The second moment and the autocovariance function of the squared errors of the GARCH model (original) (raw)

Analytic moments for GARCH processes

Conditional returns distributions generated by a GARCH process, which are important for many problems in market risk assessment and portfolio optimization, are typically generated via simulation. This paper extends previous research on analytic moments of GARCH returns distributions in several ways: we consider a general GARCH model -the GJR specification with a generic innovation distribution; we derive analytic expressions for the first four conditional moments of the forward return, of the forward variance, of the aggregated return and of the aggregated variance -corresponding moments for some specific GARCH models largely used in practice are recovered as special cases; we derive the limits of these moments as the time horizon increases, establishing regularity conditions for the moments of aggregated returns to converge to normal moments; and we demonstrate empirically that some excellent approximate predictive distributions can be obtained from these analytic moments, thus precluding the need for time-consuming simulations. JEL Code: C53 1 1 INTRODUCTION Forward-looking physical return distributions have attracted a vast academic research literature because they have a great variety of financial applications to market risk assessment and portfolio optimization techniques. Since it has been recognized that time series of asset returns are not well described by normal, independent processes. Typically, their conditional distributions are non-normal and they exhibit volatility clustering, so are not independent.

NECESSARY AND SUFFICIENT MOMENT CONDITIONS FOR THE GARCH(r,s) AND ASYMMETRIC POWER GARCH(r,s) MODELS

Econometric Theory, 2002

Although econometricians have been using Bollerslev's (1986) GARCH (r, s) model for over a decade, the higher-order moment structure of the model remains unresolved. The sufficient condition for the existence of the higherorder moments of the GARCH (r, s) model was given by Ling (1999a). This paper shows that Ling's condition is also necessary. As an extension, the necessary and sufficient moment conditions are established for Ding, Granger and Engle's (1993) asymmetric power GARCH (r, s) model. * The authors wish to acknowledge the helpful comments of the Co-Editor, Bruce Hansen, and a referee.

Recent Theoretical Results for Time Series Models with GARCH Errors

Journal of Economic Surveys, 2002

This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model, some results for stationary and nonstationary ARMA-GARCH are summarized. Various new ARCH-type models, including double threshold ARCH and GARCH, ARFIMA-GARCH, CHARMA and vector ARMA-GARCH, are also reviewed.

LECTURE NOTES ON GARCH MODELS

In these notes we present a survey of the theory of univariate and multivariate GARCH models. ARCH, GARCH, EGARCH and other possible nonlinear extensions are examined. Conditions for stationarity (weak and strong) are presented. Inference and testing is presented in the quasi-maximum likelihood framework. Multivariate parameterizations are examined in details.

NOTES AND PROBLEMS A CLOSED-FORM ESTIMATOR FOR THE GARCH (1, 1) MODEL

Econometric Theory, 2006

We propose a closed-form estimator for the linear GARCH(1,1) model. The estimator has the advantage over the often used quasi-maximum-likelihood estimator (QMLE) that it can be easily implemented, and does not require the use of any numerical optimisation procedures or the choice of initial values of the conditional variance process. We derive the asymptotic properties of the estimator, showing T ( 1)=consistency for some 2 (1; 2) when the 4th moment exists and p T -asymptotic normality when the 8th moment exists. We demonstrate that a …nite number of Newton-Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the 4th moment exists. A simulation study con…rms our theoretical results.

The GARCH(1,1)-M model: results for the densities of the variance and the mean

Insurance: Mathematics and Economics, 1999

This paper starts from the GARCH(1,1)-M model of Bollerslev [Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (1986) 307-327], and investigates the limit diffusion form as it is presented in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7-38]. The distribution for the conditional variance process is derived, and in the limit for t going to infinity is shown to coincide with the stationary distribution given in Nelson [ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7-38]. In addition it is shown how the distribution for the complete model can be arrived at; explicit calculations are given in case the conditional variance is a martingale.

A Note on the Covariance Matrix of Multivariate GARCH Models

In this paper we provide su cient condition for the positive deniteness of the covariance matrix of a multivariate GARCH model. We show that the elemtwise product of two quadratic, symmetric and positive semide nite matrices is symmetric positive semide nite as well. This condition can be used quite well in practice to guide applied work.

A CLOSED-FORM ESTIMATOR FOR THE GARCH(1,1) MODEL

Econometric Theory, 2006

We propose a closed-form estimator for the linear GARCH(1,1) model. The estimator has the advantage over the often used quasi-maximum-likelihood estimator (QMLE) that it can be easily implemented, and does not require the use of any numerical optimisation procedures or the choice of initial values of the conditional variance process. We derive the asymptotic properties of the estimator, showing T ( 1)=consistency for some 2 (1; 2) when the 4th moment exists and p T -asymptotic normality when the 8th moment exists. We demonstrate that a …nite number of Newton-Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the 4th moment exists. A simulation study con…rms our theoretical results.

Modeling Variance of Variance: The Square-Root the Affine and the CEV GARCH Models

This paper,develops,a new,econometric,framework,for investigating,how,the sensitivity of the financial market volatility to shocks varies with the volatility level. For this purpose, the paper first introduces,the square-root (SQ) GARCH model,for financial time series. It is an ARCH analogue,of the continuous-time,square-root stochastic volatility model,popularly,used in derivatives pricing and hedging.,The variance,of variance,is a linear function of the conditional,variance,in the SQGARCH and of the square of it in the GARCH. After showing some implications of this difference, the paper introduces the constant-elasticity-of-variance (CEV) GARCH model, which allows more flexible fitting of variance-of-variance dynamics. The paper develops conditions for stationarity, the existence of finite moments, β-mixing, and other properties of the conditional variance process via the general state-space Markov chains approach. In particular, the paper generalizes the strict stationarity condi...