Inference for the panel ARMA-GARCH model when both $N$ and $T$ are large

TL;DR

This paper develops a two-step estimation method for a panel ARMA-GARCH model with large N and T, addressing biases from fixed effects and initial values, and proposes bias correction techniques.

Contribution

It introduces a novel bias correction approach for estimators in large panel ARMA-GARCH models, supported by new asymptotic theory and simulations.

Findings

Estimators are asymptotically normal with rate (NT)^{-1/2}.

Biases from fixed effects and initial values are identified and corrected.

Simulation and real data demonstrate the effectiveness of the proposed methods.

Abstract

We propose a panel ARMA-GARCH model to capture the dynamics of large panel data with $N$ individuals over $T$ time periods. For this model, we provide a two-step estimation procedure to estimate the ARMA parameters and GARCH parameters stepwisely. Under some regular conditions, we show that all of the proposed estimators are asymptotically normal with the convergence rate $(NT)^{-1/2}$, and they have the asymptotic biases when both $N$ and $T$ diverge to infinity at the same rate. Particularly, we find that the asymptotic biases result from the fixed effect, estimation effect, and unobservable initial values. To correct the biases, we further propose the bias-corrected version of estimators by using either the analytical asymptotics or jackknife method. Our asymptotic results are based on a new central limit theorem for the linear-quadratic form in the martingale difference sequence, when the weight matrix is uniformly bounded in row and column. Simulations and one real example are given to demonstrate the usefulness of our panel ARMA-GARCH model.