**4.1 Parameter estimation of GARCH(p,q) model using the QL method**

The GARCH(p,q) process is defined by

$$y\_t = \mu + \xi\_t, \qquad t = 1, 2, 3, \dots, T. \tag{31}$$

The QL estimate of *μ*, *α*0, *α*1, ⋯, *αq*, *β*1, ⋯, *β<sup>q</sup>* is the solation of *GT*ð Þ¼ *θ* 0, where

*<sup>t</sup>*¼<sup>1</sup> ^*ζ<sup>t</sup>* � ^*<sup>ζ</sup>* � �<sup>2</sup>

Considering the GARCH(p,q) model given by Eqs. (31) and (32) and using the

� � <sup>¼</sup> ð Þ 0, 1 <sup>Σ</sup>�<sup>1</sup>

� �, <sup>Σ</sup>ð Þ <sup>0</sup>

*<sup>t</sup>* is the solation of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup>

*σ*^*<sup>n</sup> σt*, *yt*

0

BBBBBBBBBBBBBB@

The estimation of GARCH(1,1) model using QL and AQL methods are consid-

*T*

*t*¼1

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>β</sup>qσ*<sup>2</sup>

*t*,*n*

*t*�*j*

� � *<sup>σ</sup>*^*<sup>n</sup> yt*

� � *<sup>σ</sup>*^*n*ð Þ *<sup>σ</sup><sup>t</sup>* !

> �1 0 0 �1 <sup>0</sup> �*ξ*<sup>2</sup> *t*�1

⋮ ⋮ <sup>0</sup> �*ξ*<sup>2</sup>

<sup>0</sup> �*σ*<sup>2</sup>

⋮ ⋮ <sup>0</sup> �*σ*<sup>2</sup>

*t*�*q*

*t*�1

*t*�*q*

� � is the solation of *GT*ð Þ¼ *<sup>θ</sup>* 0. The

*ξt ζt* � �

*t*

, *σ<sup>t</sup>* � �

� � using *σ*^<sup>2</sup>

1

CCCCCCCCCCCCCCA <sup>Σ</sup>^�<sup>1</sup> *t*,*n*

*ξt ζt* � �*:*

*<sup>t</sup>*�<sup>1</sup> � <sup>⋯</sup> � ^*βqσ*^<sup>2</sup>

*<sup>t</sup>*�*q*, *t* ¼ 1, 2, 3, ⋯, *T* and

*<sup>t</sup>* , so the sequence

2 *<sup>t</sup>*�*<sup>i</sup>* ¼

*<sup>T</sup>* � <sup>1</sup> (35)

*<sup>t</sup>*,*<sup>n</sup>* <sup>¼</sup> **<sup>I</sup>**2, and ^*<sup>ξ</sup>*

� � <sup>¼</sup> 0, that is,

*:*

, where i = 1, 2, ⋯, p and j = 1, 2, ⋯,

*<sup>t</sup>*�*<sup>q</sup>*, *t* ¼ 1, 2, 3⋯, *T:* (36)

*t* � � and *yt*

� �

*<sup>ζ</sup>*Þ is an initial value in the iterative procedure.

^*ζ<sup>t</sup>* <sup>¼</sup> *<sup>σ</sup>*^<sup>2</sup>

*<sup>t</sup>* � *α*^<sup>0</sup> � *α*^<sup>1</sup>

of (AQLEF) is given by

, and *σ*^<sup>2</sup>

^*ξ* 2

and the sequence of (AQLEF):

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>α</sup>pξ*<sup>2</sup>

Second, by kernel estimation method, we find

*GT μ*0, *α*0, *α*1, ⋯, *α<sup>q</sup>*

The AQL estimate of *θ* ¼ *μ*, *α*0, *α*1, ⋯, *α<sup>q</sup>*

**4.3 Simulation studies for the GARCH(1,1) model**

q, then the AQL estimation of *σ*<sup>2</sup>

*yt*�*<sup>i</sup>* � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup>

*<sup>t</sup>* ¼ *α*<sup>0</sup> þ *α*<sup>1</sup>

*σ*^2

**87**

^*ξ* 2

*DOI: http://dx.doi.org/10.5772/intechopen.93726*

*<sup>ψ</sup>*^ <sup>¼</sup> <sup>ð</sup>*μ*^,*α*^0, *<sup>α</sup>*^1, <sup>⋯</sup>, *<sup>α</sup>*^*p*, *<sup>β</sup>*^1, <sup>⋯</sup>, *<sup>β</sup>*^*q*, *<sup>σ</sup>*^<sup>2</sup>

*<sup>t</sup>*�<sup>1</sup> � <sup>⋯</sup> � *<sup>α</sup>*^*<sup>p</sup>*

^*ξ* 2

> *σ*^2 *<sup>ζ</sup>* ¼

same argument listed under Eq. (32). First, we need to estimate *σ*<sup>2</sup>

*<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t*

*<sup>t</sup>*�*<sup>j</sup>* is the AQL estimation of *<sup>σ</sup>*<sup>2</sup>

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

<sup>Σ</sup>^*<sup>t</sup>*,*<sup>n</sup> <sup>θ</sup>*ð Þ <sup>0</sup> � � <sup>¼</sup> *<sup>σ</sup>*^*<sup>n</sup> yt*

Third, to estimate the parameters *θ*<sup>0</sup> ¼ *μ*0, *α*0, *α*1, ⋯, *α<sup>q</sup>*

� � <sup>¼</sup> <sup>X</sup>

estimation procedure will be iteratively repeated until it converges.

ered in simulation studies. The GARCH(1,1) process is defined by

Given ^*ξ*<sup>0</sup> <sup>¼</sup> 0, *<sup>θ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>μ</sup>*0, *<sup>α</sup>*<sup>00</sup> , *<sup>α</sup>*<sup>10</sup> , <sup>⋯</sup>, *<sup>α</sup><sup>p</sup>*<sup>0</sup> , *<sup>β</sup>*<sup>10</sup> , <sup>⋯</sup>, *<sup>β</sup><sup>q</sup>*<sup>0</sup>

*<sup>t</sup>*�*<sup>p</sup>* � ^*β*1*σ*^<sup>2</sup>

P*<sup>T</sup>*

*ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches*

**4.2 Parameter estimation of GARCH(p,q) model using the AQL method**

and

$$
\sigma\_t^2 = a\_0 + a\_1 \mathfrak{f}\_{t-1}^2 + \dots + a\_p \mathfrak{f}\_{t-p}^2 + \beta\_1 \sigma\_{t-1}^2 + \dots + \beta\_q \sigma\_{t-q}^2, \qquad t = 1, 2, 3, \dots, T. \tag{32}
$$

*<sup>ξ</sup><sup>t</sup>* are i.i.d with *<sup>E</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ 0 and *<sup>V</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>* ; and *ζ<sup>t</sup>* are i.i.d with *E ζ<sup>t</sup>* ð Þ¼ 0 and *<sup>V</sup> <sup>ζ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup> *<sup>ζ</sup>*. For this scenario, the martingale difference is

$$
\begin{pmatrix} \xi\_t \\ \zeta\_t \end{pmatrix} = \begin{pmatrix} \mathcal{Y}\_t - \mu \\ \sigma\_t^2 - a\_0 - a\_1 \xi\_{t-1}^2 - \dots - a\_p \xi\_{t-p}^2 - \beta\_1 \sigma\_{t-1}^2 - \dots - \beta\_q \sigma\_{t-q}^2 \end{pmatrix}.
$$

The QLEF to estimate *σ*<sup>2</sup> *<sup>t</sup>* is given by

$$G\_{(t)}\left(\sigma\_t^2\right) = (\mathbf{0}, \mathbf{1}) \begin{pmatrix} \sigma\_t^2 & \mathbf{0} \\ \mathbf{0} & \sigma\_\zeta^2 \end{pmatrix}^{-1} \begin{pmatrix} \xi\_t \\ \zeta\_t \end{pmatrix} \tag{33}$$

$$= \sigma\_\zeta^{-2} \left(\sigma\_t^2 - a\_0 - a\_1 \xi\_{t-1}^2 - \dots - a\_p \xi\_{t-p}^2 - \beta\_1 \sigma\_{t-1}^2 - \dots - \beta\_q \sigma\_{t-q}^2\right).$$

Given ^*ξ*<sup>0</sup> <sup>¼</sup> 0, initial values *<sup>ψ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>μ</sup>*0, *<sup>α</sup>*<sup>00</sup> , *<sup>α</sup>*<sup>10</sup> , <sup>⋯</sup>, *<sup>α</sup><sup>p</sup>*<sup>0</sup> , *<sup>β</sup>*<sup>10</sup> , <sup>⋯</sup>, *<sup>β</sup><sup>q</sup>*<sup>0</sup> , *σ*<sup>2</sup> *ζ*0 � �, ^*<sup>ξ</sup>* 2 *<sup>t</sup>*�*<sup>i</sup>* ¼ *yt*�*<sup>i</sup>* � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup> , and *σ*^<sup>2</sup> *<sup>t</sup>*�*<sup>j</sup>* are the QL estimations of *<sup>σ</sup>*<sup>2</sup> *t*�*j* , where i = 1, 2, ⋯, p and j = 1, 2, ⋯, q, then the QL estimation of *σ*<sup>2</sup> *<sup>t</sup>* is the solation of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t* � � <sup>¼</sup> 0,

$$\hat{\sigma}\_{t}^{2} = a\_{0} + a\_{1}\xi\_{t-1}^{2} + \cdots + a\_{p}\xi\_{t-p}^{2} + \beta\_{1}\sigma\_{t-1}^{2} + \cdots + \beta\_{q}\sigma\_{t-q}^{2}, \qquad t = 1, 2, 3\cdots, T. \tag{34}$$

The QLEF, using *σ*^<sup>2</sup> *t* � � and *yt* � �, to estimate the parameters *<sup>θ</sup>* <sup>¼</sup> *<sup>μ</sup>*, *<sup>α</sup>*0, *<sup>α</sup>*1, <sup>⋯</sup>, *<sup>α</sup>q*, *β*1, ⋯, *β<sup>q</sup>* is given by

$$\mathbf{G}\_{T}(\boldsymbol{\theta}) = \sum\_{t=1}^{T} \begin{pmatrix} -\mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \\ \mathbf{0} & -\boldsymbol{\xi}\_{t-1}^{2} \\ \vdots & \vdots \\ \mathbf{0} & -\boldsymbol{\xi}\_{t-p}^{2} \\ \mathbf{0} & -\boldsymbol{\sigma}\_{t-1}^{2} \\ \vdots & \vdots \\ \mathbf{0} & -\boldsymbol{\sigma}\_{t-q}^{2} \end{pmatrix} \begin{pmatrix} \sigma\_{t}^{2} & \mathbf{0} \\ \mathbf{0} & \sigma\_{\zeta\_{0}}^{2} \end{pmatrix}^{-1} \begin{pmatrix} \boldsymbol{\xi}\_{t} \\ \boldsymbol{\zeta}\_{t} \end{pmatrix}$$

*ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches DOI: http://dx.doi.org/10.5772/intechopen.93726*

$$\text{The QL estimate of } \mu, a\_0, a\_1, \dots, a\_q, \beta\_1, \dots, \beta\_q \text{ is the solution of } G\_T(\theta) = 0 \text{, where } \hat{\zeta}\_t = \hat{\sigma}\_t^2 = \frac{1}{T}$$

$$\hat{\zeta}\_t = \hat{\sigma}\_t^2 - \hat{\alpha}\_0 - \hat{\alpha}\_1 \hat{\xi}\_{t-1}^2 - \dots - \hat{\alpha}\_p \hat{\xi}\_{t-p}^2 - \hat{\beta}\_1 \hat{\sigma}\_{t-1}^2 - \dots - \hat{\beta}\_q \hat{\sigma}\_{t-q}^2, t = 1, 2, 3, \dots, T \text{ and }$$

$$\hat{\sigma}\_{\zeta}^2 = \frac{\sum\_{t=1}^T \left( \hat{\zeta}\_t - \overline{\hat{\zeta}} \right)^2}{T - 1} \tag{35}$$

*<sup>ψ</sup>*^ <sup>¼</sup> <sup>ð</sup>*μ*^,*α*^0, *<sup>α</sup>*^1, <sup>⋯</sup>, *<sup>α</sup>*^*p*, *<sup>β</sup>*^1, <sup>⋯</sup>, *<sup>β</sup>*^*q*, *<sup>σ</sup>*^<sup>2</sup> *<sup>ζ</sup>*Þ is an initial value in the iterative procedure.

#### **4.2 Parameter estimation of GARCH(p,q) model using the AQL method**

Considering the GARCH(p,q) model given by Eqs. (31) and (32) and using the same argument listed under Eq. (32). First, we need to estimate *σ*<sup>2</sup> *<sup>t</sup>* , so the sequence of (AQLEF) is given by

$$G\_{(t)}\left(\sigma\_t^2\right) = (\mathbf{0}, \mathbf{1})\Sigma\_{t,n}^{-1}\begin{pmatrix} \xi\_t\\ \zeta\_t \end{pmatrix}$$

Given ^*ξ*<sup>0</sup> <sup>¼</sup> 0, *<sup>θ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>μ</sup>*0, *<sup>α</sup>*<sup>00</sup> , *<sup>α</sup>*<sup>10</sup> , <sup>⋯</sup>, *<sup>α</sup><sup>p</sup>*<sup>0</sup> , *<sup>β</sup>*<sup>10</sup> , <sup>⋯</sup>, *<sup>β</sup><sup>q</sup>*<sup>0</sup> � �, <sup>Σ</sup>ð Þ <sup>0</sup> *<sup>t</sup>*,*<sup>n</sup>* <sup>¼</sup> **<sup>I</sup>**2, and ^*<sup>ξ</sup>* 2 *<sup>t</sup>*�*<sup>i</sup>* ¼ *yt*�*<sup>i</sup>* � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup> , and *σ*^<sup>2</sup> *<sup>t</sup>*�*<sup>j</sup>* is the AQL estimation of *<sup>σ</sup>*<sup>2</sup> *t*�*j* , where i = 1, 2, ⋯, p and j = 1, 2, ⋯, q, then the AQL estimation of *σ*<sup>2</sup> *<sup>t</sup>* is the solation of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t* � � <sup>¼</sup> 0, that is,

$$\hat{\sigma}\_t^2 = a\_0 + a\_1 \hat{\mathfrak{s}}\_{t-1}^2 + \dots + a\_p \mathfrak{s}\_{t-p}^2 + \beta\_1 \sigma\_{t-1}^2 + \dots + \beta\_q \sigma\_{t-q}^2, \qquad t = 1, 2, 3 \dots, T. \tag{36}$$

Second, by kernel estimation method, we find

$$
\hat{\Sigma}\_{\mathfrak{t},\mathfrak{n}}\left(\theta^{(0)}\right) = \begin{pmatrix}
\hat{\sigma}\_{\mathfrak{n}}\left(\mathcal{y}\_{\mathfrak{t}}\right) & \hat{\sigma}\_{\mathfrak{n}}\left(\mathcal{y}\_{\mathfrak{t}},\sigma\_{\mathfrak{t}}\right) \\
\hat{\sigma}\_{\mathfrak{n}}\left(\sigma\_{\mathfrak{t}},\mathcal{y}\_{\mathfrak{t}}\right) & \hat{\sigma}\_{\mathfrak{n}}\left(\sigma\_{\mathfrak{t}}\right)
\end{pmatrix}.
$$

Third, to estimate the parameters *θ*<sup>0</sup> ¼ *μ*0, *α*0, *α*1, ⋯, *α<sup>q</sup>* � � using *σ*^<sup>2</sup> *t* � � and *yt* � � and the sequence of (AQLEF):

$$G\_T(\mu\_0, a\_0, a\_1, \dots, a\_q) = \sum\_{t=1}^T \begin{pmatrix} -1 & 0 \\ \mathbf{0} & -\mathbf{1} \\ \mathbf{0} & -\boldsymbol{\xi}\_{t-1}^2 \\ \vdots & \vdots \\ \mathbf{0} & -\boldsymbol{\xi}\_{t-q}^2 \\ \mathbf{0} & -\boldsymbol{\sigma}\_{t-1}^2 \\ \vdots & \vdots \\ \mathbf{0} & -\boldsymbol{\sigma}\_{t-q}^2 \end{pmatrix} \hat{\boldsymbol{\Sigma}}\_{t,n}^{-1} \begin{pmatrix} \boldsymbol{\xi}\_t \\ \boldsymbol{\xi}\_t \end{pmatrix}.$$

The AQL estimate of *θ* ¼ *μ*, *α*0, *α*1, ⋯, *α<sup>q</sup>* � � is the solation of *GT*ð Þ¼ *<sup>θ</sup>* 0. The estimation procedure will be iteratively repeated until it converges.

#### **4.3 Simulation studies for the GARCH(1,1) model**

The estimation of GARCH(1,1) model using QL and AQL methods are considered in simulation studies. The GARCH(1,1) process is defined by

**4. Parameter estimation of GARCH(p,q) model using the QL and AQL**

In this section, we developing the estimation of GARCH model using QL and

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>β</sup>qσ*<sup>2</sup>

!

*<sup>t</sup>*�<sup>1</sup> � <sup>⋯</sup> � *<sup>α</sup>pξ*<sup>2</sup>

*<sup>t</sup>* 0 0 *σ*<sup>2</sup> *ζ*

� �

!�<sup>1</sup> *<sup>ξ</sup><sup>t</sup>*

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

*<sup>t</sup>* is the solation of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup>

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>β</sup>qσ*<sup>2</sup>

1

CCCCCCCCCCCCCCCCCA

*σ*2 *<sup>t</sup>* 0 0 *σ*<sup>2</sup> *ζ*0

!�<sup>1</sup>

*t*�*j*

*yt* ¼ *μ* þ *ξt*, *t* ¼ 1, 2, 3, ⋯, *T:* (31)

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

*ζt*

� �

*<sup>t</sup>*�<sup>1</sup> � <sup>⋯</sup> � *<sup>β</sup>qσ*<sup>2</sup>

*t* � � <sup>¼</sup> 0,

� �, to estimate the parameters *<sup>θ</sup>* <sup>¼</sup> *<sup>μ</sup>*, *<sup>α</sup>*0, *<sup>α</sup>*1, <sup>⋯</sup>, *<sup>α</sup>q*,

!

*<sup>t</sup>*�*<sup>q</sup>*, *t* ¼ 1, 2, 3, ⋯, *T:* (32)

*<sup>t</sup>* ; and *ζ<sup>t</sup>* are i.i.d with *E ζ<sup>t</sup>* ð Þ¼ 0 and

*<sup>t</sup>*�<sup>1</sup> � <sup>⋯</sup> � *<sup>β</sup>qσ*<sup>2</sup>

*t*�*q*

*:*

, *σ*<sup>2</sup> *ζ*0

, where i = 1, 2, ⋯, p and j = 1, 2,

*<sup>t</sup>*�*<sup>q</sup>*, *t* ¼ 1, 2, 3⋯, *T:* (34)

*ξt ζt* � � , ^*ξ* 2 *<sup>t</sup>*�*<sup>i</sup>* ¼

*t*�*q*

*:*

(33)

**4.1 Parameter estimation of GARCH(p,q) model using the QL method**

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

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

*<sup>ζ</sup>*. For this scenario, the martingale difference is

<sup>¼</sup> *yt* � *<sup>μ</sup>*

*<sup>t</sup>* is given by

� � <sup>¼</sup> ð Þ 0, 1 *<sup>σ</sup>*<sup>2</sup>

*<sup>t</sup>*�<sup>1</sup> � <sup>⋯</sup> � *<sup>α</sup>pξ*<sup>2</sup>

Given ^*ξ*<sup>0</sup> <sup>¼</sup> 0, initial values *<sup>ψ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>μ</sup>*0, *<sup>α</sup>*<sup>00</sup> , *<sup>α</sup>*<sup>10</sup> , <sup>⋯</sup>, *<sup>α</sup><sup>p</sup>*<sup>0</sup> , *<sup>β</sup>*<sup>10</sup> , <sup>⋯</sup>, *<sup>β</sup><sup>q</sup>*<sup>0</sup>

*<sup>t</sup>*�*<sup>j</sup>* are the QL estimations of *<sup>σ</sup>*<sup>2</sup>

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

�1 0 0 �1 <sup>0</sup> �*ξ*<sup>2</sup> *t*�1

⋮ ⋮ <sup>0</sup> �*ξ*<sup>2</sup>

<sup>0</sup> �*σ*<sup>2</sup>

⋮ ⋮ <sup>0</sup> �*σ*<sup>2</sup>

*t*�*p*

*t*�1

*t*�*q*

*<sup>t</sup>* � *<sup>α</sup>*<sup>0</sup> � *<sup>α</sup>*1*ξ*<sup>2</sup>

The GARCH(p,q) process is defined by

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>α</sup>pξ*<sup>2</sup>

*<sup>ξ</sup><sup>t</sup>* are i.i.d with *<sup>E</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ 0 and *<sup>V</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup>

*<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t*

*<sup>t</sup>* � *<sup>α</sup>*<sup>0</sup> � *<sup>α</sup>*1*ξ*<sup>2</sup>

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>α</sup>pξ*<sup>2</sup>

*t* � � and *yt*

*GT*ð Þ¼ *<sup>θ</sup>* <sup>X</sup>

*T*

0

*t*¼1

BBBBBBBBBBBBBBBBB@

*σ*2

**methods**

AQL methods.

and

*<sup>V</sup> <sup>ζ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup>

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

*ξt ζt* � �

<sup>¼</sup> *<sup>σ</sup>*�<sup>2</sup> *<sup>ζ</sup> σ*<sup>2</sup>

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

*β*1, ⋯, *β<sup>q</sup>* is given by

*yt*�*<sup>i</sup>* � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup>

*σ*^2

**86**

The QLEF to estimate *σ*<sup>2</sup>

, and *σ*^<sup>2</sup>

The QLEF, using *σ*^<sup>2</sup>

⋯, q, then the QL estimation of *σ*<sup>2</sup>

*σ*2

$$y\_t = \mu + \xi\_t, \qquad t = 1, 2, 3, \dots, T. \tag{37}$$

*<sup>β</sup>*^<sup>1</sup> <sup>¼</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.93726*

P*<sup>T</sup> <sup>t</sup>*¼1*σ*^<sup>2</sup> *<sup>t</sup>* � *α*^<sup>1</sup>

*α*^<sup>0</sup> ¼

^*ζ<sup>t</sup>* <sup>¼</sup> *<sup>σ</sup>*^<sup>2</sup>

<sup>¼</sup> <sup>X</sup> *T*

<sup>¼</sup> <sup>X</sup> *T*

<sup>¼</sup> <sup>X</sup> *T*

*<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> <sup>X</sup> *T*

*<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> <sup>X</sup> *T*

*ζ*

sequence of (AQLEF) is given by

the AQL estimation of *σ*<sup>2</sup>

� � <sup>¼</sup> 0, that is,

*<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t*

**89**

*<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t*

*σ*^2

Given ^*ξ*<sup>0</sup> <sup>¼</sup> 0, *<sup>θ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>μ</sup>*0, *<sup>α</sup>*0,0, *<sup>α</sup>*1,0, *<sup>β</sup>*1,0 � �, <sup>Σ</sup>ð Þ <sup>0</sup>

*<sup>t</sup>* ¼ *α*<sup>0</sup> þ *α*<sup>1</sup>

^*ξ* 2

*t*¼1 *σ*^2 *t*�1 ^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> �

*t*¼1 *σ*^2 *t* ^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> �

*t*¼1 ^*ξ* 4 *<sup>t</sup>*�<sup>1</sup> �

*t*¼1 *σ*^2 *t σ*^2 *<sup>t</sup>*�<sup>1</sup> �

*t*¼1 *σ*^4 *<sup>t</sup>*�<sup>1</sup> �

*Sσ*^2 *t*�1 ^*ξ* 2 *t*�1

> *Sσ*^2 *t* ^*ξ* 2 *t*�1

*S*^*ξ* 2 *t*�1 ^*ξ* 2 *t*�1

> *Sσ*^2 *t σ*^2

*Sσ*^2 *<sup>t</sup>*�1*σ*^<sup>2</sup>

*<sup>ψ</sup>*^ <sup>¼</sup> *<sup>μ</sup>*^, *<sup>α</sup>*^0, *<sup>α</sup>*^1, *<sup>σ</sup>*^<sup>2</sup>

and let

where

*Sσ*^2 *t*�1 ^*ξ* 2 *t*�1 *Sσ*^2 *t* ^*ξ* 2 *t*�1 � *S*^*<sup>ξ</sup>* 2 *t*�1 ^*ξ* 2 *t*�1 *Sσ*^2 *t σ*^2 *t*�1

*S*2 *σ*^2 *t*�1 ^*ξ* 2 *t*�1 � *S<sup>σ</sup>*^<sup>2</sup> *<sup>t</sup>*�1*σ*^<sup>2</sup> *t*�1 *S*^*ξ* 2 *t*�1 ^*ξ* 2 *t*�1

> *Sσ*^2 *t* ^*ξ* 2 *t*�1 � *β* ^1*S<sup>σ</sup>*^<sup>2</sup> *t*�1 ^*ξ* 2 *t*�1

> > *S*^*ξ* 2 *t*�1 ^*ξ* 2 *t*�1

> > > P*<sup>T</sup> <sup>t</sup>*¼1*σ*^<sup>2</sup> *t*�1 *<sup>T</sup> :* (44)

P*<sup>T</sup> t*¼1 ^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> � *<sup>β</sup>*^<sup>1</sup>

*ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches*

P*<sup>T</sup>*

*<sup>t</sup>*¼<sup>1</sup> ^*ζ<sup>t</sup>* � ^*<sup>ζ</sup>* � �<sup>2</sup>

*α*^<sup>1</sup> ¼

*σ*^2 *<sup>ζ</sup>* ¼

> ^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> � *<sup>β</sup>*^1*σ*^<sup>2</sup>

P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t*�1 P*<sup>T</sup> t*¼1 ^*ξ* 2 *t*�1 *<sup>T</sup>* ,

P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t* P*<sup>T</sup> t*¼1 ^*ξ* 2 *t*�1 *<sup>T</sup>* ,

P*<sup>T</sup> t*¼1 ^*ξ* 2 *t*�1 � �<sup>2</sup>

> P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t* P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t*�1 *<sup>T</sup>* ,

P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t*�1 � �<sup>2</sup>

� � is an initial value in the iterative procedure.

sion on assigning initial values in the QL estimation procedures, see [21, 34].

*4.3.2 Parameter estimation of GARCH(1,1) model using the AQL method*

same argument listed under (Eq. (38)). First, we need to estimate *σ*<sup>2</sup>

*t*,*n*

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

*σ*2

*<sup>t</sup>*�1, then the AQL estimation of *<sup>σ</sup>*<sup>2</sup>

� � <sup>¼</sup> ð Þ 0, 1 <sup>Σ</sup>�<sup>1</sup>

*<sup>T</sup>* ,

*<sup>T</sup> :*

The initial values might be affected the estimation results. For extensive discus-

Considering the GARCH(1,1) model given by Eqs. (37) and (38) and using the

*yt* � *μ*

2

� �

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

*t*�1

*<sup>t</sup>* is the solation of

*<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> *yt*�<sup>1</sup> � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup>

*<sup>t</sup>*�1, *t* ¼ 1, 2, 3⋯, *T:* (46)

*<sup>t</sup>* � *<sup>α</sup>*<sup>0</sup> � *<sup>α</sup>*1*ξ*<sup>2</sup>

*<sup>t</sup>*,*<sup>n</sup>* <sup>¼</sup> **<sup>I</sup>**2, ^*<sup>ξ</sup>*

*<sup>t</sup>* � *α*^<sup>0</sup> � *α*^<sup>1</sup>

*:* (42)

*<sup>t</sup>* , so the

, and *σ*^<sup>2</sup>

*<sup>t</sup>*�<sup>1</sup> is

*:* (43)

*<sup>T</sup>* � <sup>1</sup> (45)

*<sup>t</sup>*�1, *t* ¼ 1, 2, 3, ⋯, *T*,

and

$$
\sigma\_t^2 = a\_0 + a\_1 \xi\_{t-1}^2 + \beta\_1 \sigma\_{t-1} + \zeta\_t, \qquad t = 1, 2, 3, \cdots, T. \tag{38}
$$

*<sup>ξ</sup><sup>t</sup>* are i.i.d with *<sup>E</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ 0 and *<sup>V</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>* ; and *ζ<sup>t</sup>* are i.i.d with *E ζ<sup>t</sup>* ð Þ¼ 0 and *<sup>V</sup> <sup>ζ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup> *ζ*.

#### *4.3.1 Parameter estimation of GARCH(1,1) model using the QL method*

For GARCH(1,1) given by Eqs. (37) and (38), the martingale difference is

$$
\begin{pmatrix} \xi\_t \\ \zeta\_t \end{pmatrix} = \begin{pmatrix} \chi\_t - \mu \\ \sigma\_t^2 - a\_0 - a\_1 \xi\_{t-1}^2 - \beta\_1 \sigma\_{t-1}^2 \end{pmatrix} \cdot \mathbf{1}
$$

The QLEF to estimate *σ*<sup>2</sup> *<sup>t</sup>* is given by

$$G\_{(t)}\left(\sigma\_t^2\right) = (\mathbf{0}, \mathbf{1}) \begin{pmatrix} \sigma\_t^2 & \mathbf{0} \\ & \mathbf{0} & \sigma\_\zeta^2 \end{pmatrix}^{-1} \begin{pmatrix} y\_t - \mu \\ \sigma\_t^2 - a\_0 - a\_1 \mathfrak{z}\_{t-1}^2 - \beta\_1 \sigma\_{t-1}^2 \end{pmatrix} \tag{39}$$

$$= \sigma\_\zeta^{-2} \left(\sigma\_t^2 - a\_0 - a\_1 \mathfrak{z}\_{t-1}^2 - \beta\_1 \sigma\_{t-1}^2\right).$$

Given ^*ξ*<sup>0</sup> <sup>¼</sup> 0, initial values *<sup>ψ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>μ</sup>*0, *<sup>α</sup>*<sup>00</sup> , *<sup>α</sup>*<sup>10</sup> , *<sup>σ</sup>*<sup>2</sup> *ζ*0 � �, ^*<sup>ξ</sup>* 2 *<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> *yt*�<sup>1</sup> � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup> , and *σ*^<sup>2</sup> *t*�1 is the QL estimation of *σ*<sup>2</sup> *<sup>t</sup>*�1, then the QL estimation of *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>* is the solation of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t* � � <sup>¼</sup> 0,

$$
\hat{\sigma}\_t^2 = a\_0 + a\_1 \hat{\xi}\_{t-1}^2 + \beta\_1 \hat{\sigma}\_{t-1}^2, \qquad t = 1, 2, 3 \cdots, T. \tag{40}
$$

To estimate the parameters *μ*, *α*0, and *α*1, using *σ*^<sup>2</sup> *t* � � and *yt* � �, the QLEF is given by

$$G\_T(\mu, a\_0, a\_1, \beta\_1) = \sum\_{t=1}^T \begin{pmatrix} -1 & 0 \\ 0 & -1 \\ 0 & -\xi\_{t-1}^2 \\ 0 & -\sigma\_{t-1}^2 \end{pmatrix} \begin{pmatrix} \sigma\_t^2 & 0 \\ 0 & \sigma\_{\zeta\_0}^2 \end{pmatrix}^{-1}$$

$$\* \begin{pmatrix} & \quad & \quad \\ & & \quad \\ & \sigma\_t^2 - \mu & \\ & & \sigma\_t^2 - a\_0 - a\_1 \bar{\xi}\_{t-1}^2 - \beta\_1 \sigma\_{t-1}^2 \end{pmatrix}.$$

The solation of *GT μ*, *α*0, *α*1, *β*<sup>1</sup> ð Þ¼ 0 is the QL estimate of *μ*, *α*0, *α*1, and *β*1. Therefore

$$
\hat{\mu} = \sum\_{t=1}^{T} \frac{\mathcal{Y}\_t}{\hat{\sigma}\_t^2} / \sum\_{t=1}^{T} \frac{\mathbf{1}}{\hat{\sigma}\_t^2}. \tag{41}
$$

*ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches DOI: http://dx.doi.org/10.5772/intechopen.93726*

$$\hat{\boldsymbol{\beta}}\_{1} = \frac{\mathbf{S}\_{\hat{\sigma}^{2}\_{t-1}\hat{\xi}^{2}\_{t-1}}\mathbf{S}\_{\hat{\sigma}^{2}\_{t}\hat{\xi}^{2}\_{t-1}} - \mathbf{S}\_{\hat{\xi}^{2}\_{t-1}\hat{\xi}^{2}\_{t-1}}\mathbf{S}\_{\hat{\sigma}^{2}\_{t}\hat{\sigma}^{2}\_{t-1}}}{\mathbf{S}\_{\hat{\sigma}^{2}\_{t-1}\hat{\xi}^{2}\_{t-1}} - \mathbf{S}\_{\hat{\sigma}^{2}\_{t-1}\hat{\sigma}^{2}\_{t-1}}\mathbf{S}\_{\hat{\xi}^{2}\_{t-1}\hat{\xi}^{2}\_{t-1}}}}. \tag{42}$$

$$
\hat{a}\_1 = \frac{\mathbb{S}\_{\hat{\sigma}^2\_t \hat{\xi}^2\_{t-1}} - \hat{\beta}\_1 \mathbb{S}\_{\hat{\sigma}^2\_{t-1} \hat{\xi}^2\_{t-1}}}{\mathbb{S}\_{\hat{\xi}^2\_{t-1} \hat{\xi}^2\_{t-1}}}. \tag{43}
$$

$$\hat{\alpha}\_{0} = \frac{\sum\_{t=1}^{T} \hat{\sigma}\_{t}^{2} - \hat{\alpha}\_{1} \sum\_{t=1}^{T} \hat{\xi}\_{t-1}^{2} - \hat{\beta}\_{1} \sum\_{t=1}^{T} \hat{\sigma}\_{t-1}^{2}}{T}. \tag{44}$$

and let

*yt* ¼ *μ* þ *ξt*, *t* ¼ 1, 2, 3, ⋯, *T:* (37)

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*1*σt*�<sup>1</sup> <sup>þ</sup> *<sup>ζ</sup>t*, *<sup>t</sup>* <sup>¼</sup> 1, 2, 3, <sup>⋯</sup>, *<sup>T</sup>:* (38)

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

*yt* � *μ*

*ζ*0

, ^*ξ* 2

*t* � � and *yt*

1

CCCCCCCCA

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

*σ*2 *<sup>t</sup>* 0

0 @

0 *σ*<sup>2</sup> *ζ*0

*t*�1

1 A*:*

*:* (41)

*<sup>t</sup>* � *<sup>α</sup>*<sup>0</sup> � *<sup>α</sup>*1*ξ*<sup>2</sup>

*t*�1

� �

�1 0

0 �1

<sup>0</sup> �*ξ*<sup>2</sup> *t*�1

<sup>0</sup> �*σ*<sup>2</sup>

*<sup>t</sup>* � *<sup>α</sup>*<sup>0</sup> � *<sup>α</sup>*1*ξ*<sup>2</sup>

The solation of *GT μ*, *α*0, *α*1, *β*<sup>1</sup> ð Þ¼ 0 is the QL estimate of *μ*, *α*0, *α*1, and *β*1.

*yt σ*^2 *t =* X *T*

*t*¼1

1 *σ*^2 *t*

*t*�1

*yt* � *μ*

*t*�1

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

*t*�1

*<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> *yt*�<sup>1</sup> � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup>

*<sup>t</sup>*�1, *t* ¼ 1, 2, 3⋯, *T:* (40)

*<sup>t</sup>* is the solation of

1 A

�1

� �, the QLEF is

1 A

(39)

, and *σ*^<sup>2</sup> *t*�1

*:*

*<sup>t</sup>* ; and *ζ<sup>t</sup>* are i.i.d with *E ζ<sup>t</sup>* ð Þ¼ 0 and

and

*<sup>V</sup> <sup>ζ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup>

*ζ*.

*σ*2

The QLEF to estimate *σ*<sup>2</sup>

<sup>¼</sup> *<sup>σ</sup>*�<sup>2</sup> *<sup>ζ</sup> σ*<sup>2</sup>

*σ*^2

*<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t* � � <sup>¼</sup> ð Þ 0, 1

is the QL estimation of *σ*<sup>2</sup>

*<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t* � � <sup>¼</sup> 0,

given by

Therefore

**88**

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

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

*<sup>ξ</sup><sup>t</sup>* are i.i.d with *<sup>E</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ 0 and *<sup>V</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup>

*ξt ζt* � �

*4.3.1 Parameter estimation of GARCH(1,1) model using the QL method*

*σ*2

1 A

�1

� �*:*

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

*T*

0

BBBBBBBB@

*t*¼1

*σ*2

*<sup>μ</sup>*^ <sup>¼</sup> <sup>X</sup> *T*

*t*¼1

∗

0 @ 0 @

*<sup>t</sup>* is given by

*σ*2 *<sup>t</sup>* 0

Given ^*ξ*<sup>0</sup> <sup>¼</sup> 0, initial values *<sup>ψ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>μ</sup>*0, *<sup>α</sup>*<sup>00</sup> , *<sup>α</sup>*<sup>10</sup> , *<sup>σ</sup>*<sup>2</sup>

*<sup>t</sup>* ¼ *α*<sup>0</sup> þ *α*<sup>1</sup>

*GT <sup>μ</sup>*, *<sup>α</sup>*0, *<sup>α</sup>*1, *<sup>β</sup>*<sup>1</sup> ð Þ¼ <sup>X</sup>

To estimate the parameters *μ*, *α*0, and *α*1, using *σ*^<sup>2</sup>

0 @

0 *σ*<sup>2</sup> *ζ*

*<sup>t</sup>* � *<sup>α</sup>*<sup>0</sup> � *<sup>α</sup>*1*ξ*<sup>2</sup>

^*ξ* 2

For GARCH(1,1) given by Eqs. (37) and (38), the martingale difference is

*<sup>t</sup>* � *<sup>α</sup>*<sup>0</sup> � *<sup>α</sup>*1*ξ*<sup>2</sup>

*σ*2

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

*<sup>t</sup>*�1, then the QL estimation of *<sup>σ</sup>*<sup>2</sup>

� �

<sup>¼</sup> *yt* � *<sup>μ</sup>*

$$
\hat{\sigma}\_{\zeta}^{2} = \frac{\sum\_{t=1}^{T} \left(\hat{\zeta}\_{t} - \overline{\hat{\zeta}}\right)^{2}}{T - 1} \tag{45}
$$

where

^*ζ<sup>t</sup>* <sup>¼</sup> *<sup>σ</sup>*^<sup>2</sup> *<sup>t</sup>* � *α*^<sup>0</sup> � *α*^<sup>1</sup> ^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> � *<sup>β</sup>*^1*σ*^<sup>2</sup> *<sup>t</sup>*�1, *t* ¼ 1, 2, 3, ⋯, *T*, *Sσ*^2 *t*�1 ^*ξ* 2 *t*�1 <sup>¼</sup> <sup>X</sup> *T t*¼1 *σ*^2 *t*�1 ^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> � P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t*�1 P*<sup>T</sup> t*¼1 ^*ξ* 2 *t*�1 *<sup>T</sup>* , *Sσ*^2 *t* ^*ξ* 2 *t*�1 <sup>¼</sup> <sup>X</sup> *T t*¼1 *σ*^2 *t* ^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> � P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t* P*<sup>T</sup> t*¼1 ^*ξ* 2 *t*�1 *<sup>T</sup>* , *S*^*ξ* 2 *t*�1 ^*ξ* 2 *t*�1 <sup>¼</sup> <sup>X</sup> *T t*¼1 ^*ξ* 4 *<sup>t</sup>*�<sup>1</sup> � P*<sup>T</sup> t*¼1 ^*ξ* 2 *t*�1 � �<sup>2</sup> *<sup>T</sup>* , *Sσ*^2 *t σ*^2 *<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> <sup>X</sup> *T t*¼1 *σ*^2 *t σ*^2 *<sup>t</sup>*�<sup>1</sup> � P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t* P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t*�1 *<sup>T</sup>* , *Sσ*^2 *<sup>t</sup>*�1*σ*^<sup>2</sup> *<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> <sup>X</sup> *T t*¼1 *σ*^4 *<sup>t</sup>*�<sup>1</sup> � P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup>*σ*^<sup>2</sup> *t*�1 � �<sup>2</sup> *<sup>T</sup> :*

*<sup>ψ</sup>*^ <sup>¼</sup> *<sup>μ</sup>*^, *<sup>α</sup>*^0, *<sup>α</sup>*^1, *<sup>σ</sup>*^<sup>2</sup> *ζ* � � is an initial value in the iterative procedure.

The initial values might be affected the estimation results. For extensive discussion on assigning initial values in the QL estimation procedures, see [21, 34].

#### *4.3.2 Parameter estimation of GARCH(1,1) model using the AQL method*

Considering the GARCH(1,1) model given by Eqs. (37) and (38) and using the same argument listed under (Eq. (38)). First, we need to estimate *σ*<sup>2</sup> *<sup>t</sup>* , so the sequence of (AQLEF) is given by

$$G\_{(t)}\left(\sigma\_t^2\right) = (\mathbf{0}, \mathbf{1})\Sigma\_{t,n}^{-1} \begin{pmatrix} \mathcal{Y}\_t - \mu\\ \sigma\_t^2 - a\_0 - a\_1\xi\_{t-1}^2 - \beta\_1\sigma\_{t-1}^2 \end{pmatrix}.$$

Given ^*ξ*<sup>0</sup> <sup>¼</sup> 0, *<sup>θ</sup>*<sup>0</sup> <sup>¼</sup> *<sup>μ</sup>*0, *<sup>α</sup>*0,0, *<sup>α</sup>*1,0, *<sup>β</sup>*1,0 � �, <sup>Σ</sup>ð Þ <sup>0</sup> *<sup>t</sup>*,*<sup>n</sup>* <sup>¼</sup> **<sup>I</sup>**2, ^*<sup>ξ</sup>* 2 *<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> *yt*�<sup>1</sup> � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup> , and *σ*^<sup>2</sup> *<sup>t</sup>*�<sup>1</sup> is the AQL estimation of *σ*<sup>2</sup> *<sup>t</sup>*�1, then the AQL estimation of *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>* is the solation of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t* � � <sup>¼</sup> 0, that is,

$$
\hat{\sigma}\_t^2 = a\_0 + a\_1 \hat{\xi}\_{t-1}^2 + \beta\_1 \sigma\_{t-1}^2, \qquad t = 1, 2, 3 \cdots, T. \tag{46}
$$

Second, by kernel estimation method, we find

$$
\hat{\Sigma}\_{t,n}\left(\theta^{(0)}\right) = \begin{pmatrix}
\hat{\sigma}\_n\left(\mathcal{Y}\_t\right) & \mathbf{0} \\
\mathbf{0} & \hat{\sigma}\_n(\sigma\_t)
\end{pmatrix}.
$$

Third, to estimate the parameters *<sup>θ</sup>* <sup>¼</sup> *<sup>μ</sup>*, *<sup>α</sup>*0, *<sup>α</sup>*1, *<sup>β</sup>*<sup>1</sup> ð Þ using *<sup>σ</sup>*^<sup>2</sup> *t* � � and *yt* � � and the sequence of AQLEF:

$$G\_{T}(\mu, a\_{0}, a\_{1}, \beta\_{1}) = \sum\_{t=1}^{T} \begin{pmatrix} -\mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \\ \mathbf{0} & -\hat{\xi}\_{t-1}^{2} \\ \mathbf{0} & -\hat{\sigma}\_{t-1}^{2} \end{pmatrix} \hat{\Sigma}\_{t,\mu}^{-1} \begin{pmatrix} y\_{t} - \mu \\ \sigma\_{t}^{2} - a\_{0} - a\_{1} \xi\_{t-1}^{2} - \beta\_{1} \sigma\_{t-1}^{2} \end{pmatrix}.$$

The AQL estimate of *μ*, *α*0, *α*1, and *β*<sup>1</sup> is the solation of *GT μ*, *α*0, *α*1, *β*<sup>1</sup> ð Þ¼ 0. Therefore

$$\hat{\mu} = \sum\_{t=1}^{T} \frac{\mathbf{y}\_t}{\hat{\sigma}\_n(\mathbf{y}\_t)} / \sum\_{t=1}^{T} \frac{\mathbf{1}}{\hat{\sigma}\_n(\mathbf{y}\_t)}. \tag{47}$$

$$\hat{\boldsymbol{\beta}}\_{1} = \frac{\text{SS}\_{\hat{\sigma}^{2}\_{t-1}\hat{\xi}^{2}\_{t-1}}\text{SS}\_{\hat{\sigma}^{2}\_{t}\hat{\xi}^{2}\_{t-1}} - \text{SS}\_{\hat{\xi}^{2}\_{t-1}\hat{\xi}^{2}\_{t-1}}\text{SS}\_{\hat{\sigma}^{2}\_{t}\hat{\sigma}^{2}\_{t-1}}}{\text{SS}^{2}\_{\hat{\sigma}^{2}\_{t-1}\hat{\xi}^{2}\_{t-1}} - \text{SS}\_{\hat{\sigma}^{2}\_{t-1}\hat{\sigma}^{2}\_{t-1}}\text{SS}\_{\hat{\xi}^{2}\_{t-1}\hat{\xi}^{2}\_{t-1}}}. \tag{48}$$

$$
\hat{\alpha}\_1 = \frac{\text{SS}\_{\hat{\sigma}\_t^2 \hat{\xi}\_{t-1}^2} - \hat{\beta}\_1 \text{SS}\_{\hat{\sigma}\_{t-1}^2 \hat{\xi}\_{t-1}^2}}{\text{SS}\_{\hat{\xi}\_{t-1}^2 \hat{\xi}\_{t-1}^2}}. \tag{49}
$$

The estimation procedure will be iteratively repeated until it converges. For each parameter setting, T = 500 observations are simulated from the true model. We then replicate the experiment for 1000 times to obtain the mean and root mean squared errors (RMSE) for *α*^0, *α*^1, *β*^1, and *μ*^. In **Table 4**, QL denotes the

and 100 from GARCH(1,1) model. In **Table 5**, The QL and AQL estimation methods show the property of consistency, and the RMSE decreases as the sample

We generated N = 1000 independent random samples of size T = 20, 40, 60, 80,

*μ α***<sup>0</sup>** *α***<sup>1</sup>** *β***<sup>1</sup>** *μ α***<sup>0</sup>** *α***<sup>1</sup>** *β***<sup>1</sup>**

0.040 0.353 0.011 0.029 0.031 0.155 0.033 0.025

0.001 0.012 0.019 0.009 0.010 0.006 0.021 0.004

0.034 0.212 0.014 0.024 0.030 0.189 0.024 0.025

0.010 0.006 0.001 0.005 0.001 0.004 0.011 0.004

0.031 0.146 0.015 0.024 0.033 0.090 0.009 0.015

0.001 0.005 0.002 0.005 0.002 0.004 0.002 0.004

True 0.15 0.65 0.87 0.10 0.20 0.41 0.88 0.08 QL 0.149 0.779 0.865 0.074 0.199 0.461 0.912 0.057

*ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches*

*DOI: http://dx.doi.org/10.5772/intechopen.93726*

AQL 0.150 0.661 0.851 0.092 0.209 0.405 0.901 0.076

True �0.10 0.48 0.89 0.08 0.16 0.37 0.9 0.08 QL �0.101 0.556 0.902 0.058 0.159 0.434 0.922 0.058

AQL �0.110 0.486 0.891 0.0752 0.161 0.374 0.911 0.076

True 0.18 0.39 0.88 0.08 0.09 0.50 0.89 0.05 QL 0.179 0.447 0.892 0.058 0.089 0.538 0.898 0.036

AQL 0.180 0.395 0.882 0.076 0.091 0.504 0.892 0.046

*The QL and AQL estimates and the RMSE of each estimate is stated below that estimate for GARCH model.*

The second set of data is the weekly price changes of crude oil prices *Pt*. The *Pt* of Cushing, OK, West Texas Intermediate (US dollars per barrel) is considered for the period from 7/1/2000 to 10/6/2016, with 858 observations in total. The data are transformed into rates of change by taking the first difference of the logs. Thus,

The estimation of unknown parameters, (*α*0, *α*1, *β*1, *μ*), using QL and AQL are given in **Table 6**. Conclusion can be drawn based on the standardized residuals

by using GARCH (1,1):

*yt* ¼ *μ* þ *ξt*, *t* ¼ 1, 2, 3, ⋯, *T:* (52)

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>β</sup>*1*σ<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> *<sup>ζ</sup>t*, *<sup>t</sup>* <sup>¼</sup> 1, 2, 3, <sup>⋯</sup>, *<sup>T</sup>:* (53)

*<sup>t</sup>* ; and *ζ<sup>t</sup>* are i.i.d with *E ζ<sup>t</sup>* ð Þ¼ 0 and

QL estimate and AQL denotes the AQL estimate.

size increases.

**Table 4.**

and

*<sup>V</sup> <sup>ζ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup>

**91**

*ζ*.

**4.4 Empirical applications**

*yt* ¼ *log P*ð Þ�*<sup>t</sup> log P*ð Þ *<sup>t</sup>*�<sup>1</sup> and fit *yt*

*σ*2

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

*<sup>ξ</sup><sup>t</sup>* are i.i.d with *<sup>E</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ 0 and *<sup>V</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup>

$$\hat{\alpha}\_{0} = \frac{\sum\_{t=1}^{T} \frac{\hat{\sigma}\_{t}^{2}}{\hat{\sigma}\_{\text{n}}(\sigma\_{t})} - \hat{\alpha}\_{1} \sum\_{t=1}^{T} \frac{\hat{\xi}\_{t-1}^{2}}{\hat{\sigma}\_{\text{n}}(\sigma\_{t})} - \hat{\beta}\_{1} \sum\_{t=1}^{T} \frac{\hat{\sigma}\_{t-1}^{2}}{\hat{\sigma}\_{\text{n}}(\sigma\_{t})}}{\sum\_{t=1}^{T} \frac{1}{\hat{\sigma}\_{\text{n}}(\sigma\_{t})}},\tag{50}$$

and let

$$
\hat{\sigma}\_{\zeta}^{2} = \frac{\sum\_{t=1}^{T} \left(\hat{\zeta}\_{t} - \overline{\hat{\zeta}}\right)^{2}}{T - 1} \tag{51}
$$

where

^*ζ<sup>t</sup>* <sup>¼</sup> *<sup>σ</sup>*^<sup>2</sup> *<sup>t</sup>* � *α*^<sup>0</sup> � *α*^<sup>1</sup> ^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> � *<sup>β</sup>*^1*σ*^<sup>2</sup> *<sup>t</sup>*�1, *t* ¼ 1, 2, 3, ⋯, *T*, *SS<sup>σ</sup>*^<sup>2</sup> *t*�1 ^*ξ* 2 *t*�1 <sup>¼</sup> <sup>X</sup> *T t*¼1 *σ*^2 *t*�1 ^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup> *T t*¼1 1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! � <sup>X</sup> *T t*¼1 *σ*^2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup> *T t*¼1 ^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* !, *SS<sup>σ</sup>*^<sup>2</sup> *t* ^*ξ* 2 *t*�1 <sup>¼</sup> <sup>X</sup> *T t*¼1 *σ*^2 *t* ^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup> *T t*¼1 1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! � <sup>X</sup> *T t*¼1 *σ*^2 *t σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup> *T t*¼1 ^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* !, *SS*^*<sup>ξ</sup>* 2 *t*�1 ^*ξ* 2 *t*�1 <sup>¼</sup> <sup>X</sup> *T t*¼1 1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup> *T t*¼1 ^*ξ* 4 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! � <sup>X</sup> *T t*¼1 ^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* !<sup>2</sup> , *SS<sup>σ</sup>*^<sup>2</sup> *t σ*^2 *<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> <sup>X</sup> *T t*¼1 1 *σ*^*n*ð Þ *σ<sup>t</sup>* !X *T t*¼1 *σ*^2 *t σ*^2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* �<sup>X</sup> *T t*¼1 *σ*^2 *t σ*^*n*ð Þ *σ<sup>t</sup>* X *T t*¼1 *σ*^2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* , *SS<sup>σ</sup>*^<sup>2</sup> *<sup>t</sup>*�1*σ*^<sup>2</sup> *<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> <sup>X</sup> *T t*¼1 1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup> *T t*¼1 *σ*^4 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! � <sup>X</sup> *T t*¼1 *σ*^2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* !<sup>2</sup> *:*

*ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches DOI: http://dx.doi.org/10.5772/intechopen.93726*


**Table 4.**

Second, by kernel estimation method, we find

*T*

0

BBB@

*t*¼1

*<sup>β</sup>*^<sup>1</sup> <sup>¼</sup>

P*<sup>T</sup> t*¼1 *σ*^2 *t <sup>σ</sup>*^*n*ð Þ *<sup>σ</sup><sup>t</sup>* � *<sup>α</sup>*^<sup>1</sup>

*<sup>t</sup>* � *α*^<sup>0</sup> � *α*^<sup>1</sup>

*σ*^2 *t*�1 ^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup>

*σ*^2 *t* ^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup>

1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup>

1 *σ*^*n*ð Þ *σ<sup>t</sup>* !X

1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup>

*α*^<sup>0</sup> ¼

^*ζ<sup>t</sup>* <sup>¼</sup> *<sup>σ</sup>*^<sup>2</sup>

<sup>¼</sup> <sup>X</sup> *T*

<sup>¼</sup> <sup>X</sup> *T*

<sup>¼</sup> <sup>X</sup> *T*

*<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> <sup>X</sup> *T*

*<sup>t</sup>*�<sup>1</sup> <sup>¼</sup> <sup>X</sup> *T*

*t*¼1

*t*¼1

*t*¼1

*t*¼1

*t*¼1

sequence of AQLEF:

Therefore

and let

where

*SS<sup>σ</sup>*^<sup>2</sup> *t*�1 ^*ξ* 2 *t*�1

> *SS<sup>σ</sup>*^<sup>2</sup> *t* ^*ξ* 2 *t*�1

*SS*^*<sup>ξ</sup>* 2 *t*�1 ^*ξ* 2 *t*�1

> *SS<sup>σ</sup>*^<sup>2</sup> *t σ*^2

*SS<sup>σ</sup>*^<sup>2</sup> *<sup>t</sup>*�1*σ*^<sup>2</sup>

**90**

*GT <sup>μ</sup>*, *<sup>α</sup>*0, *<sup>α</sup>*1, *<sup>β</sup>*<sup>1</sup> ð Þ¼ <sup>X</sup>

<sup>Σ</sup>^*t*,*<sup>n</sup> <sup>θ</sup>*ð Þ <sup>0</sup> � �

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

Third, to estimate the parameters *<sup>θ</sup>* <sup>¼</sup> *<sup>μ</sup>*, *<sup>α</sup>*0, *<sup>α</sup>*1, *<sup>β</sup>*<sup>1</sup> ð Þ using *<sup>σ</sup>*^<sup>2</sup>

�1 0 0 �1 <sup>0</sup> �^*<sup>ξ</sup>* 2 *t*�1

<sup>0</sup> �*σ*^<sup>2</sup>

*<sup>μ</sup>*^ <sup>¼</sup> <sup>X</sup> *T*

> *SS*<sup>2</sup> *σ*^2 *t*�1 ^*ξ* 2 *t*�1

*α*^<sup>1</sup> ¼

*σ*^2 *<sup>ζ</sup>* ¼

^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> � *<sup>β</sup>*^1*σ*^<sup>2</sup>

*SSσ*^<sup>2</sup> *t*�1 ^*ξ* 2 *t*�1 *SSσ*^<sup>2</sup> *t* ^*ξ* 2 *t*�1 � *SS*^*<sup>ξ</sup>* 2 *t*�1 ^*ξ* 2 *t*�1 *SSσ*^<sup>2</sup> *t σ*^2 *t*�1

*t*¼1

*SSσ*^<sup>2</sup> *t* ^*ξ* 2 *t*�1

*t*�1

*yt σ*^*<sup>n</sup> yt* � � *<sup>=</sup>*

The AQL estimate of *μ*, *α*0, *α*1, and *β*<sup>1</sup> is the solation of *GT μ*, *α*0, *α*1, *β*<sup>1</sup> ð Þ¼ 0.

� *SSσ*^<sup>2</sup> *<sup>t</sup>*�1*σ*^<sup>2</sup> *t*�1 *SS*^*<sup>ξ</sup>* 2 *t*�1 ^*ξ* 2 *t*�1

*SS*^*<sup>ξ</sup>* 2 *t*�1 ^*ξ* 2 *t*�1

P*<sup>T</sup> t*¼1 ^*ξ* 2 *t*�1 *<sup>σ</sup>*^*n*ð Þ *<sup>σ</sup><sup>t</sup>* � *<sup>β</sup>*^<sup>1</sup>

P*<sup>T</sup> t*¼1 1 *σ*^*n*ð Þ *σ<sup>t</sup>*

P*<sup>T</sup>*

*T*

1 *σ*^*n*ð Þ *σ<sup>t</sup>*

!

1 *σ*^*n*ð Þ *σ<sup>t</sup>*

!

^*ξ* 4 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>*

*σ*^4 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>*

!

!

*t*¼1

*T*

*t*¼1

*T*

*t*¼1

*σ*^2 *t σ*^2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>*

*T*

*t*¼1

*T*

*t*¼1

*<sup>t</sup>*¼<sup>1</sup> ^*ζ<sup>t</sup>* � ^*<sup>ζ</sup>* � �<sup>2</sup>

1

CCCA <sup>Σ</sup>^�<sup>1</sup> *t*,*n*

> X *T*

> > *t*¼1

� *<sup>β</sup>*^1*SSσ*^<sup>2</sup>

*t*�1 ^*ξ* 2 *t*�1

*<sup>t</sup>*�1, *t* ¼ 1, 2, 3, ⋯, *T*,

� <sup>X</sup> *T*

� <sup>X</sup> *T*

� <sup>X</sup> *T*

� <sup>X</sup> *T*

*t*¼1

�<sup>X</sup> *T*

*t*¼1

*t*¼1

*t*¼1

*σ*^2 *t σ*^*n*ð Þ *σ<sup>t</sup>*

*t*¼1

*σ*^2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup>

*σ*^2 *t σ*^*n*ð Þ *σ<sup>t</sup>* ! <sup>X</sup>

> ^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>*

> > X *T*

> > > *t*¼1

*σ*^2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>*

!<sup>2</sup>

!<sup>2</sup>

P*<sup>T</sup> t*¼1 *σ*^2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>*

*<sup>T</sup>* � <sup>1</sup> (51)

*T*

^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>*

,

,

!

*t*¼1

^*ξ* 2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>*

!

*T*

*t*¼1

,

*:*

*σ*^2 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>* ,

<sup>¼</sup> *<sup>σ</sup>*^*<sup>n</sup> yt*

� � 0 0 *σ*^*n*ð Þ *σ<sup>t</sup>* � �

*σ*2

1 *σ*^*<sup>n</sup> yt* *:*

*<sup>t</sup>* � *<sup>α</sup>*<sup>0</sup> � *<sup>α</sup>*1*ξ*<sup>2</sup>

*t* � � and *yt*

*yt* � *μ*

� �

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

� � *:* (47)

*:* (48)

, (50)

*:* (49)

*t*�1

� � and the

*:*

*The QL and AQL estimates and the RMSE of each estimate is stated below that estimate for GARCH model.*

The estimation procedure will be iteratively repeated until it converges.

For each parameter setting, T = 500 observations are simulated from the true model. We then replicate the experiment for 1000 times to obtain the mean and root mean squared errors (RMSE) for *α*^0, *α*^1, *β*^1, and *μ*^. In **Table 4**, QL denotes the QL estimate and AQL denotes the AQL estimate.

We generated N = 1000 independent random samples of size T = 20, 40, 60, 80, and 100 from GARCH(1,1) model. In **Table 5**, The QL and AQL estimation methods show the property of consistency, and the RMSE decreases as the sample size increases.

## **4.4 Empirical applications**

The second set of data is the weekly price changes of crude oil prices *Pt*. The *Pt* of Cushing, OK, West Texas Intermediate (US dollars per barrel) is considered for the period from 7/1/2000 to 10/6/2016, with 858 observations in total. The data are transformed into rates of change by taking the first difference of the logs. Thus, *yt* ¼ *log P*ð Þ�*<sup>t</sup> log P*ð Þ *<sup>t</sup>*�<sup>1</sup> and fit *yt* by using GARCH (1,1):

$$y\_t = \mu + \xi\_t, \qquad t = 1, 2, 3, \dots, T. \tag{52}$$

and

$$
\sigma\_t^2 = a\_0 + a\_1 \xi\_{t-1}^2 + \beta\_1 \sigma\_{t-1} + \zeta\_t, \qquad t = 1, 2, 3, \cdots, T. \tag{53}
$$

*<sup>ξ</sup><sup>t</sup>* are i.i.d with *<sup>E</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ 0 and *<sup>V</sup> <sup>ξ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup> *<sup>t</sup>* ; and *ζ<sup>t</sup>* are i.i.d with *E ζ<sup>t</sup>* ð Þ¼ 0 and *<sup>V</sup> <sup>ζ</sup><sup>t</sup>* ð Þ¼ *<sup>σ</sup>*<sup>2</sup> *ζ*.

The estimation of unknown parameters, (*α*0, *α*1, *β*1, *μ*), using QL and AQL are given in **Table 6**. Conclusion can be drawn based on the standardized residuals


on different scenarios in which the conditional covariance of the error's terms are assumed to be known or unknown. Simulation studies and empirical analysis show that our proposed estimation methods work reasonably quite well for parameter estimation of ARCH and GARCH models. It will provide a robust tool for obtaining an optimal point estimate of parameters in heteroscedastic models like ARCH and

*ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches*

This chapter focuses on models in univariate, while it is desirable to consider

The author would like to acknowledge the helpful comments and suggestion of the editor. This study is conducted in the King Faisal University, Saudi Arabia, during the sabbatical year of the author from the Al Balqa Applied University,

1 Department of Mathematics, Faculty of Science, Al-Balqa Applied University,

2 Department of Quantitative Methods, School of Business, King Faisal University,

\*Address all correspondence to: raedalzghool@bau.edu.jo; ralzghool@kfu.edu.sa

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

GARCH models.

Jordan.

**Author details**

Raed Alzghool1,2

Al-Ahsa, Saudi Arabia

provided the original work is properly cited.

Salt, Jordan

**93**

**Acknowledgements**

multivariate extensions of the proposed models.

*DOI: http://dx.doi.org/10.5772/intechopen.93726*

#### **Table 5.**

*The QL and AQL estimates and the RMSE of each estimate is stated below that estimate for GARCH model with different sample size.*


#### **Table 6.**

*Estimation of μ*, *α*0, *α*1, *β*<sup>1</sup> *for the rates of change prices data.*

from the fourth column in **Table 6**, which favors the QL method and gives smaller standardized residuals, better than AQL method.

## **5. Conclusions**

In this chapter, two alternative approaches, QL and AQL, have been developed to estimate the parameters in ARCH and GARCH models. Parameter estimation for ARCH and GARCH models, which include nonlinear and non-Gaussian models is given. The estimations of unknown parameters are considered without any distribution assumptions concerning the processes involved, and the estimation is based

*ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches DOI: http://dx.doi.org/10.5772/intechopen.93726*

on different scenarios in which the conditional covariance of the error's terms are assumed to be known or unknown. Simulation studies and empirical analysis show that our proposed estimation methods work reasonably quite well for parameter estimation of ARCH and GARCH models. It will provide a robust tool for obtaining an optimal point estimate of parameters in heteroscedastic models like ARCH and GARCH models.

This chapter focuses on models in univariate, while it is desirable to consider multivariate extensions of the proposed models.
