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.
