**1. Introduction**

The autoregressive conditional heteroscedasticity (ARCH(q)) process is defined by

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

and

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

*<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>* ; 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 estimation and applications of ARCH models, see [1–19]. Moreover, ARCH models have now become the standard textbook material in econometrics and finance as exemplified by, for example, [20–23].

The generalized autoregressive conditional heteroscedasticity (GARCH(p,q)) process *yt* is defined by

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

and

$$
\sigma\_t^2 = a\_0 + a\_1 \mathfrak{f}\_{t-1}^2 + \dots + a\_p \mathfrak{f}\_{t-p}^2 + \beta\_1 \sigma\_{t-1}^2 + \dots + \beta\_q \sigma\_{t-q}^2, \qquad t = 1, 2, 3, \dots, T. \tag{4}
$$

If the sub-estimating function spaces of G*<sup>T</sup>* are considered as follows:

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

then the QLEF can be defined by

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

**G**<sup>∗</sup>

ð Þ*<sup>t</sup>* ð Þ¼ *<sup>θ</sup>* 0.

**2.2 The AQL method**

[26] proposed the AQL method. Definition 2.2.1: *Let* **G**<sup>∗</sup>

> *EG*\_ *<sup>T</sup>* � ��<sup>1</sup>

sequence estimate *θ* <sup>∗</sup>

**79**

tors, the AQL equation becomes

**G**<sup>∗</sup>

assuming that the covariance matrix Σ*<sup>t</sup>* is unknown.

ð Þ *<sup>E</sup>GTG<sup>T</sup>* <sup>0</sup> *<sup>E</sup>G*\_ <sup>0</sup>

**G**<sup>∗</sup>

*T*,*n*

**G**<sup>∗</sup>

ordinary least squares (OLS) estimator ^*θ*

(Eq. (9)) to obtain the AQL estimator ^*θ*

ARCH model using the QL and AQL methods.

expresses an AQLEF sequence. The solution of **G**<sup>∗</sup>

*<sup>T</sup>*,*<sup>n</sup>*ð Þ¼ *<sup>θ</sup>* <sup>X</sup>

*T*

*t*¼1 \_ **<sup>f</sup>***t*ð Þ*<sup>θ</sup>* <sup>Σ</sup>^�<sup>1</sup> *<sup>t</sup>*,*<sup>n</sup>* ^*<sup>θ</sup>*

is asymptotically nonnegative definite, **G**<sup>∗</sup>

tion is not always accessible. To find the QL when *Et*�<sup>1</sup> *ζtζ*<sup>0</sup>

Suppose, in probability, Σ*<sup>t</sup>*,*<sup>n</sup>* is converging to *Et*�<sup>1</sup> *ζtζ*<sup>0</sup>

*<sup>T</sup>*,*<sup>n</sup>*ð Þ¼ *<sup>θ</sup>* <sup>X</sup>

*T* � ��<sup>1</sup>

estimate *θ<sup>T</sup>*,*<sup>n</sup>* by the AQL method is the solution of the AQL equation **G**<sup>∗</sup>

*T*

*t*¼1 \_ **<sup>f</sup>***t*ð Þ*<sup>θ</sup>* <sup>Σ</sup>�<sup>1</sup>

ð Þ*<sup>t</sup>* ð Þ¼ *<sup>θ</sup>* \_

<sup>G</sup>*<sup>t</sup>* <sup>¼</sup> **<sup>W</sup>***<sup>t</sup>* **<sup>y</sup>***<sup>t</sup>* � **<sup>f</sup>***t*ð Þ*<sup>θ</sup>* � � � �

**<sup>f</sup>***t*ð Þ*<sup>θ</sup>* <sup>Σ</sup>�<sup>1</sup>

and the estimation of *θ* by the QL method is the solution of the QL equation

A limitation of the QL method is that the nature of Σ*<sup>t</sup>* may not be obtainable. A misidentified Σ*<sup>t</sup>* could result in a deceptive inference about parameter *θ*. In the next subsection, we will introduce the AQL method, which is basically the QL estimation

The QLEF (see Eqs. (6) and (7)) relies on the information of Σ*t*. Such informa-

� *<sup>E</sup>G*\_ <sup>0</sup> <sup>∗</sup> *T*,*n* � ��<sup>1</sup>

quasi-likelihood estimation function (AQLEF) sequence in G, and the AQL sequence

In this chapter, the kernel smoothing estimator of Σ*<sup>t</sup>* is suggested to find Σ*<sup>t</sup>*,*<sup>n</sup>* in the AQLEF (Eq. (8)). A wide-ranging appraisal of the Nadaray-Watson (NW) estimator-type kernel estimator is available in [27]. By using these kernel estima-

The estimation of *θ* by the AQL method is the solution to Eq. (9). Iterative techniques are suggested to solve the AQL equation (Eq. (9)). Such techniques start with the

ð Þ1

For estimation of unknown parameters in fanatical models by QL and AQL approaches, see [21, 28–33]. The next sections present the parameter estimation of

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

*<sup>T</sup>*,*<sup>n</sup> be a sequence of the EF in* G*. For all* **G***<sup>T</sup>* ∈ G*, if*

� �, which converges to *θ* under certain regular conditions.

*<sup>t</sup>* **<sup>y</sup>***<sup>t</sup>* � **<sup>f</sup>***t*ð Þ*<sup>θ</sup>* � � (7)

*t*

*<sup>T</sup>*,*<sup>n</sup>* can be denoted as the asymptotic

*<sup>t</sup>*,*<sup>n</sup>* **<sup>y</sup>***<sup>t</sup>* � **<sup>f</sup>***t*ð Þ*<sup>θ</sup>* � � (8)

ð Þ <sup>0</sup> � � **<sup>y</sup>***<sup>t</sup>* � **<sup>f</sup>***t*ð Þ*<sup>θ</sup>* � � <sup>¼</sup> <sup>0</sup>*:* (9)

. Repeat this a few times until it converges.

ð Þ <sup>0</sup> � � in the AQL equation

*<sup>T</sup>*,*<sup>n</sup>*ð Þ¼ *θ* 0 expresses the AQL

*EG*<sup>∗</sup> *T*,*nG*<sup>∗</sup> *T* <sup>0</sup> � � *EG*\_ <sup>∗</sup> <sup>0</sup>

*t* � �. Then,

� � is not accessible, Lin

*T*,*n* � ��<sup>1</sup>

*<sup>T</sup>*,*<sup>n</sup>* ¼ 0.

*<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> *t* .

The GARCH model was developed by Bollersev [24] to extend the earlier work on ARCH models by Engle [1]. For estimation and applications of GARCH models, (see, [2, 3, 6–8, 10, 11, 14]). Moreover, GARCH models have now become the standard textbook material in econometrics and finance as exemplified by, for example, [20–23].

This chapter considers estimation of ARCH and GARCH models using quasilikelihood (QL) and asymptotic quasi-likelihood (AQL) approaches. Distribution assumptions are not required of ARCH and GARCH processes by the QL method. But, the QL technique assumes knowing the first two moments of the process. However, The AQL estimation procedure is suggested when the conditional variance of process is unknown. The AQL estimation substitutes the variance and covariance by kernel estimation in QL.

This chapter is structured as follows. Section 2 introduces the QL and AQL approaches. The estimation of ARCH model using QL and AQL methods are developed in Section 3. The estimation of GARCH model using QL and AQL methods are developed in Section 4. Reports of simulation outcomes, numerical cases and applications of the methods to a daily exchange rate series, and weekly prices changes of crude oil are also presented. Summary and conclusion are given in Section 5.
