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A generalized regime-switching integer-valued GARCH(1, 1) model and its volatility forecasting

Jiyoung Leea, and Eunju Hwang1,a

aDepartment of Applied Statistics, Gachon University, Korea
Correspondence to: 1Corresponding author: Department of Applied Statistics, Gachon University, 1342 Seongnamdaero, Sujeong-gu, Seongnam-si, Gyeonggido 13120, South Korea. E-mail: ehwang@gachon.ac.kr
Received July 4, 2017; Revised December 11, 2017; Accepted December 12, 2017.
Abstract

We combine the integer-valued GARCH(1, 1) model with a generalized regime-switching model to propose a dynamic count time series model. Our model adopts Markov-chains with time-varying dependent transition probabilities to model dynamic count time series called the generalized regime-switching integer-valued GARCH(1, 1) (GRS-INGARCH(1, 1)) models. We derive a recursive formula of the conditional probability of the regime in the Markov-chain given the past information, in terms of transition probabilities of the Markovchain and the Poisson parameters of the INGARCH(1, 1) process. In addition, we also study the forecasting of the Poisson parameter as well as the cumulative impulse response function of the model, which is a measure for the persistence of volatility. A Monte-Carlo simulation is conducted to see the performances of volatility forecasting and behaviors of cumulative impulse response coefficients as well as conditional maximum likelihood estimation; consequently, a real data application is given.

Keywords : integer-valued GARCH(1, 1), regime-switching Markov-chain, forecasting, cumulative impulse response function
1. Introduction

Integer-valued time series models have attracted recent attention because big data sets of count time series, such as the number of transactions of some stock and incidences of a certain disease, are available in financial markets or epidemiology. These now need to be analyzed via a successful fitted model as sources for future predictions. Most time series data are affected by various market circumstances such as international policy and financial crisis. In particular, real world count time series data are greatly influenced by external factors, which can be represented as a multiple-regime model. This work is concerned with a successful model for count time series data with external multiple-regimes.

In this paper, we consider an integer-valued GARCH(1, 1) (INGARCH(1, 1)) model, which is a popular count time series model, and combine the model with Markov chain regime-switchings that can account for the external factors. The INGARCH model is proposed by Ferland et al. (2006) as an integer-valued analogue of the classical GARCH model, which follows the Poisson process with GARCH structured parameter. Integer-valued autoregressive (or ARMA) processes were earlier discussed by Al-Osh and Alzaid (1987), McKenzie (1988), Alzeid and Al-Osh (1990), and Du and Li (1991). Davis et al. (1999) dealt with dynamic modelings for count time series data with parameter-driven and observation-driven specifications, whereas Brandt et al. (2000) for persistent event-count time series. Fokianos et al. (2009) considered linear and nonlinear Poisson autoregressions and discussed their geometric ergodicity and likelihood-based inference. For recent progress in count time series with discussion of some possible extensions we refer to Fokianos (2011).

This work adopts the generalized regime-switching (GRS) of Gray (1996) to the INGARCH model of Ferland et al. (2006). Gray (1996) proposed the generalized regime-switching GARCH (GRS-GARCH) process for the short-term interest rate, with time-varying dependent sample path. Several authors attempted to combine the Markov-chain with the GARCH model in order to model the dynamics properties of the financial markets data. Cai (1994) and Hamilton and Susmel (1994) were the first to apply the Markov regime-switching to ARCH models to account for the possible presence of structural breaks. With a broader information set than the work of Gray (1996), Klaassen (2002), and Haas et al. (2004) enhanced the Gray’s model by allowing a higher flexibility in capturing the persistence of volatility shocks, and by introducing a new tractable approach to overcome severe estimation difficulty. Marcucci (2005) compared a set of different standard GARCH models with a group of Markov regime-switching GARCH models. For a nonstationary Markov regime-switching GARCH model applied by Gray’s model, Hong and Hwang (2016) dealt with the asymptotic normality of the renormalized volatility.

In the present work, as a dynamic count time series model, we propose the generalized regime-switching integer-valued GARCH(1, 1) (GRS-INGARCH(1, 1)) process and discuss the forecasting, the cumulative impulse response function and the conditional maximum likelihood estimation (CMLE). As main theoretical results, we first derive a recursive formula of the conditional probability of Markov-chain state given the past information, in terms of transition probabilities and the Poisson parameters of the INGARCH(1, 1) process. Secondly under assumption of known parameters in INGARCH(1, 1), the forecasting of the Poisson parameter in the GRS-INGARCH(1, 1) process is investigated, and finally the cumulative impulse response function, which is a measure of the persistence of the volatility, is discussed. A Monte-Carlo study is conducted to see the time series plots of the GRS-INGARCH(1, 1) along with the volatility, and the performance of volatility forecasting and the behavior of the cumulative impulse response functions as well as the CMLE. A real data application is given to compare the INGARCH(1, 1) and the GRS-INGARCH(1, 1) process for the numbers of visitors in Hallasan Mountain National Park.

The remainder of the paper is organized as follows. In Section 2 we present the model and main results of forecasting and the cumulative impulse response function. In Section 3 a Monte-Carlo study is given and in Section 4 a real data analysis is presented. Proofs are drawn in Section 5.

2. Model and main results

### 2.1. Generalized regime-switching integer-valued GARCH(1, 1) process

An INGARCH (1, 1) process, introduced by Ferland et al. (2006), is defined by {Xt}t∈ℤ such that

$Xt∣ℱt-1:Poisson(λt)λt=ω+αXt-1+βλt-1,$

where ω > 0, α ≥ 0, and β ≥ 0. ℱt–1 is the entire history up to time t − 1. It is known that if α + β < 1 then the INGARCH(1, 1) is stationary. Let θλ = (ω, α, β), the parameter vector, and write λt = λ(θλ, ℱt–1).

Now we apply a GRS-GARCH of Gray (1996) to the INGARCH model and consider the GRS-INGARCH( 1, 1) model in this work. To propose the GRS-INGARCH(1, 1) process, we adopt a two-state Markov chain {St : t ∈ ℤ} with St = 1, 2 and the following time-varying transition probabilities (A multiple-state can be given straightforwardly).

$Pr(St=1∣St-1=1)=Pt,Pr(St=2∣St-1=1)=1-Pt,Pr(St=2∣St-1=2)=Qt,Pr(St=1∣St-1=2)=1-Qt.$

In the GRS-INGARCH(1, 1) model, the parameter vector θλ depends on the state St and so we write θλ = θλ(St). For i = 1, 2, if St = i, let θλ(St) = (ωi, αi, βi) and

$λit=λ(θλ(St),ℱt-1)=ωi+αiXt-1+βiλt-1, if St=i.$

The GRS-INGARCH(1, 1) model with (2.1) is written as

$Xt∣ℱt-1={Poisson(λ1t),w.p. p1t,Poisson(λ2t),w.p. p2t,$

where p1t = Pr(St = 1| ℱt–1) and p2t = Pr(St = 2| ℱt–1) = 1 − p1t. For i = 1, 2, the conditional probability pit of state i, given the past information, is seen as a recursive formula in the following theorem.

Theorem 1

Givent–1, the past information at time t − 1, we have

$p1t=e-λ1,t-1λ1,t-1Xt-1p1,t-1Pt+e-λ2,t-1λ2,t-1Xt-1p2,t-1(1-Qt)e-λ1,t-1λ1,t-1Xt-1p1,t-1+e-λ2,t-1λ2,t-1Xt-1p2,t-1$

and p2t = 1 − p1t.

Remark 1

Note that for the nonstationary GRS-INGARCH(1, 1) model with a regime having explosive values in volatility, the probabilities p1t converge to a constant related with the limit of the transition probabilities, as seen in the formula of (2.2). However, for the stationary GRS-INGARCH(1, 1) models, the probabilities might be dynamics depending on the randomness of the processes. In the simulation of Section 3, we compute the probabilities p1t in stationary GRS-INGARCH(1, 1) models and see their dynamics even in the case of constant transition probabilities.

The volatility, which is the conditional mean and the conditional variance, λt = E[Xt| ℱt–1], of the GRS-INGARCH(1, 1) process given the past is expressed as $λt=P(St=1∣ℱt-1)E[Xt∣ℱt-1,St=1]+P(St=2∣ℱt-1)E[Xt∣ℱt-1,St=2]=p1tλ1t+p2tλ2t=pt′Λt$ where pt = (p1t, p2t)′ and Λt = (λ1t, λ2t)′.

$Λt=[λ1tλ2t]=[ω1ω2]+Xt-1 [α1α2]+λt-1 [β1β2]=:w+Xt-1a+λt-1b,$

where w = (ω1, ω2)′, a = (α1, α2)′, and b = (β1, β2)′. We multiply by pt′ to obtain $pt′Λt=pt′w+Xt-1pt′a+λt-1 pt′b$. Thus we write

$λt=Wt+AtXt-1+Btλt-1,$

where Wt = pt′w, At = pt′a, and $Bt=pt′b$.

It is easily shown that λt can be expressed as

$λt=Wt+∑ℓ=1t-1Wt-ℓ∏j=1ℓBt-j+1+∑ℓ=1tAt-ℓ-1∏j=1ℓBt-j+1Xt-ℓ+∏j=1tBjλ0.$

In the next section we derive the recursive formula of the forecasting of λt and the representation of the cumulative impulse response function.

2.2. Forecasting and cumulative impulse response function

On behalf of the forecasting of the GRS-INGARCH(1, 1) process, we assume that parameters w = (ω1, ω2)′, a = (α1, α2)′, and b = (β1, β2)′ are known and study the forecasting of the volatility in the GRS-INGARCH(1, 1) model. For the case of unknown parameters, we consider the CMLE of Cui and Wu (2016) for the parameters, which will be discussed in the next section. Assume that we observe {X1, …, Xt} where t is the present time and note that the Markov Chain {S1, …, St} is hidden in the observation. The process {S1, …, St} is understood with the transition probabilities and the conditional probabilities in Theorem 1.

In the following theorem we present the -step ahead of forecasting of the Poisson parameter λt+ of the GRS-INGARCH(1, 1) process, given the information ℱt at the present time t. For = 1 the forecasting is given as

$λ^t(1)=λ^t+1=Wt+1+At+1Xt+Bt+1λt$

where $Wt+1=pt+1′w,At+1=pt+1′a, Bt+1=pt+1′b$ with pt+1 = (p1,t+1, p2,t+1)′, p1,t+1 of the form as in (2.2) replaced t by t + 1, and p2,t+1 = 1 − p1,t+1.

### Theorem 2

For ℓ = 2, 3, …, let λ̂t() = λ̂t+ = E[λt+|ℱt], the ℓ-step ahead forecasting of the parameter of the Poisson of the GRS-INGARCH(1, 1) process, given the informationt at the present time t. Then

$λ^t(ℓ)=λ^t+ℓ=W˜t+ℓ+∑k=1ℓ-1W˜t+k+1∏j=2ℓ-k(A˜t+j+1+B˜t+j+1)+∏i=1ℓ-1(A˜t+i+1+B˜t+i+1)λ^t(1),$

where, for j = 2, 3, …, ℓ,

$W˜t+j=p˜t+j′w, A˜t+j=p˜t+j′a, B˜t+j=p˜t+j′b$

witht+j = (1,t+j, 2,t+j)′,

$p˜1,t+j=e-λ^1,t+j-1λ^1,t+j-1λ^t+j-1p˜1,t+j-1Pt+j+e-λ^2,t+j-1λ^2,t+j-1λ^t+j-1p˜2,t+j-1 (1-Qt+j)e-λ^1,t+j-1λ^1,t+j-1λ^t+j-1p˜1,t+j-1+e-λ^2,t+j-1λ^2,t+j-1λ^t+j-1p˜2,t+j-1λ^i,t+j-1=ωi+(αi+βi)λ^t+j-2, i=1,2$

and p̃2,t+j = 1 − 1,t+j.

Now we derive the representation for the cumulative impulse response function of the GRS-INGARCH( 1, 1) process. Impulse response function of a dynamic system is its output when presented with a brief input signal, called an impulse. An impulse response refers to the reaction of the dynamic system in response to some external change. Baillie et al. (1996) introduced fractionally integrated GARCH processes, and noted that cumulative impulse response ζ in volatility, given by the partial derivative of the volatility with respect to the prediction error for the squared observations, goes to zero as → ∞ for a class of stable GARCH processes, while ζ → ∞ as → ∞ for a class of explosive GARCH processes. In the INGARCH(1, 1) case as in our work, the cumulative impulse response ζ is given by

$ζℓ=∂λ^t(ℓ)∂ηt$

with ηt = Xtλt = XtE[Xt|ℱt–1], the prediction error for the observations Xt. The ζ measures a certain contribution of innovation ηt at time t to the -step ahead volatility. The cumulative impulse response function ζ of the GRS-INGARCH(1, 1) model is derived as:

Letting ηt = Xtλt, which is a white noise with mean zero, we can express

$Xt=Wt+(At+Bt)Xt-1+ηt-Btηt-1.$

Note that {Xt} is an ARMA(1, 1) process with time-varying coefficients, and for ≥ 2,

$λ^t(ℓ)=E[Xt+ℓ∣ℱt]=E[Wt+ℓ+(At+ℓ+Bt+ℓ)Xt+ℓ-1+ηt+ℓ-Bt+ℓ ηt+ℓ-1∣ℱt]=Wt+ℓ+(At+ℓ+Bt+ℓ)λ^t(ℓ-1)$

and thus ζ = (At+ + Bt+)ζ−1. For = 1, since

$λ^t(1)=Wt+1+At+1Xt+Bt+1λt=Wt+1+(At+1+Bt+1)Xt-Bt+1ηt$

we have ζ1 = (At+1 + Bt+1) − Bt+1 = At+1 where ∂Xt/∂ηt = 1 is used. The following result presents the general solution of the linear difference equation for the cumulative impulse response function ζ for = 1, 2, …, straightforwardly from Theorem 2, where the estimated parameters are used.

### Lemma 1

The cumulative impulse response coefficients ζ of the GRS-INGARCH(1, 1) model are given by ζ1 = At+1and for ℓ = 2, 3, …,

$ζℓ=∏j=1ℓ-1(A˜t+j+1+B˜t+j+1) At+1.$

In the next section of a Monte-Carlo simulation, we observe the behaviors of ζ in various cases of INGARCH(1, 1) and GRS-INGARCH(1, 1) processes.

2.3. The conditional maximum likelihood estimation for the generalized regime-switching GARCH(1, 1) process

Recently, Cui and Wu (2016) proposed the CMLE for INGARCH models and established the strong consistency and asymptotic normality of the CMLE. We adopt their spirit of the CMLE for the case of unknown parameters. We discuss here the CMLE of the six parameters of the GRS-INGARCH(1, 1) models, but the explicit theoretical validity remains as a future study. Let θ = (ω1, α1, β1, ω2, α2, β2)′ denote the vector of parameters, assuming that it belongs to the parameters domain Θ ⊂ (0,∞) × [0,∞)2 × (0,∞) × [0,∞)2. Suppose that we observed {X1, X2, …, Xt} with the present time t. The conditional likelihood function $L(θ)=∏j=1t[e-λjλjXj/(Xj!)]$. In practice, if the initial values λ1 and X0 are known, then the sequential λ̂j’s are obtained with λ̂j = λ̂1jp1j + λ̂2j p2j where λ̂i j and pi j, i = 1, 2, j = 1, 2, …, t, are obtained via (2.1) and (2.2), respectively, and the conditional likelihood function becomes $∏j=1t[e-λ^jλ^jXj/(Xj!)]$ and the corresponding log-likelihood function is given by

$ℒ(θ)=∑j=1tℓ^j(θ)=∑j=1t[Xj log (λ^j)-λ^j-log (Xj!)].$

The CMLE θ̂ of θ is defined as the maximizer of ℒ(θ), that is, θ̂= arg maxθ∈Θ ℒ(θ). We refer to Cui and Wu (2016) for specific conditions and asymptotic theory for the CMLE of INGARCH(p, q) models. In the next section of a Monte-Carlo study, we provide with some simulation results for the CMLE of the GRS-INGARCH(1, 1) models. Its verification of the asymptotic theory for the CMLE will be considered as a further study. A hypothesis test for strict stationarity or explosive volatility in the GRS-INGARCH(1, 1) model will also be considered as a future study, based on Lee and Noh (2013), who discussed the explosive volatility test as well as the Gaussian quasi-MLE in possibly nonstationary GARCH(1, 1) models.

3. A Monte-Carlo study

In this section a Monte-Carlo study is conducted to see time series plots of the GRS-INGARCH(1, 1) process and its volatility, and to obtain the performance of the forecasting and the cumulative impulse response. We first start with time series plot of the INGARCH(1, 1) model of Ferland et al. (2006). igure 1(a), (b) depicts the INGARCH(1, 1) process and its volatility with ω = 0.5, α = 0.3, and β = 0.5, which is a stationary case while Figure 1(c), (d) with ω = 0.1, α = 0.5, and β = 0.5001, which is a nonstationary case.

Figure 2 depicts the conditional probabilities p1t and p2t = 1 − p1t of regime 1 and regime 2, respectively, given the past information, which is given in Theorem 1, where the parameters of GRS-INGARCH( 1, 1) process are ω1 = 0.1, α1 = 0.2, β1 = 0.3, ω2 = 0.1, α2 = 0.4, β2 = 0.5, that is a stationary case. Figure 2(a), (b) use constant transition probabilities Pt = 0.4, Qt = 0.7 in (a) and Pt = 0.8, Qt = 0.4 in (b), while Figure 2(c), (d) use time-dependent transition probabilities Pt = 0.8 – 0.6/t, Qt = 0.6 + 0.1/t in (c) and Pt = 0.9 – 0.5/t2, Qt = 0.7 + 0.1/t2 in (d).

In Figures 3 and 4, we see the time series plots of the stationary GRS-INGARCH(1, 1) process and its volatility. Figure 3(a)–(d) use the constant transition probabilities Pt and Qt as in Figure 2(a), (b), while Figure 4(a)–(d) use the time-dependent transition probabilities as in Figure 2(c), (d). Figures 5 and 6 repeat the same way as Figures 3 and 4, but with one of the two regimes being a nonstationary case. Figures 3 and 4 are related to switching two stationary INGARCH(1, 1) models, while Figures 5 and 6 are related to switching a stationary INGARCH(1, 1) process and a nonstationary INGARCH(1, 1) process with the same transition probabilities as those in Figures 3 and 4.

Now we consider the CMLE in Subsection 2.3. For the GRS-INGARCH(1, 1) models with two cases of

$θ=(0.5,0.2,0.3,1,0.4,0.5),(2,0.1,0.5,0.8,0.25,0.3),$

and Pt = 0.4, Qt = 0.7, Table 1 reports the sample mean and the standard error of the 100 CMLE estimates for the sample sizes 300, 600.

Now we discuss the forecasting, given in Theorem 2, under the assumption of known parameters of the INGARCH(1, 1). Tables 2 and 3 indicate the -step ahead forecasting of volatility in the stationary GRS-INGARCH(1, 1) in case of Figure 2(a), (b), respectively, with t = 300.

Finally we observe the behaviors of the cumulative impulse response functions discussed in Lemma 1. Figure 7(a) depicts the plot of the cumulative impulse response functions for three cases: stationary (ω = 0.5, α = 0.3, β = 0.5), integrated (ω = 0.1, α = 0.5, β = 0.5) and explosive (ω = 0.1, α = 0.55, β = 0.55) cases of the INGARCH(1, 1) process, while Figure 7(b) for the stationary GRS-INGARCH(1, 1) with four combinations of transition probabilities Pt, Qt ∈ {0.2, 0.9} with INGARCH(1, 1) parameters as in Figures 3 and 4. Figure 8(a) and Figure 9(a) illustrate the stationary INGARCH(1, 1) with various parameters of α ∈ {0.1, 0.2, 0.3, 0.4} and β ∈ {0.3, 0.4, 0.5, 0.6}, respectively, whereas Figure 8(b) and Figure 9(b) the stationary GRS-INGARCH(1, 1) model with various parameters of α2 ∈ {0.1, 0.2, 0.3, 0.4} and β2 ∈ {0.15, 0.3, 0.45, 0.59}, respectively, where Pt = 0.2, Qt = 0.9 are used.

4. Real data analysis

As a real data application, we consider the number of visitors in Hallasan Mountain National Park in Jeju Island, during Jan. 2007–Dec. 2016, of which data are obtained from The Korea Tourism and Information System ( www.tour.go.kr). To deseasonalize the data we take the ratio of the monthly data to the minimum of each month during ten years. For instance, for ten data as of January (Jan), let Janmin = min{x2007,Jan, x2008,Jan, …, x2016,Jan} and take yi,Jan = xi,Jan/Janmin in the model, for i = year, that is, the deseasonalized data are given by yi, j = xi, j/mj, where mj = min{xi, j|i = year}, i = year, j = month. To adjust the INGARCH model related with Poisson distribution we renormalize the data {yi, j} by taking a(yi, jμy)/σy +a2, where μy = E[yi, j] and $σy2=Var[yi,j]$ with the approximately same mean and variance. The deseasonalized and renormailized data a(yi, jμy)/σy + a2 with a = 4 are relabeled as {Xt : t = 1, 2, …, 120} the same mean and variance 4. According to Cui and Wu (2016), the CMLE as adopted in INGARCH(1, 1) model, which minimize the likelihood function in (2.6) with λ̂t = ω + αXt–1 + βλ̂t–1 and initial λ0 = 4, is given by ω̂ = 0.00002, α̂ = 0.99984, β̂ = 0.00001. In this case, one-step forecasting of λt is λ̂t(1) = 3.9994 and one-step forecasting of visitors is given by 68522.

Now, to adopt a GRS-INGARCH(1, 1) model, we use two-state Markov chain with transition probability P = 0.4, Q = 0.7 assuming that two states are no rain and rain in Hallasan Mountain. The values of the transition probability P = 0.4, Q = 0.7 are chosen so that the limiting probabilities of no rain and rain are respectively, 1/3 and 2/3. We compute the conditional probabilities p1t, p2t in Theorem 1 with initials p1,0 = 0.33, p2,0 = 0.66 and λ0 = λ1,0 = λ2,0 = 4, and obtain the CMLE in (2.6), which is given by ω̂1 = 1.8124, α̂1 = 0.37527, β̂1 = 0.29556, ω̂2 = 0.82143, α̂2 = 0.11036, β̂2 = 0.55403. By adopting these CMLE in the GRS-INGARCH(1, 1) model, one-step forecasting of λt is λ̂t(1) = 3.2974 and one-step forecasting of visitors is given by 63056. As expected, the GRS-INGARCH( 1, 1) using Markov-chain with two weather states is more appropriate for the forecasting of the stable number of visitors in Hallasan Mountain National Park.

5. Proofs

### Proof of Theorem 1

We observe

$p1t=Pr(St=1∣ℱt-1)=Pr(St-1=1∣ℱt-1)P(St=1∣St-1=1,ℱt-1)+Pr(St-1=2∣ℱt-1)P(St=1∣St-1=2ℱt-1)=Pr(St-1=1∣ℱt-1)Pt+Pr(St-1=2∣ℱt-1)(1-Qt).$

Now we compute Pr(St–1 = j|ℱt–1) = Pr(St–1 = j|Xt–1,ℱt–2). Set Aj = {St–1 = j|ℱt–2}, j = 1, 2 and B = {Xt–1 = k|ℱt–2}, k = 0, 1, 2, 3, …, and we have

$Pr(St-1=j∣Xt-1=k,ℱt-2)=Pr(Aj∣B)=Pr(B∣Aj) Pr(Aj)∑i=12Pr(B∣Ai) Pr(Ai)=Pr(Xt-1=k∣ℱt-2,St-1=j)pj,t-1∑i=12P(Xt-1=k∣ℱt-2,St-1=i)pi,t-1.$

Since

$Pr(Xt-1=k∣ℱt-2,St-1=j)=e-λj,t-1k!λj,t-1k$

with λj,t–1 = ωj + αjXt–2 + βjλt–2, j = 1, 2, we have

$Pr(St-1=j∣Xt-1,ℱt-2)=e-λj,t-1Xt-1!λj,t-1Xt-1pj,t-1e-λ1,t-1Xt-1!λ1,t-1Xt-1p1,t-1+e-λ2,t-1Xt-1!λ2,t-1Xt-1p2,t-1=e-λj,t-1λj,t-1Xt-1pj,t-1e-λ1,t-1λ1,t-1Xt-1p1,t-1+e-λ2,t-1λ2,t-1Xt-1p2,t-1.$

Hence we obtain

$p1t=Pr(St=1∣ℱt-1)Pt+Pr(St=2∣ℱt-1)(1-Qt)=e-λ1,t-1λ1,t-1Xt-1p1,t-1Pt+e-λ2,t-1λ2,t-1Xt-1p2,t-1(1-Qt)e-λ1,t-1λ1,t-1Xt-1p1,t-1+e-λ2,t-1λ2,t-1Xt-1p2,t-1.$

### Proof of Theorem 2

Given data X1 · · · Xt and information λ1, …, λt.

Two-step ahead forecast of λt is given by

$λ^t+2=λ^t(2)=E[Wt+2+At+2Xt+1+Bt+2λt+1∣ℱt]=Wt+2+(At+2+Bt+2)λ^t(1).$

However, we estimate Wt+2, At+2, Bt+2 by means of the estimates of pi,t+2, i = 1, 2. We use the formula of pit in Theorem 1 recursively to obtain (2.4) with j = 2, and thus we get λ̂t+2 = t+2 + (Ãt+2 + t+2)λ̂t+1.

For general = 2, 3, …, the -step ahead forecast is given by

$λ^t+ℓ=λ^t(ℓ)=E[Wt+ℓ+At+ℓXt+ℓ-1+Bt+ℓλt+ℓ-1∣ℱt]=Wt+ℓ+Wt+ℓ-1(At+ℓ+Bt+ℓ)+⋯+Wt+2(At+ℓ+Bt+ℓ)(At+ℓ-1+Bt+ℓ-1)⋯(At+3+Bt+3)+(At+ℓ+Bt+ℓ)⋯(At+ℓ+Bt+ℓ)λt(1)=Wt+ℓ+∑k=1ℓ-1Wt+k+1∏j=2ℓ-k(At+j+1+Bt+j+1)+∏i=1ℓ-1(At+i+1+Bt+i+1)λ^t(1)$

provided all pi,t+j, i = 1, 2, are given. These probabilities are estimated recursively by using Theorem 1 and obtain the results along with (2.4) in Theorem 2.

Acknowledgements

Authors are very grateful for valuable comments of two referee and an editor which improve the paper considerably. This work was supported by the National Research Foundation of Korea (NRF-2015-1006133).

Figures
Fig. 1. Stationary and nonstationary INGARCH(1, 1) process Xt with volatility λt = ω + αXt–1 + βλt–1: (a), (b) ω = 0.5, α = 0.3, β = 0.5; and (c), (d) ω = 0.1, α = 0.5, β = 0.5001.
Fig. 2. Conditional probability p1t in the GRS-INGARCH(1, 1) process with ω1 = 0.1, α1 = 0.2, β1 = 0.3; ω2 = 0.1, α2 = 0.4, β2 = 0.5: (a) Pt = 0.4, Qt = 0.7; (b) Pt = 0.8, Qt = 0.4; (c) Pt = 0.8 – 0.6/t, Qt = 0.6 + 0.1/t; (d) Pt = 0.9 – 0.5/t2, Qt = 0.7 + 0.1/t2.
Fig. 3. GRS-INGARCH(1, 1) process and its volatility with ω1 = 0.1, α1 = 0.2, β1 = 0.3; ω2 = 0.1, α2 = 0.4, β2 = 0.5 and with constant transition probabilities in (a), (b) Pt = 0.4, Qt = 0.7 and in (c), (d) Pt = 0.8, Qt = 0.4.
Fig. 4. GRS-INGARCH(1, 1) process and its volatility with ω1 = 0.1, α1 = 0.2, β1 = 0.3; ω2 = 0.1, α2 = 0.4, β2 = 0.5 and with time-dependent transition probabilities in (a), (b) Pt = 0.8 – 0.6/t, Qt = 0.6 + 0.1/t and in (c), (d) Pt = 0.9 – 0.5/t2, Qt = 0.7 + 0.1/t2.
Fig. 5. GRS-INGARCH(1, 1) process and its volatility with ω1 = 0.1, α1 = 0.2, β1 = 0.3; ω2 = 0.1, α2 = 0.5, β2 = 0.5001 and with constant transition probabilities in (a), (b) Pt = 0.4, Qt = 0.7 and in (c), (d) Pt = 0.8, Qt = 0.4.
Fig. 6. GRS-INGARCH(1, 1) process and its volatility with ω1 = 0.1, α1 = 0.2, β1 = 0.3; ω2 = 0.1, α2 = 0.5, β2 = 0.5001 and with time-dependent transition probabilities (a), (b) Pt = 0.8 – 0.6/t, Qt = 0.6 + 0.1/t and in (c), (d) Pt = 0.9 – 0.5/t2, Qt = 0.7 + 0.1/t2.
Fig. 7. Cumulative impulse response functions of (a) stationary, integrated, explosive INGARCH(1, 1) and (b) stationary GRS-INGARCH(1, 1) with transition probabilities Pt, Qt ∈ {0.2, 0.9}.
Fig. 8. Cumulative impulse response functions of (a) INGARCH(1, 1) with α ∈ {0.1, 0.2, 0.3, 0.4}; fixed ω, β and (b) GRS-INGARCH(1, 1) with α2 ∈ {0.1, 0.2, 0.3, 0.4}; fixed α1, ωi, βi, i = 1, 2.
Fig. 9. Cumulative impulse response functions of (a) INGARCH(1, 1) with β ∈ {0.3, 0.4, 0.5, 0.6}; fixed ω, α and (b) GRS-INGARCH(1, 1) with β2 ∈ {0.15, 0.3, 0.45, 0.59}; fixed β1, ωi, αi, i = 1, 2.
TABLES

### Table 1

Conditional maximum likelihood estimates for GRS-INGARCH(1, 1) with Pt = 0.4, Qt = 0.7

ω1 = 0.5α1 = 0.2β1 = 0.3ω2 = 1α2 = 0.4β2 = 0.5
n = 3000.4791 (0.0122)0.2217 (0.0438)0.2592 (0.0271)0.9587 (0.0104)0.4295 (0.0396)0.4661 (0.0331)
n = 6000.4853 (0.0205)0.2186 (0.0353)0.2687 (0.0214)0.9706 (0.0272)0.4167 (0.0309)0.4654 (0.0221)
ω1 = 2α1 = 0.1β1 = 0.5ω2 = 0.8α2 = 0.25β2 = 0.3
n = 3001.9910 (0.0489)0.1271 (0.0363)0.4755 (0.0400)0.7931 (0.0489)0.2811 (0.0380)0.2795 (0.0401)
n = 6002.001 (0.0386)0.1405 (0.0105)0.4887 (0.0371)0.8026 (0.0399)0.2905 (0.0015)0.2933 (0.0383)

### Table 2

Forecasting λ̂t() of GRS-INGARCH(1, 1) with Pt = 0.4, Qt = 0.7, ω1 = 0.1, α1 = 0.2, β1 = 0.3, ω2 = 0.1, α2 = 0.4, β2 = 0.5

-stepλ̂t()-stepλ̂t()-stepλ̂t()-stepλ̂t()-stepλ̂t()
10.434110.448210.439310.432410.438
20.438120.449220.438320.431420.437
30.441130.449230.437330.432430.438
40.444140.450240.435340.434440.437
50.446150.450250.434350.438450.440
60.447160.448260.433360.442460.443
70.448170.445270.432370.443470.444
80.449180.443280.431380.441480.446
90.447190.441290.431390.441490.447
100.446200.440300.432400.439500.446

### Table 3

Forecasting λ̂t() of GRS-INGARCH(1, 1) with Pt = 0.8, Qt = 0.4, ω1 = 0.1, α1 = 0.2, β1 = 0.3, ω2 = 0.1, α2 = 0.4, β2 = 0.5

-stepλ̂t()-stepλ̂t()-stepλ̂t()-stepλ̂t()-stepλ̂t()
10.255110.278210.256310.248410.253
20.268120.277220.264320.248420.255
30.276130.283230.261330.258430.249
40.281140.279240.257340.270440.259
50.283150.272250.255350.276450.267
60.284160.263260.253360.277460.272
70.281170.261270.252370.268470.277
80.273180.264280.248380.264480.276
90.268190.262290.250390.261490.271
100.275200.256300.251400.254500.266

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