Necessary Conditions for Isolation of Special Classes of Bilinear Autoregressive Moving Average Vector (BARMAV) Models

Bilinear Autoregressive Moving Average Vector (BARMAV) Models are models aggregated with the linear and non-linear vector components of autoregressive and moving average processes. The linear part is the sum of the two vector processes, while the non-linear part is the product of the processes. From the general BARMAV models, Bilinear Autoregressive Vector (BARV) Models and Bilinear Moving Average Vector (BMAV) Models have been isolated. Under certain conditions, the models are proved to exist. Empirically, Nigerian consumer price index and inflation rate are used to test the fitness of the bilinear models. Data for the analysis are from Central Bank of Nigeria Statistical Bulletin, collected from January 2009 to December 2016 with November 2009 as the base year for each of the series. The bilinear autoregressive moving average vector models are fitted to the data. Parameters are tested and found to be significant. The adequacy of each estimated model is confirmed with ACF, PACF and descriptive statistics adopted in the paper. The plots of the actual and fitted CPI and IR have shown that models are adequate as estimates compete favourably with the actual values. The models are useful in modelling some economic and financial data that exhibit some characteristics of non-linearity.


Introduction
When dealing with classical time series models, the two popular processes that explain the behaviour of empirical data in a stationary time series are autoregressive and moving average processes. These processes are described on the basis of autocorrelation and partial autocorrelation functions of empirical data. The popular Autoregressive Moving Average (ARMA) model in time series is a model of linear relationship between a time series process X t and the lag variables of both the process and error term. The General ARMA (p,q) model is expressed in a linear form as, Where, is the time process, ∅ and are parameters of autoregressive and moving average processes respectively, is the error term.
The above model is a univariate linear time series model for the two processes from which AR or MA model can be isolated on condition that 0 0 respectively, [2,5,8,9]. The interest in this paper is to identify special classes of bilinear autoregressive moving average vector models under certain conditions. The fact is that in time series modelling, a process may be described by either AR, MA or both. In as much as AR and MA exist independently each as a univariate linear, multivariate linear, univariate bilinear; it follows that there exists multivariate bilinear model for each of the AR and MA processes under certain conditions.

Non-Linear ARMA Models
The non-linear Autoregressive Moving Average models are presented in the form,

% -A
The expansion of the above matrices gives non-linear ARMA models for , , … , % . The models are reduced Model (6) can be written as Model "7" is a multivariate non-linear Autoregressive Moving Average Model.

Isolation of Special Models
In this section, conditions for isolation of special classes of bilinear autoregressive moving average vector models are considered. Usoro [15] identified special classes of bilinear time series models. Under certain conditions BAR and BMA were identified from the mixed BARMA model. Here, we consider the multivariate case of Usoro [15,16].

Bilinear Moving Average Vector (BMAV) Model
BMAV model is given as

Test for Linearity and Bilinearity
The above test involves the parameters of both the linear and non-linear components of bilinear time series models.

Model Fitting to Empirical Data
In this section, we consider fitting the bilinear models to the empirical data. For illustration, Nigeria Consumer Price Index (CPI) and Inflation rates (IR) are fitted with the bilinear models. The procedures of fitting bilinear models to time series data are not different from the ordinary ARMA models.      The above ACF's and PACF's for the CPI and IR suggest BARMA(1,1,1,1) for each CPI and IR. The models with parameter estimates are presented as follows

Estimation of Parameters and Interpretation of Results
The above results have it that parameters of the bilinear model fitted to CPI are all significant, except for X 2t-1. For IR, evidence has it that some parameters of the linear components are significant with one of the non-linear components. This is a true indication of model fitness to the data. Further evidence is shown in the ACF and PACF of the model residual.

Conclusion
There is no gainsaying the fact that each of the AR and MA processes are combined to form mixed ARMA process. This is explained by the behaviour of the empirical data as always shown in the distribution of the autocorrelation and partial autocorrelation functions. A process that is only described by either AR or MA remains a singular process, except characterised by both. The idea about this paper is that if there exist condition(s) for isolation of AR or MA from ARMA, isolation of BAR or BMA from BARMA models, therefore, BARV and BMAV are conditionally isolated from BARMAV model. For a pure bilinear autoregressive vector model, the non-linear component of the model is the product of lagged and non-lagged + . That is the multiplication of , , … , B by + . That means each of the non-zero lags of is multiplied by zero lag of + to form the non-linear part of the model. Similarly, for a pure bilinear moving average vector model, the non-linear component of the model is the product of lagged + and non-lagged . That is the multiplication of + , … , + A by . Here, each of the non-zero lags of + is multiplied by zero lag of to form the non-linear part of the BMAV model. Empirically, see Usoro and Omekara (2008) and Usoro (2018). Empirically, the monthly consumer price index and inflation rate used in this paper are characterised by both AR and MA process. This called for adoption of "8" in the analysis. The plots of the actual and fitted CPI and IR data in figures "9" and "10" have shown that estimates compete favourably with the actual. Hence, the models are suitable in modelling time series data that exhibit some form of nonlinearity characteristics.