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

Raed Alzghool

doi:10.5772/intechopen.93726

Abstract

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.

Keywords

ARCH model
GARCH model
the quasi-likelihood
asymptotic quasi-likelihood
martingale difference
daily exchange rate series
prices changes of crude oil

chapter and author info

Author

Raed Alzghool*
- Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Jordan
- Department of Quantitative Methods, School of Business, King Faisal University, Al-Ahsa, 31982, Saudi Arabia

*Address all correspondence to: raedalzghool@bau.edu.jo;, ralzghool@kfu.edu.sa

DOI: 10.5772/intechopen.93726

From the Edited Volume

Linear and Non-Linear Financial Econometrics -Theory and PracticeEdited by Mehmet Kenan Terzioğlu

Linear and Non-Linear Financial Econometrics -Theory and Practice [Working Title]

Dr. Mehmet Kenan Terzioğlu and Prof. Gordana Djurovic

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1. Introduction

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

yt=μ+ξt,t=1,2,3,⋯,T.E1

and

σt2=α0+α1ξt−12+⋯+αqξt−q2+ζt,t=1,2,3,⋯,T.E2

ξtare i.i.d with Eξt=0and Vξt=σt2; and ζtare i.i.d with Eζt=0and Vζt=σζ2. For estimation and applications of ARCH models, see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. Moreover, ARCH models have now become the standard textbook material in econometrics and finance as exemplified by, for example, [20, 21, 22, 23].

The generalized autoregressive conditional heteroscedasticity (GARCH(p,q)) process ytis defined by

yt=μ+ξt,t=1,2,3,⋯,T.E3

and

σt2=α0+α1ξt−12+⋯+αpξt−p2+β1σt−12+⋯+βqσt−q2,t=1,2,3,⋯,T.E4

ξtare i.i.d with Eξt=0and Vξt=σt2.

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, 7, 8, 10, 11, 14]). Moreover, GARCH models have now become the standard textbook material in econometrics and finance as exemplified by, for example, [20, 21, 22, 23].

This chapter considers estimation of ARCH and GARCH models using quasi-likelihood (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.

2. The QLE and AQL methods

Let the observation equation be given by

yt=ftθ+ζt,t=1,2,3⋯,T,E5

where ζtis a sequence of martingale difference with respect to Ft, Ftdenotes the σ-field generated by yt,yt−1,⋯,y1for t≥1; that is, EζtFt−1= Et−1ζt=0, where ftθis an Ft−1measurable and θis parameter vector, which belongs to an open subset Θ∈Rd. Note that θis a parameter of interest.

2.1 The QL method

For the model given by Eq. (5), assume that Et−1ζtζt′=Σtis known. Now, the linear class GTof the estimating function (EF) can be defined by

GT=∑t=1TWtyt−ftθ

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

GT∗θ=∑t=1TḟtθΣt−1yt−ftθE6

where Wtis Ft−1-measureable and ḟtθ=∂ftθ/∂θ. Then, the estimation of θby the QL method is the solution of the QL equation GT∗θ=0(see [25]).

If the sub-estimating function spaces of GTare considered as follows:

Gt=Wtyt−ftθ

then the QLEF can be defined by

Gt∗θ=ḟtθΣt−1yt−ftθE7

and the estimation of θby the QL method is the solution of the QL equation Gt∗θ=0.

A limitation of the QL method is that the nature of Σtmay not be obtainable. A misidentified Σtcould 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 Σtis unknown.

2.2 The AQL method

The QLEF (see Eqs. (6) and (7)) relies on the information of Σt. Such information is not always accessible. To find the QL when Et−1ζtζt′is not accessible, Lin [26] proposed the AQL method.

Definition 2.2.1: LetGT,n∗be a sequence of the EF inG. For allGT∈G, if

EĠT−1EGTGT′EĠT′−1−EĠ′T,n∗−1EGT,n∗GT∗′EĠT,n∗′−1

is asymptotically nonnegative definite, GT,n∗can be denoted as the asymptotic quasi-likelihood estimation function (AQLEF) sequence in G, and the AQL sequence estimate θT,nby the AQL method is the solution of the AQL equation GT,n∗=0.

Suppose, in probability, Σt,nis converging to Et−1ζtζt′. Then,

GT,n∗θ=∑t=1TḟtθΣt,n−1yt−ftθE8

expresses an AQLEF sequence. The solution of GT,n∗θ=0expresses the AQL sequence estimate θT,n∗, which converges to θunder certain regular conditions.

In this chapter, the kernel smoothing estimator of Σtis suggested to find Σt,nin 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

GT,n∗θ=∑t=1TḟtθΣ̂t,n−1θ̂0yt−ftθ=0.E9

The estimation of θby 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 θ̂0and use Σ̂t,nθ̂0in the AQL equation (Eq. (9)) to obtain the AQL estimator θ̂1. Repeat this a few times until it converges.

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.

3. 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.

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

The ARCH(q) process is defined by

yt=μ+ξt,t=1,2,3,⋯,T.E10

and

σt2=α0+α1ξt−12+⋯+αqξt−q2+ζt,t=1,2,3,⋯,T.E11

ξtare i.i.d with Eξt=0and Vξt=σt2; and ζtare i.i.d with Eζt=0and Vζt=σζ2. For this scenario, the martingale difference is

ξtζt=yt−μσt2−α0−α1ξt−12−⋯−αqξt−q2.

The QLEF to estimate σt2is given by

Gtσt2=01σt200σζ2−1yt−μσt2−α0−α1ξt−12−⋯−αqξt−q2=σζ−2σt2−α0−α1ξt−12−⋯−αqξt−q2.E12

Given ξ̂0=0, initial values ψ0=μ0α00α10⋯αq0σζ02and ξ̂t−12=yt−1−μ02, then the QL estimation of σt2is the solution of Gtσt2=0:

σ̂t2=α0+α1ξ̂t−12+⋯+αqξ̂t−q2,t=1,2,3⋯,T.E13

The QLEF, using σ̂t2and yt, to estimate the parameters μ, α0, α1, ⋯, αqis given by

GTμα0α1⋯αq=∑t=1T−100−10−ξt−12⋮⋮0−ξt−q2σt200σζ02−1×yt−μσt2−α0−α1ξt−12−αqξt−q2.

The QL estimate of μ, α0, α1, ⋯, αqis the solution of GTμα0α1⋯αq=0, where ζ̂t=σ̂t2−α̂0−α̂1ξ̂t−12−⋯−α̂qξ̂t−q2,t=1,2,3,⋯,Tand

σ̂ζ2=∑t=1Tζ̂t−ζ̂¯2T−1E14

ψ̂=μ̂α̂0α̂1⋯αqσ̂ζ2is 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 σt2, so the sequence of (AQLEF) is given by

Gtσt2=01Σt,n−1yt−μσt2−α0−α1ξt−12−⋯−αqξt−q2

Given ξ̂0=0, θ0=μ0α0α1⋯αq, Σt,n0=I2, and ξ̂t−12=yt−1−μ02, then the AQL estimation of σt2is the solution of Gtσt2=0, that is,

σ̂t2=α0+α1ξ̂t−12+⋯+αqξ̂t−q2,t=1,2,3⋯,T.E15

Second, by kernel estimation method, we find

Σ̂t,nθ0=σ̂nytσ̂nytσtσ̂nσtytσ̂nσt.

Third, to estimate the parameters θ0=μ0α0α1⋯αqusing σ̂t2and ytand the sequence of (AQLEF):

GTμ0α0α1⋯αq=∑t=1T−100−10−ξt−12⋮⋮0−ξt−q2Σ̂t,n−1×yt−μσt2−α0−α1ξt−12−⋯−αqξt−q2.

The AQL estimate of θ0=μ0α0α1⋯αqis the solution of GTθ0=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

yt=μ+ξt,t=1,2,3,⋯,T.E16

and

σt2=α0+α1ξt−12+ζt,t=1,2,3,⋯,T.E17

ξtare i.i.d with Eξt=0and Vξt=σt2; and ζtare i.i.d with Eζt=0and Vζt=σζ2.

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

ξtζt=yt−μσt2−α0−α1ξt−12.

The QLEF to estimate σt2is given by

Gtσt2=01σt200σζ2−1yt−μσt2−α0−α1ξt−12=σζ−2σt2−α0−α1ξt−12.E18

Given ξ̂0=0, initial values ψ0=μ0α00α10σζ02and ξ̂t−12=yt−1−μ02, then the QL estimation of σt2is the solution of Gtσt2=0,

σ̂t2=α0+α1ξ̂t−12,t=1,2,3⋯,T.E19

To estimate the parameters μ, α0, and α1, using σ̂t2and yt, the QLEF is given by

GTμα0α1=∑t=1T−100−10−ξt−12σt200σζ02−1yt−μσt2−α0−α1ξt−12.

The solution of GTμα0α1=0is the QL estimate of μ, α0, and α1. Therefore

μ̂=∑t=1Tytσ̂t2/∑t=1T1σ̂t2.E20

α̂1=T∑t=1Tσ̂t2ξ̂t−12−∑t=1Tσ̂t2∑t=1Tξ̂t−12T∑t=1Tξ̂t−14−∑t=1Tξ̂t−122.E21

α̂0=∑t=1Tσ̂t2−α̂1∑t=1Tξ̂t−12T.E22

and let

σ̂ζ2=∑t=1Tζ̂t−ζ̂¯2T−1E23

where ζ̂t=σ̂t2−α̂0−α̂1ξ̂t−12,t=1,2,3,⋯,T.

ψ̂=μ̂α̂0α̂1σ̂ζ2is 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 σt2, so the sequence of (AQLEF) is given by

Gtσt2=01Σt,n−1yt−μσt2−α0−α1ξt−12

Given ξ̂0=0, θ0=μ0α0α1μ0, Σt,n0=I2and ξ̂t−12=yt−1−μ02, then the AQL estimation of σt2is the solution of Gtσt2=0, that is,

σ̂t2=α0+α1ξ̂t−12,t=1,2,3⋯,T.E24

Second, by kernel estimation method, we find

Σ̂t,nθ0=σ̂nytσ̂nytσtσ̂nσtytσ̂nσt.

Third, to estimate the parameters θ=μα0α1using σ̂t2and ytand the sequence of AQLEF:

GTμα0α1=∑t=1T−100−10−ξ̂t−1Σ̂t,n−1yt−μσt2−α0−α1ξt−12.

The AQL estimate of γ, ϕ, and μis the solution of GTμα0α1=0. Therefore

μ̂=∑t=1Tytσ̂nyt/∑t=1T1σ̂nyt.E25

α̂1=∑t=1Tσ̂t2σ̂nσt∑t=1Tξ̂t−12σ̂nσt−∑t=1T1σ̂nσt∑t=1Tξ̂t−12σ̂nσt∑t=1Tξ̂t−12σ̂nσt2−∑t=1T1σ̂nσt∑t=1Tξ̂t−14σ̂nσt.E26

α̂0=∑t=1Tσ̂t2σ̂nσt−α̂1∑t=1Tξ̂t−12σ̂nσt∑t=1T1σ̂nσt.E27

and let

σ̂ζ2=∑t=1Tζ̂t−ζ̂¯2T−1E28

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 α̂0, α̂1, and μ̂. In Table 1, QL denotes the QL estimate and AQL denotes the AQL estimate.

	α0	α1	μ	α0	α1	μ	α0	α1	μ
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
	0.001	0.010	0.006	0.001	0.010	0.006	0.001	0.010	0.006
AQL	0.009	0.989	1.30	0.009	0.989	−1.29	0.009	0.989	0.030
	0.001	0.010	0.0003	0.002	0.009	0.0003	0.001	0.009	0.0003
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
	0.001	0.0001	0.014	0.001	0.010	0.014	0.001	0.010	0.014
AQL	0.049	0.940	1.32	0.049	0.940	−1.30	0.049	0.940	0.032
	0.001	0.010	0.018	0.001	0.010	0.018	0.001	0.01	0.001
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
	0.002	0.010	0.019	0.002	0.010	0.020	0.002	0.010	0.029
AQL	0.098	0.910	1.31	0.098	0.910	−1.32	0.098	0.910	0.031
	0.002	0.010	0.012	0.002	0.010	0.021	0.001	0.010	0.001
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
	0.002	0.010	0.019	0.001	0.001	0.014	0.001	0.016	0.006
AQL	0.098	0.910	−0.031	0.051	0.949	−0.031	0.009	0.990	−0.031
	0.002	0.010	0.001	0.001	0.001	0.002	0.001	0.010	0.001

Table 1.

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

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.

		α0	α1	μ	α0	α1	μ
T = 20	True	0.010	0.980	−0.030	0.05	0.950	1.3
	QL	0.009	0.990	−0.029	0.0495	0.9485	1.300
		0.0008	0.0100	0.0319	0.0005	0.0015	0.0703
	AQL	0.009	0.990	−0.031	0.0495	0.9485	1.3107
		0.0009	0.010	0.0084	0.0005	0.0015	0.0213
T = 40	QL	0.009	0.990	−0.031	0.0495	0.9485	1.3015
		0.00089	0.010	0.0223	0.0005	0.0015	0.0492
	AQL	0.009	0.990	−0.031	0.0495	0.9485	1.3113
		0.00089	0.010	0.0039	0.0005	0.0015	0.0143
T = 60	QL	0.009	0.990	−0.029	0.0495	0.9485	1.300
		0.0009	0.010	0.0180	0.0005	0.0015	0.0404
	AQL	0.009	0.990	−0.031	0.0495	0.9485	1.311
		0.0009	0.010	0.0027	0.0005	0.0015	0.0128
T = 80	QL	0.009	0.990	−0.029	0.0490	0.9485	1.300
		0.0009	0.010	0.016	0.0005	0.0015	0.0353
	AQL	0.009	0.990	−0.310	0.0495	0.9485	1.3112
		0.0009	0.010	0.0020	0.0005	0.0015	0.0119
T = 100	QL	0.009	0.990	0.0292	0.0495	0.9485	1.3017
		0.0009	0.010	0.0142	0.0005	0.0015	0.0314
	AQL	0.009	0.990	−0.031	0.0495	0.9485	1.3111
		0.0009	0.010	0.0018	0.0005	0.0015	0.0116

Table 2.

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

3.4 Empirical applications

The first data set we analyze are the daily exchange rate of rt=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=logrt−logrt−1.

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 rtthe p-values are significant (<0.05level), so reject the null hypothesis that there are no ARCH effects and we fit ytby following models:

yt=μ+ξt,t=1,2,3,⋯,T.E29

and

σt2=α0+α1ξt−12+ζt,t=1,2,3,⋯,T.E30

ξtare i.i.d with Eξt=0and Vξt=σt2; and ζtare i.i.d with Eζt=0and Vζt=σζ2.

The estimation of unknown parameters, (α0, α1, 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.

	α̂0	α̂1	μ̂	ξ̂¯tS.dξ̂t
QL	0.1300	0.8387	−0.00012	0.00013
AQL	0.0200	0.9599	−0.00111	0.1350

Table 3.

Estimation of α0,α1,μfor the exchange rate pound/dollar data.

4. 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.

4.1 Parameter estimation of GARCH(p,q) model using the QL method

The GARCH(p,q) process is defined by

yt=μ+ξt,t=1,2,3,⋯,T.E31

and

σt2=α0+α1ξt−12+⋯+αpξt−p2+β1σt−12+⋯+βqσt−q2,t=1,2,3,⋯,T.E32

ξtare i.i.d with Eξt=0and Vξt=σt2; and ζtare i.i.d with Eζt=0and Vζt=σζ2. For this scenario, the martingale difference is

ξtζt=yt−μσt2−α0−α1ξt−12−⋯−αpξt−p2−β1σt−12−⋯−βqσt−q2.

The QLEF to estimate σt2is given by

Gtσt2=01σt200σζ2−1ξtζt=σζ−2σt2−α0−α1ξt−12−⋯−αpξt−p2−β1σt−12−⋯−βqσt−q2.E33

Given ξ̂0=0, initial values ψ0=μ0α00α10⋯αp0β10⋯βq0σζ02, ξ̂t−i2=yt−i−μ02, and σ̂t−j2are the QL estimations of σt−j2, where i = 1, 2, ⋯, p and j = 1, 2, ⋯, q, then the QL estimation of σt2is the solation of Gtσt2=0,

σ̂t2=α0+α1ξt−12+⋯+αpξt−p2+β1σt−12+⋯+βqσt−q2,t=1,2,3⋯,T.E34

The QLEF, using σ̂t2and yt, to estimate the parameters θ=μ, α0, α1, ⋯, αq, β1, ⋯, βqis given by

GTθ=∑t=1T−100−10−ξt−12⋮⋮0−ξt−p20−σt−12⋮⋮0−σt−q2σt200σζ02−1ξtζt

The QL estimate of μ, α0, α1, ⋯, αq, β1, ⋯, βqis the solation of GTθ=0, where ζ̂t=σ̂t2−α̂0−α̂1ξ̂t−12−⋯−α̂pξ̂t−p2−β̂1σ̂t−12−⋯−β̂qσ̂t−q2,t=1,2,3,⋯,Tand

σ̂ζ2=∑t=1Tζ̂t−ζ̂¯2T−1E35

ψ̂=(μ̂,α̂0,α̂1,⋯,α̂p,β̂1,⋯,β̂q,σ̂ζ2)is an initial value in the iterative procedure.

4.2 Parameter estimation of GARCH(p,q) model using the AQL method

Considering the GARCH(p,q) model given by Eqs. (31) and (32) and using the same argument listed under Eq. (32). First, we need to estimate σt2, so the sequence of (AQLEF) is given by

Gtσt2=01Σt,n−1ξtζt

Given ξ̂0=0, θ0=μ0α00α10⋯αp0β10⋯βq0, Σt,n0=I2, and ξ̂t−i2=yt−i−μ02, and σ̂t−j2is the AQL estimation of σt−j2, where i = 1, 2, ⋯, p and j = 1, 2, ⋯, q, then the AQL estimation of σt2is the solation of Gtσt2=0, that is,

σ̂t2=α0+α1ξ̂t−12+⋯+αpξt−p2+β1σt−12+⋯+βqσt−q2,t=1,2,3⋯,T.E36

Second, by kernel estimation method, we find

Σ̂t,nθ0=σ̂nytσ̂nytσtσ̂nσtytσ̂nσt.

Third, to estimate the parameters θ0=μ0α0α1⋯αqusing σ̂t2and ytand the sequence of (AQLEF):

GTμ0α0α1⋯αq=∑t=1T−100−10−ξt−12⋮⋮0−ξt−q20−σt−12⋮⋮0−σt−q2Σ̂t,n−1ξtζt.

The AQL estimate of θ=μα0α1⋯αqis the solation of GTθ=0. The estimation procedure will be iteratively repeated until it converges.

4.3 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

yt=μ+ξt,t=1,2,3,⋯,T.E37

and

σt2=α0+α1ξt−12+β1σt−1+ζt,t=1,2,3,⋯,T.E38

ξtare i.i.d with Eξt=0and Vξt=σt2; and ζtare i.i.d with Eζt=0and Vζt=σζ2.

4.3.1 Parameter estimation of GARCH(1,1) model using the QL method

For GARCH(1,1) given by Eqs. (37) and (38), the martingale difference is

ξtζt=yt−μσt2−α0−α1ξt−12−β1σt−12.

The QLEF to estimate σt2is given by

Gtσt2=01σt200σζ2−1yt−μσt2−α0−α1ξt−12−β1σt−12=σζ−2σt2−α0−α1ξt−12−β1σt−12.E39

Given ξ̂0=0, initial values ψ0=μ0α00α10σζ02, ξ̂t−12=yt−1−μ02, and σ̂t−12is the QL estimation of σt−12, then the QL estimation of σt2is the solation of Gtσt2=0,

σ̂t2=α0+α1ξ̂t−12+β1σ̂t−12,t=1,2,3⋯,T.E40

To estimate the parameters μ, α0, and α1, using σ̂t2and yt, the QLEF is given by

GTμα0α1β1=∑t=1T−100−10−ξt−120−σt−12σt200σζ02−1∗yt−μσt2−α0−α1ξt−12−β1σt−12.

The solation of GTμα0α1β1=0is the QL estimate of μ, α0, α1, and β1. Therefore

μ̂=∑t=1Tytσ̂t2/∑t=1T1σ̂t2.E41

β̂1=Sσ̂t−12ξ̂t−12Sσ̂t2ξ̂t−12−Sξ̂t−12ξ̂t−12Sσ̂t2σ̂t−12Sσ̂t−12ξ̂t−122−Sσ̂t−12σ̂t−12Sξ̂t−12ξ̂t−12.E42

α̂1=Sσ̂t2ξ̂t−12−β̂1Sσ̂t−12ξ̂t−12Sξ̂t−12ξ̂t−12.E43

α̂0=∑t=1Tσ̂t2−α̂1∑t=1Tξ̂t−12−β̂1∑t=1Tσ̂t−12T.E44

and let

σ̂ζ2=∑t=1Tζ̂t−ζ̂¯2T−1E45

where

ζ̂t=σ̂t2−α̂0−α̂1ξ̂t−12−β̂1σ̂t−12,t=1,2,3,⋯,T,Sσ̂t−12ξ̂t−12=∑t=1Tσ̂t−12ξ̂t−12−∑t=1Tσ̂t−12∑t=1Tξ̂t−12T,Sσ̂t2ξ̂t−12=∑t=1Tσ̂t2ξ̂t−12−∑t=1Tσ̂t2∑t=1Tξ̂t−12T,Sξ̂t−12ξ̂t−12=∑t=1Tξ̂t−14−∑t=1Tξ̂t−122T,Sσ̂t2σ̂t−12=∑t=1Tσ̂t2σ̂t−12−∑t=1Tσ̂t2∑t=1Tσ̂t−12T,Sσ̂t−12σ̂t−12=∑t=1Tσ̂t−14−∑t=1Tσ̂t−122T.

ψ̂=μ̂α̂0α̂1σ̂ζ2is 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].

4.3.2 Parameter estimation of GARCH(1,1) model using the AQL method

Considering the GARCH(1,1) model given by Eqs. (37) and (38) and using the same argument listed under (Eq. (38)). First, we need to estimate σt2, so the sequence of (AQLEF) is given by

Gtσt2=01Σt,n−1yt−μσt2−α0−α1ξt−12−β1σt−12

Given ξ̂0=0, θ0=μ0α0,0α1,0β1,0, Σt,n0=I2, ξ̂t−12=yt−1−μ02, and σ̂t−12is the AQL estimation of σt−12, then the AQL estimation of σt2is the solation of Gtσt2=0, that is,

σ̂t2=α0+α1ξ̂t−12+β1σt−12,t=1,2,3⋯,T.E46

Second, by kernel estimation method, we find

Σ̂t,nθ0=σ̂nyt00σ̂nσt.

Third, to estimate the parameters θ=μα0α1β1using σ̂t2and ytand the sequence of AQLEF:

GTμα0α1β1=∑t=1T−100−10−ξ̂t−120−σ̂t−12Σ̂t,n−1yt−μσt2−α0−α1ξt−12−β1σt−12.

The AQL estimate of μ, α0, α1, and β1is the solation of GTμα0α1β1=0. Therefore

μ̂=∑t=1Tytσ̂nyt/∑t=1T1σ̂nyt.E47

β̂1=SSσ̂t−12ξ̂t−12SSσ̂t2ξ̂t−12−SSξ̂t−12ξ̂t−12SSσ̂t2σ̂t−12SSσ̂t−12ξ̂t−122−SSσ̂t−12σ̂t−12SSξ̂t−12ξ̂t−12.E48

α̂1=SSσ̂t2ξ̂t−12−β̂1SSσ̂t−12ξ̂t−12SSξ̂t−12ξ̂t−12.E49

α̂0=∑t=1Tσ̂t2σ̂nσt−α̂1∑t=1Tξ̂t−12σ̂nσt−β̂1∑t=1Tσ̂t−12σ̂nσt∑t=1T1σ̂nσt,E50

and let

σ̂ζ2=∑t=1Tζ̂t−ζ̂¯2T−1E51

where

ζ̂t=σ̂t2−α̂0−α̂1ξ̂t−12−β̂1σ̂t−12,t=1,2,3,⋯,T,SSσ̂t−12ξ̂t−12=∑t=1Tσ̂t−12ξ̂t−12σ̂nσt∑t=1T1σ̂nσt−∑t=1Tσ̂t−12σ̂nσt∑t=1Tξ̂t−12σ̂nσt,SSσ̂t2ξ̂t−12=∑t=1Tσ̂t2ξ̂t−12σ̂nσt∑t=1T1σ̂nσt−∑t=1Tσ̂t2σ̂nσt∑t=1Tξ̂t−12σ̂nσt,SSξ̂t−12ξ̂t−12=∑t=1T1σ̂nσt∑t=1Tξ̂t−14σ̂nσt−∑t=1Tξ̂t−12σ̂nσt2,SSσ̂t2σ̂t−12=∑t=1T1σ̂nσt∑t=1Tσ̂t2σ̂t−12σ̂nσt−∑t=1Tσ̂t2σ̂nσt∑t=1Tσ̂t−12σ̂nσt,SSσ̂t−12σ̂t−12=∑t=1T1σ̂nσt∑t=1Tσ̂t−14σ̂nσt−∑t=1Tσ̂t−12σ̂nσt2.

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 α̂0, α̂1, β̂1, and μ̂. In Table 4, QL denotes the QL estimate and AQL denotes the AQL estimate.

	μ	α0	α1	β1	μ	α0	α1	β1
True	0.15	0.65	0.87	0.10	0.20	0.41	0.88	0.08
QL	0.149	0.779	0.865	0.074	0.199	0.461	0.912	0.057
	0.040	0.353	0.011	0.029	0.031	0.155	0.033	0.025
AQL	0.150	0.661	0.851	0.092	0.209	0.405	0.901	0.076
	0.001	0.012	0.019	0.009	0.010	0.006	0.021	0.004
True	−0.10	0.48	0.89	0.08	0.16	0.37	0.9	0.08
QL	−0.101	0.556	0.902	0.058	0.159	0.434	0.922	0.058
	0.034	0.212	0.014	0.024	0.030	0.189	0.024	0.025
AQL	−0.110	0.486	0.891	0.0752	0.161	0.374	0.911	0.076
	0.010	0.006	0.001	0.005	0.001	0.004	0.011	0.004
True	0.18	0.39	0.88	0.08	0.09	0.50	0.89	0.05
QL	0.179	0.447	0.892	0.058	0.089	0.538	0.898	0.036
	0.031	0.146	0.015	0.024	0.033	0.090	0.009	0.015
AQL	0.180	0.395	0.882	0.076	0.091	0.504	0.892	0.046
	0.001	0.005	0.002	0.005	0.002	0.004	0.002	0.004

Table 4.

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

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.

		μ	α0	α1	β1	μ	α0	α1	β1
	True	0.16	0.37	0.90	0.08	−0.10	0.48	0.89	0.08
	QL	0.17	0.42	0.89	0.07	−0.09	0.51	0.90	0.06
T = 20		0.176	0.511	0.008	0.016	0.169	0.451	0.018	0.022
	AQL	0.16	0.38	0.89	0.07	−0.10	0.47	0.90	0.07
		0.037	0.012	0.007	0.014	0.066	0.014	0.013	0.018
	QL	0.16	0.42	0.89	0.07	−0.09	0.51	0.91	0.06
		0.149	0.422	0.007	0.016	0.137	0.326	0.018	0.021
T = 40	AQL	0.16	0.38	0.89	0.07	−0.10	0.47	0.90	0.07
		0.027	0.012	0.007	0.013	0.022	0.014	0.012	0.016
	QL	0.16	0.42	0.89	0.07	−0.09	0.52	0.91	0.06
		0.121	0.289	0.007	0.018	0.119	0.307	0.018	0.021
T = 60	AQL	0.16	0.38	0.89	0.07	−0.10	0.47	0.90	0.07
		0.019	0.012	0.007	0.011	0.014	0.013	0.012	0.015
	QL	0.16	0.42	0.89	0.07	−0.10	0.51	0.90	0.06
		0.100	0.159	0.007	0.017	0.108	0.248	0.018	0.021
T = 80	AQL	0.16	0.38	0.89	0.07	−0.10	0.47	0.90	0.07
		0.012	0.012	0.007	0.011	0.012	0.013	0.012	0.015
	QL	0.16	0.42	0.89	0.07	−0.10	0.51	0.90	0.06
		0.100	0.159	0.007	0.018	0.101	0.242	0.018	0.021
T = 100	AQL	0.16	0.38	0.89	0.07	−0.10	0.47	0.90	0.07
T = 100		0.012	0.011	0.007	0.011	0.011	0.013	0.012	0.015

Table 5.

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

4.4 Empirical applications

The second set of data is the weekly price changes of crude oil prices Pt. The Ptof 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, yt=logPt−logPt−1and fit ytby using GARCH (1,1):

yt=μ+ξt,t=1,2,3,⋯,T.E52

and

σt2=α0+α1ξt−12+β1σt−1+ζt,t=1,2,3,⋯,T.E53

ξtare i.i.d with Eξt=0and Vξt=σt2; and ζtare i.i.d with Eζt=0and Vζt=σζ2.

The estimation of unknown parameters, (α0, α1, β1, μ), using QL and AQL are given in Table 6. Conclusion can be drawn based on the standardized residuals from the fourth column in Table 6, which favors the QL method and gives smaller standardized residuals, better than AQL method.

	μ̂0	α̂0	α̂1	β̂1	ξ̂¯tS.dξ̂t
QL	0.0008	0.566	0.912	0.0004	0.002
AQL	0.0089	0.630	0.972	0.041	0.185

Table 6.

Estimation of μ,α0,α1,β1for the rates of change prices data.

5. 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 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.

Acknowledgments

The author would like to acknowledge the helpful comments and suggestion of the editor. This study is conducted in the King Faisal University, Saudi Arabia, during the sabbatical year of the author from the Al Balqa Applied University, Jordan.

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

Abstract

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chapter and author info

Author

Raed Alzghool*

From the Edited Volume

1. Introduction

2. The QLE and AQL methods

2.1 The QL method

2.2 The AQL method

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

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

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

3.3 Simulation studies for the ARCH(1) model

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

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

Table 1.

Table 2.

3.4 Empirical applications

Table 3.

4. Parameter estimation of GARCH(p,q) model using the QL and AQL methods

4.1 Parameter estimation of GARCH(p,q) model using the QL method

4.2 Parameter estimation of GARCH(p,q) model using the AQL method

4.3 Simulation studies for the GARCH(1,1) model

4.3.1 Parameter estimation of GARCH(1,1) model using the QL method

4.3.2 Parameter estimation of GARCH(1,1) model using the AQL method

Table 4.

Table 5.

4.4 Empirical applications

Table 6.

5. Conclusions

Acknowledgments

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