary
series suggested pure AR process of order 3 for X_{1t}, AR process of
order 2 for X_{2t} and AR of order 1 for X_{3t}. Similarly,
autocorrelation function of the series suggested pure MA process of order 1
for X_{1t}, MA process of order 1 for X_{1t} and MA of order
2 for X_{3t}. The regression estimates obtained provide the model below
for the vector linear part.
where, γ_{1.il} = 0.130, γ_{1.i2} = 0.235, γ_{1.i3} = 0.974, γ_{2.il} = 0.224, γ_{2.i2} = 0.020, γ_{3.i1} = 0.196, λ_{1.il} = 0.153, λ_{1.i2} = 0.153, λ_{1.i3} = 189, λ_{1.i3} = 242, λ_{2.i3} = 563.
Estimate for the BIVARMA Models
The bilinear vector autoregressive moving average model consists of two
parts. The first part is the linear vector VARMA model, while the second part
comprises the sum of the non linear components from AR and MA processes. The
non linear part is product of lagged vector elements and white noise. Estimates
of the BIVARMA parameters and fits were obtained by treating Eq.
10 as an intrinsically linear model. The following parameter estimates were
obtained:
The regression estimates obtained produce the following models for the three vector elements:
As could be seen above, these models are linear in states X_{itk} but nonlinear jointly with ε_{it1} as the name bilinear implies. The estimates provided by the above models are displayed in Table 2 and are found to be good, as evidenced by the closeness between the Actual and the estimated values.
Residual Variances
After fitting the models, the calculated residual variance from Eq.
11 is 81.19. Similarly, the residual variances for the bilinear vector models
in Eq. 1214 are 15.78 for X_{1t},
21.26 for X_{2t} and 21.13 for X_{3t}.Comparatively, the residual
variances of the bilinear models are smaller than the variance obtained from
the linear vector model. This makes bilinear vector ARMA models superior to
the linear ARMA counter part.
Plots of Actual and Estimate Values
The actual and estimated values of our models presented in Table
13 are plotted in Fig. 1a, b3a,
b below. Each figure displayed contains two plots (the actual
marked by o and the estimate marked by +).
Table 1: 
Three sources of internal generated revenue (X_{1t},X_{2t},X_{3t}) 

Table 2: 
BIVARMA Estimates of the generated revenues (X_{1t},X_{2t},X_{3t}) 

Table 3: 
Estimates of the VARMA model 


Fig. 1: 
(a) Vector BILINEARARMA and (b) Vector ARMA Plots of actual
and estimates of X_{1t} 

Fig. 2: 
(a) Vector BILINEARARMA and (b) VectorARMA Plots of actual
and estimates of X_{2t} 

Fig. 3: 
(a) Vector BILINEARARMA and (b) VectorARMA Plots of actual
and estimates of X_{3t} 
The plots reveal that there is a strong marriage between the actual and the estimated values of the bilinear vector ARMA models. This is an indication of a high degree performance by the bilinear models. However, the great disparity in the actual and estimate plots of the pure vector ARMA model makes it inferior to bilinear models.
CONCLUSION
In essence, this study established the vector bilinear ARMA model and compared
its performance with vector ARMA model. From the minimum variance property and
graphical verdict shown, 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 study confirms that non linear models such as bilinear vector
ARMA are superior to pure linear vector ARMA models.
DISCUSSION
Through our practical illustrations in applying the vector bilinear models, we are led to believe that this new class of models offers exciting potential in the analysis of revenue data and opens up new vistas.
Comparatively, present result has contradicted the conclusions drawn by Imeh
(2007) and John (2008). Whereas, Imeh
(2007) and John (2008) emerged linear models as
the best; this study is in support of Maravall (1983).
However, we have to note that this study utilized a vector bilinear approach as opposed to the univariate cases prioritized by the aforementioned authors; hence the difference in the respective outcomes.
Besides, we cannot ignore the fact that choice of data may also affect our results. Since present findings are restricted to economic time series data, evidence obtained here cannot be conclusive in general. Therefore we, suggest that this approach be extended to other cases, especially the modeling of hydrologic time series that are measured at short periods of time such as hourly or daily time intervals where the fitted stochastic models must take into account unique non linear properties of the data that are caused by complex physical processes.