**5. Results and discussion**

The section presented the results emanating from the analysis and discussions of results.

**Figure 1** below presented the plot of the log of Zenith Bank returns which is the first step in financial time series analysis. The plot revealed some spikes at the early part of the return series while later the series returns became stable.

**Figure 2** below presented the plot of the cleansed log of Zenith Bank returns, this is necessary to remove any possible outlier that may be presents in the return series.

The **Table 1** below presented the descriptive statistics of the zenith bank return series. The **Table 1** revealed a maximum return as 0.338000 while minimum return as �0.405850. The average return as 0.000114 which signifies a gain in the stock for the period under study. The series is negatively skewed with high value of kurtosis. The return series is not normally distributed and the return series is stationary with presence of ARCH effects in the return series, these are typical characteristics of a financial return series [34, 35].

**Table 2** below presents the selection criteria values for daily zenith Bank stock returns based on the student and skewed student t-distributions. The log returns of

**Figure 1.** *The time plot of the log of zenith Bank returns.*

#### **Figure 2.**

The Skewed student t-distribution is given as

with �1<*λ*<1. The constants a, b and c are given as

*v* � 2 *v* � 1 � �

8 >>>>>><

*Linked Open Data - Applications,Trends and Future Developments*

>>>>>>:

*bc* <sup>1</sup> <sup>þ</sup> <sup>1</sup> *v*�2

*bc* <sup>1</sup> <sup>þ</sup> <sup>1</sup> *v*�2

*<sup>b</sup> <sup>y</sup>*�*<sup>μ</sup>* ð Þ *<sup>σ</sup>* <sup>þ</sup>*<sup>a</sup>* 1�*λ* � �2 !�*v*þ<sup>1</sup>

*<sup>b</sup> <sup>y</sup>*�*<sup>μ</sup>* ð Þ *<sup>σ</sup>* <sup>þ</sup>*<sup>a</sup>* 1þ*λ* � �2 !�*v*þ<sup>1</sup>

Where *v* is the shape parameter with 2 <*v*< ∞ and *λ* is the skewness parameter

Where *μ* and *σ* are the mean and the standard deviation of the skewed student t

The data used in this study is a secondary data that was collected from www.ca

Where Rt is stock returns; Pt is the present stock price; Pt-1 is the previous stock

The section presented the results emanating from the analysis and discussions of

**Figure 1** below presented the plot of the log of Zenith Bank returns which is the first step in financial time series analysis. The plot revealed some spikes at the early

**Figure 2** below presented the plot of the cleansed log of Zenith Bank returns, this is necessary to remove any possible outlier that may be presents in the return

The **Table 1** below presented the descriptive statistics of the zenith bank return series. The **Table 1** revealed a maximum return as 0.338000 while minimum return as �0.405850. The average return as 0.000114 which signifies a gain in the stock for the period under study. The series is negatively skewed with high value of kurtosis. The return series is not normally distributed and the return series is stationary with presence of ARCH effects in the return series, these are typical characteristics of a

**Table 2** below presents the selection criteria values for daily zenith Bank stock returns based on the student and skewed student t-distributions. The log returns of

shcraft.com under stock trend and analysis. Daily stock price was collected on

price and ln is the natural logarithm transformation. Then total observation

part of the return series while later the series returns became stable.

Zenith bank stock price from October 21st 2004 to May 8th 2017. The returns was calculated using the formula below

; *<sup>b</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>3</sup>ð Þ*<sup>λ</sup>* <sup>2</sup> � *<sup>a</sup>*<sup>2</sup>

2

2

;*<sup>c</sup>* <sup>¼</sup> <sup>Γ</sup> *<sup>v</sup>*þ<sup>1</sup>

*Rt* ¼ ln *Pt* � ln *Pt*�<sup>1</sup> (29)

, *if y* <sup>&</sup>lt; � *<sup>a</sup>*

, *if y* <sup>≥</sup> � *<sup>a</sup>*

2 � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>π</sup>*ð Þ *<sup>v</sup>* � <sup>2</sup> <sup>Γ</sup> *<sup>v</sup>*

q � �

*b*

*b*

2

*f y*ð Þ¼ ; *μ*, *σ*, *v*, *λ*

*a* ¼ 4*λc*

distribution respectively.

becomes 3070.

results.

series.

**110**

**5. Results and discussion**

financial return series [34, 35].

**4. Method of data collection**

*The time plot of the removal of possible outliers in the log of zenith Bank returns.*


#### **Table 1.**

*Descriptive statistics and unit root testing of zenith Bank stock returns.*

**Model Information criteria Std t innovation Skewed stdt innovation**

5.8908 5.8751 5.8908 5.8851

15.563 15.553 15.563 15.559

14.470 14.458 14.470 14.466

9.5866 9.5729 9.5866 9.5817

7.8258 7.8140 7.8258 7.8216

8.1226 8.1069 8.1226 8.1170

16.904 16.886 16.904 16.897

5.1428 5.1330 5.1428 5.1393

5.1296 5.1158 5.1296 5.1246

5.1221 5.1063 5.1221 5.1164

5.8467 5.8349 5.8467 5.8425

5.6197 5.6020 5.6197 5.6134

5.4227 5.4031 5.4228 5.4157

5.8752 5.8575 5.8752 5.8688

13.191 13.179 13.191 13.187

16.419 16.405 16.419 16.414

11.248 11.232 11.248 11.242

NA

8.7718 8.7541 8.7718 8.7654

> 9.4341 9.4538 9.4341 9.4412

5.1402 5.1285 5.1403 5.1360

5.1343 5.1186 5.1343 5.1286

5.0439 5.0262 5.0439 5.0375

5.6004 5.5866 5.6004 5.5954

5.9524 5.9327 5.9524 5.9453

5.8644 5.8428 5.8644 5.8567

TGARCH(2,2) Akaike

*DOI: http://dx.doi.org/10.5772/intechopen.94112*

NGARCH(1,1) Akaike

NGARCH(2,1) Akaike

NGARCH(2,2) Akaike

apARCH(1,1) Akaike

apARCH(2,1) Akaike

apARCH(2,2) Akaike

NAGARCH(1,1) Akaike

NAGARCH(2,1) Akaike

NAGARCH(2,2) Akaike

AVGARCH(1,1) Akaike

AVGARCH(2,1) Akaike

AVGARCH(2,2) Akaike

*Note: NA-Not Available.*

**Table 2.**

**113**

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*Financial Time Series Analysis via Backtesting Approach*

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Bayes Shibata Hannan-Quinn

*GARCH models and their performance on the log returns of daily log zenith Bank returns.*



**Model Information criteria Std t innovation Skewed stdt innovation**

NA NA

NA NA

NA 5.3667

NA NA

5.5110 5.5012 5.5110 5.5075

5.5493 5.5356 5.5493 5.5444

5.0584 5.0485 5.0584 5.0548

5.0853 5.0716 5.0853 5.0804

5.0196 5.0039 5.0196 5.0140

5.1474 5.1415 5.1474 5.1453

5.1527 5.1449 5.1527 5.1499

5.1496 5.1397 5.1496 5.1460

5.8914 5.8815 5.8914 5.8878

5.9253 5.9115 5.9253 5.9203 5.3530 5.3667 5.3618

NA

NA

5.0587 5.0469 5.0587 5.0545

5.0859 5.0702 5.0859 5.0802

NA

5.1498 5.1420 5.1498 5.1470

5.1526 5.1428 5.1527 5.1491

5.1547 5.1429 5.1547 5.1505

5.8920 5.8803 5.8921 5.8878

5.8819 5.8662 5.8819 5.8763

sGARCH (1,1) Akaike

sGARCH (2,1) Akaike

sGARCH(2,2) Akaike

gjrGARCH(1,1) Akaike

gjrGARCH(2,1) Akaike

gjrGARCH(2,2) Akaike

eGARCH (1,1) Akaike

eGARCH (2,1) Akaike

eGARCH (2,2) Akaike

iGARCH (1,1) Akaike

iGARCH (2,1) Akaike

iGARCH (2,2) Akaike

TGARCH(1,1) Akaike

TGARCH(2,1) Akaike

**112**

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*Linked Open Data - Applications,Trends and Future Developments*

Bayes Shibata Hannan-Quinn

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Bayes Shibata Hannan-Quinn

Bayes Shibata Hannan-Quinn

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### **Table 2.**

*GARCH models and their performance on the log returns of daily log zenith Bank returns.*


**Table 3.** *Parameter estimates and ARCH LM tests of the GARCH*

**Table 4.**

**115**

 *models.* the daily stock price of Zenith Bank returns were modeled with nine different GARCH models (sGARCH, gjrGARCH, eGARCH, iGARCH, aPARCH, TGARCH, NGARCH, NAGARCH and AVGARCH) with maximum lag of 2. Most the information criteria for the sGARCH model were not available because the model fails to converge. The lowest information criteria were associated with apARCH (2,2) with Student t-distribution followed by NGARCH(2,1) with skewed student t distribution. The caution here is that GARCH model should not be selected only based on information criteria only but the significance value of the coefficients, goodness-oftest fit and backtesting should be considered also [3]. The estimated GARCH models for the zenith bank stock with nine different GARCH models (sGARCH, gjrGARCH, eGARCH, iGARCH, aPARCH, TGARCH, NGARCH, NAGARCH and AVGARCH) shows that most of the coefficients of the fitted GARCH models were

*Financial Time Series Analysis via Backtesting Approach*

*DOI: http://dx.doi.org/10.5772/intechopen.94112*

**Models Std Skewed std**

*Persistence and half-life volatility of the GARCH models of daily log zenith Bank stock returns.*

sGARCH (1,1) NA NA NA NA sGARCH (2,1) NA NA NA NA sGARCH(2,2) NA NA 0.9783281 31.63581 gjrGARCH(1,1) 0.9945289 126.3447 NA NA gjrGARCH(2,1) 0.9939226 113.7067 NA NA gjrGARCH(2,2) NA NA NA NA eGARCH (1,1) 0.9495821 13.39848 0.9501065 13.543 eGARCH (2,1) 0.9799226 34.17597 0.9802555 34.75809 eGARCH (2,2) 0.9775862 30.57714 NA NA iGARCH (1,1) 1 infinity 1 infinity iGARCH (2,1) 1 infinity 1 infinity iGARCH (2,2) 1 infinity 1 Infinity TGARCH(1,1) 0.9463794 12.57713 0.9587135 16.43969 TGARCH(2,1) 0.9529079 14.36961 0.9506704 13.70184 TGARCH(2,2) 0.9315479 9.775345 0.9470317 12.73636 NGARCH(1,1) 0.9925847 93.1287 0.9732531 25.56687 NGARCH(2,1) 0.984207 43.54208 0.9888705 61.93282 NGARCH(2,2) 0.9704479 23.10679 0.9971636 244.0279 apARCH(1,1) 0.9759139 28.42987 NA NA apARCH(2,1) 0.9829391 40.28021 0.9853317 46.90736 apARCH(2,2) 0.9869038 52.58005 0.9513766 13.90596 NAGARCH(1,1) 0.9933088 103.2444 0.9950269 139.0335 NAGARCH(2,1) 0.9910378 76.99442 0.9942849 120.9365 NAGARCH(2,2) 0.9974602 272.5659 0.9978423 320.8989 AVGARCH(1,1) 0.9579476 16.13387 0.9315018 9.768526 AVGARCH(2,1) 0.9311321 9.714181 0.9513755 13.90564 AVGARCH(2,2) 0.9635552 18.6704 0.9633697 18.57406

**Persistence Half-life volatility Persistence Half-life volatility**
