FITTING AUTOREGRESSIVE INTEGRATED MOVING AVERAGE WITH EXOGENOUS VARIABLES MODEL ASSUMING LOGNORMAL ERROR TERM

Abstract

The conventional Autoregressive Integrated Moving Average with Exogenous Variables (arimax) model with Normal Error term and Multiple Linear Regression (MLR) require stringent assumptions of normality of error term and stationarity of the series. These models have found widespread application in multidimensional relationships among economic variables; when these assumptions are often violated in practice leading to spurious regression model with poor forecast performance. Thus, this study was designed to develop an arimax model with Lognormal Error term capable of analysing time series data even when the assumptions were violated with reasonable forecast performance. The conventional arimax (1, 0, 1) with normal error term defined as:where the lag operator B = yt−1; the parameter 1 was the coefficient of the Autoregressive model (AR), θ1 was the coefficient of Moving Average (MA), β0 was the intercept and β1 was the slope of the Regression part of the model. The proposed model was estimated by modifying the arimax (p, d, q) with lognormal error term where p is order of AR part, d is order of difference and q is order of MA part of the mixed model. The parameters were estimated using the maximum likelihood method. The choice of lognormal error term was based on the asymmetric property which overcomes non normality, the long tail and positive limit values properties overcome non stationarity. The dataset used were monthly External Reserves (Million USD), Official Exchange Rate (Naira to USD), Crude Oil Export (Million Barrel per Day) and Crude Oil Price (USD per Barrel). One hundred and twenty (120) observations were used for the modeling process. The proposed arimax (1, 0, 1) with lognormal error term ameliorate the non-normal and non-stationary assumptions. The proposed model performance was compared with conventional arimax (1, 1, 1) with normal error term and MLR model. Box-Jenkins Time Series procedure was used to model arimax (1, 1, 1) with normal error and Least Squares Estimator (LSE) technique for modeling MLR. The performance of proposed model was tested using Akaike Information Criteria (AIC), Mean Square Forecast Error (MSFE) and Loglikelihood (Loglik) values. The non normal error function was obtained as:while the loglikelihood function was: where σ2 is variance. All the series were found to be non-stationary and non-normally distributed. The Loglik values of MLR, conventional arimax (1, 1, 1) with normal error and proposed arimax (1, 0, 1) with lognormal error term were -317.41, -240.23 and 1344.47; AIC values were 5.36, 490.45 and -0.41 while MSFE values were 12.41, 12.48 and 1.77. The proposed model has the highest Loglik value, smallest AIC and smallest MSFE values when compared with conventional arimax (1, 1, 1) with normal error and MLR model. Hence, the proposed model was considered better. The autoregressive integrated moving average with exogenous variables assuming lognormal error term improved the capability of modeling time series data with better forecast performance even when the assumptions of normality of error term and stationarity of series were violated.