Parameter Estimation of the Stable GARCH(1,1)-Model

Transcription

1 WDS'09 Proceedings of Contributed Papers, Part I, , ISBN MATFYZPRESS Parameter Estimation of the Stable GARCH(1,1)-Model V. Omelchenko Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic. Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Prague, Czech Republic. Abstract. The paper aims to show methodology of parameter estimation of the stable GARCH(1,1) model. There are represented and compared 3 methods of finding estimates of their parameters. We assume that we have a stable GARCH(1,1) model with the stable symmetric innovation. We search for the estimates of parameters of the stable GARCH model under assumption that we don t know anything about parameters, that is neither α nor parameters of the model are known, and all the information must be extracted from the observations. We develop the methodology of parameter estimation by simulating a sample from the stable GARCH model and estimating its parameters and comparing estimates with theoretical values. It has been shown by simulation that we can obtain appropriate estimates of the parameters having only 1000 observations. Introduction The stable distributions are characterized by the same convolution and limit properties as the normal distributions and they can be an extension of it. The normal distribution also belongs to the family of the stable distributions as a special case and the financial models such as stable GARCH(p,q) can serve as an extension of the classical GARCH(p,q) model with the normal innovation. The classical GARCH models with the stable innovation are quiet popular in economic practice but sometimes the behavior of prices of many assets is not typical for the normal models therefore, it is reasonable to replace the normal distribution by a family of the distributions extending the normal one. There are developed 3 methodologies of parameter estimation that will be described below. We can use Two Sample Kolmogorov Smirnov test to verify if the estimates are correct. The second sample will be obtained by the simulation. Simulation will be the central point of this methodology. Definition of the Stable Distributions There are four equivalent definitions of the stable distributions. We will mention only one of them which we need for our purposes. The rest of them specifies convolution and limit properties of the stable distributions and can be derived from the definition specifying its characteristic function. Definition A random variable X has the stable distribution if its characteristic function is of the form: and E exp(i u X) = exp ( σ α u α [ 1 iβ ( [ E exp(i u X) = exp σ u 1 + iβ ( tan α 2 ) sign(u) ] ) + iµu, for α 1 ( ) ] ) 2 sign(u) ln u + iµu, for α = 1 This definition is the most important for our purposes because it gives us the explicit form of the characteristic function. The density and the distribution functions of the stable distributions do not have any explicit form with only 3 exceptions: normal distribution, Cauchy distribution and Levy distribution. The constants contained in the formula of the characteristic function are parameters of the stable distributions which uniquely determine them. The most important parameter is α which is called the tail index. α (0, 2] and if α = 2 then we deal with the normal distribution. If α < 2 then for any a α EX a = and EX a < if a < α. The parameter σ is called the scale parameter which has the properties of the standard deviation. The parameter µ is called the location parameter, it equals the mean, i.e. EX = µ if α > 1. The parameter β is called the index of asymmetry which equals zero if α = 2 and belongs to the interval [ 1, 1] if α < 2. The stable distribution with parameters α, σ, β, and µ is denoted by S α (σ, β, µ). The centralized stable distribution S α (σ, 0, 0) will be denoted by S α S. E.g 137

2 N(0, σ 2 ) = S 2 (σ/ 2, 0, 0) = S 2 S. Simulation of the Stable Distributions To simulate a random variable X with the stable distribution it is enough to simulate uniform and exponential distributions. [2] For α 1 we have: X = S α,β sin{α(v + B [ ] (1 α)/α α,β)} cos{v α(v + Bα,β )} {cos(v )} 1/α W where B α,β = arctan(β tan α 2 ) α V U ( 2, ) 2 and W exp(1). For α = 1 we compute: X = 2 { ( { ( and S α,β = 1 + β 2 tan 2 α β V ) tan V β ln ( W 2 cos V 2 + β V )} )} 1/(2α) The best way to assess the quality of the methodology of the parameter estimation of the stable distributions is to simulate the sample of the stable distribution knowing all its parameter and then to try to estimate the parameters. The Stable GARCH(1,1)-Model The stable GARCH(1,1)-model is defined as follows: h n = σ n ε n, σ n = a 0 + a 1 h n 1 + b 1 σ n 1, ε n S α (1, 0, 0) and i.i.d., for n=1,2,.. a 0, a 1 and b 1 are larger than 0 and a 1 + b 1 < 1. [4] Parameter estimation of the stable distributions The most important parameter is α and its estimation is the most complicated. The easiest task is the parameter estimation of σ and µ. But this task is easy provided that α is greater than 1. In case when all parameters are unknown we can use properties of the characteristic function. If X 1, X 2,..., X n is a random sample from S α (σ, β, µ) with unknown parameters, then the empirical characteristic function is equal to Ψ emp (t) = 1 n n exp(i t X j), t R. The parameter estimates will be obtained by minimization the function K f(α, σ, β, µ) = Ψ(α, σ, β, µ; t j ) Ψ emp (t j ) where t 1, t 2,..., t K are points located close to 0 because the closer to zero the point, the closer the value of the empirical characteristic function to the theoretical one and Ψ(α, σ, β, µ, t) = exp ( σ α u α [ 1 iβ ( ) ] ) tan α 2 sign(u) + iµu. That is the parameter estimates are obtained by solution the program: min f(α, σ, β, µ) α (0,2],σ>0,β [ 1,1],µ R The consistency of this estimator is guaranteed by the Csörgő theorem [3] and it was proven in detail in my diploma thesis supervised by Klebanov [5] Csörgő Theorem For any characteristic function C, if for the deterministic sequence T n such that the following property holds: T n lim n n = 0 then where lim n(t n ) = 0 a.s. n n (T n ) = sup C n (t) C(t), C(t) = exp (itx) df (x) t T n R C n (t) = exp (itx) df n (x) = 1 n exp(itx j ) R n 138

3 Parameter estimation of the stable GARCH(1,1)-model The parameter estimation of this model involve optimization problems. There are many approaches how to estimate parameters of this model. We will demonstrate 3 methos 1) Method based on characteristic function 2) Mtheod based on maximum likelihood 3) Method based on minimization Kolmogorov Smirnov statistic Empirical characteristic function For X S α (1, 0, 0) the characteristic function is of the form: ψ(t, α) = exp( t α ) and the empirical characteristic function is of the form ψ EMP (t) = 1 n n cos(tx j) where X 1, X 2,..., X n are the observations of the stable distribution S α (1, 0, 0) where α is unknown and t R 1. The estimate of the α parameter is obtained by minimizing the function M i=1 ψ(t i, α) ψ EMP (t i ), where t 1, t 2,..., t M are some points in the vicinity of zero because the closer the point to zero, the closer the value of the empirical characteristic function to the theoretical one at that point. When we deal with the GARCH model we can apply the similar methodology. The parameter estimations will be obtained by minimizing the function over α, a 0, a 1 and b 1 : f(α, a 0, a 1, b 1 ) = ˆε i = n exp ( ˆε j α ) ψ EMP (ˆε j ) h i a 0 + a 1 h i 1 + b 1 σ i 1, i = 1, 2,.., n It is difficult to define this function because the empirical characteristic function must be obtained by virtue of simulation of the sample from S α (1, 0, 0)-distribution. But it is easy to calculate the value of this function at any point θ = (α, a 0, a 1, b 1 ) T therefore, we can solve this problem in the discrete setup calculating it in a large number of points and choosing such α and the trinity a 0, a 1, b 1 for which the function f(,,, ) has the lowest value. Maximum Likelihood To find the parameters of the GARCH model we can use the notoriously known method of maximum likelihood. We will maximize the following function with respect to these parameters: L(α, a 0, a 1, b 1 ; h 1,..., h n ) = n ( ) σ 1 hj j S α σ j where S α ( ) is the density function of S α (1, 0, 0) which we need to approximate. We emphasize that there is not any explicit form of density nor distribution function for a general stable distribution, moreover its density function is not of the exponential type. Therefore, this method is complicated first of all from the computational point of view. The definition of such function and its optimization requires so much memory that the software in which the parameter estimation is conducted alerts that there was a lack of the memory when we compute it directly without some special numerical techniques which exceed this paper. It is brought about by the fact that the function S α ( ) is defined in the form of the Lebegue integral. And it causes many computational problems when we have a product of hundreds of such functions. But there is a solution how to settle this problem. On the one hand it is takes up a lot of memory and time but on the other hand it is easy to compute the value of this function in any point not having defined it. This function is smooth and continuous with respect to these parameters therefore, we can find the optimal estimates of the parameters having calculated a large number of points and having chosen such sample of the estimates for which the value of maximum likelihood function is the largest. [4] Minimization of Kolmogorov-Smirnov Statistic How can we verify that the sample X 1, X 2,..., X n has the stable distribution? We can first of all estimate its parameters. Suppose that ˆα is the estimate of the parameter α. We will simulate the sample Y 1, Y 2,..., Y n from Sˆα (1, 0, 0)-distribution and apply the two-sample Kolmogorov-Smirnov test on both samples. When we have the GARCH-model we know a priori that h n /(a 0 + a 1 h n 1 + b 1 σ n 1 ) 139

4 S α (1, 0, 0). Therefore the Kolmogorov-Smirnov test will be applied on h j /(a 0 + a 1 h j 1 + b 1 σ j 1 ) S α (1, 0, 0), j = 1, 2,.., n and the samle Z 1, Z 2,..., Z n obtained by simulation the sample of the size n from S α (1, 0, 0). The Kolmogorov-Smirnov test is based on the Kolmogorov-Smirnov statistic which equals: MN D M,N = M + N sup F M (x) G N (x) x Where M and N are sample sizes and F N and G N are their empirical distribution functions. When M = N it takes the form: N D N,N = 2 sup F N (x) G N (x) x Table of the results The following table specifies the exactness of the estimates of the parameters of the stable GARCH(1,1) model with 1000 observations. ML means that the estimate was obtained by maximum likelihood methodology and CFB means that it was obtained by the methodology based on the characteristic function. Table 1. Table of parameter estimates of GARCH(1,1) model for different values of the unknown parameters Method α a 0 a 1 b 1 ˆα â 0 â 1 ˆb1 ML ML ML CFB CFB CFB KSB CFB CFB CFB CFB CFB CFB CFB CFB CFB CFB CFB CFB The table below specifies deviations of the estimates from the real values of the unknown parameters for the stable GARCH(1,1) model with parameters α = 1.6, a 0 = 0.02, a 1 = 0.09, b 1 = The number of observations for every estimate is The number of estimates is 50: Table 2. Table showing how the average of the estimates converges to the real value of the parameter Parameters α a 0 a 1 a 2 Theoretical Values Mean of the Estimates Standard Deviation

5 Demonstration of the KSB Methodology The basic thought of this methodology lies in the fact that if we have obtained residuals of the stable GARCH model as follows: ɛ n = h n a 0 + a 1 h n 1 + b 1 σ n 1, for n = 1, 2,.. where a 0, a 1 and b 1 are real values of the parameters of the model then ɛ n S α (1, 0, 0) for n = 1, 2,.. where α is the parameter of the innovation of the GARCH(1,1) model. This follows from the definition of the stable GARCH model. This means that Kolmogorov Smirnov test won t refuse the stability hypothesis for this choice of the parameters. If we have another choice of parameters such that the difference between them and the real values is large then there is a low probability that the residuals will be stably distributed and this probability is even lower if we assume that the innovation has σ = 1, β = µ = 0. This assertion will be confirmed by the simulations. We will do it as follows: Assume that we have a sample h 1, h 2,..., h N from GARCH(1,1)-model with unknown parameters a 0, a 1, b 1 and α. 1) For any trinity of the parameters a 0, a 1 and b 1, where a 0 = 0.1, 0.2, 0.3,.., 1, a 1 = 0.1, 0.2, 0.3,...1, and b 1 = 0.1, 0.2, 0.3,..., 1 (i.e. we have in total 1000 of such trinities) we obtain residuals ˆς n = h n a 0+a 1h n 1+b 1σ n 1. (Or it can be in the form a 0 = 0.1q, 0.2q,.., q etc, where q 1) 2) We find the estimate of α of these residuals under assumption that the residuals are stably distributed. We calculate the α estimates by means of the methodology MLP (Maximum Likelihood Projections Methodology) which is described in detail in [5]. But we can also estimate the parameter α by means of empirical and theoretical characteristic functions. 3) Having obtained the estimate ˆα of the tail index of the residuals we check if it belongs to the interval (1,2] because otherwise this value is irrelevant. If it belongs to this interval then we simulate the sample from η Sˆα (1, 0, 0) of the size N and get two empirical distribution functions: F 1 (x) = 1 N N I(x ˆς j ) and F 2 (x) = 1 N N I(x η j ) 4) We choose such trinity a 0, a 1, b 1 and the estimate ˆα such that the value max x F 1 (x) F 2 (x) is minimal. The efficiency of this methodology will be discussed by simulations. The next table demonstrates the parameter estimates for different values of the real parameters. The number of observations is equal to 1000, i.e. the estimates were obtained from 1000 values of the stable GARCH(1,1) model: Table 3. Table of parameter estimates of GARCH(1,1) model for different values of the unknown parameters Method α a 0 a 1 b 1 ˆα â 0 â 1 ˆb1 KSB KSB KSB KSB KSB KSB KSB KSB KSB KSB KSB KSB KSB KSB The table below shows the rate of convergence of the estimates to the real values of the parameters for different number of observations ranging from 100 to The GARCH(1,1) model has the following parameters: a 0 = 0.2, a 1 = 0.4, b 1 = 0.3 and α = We calculate for any sample size 20 estimates to get the standard deviation and the mean of the estimates. 141

6 Table 4. The table specifying the rate of convergence, the number of observations is at the right side of the table, the second line represents real values of the parameters, the rest are the estimates Numb. parameter α a 0 a 1 b 1 α a 0 a 1 b 1 of Obs. Real Values St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Mean St. Dev Median St. Dev Conclusion When comparing these three methods the result will be given as follows: 1) The estimates based on empirical characteristic function are precise enough but they overestimate the tail index, i.e. in most cases its value is a bit higher that the real values. 2) The methodology based on minimization of the K-S statistic that we denote as KSB is also precise enough and consistent but it underestimates a bit the real value of the tail index α. Its advantage consists in that it calculates the parameters of the GARCH model almost 10 times faster than the first methodology. 3) This methodology based on minimization the maximum likelihood function is the slowest in sence of number of operations required to get the estimate. To estimate the Maximum Likelihood function we need first to estimate the density function which is not of the exponential form therefore, due to the approximations we will need to wait very long time for the results. There are two mistakes: the mistake of the ML methodology and the mistake of the approximation. The most important task of this paper are precise estimators based on the first two methodologies because the ML methodology is classical and is well studied analysed and described by other authors. Acknowledgement. This work was supported by the Grant Agency of the Czech Republic under grant GACR 402/09/H045 References [1] Andel, J.: The Basics of Mathematical Statistics. Matfyzpress, Praha 2005 [2] Borak S., Hrdle W. and Weron R.: Stable Distributions. SFB 649 Discussion Paper, , Humboldt University, Berlin 2005 [3] Csorgo S.: Rates of uniform convergence for the empirical characteristic function. Acta Scientiarum Mathematicarum, 1985 ( b: ) [4] Mittnik S. and Paolella M. S.: Prediction of Financial Downside-Risk. Handbook of Heavy Tailed Distributions in Finance (S. Rachev, eds.), Elsevier, Amsterdam 2003 [5] Omelchenko V.: Stable Distributions and Application to Finance. Disloma Thesis (supervisor L. Klebanov), Charles University in Prague, Faculty of Mathematics and Physics. Prague, 2007 [6] Shiryaev, A.: Essentials of Stochastic Finance. (Facts, Models, Theory). World Scientific, Singapore, 1999 [7] Silvennoinen A.: Multivariate GARCH models. Handbook of Financial Time Series.(T. G. Andersen, R. A. Davis, J.-P. Kreiss and T. Mikosch, eds.) New York: Springer. (To appear) 142