**4. The linear and non-linear GARCH models**

The presence of the ARCH effects in our variables as presented in **Table 3** endorses the use of the GARCH models. There are many types of GARCH models. We have the symmetric (linear) GARCH, which is the normal GARCH and asymmetric (nonlinear) GARCH such as exponential GARCH (EGARCH) and the Threshold GARCH (TGARCH) or Glosten, Jagannathan and Runkle GARCH (GJR-GARCH).

We started with the ARCH model formulated in two parts, the mean equation and the variance equation proposed by Engle [24] and written as:

$$Y\_t = a + \beta' X\_t + \mu\_t \tag{1}$$

**5. Empirical analysis and result discussions**

*DOI: http://dx.doi.org/10.5772/intechopen.93497*

the variables of interest from the mean equation as:

<sup>1</sup> More exposition on student-t distribution can be found in Fisher (1925).

stated as:

**177**

Having described both the symmetric and asymmetric GARCH, we expressed

*Volatility Effects of the Global Oil Price on Stock Price in Nigeria: Evidence from Linear…*

Eq. (6) expresses stock price return as a function of oil price return. Where *RSPt* is the return of the stock price over time, *α* is the constant term, *ROPt* is the returns of the oil price, *β* is the marginal effect of the oil price on the stock price while *ut* is the error term. The variance equation with the parsimonious GARCH (1,1) model is

*ht* <sup>¼</sup> *<sup>γ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>λ</sup>*1*ht*�<sup>1</sup> <sup>þ</sup> *<sup>γ</sup>*1*u*<sup>2</sup>

Where *λ*<sup>1</sup> þ *γ*<sup>1</sup> < 1 implies stationarity and *λ*<sup>1</sup> þ *γ*<sup>1</sup> >1 signifies non-stationarity of the ARCH and GARCH. The justification for the choice of GARCH (1,1) apart from being parsimonious is that the variance model depends on the most recent past variance. The use of any higher lags would result to loss of degree of freedom, information and over parameterization of the GARCH model [28]. The GARCH (1,1) model is estimated with different error distributions so as to identify the model with minimum variance using the Schwarz criterion (SC) and the log likelihood. The GARCH model with the minimum variance represents the model with minimum asset risk. The result of the of the GARCH (1,1) model with different error distributions is presented in **Table 4** (See the Appendix 1 for the log likelihood of the distributions). It can be observed from the **Table 4** that all the GARCH (1,1) result with the different errors are stationary given that their parameter values of *λ*<sup>1</sup> þ *γ*<sup>1</sup> <1. In addition, the previous period of volatility of all the error distributions have significant effects on the current conditional volatility. For the GARCH (1,1) with normal distribution error, the sum of the coefficients of the ARCH and GARCH [the sum of the residual square and Garch(�1)] are positive and statistically significant at 0.05% with a value of 0.9037. The value is less than 1, which satisfies the stability condition of the GARCH process. That of the<sup>1</sup> student-t error distribution is 0.8473 and 0.8731 for the generalized error distribution model. The result suggests that the persistence of volatility effects of oil price on stock price is large for Nigeria (the volatility clustering in **Figure 2** equally suggests the persistence of volatility movement of the two series). The large volatility for Nigeria is supported by previous study done by Uyaebo et al. [8] done for six selected countries with Nigeria inclusive. For the GARCH (1,1), the error distribution for the student-t error distribution is 0.85%, 0.87% for generalized error distribution, and there is highest value of 0.90% for normal distribution. The mean equation, on the other hand, implies that 1% change in oil price affects the stock price by 0.13% for the GARCH (1,1) using normal distribution and the generalized error distribution while it is a bit higher at 0.14% for student-t error distribution. However, in terms of the model with goodness of fit and with minimum variance, the GARCH (1,1) model with student-t error distribution behaves optimally with minimum SC value of �2.56 and with the highest log likelihood value of 327.18. The implication of the optimality of the student-t error distribution implies that stock price returns in Nigeria is unpredictable and volatile because of the effect of the global oil price. We therefore conclude here that GARCH (1,1) process with student-t error distribution

*RSPt* ¼ *α* þ *βROPt* þ *ut* (6)

*<sup>t</sup>*�<sup>1</sup> (7)

Eq.(1) is the mean equation, where *Yt* is a column vector of response variables, *α* is the constant term, *β*<sup>0</sup> is a row vector of unknown parameters, *Xt* is a column vector of explanatory variables and *μ<sup>t</sup>* is a column vector of random error terms with *μ<sup>t</sup>* ¼ *zt* ffiffiffiffi *ht* <sup>p</sup> . Where *zt*≈ð Þ 0, *ht* and *ht* is a scaling factor. The variance equation of the ARCH model on the other hand in general term is stated as:

$$h\_t = \chi\_0 + \sum\_{i=1}^{q} \chi\_i u\_{t-i}^2 \tag{2}$$

The limitation of the ARCH model is that it is more of a moving average (MA) model where the variance is only responding to the errors. The autoregressive (AR) parts of the model are not captured, hence the use of more superior model like the GARCH model propounded by Bollerslev [25]. The mean equation still remains the same while the variance equation in general term is written a bit differently from the ARCH model as:

$$h\_t = \gamma\_0 + \sum\_{i=1}^p \lambda\_i h\_{t-i} + \sum\_{i=1}^q \gamma\_i u\_{t-i}^2 \tag{3}$$

The GARCH model equally has its own deficiency; it cannot accounts for the impacts of news and events that can have asymmetric effects on financial assets. For instance, investors would react differently to the occurrence of good or bad news to financial assets or the market. Whenever bad news happen in the financial market, the volatility is usually higher and larger than a state of tranquility. To address such asymmetric effects, non-linear or asymmetric GARCH models such as TGARCH and EGARCH are propounded. The TGARCH model propounded by Zokoian [26] can be stated in its general form as:

$$h\_t = \gamma\_0 + \sum\_{i=1}^p \lambda\_i h\_{t-i} + \sum\_{i=1}^q (\phi\_i + \eta\_i D\_{t-i}) u\_{t-i}^2 \tag{4}$$

Where *Dt*�*<sup>i</sup>* = 1 is bad news for *ut* <0 and 0 otherwise, *β<sup>i</sup>* measures good news, *η<sup>i</sup>* denotes the asymmetry or leverage term, *η<sup>i</sup>* >0 implies asymmetry, while *η<sup>i</sup>* ¼ 0means symmetry. If *η<sup>i</sup>* is found to be significant and positive, then negative shocks have larger impacts on the conditional variance, *ht* than the positive shocks. Another asymmetric GARCH model is EGARCH propounded by Nelson [27] described in logarithm form as:

$$\log\left(h\_t\right) = \gamma\_0 + \sum\_{i=1}^p \beta\_i \left| \frac{u\_{t-i}}{\sqrt{h\_{t-i}}} \right| + \sum\_{i=1}^q \gamma\_i \frac{u\_{t-i}}{\sqrt{h\_{t-i}}} + \sum\_{i=1}^m a\_i \log\left(h\_{t-i}\right) \tag{5}$$

where good news is denoted by positive value of *ut*�*<sup>i</sup>* with total effect as 1 þ *γ<sup>i</sup>* ð Þ *ut*�*<sup>i</sup>* j j and bad news given by *ut*�*<sup>i</sup>* being negative with total effect as 1 � *γ<sup>i</sup>* ð Þ *ut*�*<sup>i</sup>* j j. If *γ<sup>i</sup>* < 0 then bad news is assumed to have higher effects on volatility than good news. There is symmetry if *γ<sup>i</sup>* ¼ 0 and there is asymmetry if *γ<sup>i</sup>* 6¼ 0*:* In short, *γ*<sup>0</sup> is the constant term, *β<sup>i</sup>* measure the ARCH effect, *γ*<sup>1</sup> measures the leverage effect and lastly, *α<sup>i</sup>* account for the GARCH effect.

*Volatility Effects of the Global Oil Price on Stock Price in Nigeria: Evidence from Linear… DOI: http://dx.doi.org/10.5772/intechopen.93497*
