Fitting Autoregressive Integrated Moving Average with Exogenous Variables Model with Lognormal Error Term

Abstract

The conventional Autoregressive Integrated Moving Average with Exogenous Variables (ARIMAX (p, d, q)) modsel with Normal Error term requires stringent assumptions of normality of error term and stationarity of the series. These models have found widespread application in multidimensional relationships among economic variables; these assumptions are often violated in practice leading to spurious regression model with poor forecast performance. Thus, this paper is designed to develop an ARIMAX (p, d, q) model with Lognormal Error term capable of analysing time series data even when the assumptions were violated with reasonable forecast performance. 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. Box-Jenkins Time Series procedure was used to model ARIMAX (1, 1, 1) with normal error. The performance of proposed model was tested using Akaike Information Criteria (AIC), Mean Square Forecast Error (MSFE) and Loglikelihood (Loglik) values. The Loglik values of conventional ARIMAX (1, 1, 1) with normal error and proposed ARIMAX (1, 0, 1) with lognormal error term were  -240.23 and 1344.47; AIC values were  490.45 and -0.41 while MSFE values were 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, hence, the proposed  model is considered better. The autoregressive integrated moving average with exogenous variables with 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.