**2.2 The AQL method**

and

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

example, [20–23].

*<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>

covariance by kernel estimation in QL.

**2. The QLE and AQL methods**

the *σ*-field generated by **y***<sup>t</sup>*

**2.1 The QL method**

**78**

Let the observation equation be given by

open subset Θ ∈*R<sup>d</sup>*. Note that *θ* is a parameter of interest.

For the model given by Eq. (5), assume that *Et*�<sup>1</sup> *ζtζ*<sup>0</sup>

**G**<sup>∗</sup>

by the QL method is the solution of the QL equation **G**<sup>∗</sup>

where **<sup>W</sup>***<sup>t</sup>* is <sup>F</sup>*<sup>t</sup>*�1-measureable and \_

linear class G*<sup>T</sup>* of the estimating function (EF) can be defined by

<sup>G</sup>*<sup>T</sup>* <sup>¼</sup> <sup>X</sup> *T*

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

*t*¼1

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

and the quasi-likelihood estimation function (QLEF) can be defined by

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

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

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

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

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

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.

where *ζ<sup>t</sup>* is a sequence of martingale difference with respect to F*t*, F*<sup>t</sup>* denotes

where **f***t*ð Þ*θ* is an F*<sup>t</sup>*�<sup>1</sup> measurable and *θ* is parameter vector, which belongs to an

**y***<sup>t</sup>* ¼ **f***t*ð Þþ *θ ζt*, *t* ¼ 1, 2, 3⋯, *T*, (5)

, **<sup>y</sup>***<sup>t</sup>*�<sup>1</sup>, <sup>⋯</sup>, **<sup>y</sup>**<sup>1</sup> for *<sup>t</sup>*≥1; that is, *<sup>E</sup> <sup>ζ</sup><sup>t</sup>* ð Þ jF*<sup>t</sup>*�<sup>1</sup> <sup>=</sup> *Et*�<sup>1</sup> *<sup>ζ</sup><sup>t</sup>* ð Þ¼ 0,

*t*

� � <sup>¼</sup> <sup>Σ</sup>*<sup>t</sup>* is known. Now, the

*<sup>t</sup>* **<sup>y</sup>***<sup>t</sup>* � **<sup>f</sup>***t*ð Þ*<sup>θ</sup>* � � (6)

*<sup>T</sup>* ð Þ¼ *θ* 0 (see [25]).

**<sup>f</sup>***t*ð Þ¼ *<sup>θ</sup> <sup>∂</sup>***f***t*ð Þ*<sup>θ</sup> <sup>=</sup>∂θ*. Then, the estimation of *<sup>θ</sup>*

*<sup>t</sup>*�*q*, *t* ¼ 1, 2, 3, ⋯, *T:* (4)

*σ*2

The QLEF (see Eqs. (6) and (7)) relies on the information of Σ*t*. Such information is not always accessible. To find the QL when *Et*�<sup>1</sup> *ζtζ*<sup>0</sup> *t* � � is not accessible, Lin [26] proposed the AQL method.

Definition 2.2.1: *Let* **G**<sup>∗</sup> *<sup>T</sup>*,*<sup>n</sup> be a sequence of the EF in* G*. For all* **G***<sup>T</sup>* ∈ G*, if*

$$\left(\left(\dot{\mathbf{E}}\dot{\mathbf{G}}\_{T}\right)^{-1}\left(\mathbf{E}\mathbf{G}\_{T}\mathbf{G}\_{T}\right)'\left(\dot{\mathbf{E}}\dot{\mathbf{G}}\_{T}^{'}\right)^{-1}-\left(\dot{\mathbf{E}}\dot{\mathbf{G}}\_{T,n}^{'}\right)^{-1}\left(\mathbf{E}\mathbf{G}\_{T,n}^{\*}\mathbf{G}\_{T}^{\*'}\right)\left(\dot{\mathbf{E}}\dot{\mathbf{G}}\_{T,n}^{\*'}\right)^{-1}\right)$$

is asymptotically nonnegative definite, **G**<sup>∗</sup> *<sup>T</sup>*,*<sup>n</sup>* can be denoted as the asymptotic quasi-likelihood estimation function (AQLEF) sequence in G, and the AQL sequence estimate *θ<sup>T</sup>*,*<sup>n</sup>* by the AQL method is the solution of the AQL equation **G**<sup>∗</sup> *<sup>T</sup>*,*<sup>n</sup>* ¼ 0.

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

$$\mathbf{G}\_{T,n}^\*(\theta) = \sum\_{t=1}^T \dot{\mathbf{f}}\_t(\theta) \Sigma\_{t,n}^{-1} (\mathbf{y}\_t - \mathbf{f}\_t(\theta)) \tag{8}$$

expresses an AQLEF sequence. The solution of **G**<sup>∗</sup> *<sup>T</sup>*,*<sup>n</sup>*ð Þ¼ *θ* 0 expresses the AQL sequence estimate *θ* <sup>∗</sup> *T*,*n* � �, which converges to *θ* under certain regular conditions.

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 estimators, the AQL equation becomes

$$\mathbf{G}\_{T,n}^\*(\theta) = \sum\_{t=1}^T \dot{\mathbf{f}}\_t(\theta) \hat{\Sigma}\_{t,n}^{-1} \left(\hat{\theta}^{(0)}\right) \left(\mathbf{y}\_t - \mathbf{f}\_t(\theta)\right) = \mathbf{0}.\tag{9}$$

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 ordinary least squares (OLS) estimator ^*θ* ð Þ <sup>0</sup> and use <sup>Σ</sup>^*<sup>t</sup>*,*<sup>n</sup>* ^*<sup>θ</sup>* ð Þ <sup>0</sup> � � in the AQL equation (Eq. (9)) to obtain the AQL estimator ^*θ* ð Þ1 . Repeat this a few times until it converges.

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 ARCH model using the QL and AQL methods.
