On Exploring the Estimating Parameter of Exponential Generalized Autoregressive Conditional Heteroscedasticity (EGARCH) Models

1. Introduction

At some cases, time series data showing behavior differently, linear and non-linear. Stock market data, monetary data, and economic data, for example, often behave as non linear, while other demand in production line data almost often behave in linear form. The other view comes up due to their heteroscedasticity. To grasp this characteristic into the model, in order to overcome the problem, one not only has to involve a special approach called Autoregressive Conditional Heteroscedasticity (ARCH) introduced firstly by Engle (1982) and Bollerslev (1986), but also has to considering Generalized Autoregressive Conditional Heteroscedasticity (GARCH). Even family of GARCH, such as Exponential GARCH (Nelson, 1991) is frequently used to transform the variance of data leading not constant.

Before forecasting step, especially when EGARCH is used, the requirement of parameter estimation for making the model closer to the pattern of data has to be made. With the intention that estimator can minimize risk. Some method frequently used to estimate parameter, which often give estimator coming near truth, are maximum likelihood estimation, least square method, and moment method.

Many researchers have studied about these methods in estimating parameter model. Some of them are Cryer (1986) who had succeed to use method of moment to estimate parameter time series model; Box, Jenkins, and Reinsel (1994) who had carried out in estimating parameter using maximum likelihood estimation, least squarer estimates, and Bayesian estimation; and Hilmer and Tiao (1979) who had been certified in using likelihood function in estimating the stationerity of ARIMA model.

This paper deals with estimating parameter model using maximum likelihood estimation couple with its statistics tests for Exponential GARCH (EGARCH) model.

2. ARIMA Box-Jenkins Models

In some time series modeling, series data are assumed to be stationer, and therefore, there are no trend in mean and variance. But, in some fact, when the data follow the stochastic change, time series analysis has to consider the process by expressing it into a random variable with the density function . Where would be a time series data of having value at a random variable from such probability distribution (Wei, 1990).

A popular method developed for analyzing this time series data is ARIMA Box-Jenkins methods introduced by Box-Jenkins (Box, et. al, 1994). But this model is still claimed stay on with the existence of its residual assumption to be white noise and normally distributed. The advance and general model of ARIMA Box-Jenkins is a mixed autoregressive order p which moving average order q and differencing d can be written as

, (1)

where dan .

Even the innovation and modification of d in (1) to be altered from integer to fractional and real number, and therefore, changing its name from ARIMA to ARFIMA model (Autoregressive Fractional Integrated Moving Average), the assumption above can not always be granted.

3. ARCH, GARCH, and EGARCH Models

Autoregressive Conditional Heteroscedasticity (ARCH) process introduced in Engle (1982) allows the conditional variance to change over time as a function of past errors laving the unconditional variance constant. ARCH modeling is a method to model time series data having heteroscedasticity properties in the error variance or error variance a conditional function (Lubrano and Bauwens, 1998).

ARCH models is an ARIMA Box-Jenkins models with white noise residual, but the square plot of residual show the existence of change of variance. Doing with this kind of data, Engle (1982) proposed to conduct an addition modeling of square of residual, and then draw the conclusion added by the information of the change matching with time change.

Consider that we have models:

(2)

where f is at least twice continuously differentiable function of , with . Where error process is parameterized as:

, t = 1, 2, ..., T. (3)

Engle (1982) ARCH (q) models can be defined as follow:

, (4)

where , , and where is a sequence of independent identically distributed random variable with zero mean and unit variance or .

Bollerslev (1986) extended the ARCH model to the Generalized Autoregressive Conditional Heteroscedasticity (GARCH). GARCH models value of in (3) by writing as

, (5)

where. If p = 0 and q = 1, then this model can be written as GARCH(0,1) which constitute to ARCH(1) model, and if = 0, then GARCH (p,q) equivalent to ARCH (q).

One of family GARCH model is Exponential GARCH. This model was firstly proposed by Nelson (1991). EGARCH model can be written based on (3) or by changing (5) into the exponential form and by adding a new parameter which representing the linear combination of . A model EGARCH(p,q), therefore, can be represented as

, (6)

where , such that:

(7)

Model (7) is defined as EGARCH(p,q) model (Nelson, 1991).

Data which follows EGARCH process can be identified by calculating its sample ACF and PACF from its square residual using Lagrange Multiplier (LM) test. The LM test can be carried out in TR² form as follows:

1. Estimate the parameter of the EGARCH (p,q) model and compute the square standardized residuals , t = 1, …, T, and the residual sum of square

2. Regress on and and compute the sum of squared residuals,

3. Compute the value of the test statistic

that has an asymptotic distribution with 2r degree of freedom under the null hypothesis.

4. Estimator of EGARCH Models

Suppose we have model (2) and (6). Then we can estimate parameter variance EGARCH(p,q) using Maximum Likelihood Estimation (MLE) method. Assuming that is normally distributed, then log likelihood function for EGARCH(p,q) can be written as

, (8)

with . (9)

where are parameters of .

Estimating parameters in (8) can not be done analytically, because they do not have close form functions. To do so, parameter estimation is done numerically from the second order conditions as arranger Hessian matrix. Some other methods, such as Newtons method, Quasi Newton method and BHHH Algorithm, are able to be used to estimate these parameters numerically (Lange, 1999).

Suppose that is an vactor of parameters to be estimated. Let denote the gradient vector of the log likelihood function at and let denite -1 times the matrix of second derivatives of the log likelihood function, then:

and

Consider approximating with a second order Taylor series around .

(10)

Setting the derivative of (10) with respect to equal to zero result in:

or (11)

(12)

One could next calculate the gradient and Hessian at and use the these to find a new estimate and continu iterating in this fashion. The th step in the iteration updates the estimate of by using the formula:

(13)

Process (13) can be calculate with numerical optimization using Quali Newton method (Lange, 1999):

(14)

If = in (15), then:

(15)

Where and

After estimating parameters of EGARCH model, a further step is evaluating the significance of parameter model. Suppose is parameter estimation EGARCH model, then hypothesis test is: H₀ : = 0 ( not significant) versus H₁ : ¹ 0 ( significant), with test statistics is , and rejection area define Reject H_0­ if where n_p­ is number of parameter.

5. Conclusion

When the high volatility and heteroscedasticity condition stay on in time series data, ARCH, GARCH, and EGARCH model could be used to model it. Maximum likelihood estimation and hypothesis test using t-statistic are applicable for estimating the parameter of these models, especially for EGARCH model.

6. References

Bollerslev, T. 1986.“Generalized Autoregressive Conditional Heteroscedasticity.” Journal of Econometrics, 31, 307–27.

Box, P.E.G, Jenkins M.G and Reinsel C. G., 1994. Time Series Analysis: Forecasting and Control, New Jersey, Prentice Hall.

Cryer, J.D., 1986. Time Series Analysis. PWS-KENT Publishing Company. Boston.

Engle, R. F., 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation”, Econometrica 50, 987-1007.

Hamilton, J.D., 1994. Time Series Analysis, Princeton, New Jersey.

Hillmer,S.C. and Tiao, G.C. 1979. ”Likelihood function of stationary multiple autoregressive moving average models”. Journal of the American Statistician Association. 74, 652-660.

Lange, K., (1999). “Numerical Analysis for Statisticians”. Springer-Verlag, New York.

Laws, J. and Andrew, G., 2000. “Ferecasting Stock Market Volatility and The Application of Volatility Trading Models”, Paper presented at CIBEF (Centre for Internationa Banking, Economics, and Finance, JMU, John Foster Building, 98 Mounth Pleasant, Liverpool L3 5UZ, November 1-16.

Lubrano and Bauwens, 1998. “Bayesian Inference on GARCH models using the Gibbs Sampler”, Journal of Econometrics, 1, C23 – C46.

Nelson, D. B., 1991. “Conditional Heteroskedasticity in Asset Returns: A New Approach.” Econometrica 59 2: 347-70

Wei W.S, 1990. Time Series Analysis, Univariate and Multivariate Methods. Addison-Wesley Publishing Company, New York.