American Journal of Mathematics and Statistics, (): - DOI:./j.ajms.0 Comparative Analysis of Linear and Bilinear ime Series Models Usoro Anthony E. Department of Mathematics and Statistics, Akwa Ibom State University, Mkpat Enin, Nigeria Abstract his paper considered alication of linear and bilinear time series models in estimating revenue data. he data used were monthly revenue, which comprised the allocation from Federal Government and internally generated revenue of a Local Government Council in Akwa Ibom State. he motivation behind the comparison between the revenue estimates of linear and bilinear models was to find out if the assertions of [] and [] could be alicable to Nigerian system of revenue generation, by using available data at a Local Government level. he aim was to choose more suitable time series model for making projection and proposing feasible targets to revenue generation units or Departments of government establishments. Ordinary least squares method was adopted to estimate the parameters of both linear and bilinear models. From the empirical findings, it was observed that bilinear model fitted the revenue data better than the linear model with standard error difference of.. his affirms the fact that bilinear time series models are more suitable in modeling revenue series, considering the dynamic nature of the time series data. Since the improvement and superiority of bilinear models over linear models are established, this paper recommends forecast of revenue series with bilinear models. Keywords Autoregressive model, Linear model and Bilinear model. Introduction In univariate time series, a series is modeled only in terms of its own past values and some disturbance. A particular set of data assumed by a variable, say, X t, where t represents time, is modeled on the basis of its previous time observations. For instance, every government budget is prepared taking into consideration the past budget figures, be it at the Local, State or Federal Government level. In time series, such expression can be written as X t = f(x t-, X t-,x t-,,x t-p, e t ). he X t is a variable, which assumes values at the present time period for which forecast can be made, X t-, X t-,x t-,,x t-p are the distributed lag variables which represent time past values. he e t is the random error assumed to be zero, Johnston and []. If, for example, one specified a linear function with one lag and a white noise disturbance, the result would be the first-order autoregressive AR () process of the form, X t = αx t- + e t. he above model shows that the data series represented by x t is a function of its one period lag. he general form of the model is X t = α X t- + α X t- + α X t- +,, + α p X t-p + e t. his is a pure autoregressive AR (p) model, with e t as a * esponding author: toskila@yahoo.com (Usoro Anthony E.) Published online at http://journal.sapub.org/ajms Copyright Scientific & Academic Publishing. All Rights Reserved disturbance term. he above model explains the fact, for forecast to be made on the variable, X t, there must be a relationship between the values of the variable at the present time period and previous time. herefore, the time series model which expresses such relationship is a linear time series model. In this paper, we also consider bilinear time series model. Bilinear time series model is a model which comprises linear and nonlinear components of the two processes. In time series, the two processes, which make a complete bilinear time series model, are autoregressive and moving average processes. Any of the two processes can be identified in the distribution of autocorrelation and partial autocorrelation functions. For a process to follow autoregressive, the autocorrelation function must exhibit exponential decay or sine wave pattern, while there is a spike or cut- off within the first two lags of partial autocorrelation function. In moving average process, partial autocorrelation function exhibits exponential decay or sine wave pattern, while there is a cut-off or spike within the first two lags of autocorrelation function. hese two processes separately form linear model, called AUOREGRESSIVE MOVING AVERAGE MODEL, ARMA (p,q). In bilinear time series model, there are two parts which make up a complete model. hese include linear and nonlinear parts. he nonlinear part is the product of the two processes. Model shown below is an example of a complete bilinear time series model. Model is a complete linear model of autoregressive and moving average processes.
Usoro Anthony E.: Comparative Analysis of Linear and Bilinear ime Series Models. Review he introduction of some nonlinear models, such as logarithmic, quadratic, bilinear time series models to some data is due to the fact that some data assume nonlinearity component. he exhibition of non-linearity property is due to certain occurrences, which some are not under human control. However, there are some events whose occurrences are characterized or generalized by human, but still assume non-linearity. [] asserted that some of the microeconomic and financial data are not linear due to its dynamic behaviour. According to them, classical linear models are not aropriate for modeling such non-linear series. In most cases, nonlinear forecast is superior to linear forecast. [] investigated the effect of alying different number of input nodes, activation functions and pre-processing techniques on the performance of back propagation (BP) network in time series revenue forecasting. In the study, several preprocessing techniques were presented to remove the non-stationarity in the time series and their effect on Artificial Neutral Network (ANN) model. he study compared the use of logarithmic function and new proposed Artificial Neutral Network (ANN) model. From the empirical findings, it showed that ANN model, which consisted of small number of inputs and smaller corresponding network structure, produced accurate forecast result although it suffered from slow convergence. [] reported on the forecast accuracy between time series models and management and analysts. All comparisons were carried out not only on the basis of prediction errors, but also by an analysis of the forecasting bias involved, and the specification of uncertainty. he data used were representative sample of companies listed on the Amsterdam Stock Exchange. he findings revealed improvement on the use of time series models. [] used a bilinear model to forecast Spanish monetary data and reported a near % improvement in one-step ahead mean square forecast errors over several ARMA alternatives. [] stated the general Bilinear Autoregressive Moving Average model of order (p, q, P, Q) denoted by BARMA (p, q, P, Q) as PP QQ XX tt = ii= αα ii XX tt ii + ii= ββ jj tt jj + ii= jj = γγ iiii XX tt ii tt jj () X t-i is the distributed lag component of autoregressive process with coefficient α i. Є t-j is the distributed lag component of moving average process with coefficient β j. he sum of the two processes makes up the linear part of model. he nonlinear component is the product of the two processes. he model is thus linear in the X s and also in the Є s separately, but not in both as product of the two series. In the general ARMA model, XX tt = ii= αα ii XX tt ii + ii= ββ jj tt jj + tt () Where, the first part on the R.H.S is the autoregressive process, while the second is the moving average process, where Є t is strict white noise, CASE : If j = 0, the model becomes XX tt = ii= αα ii XX tt ii + tt () his implies ββ 0 = 0 Model is a pure autoregressive model. CASE : If i=0, the model becomes XX tt = ii= ββ jj tt jj + tt () his implies αα 0 = 0 Model is a pure moving average model. From model, XX tt = αα ii XX tt ii + ββ jj tt jj + γγ iiii XX tt ii tt jj ii= ii= If j=0, the above model becomes PP QQ ii= jj = PP XX tt = ii= αα ii XX tt ii + ii= XX tt ii tt + tt () his implies ββ 0 = 0 he model is Bilinear Autoregressive (BAR) model. his is so because the component of the moving average process is eliminated from the general model. he in the models is the usual white noise, assumed to be independent and identically distributed with mean zero and variance δ e, []. Normally, the values of Є t, being the error values are obtained as deviation of estimates from the original series. X t is a variable which represents the original values of the revenue series. While, X t-, X t-, X t- are the time lag of variable X t. In this paper, we aly Pure Autoregressive (AR) model () and Bilinear Autoregressive (BAR) model () to fit the revenue data for comparison of estimates between the two models.. Method of Estimation In this paper, the initial process was to obtain the autocorrelation and partial autocorrelation functions of the revenue series, which is represented by X t. his was done as a procedure to the choice and order of the time series model that is alicable to the data set. he distribution of the autocorrelation function exhibited exponential decay, while partial autocorrelation function indicated a spike at lag, as shown in the autocorrelation and partial autocorrelation functions in section below. his distribution suggested autoregressive model of order. he model is expresses as X t = α X t- + α X t- + Є t. Є t is the residual, expected to be zero. t
American Journal of Mathematics and Statistics, (): - AUOCORLAION OF ORIGINAL SERIES Xt Autocorrelation 0. 0. -0. -0. - - - 0 0.0 0 0. 0. 0. 0. 0. 0............0....... 0. 0. 0. 0. 0.......0....... 0. 0. 0. 0. 0. 0.......... 0 0 0. -0...0. PARIAL AUOCORRELAION FUNCION OF Xt Partial Autocorrelation 0. 0. -0. -0. - - - 0 0.0 0. 0. - -..00.. 0. - 0. - - -0. - 0. - 0. 0.. -0. -.0 -. -0. -0 - - 0. - - - -0. -0.. 0. 0. 0-0. - - 0. -.. Estimates of Linear and Bilinear Models Linear Model: From the autocorrelation and partial autocorrelation functions, the revenue series X t is a function of X t- and X t-. his implies the revenue series at the present time period depends on two period lags of itself. herefore, regression of X t on X t- and X t- provides the following estimated model for AR (, 0) process: X t = 0X t- + X t-, where 0 and are the coefficients of X t- and X t- respectively. he estimated values from the above regression model are the LEX t values in the Aendix, and the graphs of the actual and estimates are shown in ime Plot.
Usoro Anthony E.: Comparative Analysis of Linear and Bilinear ime Series Models IME PLO PLOS OF ACUAL AND ESIMAES OF REVENUE SERIES 00 00 00 Xt 00 00 0 0 0 Index 0 0 0 0 PLO OF ACUAL IN BLACK CIRCLE PLO OF ESIMAES IN BLACK PLUS ACF PLO ACF OF RESIDUAL OF LINEAR MODEL Autocorrelation 0. 0. -0. -0. - - - -0. - - -. -. -0. 0. 0. 0. -0. - -0.......... - - 0. 0. -. -. -0... -0. -0 0...0..0...0. - - - 0. -0. - 0. 0. 0.......... - 0. -0..0. 0. Bilinear Model: his model contains two parts. he linear part, which is the sum of the autoregressive and moving average processes, as shown in the linear model. he second part is always obtained by taking the product of the two processes (autoregressive and moving average processes). But, since the effect of moving average is suressed as shown the autocorrelation and partial autocorrelation functions, model becomes alicable to the data. he components of the nonlinear part of the model are obtained by taking the products of values of Є t (residual values) and X t- and also Є t and X t- to form X t- Є t and X t- Є t in the bilinear model, which is expressed as BAR (, 0,, 0). he two zeros in the BAR indicate that both the linear and nonlinear parts of the moving average have zero order. his means the model is a bilinear autoregressive with the order of for the linear and nonlinear parts. he parameters are estimated using ordinary least squares method. herefore, the regression of X t on X t-, X t-, X t- Є t and X t- Є t produces the following model:
American Journal of Mathematics and Statistics, (): - X t = 0X t- + 0.X t- + 0X t- Є t + 0X t- Є t, where 0, 0., 0, and 0 are the coefficients of X t-, X t-, X t- Є t and X t- Є t respectively. he estimates from the above regression model are the BEX t values in the Aendix, and the graphs of the actual and estimates are shown in ime Plot. IME PLO PLOS OF ACUAL AND ESIMAES OF REVENUE SERIES 00 00 00 Xt 00 00 0 0 0 Index 0 0 0 0 PLO OF ACUAL IN BLACK CIRCLE PLO OF ESIMAES IN BLACK PLUS ACF PLO ACF OF RESIDUAL OF BILINEAR MODEL Autocorrelation 0. 0. -0. -0. - - - 0. -0. -.. 0.. 0. - -0. 0. -......0... - - 0-0. - -0. -0. -0. - -.....0.0..0. 0. - - - -0. -.. -. 0. -0. -0. -.0-0..0........ 0.. -0...0. Results Interpretation In fitting the models to the revenue data, autocorrelation and partial-autocorrelation functions were plotted. he distribution suggested autoregressive model of order two. he linear model was estimated, with standard error of., and the values from the model were plotted on the same graph with the actual data, as shown in ime Plot. he ACF Plot is the autocorrelation function of the residual obtained from the estimated linear time series model. From the autocorrelation function, it is observed that there is some level of correlation among the residual values. his correlation implies the residual e t is not independent and identically distributed. Secondly, the bilinear model was
0 Usoro Anthony E.: Comparative Analysis of Linear and Bilinear ime Series Models estimated with standard error of., and the values from the model were plotted on the same graph with the actual revenue data, as shown in ime Plot. he ACF Plot is the autocorrelation function of the residual obtained from the estimated bilinear time series model. From the function, it is observed that there is no autocorrelation among the residual values. his explains the fact that the residual e t is independent and identically distributed. hat is, e t is a white noise process, which explains that the bilinear fitted the data set better than linear model.. Conclusions he reason for the alication of both linear and bilinear models was to compare the estimates from the two models through the time plot and autocorrelation plots of the two sets of residual values. In time series modeling, one of the aims is to suggest a suitable and best among series of models for estimation and forecast of future values of the observations under consideration. he estimates from the two models have shown that bilinear models are more suitable for the forecast of revenue series. Aendix able of Actual and Estimated Revenue Data S/N ACX t LEX t BEX t S/N ACX t LEX t BEX t S/N ACX t LEX t BEX t - -....... - -..0.0.. 0.. 0.....0.0... 0...0...... 0...0.. 0....... 0................... 0.....0.... 0... 0.......0....0. 0..............0....0.0.0...0.. 0.......0..0.0....0.......0...0.......0.0.00.. 0. 0....0 0..0..0.. 0 0...0 0..0....0..... 0..00... 0.....0...0 00.................0.. 0. 0.0....0....0...... 0...........0......0.. 0.0.. 0............ 0....... 00..... 0........0 0...........0.. 0.... 00...0...........0...0.00.............0. 0.0..0 0..0.... ACX t = Actual Revenue Data LEX t = Linear Estimates BEX t = Bilinear Estimates
American Journal of Mathematics and Statistics, (): - REFERENCES [] David A. Kodde and Hein Schreuder (): Forecasting Corporate Revenue and Profit: ime Series Models Versus Management and Analysts. Journal of Business Finance and Accounting; Volume, Issue, pages -. DOI./j.-..tb00.x. [] Granger, C. W. J. and Anderson, A. P. (): Introduction to bilinear time series models, Vindenhoeck and Ruprecht. [] Johnston, J. and Dinardo, J. (): Econometric Methods, McGraw-Hill Companies, Inc, New York. [] Maravall, A. (): An alication of nonlinear time series forecasting. Journal of Business and Economic Statistics;, -. [] Siti M. Shamsuddin, Roselina Sallehuddin and Norfadzila M. Yusof (0): Artificial Neutral Network ime Series Modelling For Revenue Forecasting. Chiang Mai Journal of Science; (): -, www.science.cmu.ac.th/journal-sci ence/josci.html. [] Subba Rao,. and Gabr, M. M. (): An Introduction to Bispectral Analysis and Bilinear ime Series Models. Lecture Notes on Statistics No. Springer Verlag. [] Usoro, A. E. and Omekara. C. O. (): Necessary Conditions for Isolation of Special Cases from the General Bilinear ime Series Model. th Annual Conference of Nigerian Statistical Association,.