Open access peer-reviewed chapter - ONLINE FIRST

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

By Raed Alzghool

Submitted: June 11th 2020Reviewed: August 25th 2020Published: December 22nd 2020

DOI: 10.5772/intechopen.93726

Downloaded: 20

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

1. Introduction

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

yt=μ+ξt,t=1,2,3,,T.E1

and

σt2=α0+α1ξt12++αqξtq2+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+α1ξt12++αpξtp2+β1σt12++βqσtq2,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,yt1,,y1for t1; that is, EζtFt1= Et1ζt=0, where ftθis an Ft1measurable and θis parameter vector, which belongs to an open subset ΘRd. Note that θis a parameter of interest.

2.1 The QL method

For the model given by Eq. (5), assume that Et1ζtζt=Σtis known. Now, the linear class GTof the estimating function (EF) can be defined by

GT=t=1TWtytftθ

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

GTθ=t=1TḟtθΣt1ytftθE6

where Wtis Ft1-measureable and ḟtθ=ftθ/θ. Then, the estimation of θby the QL method is the solution of the QL equation GTθ=0(see [25]).

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

Gt=Wtytftθ

then the QLEF can be defined by

Gtθ=ḟtθΣt1ytftθE7

and the estimation of θby the QL method is the solution of the QL equation Gtθ=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 Et1ζtζtis not accessible, Lin [26] proposed the AQL method.

Definition 2.2.1: LetGT,nbe a sequence of the EF inG. For allGTG, if

EĠT1EGTGTEĠT1EĠT,n1EGT,nGTEĠT,n1

is asymptotically nonnegative definite, GT,ncan 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 Et1ζtζt. Then,

GT,nθ=t=1TḟtθΣt,n1ytftθE8

expresses an AQLEF sequence. The solution of GT,nθ=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,nθ=t=1TḟtθΣ̂t,n1θ̂0ytftθ=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+α1ξt12++αqξtq2+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=ytμσt2α0α1ξt12αqξtq2.

The QLEF to estimate σt2is given by

Gtσt2=01σt200σζ21ytμσt2α0α1ξt12αqξtq2=σζ2σt2α0α1ξt12αqξtq2.E12

Given ξ̂0=0, initial values ψ0=μ0α00α10αq0σζ02and ξ̂t12=yt1μ02, then the QL estimation of σt2is the solution of Gtσt2=0:

σ̂t2=α0+α1ξ̂t12++αqξ̂tq2,t=1,2,3,T.E13

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

GTμα0α1αq=t=1T10010ξt120ξtq2σt200σζ021×ytμσt2α0α1ξt12αqξtq2.

The QL estimate of μ, α0, α1, , αqis the solution of GTμα0α1αq=0, where ζ̂t=σ̂t2α̂0α̂1ξ̂t12α̂qξ̂tq2,t=1,2,3,,Tand

σ̂ζ2=t=1Tζ̂tζ̂¯2T1E14

ψ̂=μ̂α̂0α̂1αqσ̂ζ2is 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,n1ytμσt2α0α1ξt12αqξtq2

Given ξ̂0=0, θ0=μ0α0α1αq, Σt,n0=I2, and ξ̂t12=yt1μ02, then the AQL estimation of σt2is the solution of Gtσt2=0, that is,

σ̂t2=α0+α1ξ̂t12++αqξ̂tq2,t=1,2,3,T.E15

Second, by kernel estimation method, we find

Σ̂t,nθ0=σ̂nytσ̂nytσtσ̂nσtytσ̂nσt.

Third, to estimate the parameters θ0=μ0α0α1αqusing σ̂t2and ytand the sequence of (AQLEF):

GTμ0α0α1αq=t=1T10010ξt120ξtq2Σ̂t,n1×ytμσt2α0α1ξt12αqξtq2.

The AQL estimate of θ0=μ0α0α1αqis 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+α1ξt12+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=ytμσt2α0α1ξt12.

The QLEF to estimate σt2is given by

Gtσt2=01σt200σζ21ytμσt2α0α1ξt12=σζ2σt2α0α1ξt12.E18

Given ξ̂0=0, initial values ψ0=μ0α00α10σζ02and ξ̂t12=yt1μ02, then the QL estimation of σt2is the solution of Gtσt2=0,

σ̂t2=α0+α1ξ̂t12,t=1,2,3,T.E19

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

GTμα0α1=t=1T10010ξt12σt200σζ021ytμσt2α0α1ξt12.

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

μ̂=t=1Tytσ̂t2/t=1T1σ̂t2.E20
α̂1=Tt=1Tσ̂t2ξ̂t12t=1Tσ̂t2t=1Tξ̂t12Tt=1Tξ̂t14t=1Tξ̂t122.E21
α̂0=t=1Tσ̂t2α̂1t=1Tξ̂t12T.E22

and let

σ̂ζ2=t=1Tζ̂tζ̂¯2T1E23

where ζ̂t=σ̂t2α̂0α̂1ξ̂t12,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,n1ytμσt2α0α1ξt12

Given ξ̂0=0, θ0=μ0α0α1μ0, Σt,n0=I2and ξ̂t12=yt1μ02, then the AQL estimation of σt2is the solution of Gtσt2=0, that is,

σ̂t2=α0+α1ξ̂t12,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=1T10010ξ̂t1Σ̂t,n1ytμσt2α0α1ξt12.

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

μ̂=t=1Tytσ̂nyt/t=1T1σ̂nyt.E25
α̂1=t=1Tσ̂t2σ̂nσtt=1Tξ̂t12σ̂nσtt=1T1σ̂nσtt=1Tξ̂t12σ̂nσtt=1Tξ̂t12σ̂nσt2t=1T1σ̂nσtt=1Tξ̂t14σ̂nσt.E26
α̂0=t=1Tσ̂t2σ̂nσtα̂1t=1Tξ̂t12σ̂nσtt=1T1σ̂nσt.E27

and let

σ̂ζ2=t=1Tζ̂tζ̂¯2T1E28

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.

α0α1μα0α1μα0α1μ
True0.0100.9801.300.0100.980−1.300.0100.9800.030
QL0.0090.9891.2990.0090.989−1.300.0090.9890.029
0.0010.0100.0060.0010.0100.0060.0010.0100.006
AQL0.0090.9891.300.0090.989−1.290.0090.9890.030
0.0010.0100.00030.0020.0090.00030.0010.0090.0003
True0.0500.9501.300.0500.950−1.300.050.9500.030
QL0.0490.9491.290.0490.940−1.300.0490.940.029
0.0010.00010.0140.0010.0100.0140.0010.0100.014
AQL0.0490.9401.320.0490.940−1.300.0490.9400.032
0.0010.0100.0180.0010.0100.0180.0010.010.001
True0.100.901.300.100.90−1.300.100.900.030
QL0.0980.9101.290.0980.910−1.300.0980.9100.023
0.0020.0100.0190.0020.0100.0200.0020.0100.029
AQL0.0980.9101.310.0980.910−1.320.0980.9100.031
0.0020.0100.0120.0020.0100.0210.0010.0100.001
True0.10.90−0.030.050.95−0.030.010.98−0.03
QL0.0980.910−0.0310.0510.949−0.0300.0090.990−0.030
0.0020.0100.0190.0010.0010.0140.0010.0160.006
AQL0.0980.910−0.0310.0510.949−0.0310.0090.990−0.031
0.0020.0100.0010.0010.0010.0020.0010.0100.001

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.

α0α1μα0α1μ
T = 20True0.0100.980−0.0300.050.9501.3
QL0.0090.990−0.0290.04950.94851.300
0.00080.01000.03190.00050.00150.0703
AQL0.0090.990−0.0310.04950.94851.3107
0.00090.0100.00840.00050.00150.0213
T = 40QL0.0090.990−0.0310.04950.94851.3015
0.000890.0100.02230.00050.00150.0492
AQL0.0090.990−0.0310.04950.94851.3113
0.000890.0100.00390.00050.00150.0143
T = 60QL0.0090.990−0.0290.04950.94851.300
0.00090.0100.01800.00050.00150.0404
AQL0.0090.990−0.0310.04950.94851.311
0.00090.0100.00270.00050.00150.0128
T = 80QL0.0090.990−0.0290.04900.94851.300
0.00090.0100.0160.00050.00150.0353
AQL0.0090.990−0.3100.04950.94851.3112
0.00090.0100.00200.00050.00150.0119
T = 100QL0.0090.9900.02920.04950.94851.3017
0.00090.0100.01420.00050.00150.0314
AQL0.0090.990−0.0310.04950.94851.3111
0.00090.0100.00180.00050.00150.0116

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=logrtlogrt1.

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+α1ξt12+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.

α̂0α̂1μ̂ξ̂¯tS.dξ̂t
QL0.13000.8387−0.000120.00013
AQL0.02000.9599−0.001110.1350

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+α1ξt12++αpξtp2+β1σt12++βqσtq2,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=ytμσt2α0α1ξt12αpξtp2β1σt12βqσtq2.

The QLEF to estimate σt2is given by

Gtσt2=01σt200σζ21ξtζt=σζ2σt2α0α1ξt12αpξtp2β1σt12βqσtq2.E33

Given ξ̂0=0, initial values ψ0=μ0α00α10αp0β10βq0σζ02, ξ̂ti2=ytiμ02, and σ̂tj2are the QL estimations of σtj2, where i = 1, 2, , p and j = 1, 2, , q, then the QL estimation of σt2is the solation of Gtσt2=0,

σ̂t2=α0+α1ξt12++αpξtp2+β1σt12++βqσtq2,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=1T10010ξt120ξtp20σt120σtq2σt200σζ021ξtζt

The QL estimate of μ, α0, α1, , αq, β1, , βqis the solation of GTθ=0, where ζ̂t=σ̂t2α̂0α̂1ξ̂t12α̂pξ̂tp2β̂1σ̂t12β̂qσ̂tq2,t=1,2,3,,Tand

σ̂ζ2=t=1Tζ̂tζ̂¯2T1E35

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,n1ξtζt

Given ξ̂0=0, θ0=μ0α00α10αp0β10βq0, Σt,n0=I2, and ξ̂ti2=ytiμ02, and σ̂tj2is the AQL estimation of σtj2, 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+α1ξ̂t12++αpξtp2+β1σt12++βqσtq2,t=1,2,3,T.E36

Second, by kernel estimation method, we find

Σ̂t,nθ0=σ̂nytσ̂nytσtσ̂nσtytσ̂nσt.

Third, to estimate the parameters θ0=μ0α0α1αqusing σ̂t2and ytand the sequence of (AQLEF):

GTμ0α0α1αq=t=1T10010ξt120ξtq20σt120σtq2Σ̂t,n1ξtζt.

The AQL estimate of θ=μα0α1αqis 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+α1ξt12+β1σt1+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=ytμσt2α0α1ξt12β1σt12.

The QLEF to estimate σt2is given by

Gtσt2=01σt200σζ21ytμσt2α0α1ξt12β1σt12=σζ2σt2α0α1ξt12β1σt12.E39

Given ξ̂0=0, initial values ψ0=μ0α00α10σζ02, ξ̂t12=yt1μ02, and σ̂t12is the QL estimation of σt12, then the QL estimation of σt2is the solation of Gtσt2=0,

σ̂t2=α0+α1ξ̂t12+β1σ̂t12,t=1,2,3,T.E40

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

GTμα0α1β1=t=1T10010ξt120σt12σt200σζ021ytμσt2α0α1ξt12β1σt12.

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

μ̂=t=1Tytσ̂t2/t=1T1σ̂t2.E41
β̂1=Sσ̂t12ξ̂t12Sσ̂t2ξ̂t12Sξ̂t12ξ̂t12Sσ̂t2σ̂t12Sσ̂t12ξ̂t122Sσ̂t12σ̂t12Sξ̂t12ξ̂t12.E42
α̂1=Sσ̂t2ξ̂t12β̂1Sσ̂t12ξ̂t12Sξ̂t12ξ̂t12.E43
α̂0=t=1Tσ̂t2α̂1t=1Tξ̂t12β̂1t=1Tσ̂t12T.E44

and let

σ̂ζ2=t=1Tζ̂tζ̂¯2T1E45

where

ζ̂t=σ̂t2α̂0α̂1ξ̂t12β̂1σ̂t12,t=1,2,3,,T,Sσ̂t12ξ̂t12=t=1Tσ̂t12ξ̂t12t=1Tσ̂t12t=1Tξ̂t12T,Sσ̂t2ξ̂t12=t=1Tσ̂t2ξ̂t12t=1Tσ̂t2t=1Tξ̂t12T,Sξ̂t12ξ̂t12=t=1Tξ̂t14t=1Tξ̂t122T,Sσ̂t2σ̂t12=t=1Tσ̂t2σ̂t12t=1Tσ̂t2t=1Tσ̂t12T,Sσ̂t12σ̂t12=t=1Tσ̂t14t=1Tσ̂t122T.

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,n1ytμσt2α0α1ξt12β1σt12

Given ξ̂0=0, θ0=μ0α0,0α1,0β1,0, Σt,n0=I2, ξ̂t12=yt1μ02, and σ̂t12is the AQL estimation of σt12, then the AQL estimation of σt2is the solation of Gtσt2=0, that is,

σ̂t2=α0+α1ξ̂t12+β1σt12,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=1T10010ξ̂t120σ̂t12Σ̂t,n1ytμσt2α0α1ξt12β1σt12.

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

μ̂=t=1Tytσ̂nyt/t=1T1σ̂nyt.E47
β̂1=SSσ̂t12ξ̂t12SSσ̂t2ξ̂t12SSξ̂t12ξ̂t12SSσ̂t2σ̂t12SSσ̂t12ξ̂t122SSσ̂t12σ̂t12SSξ̂t12ξ̂t12.E48
α̂1=SSσ̂t2ξ̂t12β̂1SSσ̂t12ξ̂t12SSξ̂t12ξ̂t12.E49
α̂0=t=1Tσ̂t2σ̂nσtα̂1t=1Tξ̂t12σ̂nσtβ̂1t=1Tσ̂t12σ̂nσtt=1T1σ̂nσt,E50

and let

σ̂ζ2=t=1Tζ̂tζ̂¯2T1E51

where

ζ̂t=σ̂t2α̂0α̂1ξ̂t12β̂1σ̂t12,t=1,2,3,,T,SSσ̂t12ξ̂t12=t=1Tσ̂t12ξ̂t12σ̂nσtt=1T1σ̂nσtt=1Tσ̂t12σ̂nσtt=1Tξ̂t12σ̂nσt,SSσ̂t2ξ̂t12=t=1Tσ̂t2ξ̂t12σ̂nσtt=1T1σ̂nσtt=1Tσ̂t2σ̂nσtt=1Tξ̂t12σ̂nσt,SSξ̂t12ξ̂t12=t=1T1σ̂nσtt=1Tξ̂t14σ̂nσtt=1Tξ̂t12σ̂nσt2,SSσ̂t2σ̂t12=t=1T1σ̂nσtt=1Tσ̂t2σ̂t12σ̂nσtt=1Tσ̂t2σ̂nσtt=1Tσ̂t12σ̂nσt,SSσ̂t12σ̂t12=t=1T1σ̂nσtt=1Tσ̂t14σ̂nσtt=1Tσ̂t12σ̂nσt2.

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.

μα0α1β1μα0α1β1
True0.150.650.870.100.200.410.880.08
QL0.1490.7790.8650.0740.1990.4610.9120.057
0.0400.3530.0110.0290.0310.1550.0330.025
AQL0.1500.6610.8510.0920.2090.4050.9010.076
0.0010.0120.0190.0090.0100.0060.0210.004
True−0.100.480.890.080.160.370.90.08
QL−0.1010.5560.9020.0580.1590.4340.9220.058
0.0340.2120.0140.0240.0300.1890.0240.025
AQL−0.1100.4860.8910.07520.1610.3740.9110.076
0.0100.0060.0010.0050.0010.0040.0110.004
True0.180.390.880.080.090.500.890.05
QL0.1790.4470.8920.0580.0890.5380.8980.036
0.0310.1460.0150.0240.0330.0900.0090.015
AQL0.1800.3950.8820.0760.0910.5040.8920.046
0.0010.0050.0020.0050.0020.0040.0020.004

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.

μα0α1β1μα0α1β1
True0.160.370.900.08−0.100.480.890.08
QL0.170.420.890.07−0.090.510.900.06
T = 200.1760.5110.0080.0160.1690.4510.0180.022
AQL0.160.380.890.07−0.100.470.900.07
0.0370.0120.0070.0140.0660.0140.0130.018
QL0.160.420.890.07−0.090.510.910.06
0.1490.4220.0070.0160.1370.3260.0180.021
T = 40AQL0.160.380.890.07−0.100.470.900.07
0.0270.0120.0070.0130.0220.0140.0120.016
QL0.160.420.890.07−0.090.520.910.06
0.1210.2890.0070.0180.1190.3070.0180.021
T = 60AQL0.160.380.890.07−0.100.470.900.07
0.0190.0120.0070.0110.0140.0130.0120.015
QL0.160.420.890.07−0.100.510.900.06
0.1000.1590.0070.0170.1080.2480.0180.021
T = 80AQL0.160.380.890.07−0.100.470.900.07
0.0120.0120.0070.0110.0120.0130.0120.015
QL0.160.420.890.07−0.100.510.900.06
0.1000.1590.0070.0180.1010.2420.0180.021
T = 100AQL0.160.380.890.07−0.100.470.900.07
0.0120.0110.0070.0110.0110.0130.0120.015

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=logPtlogPt1and fit ytby using GARCH (1,1):

yt=μ+ξt,t=1,2,3,,T.E52

and

σt2=α0+α1ξt12+β1σt1+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.

μ̂0α̂0α̂1β̂1ξ̂¯tS.dξ̂t
QL0.00080.5660.9120.00040.002
AQL0.00890.6300.9720.0410.185

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.

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Raed Alzghool (December 22nd 2020). ARCH and GARCH Models: Quasi-Likelihood and Asymptotic Quasi-Likelihood Approaches [Online First], IntechOpen, DOI: 10.5772/intechopen.93726. Available from:

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