On approximate pseudo-maximum likelihood estimation for LARCH-processes

TL;DR

This paper develops and analyzes an approximate pseudo-maximum likelihood estimation method for LARCH processes, providing asymptotic properties and addressing computational challenges.

Contribution

It introduces a new estimation approach for LARCH processes with long-range dependence, deriving asymptotic distributions and handling practical computation.

Findings

Asymptotic normality of the estimator with $\

Slower convergence rate for the computable estimator compared to the ideal case.

Abstract

Linear ARCH (LARCH) processes were introduced by Robinson [J. Econometrics 47 (1991) 67--84] to model long-range dependence in volatility and leverage. Basic theoretical properties of LARCH processes have been investigated in the recent literature. However, there is a lack of estimation methods and corresponding asymptotic theory. In this paper, we consider estimation of the dependence parameters for LARCH processes with non-summable hyperbolically decaying coefficients. Asymptotic limit theorems are derived. A central limit theorem with $\sqrt{n}$-rate of convergence holds for an approximate conditional pseudo-maximum likelihood estimator. To obtain a computable version that includes observed values only, a further approximation is required. The computable estimator is again asymptotically normal, however with a rate of convergence that is slower than $\sqrt{n}.$