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Journal of Financial Econometrics Advance Access originally published online on August 23, 2006
Journal of Financial Econometrics 2006 4(4):617-635; doi:10.1093/jjfinec/nbl002
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© The Author 2006. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org.

Sample and Implied Volatility in GARCH Models

Lajos Horváth
     University of Utah

Piotr Kokoszka
     Utah State University

Ricardas Zitikis
     University of Western Ontario

Address correspondence to Piotr Kokoszka, Department of Mathematics and Statistics, Utah State University, 3900 Old Main Hill, Logan, UT 84322-3900, or e-mail: piotr.kokoszka{at}usu.edu.

The unconditional variance of various GARCH-type models is a function h({theta}) of the parameter vector {theta} which is estimated by Formula. For most models used in practice, closed-form expressions of h(·) have been found. On the contrary, the unconditional variance can be estimated by the sample variance Formula2. This article establishes the asymptotic distributions of the differences Formula2 – h({theta}) and Formula2 – h(Formula) for broad classes of GARCH-type models. Even though both limit distributions are normal, the asymptotic variances are not equal. Potential practical consequences of these results are discussed.

KEYWORDS: GARCH processes, statistical hypothesis test, volatility

Received July 12, 2005; revised July 17, 2006; accepted July 20, 2006


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