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

## 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

and

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

and

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

where

### 2.1 The QL method

For the model given by Eq. (5), assume that

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

where

If the sub-estimating function spaces of

then the QLEF can be defined by

and the estimation of

A limitation of the QL method is that the nature of

### 2.2 The AQL method

The QLEF (see Eqs. (6) and (7)) relies on the information of

Definition 2.2.1: *Let**be a sequence of the EF in**. For all**, if*

is asymptotically nonnegative definite,

Suppose, in probability,

expresses an AQLEF sequence. The solution of

In this chapter, the kernel smoothing estimator of

The estimation of

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

and

The QLEF to estimate

Given

The QLEF, using

The QL estimate of

### 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

Given

Second, by kernel estimation method, we find

Third, to estimate the parameters

The AQL estimate of

### 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

and

#### 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

The QLEF to estimate

Given

To estimate the parameters

The solution of

and let

where

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

Given

Second, by kernel estimation method, we find

Third, to estimate the parameters

The AQL estimate of

and let

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

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 |

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.

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 |

### 3.4 Empirical applications

The first data set we analyze are the daily exchange rate of

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

and

The estimation of unknown parameters, (

QL | 0.1300 | 0.8387 | −0.00012 | 0.00013 |

AQL | 0.0200 | 0.9599 | −0.00111 | 0.1350 |

## 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

and

The QLEF to estimate

Given

The QLEF, using

The QL estimate of

### 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

Given

Second, by kernel estimation method, we find

Third, to estimate the parameters

The AQL estimate of

### 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

and

#### 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

The QLEF to estimate

Given

To estimate the parameters

The solation of

and let

where

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

Given

Second, by kernel estimation method, we find

Third, to estimate the parameters

The AQL estimate of

and let

where

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

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 |

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.

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 |

0.012 | 0.011 | 0.007 | 0.011 | 0.011 | 0.013 | 0.012 | 0.015 |

### 4.4 Empirical applications

The second set of data is the weekly price changes of crude oil prices

and

The estimation of unknown parameters, (

QL | 0.0008 | 0.566 | 0.912 | 0.0004 | 0.002 |

AQL | 0.0089 | 0.630 | 0.972 | 0.041 | 0.185 |

## 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.