Some Improvements in Numerical Evaluation of Symmetric Stable Density and Its Derivatives

Abstract

We propose improvements in numerical evaluation of symmetric stable density and its partial derivatives with respect to the parameters. They are useful for more reliable evaluation of maximum likelihood estimator and its standard error. Numerical values of the Fisher information matrix of symmetric stable distributions are also given. Our improvements consist of modification of the method of Nolan (1997) for the boundary cases, i.e., in the tail and mode of the densities and in the neighborhood of the Cauchy and the normal distributions.

Estimating GARCH-type models with symmetric stable innovations: Indirect inference versus maximum likelihood

“…However, this alternative comes at a large cost of estimation, due to the absence of closed form density function and of moments for most of the parameter values. As a solution to this problem, we apply the maximum likelihood approach developed by Nolan (1997) and Matsui and Takemura (2006) and the indirect inference method introduced by Gouriéroux et al (1993).…”

“…Although we did not directly use this package, it extensively inspired us in writing our procedures. Matsui and Takemura (2006) provides improvements to Nolan's (1997) approach that help to estimate the parameters of the symmetric stable distribution at the boundary cases, i.e., when the underlying random variable approaches zero or ∞ and α is near the value 1 or 2. Thus, when x → 0 or x → ∞ and α = 1, they derive specific expressions of f (z t ; α, 1, 0) based on asymptotic expansions as stated in Sections 2.4 and 2.5 of Zolotarev (1986), while for the case α = 1 and α = 2, they use Taylor expansions of f (z t ; α, 1, 0) around these values by giving specific expressions for the partial derivatives of the function with respect to α.…”

“…Both procedures of Nolan (1997) and Matsui and Takemura (2006) are derived for ''homoscedastic'' random variables. For our purposes, we adapt these procedures to incorporate the conditional scale by means of the (T)GARCH specifications, as described in Section 3.…”

“…This might be due to the difficulty of implementing the ML approach to estimate the parameters of the stable distribution, given that the distribution has a closed-form density function only for very specific values of α. Nolan (1997) and Matsui and Takemura (2006) implement the ML approach by numerically evaluating the stable density function and its derivatives in order to estimate the parameters specific to the stable distribution. In this paper we adapt their approach and integrate the GARCH and TGARCH models in the estimation procedure.…”

“…The choice of the auxiliary model is motivated by the fact that there is a rather natural correspondence between the two models: besides having the same number of parameters and common GARCH-type specifications for the conditional heteroscedasticity, the degrees of freedom in the Student's t distribution is the direct counterpart of the parameter of stability or characteristic exponent in the stable distribution, as both measure the tail-thickness of the distribution. In what regards the ML implementation, we apply the method described by Nolan (1997) and Matsui and Takemura (2006) based on the numerical evaluation of the symmetric stable density and its derivatives for a wide range of parameter constellations. Furthermore, we adapt their procedure to estimate both the parameters of the symmetric α-stable distribution and of the GARCH specifications.…”