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This paper introduces a Nearly Unstable INteger-valued AutoRegressiveConditional Heteroskedasticity (NU-INARCH) process for dealing with count timeseries data. It is proved that a proper normalization of the NU-INARCH processendowed with a Skorohod topology weakly converges to a Cox-Ingersoll-Rossdiffusion. The asymptotic distribution of the conditional least squaresestimator of the correlation parameter is established as a functional ofcertain stochastic integrals. Numerical experiments based on Monte Carlosimulations are provided to verify the behavior of the asymptotic distributionunder finite samples. These simulations reveal that the nearly unstableapproach provides satisfactory and better results than those based on thestationarity assumption even when the true process is not that close tonon-stationarity. A unit root test is proposed and its Type-I error and powerare examined via Monte Carlo simulations. As an illustration, the proposedmethodology is applied to the daily number of deaths due to COVID-19 in theUnited Kingdom.

Nearly Unstable Integer-Valued ARCH Process and Unit Root Testing