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An ARCH-in-Mean Model without Intercept

Author

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  • Hafner, Christian

    (Université catholique de Louvain, LIDAM/ISBA, Belgium)

  • Preminger, Arie

Abstract

This paper introduces an ARCH-in-mean model without intercept, which is characterized by nonstationarity irrespective of the parameters. A critical parameter region is derived for which the volatility process behaves like a random walk. We establish the asymptotic properties of the quasi maximum likelihood estimator (QMLE) for the volatility and risk premium parameters. It is shown that the existence of the second and fourth order moment of the innovations is sufficient to derive the consistency and the asymptotic normality of the QMLE, respectively. The variance of the estimator of the volatility parameter does not depend on the underlying process. We further suggest a test for model stability and a test for the risk premium, which is non-standard as the parameter is on the boundary under the null. The finite sample performance of the estimator and the stability test are investigated in a simulation study. Finally, the model is applied to a monthly stock index series.

Suggested Citation

  • Hafner, Christian & Preminger, Arie, 2026. "An ARCH-in-Mean Model without Intercept," LIDAM Discussion Papers ISBA 2026017, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    • Handle: RePEc:aiz:louvad:2026017
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