**2. The QLE and AQL methods**

Let the observation equation be given by

$$\mathbf{y}\_t = \mathbf{f}\_t(\theta) + \zeta\_t, \qquad t = 1, 2, 3\cdots, T,\tag{5}$$

where *ζ<sup>t</sup>* is a sequence of martingale difference with respect to F*t*, F*<sup>t</sup>* denotes the *σ*-field generated by **y***<sup>t</sup>* , **<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, 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 open subset Θ ∈*R<sup>d</sup>*. Note that *θ* is a parameter of interest.

### **2.1 The QL method**

For the model given by Eq. (5), assume that *Et*�<sup>1</sup> *ζtζ*<sup>0</sup> *t* � � <sup>¼</sup> <sup>Σ</sup>*<sup>t</sup>* is known. Now, the linear class G*<sup>T</sup>* of the estimating function (EF) can be defined by

$$\mathcal{G}\_T = \left\{ \sum\_{t=1}^T \mathbf{W}\_t \left( \mathbf{y}\_t - \mathbf{f}\_t(\theta) \right) \right\}$$

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

$$\mathbf{G}\_T^\*(\theta) = \sum\_{t=1}^T \dot{\mathbf{f}}\_t(\theta) \Sigma\_t^{-1} \left( \mathbf{y}\_t - \mathbf{f}\_t(\theta) \right) \tag{6}$$

where **<sup>W</sup>***<sup>t</sup>* is <sup>F</sup>*<sup>t</sup>*�1-measureable and \_ **<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>* by the QL method is the solution of the QL equation **G**<sup>∗</sup> *<sup>T</sup>* ð Þ¼ *θ* 0 (see [25]).

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

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

$$\mathcal{G}\_t = \left\{ \mathbf{W}\_t \left( \mathbf{y}\_t - \mathbf{f}\_t(\theta) \right) \right\}.$$

then the QLEF can be defined by

$$\mathbf{G}\_{(t)}^{\*}(\theta) = \dot{\mathbf{f}}\_{t}(\theta)\Sigma\_{t}^{-1}(\mathbf{y}\_{t} - \mathbf{f}\_{t}(\theta))\tag{7}$$

and the estimation of *θ* by the QL method is the solution of the QL equation **G**<sup>∗</sup> ð Þ*<sup>t</sup>* ð Þ¼ *<sup>θ</sup>* 0.

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