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

Abstract

This paper provides detailed proofs of all Lemmas contained in “Closed-Form Estimation of Finite-Order ARCH Models: Asymptotic Theory and Finite-Sample Performance.” Specifically, regular variation of ARCH(1) and threshold ARCH(1) processes is established when the, respective, model’s rescaled errors are skewed. Based on this result, (weak) distributional convergence of sums of point processes constructed from these ARCH sequences is demonstrated and forms the basis for (weak) distributional convergence of the cross-order covariances implied for these same sequences. Higher-order moment existence and identification conditions are also provided for ARCH(p) processes and IV estimators for these processes, respectively. Lastly, asymptotic properties of OLS estimators for the ARCH(1), threshold ARCH(1) and ARCH(p) models are developed, where these results are not included in the main paper.

Keywords: ARCH, Threshold ARCH, closed form, two stage least squares, instrumental variables, heavy tails, regular variation

JEL Classification: C13, C22, C58