October 2016 (Revised July 2017)

Closed-Form Estimation of Finite-Order ARCH Models: Asymptotic Theory and Finite-Sample Performance

Todd Prono


Covariances between contemporaneous squared values and lagged levels form the basis for closed-form instrumental variables estimators of ARCH processes. These simple estimators rely on asymmetry for identification (either in the model's rescaled errors or the conditional variance function) and apply to threshold ARCH(1) and ARCH(p) with p < ∞ processes. Limit theory for these estimators is established in the case where the ARCH processes are regularly varying with a well-defined third and sixth moment of the raw returns and rescaled errors, respectively. The resulting limits are highly non-normal in empirically relevant cases, with slow rates of convergence relative to the thin-tailed √n-case. Nevertheless, Monte Carlo studies of a heavy-tailed ARCH(1) process show the simple IV estimator to outperform standard QMLE in (relatively) small samples when the data are (heavily) skewed. Methods for determining confidence intervals for the ARCH estimates are also discussed.

Revised Appendix (PDF)

Accessible materials (.zip)

Original: Full Paper (PDF) | Accessible materials (.zip) | Appendix (PDF)

Original DOI: https://doi.org/10.17016/FEDS.2016.083

Keywords: ARCH, Closed form estimation, Heavy tails, Instrumental variables, Regular variation, Three-step estimation

DOI: https://doi.org/10.17016/FEDS.2016.083r1

PDF: Full Paper

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Last Update: June 19, 2020