**3.1 Parameter estimation of ARCH(q) model using the QL method**

The ARCH(q) process is defined by

$$y\_t = \mu + \xi\_t, \qquad t = 1, 2, 3, \dots, T. \tag{10}$$

The QL estimate of *μ*, *α*0, *α*1, ⋯, *α<sup>q</sup>* is the solution of *GT μ*, *α*0, *α*1, ⋯, *α<sup>q</sup>*

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

P*<sup>T</sup>*

� � is an initial value in the iterative procedure.

For ARCH(q) model given by Eqs. (10) and (11) and using the same argument

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

*t*

� � *<sup>σ</sup>*^*<sup>n</sup> yt*

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

^*ξ* 2

*σ*^*<sup>n</sup> σt*, *yt*

*T*

0

BBBBBBBB@

*t*¼1

*yt* � *μ*

The estimation of ARCH(1) model using QL and AQL methods are considered in

!

**3.2 Parameter estimation of ARCH(q) model using the AQL method**

*t*,*n*

� �, Σð Þ <sup>0</sup>

*<sup>t</sup>*�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *<sup>α</sup><sup>q</sup>*

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

� � <sup>¼</sup> <sup>X</sup>

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

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

�

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

**3.3 Simulation studies for the ARCH(1) model**

simulation studies. The ARCH(1) process is defined by

*σ*2

estimation procedure will be iteratively repeated until it converges.

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

^*ξ* 2 *σ*2

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

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

*yt* � *μ*

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

� � <sup>¼</sup> 0, that is,

!

^*ξ* 2

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>

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

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

^*ξ* 2

*ζ*

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

listed under Eq. (11). First, to estimate *σ*<sup>2</sup>

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, *<sup>α</sup>*1, <sup>⋯</sup>, *<sup>α</sup><sup>q</sup>*

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

Second, by kernel estimation method, we find

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

*σ*^2

and the sequence of (AQLEF):

**81**

AQL estimation of *σ*<sup>2</sup>

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

*σ*^2 *<sup>ζ</sup>* ¼ � � <sup>¼</sup> 0,

*<sup>T</sup>* � <sup>1</sup> (14)

*<sup>t</sup>* , so the sequence of (AQLEF) is given by

*t*�*q*

*<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>*�*<sup>q</sup>*, *t* ¼ 1, 2, 3⋯, *T:* (15)

*:*

1

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

*:*

*t*�*q*

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

*yt* ¼ *μ* þ *ξt*, *t* ¼ 1, 2, 3, ⋯, *T:* (16)

*t* � � and *yt*

, then the

� �

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

2

, *σ<sup>t</sup>* � �

� � using *σ*^<sup>2</sup>

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

⋮ ⋮ <sup>0</sup> �*ξ*<sup>2</sup> *t*�*q*

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

and

$$
\sigma\_t^2 = a\_0 + a\_1 \mathfrak{f}\_{t-1}^2 + \dots + a\_q \mathfrak{f}\_{t-q}^2 + \zeta\_t, \qquad t = 1, 2, 3, \dots, T. \tag{11}
$$

*<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

$$
\begin{pmatrix} \xi\_t \\ \zeta\_t \end{pmatrix} = \begin{pmatrix} \mathbb{y}\_t - \mu \\ \sigma\_t^2 - a\_0 - a\_1 \xi\_{t-1}^2 - \dots - a\_q \xi\_{t-q}^2 \end{pmatrix}.
$$

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

$$\mathcal{G}\_{(t)}\left(\sigma\_t^2\right) = (0, 1) \begin{pmatrix} \sigma\_t^2 & 0\\ & \\ 0 & \sigma\_\zeta^2 \end{pmatrix}^{-1} \begin{pmatrix} y\_t - \mu\\ \sigma\_t^2 - a\_0 - a\_1 \mathfrak{k}\_{t-1}^2 - \dots - a\_q \mathfrak{k}\_{t-q}^2 \end{pmatrix} \tag{12}$$

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

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>q</sup>*<sup>0</sup> , *<sup>σ</sup>*<sup>2</sup> *ζ*0 � � and ^*<sup>ξ</sup>* 2 *<sup>t</sup>*�<sup>1</sup> ¼ *yt*�<sup>1</sup> � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup> , then the QL estimation of *σ*<sup>2</sup> *<sup>t</sup>* is the solution of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t* � � <sup>¼</sup> 0:

$$
\hat{\sigma}\_t^2 = a\_0 + a\_1 \hat{\xi}\_{t-1}^2 + \dots + a\_q \hat{\xi}\_{t-q}^2, \qquad t = 1, 2, 3 \dots, T. \tag{13}
$$

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

$$\mathbf{G}\_{T}(\mu, a\_{0}, a\_{1}, \cdots, a\_{q}) = \sum\_{t=1}^{T} \begin{pmatrix} -\mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \\ \mathbf{0} & -\xi\_{t-1}^{2} \\ \vdots & \vdots \\ \mathbf{0} & -\xi\_{t-q}^{2} \end{pmatrix} \begin{pmatrix} \sigma\_{t}^{2} & \mathbf{0} \\ \mathbf{0} & \sigma\_{\zeta\_{0}}^{2} \end{pmatrix}^{-1}$$
 
$$\times \begin{pmatrix} \boldsymbol{\gamma}\_{t} - \boldsymbol{\mu} \\ \sigma\_{t}^{2} - a\_{0} - a\_{1}\xi\_{t-1}^{2} - a\_{q}\xi\_{t-q}^{2} \end{pmatrix}.$$

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

The QL estimate of *μ*, *α*0, *α*1, ⋯, *α<sup>q</sup>* is the solution of *GT μ*, *α*0, *α*1, ⋯, *α<sup>q</sup>* � � <sup>¼</sup> 0, 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>⋯</sup> � *<sup>α</sup>*^*<sup>q</sup>* ^*ξ* 2 *<sup>t</sup>*�*q*, *t* ¼ 1, 2, 3, ⋯, *T* and *σ*^2 *<sup>ζ</sup>* ¼ P*<sup>T</sup> <sup>t</sup>*¼<sup>1</sup> ^*ζ<sup>t</sup>* � ^*<sup>ζ</sup>* � �<sup>2</sup> *<sup>T</sup>* � <sup>1</sup> (14)

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

#### **3.2 Parameter estimation of ARCH(q) model using the AQL method**

For ARCH(q) model given by Eqs. (10) and (11) and using the same argument listed under Eq. (11). First, to estimate *σ*<sup>2</sup> *<sup>t</sup>* , so the sequence of (AQLEF) is given by

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

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, *<sup>α</sup>*1, <sup>⋯</sup>, *<sup>α</sup><sup>q</sup>* � �, Σð 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>1</sup> <sup>¼</sup> *yt*�<sup>1</sup> � *<sup>μ</sup>*<sup>0</sup> � �<sup>2</sup> , then the AQL estimation of *σ*<sup>2</sup> *<sup>t</sup>* is the solution of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t* � � <sup>¼</sup> 0, that is,

$$
\hat{\sigma}\_t^2 = a\_0 + a\_1 \hat{\mathfrak{z}}\_{t-1}^2 + \dots + a\_q \hat{\mathfrak{z}}\_{t-q}^2, \qquad t = 1, 2, 3 \dots, T. \tag{15}
$$

Second, by kernel estimation method, we find

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

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):

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

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

#### **3.3 Simulation studies for the ARCH(1) model**

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

$$y\_t = \mu + \xi\_t, \qquad t = 1, 2, 3, \dots, T. \tag{16}$$

**3. Parameter estimation of ARCH(q) model using the QL and AQL**

**3.1 Parameter estimation of ARCH(q) model using the QL method**

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

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

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

*σ*2

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

1 A

�1

*<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><sup>q</sup>*

*T*

0

BBBBBBBBBB@

*yt* � *μ*

*t*¼1

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

0 @

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

*σ*2

� �

^*ξ* 2

�1 0

0 �1

<sup>0</sup> �*ξ*<sup>2</sup> *t*�1

⋮ ⋮

<sup>0</sup> �*ξ*<sup>2</sup> *t*�*q*

*<sup>t</sup>*�<sup>1</sup> � *<sup>α</sup>qξ*<sup>2</sup>

The ARCH(q) process is defined by

*<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>

*ξt ζt*

¼

*σ*2 *<sup>t</sup>* 0

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

0 @

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

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

*t* � � and *yt*

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

� � <sup>¼</sup> <sup>X</sup>

�

0 @

*σ*2

0 *σ*<sup>2</sup> *ζ*

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>q</sup>*<sup>0</sup> , *<sup>σ</sup>*<sup>2</sup>

^*ξ* 2

!

*σ*2

The QLEF to estimate *σ*<sup>2</sup>

*σ*^2

The QLEF, using *σ*^<sup>2</sup>

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

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

given by

**80**

In this section, we will develop the estimation of ARCH model using QL and

*yt* ¼ *μ* þ *ξt*, *t* ¼ 1, 2, 3, ⋯, *T:* (10)

*yt* � *μ*

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

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

� �

!

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

*yt* � *μ*

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

*<sup>t</sup>*�*<sup>q</sup>* <sup>þ</sup> *<sup>ζ</sup>t*, *<sup>t</sup>* <sup>¼</sup> 1, 2, 3, <sup>⋯</sup>, *<sup>T</sup>:* (11)

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

*t*�*q*

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

*:*

*ζ*0

*t*�*q*

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

1

CCCCCCCCCCA

*t*�*q*

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

0 @

1 A*:* *:*

*t*�*q*

and ^*ξ* 2 *<sup>t</sup>*�<sup>1</sup> ¼

> *t* � � <sup>¼</sup> 0:

> > 1 A

�1

*<sup>t</sup>*�*<sup>q</sup>*, *t* ¼ 1, 2, 3⋯, *T:* (13)

1 A

(12)

**methods**

AQL methods.

and

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

and

$$
\sigma\_t^2 = a\_0 + a\_1 \xi\_{t-1}^2 + \zeta\_t, \qquad t = 1, 2, 3, \cdots, T. \tag{17}
$$

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

(AQLEF) is given by

AQL estimation of *σ*<sup>2</sup>

sequence of AQLEF:

*α*^<sup>1</sup> ¼

and let

**83**

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

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

> > *α*^<sup>0</sup> ¼

*ζ*

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

� � is an initial value in the iterative procedure.

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

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

*3.3.2 Parameter estimation of ARCH(1) model using the AQL method*

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

*<sup>t</sup>* is the solution of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<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 *<sup>θ</sup>* <sup>¼</sup> ð Þ *<sup>μ</sup>*, *<sup>α</sup>*0, *<sup>α</sup>*<sup>1</sup> using *<sup>σ</sup>*^<sup>2</sup>

0

BB@

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

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

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

> > *σ*^2 *<sup>ζ</sup>* ¼

*t*¼1

X *T*

*t*¼1

^*ξ* 2

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

argument listed under Eq. (17). First, we need to estimate *σ*<sup>2</sup>

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

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, *<sup>α</sup>*1, *<sup>μ</sup>*<sup>0</sup> ð Þ, <sup>Σ</sup>ð Þ <sup>0</sup>

*σ*^2

*GT*ð Þ¼ *μ*, *α*0, *α*<sup>1</sup>

Second, by kernel estimation method, we find

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

Considering the ARCH(1) model given by Eqs. (16) and (17) and using the same

*σ*2

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

� � *<sup>σ</sup>*^*<sup>n</sup> yt*

!

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

1

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

X *T*

1 *σ*^*<sup>n</sup> yt*

> *t*¼1 ^*ξ* 4 *t*�1 *σ*^*n*ð Þ *σ<sup>t</sup>*

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

*t*¼1

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

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

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

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

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

P*<sup>T</sup>*

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

*σ*^*<sup>n</sup> σt*, *yt*

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

*yt* � *μ*

� �

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

� � <sup>¼</sup> 0, that is,

2

*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:* (24)

*:*

*t* � � and *yt*

*yt* � *μ*

� �*:*

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

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

*<sup>T</sup>* � <sup>1</sup> (28)

*t*�1

� � *:* (25)

� � *:* (26)

*:* (27)

, *σ<sup>t</sup>* � �

*σ*2

*t*,*n*

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

, then the

� � and the

*<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*.

#### *3.3.1 Parameter estimation of ARCH(1) model using the QL method*

For ARCH(1) given by Eqs. (16) and (17), the martingale difference is

$$
\begin{pmatrix} \xi\_t \\ \zeta\_t \end{pmatrix} = \begin{pmatrix} \wp\_t - \mu \\ \sigma\_t^2 - a\_0 - a\_1 \xi\_{t-1}^2 \end{pmatrix}.
$$

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

$$\begin{split} G\_{(t)}\left(\sigma\_{t}^{2}\right) &= (\mathbf{0}, \mathbf{1}) \begin{pmatrix} \sigma\_{t}^{2} & \mathbf{0} \\ \mathbf{0} & \sigma\_{\zeta}^{2} \end{pmatrix}^{-1} \begin{pmatrix} y\_{t} - \mu \\ \sigma\_{t}^{2} - a\_{0} - a\_{1}\xi\_{t-1}^{2} \end{pmatrix} \\ &= \sigma\_{\zeta}^{-2} \left(\sigma\_{t}^{2} - a\_{0} - a\_{1}\xi\_{t-1}^{2}\right). \end{split} \tag{18}$$

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 � � and ^*<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> , then the QL estimation of *σ*<sup>2</sup> *<sup>t</sup>* is the solution of *<sup>G</sup>*ð Þ*<sup>t</sup> <sup>σ</sup>*<sup>2</sup> *t* � � <sup>¼</sup> 0,

$$
\hat{\sigma}\_t^2 = a\_0 + a\_1 \hat{\xi}\_{t-1}^2, \qquad t = 1, 2, 3 \cdots, T. \tag{19}
$$

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

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

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

$$\hat{\mu} = \sum\_{t=1}^{T} \frac{\mathcal{Y}\_t}{\hat{\sigma}\_t^2} / \sum\_{t=1}^{T} \frac{\mathbf{1}}{\hat{\sigma}\_t^2}. \tag{20}$$

$$\hat{\alpha}\_{1} = \frac{T\sum\_{t=1}^{T} \hat{\sigma}\_{t}^{2} \hat{\xi}\_{t-1}^{2} - \sum\_{t=1}^{T} \hat{\sigma}\_{t}^{2} \sum\_{t=1}^{T} \hat{\xi}\_{t-1}^{2}}{T\sum\_{t=1}^{T} \hat{\xi}\_{t-1}^{4} - \left(\sum\_{t=1}^{T} \hat{\xi}\_{t-1}^{2}\right)^{2}}. \tag{21}$$

$$\hat{a}\_0 = \frac{\sum\_{t=1}^T \hat{\sigma}\_t^2 - \hat{a}\_1 \sum\_{t=1}^T \hat{\xi}\_{t-1}^2}{T}. \tag{22}$$

and let

$$
\hat{\sigma}\_{\zeta}^{2} = \frac{\sum\_{t=1}^{T} \left(\hat{\zeta}\_{t} - \overline{\hat{\zeta}}\right)^{2}}{T - 1} \tag{23}
$$

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>*�1, *t* ¼ 1, 2, 3, ⋯, *T*. *ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches DOI: http://dx.doi.org/10.5772/intechopen.93726*

*<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].

## *3.3.2 Parameter estimation of ARCH(1) model using the AQL method*

Considering the ARCH(1) model given by Eqs. (16) and (17) and using the same argument listed under Eq. (17). First, we need to estimate *σ*<sup>2</sup> *<sup>t</sup>* , so the sequence of (AQLEF) is given by

$$G\_{(t)}\left(\sigma\_t^2\right) = (\mathbf{0}, \mathbf{1})\Sigma\_{\mathbf{t}, \mathbf{u}}^{-1} \binom{\mathcal{Y}\_\mathbf{t} - \mu}{\sigma\_\mathbf{t}^2 - a\_0 - a\_1 \mathfrak{s}\_{t-1}^2}$$

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

$$
\hat{\sigma}\_t^2 = a\_0 + a\_1 \hat{\xi}\_{t-1}^2, \qquad t = 1, 2, 3 \cdots, T. \tag{24}
$$

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

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

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

$$\hat{\mu} = \sum\_{t=1}^{T} \frac{\mathcal{Y}\_t}{\hat{\sigma}\_n(\mathcal{Y}\_t)} / \sum\_{t=1}^{T} \frac{1}{\hat{\sigma}\_n(\mathcal{Y}\_t)}. \tag{25}$$

$$\hat{\alpha}\_{1} = \frac{\left(\sum\_{t=1}^{T} \frac{\dot{\sigma}\_{t}^{2}}{\dot{\sigma}\_{n}(\sigma\_{t})}\right)\left(\sum\_{t=1}^{T} \frac{\dot{\xi}\_{t-1}^{2}}{\dot{\sigma}\_{n}(\sigma\_{t})}\right) - \left(\sum\_{t=1}^{T} \frac{1}{\dot{\sigma}\_{n}(\sigma\_{t})}\right)\left(\sum\_{t=1}^{T} \frac{\dot{\xi}\_{t-1}^{2}}{\dot{\sigma}\_{n}(\sigma\_{t})}\right)}{\left(\sum\_{t=1}^{T} \frac{\dot{\xi}\_{t-1}^{2}}{\dot{\sigma}\_{n}(\sigma\_{t})}\right)^{2} - \left(\sum\_{t=1}^{T} \frac{1}{\dot{\sigma}\_{n}(\sigma\_{t})}\right)\left(\sum\_{t=1}^{T} \frac{\dot{\xi}\_{t-1}^{4}}{\dot{\sigma}\_{n}(\sigma\_{t})}\right)}}. \tag{26}$$

$$\hat{\alpha}\_{0} = \frac{\left(\sum\_{t=1}^{T} \frac{\dot{\sigma}\_{t}^{2}}{\hat{\sigma}\_{n}(\sigma\_{t})}\right) - \hat{\alpha}\_{1} \left(\sum\_{t=1}^{T} \frac{\dot{\xi}\_{t-1}^{2}}{\hat{\sigma}\_{n}(\sigma\_{t})}\right)}{\sum\_{t=1}^{T} \frac{1}{\hat{\sigma}\_{n}(\sigma\_{t})}}. \tag{27}$$

and let

$$
\hat{\sigma}\_{\zeta}^{2} = \frac{\sum\_{t=1}^{T} \left(\hat{\zeta}\_{t} - \overline{\hat{\zeta}}\right)^{2}}{T - 1} \tag{28}
$$

and

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

*ζ*.

The QLEF to estimate *σ*<sup>2</sup>

the QL estimation of *σ*<sup>2</sup>

*GT*ð Þ¼ *μ*, *α*0, *α*<sup>1</sup>

and let

**82**

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

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

*σ*2

*<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>* <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*

*3.3.1 Parameter estimation of ARCH(1) model using the QL method*

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

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

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

^*ξ* 2

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

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

*t*¼1

P*<sup>T</sup>*

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

1 CA *<sup>σ</sup>*<sup>2</sup>

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

*t*¼1

*<sup>t</sup>*�<sup>1</sup> � <sup>P</sup>*<sup>T</sup>*

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

1 *σ*^2 *t*

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

> *t*¼1 ^*ξ* 2 *t*�1

P*<sup>T</sup> t*¼1 ^*ξ* 2 *t*�1

*yt σ*^2 *t =* X *T*

*ξt ζt* � �

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

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

0

B@

*<sup>α</sup>*^<sup>1</sup> <sup>¼</sup> *<sup>T</sup>*

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

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

*T* P*<sup>T</sup> t*¼1 ^*ξ* 4

*α*^<sup>0</sup> ¼

*σ*^2 *<sup>ζ</sup>* ¼

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

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

*σ*^2

X *T*

*t*¼1

For ARCH(1) given by Eqs. (16) and (17), the martingale difference is

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

!�<sup>1</sup>

<sup>¼</sup> *yt* � *<sup>μ</sup> σ*2

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

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

� �*:*

� �

*t* � � <sup>¼</sup> 0,

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

!�<sup>1</sup>

� �

*t*�1

*σ*2

*t*�1

*ζ*0

*t* � � and *yt*

*:*

*yt* � *μ*

!

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

and ^*ξ* 2

*<sup>t</sup>*�1, *t* ¼ 1, 2, 3⋯, *T:* (19)

*σ*2

*t*�1

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

*yt* � *μ*

� �

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

*:* (20)

� �<sup>2</sup> *:* (21)

*<sup>T</sup> :* (22)

*<sup>T</sup>* � <sup>1</sup> (23)

� �, the QLEF is given by

*t*�1

*:*

(18)

, then

*<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>:* (17)

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

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, and *μ*^. In **Table 1**, 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 ARCH(1) model. In **Table 2**, the QL and AQL estimation methods show the property of consistency, the RMSE decreases as the sample size increases.

#### **3.4 Empirical applications**

The first data set we analyze are the daily exchange rate of *rt* ¼ *AUD=USD* (Australian dollar/US dollar) for the period from 5/6/2010 to 5/5/2016, 1590 observations in total. The ARCH model (Eqs. (16) and (17)) is used to model *yt* ¼ *log r*ð Þ�*<sup>t</sup> log r*ð Þ *<sup>t</sup>*�<sup>1</sup> .

We used the S + FinMetrics function archTest to carry out Lagrange multiplier (ML) test for the presence of ARCH effects in the residuals (see [35]). For *rt* the pvalues are significant (<0*:*05 level), so reject the null hypothesis that there are no ARCH effects and we fit *yt* by following models:

$$y\_t = \mu + \xi\_t, \qquad t = 1, 2, 3, \dots, T. \tag{29}$$

and

*different sample size.*

**Table 2.**

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

**Table 3.**

**85**

*ζ*.

*σ*2

residuals, better than AQL method.

*<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>

*Estimation of α*0, *α*1, *μ for the exchange rate pound/dollar data.*

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

*<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>:* (30)

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

0.0008 0.0100 0.0319 0.0005 0.0015 0.0703

0.0009 0.010 0.0084 0.0005 0.0015 0.0213

0.00089 0.010 0.0223 0.0005 0.0015 0.0492

0.00089 0.010 0.0039 0.0005 0.0015 0.0143

0.0009 0.010 0.0180 0.0005 0.0015 0.0404

0.0009 0.010 0.0027 0.0005 0.0015 0.0128

0.0009 0.010 0.016 0.0005 0.0015 0.0353

0.0009 0.010 0.0020 0.0005 0.0015 0.0119

0.0009 0.010 0.0142 0.0005 0.0015 0.0314

0.0009 0.010 0.0018 0.0005 0.0015 0.0116

QL 0.009 0.990 �0.029 0.0495 0.9485 1.300

AQL 0.009 0.990 �0.031 0.0495 0.9485 1.3107

AQL 0.009 0.990 �0.031 0.0495 0.9485 1.3113

AQL 0.009 0.990 �0.031 0.0495 0.9485 1.311

AQL 0.009 0.990 �0.310 0.0495 0.9485 1.3112

AQL 0.009 0.990 �0.031 0.0495 0.9485 1.3111

T = 20 True 0.010 0.980 �0.030 0.05 0.950 1.3

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

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

T = 40 QL 0.009 0.990 �0.031 0.0495 0.9485 1.3015

T = 60 QL 0.009 0.990 �0.029 0.0495 0.9485 1.300

T = 80 QL 0.009 0.990 �0.029 0.0490 0.9485 1.300

T = 100 QL 0.009 0.990 0.0292 0.0495 0.9485 1.3017

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

The estimation of unknown parameters, (*α*0, *α*1, *μ*), using QL and AQL are given in **Table 3**. Conclusion can be drawn based on the standardized residuals from the fourth column in **Table 3**, which favors the QL method, gives smaller standardized

QL 0.1300 0.8387 �0.00012 0.00013 AQL 0.0200 0.9599 �0.00111 0.1350

*α*^**<sup>0</sup>** *α*^**<sup>1</sup>** *μ*^ ^*ξ<sup>t</sup>*

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

*<sup>S</sup>:<sup>d</sup>* ^*<sup>ξ</sup>* ð Þ*<sup>t</sup>*


**Table 1.** *The QL and AQL estimates and the RMSE of each estimate is stated below that estimate for ARCH model.*


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

#### **Table 2.**

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, and *μ*^. In **Table 1**, QL denotes the QL

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

We generated N = 1000 independent random samples of size T = 20, 40, 60, 80, and 100 from ARCH(1) model. In **Table 2**, the QL and AQL estimation methods show the property of consistency, the RMSE decreases as the sample size increases.

The first data set we analyze are the daily exchange rate of *rt* ¼ *AUD=USD* (Australian dollar/US dollar) for the period from 5/6/2010 to 5/5/2016, 1590 observations in total. The ARCH model (Eqs. (16) and (17)) is used to model

We used the S + FinMetrics function archTest to carry out Lagrange multiplier (ML) test for the presence of ARCH effects in the residuals (see [35]). For *rt* the pvalues are significant (<0*:*05 level), so reject the null hypothesis that there are no

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

0.001 0.010 0.006 0.001 0.010 0.006 0.001 0.010 0.006

0.001 0.010 0.0003 0.002 0.009 0.0003 0.001 0.009 0.0003

0.001 0.0001 0.014 0.001 0.010 0.014 0.001 0.010 0.014

0.001 0.010 0.018 0.001 0.010 0.018 0.001 0.01 0.001

0.002 0.010 0.019 0.002 0.010 0.020 0.002 0.010 0.029

0.002 0.010 0.012 0.002 0.010 0.021 0.001 0.010 0.001

0.002 0.010 0.019 0.001 0.001 0.014 0.001 0.016 0.006

0.002 0.010 0.001 0.001 0.001 0.002 0.001 0.010 0.001

True 0.010 0.980 1.30 0.010 0.980 �1.30 0.010 0.980 0.030 QL 0.009 0.989 1.299 0.009 0.989 �1.30 0.009 0.989 0.029

AQL 0.009 0.989 1.30 0.009 0.989 �1.29 0.009 0.989 0.030

True 0.050 0.950 1.30 0.050 0.950 �1.30 0.050 .950 0.030 QL 0.049 0.949 1.29 0.049 0.940 �1.30 0.049 0.94 0.029

AQL 0.049 0.940 1.32 0.049 0.940 �1.30 0.049 0.940 0.032

True 0.10 0.90 1.30 0.10 0.90 �1.30 0.10 0.90 0.030 QL 0.098 0.910 1.29 0.098 0.910 �1.30 0.098 0.910 0.023

AQL 0.098 0.910 1.31 0.098 0.910 �1.32 0.098 0.910 0.031

True 0.1 0.90 �0.03 0.05 0.95 �0.03 0.01 0.98 �0.03 QL 0.098 0.910 �0.031 0.051 0.949 �0.030 0.009 0.990 �0.030

AQL 0.098 0.910 �0.031 0.051 0.949 �0.031 0.009 0.990 �0.031

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

*yt* ¼ *μ* þ *ξt*, *t* ¼ 1, 2, 3, ⋯, *T:* (29)

by following models:

estimate and AQL denotes the AQL estimate.

**3.4 Empirical applications**

*yt* ¼ *log r*ð Þ�*<sup>t</sup> log r*ð Þ *<sup>t</sup>*�<sup>1</sup> .

ARCH effects and we fit *yt*

**Table 1.**

**84**

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

and

$$
\sigma\_t^2 = a\_0 + a\_1 \mathfrak{f}\_{t-1}^2 + \zeta\_t, \qquad t = 1, 2, 3, \cdots, T. \tag{30}
$$

*<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, *μ*), using QL and AQL are given in **Table 3**. Conclusion can be drawn based on the standardized residuals from the fourth column in **Table 3**, which favors the QL method, gives smaller standardized residuals, better than AQL method.


**Table 3.**

*Estimation of α*0, *α*1, *μ for the exchange rate pound/dollar data.*
