# Journal of Time Series Analysis

## On Asymptotic Theory for ARCH (∞) Models

### Journal Article

Autoregressive conditional heteroskedasticity (ARCH)( $\infty$) models nest a wide range of ARCH and generalized ARCH models including models with long memory in volatility. Existing work assumes the existence of second moments. However, the fractionally integrated generalized ARCH model, one version of a long memory in volatility model, does not have finite second moments and rarely satisfies the moment conditions of the existing literature. This article weakens the moment assumptions of a general ARCH( $\infty$) class of models and develops the theory for consistency and asymptotic normality of the quasi‐maximum likelihood estimator.

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