Open access peer-reviewed chapter

Financial Time Series Analysis via Backtesting Approach

By Monday Osagie Adenomon

Submitted: March 24th 2020Reviewed: September 18th 2020Published: October 27th 2020

DOI: 10.5772/intechopen.94112

Downloaded: 83

Abstract

This book chapter investigated the place of backtesting approach in financial time series analysis in choosing a reliable Generalized Auto-Regressive Conditional Heteroscedastic (GARCH) Model to analyze stock returns in Nigeria. 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 (standard GARCH (sGARCH), Glosten-Jagannathan-Runkle GARCH (gjrGARCH), Exponential GARCH (Egarch), Integrated GARCH (iGARCH), Asymmetric Power Autoregressive Conditional Heteroskedasticity (ARCH) (apARCH), Threshold GARCH (TGARCH), Non-linear GARCH (NGARCH), Nonlinear (Asymmetric) GARCH (NAGARCH) and The Absolute Value GARCH (AVGARCH) with maximum lag of 2. Most the information criteria for the sGARCH model were not available due to lack of convergence. 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 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 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. This chapter recommended the backtesting approach to selecting reliable GARCH model.

Keywords

  • financial
  • time series
  • backtesting
  • GARCH
  • ARCH-LM

1. Introduction

Time series is a series of observation collected with respect to time. The time could be in minutes, hours, daily, weekly, monthly, yearly etc. Time series data can be seen and applied in all fields of endeavors such as engineering, geophysics, business, economics, finance, agriculture, medical sciences, social sciences, meteorology, quality control etc. [1] but this chapter focused on financial time series analysis.

In the field of time series analysis, Autoregressive Moving Average (ARMA) and Autoregressive Integrated Moving Average (ARIMA) models are popular and excellent for modeling and forecasting univariate time series data as proposed by Box and Jenkins in 1970 but many times these models failed in analyzing and forecasting financial time series [2], this is because the ARMA and ARIMA models are used to model conditional expectation of a process but in ARMA model, the conditional variance is constant. This means that ARMA model cannot capture process with time-varying conditional variance (volatility) which is mostly common with economic and financial time series data [3].

In economic and financial time series literatures, time-varying is more common than constant volatility, and accurate modeling of time volatility is of great importance in financial time series analysis by financial econometricians [4]. In practice, financial time series contains uncertainty, volatility, excess kurtosis, high standard deviation, high skewness and sometimes non normality [3]. Therefore, to model and capture properly the characteristics of financial time series models such as Auto-Regressive Conditional Heteroscedastic (ARCH), Generalized Auto-Regressive Conditional Heteroscedastic (GARCH), multivariate GARCH, Stochastic volatitlity (SV) and various variants of the models have been proposed to handle these characteristics of financial time series [5]. This chapter would focus on univariate GARCH models. In practice, the backtesting approach compliment the estimated GARCH model, in order to select a reliable GARCH model useful for real life application.

This book chapter aimed at obtaining reliable GARCH model via backtesting approach using daily Zenith bank Nigeria plc stock returns.

2. Empirical literature reviews of previous studies

Emenogu, et al. [3] modeled and forecasted the Guaranty Trust (GT) Bank daily stock returns from January 22,001 to May 82,017 data set collected from a secondary source. The ARMA-GARCH models, persistence, half-life and backtesting were used to analyzed the collected data using student t and skewed student t-distributions, and the analyses are carried out R environment using rugarch and performanceAnaytics Packages. The study revealed that using the lowest information criteria values only could be misleading rather we added the use of backtesing. The ARMA(1,1)-GARCH(1,1) models fitted exhibited high persistency in the daily stock returns while the days it takes for mean-reverting of the models is about 5 days, but unfortunately the models failed backtesting. The results further revealed ARMA(1,1)-eGARCH (2,2) model with student t distribution provides a suitable model for evaluating the GT bank stock returns among the competing models while it takes less than 30 days for the persistence volatility to return back to its average value of the stock returns. They recommended that researchers should adopt backtesting approach while fitting GARCH models while GT bank stocks investor should be assured that no matter the fluctuations in the stock market, the GT bank stock returns has the ability to returns to its mean price return.

Asemota and Ekejiuba [6] examined the volatility of banks equity weekly returns for six banks (coded B1 to B6) using GARCH models. Results reveal the presence of ARCH effect in B2 and B3 equity returns. In addition, the estimated models could not find evidence of leverage effect. On evaluating the estimated models using standard criteria, EGARCH (1, 1) and CGARCH (1, 1) model in Student’s t-distribution are adjudged the best volatility models for B2 and B3 respectively. The study recommends that in modeling stock market volatility, variants of GARCH models and alternative error distribution should be considered for robustness of results. The study also recommended for adequate regulatory effort by the CBN over commercial banks operations that will enhance efficiency of their stocks performance and reduce volatility aimed at boosting investors’ confidence in the banking sector.

Adigwe, et al. [7], examined the effect of stock market development on Nigeria’s economic growth. The objective of the study was to determine if stock market development significantly impact on the country’s economic growth. Secondary data were employed for the study covering 1985 to 2014. Ordinary Least Square (OLS) econometric technique was used for the time series analysis in which variations in economic growth was regressed on market capitalisation ratio to GDP, value of stock traded ratio to GDP, trade openness and inflation rate. The analysis revealed that stock market has the potentials of growth inducing, but has not contributed meaningfully to Nigerian economic growth, since only 26.5% of variations in economic growth were explained by the stock market development variables. Based on this, they suggested for an encouragement of more investors in the market, improvement in the settlement system and ensuring investors’ confidence in the market.

Yaya, et al. [8] examined the application of nonlinear Smooth Transition- Generalized Autoregressive Conditional Heteroscedasticity (ST-GARCH) model of Hagerud on prices of banks’ shares in Nigeria. The methodology was informed by the failure of the conventional GARCH model to capture the asymmetric properties of the banks’ daily share prices. The asymmetry and non-linearity in the model dynamics make it useful for generating nonlinear conditional variance series. From the empirical analysis, we obtained the conditional volatility of each bank’s share price return. The highest volatility persistence was observed in Bank 6, while Bank 12 had the least volatility. Evidently, about 25% of the investigated banks exhibited linear volatility behavior, while the remaining banks showed nonlinear volatility specifications. Given the level of risk associated with investment in stocks, investors and financial analysts could consider volatility modeling of bank share prices with variants of the ST-GARCH models. The impact of news is an important feature that relevant agencies could study so as to be guided while addressing underlying issues in the banking system.

Emenike and Aleke [9] studied the daily closing prices of the Nigerian stocks from January 1996 to December 2011 used the asymmetric GARCH variants. Their result showed strong evidence of asymmetric effects in the stock returns and therefore proposed EGARCH as performing better than other asymmetric rivals.

Arowolo [10] examined the forecasting properties of linear GARCH model for daily closing stocks prices of Zenith bank Plc in the Nigerian Stock Exchange. The Akaike and Bayesian Information Criteria (AIC and BIC) techniques were used to obtain the order of the GARCH (p,q) that best fit the Zenith Bank return series. The information criteria identified GARCH (1,2) as the appropriate model. His result further supported the claim that financial data are leptokurtic.

Emenike and Ani [11], examined the nature of volatility of stock returns in the Nigerian banking sector using GARCH models. Individual bank indices and the All-share Index of the Nigerian Stock Exchange were evaluated for evidence of volatility persistence, volatility asymmetry and fat tails using data from 3 January 2006 to 31 December 2012. Results obtained from GARCH models suggest that stock returns volatility of the Nigerian banking sector move in cluster and that volatility persistence is high for the sample period. The results also indicate that stock returns distribution of the banking sector is leptokurtic and that sign of the innovations have insignificant influence on the volatility of stock returns of the banks. Finally, the findings of this study show that the degree of volatility persistence is higher for the All Share Index than for most of the banks.

Abubakar and Gani [12] re-examined the long run relationship between financial development indicators and economic growth in Nigeria over the period 1970–2010. The study employed the Johansen and Juselius (1990) approach to cointegration and Vector Error Correction Modeling (VECM). Their findings revealed that in the long-run, liquid liabilities of commercial banks and trade openness exert significant positive influence on economic growth, conversely, credit to the private sector, interest rate spread and government expenditure exert significant negative influence. The findings implied that, credit to the private sector is marred by the identified problems and government borrowing and high interest rate are crowding out investment and growth. The policy implications are financial reforms in Nigeria should focus more on deepening the sector in terms of financial instruments so that firms can have alternatives to banks’ credit which proved to be inefficient and detrimental to growth, moreover, government should inculcate fiscal discipline.

3. Model specification

This study focuses on the GARCH models that are robust for forecasting the volatility of financial time series data; so GARCH model and some of its extensions are presented in this section.

3.1 Autoregressive conditional heteroskedasticity (ARCH) family model

Every ARCH or GARCH family model requires two distinct specifications, namely: the mean and the variance Equations [13]. The mean equation for a conditional heteroskedasticity in a return series, ytis given by

yt=Et1yt+εtE1

where εt=φtσt.

The mean equation in Eq. (1) also applies to other GARCH family models. Et1.is the expected value conditional on information available at time t-1, while εtis the error generated from the mean equation at time t and φtis the sequence of independent and identically distributed random variables with zero mean and unit variance.

The variance equation for an ARCH(p) model is given by

σt2=ω+α1at12++αpatp2E2

It can be seen in the equation that large values of the innovation of asset returns have bigger impact on the conditional variance because they are squared, which means that a large shock tends to follow another large shock and that is the same way the clusters of the volatility behave. So the ARCH(p) model becomes:

at=σtεt,σt2=ω+α1at12++αpatp2E3

Where εtN(0,1) iid, ω> 0 and αi0for i> 0. In practice, εtis assumed to follow the standard normal or a standardized student-t distribution or a generalized error distribution [14].

3.2 Asymmetric power ARCH

According to Rossi [15], the asymmetric power ARCH model proposed by [16] given below forms the basis for deriving the GARCH family models.

Given that:

r=μ+at,εt=σtεt,εtN01σtδ=ω+i=1pαiatiγiatiδ+j=1qβjσtjδ,E4

where

ω>0,δ0αi0i=1,2,,p1<γi<1i=1,2,,pβj>0j=1,2,,q

This model imposes a Box-Cox transformation of the conditional standard deviation process and the asymmetric absolute residuals. The leverage effect is the asymmetric response of volatility to positive and negative “shocks”.

3.3 Standard GARCH(p, q) model

The mathematical model for the sGARCH(p,q) model is obtained from Eq. (4) by letting δ=2and γi=0, i=1,,pto be:

at=σtεt,σt2=ω+i=1pαiati2+j=1qβjσtj2E5

Where at=rtμt(rtis the continuously compounded log return series), and.

εtN(0,1) iid, the parameter αiis the ARCH parameter and βjis the GARCH parameter, and ω> 0, αi0, βj0, and i=1maxpqαi+βi< 1, [17].

The restriction on ARCH and GARCH parameters αiβjsuggests that the volatility (ai) is finite and that the conditional standard deviation (σi) increases. It can be observed that if q = 0, then the model GARCH parameter (βj) becomes extinct and what is left is an ARCH(p) model.

To expatiate on the properties of GARCH models, the following representation is necessary:

Let ηt=at2σt2so that σt2=at2ηt. By substituting σti2=ati2ηti, (i = 0,. .., q) into Eq. (3), the GARCH model can be rewritten as

at=α0+i=1maxpqαi+βiati2+ηtj=1qβjηtj,E6

It can be seen that {ηt} is a martingale difference series (i.e., E(ηt) = 0 and.

covηtηtj=0, for j ≥ 1). However, {ηt} in general is not an iid sequence.

A GARCH model can be regarded as an application of the ARMA idea to the squared series at2. Using the unconditional mean of an ARMA model, results in this

Eat2=α01i=1maxpqαi+βi

provided that the denominator of the prior fraction is positive [14].

When p = 1 and q = 1, we have GARCH(1, 1) model given by:

at=σtεt,σt2=ω+α1at12+β1σt12,E7

3.4 GJR-GARCH(p, q) model

The Glosten-Jagannathan-Runkle GARCH (GJRGARCH) model, which is a model that attempts to address volatility clustering in an innovation process, is obtained by letting δ=2.

When δ=2and 0γi<1,

σt2=ω+i=1pαiεtiγiεti2+j=1qβjσtj2=ω+i=1pαiεti2+γi2εt122γiεtiεti+j=1qβjσtj2E8
σt2=ω+i=1pαi21+γi2εti2+j=1qβjσtj2,εti<0ω+i=1pαi1γi2εti2+j=1qβjσtj2,εti>0

i.e.;

σt2=ω+i=1pαi1γi2εti2+i=1pαi1+γi21γi2Siεti2+j=1qβjσtj2σt2=ω+i=1pαi1γi2εt12+j=1qβjσt12+i=1p4αiγiSiεti2

where Si=1ifεti<00ifεti0,

Now define

αi=αi1γi2andγi=4αiγi,

then

σt2=ω+i=1pαi1γi2εti2+j=1qβjσti2+i=1pγiSiεt12E9

Which is the GJRGARCH model [15].

But when 1γi<0,

Then recall Eq. (8)

σt2=ω+i=1pαiεtiγiεti2+j=1qβjσtj2=ω+i=1pαiεti2+γi2εt122γiεtiεti+j=1qβjσtj2
σt2=ω+i=1pαi21γi2εti2+j=1qβjσtj2,εti>0ω+i=1pαi1+γi2εti2+j=1qβjσtj2,εti<0
σt2=ω+i=1pαi1+γi2εti2+j=1qβjσtj2+i=1pαi1γi21+γi2Si+εti2=ω+i=1pαi1+γi2εti2+j=1qβjσtj2+i=1pαi1+γi22γi1γi22γiSi+εti2

Where

Si+=1ifεti>00ifεti0

also define

αi=αi1+γi2andγi=4αiγi,

then

σt2=ω+i=1pαiεti2+j=1qβjσti2+i=1pγiSi+εt12E10

which allows positive shocks to have a stronger effect on volatility than negative shocks [15]. But when p=q=1, the GJRGARCH(1,1) model will be written as

σt2=ω+αεt2+γSiεt12+βσt12.E11

3.5 IGARCH(1, 1) model

The integrated GARCH (IGARCH) models are unit- root GARCH models. The IGARCH (1, 1) model is specified in Grek [18] as

at=σtεt;σt2=α0+β1σt12+1β1at12E12

Where εt∼ N(0, 1) iid, and 0<β1<1, Ali (2013) used αito denote 1βi.

The model is also an exponential smoothing model for the {at2} series. To see this, rewrite the model as.

σt2=1β1at12+β1σt12.=1β1at12+β11βat22+β1σt22=1β1at12+1β1β1at22+β12σt22.E13

By repeated substitutions, we have

σt2=1β1at12+β1at22+β12at33+,E14

which is the well-known exponential smoothing formation with β1being the discounting factor [14].

3.6 TGARCH(p, q) model

The Threshold GARCH model is another model used to handle leverage effects, and a TGARCH(p, q) model is given by the following:

σt2=α0+i=1pαi+γiNt=iati2+j=1qβjσtj2,E15

where Ntiis an indicator for negative ati, that is,

Nti=1ifati<0,0ifati0,.

and αi, γi, and βjare nonnegative parameters satisfying conditions similar to those of GARCH models, [14]. When p=1,q=1, the TGARCH(1, 1) model becomes:

σt2=ω+α+γNt1at12+βσt12E16

3.7 NGARCH(p, q) model

The Nonlinear Generalized Autoregressive Conditional Heteroskedasticity (NGARCH) Model has been presented variously in literature by the following scholars [19, 20, 21]. The following model can be shown to represent all the presentations:

ht=ω+i=1qαiεti2+i=1qγiεti+j=1pβjhtjE17

Where htis the conditional variance, and ω, βand αsatisfy ω>0, β0and α0.

Which can also be written as

σt=ω+i=1qαiεti2+i=1qγiεti+j=1pβjσtjE18

3.8 The exponential generalized autoregressive conditional heteroskedasticity (EGARCH) model

The EGARCH model was proposed by Nelson [22] to overcome some weaknesses of the GARCH model in handling financial time series pointed out by [23], In particular, to allow for asymmetric effects between positive and negative asset returns, he considered the weighted innovation

gεt=θεt+γεtEεt,E19

where θand γare real constants. Both εtand εtEεtare zero-mean iid sequences with continuous distributions. Therefore, Egεt=0. The asymmetry of gεtcan easily be seen by rewriting it as

gεt=θ+γεtγEεtifεt0,θγεtγEεtifεt<0.E20

An EGARCH(m, s) model, according to Dhamija and Bhalla [24] can be written as

at=σtεt,lnσt2=ω+i=1sαiati+θiatiσti+j=1mβjlnσti2,E21

Which specifically results in EGARCH (1, 1) being written as

at=σtεt
lnσt2=ω+αat1E(at1)+θat1+βlnσt12E22

where at1Eat1are iidand have mean zero. When the EGARCH model has a Gaussian distribution of error term, then Eεt=2/π, which gives:

lnσt2=ω+αat12/π+θat1+βlnσt12E23

3.9 The absolute value GARCH (AVGARCH)

An asymmetric GARCH (AGARCH), according to Ali [25] is simply

at=σtεt;σ2=ω+i=1pαiεtib2+j=1qβjσtj2,E24

While the absolute value generalized autoregressive conditional heteroskedasticity (AVGARCH) model is specified as:

at=σtεt;σ2=ω+i=1pαiεti+bcεti+b2+j=1qβjσtj2E25

3.10 Nonlinear (asymmetric) GARCH, or N(a)GARCH or NAGARCH

NAGARCH plays key role in option pricing with stochastic volatility because, as we shall see later on, NAGARCH allows you to derive closed-form expressions for European option prices in spite of the rich volatility dynamics. Because a NAGARCH may be written as

σt+12=ω+ασt2ztδ2+βσt2E26

And if ztIIDN01, ztis independent of σt2as σt2is only a function of an infinite number of past squared returns, it is possible to easily derive the long run, unconditional variance under NGARCH and the assumption of stationarity:

Eσt+12=σ¯2=ω+αEσt2ztδ2+βEσt2=ω+αEσt2Ezt2+δ22δzt+βEσt2=ω+ασ¯21+δ2+βσ¯2E27

Where σ¯2=Eσt2and Eσt2=Eσt+12because of stationary. Therefore

σ¯21α1+δ2+β=ωσ¯2=ω1α1+δ2+βE28

Which exists and positive if and only if α1+δ2+β<1. This has two implications:

  1. The persistence index of a NAGARCH(1,1) is α1+δ2+βand not simply α+β;

  2. a NAGARCH(1,1) model is stationary if and only if α1+δ2+β<1.

See details in [22].

3.11 Persistence

The low or high persistency in volatility exhibited by financial time series can be determined by the GARCH coefficients of a stationary GARCH model. The persistence of a GARCH model can be calculated as the sum of GARCH (β1) and ARCH (α1) coefficients that is α+β1. In most financial time series, it is very close to one (1) [26, 27]. Persistence could take the following conditions:

If α+β1<1: The model ensures positive conditional variance as well as stationary.

If α+β1=1: we have an exponential decay model, then the half-life becomes infinite. Meaning the model is strictly stationary.

If α+β1>1: The GARCH model is said to be non-stationary, meaning that the volatility ultimately detonates toward the infinitude [27]. In addition, the model shows that the conditional variance is unstable, unpredicted and the process is non-stationary [28].

3.12 Half-life volatility

Half-life volatility measures the mean reverting speed (average time) of a stock price or returns. The mathematical expression of half-life volatility is given as

HalfLife=ln0.5lnα1+β2

It can be noted that the value of α+β1influences the mean reverting speed [27], which means that if the value of α+β1is closer to one (1), then the volatility shocks of the half-life will be longer.

3.13 Backtesting

Financial risk model evaluation or backtesting is an important part of the internal model’s approach to market risk management as put out by Basle Committee on Banking Supervision [29]. Backtesting is a statistical procedure where actual profits and losses are systematically compared to corresponding VaR estimates [30]. This book chapter adopted Backstesting techniques of [29]; The test was implemented in R using rugarch package and this test considered both the unconditional (Kupiec) and conditional (Christoffersen) coverage tests for the correct number of exceedances (see details in [31, 32].

The unconditional (Kupiec) test also refer to as POF-test (Proportion of failure) with its null hypothesis given as

H0:p=p̂=yT

Here y is the number of exceptions and T is the number of observations and k is the confidence level. The test is given as

LRPOF=2ln1pTypy[1ypTyyTy.

Under the null hypothesis that the model is correct and LRPOFis asymptotically chi-squared (χ2) distributed with degree of freedom as one (1). If the value of the LRPOFstatistic is greater than the critical value (or p-value<0.01 for 1% level of significant or p-value<0.05 for 5% level of significant) the null hypothesis is rejected and the model then is inaccurate.

The Christoffersen’s Interval Forecast Test combined the independence statistic with the Kupiec’s POF test to obtained the joint test [30, 31]. This test examined the properties of a good VaR model, the correct failure rate and independence of exceptions, that is condition coverage (cc). the conditional coverage (cc) is given as

LRcc=LRPOF+LRind

Where

LRind=i=2n2lnp1pui11ui11uiui12lnp1pu11u11uu1

Where uiis the time between exceptions I and i-1 while u is the sum of ui.

If the value of the LRccstatistic is greater than the critical value (or p-value<0.01 for 1% level of significant or p-value<0.05 for 5% level of significant) the null hypothesis is rejected and that leads to the rejection of the model.

3.14 Distributions of GARCH models

In this study we employed two innovations namely student t and skewed student t distributions they can account for excess kurtosis and non-normality in financial returns [28, 33].

The student t-distribution is given as

fy=Γv+12Γv21+y2vv+12;<y<

The Skewed student t-distribution is given as

fyμσvλ=bc1+1v2byμσ+a1λ2v+12,ify<abbc1+1v2byμσ+a1+λ2v+12,ifyab

Where vis the shape parameter with 2<v<and λis the skewness parameter with 1<λ<1. The constants a, b and c are given as

a=4λcv2v1;b=1+3λ2a2;c=Γv+12πv2Γv2

Where μandσare the mean and the standard deviation of the skewed student t distribution respectively.

4. Method of data collection

The data used in this study is a secondary data that was collected fromwww.cashcraft.comunder stock trend and analysis. Daily stock price was collected on Zenith bank stock price from October 21st 2004 to May 8th 2017.

The returns was calculated using the formula below

Rt=lnPtlnPt1E29

Where Rt is stock returns; Pt is the present stock price; Pt-1 is the previous stock price and ln is the natural logarithm transformation. Then total observation becomes 3070.

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 1.

The time plot of the log of zenith Bank returns.

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.

Figure 2.

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

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].

StatisticValue
Mean0.000114
Median0.000000
Maximum0.338000
Minimum−0.405850
Std. Dev.0.027600
Skewness−1.267452
Kurtosis33.76662
Jarque-Bera121905.9 (p = 0.00000)
Number of Observation3070
Unit root testing
ADF−47.11172 (p = 0.0000)
DF-GLS−1.842682
PP−46.52078 (p = 0.0000)
ARCH test
Chi-squared = 123.05, df = 12, p-value <2.2e-16

Table 1.

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

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 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-of-test 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 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.

ModelInformation criteriaStd t innovationSkewed stdt innovation
sGARCH (1,1)Akaike
Bayes
Shibata
Hannan-Quinn
NANA
sGARCH (2,1)Akaike
Bayes
Shibata
Hannan-Quinn
NANA
sGARCH(2,2)Akaike
Bayes
Shibata
Hannan-Quinn
NA−5.3667
−5.3530
−5.3667
−5.3618
gjrGARCH(1,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.5110
−5.5012
−5.5110
−5.5075
NA
gjrGARCH(2,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.5493
−5.5356
−5.5493
−5.5444
NA
gjrGARCH(2,2)Akaike
Bayes
Shibata
Hannan-Quinn
NANA
eGARCH (1,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.0584
−5.0485
−5.0584
−5.0548
−5.0587
−5.0469
−5.0587
−5.0545
eGARCH (2,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.0853
−5.0716
−5.0853
−5.0804
−5.0859
−5.0702
−5.0859
−5.0802
eGARCH (2,2)Akaike
Bayes
Shibata
Hannan-Quinn
−5.0196
−5.0039
−5.0196
−5.0140
NA
iGARCH (1,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.1474
−5.1415
−5.1474
−5.1453
−5.1498
−5.1420
−5.1498
−5.1470
iGARCH (2,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.1527
−5.1449
−5.1527
−5.1499
−5.1526
−5.1428
−5.1527
−5.1491
iGARCH (2,2)Akaike
Bayes
Shibata
Hannan-Quinn
−5.1496
−5.1397
−5.1496
−5.1460
−5.1547
−5.1429
−5.1547
−5.1505
TGARCH(1,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.8914
−5.8815
−5.8914
−5.8878
−5.8920
−5.8803
−5.8921
−5.8878
TGARCH(2,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.9253
−5.9115
−5.9253
−5.9203
−5.8819
−5.8662
−5.8819
−5.8763
TGARCH(2,2)Akaike
Bayes
Shibata
Hannan-Quinn
−5.8908
−5.8751
−5.8908
−5.8851
−5.8752
−5.8575
−5.8752
−5.8688
NGARCH(1,1)Akaike
Bayes
Shibata
Hannan-Quinn
−15.563
−15.553
−15.563
−15.559
−13.191
−13.179
−13.191
−13.187
NGARCH(2,1)Akaike
Bayes
Shibata
Hannan-Quinn
−14.470
−14.458
−14.470
−14.466
−16.419
−16.405
−16.419
−16.414
NGARCH(2,2)Akaike
Bayes
Shibata
Hannan-Quinn
−9.5866
−9.5729
−9.5866
−9.5817
−11.248
−11.232
−11.248
−11.242
apARCH(1,1)Akaike
Bayes
Shibata
Hannan-Quinn
−7.8258
−7.8140
−7.8258
−7.8216
NA
apARCH(2,1)Akaike
Bayes
Shibata
Hannan-Quinn
−8.1226
−8.1069
−8.1226
−8.1170
−8.7718
−8.7541
−8.7718
−8.7654
apARCH(2,2)Akaike
Bayes
Shibata
Hannan-Quinn
−16.904
−16.886
−16.904
−16.897
9.4341
9.4538
9.4341
9.4412
NAGARCH(1,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.1428
−5.1330
−5.1428
−5.1393
−5.1402
−5.1285
−5.1403
−5.1360
NAGARCH(2,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.1296
−5.1158
−5.1296
−5.1246
−5.1343
−5.1186
−5.1343
−5.1286
NAGARCH(2,2)Akaike
Bayes
Shibata
Hannan-Quinn
−5.1221
−5.1063
−5.1221
−5.1164
−5.0439
−5.0262
−5.0439
−5.0375
AVGARCH(1,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.8467
−5.8349
−5.8467
−5.8425
−5.6004
−5.5866
−5.6004
−5.5954
AVGARCH(2,1)Akaike
Bayes
Shibata
Hannan-Quinn
−5.6197
−5.6020
−5.6197
−5.6134
−5.9524
−5.9327
−5.9524
−5.9453
AVGARCH(2,2)Akaike
Bayes
Shibata
Hannan-Quinn
−5.4227
−5.4031
−5.4228
−5.4157
−5.8644
−5.8428
−5.8644
−5.8567

Table 2.

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

Note: NA-Not Available.

ModelStudent t distributionSkewed Student t distribution
eGARCH (1,1)Robust Standard Errors:Robust Standard Errors:
EstimateStd. Errort valuePr(>|t|)EstimateStd. Errort valuePr(>|t|)
omega−0.357260.021440–16.66320.00000omega−0.353940.020239–17.48790.00000
alpha1-0.103240.065840-1.56810.11687alpha1-0.105970.065963-1.60650.10817
beta10.949580.001992476.77290.00000beta10.950110.001801527.64480.00000
gamma10.534170.0803676.64670.00000gamma10.530180.0792616.68910.00000
shape2.644390.3470587.61940.00000skew1.024390.01225383.60130.00000
shape2.646810.3459657.65050.00000
Weighted Ljung-Box Test on Standardized ResidualsWeighted Ljung-Box Test on Standardized Residuals
------------------------------------------------------------------------
statisticp-valuestatisticp-value
Lag [1]19.231.160e-05Lag [1]18.991.314e-05
Lag[2*(p+q)+(p+q)-1] [2]19.286.872e-06Lag[2*(p+q)+(p+q)-1] [2]19.047.944e-06
Lag[4*(p+q)+(p+q)-1] [5]19.951.999e-05Lag[4*(p+q)+(p+q)-1] [5]19.712.335e-05
d.o.f=0d.o.f=0
H0:No serial correlationH0: No serial correlation
Weighted ARCH LM TestsWeighted ARCH LM Tests
------------------------------------------------------------------------
StatisticShapeScaleP-ValueStatisticShapeScaleP-Value
ARCH Lag [3]0.021730.5002.0000.8828ARCH Lag [3]0.021000.5002.0000.8848
ARCH Lag [5]0.045701.4401.6670.9957ARCH Lag [5]0.044171.4401.6670.9959
ARCH Lag [7]0.064982.3151.5430.9998ARCH Lag [7]0.062762.3151.5430.9998

Table 3.

Parameter estimates and ARCH LM tests of the GARCH models.

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.

ModelsStdSkewed std
PersistenceHalf-life volatilityPersistenceHalf-life volatility
sGARCH (1,1)NANANANA
sGARCH (2,1)NANANANA
sGARCH(2,2)NANA0.978328131.63581
gjrGARCH(1,1)0.9945289126.3447NANA
gjrGARCH(2,1)0.9939226113.7067NANA
gjrGARCH(2,2)NANANANA
eGARCH (1,1)0.949582113.398480.950106513.543
eGARCH (2,1)0.979922634.175970.980255534.75809
eGARCH (2,2)0.977586230.57714NANA
iGARCH (1,1)1infinity1infinity
iGARCH (2,1)1infinity1infinity
iGARCH (2,2)1infinity1Infinity
TGARCH(1,1)0.946379412.577130.958713516.43969
TGARCH(2,1)0.952907914.369610.950670413.70184
TGARCH(2,2)0.93154799.7753450.947031712.73636
NGARCH(1,1)0.992584793.12870.973253125.56687
NGARCH(2,1)0.98420743.542080.988870561.93282
NGARCH(2,2)0.970447923.106790.9971636244.0279
apARCH(1,1)0.975913928.42987NANA
apARCH(2,1)0.982939140.280210.985331746.90736
apARCH(2,2)0.986903852.580050.951376613.90596
NAGARCH(1,1)0.9933088103.24440.9950269139.0335
NAGARCH(2,1)0.991037876.994420.9942849120.9365
NAGARCH(2,2)0.9974602272.56590.9978423320.8989
AVGARCH(1,1)0.957947616.133870.93150189.768526
AVGARCH(2,1)0.93113219.7141810.951375513.90564
AVGARCH(2,2)0.963555218.67040.963369718.57406

Table 4.

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

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 t-distribution 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.

ModelDistributionsAlphaExpected ExceedActual VaR ExceedUnconditional Coverage (Kupiec)
H0: Correct Exceedances
Conditional Coverage (Christoffersen)
H0: Correct Exceedances and independence of Failure
eGARCH(1,1)Student t1%10.710LR.uc Statistic: 0.047
LR.uc Critical: 6.635
LR.uc p-value: 0.828
Reject Null: NO
LR.cc Statistic: 0.236
LR.cc Critical: 9.21
LR.cc p-value: 0.889
Reject Null: NO
5%53.567LR.uc Statistic: 3.332
LR.uc Critical: 3.841
LR.uc p-value: 0.068
Reject Null: NO
LR.cc Statistic: 3.497
LR.cc Critical: 5.991
LR.cc p-value: 0.174
Reject Null: NO
Skewed student t1%10.710LR.uc Statistic: 0.047
LR.uc Critical: 6.635
LR.uc p-value: 0.828
Reject Null: NO
LR.cc Statistic: 0.236
LR.cc Critical: 9.21
LR.cc p-value: 0.889
Reject Null: NO
5%53.574LR.uc Statistic: 7.425
LR.uc Critical: 3.841
LR.uc p-value: 0.006
Reject Null: YES
LR.cc Statistic: 7.428
LR.cc Critical: 5.991
LR.cc p-value: 0.024
Reject Null: YES
NGARCH(1,1)Student t1%10.776LR.uc Statistic: 171.505
LR.uc Critical: 6.635
LR.uc p-value: 0
Reject Null: YES
LR.cc Statistic: 175.258
LR.cc Critical: 9.21
LR.cc p-value: 0
Reject Null: YES
5%53.5135LR.uc Statistic: 93.627
LR.uc Critical: 3.841
LR.uc p-value: 0
Reject Null: YES
LR.cc Statistic: 101.753
LR.cc Critical: 5.991
LR.cc p-value: 0
Reject Null: YES
NGARCH(2,1)Skewed student t1%10.774LR.uc Statistic: 163.466
LR.uc Critical: 6.635
LR.uc p-value: 0
Reject Null: YES
LR.cc Statistic: 171.614
LR.cc Critical: 9.21
LR.cc p-value: 0
Reject Null: YES
5%53.5141LR.uc Statistic: 106.038
LR.uc Critical: 3.841
LR.uc p-value: 0
Reject Null: YES
LR.cc Statistic: 111.739
LR.cc Critical: 5.991
LR.cc p-value: 0
Reject Null: YES
apARCH(2,2)Student t1%NANANANA
5%NANANANA
TGARCH(2,1)Student t1%10.731LR.uc Statistic: 25.744
LR.uc Critical: 6.635
LR.uc p-value: 0
Reject Null: YES
LR.cc Statistic: 25.755
LR.cc Critical: 9.21
LR.cc p-value: 0
Reject Null: YES
5%53.592LR.uc Statistic: 24.225
LR.uc Critical: 3.841
LR.uc p-value: 0
Reject Null: YES
LR.cc Statistic: 24.823
LR.cc Critical: 5.991
LR.cc p-value: 0
Reject Null: YES

Table 5.

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

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.

6. Conclusions

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 fromwww.cashcraft.comunder 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 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.

Acknowledgments

I wish to acknowledge my PhD student and M.Sc. students that have worked under my supervision in the area of financial time series analysis.

Conflict of interest

The Author declares no conflict of interest.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Monday Osagie Adenomon (October 27th 2020). Financial Time Series Analysis via Backtesting Approach, Linked Open Data - Applications, Trends and Future Developments, Kingsley Okoye, IntechOpen, DOI: 10.5772/intechopen.94112. Available from:

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