Elsevier

Econometrics and Statistics

Volume 15, July 2020, Pages 67-83
Econometrics and Statistics

Stable Randomized Generalized Autoregressive Conditional Heteroskedastic Models

https://doi.org/10.1016/j.ecosta.2018.11.002Get rights and content

Abstract

The class of Randomized Generalized Autoregressive Conditional Heteroskedastic (R-GARCH) models represents a generalization of the GARCH models, adding a random term to the volatility with the purpose to better accommodate the heaviness of the tails expected for returns in the financial field. In fact, it is assumed that this term has stable distribution. Allowing both, returns and volatility, to have stable distribution, a new class of models to describe volatility arises: Stable Randomized Generalized Autoregressive Conditional Heteroskedastic Models (SR-GARCH). The indirect inference method is proposed to estimate the SR-GARCH parameters, theoretical results concerning dependence structure are obtained. Simulations and an empirical application are presented.

Introduction

The main motivation for introducing the α-stable distribution family is the fact that this class allows us to generalize the central limit theorem. Taking Sn=X1++Xn as the sum of n independent random variables Xi we know that the central limit theorem requires finite variance in order to achieve convergence to the normal distribution. Empirical studies about returns indicate that this assumption may be too restrictive. In fact, tails that are much heavier than expected for residuals, when compared to the normal case, often appear in these studies. Given that these ideas arise naturally when we analyze real data that present changing conditional variance and heavy tails, we may replace the assumption of the finiteness of the variance with a much less restrictive one concerning the regular behavior of the tails (Gnedenko and Kolmogorov, 1954).

An important property of the class of stable distributions is the fact that the normal distribution is a special case thereof. Moreover, this class allows asymmetry, tails much heavier than other popular distributions (as Student’s t) and it is closed under linear combinations. It is important to note that the closeness under linear combinations makes α-stable distributions suitable for portfolio analysis and asset allocation modeling, since a portfolio resulting from the aggregation of a number of stocks having α-stable distribution will have α-stable distribution as well. Another important characteristic of stable distributions is its ability to accommodate the leptokurtic feature present in financial data.

One of the reasons why stable distributions are not so popular among practitioners, as other distributions, is the absence of a simple closed-form density function. This prevents the use of the maximum likelihood estimation method. Also, the non-existence of moments of order greater than two prevents the use of the method of moments. Methods based on quantiles, McCulloch (1986), or on the empirical characteristic function, Koutrouvelis (1981), constitutes the initial alternative for the estimation of parameters of stable distributions available in the literature.

The cause of the heavy-tailedness of asset returns may be conditional heteroskedasticity. For instance, De Vries (1991) analyzes the relation between generalized autoregressive conditional heteroskedastic (GARCH) models and α-stable distributions. From a practical standpoint, GARCH models with t-distributed innovations are usually used to accommodate for the excess of kurtosis. More recently, a generalization of GARCH models was proposed by Nowicka-Zagrajek and Weron. (2001), named randomized generalized autoregressive conditional heteroskedastic (R-GARCH) models. R-GARCH models allow α-stable distributions in the volatility equation replacing the constant factor. Consequently the asset returns are stationary with α-stable distribution.

As we know, in contrast to the estimation, simulating an α-stable random variable is quite straightforward. The indirect estimation method may be the solution to overcome these difficulties, related to the absence of a simple closed-form for the density function of a stable distribution, since this method is an intensive computationally simulation based method. This approach was successfully employed to estimate α-stable parameters and the parameters of ARMA processes with α-stable distributions by Lombardi and Calzolari (2008). Also, it was successfully employed to estimate the parameters of a randomized generalized autoregressive conditional heteroskedastic model by Sampaio and Morettin (2015).

In this paper, after a short review of the major properties of α-stable distributions, we will introduce the stable randomized generalized autoregressive conditional heteroskedastic (SR-GARCH) models and their properties as well as an indirect estimation approach. We will then examine the finite sample properties of the estimator by means of a detailed simulation study. Moreover, an empirical application is also provided.

As we pointed out before, stable distributions arises immediately as an extension of normal distributions and it is complicated to work with the density function. In contrast to the density function there is a simple way to obtain the characteristic function. Actually the characteristic function is obtained from the Lévy–Khintchine theorem, which makes the correspondence with the initial definition of stable distributions. The first definition and detailed properties are available in Samorodnitsky and Taqqu (1994). The characteristic function of an α-stable distribution is given bylnΦX(t)={itμσα|t|α[1iβsgn(t)tan(πα2)],ifα1,itμσ|t|[1+iβsgn(t)ln(t)],ifα=1,and depends on four parameters: α ∈ (0, 2], measuring the tail thickness (thicker tails for smaller values of the parameter), β[1,1] determining the degree and sign of asymmetry, σ > 0 (scale) and μR (location). The distribution will be denoted as Sα(σ, β, μ).

The expression (1.1) generalizes the characteristic function of some important known distributions: α=2, corresponding to the normal distribution (in this case β becomes unidentified), α=1 and β=0, yielding the Cauchy distribution, and α=1, β=±1 for the Lévy distribution.

Remember that estimation methods such as maximum likelihood and the method of moments cannot be applied since we do not have a simple closed-form for the density function available and moments of order greater or equal than α do not exist whenever α ≠ 2. The alternative methods mentioned above (based on quantiles or on the empirical characteristic function) are only suitable for the estimation of the distribution parameters and its use in estimating more complex models (stochastic models, time series, etc.) involving α-stable would require a two-step approach.

Over the last years we have seen an impressive increase on processing power of computers which is making possible the use of computationally-intensive estimation methods. In particular, two methods have become popular: approximating the density with the FFT (fast Fourier transform) of the characteristic function, Mittnik et al. (1999), or with numerical quadrature, Nolan (1997). However, both these approximations have quite poor accuracy for small values of α because of the spikedness of the density function. Another method that is benefited by the increased computational power is the Bayesian approach: simulation-based MCMC samplers have been proposed by Buckle (1995); Qiou and Ravishanker (1998) and Lombardi (2007).

Weron and Weron (1995) proposed an algorithm that makes quite straightforward to simulate stably distributed pseudo-random numbers. That is the reason why we stressed the advance in computing as a promise for simulation-based approaches. Let W be a random variable with exponential distribution of mean 1 and let U be an uniformly distributed random variable on [π/2,π/2]. Furthermore, let ζ=arctan(βtan(πα2)/α) and η=[1+β2tan2(πα2)]1/(2α). ThenX={ηsinα(ζ+U)(cosU)1/α[cos(Uα(U+ζ))W]1ααifα1,2π[(π/2+βU)tanU]βlnπ2WcosUπ2+βUifα=1,has Sα(1, β, 0) distribution. We can obtain a random variable with the general Sα(σ, β, μ) distribution by means of the standardization formula:Z={σX+μ,ifα1,σX+2πβσlogσ+μ,ifα=1.We conclude by saying that the indirect inference (a simulation-based inference approach detailed below) is an interesting alternative to the estimation of more complex models involving stable distributions.

Section snippets

SR-GARCH models

It is known, from statistical literature, that financial data, audio signals, among others, are heavy-tailed. Distribution that are able to capture this property becomes of paramount importance. The R-GARCH models proposed by Nowicka-Zagrajek and Weron. (2001), appears as an alternative to that effect, however, another very present characteristic, especially in financial data, is the fact that these real data are leptokurtic.

The Student’s t distribution was presented as an alternative to

Indirect inference for SR-GARCH process

The main purpose of this section is to introduce the indirect inference approach to estimate the parameters of a SR-GARCH model. As we observed some interesting asymptotic properties for the model SR-GARCH(1,1,0) we will illustrate the idea for this model.rt=htλϵt,t=0,±1,±2,,ht=θ1ηt1+ϕ1ht1,where the innovations ϵt are i.i.d. Sλ(σ, 0, 0), the innovations ηt are i.i.d. stable random variables distributed as (2.6) and {ϵt} and {ηt} are independent.

As we have noticed, α-stable distributions do

Conclusions

We have proposed the autocovariation, as defined by Kokoszka and Taqqu (1994), as an appropriate measure of dependence for Stable Randomized Generalized Autoregressive Conditional Heteroskedastic Models and we derived some limit results. After that, we have proposed the indirect estimation method for the family of SR-GARCH. The results look promising, but theoretical results are difficult to obtain if we increase the order q of the process.

Currently we are involved in a research where we

Acknowledgments

The authors would like to thank two referees for comments that improved substantially the text.

Funding

The authors acknowledge the partial support of grants from CNPq (JMS) and Fapesp 2013/00506-1 (PAM).

References (24)

  • C. Gouriéroux et al.

    Simulation-Based Econometric Methods

    (1996)
  • C. Gouriéroux et al.

    Indirect inference

    J. Appl. Econ.

    (1993)
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