Nearly Unstable Integer-Valued ARCH Process and Unit Root Testing

TL;DR

This paper develops a nearly unstable integer-valued ARCH process for count data, proving its convergence to a Cox-Ingersoll-Ross diffusion, and introduces a unit root test with applications to COVID-19 death counts.

Contribution

It introduces the NU-INARCH process, establishes its asymptotic properties, and proposes a new unit root test for count time series data.

Findings

The NU-INARCH process converges weakly to a Cox-Ingersoll-Ross diffusion.

The asymptotic distribution of the estimator is characterized as a stochastic integral.

The proposed unit root test performs well in simulations and real data application.

Abstract

This paper introduces a Nearly Unstable INteger-valued AutoRegressive Conditional Heteroskedasticity (NU-INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU-INARCH process endowed with a Skorohod topology weakly converges to a Cox-Ingersoll-Ross diffusion. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution under finite samples. These simulations reveal that the nearly unstable approach provides satisfactory and better results than those based on the stationarity assumption even when the true process is not that close to non-stationarity. A unit root test is proposed and its Type-I error and power are examined via Monte Carlo simulations. As an illustration, the proposed methodology is applied to the daily number of deaths due to COVID-19 in the United Kingdom.