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

Abstract

Strong consistency and (weak) distributional convergence to highly non-Gaussian limits are established for closed-form, two stage least squares (TSLS) estimators for a class of ARCH(p) models, with special attention paid to the ARCH(1) and Threshold ARCH(1) cases. Conditions for these results include (relatively) mild moment existence criteria that enjoy empirical support among many financial returns. These conditions are not shared by competing estimators like OLS. Identification of the TSLS estimators depends on asymmetry, either in the model's rescaled errors or in the conditional variance function. Monte Carlo studies reveal TSLS estimation can sizably outperform quasi maximum likelihood (QML) and compare favorably to recently proposed two step estimators designed to enhance the efficiency of QML.

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

JEL Classification: C13, C22, C58