A Note on GARCH(1,1) Estimation via Different Estimation Methods

PETER PRINC, VÁCLAV ŠKOLUDA Department of Econometrics

University of Economics W. Churchill Sq. 4, 13067, Prague 3

CZECH REPUBLIC peter.princ@vse.cz, vaclav.skoluda@vse.cz

Abstract: - Generalized autoregressive conditional heteroscedasticity (GARCH) model has earned large success for its competitiveness and parsimony in the financial econometric literature. GARCH model is based on the infinite ARCH specification term that reduces the number of estimated parameters from infinity to two. This paper analyzes estimation bias for different GARCH specification in various sample sizes. Furthermore, we employ generalized method of moments and maximum likelihood framework for estimation evaluation. We show that selected estimation methods yields to different biases under various sample sizes assumption. Finally, our results suggest that maximum likelihood estimation gives better estimates than generalized method of moments Key-Words: GARCH, generalized method of moments, maximum likelihood estimation, volatility, contour plot, sample size 1 Introduction Volatility indicates a period of time series that is associated with high variability or increasing variance. This phenomenon plays an important role in the modeling and analysis of financial time series. Volatility is a key element of procedures that assess the overall risk of financial assets. Predictions from volatility models are used in the investment decision-making processes for capturing the potential risk of potential investment portfolio, in the analysis of VaR models or models of options.

Financial time series are typified by volatility clustering, leptokurtic distributions and so-called leverage effect [1]. Such observations led to the wider use of the models with conditional variance. Engle [3] suggested in his pioneering work to model time-varying conditional variance as autoregressive conditional heteroscedasticity (ARCH) process with lagged random errors. The latter work has demonstrated the usefulness of higher-level ARCH process in dynamic nature representation of the conditional variance. Generalized ARCH model (GARCH) proposed by Bollerslev [2] is based on the infinite ARCH specification term that reduces the number of estimated parameters from infinity to two. Both ARCH and GARCH models are efficient to describe the volatility clustering in financial time series, however fail to model the leverage effect. Another issue that GARCH model cannot fully capture is the leptokurtosis. It means that probability

distribution of the process is more kurtic with thicker tails than with normal distribution. The way how to overcome these problems is the use of random errors from non-normal distribution.

Suitable probability function for empirical financial time series should therefore have the following characteristics. The distribution must be flexible to capture the intended range of shapes of selected process. Skewness and kurtosis in the time series of financial asset returns should be modeled according to the parameters of the distribution shape.

2 Methodology 2.1 GARCH(1,1) model GARCH(1,1) model proposed by Bollerslev [2] is formulated as t tw ε= t t teε σ= 2 2 2

1 1t t tσ ω αε βσ− −= + + , where the conditional volatility 2

tσ of tw is set on the information known at the time t-1and te is distributed iid with zero mean and unit variance. The covariance-stationary condition for GARCH(1,1) process is 1α β+ < and the level of persistence is given by α β+ . Parameter estimates

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from QMLE are consistent and asymptotically normal under conditions investigated by Ling and McAleer [6].

2.2 Maximum likelihood estimation Maximum likelihood estimation (MLE) has gained very high popularity as a method for GARCH model estimation. This method includes maximization of the likelihood function that is formulated on the normal error assumption. Under suitable regularity conditions, ML estimates have been found consistent and asymptotically normally distributed. However, the appropriate regularity conditions have been only identified for simple GARCH models [5]. Therefore, it is generally assumed in practice that these conditions are fulfilled. The log-likelihood function for normally distributed standardized errors is

22

21

1 1( ; ) log(2 ) log( ) ,2 2 2

Tt

tt t

wl wθ ϖ σσ=

= − − −∑

where w is appropriate financial time series and θ is parameters space.

2.3 Generalized method of moments

The idea of generalized method of moments (GMM) was firstly introduced by Lars Hansen in 1982 [4] and it is nowadays used for model parameters estimation and test model specification. The GMM framework is applied in various fields of study1 such as finance, macroeconomics, agricultural economics, environmental economics and labour economics. Unlike the ML estimation, GMM estimation does not stand on the basis of the joint probability distribution of the data known as the likelihood function. The GMM is useful in circumstances when the distributional assumption is not precisely known or the likelihood function is difficult for numerical evaluation. Let us outline the assumption for GARCH(1,1) GMM estimation

2 2(( ) ) 0,t t tE sε σ− =

where 2 21 1(1, , )t t ts ε σ− −= . Under this assumption

the parameters are estimated by choosing vector ' ( , , )θ ω α β= that minimizes

1 For further discussions of GMM study areas, see Hall (2005).

'1 1

ˆ( ( ; )) ( ( ; )),t tg I S g Iθ θ− −

where

1 2 21

1( ; ) ( )

T

t t t tt

g I T sθ ε σ−−

=

= − ∑

and the matrix S is positive-definite weighting matrix constructed as a Newey-West estimator [7].

3 Simulation Study The question of adequate sample size for maximum likelihood estimation in the field of volatility models is not rigorously described in the current literature. Parameter estimation is more complex in the small sample size, with many GARCH parameters and under different error distribution assumptions. The latter involves employment of the quasi maximum likelihood method. In our study we therefore examine two GARCH(1,1) processes with different specification. Table 1 and Table 2 contain the simulation results from GMM and ML estimation. It is evident that maximum likelihood estimation method yields smaller sample biases than the GMM estimation. However, GMM estimation is convenient for ω and α estimates. For the estimation of β parameters the GMM method performs poorly and the estimate is highly negatively biased. The enlargement of the sample size yields better asymptotic performance from the statistical point of view. This fact is shown in tables 1 and 2 in smaller standard errors of estimates for larger sample sizes and in smaller bias for larger sample sizes as well. For these reasons we do not recommend using GMM estimation framework for GARCH(1,1) model.

We further take a closer look on each estimation method via the optimization contour plots. Figures 1, 2, 3 shows objective function optimum in each parameter’s pair. It is obvious that the reason behind the ML bias in small sample size arises from the objective function incapability for true parameter targeting. In a similar manner we conclude that the reason for GMM bias lies in the long narrow optimum area that is stretched almost orthogonally on the 45 degree plane line.

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Generalized method of moments estimation

Average Value of Estimates

Sample Size ω α β 500 0.0194 0.0359 0.2774

1000 0.0171 0.0484 0.3549 3000 0.0155 0.0456 0.4078 5000 0.0131 0.0502 0.4937

True values 0.01 0.05 0.6 500 0.0954 0.0780 0.3428

1000 0.0771 0.0920 0.4424 3000 0.0540 0.0951 0.5793 5000 0.0524 0.0997 0.5867

True values 0.05 0.1 0.6 Standard Error of Estimates

500 0.0059 0.0470 0.1960 1000 0.0039 0.0355 0.1246 3000 0.0024 0.0206 0.0796 5000 0.0047 0.0170 0.1683 500 0.0228 0.0533 0.1416

1000 0.0249 0.0386 0.1563 3000 0.0114 0.0378 0.1149 5000 0.0114 0.0210 0.0879

Maximum likelihood estimation

Average Value of Estimates

Sample Size ω α β 500 0.0106 0.0416 0.5839 1000 0.0124 0.0529 0.5802 3000 0.0108 0.0492 0.5700 5000 0.0105 0.0509 0.5829

True values 0.01 0.05 0.6 500 0.0562 0.0860 0.5733 1000 0.0581 0.0978 0.5519 3000 0.0492 0.0974 0.6073 5000 0.0505 0.1013 0.5965

True values 0.05 0.1 0.6

Maximum likelihood estimation

Standard Error of Estimates

Sample Size ω α β 500 0.0009 0.0361 0.2143 1000 0.0008 0.0289 0.2042 3000 0.0004 0.0163 0.1706 5000 0.0004 0.0127 0.1445 500 0.0092 0.0425 0.2050 1000 0.0069 0.0322 0.1580 3000 0.0039 0.0177 0.0967 5000 0.0030 0.0140 0.0738

Table 2: ML estimation results

Fig. 1: Contour plots for GMM estimation

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Fig. 2: Contour plots for GMM and ML estimation

Fig. 3: Contour plot for ML estimation

4 Conclusions This paper has given an account of and the reasons for the appropriateness of maximum likelihood estimation framework for GARCH(1,1) model. We scrutinize generalized method of moments as well as maximum likelihood estimation framework for GARCH(1,1) model and found several drawbacks of each method with objective function inspection. Both methods are biased for small sample sizes, i.e. 500 observations. However these biases are outcomes from different problems. Maximum likelihood objective function is hard to specify for small sample sizes and therefore the objective function does not target the true parameter value. Generalized method of moments drawback lies in long and narrow optimum area that is almost orthogonal to 45 degree plane line. This can easily effect the jump in parameter’s space to the parameter value that is noticeably different from the true value. 5 Acknowledgements The research project was supported by Grant No. IGA F4/1/2012, Faculty of Informatics and Statistics, University of Economics, Prague. References:

[1] Black, F.: Studies of stock market volatility changes. In: Proceedings of the American Statistical Association, Business and Economic Statistics Section. 1976, 177–81

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[2] Bollerslev, T.: Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31 (1986), 307–327.

[3] Engle, R.F., Bollerslev, T.: Modeling the persistence of conditional variance. Econometric Reviews 5 (1986), 1-50.

[4] Hansen, L.P.: Large sample properties of Generalized Method of Moments estimators. Econometrica 50 (1982), 1029-54.

[5] Lee, S. & Hansen, B.: Asymptotic Theory for the GARCH(1,1) Quasi- Maximum Likelihood Estimator. Econometric Theory, 1993, 10, 29-52

[6] Ling, S. and McAleer, M.: Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19 (2003), 280-310.

[7] Newey, W.K. and West, K.D.: A Simple, Positive-Definite, Heteroskedasticity and Autocor- relation Consistent Covariance Matrix. Econometrica, 55 (1987), 703–708.

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