INTRODUCTION
Most time series analysts assume linearity and stationarity,
for technical convenience, when analyzing macroeconomic and financial
time series data (Franses, 1998). However, evidence of nonlinearity which
is usually found in the dynamic behaviour of such data implies that classical
linear models are not appropriate for modelling these series (Subba Rao
and Gabr, 1984). In most cases, nonlinear forecast is more superior to
linear forecast. 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 ARMA alternatives. There is nogainsaying
the fact that most of the economic or financial data assume fluctuations
due to certain factors. That is why the use of nonlinear models in forecast
gives higher precision than linear models.
According to Granger and Anderson (1978), the general
Bilinear Autoregressive Moving Average model of order (p,q,P,Q), denoted
by BARMA (p,q,P,Q) takes the form
where, ε_{t} is strict white noise. The
model is thus linear in the X`s and also in the ε`s separately, but
not in both. It obviously includes all the ARMA (p,q) models as a special
case. At this point, it is convenient to give names to several subclasses
of the general model.
The complete bilinear model with p = q = 0 is
This can be written in matrix form as
X_{t} = (ε_{ti}, ...,ε_{tQ})β(X_{t1},
X_{t2}, ..., X_{tp})’+ ε_{t} 
Where, β is the (QxP) matrix of coefficients
β={ },
k = 1,...,Q;
= 1,...,P 
If
= 0, for all k > ,
the model is called superdiagonal.
Bibi and Oyet (1991) defined a process (X_{t})_{t€z}
on a probability space (Ω,ξ,P) as a time varying bilinear process
of order (p,q,P,Q) and denoted by BL(p,q,P,Q), if it satisfies the following
stochastic difference Equation:
where, (a_{i,t}(a))_{1≤i≤p}, (c_{j,t}(c))_{1≤j≤q
}, (b_{ij,t}(b))_{1≤i≤P, 1≤j≤Q } are
timevarying coefficients which depend on finite dimensional unknown parameter
vectors a, c and b, respectively. The sequence (ε_{t})_{t∈z
}is a heteroscedastic white noise process. That is, (ε_{t})_{t∈z}
is a sequence of independent random variables, not necessarily identically
distributed, with mean zero and variance σ_{t}^{2}.
Moreover ε_{t } is independent of past X_{t}. The
initial values X_{t}, t < 1, and ε_{t, t<1 }are
assumed to be equal to zero.
Boonchai and Eivind (2005) stated the general form of
a multivariate bilinear time series model as:
X_{t} = Σ_{Ai}X_{ti}
+ ΣM_{j}e_{tj} + ΣΣΣB_{dij}X_{ti}e_{dtj}
+ e_{t} 
Here the state X_{t} and noise e_{t}
are nvectors and the coefficients A_{i}, M_{j} and B_{dij
}= 0, we have the class of wellknown vector ARMA models. The bilinear
models include additional product terms B_{dij}X_{ti}e_{dtj};
as the name indicates these models are linear in state X_{t} and
in noise e_{t }separately, but not jointly. From a theoretical
point of view, it is therefore natural to consider bilinear models in
the process of extending linear theory to nonlinear cases. According
to Boonchai and Eivind (2005) a particular reason for introducing bilinear
time series in population dynamics, is that they are suitable for modeling
environmental noise. One may start with a deterministic system with (constant)
parameters that describe conditions that depend on a fluctuating environment.
The idea is to replace them by stochastic parameters. Boonchai and Eivind
(2005) made extension first to univariate and then to multivariate bilinear
models. The main results give conditions for stationarity, ergodicity,
invertibility and consistency of least square estimates.
In this research, we are interested in estimation of
Bilinear Autoregressive Vector (BARV) models. We consider three vectors,
which consist of a response and two predictor vectors. The data source
is a monthly generated revenue (for a period of ten years) of a Local
Government Area in Nigeria.
METHODS OF ESTIMATION
Linear Model
The general multivariate analogue to the univariate Autoregressive
Moving Average (ARMA) model for the vectors is:
where, X_{it} = (X_{1t},X_{2t},...,X_{nt})
are vectors, γ_{a.if} are matrices of coefficients of the
autoregressive parameters, Є_{jt} are the vector white noise,
λ_{b.jh} are matrices of coefficients of the moving average
parameters, (r = n).
NonLinear Model
The nonlinear models for X_{1t}, X_{2t}, X_{3t},...,X_{nt}
is:
Where,
X_{it} = ( X_{1t}, X_{2t}, .
. . ,X_{nt}), β_{ab.ij} are the matrices of coefficients
of the respective vector product series.
Bilinear Autoregressive Vector Model
The general BARV model may be written in the form:
Where,
Vectors and coefficients are as described above.
RESULTS AND DISCUSSION
Estimates for BARV Models
The distribution of autocorrelation and partial autocorrelation functions
of the non stationary series suggested pure autoregressive process of
order 3 for X_{1t }, autoregressive process of order 2 for X_{2t}
and autoregressive process of order 1 for X_{3t}. The vector autoregressive
bilinear process is a process which consists of two parts. The first part
is a pure autoregressive process of the vector series, while the second
part is the product of the vector series and white noise. The regression
estimates obtained provides the following models for the three vector
series:

0.661X_{1t1}–0.184X_{2t1} + 0.0148X_{1t2}
+ 0.386X_{2t2} + 0.205X_{1t3
}+ Є_{1t }+ 0.00165X_{1t1}Є_{1t0}
+ 0.00130X_{1t2}Є_{1t0 }+ 0.000546X_{1t3}Є_{1t0
}0.00372X_{1t1}Є_{2t0}–0.00080X_{1t2}Є_{2t0}–
0.000225X_{1t3}Є_{2t0
}0.00088X_{2t1}Є_{1t0} + 0.00067X_{2t2}Є_{1t0}
+ 0.00410X_{2t1}Є_{2t0
}+ 0.00228X_{2t2}Є_{2t0} 
(5) 
From model (5)

γ_{1.11 = }0.661, γ_{1.12 =}
0.184, γ_{2.11 = }0.0148, γ_{2.12 =} 0.386,
γ_{3.11 = }0.205 β_{10.11 = }0.00165,
β_{20.11 = }0.00130, β_{30.11 = }0.000546,
β_{10.12 }=  0.00372 β_{20.12 }= 
0.00080, β_{30.12 }=  0.000225, β_{10.21 }=
 0.00088, β_{20.21 }= 0.00067 and β_{10.22
}= 0.00410, β_{20.22 }= 0.00228. 

0.194X_{1t1} + 0.202X_{2t1}
+ 0.0824X_{1t2} + 0.290X_{2t2} + 0.120X_{1t3
}+ Є_{2t} + 0.00027X_{1t1}Є_{1t0}
+ 0.00124X_{1t2}Є_{1t0 } 0.000171X_{1t3}Є_{1t0
} 0.00161X_{1t1}Є_{2t0} – 0.00148X_{1t2}Є_{2t0}
+ 0.000557X_{1t3}Є_{2t0
}+ 0.00047X_{2t1}Є_{1t0}  0.00283X_{2t2}Є_{1t0}
+ 0.00189X_{2t1}Є_{2t0
}+ 0.00667X_{2t2}Є_{2t0}

(6) 
From model (6)

γ_{1.21 = }0.194, γ_{1.22 =}
0.202, γ_{2.21 = }0.0824, γ_{2.22 =} 0.290,
γ_{3.21 = }0.120 β_{10.11 = }0.00027,
β_{20.11 = }0.00124, β_{30.11 = }0.000171,
β_{10.12 }=  0.00161 β_{20.12 }= 
0.00148, β_{30.12 }= 0.000227, β_{10.21 }=
0.00047, β_{20.21 }=  0.00283 and β_{10.22
}= 0.00189, β_{20.22 }= 0.00667. 

0.466X_{1t1} – 0.385X_{2t1}
 0.0676X_{1t2} + 0.0965X_{2t2} + 0.0851X_{1t3
}+ Є_{3t} + 0.00138X_{1t1}Є_{1t0}
+ 0.000058X_{1t2}Є_{1t0 }+ 0.000717X_{1t3}Є_{1t0
} 0.00211X_{1t1}Є_{2t0} + 0.00069X_{1t2}Є_{2t0}
– 0.000782X_{1t3}Є_{2t0
} 0.00135X_{2t1}Є_{1t0} + 0.00350X_{2t2}Є_{1t0}
+ 0.00220X_{2t1}Є_{2t0
}+ 0.00439X_{2t2}Є_{2t0}. 
(7) 
From model (7)

γ_{1.31 = }0.466, γ_{1.32 =}
 0.385, γ_{2.31 = } 0.0676, γ_{2.32 =}
0.0965, γ_{3.31 = }0.0851 β_{10.11 = }0.00138,
β_{20.11 = }0.000058, β_{30.11 = }0.000717,
β_{10.12 }=  0.00211 β_{20.12 }= 0.00069,
β_{30.12 }=  0.000782, β_{10.21 }=  0.00135
β_{20.21 }= 0.00350 and β_{10.22 }= 0.00220,
β_{20.22 }=  0.00439 
The first set of estimates in models 57 forms the parameter
estimates of the linear part, while the second set are the parameter estimates
of interactive products of vectors.
The vector models for X_{1t}, X_{2t} and X_{3t} are
used to obtain estimates, which are shown in Appendix 2. The
actual and estimated values in Appendices 1 and 2
are for each vector in Fig. 13.

Fig. 1: 
Plots of actual and estimates of a response vector
X_{1t} 

Fig. 2: 
Plots of actual and estimates of a predictor vector
X_{2t} 

Fig. 3: 
Plots of actual and estimates of the second predictor
vector X_{3t} 
CONCLUSIONS
There is no gainsaying the fact some series, especially,
revenue series assume not only linear component, but both linear and nonlinear
components. This is so because of the random nature of observations assume
by certain processes. It is in this regards that bilinear multivariate
time series models were developed. The Bilinear Autoregressive Vector
Models established in this paper provide better estimates for most nonstationary
revenue series than pure linear models.
Appendix 1: 
Actual internally generated revenue series represented
by three vectors 

Appendix 2: 
Regression estimates from bilinear autoregressive vector
models 
