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

This chapter considers estimation of autoregressive conditional heteroscedasticity (ARCH) and the generalized autoregressive conditional heteroscedasticity (GARCH) models using quasi-likelihood (QL) and asymptotic quasi-likelihood (AQL) approaches. The QL and AQL estimation methods for the estimation of unknown parameters in ARCH and GARCH models are developed. Distribution assumptions are not required of ARCH and GARCH processes by QL method. Nevertheless, 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. Reports of simulation outcomes, numerical cases, and applications of the methods to daily exchange rate series and weekly prices ’ changes of crude oil are presented.

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.

The QLE and AQL methods
Let the observation equation be given by where ζ t is a sequence of martingale difference with respect to F t , F t denotes the σ-field generated by y t , y tÀ1 , ⋯, y 1 for t ≥ 1; that is, E ζ t jF tÀ1 ð Þ= E tÀ1 ζ t ð Þ ¼ 0, where f t θ ð Þ is an F tÀ1 measurable and θ is parameter vector, which belongs to an open subset Θ ∈ R d . Note that θ is a parameter of interest.

The QL method
For the model given by Eq. (5), assume that E tÀ1 ζ t ζ 0 t À Á ¼ Σ t is known. Now, the linear class G T of the estimating function (EF) can be defined by and the quasi-likelihood estimation function (QLEF) can be defined by where W t is F tÀ1 -measureable and _ f t θ ð Þ ¼ ∂f t θ ð Þ=∂θ. Then, the estimation of θ by the QL method is the solution of the QL equation G * T θ ð Þ ¼ 0 (see [25]).
If the sub-estimating function spaces of G T are considered as follows: then the QLEF can be defined by 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 t may not be obtainable. A misidentified Σ t 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 t is unknown.

The AQL method
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 E tÀ1 ζ t ζ 0 t À Á is not accessible, Lin [26] proposed the AQL method.
is asymptotically nonnegative definite, G * T,n can be denoted as the asymptotic quasi-likelihood estimation function (AQLEF) sequence in G, and the AQL sequence estimate θ T,n by the AQL method is the solution of the AQL equation G * T,n ¼ 0. Suppose, in probability, Σ t,n is converging to E tÀ1 ζ t ζ 0 t À Á . Then, expresses an AQLEF sequence. The solution of G * T,n θ ð Þ ¼ 0 expresses the AQL sequence estimate θ * T,n È É , which converges to θ under certain regular conditions. In this chapter, the kernel smoothing estimator of Σ t is suggested to find Σ t,n 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 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) estimatorθ For estimation of unknown parameters in fanatical models by QL and AQL approaches, see [21,[28][29][30][31][32][33]. The next sections present the parameter estimation of ARCH model using the QL and AQL methods.

Parameter estimation of ARCH(q) model using the QL and AQL methods
In this section, we will develop the estimation of ARCH model using QL and AQL methods.

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 2 t , so the sequence of (AQLEF) is given by Second, by kernel estimation method, we find Third, to estimate the parameters θ 0 ¼ μ 0 , α 0 , α 1 , ⋯, α q À Á usingσ 2 t È É and y t È É and the sequence of (AQLEF): The estimation procedure will be iteratively repeated until it converges.

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

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 2 t , so the sequence of (AQLEF) is given by Second, by kernel estimation method, we find and y t È É and the sequence of AQLEF: The AQL estimate of γ, ϕ, and μ is the solution of G T μ, α 0 , α 1 ð Þ¼0. Thereforê 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) forα 0 ,α 1 , andμ. 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.

Empirical applications
The first data set we analyze are the daily exchange rate of r t 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 y t ¼ log r t ð Þ À log r tÀ1 ð Þ. 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 r t the pvalues are significant ( < 0:05 level), so reject the null hypothesis that there are no ARCH effects and we fit y t È É by following models: and 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.

Parameter estimation of GARCH(p,q) model using the QL and AQL methods
In this section, we developing the estimation of GARCH model using QL and AQL methods.

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

Empirical applications
The second set of data is the weekly price changes of crude oil prices P t . The P t 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, y t ¼ log P t ð Þ À log P tÀ1 ð Þand fit y t È É by using GARCH (1,1): and The estimation of unknown parameters, (α 0 , α 1 , β 1 , μ), using QL and AQL are given in Table 6 from the fourth column in Table 6, which favors the QL method and gives smaller standardized residuals, better than AQL method.

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  Table 6. Estimation of μ, α 0 , α 1 , β 1 for the rates of change prices data.
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.