First-order nearly unstable continuous autoregressive processes have been well explored in the literature, see for example Chan and Wei (1987), Phillips (1987), Chan, Ing and Zhang (2019), and the references therein. In these works, it is assumed that the model approaches the non-stationarity region as the sample size increases. More specifically, a nearly unstable continuous process is defined by
is a white noise and, for .
In the past few years, nearly unstable discrete processes have emerged based on the INteger-valued AutoRegressive (INAR) approach (McKenzie, 1985; Al-Osh and Alzaid, 1987). The first attempt on this subject was due to Ispány, Pap and Van Zuijlen (2003). More specifically, a nearly unstable INAR(1) process is defined by
where is the thinning operator proposed by Steutel and van Harn (1979), given by with , for , and
is a sequence of independent and identically distributed (iid) random variables withbeing independent of the counting series for all , for . These authors assumed that approaches 1 (non-stationarity) when as given in Chan and Wei (1987) in the continuous context. By assuming is known, the conditional least squares (CLS) estimator of was explored by Ispány, Pap and Van Zuijlen (2003)
. They showed that, under nearly non-stationarity and assuming finite second moment for
, the CLS estimator weakly converges to a normal distribution at the rate. Other related works dealing with nearly unstable INAR (Galton-Watson/branching) processes are due to Wei and Winnicki (1990), Winnicki (1991), Ispány, Pap and Van Zuijlen (2005), Rahimov (2007), Rahimov (2008), Drost, Van Den Akker and Werker (2009), Rahimov (2009), Barczy, Ispány and Pap (2011), Ispány, Körmendi and Pap (2014), Barczy, Ispány and Pap (2014), Guo and Zhang (2014), and Barczy, Körmendi and Pap (2016). Practical situations demonstrating evidence of a nearly unstable INAR model are discussed for instance by Hellström (2001).
Another popular way for dealing with count time series data is the INteger-valued Genenalized AutoRegressive Conditional Heterokedastic (INGARCH) models by Ferland, Latour and Oraichi (2006), Fokianos, Rahbek and Tjøstheim (2009), Fokianos and Fried (2010), Zhu (2011), Fokianos and Tjøstheim (2011), Zhu (2012), Christou and Fokianos (2015), Gonçalves et al. (2015), Davis and Liu (2016), Silva and Barreto-Souza (2019), Weiß et al. (2020), which constitute in some sense an integer-valued counterpart of the classical GARCH models by Bollerslev (1986). The INGARCH methodology is the focus of this paper. Like the existing literature on nearly unstable continuous and INAR processes that assumes first-order autoregressive dependence, in this paper we consider the first-order autoregressive version of the INGARCH approach, which is known as INARCH(1) (INteger-valued AutoRegressive Conditional Heteroskedasticity).
Our chief goal in this paper is to introduce a Nearly Unstable INARCH (denoted by NU-INARCH) process for dealing with count time series data. To the best of our knowledge, this is the first time that a nearly unstable count time series model is being proposed based on the INARCH approach; all existing nearly unstable discrete processes in the literature consider the INAR approach. We establish the weak convergence of the NU-INARCH process (when properly normalized) endowed with a Skorohod topology. With this result at hand, we derive the asymptotic distribution of the conditional least squares estimator of the correlation parameter as a functional of certain stochastic integrals. An equally important contribution of this paper is to develop a unit root test (URT) for the INARCH model, where the asymptotic distribution of the statistics under the null hypothesis is provided. Note that although URTs are well explored in the continuous case, only sporadic results are available for the discrete case. A few works dealing with this relevant problem, based on the INAR approach, are due toHellström (2001) and Drost, Van Den Akker and Werker (2009).
The paper is organized as follows. In Section 2, the NU-INARCH model is introduced and a fluctuation theorem is established, which involves the Cox-Ingersoll-Ross diffusion process. The asymptotic distribution of the CLS estimator for the correlation parameter is derived in Section 3 under the nearly unstable and stationarity assumptions. Section 4
provides simulated results about the asymptotic distribution of the CLS estimator under both nearly unstable and stationary approaches and also compares them in terms of confidence interval coverages. A unit root test for the INARCH process is proposed in Section5 and its performance is evaluated via Monte Carlo simulations. An empirical application about the daily number of deaths due to COVID-19 in the United Kingdom, which exhibits a nearly unstable/non-stationary behavior, is provided in Section 6. Concluding remarks and future research are addressed in Section 7.
2 Model and the Fluctuation Theorem
In this section, we define the nearly unstable INARCH process and obtain its weak convergence (under a proper normalization) in the space of the non-negative càdlàg functions endowed with the Skorokhod topology.
We say that a sequence is a first-order nearly unstable integer-valued ARCH process (in short NU-INARCH) if
for , where , , and , with , and (constant starting value).
In the next proposition, we provide the mean, variance, and autocorrelation function of the NU-INARCH process. These results will be important to establish the proper normalization in order to obtain a non-trivial limit for the counting process.
Let be a nearly unstable INARCH process. Then, its marginal mean and variance, and autocorrelation function are given respectively by
We have that . By using recursion times, we obtain the result for the marginal mean. For the variance, it follows that
Finally, for , the autocorrelation function becomes
where we have used in the third equality the fact that since for . ∎
From Proposition 2.2, we have that and . We then define the normalized process and obtain that , for . In the following theorem, we establish the weak convergence of the process as . We introduce some notation before presenting such a result. Denote by the space of the non-negative càdlàg (right continuous with left limits) functions on and the space of infinitely differentiable functions on having compact supports.
The stochastic process weakly converges in endowed with the Skorokhod topology to a diffusion process given by the solution of the stochastic differential equation
and , as , where is a standard Brownian motion.
We have that , with ; we here denote and . In particular, almost surely. Note that
is a Markov chain assuming values in. For , define . From Theorem 6.5 in Chapter 1 and Corollary 8.9 in Chapter 4 of Ethier and Kurtz (1986), to obtain the desired result, it is enough to show that
with , where and denote the first and second derivatives of , respectively.
For , we have that
Note that Equation (7) also holds for . Further, we can write
To show the case , we argue as in the proof of Theorem 3.1 in Chapter 9 of Ethier and Kurtz (1986). Then, the result follows by showing that for any convergent sequence , where is allowed. Without loss of generality, suppose that the support of is contained in the interval , for constant . For and , it folllows that and therefore the integral involved in equals 0 under the region ( for ), that is . Define for , , for , and . Hence, it follows that
Further, we have that . Consider , then and . These results give us that the right-hand side of (10) goes to 0 as . We obtain the same conclusion when since and , and hence . Suppose now that . We can establish the weak convergence of
via its characteristic function as follows:
as . Therefore, . Hence, the integrand in
is bounded above by an integrable random variable. Further, this integrand converges in probability to 0 since. We then apply the Dominated Convergence Theorem to conclude that .
For the case , it follows that
as . In a similar fashion, for , it can be shown that , which concludes the proof. ∎
3 Conditional Least Squares
In this section, we provide the asymptotic distribution of the conditional least squares estimator of for the nearly unstable INARCH process. The parameter is assumed to be known. This can be seen as a nuisance parameter since our main interest relies on the parameter that controls the dependence in the model. In the empirical illustration, we discuss how to deal with the unknown case.
The CLS estimator of is obtained by minimizing the -function given by . Hence, we obtain explicitly the CLS estimator of , say , which is given by
We begin by deriving the asymptotic distribution of under the stationary assumption, where we denote the count time series by (no need for the superscript ). This case will be contrasted to the nearly unstable INARCH process through simulation in the following section.
Assume that is a trajectory from a stationary Poisson INARCH(1) model, that is . Then, the CLS estimator given in (11) satisfies
as , where
From Fokianos, Rahbek and Tjøstheim (2009), we have that is strictly stationary and ergodic since . Hence, we can use Theorem 3.2 from Tjøstheim (1986) to establish the asymptotic normality of the CLS estimator . The other conditions necessary to obtain this weak convergence can be straightforwardly checked in our case and therefore are omitted. Applying this theorem, we get that the asymptotic variance, say , assumes the form , with and . Explicit expression for the marginal moments of a Poisson INARCH(1) model are given in Weiß (2010). Using these results and the notation considered there with , for , we obtain and . Direct algebric manipulations conclude the proof. ∎
From now on assume that is a nearly unstable INARCH process as given in Definition 2.1. Define , , and , for and , where denotes the integer-part of . Like in the nearly unstable INAR process by Ispány, Pap and Van Zuijlen (2003), we can express as
In the following lemma, we provide the asymptotic behavior of the autocovariance function of the process ; note that . This will be important to identify the proper normalization of in (12) yielding a non-trivial weak limit.
For , we have that , where for , , and denoting that for real sequences and .
It is straightforward that and for . Further, , where the last equality follows from the expression of the covariance given in Proposition 2.2. After using the expression of the variance given in that lemma, we obtain that .
From the above results and Proposition 2.2, we obtain that
Let be the diffusion process given in (3). Then, the CLS estimator satisfy the following weak convergence
as , where , for , with .
Define , for . We have that
where both numerator and denominator have the same order of magnitude .
For , it follows that
and then can be expressed by
Define the functions () and mapping into as and . Hence, it follows that . Using the fact that the CIR process has almost sure continuous trajectories and similar arguments given in the proof of Proposition 4.1 of Ispány, Pap and Van Zuijlen (2003), we obtain that weakly converges to as .
In particular, we have that weakly converges to . From the definition of , we have that and, therefore, . In other words, . The above results and the continuous mapping theorem give us that .
The above arguments are straightforwardly extended to establish the joint weak convergence
in as . Then, the desired result given in (13) is obtained by applying the continuous mapping theorem. ∎
4 Simulated Experiments
In this section, we present simulated results illustrating the behavior of the asymptotic distributions of the normalized CLS estimator under the nearly unstable and stable cases. All the numerical results of this paper were obtained by using the statistical software R (R Development Core Team, 2021). We conduct Monte Carlo simulations with 10000 replications, where we generate Poisson INARCH(1) trajectories with , , and initially a sample size of . Note that the chosen values for here indicate nearly unstable count processes. For each replication, we compute the CLS estimate of using (11) and then its standardized estimate as and according to the nearly unstable (Theorem 3.3) and stable/stationary (Theorem 3.1) cases, respectively.
A generator from the asymptotic distribution given on the right-hand side of (13
) was implemented, where the stochastic integrals are approximately evaluated via type-Riemann integrals. Hence, for instance, we can obtain its quantiles and also plot the associated density function by generating samples and then applying a non-parametric density estimator (here the Gaussian kernel is considered), which are important for what follows. We present the histograms and qq-plots of the standardized CLS estimates along with their associated asymptotic density/quantiles under the stable and nearly unstable cases in Figures2 and 2, respectively. From Figure 2, it is evident that the normal approximation is not adequate and it is worsening when gets closer to 1, which is expected since these results are based on stationarity. On the other hand, the histograms and qq-plots regarding the nearly unstable approximation given in Figure 2 show an excellent agreement between the empirical standardized estimates and the theoretical asymptotic distribution for all scenarios.
A natural question is what happens when is not close to 1. To address this point, we run additional simulations with