**Table 4.**

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

**Model**

**114**

**eGARCH (1,1)**

Robust Standard Errors: Estimate

0.35726

0.021440

–16.6632

0.00000

omega

omega alpha1

beta1 gamma1

shape Weighted Ljung-Box Test on

Standardized

——————————————————————————————————

Lag [1] Lag[2\*(p+q)+(p+q)-1]

Lag[4\*(p+q)+(p+q)-1]

d.o.f=0 H0: No serial correlation

Weighted ARCH LM Tests

——————————————————————————————————

Statistic

ARCH Lag [3] ARCH Lag [5] ARCH Lag [7]

> **Table 3.**

*Parameter*

 *estimates and ARCH LM tests of the GARCH models.*

0.06498

 2.315

0.04570

1.440

1.667 1.543

0.9998

ARCH Lag [7]

0.06276

 2.315

0.9957

ARCH Lag [5]

0.04417

 1.440

0.02173

0.500

2.000

0.8828

ARCH Lag [3]

0.02100

 0.500

 Shape

Scale

P-Value

 [5]

 [2]

statistic

19.23 19.28 19.95

1.999e-05

d.o.f=0

H0: No serial correlation

Weighted ARCH LM Tests

——————————————————————————————————

Statistic

 Shape

Scale 2.000 1.667 1.543

0.9998

0.9959

0.8848

P-Value

Lag[4\*(p+q)+(p+q)-1]

 [5]

6.872e-06

Lag[2\*(p+q)+(p+q)-1]

 [2]

1.160e-05

 Lag [1]

 p-value

 Residuals

2.64439

0.347058

 7.6194

0.53417

0.080367

 6.6467

0.94958

0.001992

 476.7729


0.065840


0.11687

0.00000 0.00000 0.00000

skew shape Weighted Ljung-Box Test on

Standardized

——————————————————————————————————

statistic

18.99 19.04

19.71

2.335e-05

7.944e-06

1.314e-05

 p-value

 Residuals

2.64681

0.345965

 7.6505

1.02439

0.012253

 83.6013

gamma1

0.53018

0.079261

 6.6891

beta1

0.95011

0.001801

 527.6448

 alpha1

 Std. Error

 t value

 Pr(>|t|)

**Student t distribution**

**Skewed Student t distribution**

Robust Standard Errors: Estimate

0.35394


0.065963

 -1.6065

0.020239

–17.4879

0.00000

0.10817

0.00000

0.00000

0.00000

0.00000

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

 Std. Error

 t value

 Pr(>|t|)


not significant at 5% level except for eGARCH (1,1) model that provided significant coefficients in most cases. In the overall, most of the estimated GARCH models revealed absence of serial correlation in the error terms and absence of ARCH effects in the residuals. Because of limited space, we presented only the result of

Persistence of GARCH model measure whether the estimated GARCH model is stable or not as shown in **Table 4** above. In financial time series literature it should be less than 1 [3, 36]. Most of the models are stable except for iGARCH model. The half-life measure how long it will take for mean-reversion of the stock returns. The

The **Table 5** above presented the backtesting test of some selected GARCH model. The backtesting result of the apARCH (2,2) was not available while

eGARCH(1,1) with Skewed student t-distribution, NGARCH(1,1), NGARCH(2,1), and TGARCH (2,1) failed the backtesting but eGARCH (1,1) with student tdistribution passed the backtesting approach which is supported by the results in **Table 5** above. Therefore with the backtesting approach, eGARCH(1,1) with student t-distribution emerged the superior model for modeling Zenith Bank stock returns in Nigeria [30, 31]. This chapter recommended the backtesting approach to

This book chapter investigated the place of backtesting approach in financial time series analysis in choosing a reliable GARCH Model for analyzing stock returns. To achieve this, The chapter used a secondary data that was collected from www.cashcraft.com under stock trend and analysis. Daily stock price was collected on Zenith bank stock price from October 21st 2004 to May 8th 2017. The chapter used 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 could to converged. 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-of-test fit and backtesting should be considered also [3].

The backtesting result of the apARCH (2,2) was not available while eGARCH (1,1) with Skewed student t distribution, NGARCH(1,1), NGARCH(2,1), and TGARCH (2,1) failed the backtesting but eGARCH (1,1) with student t distribution

I wish to acknowledge my PhD student and M.Sc. students that have worked

passed the backtesting approach. Therefore with the backtesting approach, eGARCH(1,1) with student distribution emerged the superior model for modeling Zenith Bank stock returns in Nigeria [30, 31]. This chapter recommended the

backtesting approach to selecting reliable GARCH model.

The Author declares no conflict of interest.

under my supervision in the area of financial time series analysis.

result revealed an average of 10 days for mean-reversion to take place.

selecting reliable GARCH model for estimating stock returns in Nigeria.

eGARCH (1,1) model in **Table 3** above.

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

*Financial Time Series Analysis via Backtesting Approach*

**6. Conclusions**

**Acknowledgements**

**Conflict of interest**

**117**

*Note: uc.LRstat: the unconditional coverage test likelihood-ratio statistic; uc.critical: the unconditional coverage test critical value; uc.LRp: the unconditional coverage test p-value; cc.LRstat: the conditional coverage test likelihood-ratio statistic; cc. critical: the conditional coverage test critical value; cc.LRp: the conditional coverage test p-value; NA: not available.*

#### **Table 5.**

*Backtesting of the GARCH models: GARCH roll forecast (backtest length: 1070) for the log daily zenith Bank stock returns.*

*Financial Time Series Analysis via Backtesting Approach DOI: http://dx.doi.org/10.5772/intechopen.94112*

not significant at 5% level except for eGARCH (1,1) model that provided significant coefficients in most cases. In the overall, most of the estimated GARCH models revealed absence of serial correlation in the error terms and absence of ARCH effects in the residuals. Because of limited space, we presented only the result of eGARCH (1,1) model in **Table 3** above.

Persistence of GARCH model measure whether the estimated GARCH model is stable or not as shown in **Table 4** above. In financial time series literature it should be less than 1 [3, 36]. Most of the models are stable except for iGARCH model. The half-life measure how long it will take for mean-reversion of the stock returns. The result revealed an average of 10 days for mean-reversion to take place.

The **Table 5** above presented the backtesting test of some selected GARCH model. The backtesting result of the apARCH (2,2) was not available while eGARCH(1,1) with Skewed student t-distribution, NGARCH(1,1), NGARCH(2,1), and TGARCH (2,1) failed the backtesting but eGARCH (1,1) with student tdistribution passed the backtesting approach which is supported by the results in **Table 5** above. Therefore with the backtesting approach, eGARCH(1,1) with student t-distribution emerged the superior model for modeling Zenith Bank stock returns in Nigeria [30, 31]. This chapter recommended the backtesting approach to selecting reliable GARCH model for estimating stock returns in Nigeria.
