Predictive Model with Square-Root Variance Stabilizing Transformation for Nigeria Crude Oil Export to America

In the last few decades, crude oil has claimed the topmost position in Nigerian export list, constituting a very fundamental change in the structure of Nigerian international trade. In this study, secondary data on monthly crude oil export to the United States was obtained from the Energy Information Administration (EIA) database. Using the Box-Jenkins (ARIMA) methodology, the results showed that Seasonal ARIMA (0, 1, 1) (1, 0, 1)12 model had the least information criteria after the data was Square-Root transformed and non-seasonally first differenced in order to achieve series stationarity. The diagnostic tests on the selected model residuals revealed the residuals are normally distributed uncorrelated random shocks.


Introduction
Crude oil is considered the major source of energy in Nigeria and the world in general. Crude oil being the mainstay of the Nigerian economy plays a vital role in shaping the economic and political destiny of the country. Nigeria exports most of its crude oil to countries like India, United States of America, Brazil, The Netherland, United Kingdom and Spain [1]. As recently as 2010, Nigeria provided about 10 percent of over-all United States oil imports and ranked as the fifth-largest source of oil imports in the United States. However, Nigeria crude oil export to the United States has recently being declining as a result of the boom in shale oil and lifting of the United States 40-years ban on crude oil exports by the U.S. Senate [2]. Therefore, the ultimate aim of this study is to construct a statistical model that could be used to monitor the export pattern of crude oil export from Nigeria to the United States. Using this model, forecast of future values of crude oil export to the United States can be obtained. A lot of studies have been carried out using time series ARIMA modelling approach to identify patterns and appropriate models. Adubisi [3] used ARIMA procedure in modelling the growth pattern of reserve currency in Nigeria, Iwueze et al; [4], modelled the Nigeria external money reserves with Autoregressive Integrated Moving average model, ARIMA modelling approach was used to model yearly exchange rates between USD/KZT, EUR/KZT and SGD/KZT, and the actual data compared with developed forecasts by Daniya [5], Kumar and Anand [6] used ARIMA modelling approach to forecast sugarcane production in Indian. Adubisi and Jolayemi [7] used ARIMA-intervention analysis modelling approach to evaluate and estimate the impact of the financial crisis on Nigeria Crude oil export. Furthermore, Smart [8] explored the feasibility for application of Box-Jenkins Approach (ARIMA) in modelling and forecasting maternal mortality Ratios (MMR) and Adubisi et al; [9] used the seasonal ARIMA to model the Nigeria money in circulation series and also produced a three years forecast values using the fitted model.

Material
The data used for this study are secondary data on Nigeria monthly crude oil export to the United States obtained from the Energy Information Administration (EIA) database for twenty-three consecutive years, from January 1993 to December 2015 [10].

Power Transformation
The parametric family of transformations from the original series to a transformed series was originally proposed by Box and Cox [11]. The power transformation is a continuously varying function with respect to the power parameter lambda is the Geometric mean of the data. It generalizes both the square root and the log transformation and admits a likelihood ratio test to select the best fitting parameter. The one-parameter Box-Cox transformation is expressed as For more details on Box-Cox variance stabilization transformation procedures, see Box and Cox [11], Yan [12], Carroll and Ruppert [13], Nishili [14], Sakia [15] and Bickel et al; [16]. The various transformation parameter lambda ( ) λ values and the appropriate transformation attached to each are summarized in the Table 1. Lambda ( ) x Source: Minitab (2010).

Box-Jenkins Methodology
The Autoregressive integrated moving average (ARIMA) model procedure popularized by Box and Jenkins [17] and Box et al. [18]. The ARIMA ( , , ) p d q model which contain both the non-seasonal and the seasonal parameters is written as represent the non-seasonal and seasonal autoregressive coefficients parameters with the roots within the unit circle , represent the non-seasonal and seasonal moving-average coefficients parameters with the roots within the unit circle is the regular differencing which is applied to remove the stochastic trend in the series, (1 ) is the seasonal differencing applied to remove the series seasonal effects and t ε is the white noise error i.e. More details on seasonal ARIMA can be found in Box and Jenkins [17], Box et al; [18], Pankratz [19].

Unit Root Tests
The Augmented Dickey-Fuller (ADF) test and the KPSS tests are performed to determine if the series contains a unit root. The tests are based on the assumption that the time series data t Y follows a random walk. The Augmented Dickey-Fuller (ADF) test, corresponding to modelling a random walk pattern with drift around a stochastic trend The expression against the alternative ( 1 ρ < or 0 ∂ < ). When the p-value is greater than the alpha, this would lead to none rejection of the null hypothesis. The KPSS test with a random walk The procedure has a null hypothesis of stationary series and an alternative of non-stationary series. A p-value less than alpha ( )

Results and Discussion
The series plot in Figure 1 depicts sharp peaks and toughs in the crude oil export series, suggesting some influence and also the variance was observed not to be stable over the periods used in this study. The non-stationarity claim from the series plot was affirmed from the slow decay in the autocorrelation function (ACF) of the data while significant spikes at lag 1 and 2 of the partial-autocorrelation function (PACF) are noticed in Figure 2. Therefore, the data requires some form of transformation and differencing to make it stable (stationary).  (2) with the help of Minitab. The series was also non-seasonally differenced to achieve stationarity. Figure 3, depicts the non-seasonal first-order differenced crude oil export series. The results of the ADF and KPSS unit root tests in Table 2, also indicates that the data is stationary after the square root transformation and the non-seasonal first-order difference was applied on the series. The decay in the correlogram of the differenced transformed data in Figure 4, when compared with the 95% confidence limits 2 0.35 n The PACF decays faster towards zero while the ACF cuts off after the lag 1 with a significant spike at lag 12. The various tentative Seasonal ARIMA model structures extracted from the correlogram plots in Figure 4 are presented in Table 3. The Seasonal ARIMA (0, 1, 1) (1, 0, 1) 12 was found to fit the series based on the AIC, AICc and BIC selection criteria values. The model parameter estimates in Table 4 are statistically significant with the t-values greater than 2 ± . The parameters also satisfy the stationarity and invertibility conditions of the ARIMA methodology. The seasonal ARIMA (0, 1, 1) (1, 0, 1) 12 model expressed in backshift format using the parameters in Table 4, is presented as 12 12 12 12 (1 0.9518 ) The fitted seasonal ARIMA model from equation (10) is mathematically expressed as 12 1 12 13 0.9518 0.5507 0.8690 0.4785 The model adequacy check performed using the correlogram plots of the model residuals coupled with other objective tests like the Box-Ljung test [22], Shapiro-Wilk Normality test [23] and the ARCH-LM test [24] are presented in Table 5. The tests results failed to reject the null hypothesis at the 5% level of significance confirming that the residuals are normally distributed with no autocorrelation and no conditional heteroscedasticity (ARCH) effects. It implies that the residuals from the fitted seasonal ARIMA model are Gaussian white noise. The Gaussian white noise residuals is clearly portrayed in Figure 5 by the randomness of the residuals, non-significant spikes in the ACF residuals plot and the probability values falling above the 0.05 limit in the probability plot.

Conclusion
In this study, the crude oil export to the United States was modelled using the Box-Jenkins (ARIMA) modelling procedures. The data required square-root transformation and differencing to achieve series stationarity. The transformed series was subjected to the Box and Jenkins iterative procedure for ARIMA model building. The results from analysis showed that the appropriate model for the series is the Seasonal ARIMA (0, 1, 1) (1, 0, 1) 12 model. The model adequacy tests confirmed that the model residuals are normally distributed uncorrelated random shocks. This model is therefore recommended for use in the forecast of Nigeria crude oil export to the United States, until proving otherwise.