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Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series



A.E. Usoro and C.O. Omekara
 
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ABSTRACT

This study was motivated by the need to establish multivariate time series models for pure autoregressive vector series which assume both linear and nonlinear components. General Bilinear Autoregressive Vector (BARV) models were established. The three vector series namely, a response vector (X1t) and predictor vectors (X2t) and (X3t) used for the modelling called for trivariate time series models as a special case of multivariate time series models and estimates obtained from the models. The finding in this study is the isolation of multivariate bilinear models for a pure autoregressive process based on the distribution of autocorrelation and partial autocorrelation functions of the series from mixed models. This has been achieved as the models were used for the estimation of the vector series. These prove reality of the BARV models established.

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  How to cite this article:

A.E. Usoro and C.O. Omekara, 2008. Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series. Asian Journal of Mathematics & Statistics, 1: 50-56.

DOI: 10.3923/ajms.2008.50.56

URL: https://scialert.net/abstract/?doi=ajms.2008.50.56

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 one-step ahead mean square forecast errors over several ARMA alternatives. There is no-gainsaying 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

Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series

where, εt 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

Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series
(1)

This can be written in matrix form as

Xt = (εt-i, ...,εt-Q)β(Xt-1, Xt-2, ..., Xt-p)’+ εt

Where, β is the (QxP) matrix of coefficients

β={Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series}, k = 1,...,Q; Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series = 1,...,P

If Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series = 0, for all k > Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series, the model is called superdiagonal.

Bibi and Oyet (1991) defined a process (Xt)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:

Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series

where, (ai,t(a))1≤i≤p, (cj,t(c))1≤j≤q , (bij,t(b))1≤i≤P, 1≤j≤Q are time-varying coefficients which depend on finite dimensional unknown parameter vectors a, c and b, respectively. The sequence (εt)t∈z is a heteroscedastic white noise process. That is, (εt)t∈z is a sequence of independent random variables, not necessarily identically distributed, with mean zero and variance σt2. Moreover εt is independent of past Xt. The initial values Xt, t < 1, and εt, 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:

Xt = ΣAiXt-i + ΣMjet-j + ΣΣΣBdijXt-iedt-j + et

Here the state Xt and noise et are n-vectors and the coefficients Ai, Mj and Bdij = 0, we have the class of well-known vector ARMA models. The bilinear models include additional product terms BdijXt-iedt-j; as the name indicates these models are linear in state Xt and in noise et 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 non-linear 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:

Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series
(2)

where, Xit = (X1t,X2t,...,Xnt) are vectors, γa.if are matrices of coefficients of the autoregressive parameters, Єjt are the vector white noise, λb.jh are matrices of coefficients of the moving average parameters, (r = n).

Non-Linear Model
The non-linear models for X1t, X2t, X3t,...,Xnt is:

Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series
(3)

Where,

Xit = ( X1t, X2t, . . . ,Xnt), βab.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:

Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series
(4)

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 X1t , autoregressive process of order 2 for X2t and autoregressive process of order 1 for X3t. 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:

X1t=

0.661X1t-1–0.184X2t-1 + 0.0148X1t-2 + 0.386X2t-2 + 0.205X1t-3
+ Є1t + 0.00165X1t-1Є1t-0 + 0.00130X1t-2Є1t-0 + 0.000546X1t-3Є1t-0
-0.00372X1t-1Є2t-0–0.00080X1t-2Є2t-0– 0.000225X1t-3Є2t-0
-0.00088X2t-1Є1t-0 + 0.00067X2t-2Є1t-0 + 0.00410X2t-1Є2t-0
+ 0.00228X2t-2Є2t-0
(5)

From model (5)

γ1.11 = 0.661, γ1.12 = 0.184, γ2.11 = 0.0148, γ2.12 = 0.386, γ3.11 = 0.205
β10.11 = 0.00165, β20.11 = 0.00130, β30.11 = 0.000546, β10.12 = - 0.00372
β20.12 = - 0.00080, β30.12 = - 0.000225, β10.21 = - 0.00088,
β20.21 = 0.00067 and β10.22 = 0.00410, β20.22 = 0.00228.

X2t=

0.194X1t-1 + 0.202X2t-1 + 0.0824X1t-2 + 0.290X2t-2 + 0.120X1t-3
+ Є2t + 0.00027X1t-1Є1t-0 + 0.00124X1t-2Є1t-0 - 0.000171X1t-3Є1t-0
- 0.00161X1t-1Є2t-0 – 0.00148X1t-2Є2t-0 + 0.000557X1t-3Є2t-0
+ 0.00047X2t-1Є1t-0 - 0.00283X2t-2Є1t-0 + 0.00189X2t-1Є2t-0
+ 0.00667X2t-2Є2t-0

(6)

From model (6)

γ1.21 = 0.194, γ1.22 = 0.202, γ2.21 = 0.0824, γ2.22 = 0.290, γ3.21 = 0.120
β10.11 = 0.00027, β20.11 = 0.00124, β30.11 = -0.000171, β10.12 = - 0.00161
β20.12 = - 0.00148, β30.12 = 0.000227, β10.21 = 0.00047,
β20.21 = - 0.00283 and β10.22 = 0.00189, β20.22 = 0.00667.

X3t=

0.466X1t-1 – 0.385X2t-1 - 0.0676X1t-2 + 0.0965X2t-2 + 0.0851X1t-3
+ Є3t + 0.00138X1t-1Є1t-0 + 0.000058X1t-2Є1t-0 + 0.000717X1t-3Є1t-0
- 0.00211X1t-1Є2t-0 + 0.00069X1t-2Є2t-0 – 0.000782X1t-3Є2t-0
- 0.00135X2t-1Є1t-0 + 0.00350X2t-2Є1t-0 + 0.00220X2t-1Є2t-0
+ 0.00439X2t-2Є2t-0.

(7)

From model (7)

γ1.31 = 0.466, γ1.32 = - 0.385, γ2.31 = - 0.0676, γ2.32 = 0.0965, γ3.31 = 0.0851
β10.11 = 0.00138, β20.11 = 0.000058, β30.11 = 0.000717, β10.12 = - 0.00211
β20.12 = 0.00069, β30.12 = - 0.000782, β10.21 = - 0.00135
β20.21 = 0.00350 and β10.22 = 0.00220, β20.22 = - 0.00439

The first set of estimates in models 5-7 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 X1t, X2t and X3t 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. 1-3.

Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series
Fig. 1: Plots of actual and estimates of a response vector X1t

Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series
Fig. 2: Plots of actual and estimates of a predictor vector X2t

Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series
Fig. 3: Plots of actual and estimates of the second predictor vector X3t

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 non-stationary revenue series than pure linear models.

Appendix 1: Actual internally generated revenue series represented by three vectors
Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series

Appendix 2: Regression estimates from bilinear autoregressive vector models
Image for - Bilinear Autoregressive Vector Models and Their Application to Estimation of Revenue Series

REFERENCES

1:  Franses, P.H., 1998. Time series models for business and economic forecast. J. Time Ser. Annal., 16: 509-529.

2:  Granger, C.W.J. and A.P. Anderson, 1978. Introduction to bilinear time series models. Vandenhoeck and Ruprecht.

3:  Maravall, A., 1983. An application of nonlinear time series forecasting. J. Bus. Econ. Stat., 1: 66-74.
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4:  Subba Rao, T. and M.M. Gabr, 1984. An introduction to Bispectral analysis and bilinear time series models. Lecture Notes in Statistics 24. Spring-Verlag, New York.

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