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Threshold-asymmetric volatility models for integer-valued time series

Deok Ryun Kima,b, Jae Eun Yoona, and Sun Young Hwang1,a

aDepartment of Statistics, Sookmyung Women’s University, Korea, bInternational Vaccine Institute, Korea
Correspondence to: 1Department of Statistics, Sookmyung Women’s University, Cheongpa-ro 47-gil 100, Seoul 04310, Korea. E-mail: shwang@sookmyung.ac.kr
Received January 9, 2019; Revised February 22, 2019; Accepted March 5, 2019.
Abstract

This article deals with threshold-asymmetric volatility models for over-dispersed and zero-inflated time series of count data. We introduce various threshold integer-valued autoregressive conditional heteroscedasticity (ARCH) models as incorporating over-dispersion and zero-inflation via conditional Poisson and negative binomial distributions. EM-algorithm is used to estimate parameters. The cholera data from Kolkata in India from 2006 to 2011 is analyzed as a real application. In order to construct the threshold-variable, both local constant mean which is time-varying and grand mean are adopted. It is noted via a data application that threshold model as an asymmetric version is useful in modelling count time series volatility.

Keywords : count data, integer-valued time series, threshold integer-valued ARCH, volatility
1. Introduction

Over the past three decades, there has been increasing interest in modeling integer-valued time series because of the broad range of potential applicability to epidemiology (Cardinal et al., 1999; Yoon and Hwang, 2015a, 2015b), social science (McCabe and Martin, 2005; Truong et al., 2017), experimental biology (Zhou and Basawa, 2005; Bartlett and McCormick, 2017), environmental science (Thyregod et al., 1999; Pavlopoulos and Karlis, 2008), and economics (Freeland and McCabe, 2004; Quoreshi, 2014).

The first order integer-valued autoregressive model (INAR(1)) with Poisson distribution has been introduced by McKenzie (1985). Alzaid and Al-Osh (1990) extended it to pth-order model (INAR(p)) and Al-Osh and Alzaid (1988) introduced a qth-order integer-valued moving average model (INMA(q)). A integer-valued ARMA models for dependent sequences of Poisson counts was investigated by McKenzie (1988).

Although modeling time series of count data with Poisson distribution is a useful tool, in practice, over-dispersion and zero-inflation in the time series is easily led to a violation of major assumptions that the variance is equal to the mean, and the parameters are to be positive. Ferland et al. (2006) proposed an integer-valued GARCH model to study over-dispersed counts, and Fokianos and Fried (2010), Weiß (2010), Zhu and Wang (2010), Zhu (2011, 2012a), and Yoon and Hwang (2015a, 2015b) made further contributions to the literature. Zhu (2012b) extended the model to address both overdispersion and zero inflation phenomenon in count data. Yoon and Hwang (2015b) presented a data application of the zero-inflated model via conditional Poisson and negative binomial marginals. Wang et al. (2014) proposed a self-exited threshold integer-valued Poisson autoregression model (SETPAR) which allows for more general modeling framework including the possibility of negative serial dependence in the time series of count data.

In this paper we study conditional variance (volatility) for over-dispersion, zero-inflation, and serial dependence of count time series data. The organization of this paper is as follows. Section 2 re-introduces existing models as threshold integer-valued analogue of the autoregressive conditional heterosckedastic (ARCH) model by adding threshold-asymmetric effects to the models. It is noted that innovation follows either Poisson distribution or negative binomial distribution. Their estimation method is discussed in Section 3. Section 4 illustrates appropriate threshold model building strategies via applying proposed threshold-models to actual, highly skewed, zero-inflated, and serially correlated data example of cholera disease in Kolkata in India from 2006 to 2011 (Ali et al., 2016). Concluding remarks are presented in Section 5.

2. Various integer-valued threshold-asymmetric ARCH models

### 2.1. INTARCH(1) model

The first-order integer-valued ARCH (INARCH(1), for short) model (Ferland, 2006) is defined as a conditional Poisson model defined by

$Xt∣Ft-1~Poisson(λt), λt=α0+α1Xt-1,$

where Ft−1 denotes the information available up to time t − 1. The parameters α0 and α1 are positive. The model (2.1) may be extended to have a two-regime structure of the conditional mean process according to the magnitude of the observation. A threshold model based on (2.1), so called, the first-order integer-valued threshold ARCH (INTARCH(1), for short) model is defined as (Wang et al., 2014)

$Xt∣Ft-1~Poisson(λt), λt=α0+α1Xt-1(r)+α2Xt-1(l),Xt-1(r)={Xt-1,if Xt-1>mt,0,if Xt-1≤mt, Xt-1(l)={Xt-1,if Xt-1≤mt,0,if Xt-1>mt,$

where the parameters α0, α1, and α2 are positive, and the initial value X0 = x0 is fixed. Here, mt is a (time varying) threshold variable that determines the dynamic switching mechanism of the model. The dynamics of the process is governed by a two-regime scheme. Specifically, if Xt−1 > mt then we say Xt lies in the upper regime, otherwise, Xt belongs to the lower regime. Various choices of the threshold variable have been used in applications (Wu and Chen, 2007). In the real data example, we employ two threshold variables: one is simply grand mean of the entire time series and the other is the local constant mean which is time-varying.

### 2.2. NB-INTARCH(1) model

To accommodate over-dispersion in the data, one may consider the model for which negative binomial distribution is used to model the process. The first-order integer-valued negative binomial ARCH (NB-INARCH(1), for short) model (Zhu, 2011; Yoon and Hwang, 2015a) is defined as

$Xt∣Ft-1~NB(r,pt), λt=1-ptpt=α0+α1Xt-1,$

where the parameter r is a positive integer. We propose the following threshold-asymmetric model as a generalization of (2.3).

$Xt∣Ft-1~NB(r,pt), λt=1-ptpt=α0+α1Xt-1(r)+α2Xt-1(l),$

where $Xt-1(r)$ and $Xt-1(l)$ are two threshold values defined by (2.2). The model (2.4) is referred to as negative binomial INTARCH(1), denoted as NB-INTARCH(1).

### 2.3. ZIP-INTARCH(1) model

In order to capture zero-inflation in the count data, Zhu (2012a) and Yoon and Hwang (2015b) investigated the following first-order zero-inflated Poisson ARCH (ZIP-INARCH(1)) model which is formulated as

$Xt∣Ft-1~ZIP(λt,w), λt=α0+α1Xt-1,$

where ZIP(λt,w) is defined as the following probability mass function (pmf)

$P(X=k)=wδk,0+(1-w)λke-λk!, k=0,1,2,…,$

where δk,0 is 1 when k = 0 and is zero when k ≠ 0. The parameter 0 < w < 1 determines severity of zero-inflation. If w = 0, then the model reduces to the Poisson INARCH(1) model defined in (2.1). The ZIP-INARCH(1) model is made to be threshold-asymmetric via

$Xt∣Ft-1~ZIP(λt,w), λt=α0+α1Xt-1(r)+α2Xt-1(l),$

where $Xt-1(r)$ and $Xt-1(l)$ are two threshold values as defined in (2.2). The model (2.6) is regarded as zero-inflated Poisson INTARCH(1), that is, ZIP-INTARCH(1) model.

### 2.4. ZINB-INTARCH(1) model

With replacing Poisson by negative binomial distribution in (2.5), the ZIP-INARCH(1) model becomes the following first-order zero-inflated negative binomial INARCH (ZINB-INARCH(1), for short) model which was considered in Zhu (2012a) and Yoon and Hwang (2015b), given as

$Xt∣Ft-1~ZINB(λt,a,w), λt=1-ptpt=α0+α1Xt-1,$

where ZINB(λt, a,w) is defined by

$P(X=k)=wδk,0+(1-w)Γ(k+λ1-ca)k!Γ(λ1-ca)(11+aλc)λ1-ca(aλc1+aλc)k, k=0,1,2,…,$

where 0 < w < 1, λ > 0, the dispersion parameter a > 0, and Γ denotes the standard Gamma function. The index c (= 0, 1) identifies the particular form of the underlying NB distribution (Ridout et al., 2001). For c = 0, this particular distribution is denoted by ZINB1(λt, a,w) and the case of c = 1 refers to ZINB2(λt, a,w). The ZINB-INARCH(1) model is now equipped with the threshold-asymmetry by using the equation

$Xt∣Ft-1~ZINB(λt,a,w), λt=1-ptpt=α0+α1Xt-1(r)+α2Xt-1(l).$

Here $Xt-1(r)$ and $Xt-1(l)$ are again two threshold values described in (2.2). The model (2.9) can be referred to as zero-inflated negative binomial INTARCH(1), abbreviated as ZINB-INTARCH(1).

3. Estimation of parameters

For each model discussed in Section 2, we use EM algorithm to estimate the parameters following the method proposed by Zhu (2012a). See also Yoon and Hwang (2015a, 2015b) for the application of EM algorithm in the context of count time series. Since general steps are the same except conditional log-likelihood function, first-order and second-order derivatives of the log-likelihood function with respect to parameters, we discuss INTARCH(1) case only, defined by (2.2). Let θ = (α0, α1, α2).

The likelihood function of Poisson distribution in INTARCH(1) model is

$L(θ)=∏t=1nλte-λtXt.$

The conditional log-likelihood function is

$l(θ)=log L(θ)=∑t=1nXt log λt-λt-log(Xt!).$

The first derivative of the log-likelihood with respect to θ = (α0, α1, α2) is given by

$∂l(θ)∂θi={Xtλt-1}∂λt∂θi,$

while the second derivative is

$∂2l(θ)∂θi∂θj={Xtλt-1}∂2λt∂θi∂θj-{Xtλt2}∂λt∂θi∂λt∂θj.$

The iterative EM procedure estimates the parameter θ = (α0, α1, α2) by maximizing the log-likelihood function. It consists of an E step and M step described as follows:

• E step: Given initial values of θ(0) and X0, calculate λt. And in case of zero-inflation model, the missing data are replaced by their conditional expectation which is given by w/(w + (1 − w)eλt) and zero respectively, according to Xt = 0 and Xt = 1, 2, . . . (Zhu, 2012a, 2012b). In the subsequent iteration, the estimated values in M step are used to calculate λt and the conditional expectation of missing data in case of zero-inflation model.

• M step: The estimation of θ can be obtained by maximizing the log-likelihood function. Starting with initial values θ(0), the values of θ in the subsequent iteration can be obtained as

$θ(i+1)=θ(i)-{∂2l∂θ∂θT|θ(i)}-1∂l∂θ|θ(i),$

where θ(i) is the value in the ith iteration. The missing data and λt are set from the previous E step of the EM procedure.

The estimates θ = (α0, α1, α2) are obtained by iterating these two steps until convergence. In the data analysis presented in the next section, the convergence criterion of the EM procedure is given as |(θ(i+1)θ(i))/θ(i)| ≤ 10−5.

4. Real data analysis: the cholera data from Kolkata in India

In this section, via real data application, we illustrate INTARCH, NB-INTARCH, ZIP-INTARCH, and ZINB-INTARCH models which are defined in Section 3. We consider time series of weekly cholera cases from Kolkata in India, consisting of 260 observations starting from 39th-week of 2006 to 38th-week of 2011. Figure 1 shows the time series plots and histograms for the entire period (in upper panel), 1st half period (in middle panel) and 2nd half period (in lower panel). For entire period, empirical mean and variance of the data are 2.58 and 27.51, respectively.

A histogram of the series shows there are 94 zeros which is 36% of the series. The zero-inflation index (zi) defined by Puig and Valero (2006) to measure the departure from the Poisson model is calculated using the formula

$zi=1+log(p0)μ,$

where p0 is the proportion of zero’s and μ is the mean. It is noted that zi is zero if the count time series data is Poisson-distributed and zi > 0 if the count time series data is zero-inflated. The zero-inflation index in the entire period is 0.61, which indicates there is a zero inflation. Similarly, the time series of 1st half period and 2nd half period also have high zero-inflation index, 0.57 and 0.62, respectively. However, when comparing these two period, the 1st half period shows lower mean (1.85) and variance (7.07) but higher proportion of zero counts (45%), while the second half period shows higher mean (3.3) and variance (47.1) but lower proportion of zero counts (28%).

The sample autocorrelation and partial autocorrelation function of the series in entire period, 1st half period, and 2nd half period are plotted in Figure 2 from which it is noted that there exists serial dependency. As in Tables 13 below, we consider various models including INTARCH(1), NB-INTARCH( 1), ZIP-INTARCH(1) and ZINB-INTARCH(1). Each model is fitted to the entire period and then is fitted further separately to two subset periods: first half and second half period.

The initial values $θ(0)=(α0(0),α1(0),α2(0))$ are randomly selected from the uniform distribution over the unit interval (0,1). Initial values for w = 0.5 and a = 0.1 are chosen arbitrary and first observation is taken as the initial value for X0. We then iterate the estimation procedure by using the previous estimated value as the next initial value and we stop the procedure when the convergence criteria |(θ(i+1)θ(i))/θ(i)| ≤ 10−5 is met. Motivated by incubation period (Azman et al., 2013) and transmission period (Ali et al., 2016), the most recent 4 time points (which is 4 weeks (28 days)) is considered as local constant mean mt which is used as a threshold variable. Specifically,

$mt=[average of (Xt-4+Xt-3+Xt-2+Xt-1)+0.5],$

where [x] denotes the greatest integer function not exceeding x.

The results of model fitting for entire period and two subset periods are summarized in Tables 13, respectively. It is noted that NB is further splitted as NB1 and NB2 according to the index c = 0 and c = 1, respectively, defined in the pmf equation (2.8). In conclusion, negative binomial models and zero-inflated negative binomial models accommodating over-dispersion and zero-inflation are best fitted both in the entire period data and in each subset period data. Considering Table 1, based on Akaike information criterion (AIC) and Bayesian information criterion (BIC), we find that negative binomial model and zero-inflated negative binomial models are more appropriate. Among the models, ZINB1-INTARCH(1) model shows that a substantial improvement occurs when using the local constant threshold instead of grand mean threshold. To assess the adequacy of the “threshold” ZINB1-INTARCH(1) model over the “non-threshold” ZINB1-INARCH(1) model, we use the following likelihood ratio test (LRT) statistic

$LRT=-2 log L(θ*)-[-2 log L(θ★)],$

where log L(θ*) and log L(θ) denotes log-likelihood function of ZINB1-INARCH(1) model and log-likelihood of ZINB1-INTARCH(1) model. Note that ZINB1-INARCH(1) is nested in ZINB1-INTARCH(1). Due to Self and Liang (1987), the asymptotic distribution of LRT is given by the 50 : 50 mixture of the constant zero and the χ2(1) distribution under the null. It is noted that LRT is calculated as 3.6 which is slightly lower than 3.84 which is the upper 5 percent of χ2(1) distribution. Consequently, the adequacy of ZINB1-INTARCH(1) over ZINB1-INARCH(1) is highly significant with the p-value given by P(null distribution exceeds 3.6) = P(χ2(1) > 6.2). Another careful examination of Table 1 indicates that NB2-INTARCH(1) and ZINB2-INTARCH(1) using local constant are appropriate models. This confirms that there exist both over-dispersion and zero-inflation in the entire data.

In each subset period (see Tables 2 and 3), it is seen overall that negative binomial models and zero-inflated negative binomial models are appropriate. See the ZINB1-INTARCH(1) model fitted in the 1st half period series and observe that a substantial improvement is obtained when using local constant threshold rather than using grand mean threshold. However, in the Table 3 (2nd half period), the ZINB2-INTARCH(1) model is improved by using grand mean threshold instead of local constant threshold. This might be partly due to the existence of high peak season in 2nd half period.

5. Concluding remarks

In this paper we have discussed various threshold-asymmetric (ARCH-type conditionally heteroscedastic) volatility models to analyze integer-valued count time series. Over-dispersion and zero-inflation are accommodated using negative binomial distributions. The EM method is adopted to estimate parameters. Two threshold variables, viz., grand mean and local constant mean, are considered in various threshold models. It is noted that the local constant mean works usually better than the grand mean while the grand mean seems better than the local constant mean in case when high peak season is prominent in short time period (see 2nd half period). We have compared models using likelihood-based approaches of AIC, BIC, and log-likelihood. Non-likelihood approaches such as forecasting error evaluations via parametric bootstrap can also be implemented to compare various threshold models and this task is now under investigation.

Acknowledgement

We are grateful for sharing the data in this paper to International Vaccine Institute (IVI), Seoul, Korea and the National Institute of Cholera and Enteric Disease (NICED), Kolkata in India who jointly owned the data. We thank the two anonymous referees for careful reading of the paper. SY Hwang’s work was supported by a grant from the National Research Foundation of Korea (NRF-2018R1A2B2004157).

Figures
Fig. 1. Weekly cholera cases series and histogram of cholera cases for entire period (n = 260; upper panel), 1st half period (n = 130; middle panel), 2nd half period (n = 130; lower panel).
Fig. 2. Sample autocorrelation functions and sample partial autocorrelation functions of the series for entire period (upper panel), 1st half period (middle panel) and 2nd half period (lower panel).
TABLES

### Table 1

Parameter estimates: entire period

ModelsThreshold value$α^0$$α^1$$α^2$AICBIC$-2logL(θ^)$
INARCH(1)0.796030.691631129.81136.91125.8
INTARCH(1)Grand mean0.798510.692580.685771131.81136.91125.8
INTARCH(1)Local constant0.766210.666310.774971130.01135.21124.0
NB1-INARCH(1)0.734100.683640.99999959.0964.1953.0
NB1-INTARCH(1)Grand mean0.733740.683500.684500.99999961.0964.1953.0
NB1-INTARCH(1)Local constant0.704560.656370.770280.99999959.2962.3951.2
NB2-INARCH(1)0.817570.666940.69629957.5962.6951.5
NB2-INTARCH(1)Grand mean0.807860.652850.698340.69632959.4962.5951.4
NB2-INTARCH(1)Local constant0.784860.588350.805450.68720957.7960.8949.7
ZIP-INARCH(1)0.196941.153420.696441097.11102.21091.1
ZIP-INTARCH(1)Grand mean0.196391.130520.693510.736031098.91102.01090.9
ZIP-INTARCH(1)Local constant0.189671.076840.672180.818241096.71099.81088.7
ZNB1-INARCH(1)0.157090.995360.700460.99999971.8974.9963.8
ZNB1-INTARCH(1)Grand mean0.156340.964850.694850.758730.99999973.2974.3963.2
ZNB1-INTARCH(1)Local constant0.151120.925170.668990.835020.99999970.2971.3960.2
ZNB2-INARCH(1)0.000010.817590.666940.69626959.5962.6951.5
ZNB2-INTARCH(1)Grand mean0.000010.807870.652850.698350.69630961.4962.5951.4
ZNB2-INTARCH(1)Local constant0.000010.784870.588350.805460.68717959.7960.8949.7

AIC = Akaike information criterion; BIC = Bayesian information criterion.

### Table 2

Parameter estimates: 1st half period

ModelsThreshold value$α^0$$α^1$$α^2$AICBIC$-2logL(θ^)$
INARCH(1)0.746510.59011516.7522.4512.7
INTARCH(1)Grand mean0.652360.569240.96502514.6518.3508.6
INTARCH(1)Local constant0.672050.521230.84128513.2516.9507.2
NB1-INARCH(1)0.592220.595790.99999431.2434.9425.2
NB1-INTARCH(1)Grand mean0.574160.585830.699740.99999432.9434.6424.9
NB1-INTARCH(1)Local constant0.563990.546260.746370.99999431.9433.6423.9
NB2-INARCH(1)0.616440.762960.99596438.5442.2432.5
NB2-INTARCH(1)Grand mean0.567810.666141.163790.96519438.6440.3430.6
NB2-INTARCH(1)Local constant0.575470.599341.072060.95667438.1439.8430.1
ZIP-INARCH(1)0.396962.156900.33124488.1491.8482.1
ZIP-INTARCH(1)Grand mean0.369111.688790.409660.90624488.4490.1480.4
ZIP-INTARCH(1)Local constant0.368961.731600.359800.69876486.9488.6478.9
ZNB1-INARCH(1)0.028760.618590.611600.99999433.1434.8425.1
ZNB1-INTARCH(1)Grand mean0.033450.602150.601880.735990.99999434.8434.5424.8
ZNB1-INTARCH(1)Local constant0.058600.602910.555090.875660.99999432.9432.6422.9
ZNB2-INARCH(1)0.000010.616460.762940.99593440.5442.2432.5
ZNB2-INTARCH(1)Grand mean0.000020.567830.666141.163820.96514440.6440.3430.6
ZNB2-INTARCH(1)Local constant0.000030.575500.599341.072070.95660440.1439.8430.1

AIC = Akaike information criterion; BIC = Bayesian information criterion.

### Table 3

Parameter estimates: 2nd half period

ModelsThreshold value$α^0$$α^1$$α^2$AICBIC$-2logL(θ^)$
INARCH(1)0.798020.75262585.8591.5581.8
INTARCH(1)Grand mean1.065860.800770.26593571.6575.3565.6
INTARCH(1)Local constant0.798660.753080.75105587.8591.5581.8
NB1-INARCH(1)0.765930.741090.99999513.6517.3507.6
NB1-INTARCH(1)Grand mean1.023770.787680.284800.99999511.4513.1503.4
NB1-INTARCH(1)Local constant0.760670.737350.753710.99999515.5517.2507.5
NB2-INARCH(1)0.878970.668710.46770500.5504.3494.5
NB2-INTARCH(1)Grand mean1.065720.792070.270630.40640495.7497.4487.7
NB2-INTARCH(1)Local constant0.863670.640010.718020.46731502.4504.1494.4
ZIP-INARCH(1)0.119340.978240.77646575.9579.6569.9
ZIP-INTARCH(1)Grand mean0.106991.279580.805410.24493565.5567.2557.5
ZIP-INTARCH(1)Local constant0.120720.978210.774810.78243577.9579.6569.9
ZNB1-INARCH(1)0.101860.918770.765170.99999519.5521.2511.5
ZNB1-INTARCH(1)Grand mean0.081831.180100.797310.273440.99999516.6516.3506.6
ZNB1-INTARCH(1)Local constant0.101630.907720.759330.787650.99999521.3521.0511.3
ZNB2-INARCH(1)0.000010.878970.668730.46768502.5504.3494.5
ZNB2-INTARCH(1)Grand mean0.000021.065750.792080.270630.40637497.7497.4487.7
ZNB2-INTARCH(1)Local constant0.000010.863660.640040.718030.46729504.4504.1494.4

AIC = Akaike information criterion; BIC = Bayesian information criterion.

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