**5. Empirical analysis and result discussions**

We started with the ARCH model formulated in two parts, the mean equation

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

*q*

*i*¼1

*<sup>λ</sup>iht*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup>

*q*

*γiu*<sup>2</sup>

ð Þ *<sup>ϕ</sup><sup>i</sup>* <sup>þ</sup> *<sup>η</sup>iDt*�*<sup>i</sup> <sup>u</sup>*<sup>2</sup>

*i*¼1

*i*¼1

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

<sup>p</sup> . Where *zt*≈ð Þ 0, *ht* and *ht* is a scaling factor. The variance equation of

*γiu*<sup>2</sup>

*Xt* þ *μ<sup>t</sup>* (1)

*<sup>t</sup>*�*<sup>i</sup>* (2)

*<sup>t</sup>*�*<sup>i</sup>* (3)

*<sup>t</sup>*�*<sup>i</sup>* (4)

*α<sup>i</sup>* log ð Þ *ht*�*<sup>i</sup>* (5)

*Yt* ¼ *α* þ *β*<sup>0</sup>

*ht* <sup>¼</sup> *<sup>γ</sup>*<sup>0</sup> <sup>þ</sup><sup>X</sup>

and the variance equation proposed by Engle [24] and written as:

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

the ARCH model on the other hand in general term is stated as:

*ht* <sup>¼</sup> *<sup>γ</sup>*<sup>0</sup> <sup>þ</sup><sup>X</sup>

*p*

*i*¼1

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]

*<sup>λ</sup>iht*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup>

*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]

*q*

*i*¼1

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>*

*i*¼1 *γi ut*�*i* ffiffiffiffiffiffiffiffi *ht*�*<sup>i</sup>* <sup>p</sup> <sup>þ</sup>X*<sup>m</sup>*

with *μ<sup>t</sup>* ¼ *zt*

the ARCH model as:

can be stated in its general form as:

described in logarithm form as:

**176**

log ð Þ¼ *ht <sup>γ</sup>*<sup>0</sup> <sup>þ</sup><sup>X</sup>

*ht* <sup>¼</sup> *<sup>γ</sup>*<sup>0</sup> <sup>þ</sup><sup>X</sup>

*p*

*i*¼1 *βi*

effect and lastly, *α<sup>i</sup>* account for the GARCH effect.

*p*

*i*¼1

denotes the asymmetry or leverage term, *η<sup>i</sup>* >0 implies asymmetry, while

*ut*�*i* ffiffiffiffiffiffiffiffi *ht*�*<sup>i</sup>* p � � � �

� � � � þ<sup>X</sup> *q*

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

ffiffiffiffi *ht*

Having described both the symmetric and asymmetric GARCH, we expressed the variables of interest from the mean equation as:

$$RSP\_t = \alpha + \beta ROP\_t + u\_t \tag{6}$$

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 stated as:

$$h\_t = \chi\_0 + \lambda\_1 h\_{t-1} + \chi\_1 u\_{t-1}^2 \tag{7}$$

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

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


is the best selection model for financial investors when taking decisions on the

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

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

Due to the limitation of the standard ARCH and GARCH model of not capable of capturing news, events and incidents that result in asymmetric impacts on financial assets in financial markets, the use of TGARCH and EGARCH that are more superior in accounting for good and bad news (the asymmetric and non-linear effects) became popular. The GARCH model usually treats the innovation in absolute term with the squared residual. However, with the TGARCH and EGARCH, the residual is decomposed into negative effects (*ut*�*<sup>i</sup>* < 0Þ and the positive effects (*ut*�*<sup>i</sup>* >0Þ. The parsimonious TGARCH (1,1) model can be written as: *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>

*<sup>t</sup>*�<sup>1</sup>*Dt*�<sup>1</sup> and the result presented in **Table 5** with the error distributions. The marginal effects of oil price on stock price is almost similar with the GARCH (1,1) result with almost 0.13 at 1% significance level for all the error distributions. Also, the GARCH effect is significant at 1% for all the error distributions, suggesting significant effects of past conditional volatility on the current volatility. This implies volatility effects of oil price on stock price in Nigeria. For TGARCH (1,1) model of the normal distribution, we found the positive effect (good news) to be insignificant with coefficient value of 0.02% while that of the negative effect (bad news) is significant at 5% with coefficient value of 0.26% (sum of 0.0176 and 0.2454). The difference between the positive effect and negative effect is 0.2454, which is the leverage effect. The result shows presence of leverage effect and negative effect of oil price has more significant impact on stock prices than positive effect. In the same vein, the positive and negative effects of the TGARCH (1,1) model using the student-t error distribution are 0.0373 and 0.1341, respectively, though the negative effect is not significant like the TGARCH (1,1) normal distribution. The negative effect also has larger effect of 0.13% than the positive effect with 0.04%. This finding supports previous study in Nigeria by Salisu [6] that also found bad news to have large effect than good news in oil market. The TGARCH (1,1) for the generalized error distribution also show asymmetric effect though the negative effect is also not significant. The negative effect has coefficient value of 0.1940 while the positive effect is 0.0276. In overall, similar to the GARCH (1,1) model, the student-t error distribution is also found to have the minimum variance with SC value of �2.54 and the maximum log likelihood value of 327.98. We, therefore, conclude

*<sup>t</sup>*�<sup>1</sup> þ

volatility effects of oil price on stock price in Nigeria.

*Graph of the monthly returns of oil price and stock price over the period of study.*

*α*1*u*<sup>2</sup>

**179**

**Figure 2.**

#### **Table 4.**

*GARCH (1,1) results of stock prices and oil prices volatility.*

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

#### **Figure 2.**

**Dependent variable: stock price**

Log likelihood 321.37 Schwarz criterion 2.53

Log likelihood 327.18 Schwarz criterion 2.56

Log likelihood 326.01 Schwarz criterion 2.54

*\*\*\*, \*\*, \* represent significance level at 1, 2, and 10%, respectively.*

*GARCH (1,1) results of stock prices and oil prices volatility.*

Variables Coefficient Std. error z-Statistics Prob. dlogoilprice 0.1257 0.0265 4.7473 0.00\*\*\* Constant 0.0117 0.0046 2.5352 0.01\*\*\*

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

Variables Coefficient Std. error z-Statistics Prob. Constant 0.0005 0.0003 1.8414 0.07\* residual square 0.1562 0.0699 2.2362 0.03\*\* Garch(1) 0.7475 0.101 6.7968 0.00\*\*\*

Variables Coefficient Std. error z-Statistics Prob. dlogoilprice 0.1384 0.0324 4.2693 0.00\*\*\* Constant 0.0079 0.004 2.013 0.04\*\*

Variables Coefficient Std. error z-Statistics Prob. Constant 0.0007 0.0006 1.2456 0.21 Residual square 0.0995 0.0769 1.2943 0.19 Garch(1) 0.7478 0.174 4.2983 0.00\*\*\*

Variables Coefficient Std. error z-Statistics Prob. dlogoilprice 0.1286 0.0315 4.0756 0.00\*\*\* Constant 0.0084 0.0039 2.1526 0.03\*\*

Variables Coefficient Std. error z-Statistics Prob. Constant 0.0006 0.0005 1.3042 0.19 Residual square 0.1230 0.0889 1.3836 0.17 Garch(1) 0.7501 0.1619 4.6326 0.00\*\*\*

**Normal Dist. Mean equation**

**Variance equation**

**Student t dist. Mean equation**

**Variance equation**

**Generalized error Mean equation**

**Variance equation**

**Table 4.**

**178**

*Graph of the monthly returns of oil price and stock price over the period of study.*

is the best selection model for financial investors when taking decisions on the volatility effects of oil price on stock price in Nigeria.

Due to the limitation of the standard ARCH and GARCH model of not capable of capturing news, events and incidents that result in asymmetric impacts on financial assets in financial markets, the use of TGARCH and EGARCH that are more superior in accounting for good and bad news (the asymmetric and non-linear effects) became popular. The GARCH model usually treats the innovation in absolute term with the squared residual. However, with the TGARCH and EGARCH, the residual is decomposed into negative effects (*ut*�*<sup>i</sup>* < 0Þ and the positive effects (*ut*�*<sup>i</sup>* >0Þ. The parsimonious TGARCH (1,1) model can be written as: *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> *<sup>t</sup>*�<sup>1</sup> þ *α*1*u*<sup>2</sup> *<sup>t</sup>*�<sup>1</sup>*Dt*�<sup>1</sup> and the result presented in **Table 5** with the error distributions. The marginal effects of oil price on stock price is almost similar with the GARCH (1,1) result with almost 0.13 at 1% significance level for all the error distributions. Also, the GARCH effect is significant at 1% for all the error distributions, suggesting significant effects of past conditional volatility on the current volatility. This implies volatility effects of oil price on stock price in Nigeria. For TGARCH (1,1) model of the normal distribution, we found the positive effect (good news) to be insignificant with coefficient value of 0.02% while that of the negative effect (bad news) is significant at 5% with coefficient value of 0.26% (sum of 0.0176 and 0.2454). The difference between the positive effect and negative effect is 0.2454, which is the leverage effect. The result shows presence of leverage effect and negative effect of oil price has more significant impact on stock prices than positive effect. In the same vein, the positive and negative effects of the TGARCH (1,1) model using the student-t error distribution are 0.0373 and 0.1341, respectively, though the negative effect is not significant like the TGARCH (1,1) normal distribution. The negative effect also has larger effect of 0.13% than the positive effect with 0.04%. This finding supports previous study in Nigeria by Salisu [6] that also found bad news to have large effect than good news in oil market. The TGARCH (1,1) for the generalized error distribution also show asymmetric effect though the negative effect is also not significant. The negative effect has coefficient value of 0.1940 while the positive effect is 0.0276. In overall, similar to the GARCH (1,1) model, the student-t error distribution is also found to have the minimum variance with SC value of �2.54 and the maximum log likelihood value of 327.98. We, therefore, conclude

that news, information and events on oil prices are very significant to stock price

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

model. The parsimonious EGARCH (1,1) is also specified as: log ð Þ¼ *ht γ*<sup>0</sup> þ

In order to have a robust estimation and result, the EGARCH, another asymmetric or non-linear model, is considered to compare its result with the TGARCH

p þ *α*<sup>1</sup> log ð Þ *ht*�<sup>1</sup> . The result of the EGARCH (1,1) model is

presented in **Table 6**. Looking at the mean equation of the EGARCH (1,1) result with the normal distribution, we found oil price to have 0.17% significant effect on stock price in Nigeria at 1% significance level. The ARCH and the leverage term are not significant while the GARCH terms are significant at 10%. For the ARCH term,

Variables Coefficient Std. error z-Statistics Prob. dlogoilprice 0.1700 0.0285 5.9670 0.00\*\*\* Constant 0.0053 0.0050 1.0484 0.29

Variables Coefficient Std. error z-Statistics Prob. Constant �7.8302 1.4262 �5.4903 0.00\*\*\* Residual square 0.0112 0.1446 0.0773 0.94 Leverage term 0.1330 0.0850 1.5640 0.12 Garch(�1) �0.4560 0.2709 �1.6831 0.09\*

Variables Coefficient Std. error z-Statistics Prob. dlogoilprice 0.1375 0.0342 4.0190 0.00\*\*\* Constant 0.0091 0.004 2.2437 0.02\*\*

Variables Coefficient Std. error z-Statistics Prob. Constant �1.0114 0.6257 �1.6165 0.11 Residual square 0.2042 0.1322 1.5447 0.12 Leverage term �0.0567 0.0783 �0.7248 0.47 Garch(�1) 0.8421 0.1064 7.9128 0.00\*\*\*

Variables Coefficient Std. error z-Statistics Prob. dlogoilprice 0.1257 0.0334 3.7506 0.00\*\*\* Constant 0.0090 0.0040 2.2599 0.04\*\*

volatility in Nigeria.

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

� <sup>þ</sup> *<sup>γ</sup>*<sup>1</sup> *ut*�<sup>1</sup> ffiffiffiffiffiffi *ht*�<sup>1</sup>

**Dependent variable: stock price**

Log likelihood 307.63 Schwarz criterion �2.44

Log likelihood 327.20 Schwarz criterion �2.53

*<sup>β</sup>*<sup>1</sup> *ut*�<sup>1</sup> ffiffi *h* p *t*�1

� �

**Normal dist. Mean equation**

**Variance equation**

**Student t dist. Mean equation**

**Variance equation**

**Generalized error dist. Mean equation**

**181**

� � �


#### **Table 5.**

*TGARCH (1,1) results of stock prices and oil prices volatility.*

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

that news, information and events on oil prices are very significant to stock price volatility in Nigeria.

In order to have a robust estimation and result, the EGARCH, another asymmetric or non-linear model, is considered to compare its result with the TGARCH

model. The parsimonious EGARCH (1,1) is also specified as: log ð Þ¼ *ht γ*<sup>0</sup> þ

*<sup>β</sup>*<sup>1</sup> *ut*�<sup>1</sup> ffiffi *h* p *t*�1 � � � � � � <sup>þ</sup> *<sup>γ</sup>*<sup>1</sup> *ut*�<sup>1</sup> ffiffiffiffiffiffi *ht*�<sup>1</sup> p þ *α*<sup>1</sup> log ð Þ *ht*�<sup>1</sup> . The result of the EGARCH (1,1) model is

presented in **Table 6**. Looking at the mean equation of the EGARCH (1,1) result with the normal distribution, we found oil price to have 0.17% significant effect on stock price in Nigeria at 1% significance level. The ARCH and the leverage term are not significant while the GARCH terms are significant at 10%. For the ARCH term,


**Dependent variable: stock price**

Log likelihood 324.67 Schwarz criterion 2.53

Log likelihood 327.98 Schwarz criterion 2.54

Log likelihood 327.62 Schwarz criterion 2.53 *\*\*\*, \*\*, \* represent significance level at 1, 2, and 10%, respectively.*

*TGARCH (1,1) results of stock prices and oil prices volatility.*

Variables Coefficient Std. error z-Statistics Prob. dlogoilprice 0.1271 0.0293 4.3383 0.00\*\*\* Constant 0.0106 0.0045 2.3545 0.02\*\*

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

Variables Coefficient Std. error z-Statistics Prob. Constant 0.0008 0.0004 1.9647 0.05\* Residual square 0.0176 0.0726 0.2424 0.81 resid square(resid(1) > 0 0.2454 0.123 1.9957 0.04\*\* Garch(1) 0.6862 0.1319 5.2028 0.00\*\*\*

Variables Coefficient Std. error z-Statistics Prob. dlogoilprice 0.1393 0.0338 4.1172 0.00\*\*\* Constant 0.0080 0.0040 2.0086 0.04\*\*

Variables Coefficient Std. error z-Statistics Prob. Constant 0.0008 0.0006 1.3834 0.17 Residual square 0.0373 0.0886 0.4214 0.67 Resid square(resid(1) > 0 0.1341 0.1327 1.0100 0.31 Garch(1) 0.7111 0.1838 3.8685 0.00\*\*\*

Variables Coefficient Std. error z-Statistics Prob. dlogoilprice 0.1305 0.03335 3.8936 0.00\*\*\* Constant 0.0082 0.0040 2.0685 0.04\*\*

Variables Coefficient Std. error z-Statistics Prob. Constant 0.0008 0.0005 1.4810 0.13 Residual square 0.0276 0.0902 0.3060 0.75 Resid square(resid(1) > 0 0.1940 0.1496 1.2969 0.19 Garch(1) 0.6974 0.1766 3.9499 0.00\*\*\*

**Normal Dist. Mean equation**

**Variance equation**

**Student t dist. Mean equation**

**Variance equation**

**Generalized Error dist. Mean equation**

**Variance equation**

**Table 5.**

**180**


Appendix 4 showing rejection of the null hypothesis of presence of serial correlation

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

In this study we examined the volatility effects of oil price behavior on stock price in Nigeria from the first month of year 2000 to the fourth month of year 2020 using both standard and asymmetric GARCH. Before performing the GARCH, TGARCH and EGARCH, we carried out some preliminary tests such as the ARCH tests for heteroscedasticity, unit root test for stationary test and all the tests show evidence of volatility clustering which necessitate the use of GARCH process on the variables. The standard GARCH was first done and the model with student-t distribution showed goodness of fit. We proceeded to use the non-linear GARCH models such as the TGARCH and EGARCH to account for news, events and information that can filter into the oil market and thereby create asymmetric behavior in the financial market. The non-linear GARCH models also confirm the student-t distribution as the best model for traders in the financial market in Nigeria. In this study, we found oil price volatility to be a significant predictor of stock price returns. Secondly, our study showed that the volatility movement is high and persist over the study period. Also, we found leverage effects in stock price response to oil price. Bad news tends to increase volatility than good news. One of the implications of the findings of this study is that oil price volatility should be considered in the prediction of stock price returns by investors and financial analyst in Nigeria. In addition, the finding implies that most of the investors in the financial market are risk averse; this is because they are more sensitive in their asset decisions to bad news than to good news. This study concludes that bad news have much effects on investors than

good news in the movement of oil price effect to stock price returns.

**A.1 The probability density function of normal distribution is**

2

<sup>2</sup> log ð Þ� *π ν*ð Þ � <sup>2</sup> <sup>1</sup>

and its log likelihood function in GARCH term is:

ln 2ð Þ� *<sup>π</sup> <sup>n</sup>*

*f y*ð Þ¼ , *ν*

Its log likelihood function in GARCH term is:

� log <sup>Γ</sup> *<sup>ν</sup>* 2 h i � � � <sup>1</sup>

� *n* 2 *f x*ð Þ¼ <sup>j</sup>*μ*, *<sup>σ</sup>* <sup>1</sup>

*σ* ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup> *<sup>e</sup>*

ln ð Þ� *<sup>h</sup>* <sup>1</sup>

**A.2 The probability density function of the student-t distribution is:**

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *π ν*ð Þ � <sup>2</sup> <sup>p</sup> <sup>Γ</sup> *<sup>ν</sup>*

> 2 X*n j*¼1

2*h* X*n j*¼1

Γ *<sup>ν</sup>*þ<sup>1</sup> 2

> <sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>ν</sup>*<sup>2</sup> *ν*�2 � �*<sup>ν</sup>*þ<sup>1</sup> 2

�ð Þ *<sup>x</sup>*�*<sup>μ</sup>* <sup>2</sup>

*<sup>x</sup> <sup>j</sup>* � *<sup>μ</sup>* � �<sup>2</sup>

log ð Þþ *ht* ð Þ *<sup>ν</sup>* <sup>þ</sup> <sup>1</sup> log 1 <sup>þ</sup> *<sup>ε</sup>*<sup>2</sup>

� � � � (11)

<sup>2</sup>*σ*<sup>2</sup> (8)

*:* (9)

*t ht*ð Þ *ν* � 2 (10)

with p-values greater than 0.05.

**A. Appendix**

log <sup>Γ</sup> *<sup>ν</sup>* <sup>þ</sup> <sup>1</sup> 2 � � � �

**183**

**written as:**

**6. Conclusion and policy implications**

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

#### **Table 6.**

*EGARCH (1,1) results of stock prices and oil prices volatility.*

the result shows a positive relationship between the shock of the oil price and the volatility of stock price returns. Also, the leverage effect is positive meaning that good news prevails over bad news in the oil market on stock price volatility. Negative effect is found between the past volatility and the future. The past volatility negatively predicts the future volatility at 10% significance level. We further examine the EGARCH (1,1) result with the student-t distribution and we found the marginal effect of oil price on stock price returns to be 0.14%, lower than the 0.17% of the EGARCH (1,1) model with normal distribution. Similar to the result of the normal distribution, the ARCH and the leverage term are also not significant only the GARCH term is significant at 1%. The ARCH term shows a positive relationship between the oil price shocks and the stock price volatility returns. 1% increase in oil price shock, stock price fluctuates by 0.20%. The leverage effects on the other hand are negative. This implies that 1% increase in the negative shocks in the oil price; it reduces the stock price returns by 0.06%. The GARCH term is significant at 1% level suggesting that the previous volatility predicts significantly the future volatility in the effect of oil price volatility on stock price returns. A 1% increase in past volatility leads to 0.84% increase in future volatility significantly at 1% level. Lastly, we examine the EGARCH (1,1) result using the generalized error distribution and we found the marginal effect of oil price volatility on stock price returns to be 0.13% at 1% significance level. The result of the ARCH, leverage and GARCH term of the generalized error term is similar to that of the student-t distribution. The ARCH term shows that 1% increase in the oil price shock insignificantly increases the stock price returns by 0.22%. The leverage effect also shows prevalence of bad news with 1% increase in bad news in the oil market reducing stock price returns by 0.08%. The GARCH term is significant with 0.85% future volatility increase resulting from 1% increase in past volatility in relation to the effect of oil price on the stock price in Nigeria. Of all the distributions, the EGARCH (1,1) of the student-t distribution is found to be the best model with minimum variance looking at the SC and likelihood. The EGARCH (1,1) with student-t distribution has SC with minimum value of 2.53 and likelihood maximum value of 327.20. We therefore conclude that both the standard GARCH and non-linear GARCH process driven by the student-t distribution is the best selection model for investors for valuing the volatility effect of oil price on stock price in Nigeria. Finally, considering the diagnostic tests of our model, the serial correlation for all the error distributions used are presented at the

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

Appendix 4 showing rejection of the null hypothesis of presence of serial correlation with p-values greater than 0.05.
