GARCH Model Estimation Using Estimated Quadratic Variation

Abstract

We consider estimates of the parameters of Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models obtained using auxiliary information on latent variance which may be available from higher-frequency data, for example from an estimate of the daily quadratic variation such as the realized variance. We obtain consistent estimators of the parameters of the infinite Autoregressive Conditional Heteroskedasticity (ARCH) representation via a regression using the estimated quadratic variation, without requiring that it be a consistent estimate; that is, variance information containing measurement error can be used for consistent estimation. We obtain GARCH parameters using a minimum distance estimator based on the estimated ARCH parameters. With Least Absolute Deviations (LAD) estimation of the truncated ARCH approximation, we show that consistency and asymptotic normality can be obtained using a general result on LAD estimation in truncated models of infinite-order processes. We provide simulation evidence on small-sample performance for varying numbers of intra-day observations.

Keywords:

• GARCH
• High-frequency data
• LAD
• Quadratic variation

JEL Classification:

• C22

ACKNOWLEDGMENTS

We are grateful to Ilze Kalnina, John Maheu, Nour Meddahi, Tom McCurdy and Eric Renault, and to participants in Econometric Society and Canadian Econometric Study Group meetings, for valuable comments and discussions, and to Roger Koenker for kindly providing his Matlab code for estimation of constrained quantile regressions.

Notes

¹We will refer in most instances to estimated quadratic variation, rather than to realized variance, to emphasize that estimators of the quadratic variation other than the realized variance are also admissible, and may indeed produce superior results. The realized variance is nonetheless one admissible estimator, and we use it in the simulation experiments below.

²Definitions of strong, semistrong, and weak GARCH given in Drost and Nijman (1993). In strong GARCH, {ϵ _t } is such that z _t  ≡ ϵ _t /σ _t  ∼ IID(0, 1); semistrong GARCH holds where {ϵ _t } is such that E[ϵ _t |ϵ _t−1,…] = 0 and weak GARCH holds where {ϵ _t } is such that P[ϵ _t |ϵ _t−1,…] = 0 and where denotes the best linear predictor of given a constant and past values of both ϵ _t and To obtain consistent estimation by QML, it will be necessary that the process is semistrong GARCH: the standard quasi-ML estimator of the GARCH model will in general be inconsistent in weak GARCH models (as noted by Meddahi and Renault, 2004, and Francq and Zakoïan, 2000).

³This distribution allows us to generate many replications without encountering any instances of unstable squared returns; low degrees of freedom would at least occasionally produce explosive sequences of squared returns.

⁴For any matrix X, max |X| denotes in this article the absolute value of the largest component of the matrix.

⁵For example, in the GARCH(1,1), γ _l  = β ^l−1α, so that for and k = cln T we have