Fourier Autoregressive Moving Average Model for Complex Time series Datasets

Abstract

Time series datasets that are non-stationary and seasonal are often analysed using various techniques. Due to structural changes, many time series datasets have become increasingly complex. Therefore, there is a need to develop a model capable of analysing irregular variation, cyclical, seasonal and periodic variation in time series datasets. The Fourier Autoregressive Moving Average model (FARMA) was developed to analyse complex time series datasets. Time plots were used to determine the variation pattern, correlogram and magnitude plots were used to identify model order and estimation was carried out using Maximum likelihood (ML) estimation method, Diagnostic checking was achieved base on the Durbin Watson statistical test, histogram, Autocorrelation function (ACF) and Partial Autocorrelation (PACF) plot of the residuals. The model was validated using coefficient of determination (R²) and adjusted coefficient of determination ( ) while forecast evaluation was based on the least values of Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE). The Nigerian exchange rate was considered and the time plot indicated the presence of trend, cyclical, seasonality and irregular variations simultaneously. The FARMA model was estimated using ML estimation method for fitting the coefficient of the model. The model were diagnosed using Durbin Watson statistic, autocorrelation and partial autocorrelation function, histogram of the residual.  The results obtained indicated that the Fourier Autoregressive moving average model performed better than ARMA(2,2,3),  SARIMA(2,2,1)(2,2,3)₁₂ and FAR(6) models in terms of coefficient of Determination and Adjusted coefficient of Determination (0.9650,0.9574,0.7825,0.7975) and (0.9610,0.9540,0.7729 0.7934) for Nigeria exchange rate The optimum FARMA component model was established using Akaike information Criteria( AIC) and Bayesian information criteria (BIC) values with (10.3948 and 10.674) for Nigeria exchange rate which signifies the minimum value of the information criteria. The efficiency of the FARMA(6) model was ascertained by modeling and forecasting Nigeria exchange rate datasets. The forecast evaluation of the proposed model was (155.0275, 263.3046 and 42.1168) for MAE, RMSE and MAPE respectively for Nigeria exchange rate, the smallest value of FARMA(6) model makes it better model compared with existing fitted model. Therefore, the proposed FARMA model is capable of modelling and forecasting time series datasets that exhibits seasonal, cyclical and periodic and irregular structure simultaneously. Fourier autoregressive moving average model is applicable in other field like environmental data, climatic data and sales data. It is recommended that the model can be applied in other sphere of life.