Prediction of α‐stable GARCH and ARMA‐GARCH‐M models

• Published date: 01 November 2017
• Date: 01 November 2017
• DOI: http://doi.org/10.1002/for.2477
• Author: Mohammad Mohammadi

Received: 25 October 2016 Revised: 8 February 2017 Accepted: 12 April 2017

DOI: 10.1002/for.2477

RESEARCH ARTICLE

Prediction of -stable GARCH and ARMA-GARCH-M models

Mohammad Mohammadi1

1Department of Statistics, Behbahan Khatam

Alanbia University of Technology,Iran

Correspondence

Mohammad Mohammadi, Department of

Statistics, Faculty of Basic Science,

Behbahan Khatam Alanbia University of

Technology,Iran.

Email: mohammadi@bkatu.ac.ir

Abstract

The best prediction of generalized autoregressive conditional heteroskedasticity

(GARCH) models with -stable innovations, -stable power-GARCH models and

autoregressive moving average (ARMA) models with GARCH in mean effects

(ARMA-GARCH-M) are proposed. We present a sufficient condition for stationar-

ity of -stable GARCH models. The prediction methods are easy to implement in

practice. The proposed prediction methods are applied for predicting future values

of the daily SP500 stock market and wind speed data.

KEYWORDS

GARCH-M model, -stable distribution, conditional expectation, prediction, volatility

1INTRODUCTION

Generalized autoregressive conditional heteroskedasticity

(GARCH) processes are widely used for modeling financial

data at regular intervals on stocks, currency investments,

and other assets. The GARCH model was first introduced by

Bollerslev (1986) as a generalization of the ARCH model,

which was introduced byEngle (1982). There are typical vari-

ants of GARCH models suchas exponential GARCH (Nelson,

1991), Glosten–Jagannathan–Runkle GARCH (Glosten,

Jagannathan, & David, 1993), nonlinear GARCH (Engle

& Ng. V. K., 1993), long memory GARCH (Conrad &

Karanasos, 2006), and stable mixture GARCH (Broda, Haas,

Krause, Paolella, & Steude, 2013), to name a few. Typical

variants of GARCH models have been used for modeling

data arising in different fields, such as econometrics, signal

processing, and hydrology. See Bollerslev (2008) for typical

variants of GARCH models and their applications. Regarding

the wide application of typical variants of GARCH models,

prediction of such models is an important issue.

The process {Zt,t∈Z}is called GARCH(r,s)if it satisfies

the following systems:

Zt=tt(1)

and

2

t=+

r



i=1

i2

t−i+

s



i=1

iZ2

t−i,(2)

where {t}is an independent and identically distributed

(i.i.d.) sequence, >0, and {i}and {i}are two sequences

of non-negative numbers with at leat one nonzero i.Infinan-

cial contexts, {Zt}and {t}are referred as return and volatility

processes. Bollerslev (1986) shows that the GARCH process

in Equations 1 and 2 is stationary if r

i=1i+E(2

0)s

i=1i<1.

Also, a necessary and sufficient condition for stationarity of

GARCH processes is given in Bougerol and Picard (1992).

We say t hat {Zt}is an -stable GARCH process with ∈

(1,2]if it satisfies Equations 1 and 2, where {t}is an

i.i.d. sequence of -stable random variables. In Section 2,

we present a sufficient condition for stationarity of -stable

GARCH models. For an application of -stable GARCH

models in econometrics, see Rachev, Stoyanov, Biglova and

Fabozzi (2005). When =2, {t}is an i.i.d. sequence

of Gaussian random variables. For elementary properties of

-stable distributions see Samorodnitsky and Taqqu (1994).

The -stable GARCH models are different from -stable

power-GARCH models, which wereconsidered by Panorska,

Mittnik, and Rachev (1995) and Mittnik, Paolella, and Rachev

(1998), among many others.

Using the same procedure as in Baillie and Bollerslev

(1992) one can find prediction of an ARMA model with

-stable power-GARCH innovations. Since E(2

t)=∞for

∈(0,2)the method presented in Baillie and Bollerslev

cannot be used for predicting -stable GARCH models. In

this paper, we present a method for estimating the conditional

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