**Table 2.**

returns with fixed interest and less sensitive to financial news and risk. Also, the choice of Brent spot oil price as against West Texas Intermediate (WTI) oil price is because the Nigeria's oil export is usually measured and priced in Brent oil market while the WTI is a bench mark for North American market. The stock price is measured as the monthly equity investment of ASPI in Naira on Nigerian Stock exchange while the Brent oil price is the monthly global oil price in US dollar per barrel (pbl) in the international oil market. The returns of both stock price and oil price were generated through the log of difference (*d* log*X*) of the series which can

The statistical distributions of the 252 monthly observations of stock price and oil price with their returns used in this study are presented in **Table 1**. The average monthly observation of the oil price returns is �0.0013%, which implies that there were losses and low returns on oil revenue during the period of study. The high difference between the maximum oil price of \$US132.72 and the minimum value of \$US18.38 confirms the high volatile nature of the oil price. For the stock price returns, the minimum value is negative with a value of �0.3659. This implies that the stock price returns is less volatile than the oil price returns with minimum value of �0.55%. Although, there is also a large difference between the maximum value of the stock price with N65652.38 in billion and the minimum values of N5892.8 billion. The variability is just lower compared to that of the oil price. The standard deviation, skewness and kurtosis greater than zero imply that distribution is not normally distributed except for both returns that are close to zero and being normal. The positive skewness of 0.39% and 0.55% for oil price and stock price imply that their distributions are skewed to the right. On the other hand, the negative skewness of �1.75% and �0.47% for oil price returns and stock price returns imply that their distributions are skewed to the left. Furthermore, the kurtosis of oil price with value of 2.09% and the stock price with value 3.51 imply normal distribution because the values are less than 3. However for the returns, the kurtosis value of 9.08 and 7.71% for both oil price returns and stock price returns denote leptokurtic characteristic. Lastly, the null hypothesis for Jarque-Bera is that the data is normally distributed, however, with the probability value of 0.00 less than 0.05% in **Table 1**, then the null hypothesis is rejected and the alternative hypothesis that the data are

**Statistics Oil price Oil price returns Stock price Stock price returns** Mean 64.35 �0.0013 27315.4 0.0056 Median 61.96 0.0164 26011.64 0.0024 Maximum 132.72 0.1979 65652.38 0.3235 Minimum 18.38 �0.5548 5892.8 �0.3659 Std-dev 29.91 0.1035 11756.38 0.0708 Skewness 0.39 �1.7543 0.55 �0.4734 Kurtosis 2.09 9.0837 3.51 7.71 Jarque-Bera 14.69 499.37 14.74 233.47 Prob. 0.00 0.00 0.00 0.00 Observation 252 252 252 252

be mathematically written as: *d* log*X* ¼ log*X* � log*X*ð Þ �1 .

*Linear and Non-Linear Financial Econometrics - Theory and Practice*

**3.1 Descriptive statistics**

**Table 1.**

**174**

*Descriptive analysis.*

*Results of the unit root tests.*


#### **Table 3.**

*Breusch-pagan-Godfrey test.*

not normally distributed is accepted. It is evident that the statistical properties of the variables used in this study can be described as fat tailed, leptokurtic and deviated from normal distribution which is typical of financial time series, risks and returns.

#### **3.2 Preliminary test**

The first exercise after the descriptive analysis is to verify the stationary properties of the variables used in the analysis and then test for the ARCH effect on the variables. Once the variables are stationary and ARCH effect is present, then we can proceed to estimate the GARCH models. The Augmented Dickey Fuller [22] and the Philips-Perron [23] tests were conducted and the results shown in **Table 2**. The unit root results show that both oil price returns and stock price returns are stationary at levels. The stationarity of the returns of the variable of interest is one of the conditions for carrying out the GARCH process.

The final preliminary test is to test for ARCH effects using Breusch-Pagan-Godfrey method of Engle [24] to verify the presence of heteroscedasticity and proceed to the GARCH process. The heteroscedasticity test presented in **Table 3** shows the presence of heteroscedasticity, which means that the variance is not constant over time (see also Appendix 5 for additional evidence of heteroscedasticity with the fat tail of the histogram distribution). The null hypothesis is that there is no presence of heteroscedasticity in the returns series. And since the probability value is less than 0.05%, then the null hypothesis is rejected and the alternative hypothesis of presence of ARCH effects or heteroscedasticity is accepted.
