**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, *yt* is given by

$$\mathbf{y}\_t = E\_{t-1}(\mathbf{y}\_t) + \mathbf{e}\_t \tag{1}$$

Given that:

where

is necessary: Let *<sup>η</sup><sup>t</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>2</sup>

cov *ηt*, *η<sup>t</sup>*�*<sup>j</sup>*

squared series *a*<sup>2</sup>

**103**

*r* ¼ *μ* þ *at*, *ε<sup>t</sup>* ¼ *σtεt*, *ε<sup>t</sup>* � *N*ð Þ 0, 1

*Financial Time Series Analysis via Backtesting Approach*

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

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

*p*

*i*¼1

*α<sup>i</sup> at*�*<sup>i</sup>* j j � *γ* ð Þ *iat*�*<sup>i</sup>*

*α<sup>i</sup>* ≥0 *i* ¼ 1, 2, … , *p* �1<*γ<sup>i</sup>* <1 *i* ¼ 1, 2, … , *p β <sup>j</sup>* >0 *j* ¼ 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

The mathematical model for the sGARCH(p,q) model is obtained from Eq. (4)

*p*

*αia*<sup>2</sup>

*<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q*

*j*¼1 *β j σ*2

*<sup>i</sup>*¼<sup>1</sup> *<sup>α</sup><sup>i</sup>* <sup>þ</sup> *<sup>β</sup><sup>i</sup>* ð Þ <sup>&</sup>lt; 1, [17].

*<sup>t</sup>*�*<sup>i</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>2</sup>

*q*

*j*¼1

*<sup>t</sup>*�*<sup>j</sup>* (5)

� � suggests that the

*<sup>t</sup>*�*<sup>i</sup>* � *<sup>η</sup><sup>t</sup>*�*<sup>i</sup>*

*β <sup>j</sup>η<sup>t</sup>*�*<sup>j</sup>*, (6)

, (*i* = 0*,*. *.., q*)

*i*¼1

Where *at* ¼ *rt* � *μ<sup>t</sup>* (*rt* is the continuously compounded log return series), and. *εt*� *N* (0,1) *iid*, the parameter *α<sup>i</sup>* is the ARCH parameter and *β <sup>j</sup>* is the GARCH

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 (*β <sup>j</sup>*) becomes

To expatiate on the properties of GARCH models, the following representation

*<sup>α</sup><sup>i</sup>* <sup>þ</sup> *<sup>β</sup><sup>i</sup>* ð Þ*a*<sup>2</sup>

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

� � <sup>¼</sup> 0, for *<sup>j</sup>* <sup>≥</sup> 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

*<sup>t</sup>* � *<sup>η</sup>t*. By substituting *<sup>σ</sup>*<sup>2</sup>

*<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> *<sup>η</sup><sup>t</sup>* �<sup>X</sup>

*<sup>t</sup>* . Using the unconditional mean of an ARMA model, results in this

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

asymmetric response of volatility to positive and negative "shocks".

*ω*>0, *δ*≥ 0

*<sup>δ</sup>* þ<sup>X</sup> *q*

*j*¼1 *β j σδ t*�*j* , (4)

*σδ*

**3.3 Standard GARCH(p, q) model**

by letting *δ* ¼ 2 and *γ<sup>i</sup>* ¼ 0, *i* ¼ 1, … , *p* to be:

*at* <sup>¼</sup> *<sup>σ</sup>tεt*, *<sup>σ</sup>*<sup>2</sup>

parameter, and *<sup>ω</sup>* <sup>&</sup>gt; 0, *<sup>α</sup><sup>i</sup>* <sup>≥</sup> 0, *<sup>β</sup> <sup>j</sup>* <sup>≥</sup> 0, and <sup>P</sup>max ð Þ *<sup>p</sup>*,*<sup>q</sup>*

extinct and what is left is an ARCH(p) model.

*<sup>t</sup>* so that *σ*<sup>2</sup>

*at* ¼ *α*<sup>0</sup> þ

into Eq. (3), the GARCH model can be rewritten as

*<sup>t</sup>* � *<sup>σ</sup>*<sup>2</sup>

The restriction on ARCH and GARCH parameters *αi*, *β <sup>j</sup>*

*<sup>t</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>2</sup>

max<sup>X</sup> ð Þ *p*, *q*

*i*¼1

where *ε<sup>t</sup>* ¼ *φtσt*.

The mean equation in Eq. (1) also applies to other GARCH family models. *Et*�<sup>1</sup>ð Þ*:* is the expected value conditional on information available at time *t-1*, while *ε<sup>t</sup>* is the error generated from the mean equation at time t and *φ<sup>t</sup>* is 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

$$
\sigma\_t^2 = \alpha + a\_1 a\_{t-1}^2 + \dots + a\_p a\_{t-p}^2 \tag{2}
$$

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:

$$\mathfrak{a}\_{t} = \sigma\_{t}\varepsilon\_{t}, \quad \sigma\_{t}^{2} = \alpha + a\_{1}a\_{t-1}^{2} + \dots + a\_{p}a\_{t-p}^{2} \tag{3}$$

Where *ε<sup>t</sup>* � *N* (0,1) *iid*, *ω* > 0 and *α<sup>i</sup>* ≥ 0 for *i* > 0. In practice, *ε<sup>t</sup>* is 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:

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

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

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

**3.1 Autoregressive conditional heteroskedasticity (ARCH) family model**

conditional heteroskedasticity in a return series, *yt* is given by

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

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *<sup>α</sup>*1*a*<sup>2</sup>

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

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *<sup>α</sup>*1*a*<sup>2</sup>

Where *ε<sup>t</sup>* � *N* (0,1) *iid*, *ω* > 0 and *α<sup>i</sup>* ≥ 0 for *i* > 0. In practice, *ε<sup>t</sup>* is assumed to follow the standard normal or a standardized student-*t* distribution or a generalized

According to Rossi [15], the asymmetric power ARCH model proposed by [16]

given below forms the basis for deriving the GARCH family models.

*σ*2

*at* <sup>¼</sup> *<sup>σ</sup>tεt*, *<sup>σ</sup>*<sup>2</sup>

Every ARCH or GARCH family model requires two distinct specifications, namely: the mean and the variance Equations [13]. The mean equation for a

*yt* ¼ *Et*�<sup>1</sup> *yt*

The mean equation in Eq. (1) also applies to other GARCH family models. *Et*�<sup>1</sup>ð Þ*:* is the expected value conditional on information available at time *t-1*, while *ε<sup>t</sup>* is the error generated from the mean equation at time t and *φ<sup>t</sup>* is the sequence of independent and identically distributed random variables with zero mean and unit

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>α</sup>pa*<sup>2</sup>

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> … <sup>þ</sup> *<sup>α</sup>pa*<sup>2</sup>

<sup>þ</sup> *<sup>ε</sup><sup>t</sup>* (1)

*<sup>t</sup>*�*<sup>p</sup>* (2)

*<sup>t</sup>*�*<sup>p</sup>* (3)

fiscal discipline.

**3. Model specification**

are presented in this section.

where *ε<sup>t</sup>* ¼ *φtσt*.

variance.

becomes:

**102**

error distribution [14].

**3.2 Asymmetric power ARCH**

$$\begin{aligned} r &= \mu + a\_t, \\ \varepsilon\_t &= \sigma\_t \varepsilon\_t, \\ \varepsilon\_t &\sim N(0, 1) \\ \sigma\_t^\delta &= \omega + \sum\_{i=1}^p a\_i \left( |a\_{t-i}| - \gamma\_i a\_{t-i} \right)^\delta + \sum\_{j=1}^q \beta\_j \sigma\_{t-j}^\delta, \end{aligned} \tag{4}$$

where

$$\begin{aligned} &a>0, & \delta \ge 0\\ &a\_i \ge 0 & &i = 1,2,\dots,p\\ &-1 < \gamma\_i < 1 & i = 1,2,\dots,p\\ &\beta\_j > 0 & &j = 1,2,\dots,q \end{aligned}$$

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* ¼ 2 and *γ<sup>i</sup>* ¼ 0, *i* ¼ 1, … , *p* to be:

$$\mathfrak{a}\_{t} = \sigma\_{t}\mathfrak{e}\_{t}, \quad \sigma\_{t}^{2} = \omega + \sum\_{i=1}^{p} a\_{i}\mathfrak{a}\_{t-i}^{2} + \sum\_{j=1}^{q} \beta\_{j}\sigma\_{t-j}^{2} \tag{5}$$

Where *at* ¼ *rt* � *μ<sup>t</sup>* (*rt* is the continuously compounded log return series), and. *εt*� *N* (0,1) *iid*, the parameter *α<sup>i</sup>* is the ARCH parameter and *β <sup>j</sup>* is the GARCH

parameter, and *<sup>ω</sup>* <sup>&</sup>gt; 0, *<sup>α</sup><sup>i</sup>* <sup>≥</sup> 0, *<sup>β</sup> <sup>j</sup>* <sup>≥</sup> 0, and <sup>P</sup>max ð Þ *<sup>p</sup>*,*<sup>q</sup> <sup>i</sup>*¼<sup>1</sup> *<sup>α</sup><sup>i</sup>* <sup>þ</sup> *<sup>β</sup><sup>i</sup>* ð Þ <sup>&</sup>lt; 1, [17].

The restriction on ARCH and GARCH parameters *αi*, *β <sup>j</sup>* � � suggests 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 (*β <sup>j</sup>*) 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 *<sup>η</sup><sup>t</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> *<sup>t</sup>* � *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>* so that *σ*<sup>2</sup> *<sup>t</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> *<sup>t</sup>* � *<sup>η</sup>t*. By substituting *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>*�*<sup>i</sup>* <sup>¼</sup> *<sup>a</sup>*<sup>2</sup> *<sup>t</sup>*�*<sup>i</sup>* � *<sup>η</sup><sup>t</sup>*�*<sup>i</sup>* , (*i* = 0*,*. *.., q*) into Eq. (3), the GARCH model can be rewritten as

$$a\_t = a\_0 + \sum\_{i=1}^{\max(p,q)} (a\_i + \beta\_i) a\_{t-i}^2 + \eta\_t - \sum\_{j=1}^q \beta\_j \eta\_{t-j},\tag{6}$$

It can be seen that {*ηt*} is a martingale difference series (i.e., *E(ηt*) = 0 and. cov *ηt*, *η<sup>t</sup>*�*<sup>j</sup>* � � <sup>¼</sup> 0, for *<sup>j</sup>* <sup>≥</sup> 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 *a*<sup>2</sup> *<sup>t</sup>* . Using the unconditional mean of an ARMA model, results in this *Linked Open Data - Applications,Trends and Future Developments*

$$\operatorname{E}\left(a\_i^2\right) = \frac{a\_0}{1 - \sum\_{i=1}^{\max\left(p, q\right)} (a\_i + \beta\_i)}$$

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:

$$\begin{aligned} a\_t &= \sigma\_t \varepsilon\_t, \\ \sigma\_t^2 &= \omega + a\_1 a\_{t-1}^2 + \beta\_1 \sigma\_{t-1}^2, \end{aligned} \tag{7}$$

*σ*2

*σ*2 *t* ¼

*p*

*i*¼1

*i*¼1

*σ*2

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

<sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup> *p*

also define

then

Where

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

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

<sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup> *p*

> 8 >>><

> >>>:

*<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup>

*<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup>

*σ*2

**3.5 IGARCH(1, 1) model**

rewrite the model as.

**105**

*p*

*Financial Time Series Analysis via Backtesting Approach*

*i*¼1

*i*¼1

*<sup>ω</sup>* <sup>þ</sup> <sup>P</sup> *p*

*<sup>ω</sup>* <sup>þ</sup> <sup>P</sup> *p*

*i*¼1 *α*2

*i*¼1

*ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q*

*ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q*

*α*∗

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

*p*

*i*¼1 *α*∗ *<sup>i</sup> ε*<sup>2</sup>

*σ*2

IGARCH (1, 1) model is specified in Grek [18] as

*σ*2

*at* <sup>¼</sup> *<sup>σ</sup>tεt*; *<sup>σ</sup>*<sup>2</sup>

*<sup>t</sup>* <sup>¼</sup> <sup>1</sup> � *<sup>β</sup>*<sup>1</sup> ð Þ*a*<sup>2</sup>

<sup>¼</sup> <sup>1</sup> � *<sup>β</sup>*<sup>1</sup> ð Þ*a*<sup>2</sup>

<sup>¼</sup> <sup>1</sup> � *<sup>β</sup>*<sup>1</sup> ð Þ*a*<sup>2</sup>

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *αε*<sup>2</sup>

The model is also an exponential smoothing model for the {*a*<sup>2</sup>

*α<sup>i</sup> εt*�*<sup>i</sup>* j j � *γ<sup>i</sup>* ð Þ *εt*�*<sup>i</sup>*

*<sup>α</sup><sup>i</sup> <sup>ε</sup>t*�*<sup>i</sup>* j j<sup>2</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup>

*<sup>i</sup>* <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup>

*<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup>

*j*¼1

*j*¼1

*S*þ

*β <sup>j</sup>σ*<sup>2</sup>

*β <sup>j</sup>σ*<sup>2</sup>

*i ε*2

*ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> <sup>P</sup> *q*

*ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> <sup>P</sup> *q*

*<sup>t</sup>*�*<sup>j</sup>* <sup>þ</sup><sup>X</sup> *p*

*<sup>t</sup>*�*<sup>j</sup>* <sup>þ</sup><sup>X</sup> *p*

*<sup>i</sup>* <sup>¼</sup> 1 if *<sup>ε</sup><sup>t</sup>*�*<sup>i</sup>* <sup>&</sup>gt; <sup>0</sup> 0 if *ε<sup>t</sup>*�*<sup>i</sup>* ≤0

�

*<sup>i</sup>* <sup>¼</sup> *<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> and *<sup>γ</sup>* <sup>∗</sup>

*<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q*

shocks [15]. But when *p* ¼ *q* ¼ 1, the GJRGARCH(1,1) model will be written as

*j*¼1 *β j σ*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *p*

which allows positive shocks to have a stronger effect on volatility than negative

*<sup>t</sup>* <sup>þ</sup> *<sup>γ</sup>Siε*<sup>2</sup>

The integrated GARCH (IGARCH) models are unit- root GARCH models. The

*<sup>t</sup>* <sup>¼</sup> *<sup>α</sup>*<sup>0</sup> <sup>þ</sup> *<sup>β</sup>*1*σ*<sup>2</sup>

Where *ε<sup>t</sup>* � N(0, 1) *iid*, and 0< *β*<sup>1</sup> < 1, Ali (2013) used *α<sup>i</sup>* to denote 1 � *βi*.

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*1*σ*<sup>2</sup>

*t*�1*:*

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*<sup>1</sup> ð Þ <sup>1</sup> � *<sup>β</sup> <sup>a</sup>*<sup>2</sup>

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>β</sup>*<sup>1</sup> ð Þ*β*1*a*<sup>2</sup>

*i*¼1

*i*¼1

*<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup>

*<sup>i</sup>* ¼ �4*αiγi*,

*i*¼1 *γ* ∗ *<sup>i</sup> S*<sup>þ</sup> *i ε*2

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *βσ*<sup>2</sup>

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>1</sup> � *<sup>β</sup>*<sup>1</sup> ð Þ*a*<sup>2</sup>

*<sup>t</sup>*�<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*1*σ*<sup>2</sup>

*<sup>t</sup>*�<sup>2</sup> <sup>þ</sup> *<sup>β</sup>*<sup>2</sup> 1*σ*2 *t*�2*:*

� �

*t*�2

<sup>2</sup> þ<sup>X</sup> *q*

*j*¼1

*<sup>t</sup>*�<sup>1</sup> � <sup>2</sup>*γ<sup>i</sup> <sup>ε</sup>t*�*<sup>i</sup>* j j*εt*�*<sup>i</sup>* � � þ<sup>X</sup>

> *j*¼1 *β j σ*2 *t*�*j*

*j*¼1 *β j σ*2 *t*�*j*

*β <sup>j</sup>σ*<sup>2</sup> *t*�*j*

*q*

*j*¼1 *β j σ*2 *t*�*j*

, *εt*�*<sup>i</sup>* > 0

, *εt*�*<sup>i</sup>* < 0

*<sup>i</sup>* � <sup>2</sup>*γ<sup>i</sup>* � <sup>1</sup> � *<sup>γ</sup>*<sup>2</sup>

� �*S*<sup>þ</sup>

*i ε*2 *t*�*i*

*<sup>i</sup>* � 2*γ<sup>i</sup>*

*<sup>t</sup>*�<sup>1</sup> (10)

*<sup>t</sup>*�<sup>1</sup>*:* (11)

*<sup>t</sup>*�<sup>1</sup> (12)

*<sup>t</sup>* } series. To see this,

(13)

*i ε*2 *t*�*i*

*<sup>α</sup><sup>i</sup>* <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> � <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> n o*S*<sup>þ</sup>

### **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* ¼ 2.

When *δ* ¼ 2 and 0 ≤*γ<sup>i</sup>* <1,

$$\begin{split} \sigma\_{t}^{2} &= \boldsymbol{\alpha} + \sum\_{i=1}^{p} a\_{i} (|\boldsymbol{e}\_{t-i}| - \gamma\_{i} \boldsymbol{e}\_{t-i})^{2} + \sum\_{j=1}^{q} \beta\_{j} \sigma\_{t-j}^{2} \\ &= \boldsymbol{\alpha} + \sum\_{i=1}^{p} a\_{i} \left( |\boldsymbol{e}\_{t-i}|^{2} + \gamma\_{i}^{2} \boldsymbol{e}\_{t-1}^{2} - 2\gamma\_{i} |\boldsymbol{e}\_{t-i}| \boldsymbol{e}\_{t-i} \right) + \sum\_{j=1}^{q} \beta\_{j} \sigma\_{t-j}^{2} \\ \sigma\_{t}^{2} &= \begin{cases} \boldsymbol{\alpha} + \sum\_{i=1}^{p} a\_{i}^{2} (\mathbf{1} + \boldsymbol{\gamma}\_{i})^{2} \boldsymbol{e}\_{t-i}^{2} + \sum\_{j=1}^{q} \beta\_{j} \sigma\_{t-j}^{2}, & \boldsymbol{e}\_{t-i} < \mathbf{0} \\\\ \boldsymbol{\alpha} + \sum\_{i=1}^{p} a\_{i} (\mathbf{1} - \boldsymbol{\gamma}\_{i})^{2} \boldsymbol{e}\_{t-i}^{2} + \sum\_{j=1}^{q} \beta\_{j} \sigma\_{t-j}^{2}, & \boldsymbol{e}\_{t-i} > \mathbf{0} \end{cases} \end{split} \tag{8}$$

i.e.;

$$\begin{aligned} \sigma\_t^2 &= \alpha + \sum\_{i=1}^p a\_i (1 - \gamma\_i)^2 e\_{t-i}^2 + \sum\_{i=1}^p a\_i \left\{ (1 + \gamma\_i)^2 - (1 - \gamma\_i)^2 \right\} S\_i^- e\_{t-i}^2 + \sum\_{j=1}^q \beta\_j \sigma\_{t-j}^2, \\\ \sigma\_t^2 &= \alpha + \sum\_{i=1}^p a\_i (1 - \gamma\_i)^2 e\_{t-1}^2 + \sum\_{j=1}^q \beta\_j \sigma\_{t-1}^2 + \sum\_{i=1}^p 4a\_i \gamma\_i S\_i^- \varepsilon\_{t-i}^2 \\\ \text{where } S\_i^- &= \begin{cases} 1 & \text{if } \varepsilon\_{t-i} < 0 \\ 0 & \text{if } \varepsilon\_{t-i} \ge 0 \end{cases} \end{aligned}$$

Now define

$$a\_i^\* = a\_i(1 - \gamma\_i)^2 \text{ and } \gamma\_i^\* = 4a\_i\gamma\_i,$$

then

$$
\sigma\_t^2 = \omega + \sum\_{i=1}^p a\_i (\mathbf{1} - \boldsymbol{\gamma}\_i)^2 \boldsymbol{e}\_{t-i}^2 + \sum\_{j=1}^q \beta\_j \sigma\_{t-i}^2 + \sum\_{i=1}^p \boldsymbol{\gamma}\_i^\* \, \mathbf{S}\_i^- \, \mathbf{e}\_{t-1}^2 \tag{9}
$$

Which is the GJRGARCH model [15]. But when �1≤*γ<sup>i</sup>* <0, Then recall Eq. (8)

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

*σ*2 *<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup> *p i*¼1 *α<sup>i</sup> εt*�*<sup>i</sup>* j j � *γ<sup>i</sup>* ð Þ *εt*�*<sup>i</sup>* <sup>2</sup> þ<sup>X</sup> *q j*¼1 *β <sup>j</sup>σ*<sup>2</sup> *t*�*j* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup> *p i*¼1 *<sup>α</sup><sup>i</sup> <sup>ε</sup>t*�*<sup>i</sup>* j j<sup>2</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup> *i ε*2 *<sup>t</sup>*�<sup>1</sup> � <sup>2</sup>*γ<sup>i</sup> <sup>ε</sup>t*�*<sup>i</sup>* j j*εt*�*<sup>i</sup>* � � þ<sup>X</sup> *q j*¼1 *β j σ*2 *t*�*j σ*2 *t* ¼ *<sup>ω</sup>* <sup>þ</sup> <sup>P</sup> *p i*¼1 *α*2 *<sup>i</sup>* <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> *ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> <sup>P</sup> *q j*¼1 *β j σ*2 *t*�*j* , *εt*�*<sup>i</sup>* > 0 *<sup>ω</sup>* <sup>þ</sup> <sup>P</sup> *p i*¼1 *<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> *ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> <sup>P</sup> *q j*¼1 *β j σ*2 *t*�*j* , *εt*�*<sup>i</sup>* < 0 8 >>>< >>>: *σ*2 *<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup> *p i*¼1 *<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> *ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q j*¼1 *β <sup>j</sup>σ*<sup>2</sup> *<sup>t</sup>*�*<sup>j</sup>* <sup>þ</sup><sup>X</sup> *p i*¼1 *<sup>α</sup><sup>i</sup>* <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> � <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> n o*S*<sup>þ</sup> *i ε*2 *t*�*i* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup> *p i*¼1 *<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> *ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q j*¼1 *β <sup>j</sup>σ*<sup>2</sup> *<sup>t</sup>*�*<sup>j</sup>* <sup>þ</sup><sup>X</sup> *p i*¼1 *<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup> *<sup>i</sup>* � <sup>2</sup>*γ<sup>i</sup>* � <sup>1</sup> � *<sup>γ</sup>*<sup>2</sup> *<sup>i</sup>* � 2*γ<sup>i</sup>* � �*S*<sup>þ</sup> *i ε*2 *t*�*i*

Where

E *a*<sup>2</sup> *t*

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

� � <sup>¼</sup> *<sup>α</sup>*<sup>0</sup>

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:

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *<sup>α</sup>*1*a*<sup>2</sup>

The Glosten-Jagannathan-Runkle GARCH (GJRGARCH) model, which is a model that attempts to address volatility clustering in an innovation process, is

*i ε*2

*ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> <sup>P</sup> *q*

*ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> <sup>P</sup> *q*

� �

*at* ¼ *σtεt*,

*α<sup>i</sup> ε<sup>t</sup>*�*<sup>i</sup>* j j � *γ<sup>i</sup>* ð Þ *ε<sup>t</sup>*�*<sup>i</sup>*

*<sup>α</sup><sup>i</sup> <sup>ε</sup><sup>t</sup>*�*<sup>i</sup>* j j<sup>2</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup>

*<sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup>

*<sup>α</sup><sup>i</sup>* <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup>

*i*¼1

*j*¼1 *β j σ*2 *<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>P</sup> *p*

*<sup>i</sup>* <sup>¼</sup> *<sup>α</sup><sup>i</sup>* <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> and *<sup>γ</sup>* <sup>∗</sup>

*ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *q*

*j*¼1 *β j σ*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup><sup>X</sup> *p*

*σ*2

**3.4 GJR-GARCH(p, q) model**

When *δ* ¼ 2 and 0 ≤*γ<sup>i</sup>* <1,

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

<sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup> *p*

> 8 >>>><

> >>>>:

*p*

*i*¼1

*i*¼1

*<sup>ω</sup>* <sup>þ</sup> <sup>P</sup> *p*

*<sup>ω</sup>* <sup>þ</sup> <sup>P</sup> *p*

*<sup>α</sup><sup>i</sup>* <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup>

*<sup>α</sup><sup>i</sup>* <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup>

*<sup>i</sup>* <sup>¼</sup> 1 if *<sup>ε</sup><sup>t</sup>*�*<sup>i</sup>* <sup>&</sup>lt; <sup>0</sup> 0 if *ε<sup>t</sup>*�*<sup>i</sup>* ≥0

*i*¼1 *α*2

*i*¼1

*ε*2 *<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> <sup>P</sup> *p*

*ε*2 *<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>P</sup> *q*

*α*∗

*p*

*i*¼1

,

*<sup>α</sup><sup>i</sup>* <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup>

obtained by letting *δ* ¼ 2.

*σ*2

*σ*2 *<sup>t</sup>* ¼

i.e.;

*σ*2

*σ*2

where *S*�

then

**104**

Now define

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> <sup>P</sup>

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> <sup>P</sup>

*p*

*i*¼1

*p*

*i*¼1

(

*σ*2

But when �1≤*γ<sup>i</sup>* <0, Then recall Eq. (8)

*<sup>t</sup>* <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

Which is the GJRGARCH model [15].

<sup>1</sup> � <sup>P</sup>max ð Þ *<sup>p</sup>*,*<sup>q</sup>*

*<sup>i</sup>*¼<sup>1</sup> *<sup>α</sup><sup>i</sup>* <sup>þ</sup> *<sup>β</sup><sup>i</sup>* ð Þ

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*1*σ*<sup>2</sup>

<sup>2</sup> þ<sup>X</sup> *q*

*j*¼1

*<sup>t</sup>*�<sup>1</sup> � <sup>2</sup>*γ<sup>i</sup> <sup>ε</sup><sup>t</sup>*�*<sup>i</sup>* j j*ε<sup>t</sup>*�*<sup>i</sup>*

*j*¼1 *β <sup>j</sup>σ*<sup>2</sup> *t*�*j*

*j*¼1 *β j σ*2 *t*�*j*

*<sup>α</sup><sup>i</sup>* <sup>1</sup> <sup>þ</sup> *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> � <sup>1</sup> � *<sup>γ</sup><sup>i</sup>* ð Þ<sup>2</sup> n o

*i*¼1

4*αiγiS*� *<sup>i</sup> ε*<sup>2</sup> *t*�*i*

*<sup>i</sup>* ¼ 4*αiγi*,

*i*¼1 *γ* ∗ *<sup>i</sup> S*� *i ε*2

*β <sup>j</sup>σ*<sup>2</sup> *t*�*j*

> þ<sup>X</sup> *q*

, *ε<sup>t</sup>*�*<sup>i</sup>* < 0

, *ε<sup>t</sup>*�*<sup>i</sup>* > 0

*S*� *<sup>i</sup> ε*<sup>2</sup>

*<sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> <sup>P</sup> *q*

*j*¼1 *β j σ*2 *t*�*j*

*<sup>t</sup>*�<sup>1</sup> (9)

*j*¼1 *β j σ*2 *t*�*j*

*t*�1,

(7)

(8)

$$\mathbb{S}\_{i}^{+} = \begin{cases} \mathbf{1} & \text{if } e\_{t-i} > \mathbf{0} \\ \mathbf{0} & \text{if } e\_{t-i} \le \mathbf{0} \end{cases}$$

also define

$$a\_i^\* = a\_i(1 + \gamma\_i)^2 \text{ and } \gamma\_i^\* = -4a\_i\gamma\_i,$$

then

$$
\sigma\_t^2 = \omega + \sum\_{i=1}^p a\_i^\* \, \varepsilon\_{t-i}^2 + \sum\_{j=1}^q \beta\_j \sigma\_{t-i}^2 + \sum\_{i=1}^p \gamma\_i^\* \, \mathbb{S}\_i^+ \, \varepsilon\_{t-1}^2 \tag{10}
$$

which allows positive shocks to have a stronger effect on volatility than negative shocks [15]. But when *p* ¼ *q* ¼ 1, the GJRGARCH(1,1) model will be written as

$$
\sigma\_t^2 = \alpha + a e\_t^2 + \chi \mathbb{S}\_i e\_{t-1}^2 + \beta \sigma\_{t-1}^2. \tag{11}
$$

#### **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

$$\boldsymbol{a}\_{t} = \sigma\_{t}\boldsymbol{e}\_{t}; \sigma\_{t}^{2} = \boldsymbol{a}\_{0} + \boldsymbol{\beta}\_{1}\sigma\_{t-1}^{2} + (\mathbf{1} - \boldsymbol{\beta}\_{1})\boldsymbol{a}\_{t-1}^{2} \tag{12}$$

Where *ε<sup>t</sup>* � N(0, 1) *iid*, and 0< *β*<sup>1</sup> < 1, Ali (2013) used *α<sup>i</sup>* to denote 1 � *βi*.

The model is also an exponential smoothing model for the {*a*<sup>2</sup> *<sup>t</sup>* } series. To see this, rewrite the model as.

$$\begin{split} \sigma\_{t}^{2} &= (\mathbf{1} - \boldsymbol{\beta}\_{1}) \boldsymbol{a}\_{t-1}^{2} + \boldsymbol{\beta}\_{1} \sigma\_{t-1}^{2} . \\ &= (\mathbf{1} - \boldsymbol{\beta}\_{1}) \boldsymbol{a}\_{t-1}^{2} + \boldsymbol{\beta}\_{1} \Big[ (\mathbf{1} - \boldsymbol{\beta}) \boldsymbol{a}\_{t-2}^{2} + \boldsymbol{\beta}\_{1} \sigma\_{t-2}^{2} \Big] \\ &= (\mathbf{1} - \boldsymbol{\beta}\_{1}) \boldsymbol{a}\_{t-1}^{2} + (\mathbf{1} - \boldsymbol{\beta}\_{1}) \boldsymbol{\beta}\_{1} \boldsymbol{a}\_{t-2}^{2} + \boldsymbol{\beta}\_{1}^{2} \sigma\_{t-2}^{2} . \end{split} \tag{13}$$

By repeated substitutions, we have

$$
\sigma\_t^2 = (1 - \beta\_1) \left( a\_{t-1}^2 + \beta\_1 a\_{t-2}^2 + \beta\_1^2 a\_{t-3}^3 + \cdots \right),
\tag{14}
$$

where *θ* and *γ* are real constants. Both *ε<sup>t</sup>* and *ε<sup>t</sup>* j j � *E ε<sup>t</sup>* ð Þ j j are zero-mean iid sequences with continuous distributions. Therefore, *E g*ð Þ *ε<sup>t</sup>* ½ �¼ 0. The asymmetry of

> ð Þ *θ* þ *γ ε<sup>t</sup>* � *γE ε<sup>t</sup>* ð Þ j j if *ε<sup>t</sup>* ≥0, ð Þ *θ* � *γ ε<sup>t</sup>* � *γE ε<sup>t</sup>* ð Þ j j if *ε<sup>t</sup>* <0*:*

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

*at*�*<sup>i</sup>* j j þ *θiat*�*<sup>i</sup> σt*�*<sup>i</sup>*

*at* ¼ *σtε<sup>t</sup>*

� � <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *<sup>α</sup>* ½j j *at*�<sup>1</sup> � *E a* ð Þ� j j *<sup>t</sup>*�<sup>1</sup> Þ þ *<sup>θ</sup>at*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>* ln *<sup>σ</sup>*<sup>2</sup>

2*=π* � � h i <sup>p</sup> <sup>þ</sup> *<sup>θ</sup>at*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>* ln *<sup>σ</sup>*<sup>2</sup>

*<sup>α</sup>i*j j *<sup>ε</sup><sup>t</sup>*�*<sup>i</sup>* � *<sup>b</sup>* <sup>2</sup> <sup>þ</sup><sup>X</sup>

*<sup>α</sup>i*ð Þ j j *<sup>ε</sup><sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> *<sup>b</sup>* � *<sup>c</sup>* ð Þ *<sup>ε</sup><sup>t</sup>*�*<sup>i</sup>* <sup>þ</sup> *<sup>b</sup>* <sup>2</sup> <sup>þ</sup><sup>X</sup>

*<sup>t</sup>*ð Þ *zt* � *<sup>δ</sup>* <sup>2</sup> <sup>þ</sup> *βσ*<sup>2</sup>

*<sup>t</sup>* as *σ*<sup>2</sup>

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

*q*

*j*¼1 *β j σ*2 *t*�*j*

*q*

*j*¼1 *β j σ*2

where j j *at*�<sup>1</sup> � *E a* ð Þ j j *<sup>t</sup>*�<sup>1</sup> are iid and have mean zero. When the EGARCH

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

*p*

*i*¼1

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

European option prices in spite of the rich volatility dynamics. Because a

*<sup>t</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *ασ*<sup>2</sup>

an infinite number of past squared returns, it is possible to easily derive the long run, unconditional variance under NGARCH and the assumption of

While the absolute value generalized autoregressive conditional heteroske-

þX*<sup>m</sup> j*¼1

� � � (22)

*β <sup>j</sup>* ln *σ*<sup>2</sup> *t*�*i* (20)

� �, (21)

2*=π* p , which gives:

� � (23)

, (24)

*<sup>t</sup>*�*<sup>j</sup>* (25)

*<sup>t</sup>* (26)

*<sup>t</sup>* is only a function of

*t*�1

*t*�1

*g*ð Þ *ε<sup>t</sup>* can easily be seen by rewriting it as

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

*at* <sup>¼</sup> *<sup>σ</sup>tεt*, ln *<sup>σ</sup>*<sup>2</sup>

ln *σ*<sup>2</sup> *t*

ln *σ*<sup>2</sup> *t*

**3.9 The absolute value GARCH (AVGARCH)**

dasticity (AVGARCH) model is specified as:

*at* <sup>¼</sup> *<sup>σ</sup>tεt*; *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

NAGARCH may be written as

stationarity:

**107**

*g*ð Þ¼ *ε<sup>t</sup>*

*Financial Time Series Analysis via Backtesting Approach*

� � <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup>X*<sup>s</sup>*

*i*¼1 *αi*

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

model has a Gaussian distribution of error term, then *<sup>E</sup> <sup>ε</sup><sup>t</sup>* ð Þ¼ j j ffiffiffiffiffiffiffi

� � <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup> *<sup>α</sup>* j j *at*�<sup>1</sup> � ffiffiffiffiffiffiffi

*at* <sup>¼</sup> *<sup>σ</sup>tεt*; *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup>* <sup>þ</sup><sup>X</sup>

*p*

*i*¼1

*σ*2

And if *zt* � *IIDN*ð Þ 0, 1 , *zt* is independent of *<sup>σ</sup>*<sup>2</sup>

*t*

(

which is the well-known exponential smoothing formation with *β*<sup>1</sup> being 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:

$$
\sigma\_t^2 = a\_0 + \sum\_{i=1}^p (a\_i + \gamma\_i N\_{t=i}) a\_{t-i}^2 + \sum\_{j=1}^q \beta\_j \sigma\_{t-j}^2,\tag{15}
$$

where *Nt*�*<sup>i</sup>* is an indicator for negative *at*�*<sup>i</sup>*, that is,

$$N\_{t-i} = \begin{cases} \mathbf{1} & \text{if } a\_{t-i} < \mathbf{0}, \\ \mathbf{0} & \text{if } a\_{t-i} \ge \mathbf{0}, \end{cases}$$

and *αi*, *γ<sup>i</sup>* , and *β <sup>j</sup>* are nonnegative parameters satisfying conditions similar to those of GARCH models, [14]. When *p* ¼ 1, *q* ¼ 1, the TGARCH(1, 1) model becomes:

$$
\sigma\_t^2 = \alpha + (a + \gamma \mathcal{N}\_{t-1}) a\_{t-1}^2 + \beta \sigma\_{t-1}^2 \tag{16}
$$

### **3.7 NGARCH(p, q) model**

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

$$h\_t = \alpha + \sum\_{i=1}^{q} a\_i \varepsilon\_{t-i}^2 + \sum\_{i=1}^{q} \gamma\_i \varepsilon\_{t-i} + \sum\_{j=1}^{p} \beta\_j h\_{t-j} \tag{17}$$

Where *ht* is the conditional variance, and *ω*, *β* and *α* satisfy *ω* >0, *β* ≥ 0 and *α*≥ 0. Which can also be written as

$$
\sigma\_t = \alpha + \sum\_{i=1}^{q} a\_i \varepsilon\_{t-i}^2 + \sum\_{i=1}^{q} \gamma\_i \varepsilon\_{t-i} + \sum\_{j=1}^{p} \beta\_j \sigma\_{t-j} \tag{18}
$$
