2. Observation-driven ZIB models

### 2.1. ZIB models

incorrect inferential conclusions. Developing a general modeling framework that accounts for

In characterizing data comprised of counts with excess zeroes, two types of models are commonplace: a model that assumes a Poisson mixture distribution, and a model that assumes a binomial mixture distribution. A considerable literature exists for regression models based on the zero-inflated Poisson (ZIP) distribution to deal with count data that are independently distributed [1]. Many researchers have extended the classical ZIP model to analyze repeated measures data by incorporating independent random effects, as these can account for within-subject correlation and between-subject heterogeneity [2, 3]. To deal with count time series with excess zeros, some researchers have proposed parameter-driven ZIP models that accommodate the temporal dynamics by incorporating correlated random effects, which can be represented by a latent autoregressive process [4, 5]. However, for data arising from a binomial mixture distribution, a survey of the literature for analogous frameworks reflects an absence of work dealing with binomial time series with excess zeros. To handle such data, we propose two general classes of models: a class of observation-driven ZIB (ODZIB) models, and a class of parameter-driven ZIB (PDZIB) models. The inspiration for the two proposed modeling frameworks arises from the work of Hall [6], Yau et al. [4],

Depending on how the temporal correlation is conceptualized, count time series models can be classified as either observation-driven or parameter-driven [8]. For the former, serial correlation is characterized by specifying that the conditional mean of the current response depends explicitly on its past values [9–14]. For the latter, such correlation is characterized through an unobservable underlying process [15–19]. In this chapter, we employ the partial likelihood framework to formulate the ODZIB model, as this largely simplifies parameter estimation with negligible loss of information. The ODZIB model can be viewed as an extension of the observation-driven binomial model [20]. Such a model is often fit using standard statistical software available for classical ZIB regression models. For the PDZIB model, we employ a state-space approach, as this framework allows for the investigation of the underlying latent processes that govern the temporal correlation and zero inflation. Due to the non-Gaussian distribution of the count response, and the nonlinear nature of modeling its conditional mean, traditional state-space methods using the Kalman filter and the Kalman smoother are not available for parameter estimation. We thereby adopt a Monte Carlo Expectation Maximization (MCEM) algorithm based on the

The remainder of the chapter is organized as follows. In Section 2, we briefly introduce a class of observation-driven models for a zero-inflated count time series that arises from a binomial mixture. Section 3 proposes a class of parameter-driven models in the statespace framework, and presents the MCEM algorithm devised to fit such models. A comprehensive simulation study is provided in Section 4. In Section 5, we illustrate the proposed methodology through a practical application. Section 6 concludes with a brief

these characteristics poses a daunting challenge.

particle filter [21] and the particle smoother [22].

and Yang et al. [5, 7].

128 Time Series Analysis and Applications

discussion.

A popular approach for modeling independent zero-inflated binomial data is the ZIB model proposed by Hall [6]. This model assumes that data are generated from a mixture distribution, comprised of a binomial distribution and a degenerate distribution at zero. For response variable Y, let yi denote the observation for subject i, i = 1, 2, …, n. The probability mass function for the ZIB model is defined as follows:

$$f(y\_i|\pi\_i, \omega\_i) = \begin{cases} \omega\_i + (1 - \omega\_i)(1 - \pi\_i)^{n\_i} & \text{if } y\_i = 0, \\\\ (1 - \omega\_i) \binom{n\_i}{y\_i} \pi\_i^{y\_i} (1 - \pi\_i)^{n\_i - y\_i} & \text{if } y\_i > 0. \end{cases} \tag{1}$$

Here, ω<sup>i</sup> is the zero-inflation parameter, and π<sup>i</sup> is the intensity parameter representing the probability of success, both modeled via logit link functions:

$$\mathbf{1} \text{logit}(\omega\_i) = \mathbf{x}\_{i1}^T \mathbf{y}\_{\prime} \tag{2}$$

$$\mathbf{1}\operatorname{logit}(\pi\_i) = \mathbf{x}\_{i2}^T \boldsymbol{\beta}.\tag{3}$$

In the preceding, xi<sup>1</sup> and xi<sup>2</sup> are sets of explanatory variables for the corresponding vectors of regression coefficients γ and β. The Expectation Maximization (EM) algorithm or the Newton-Raphson method can be used to obtain the parameter estimates.

### 2.2. Observation-driven ZIB models

In this section, we introduce an autoregressive model for binomial time series with excess zeros based on an observation-driven approach. We retain the same model structure as that introduced in Section 2.1 to account for the binomial mixture, yet we employ lagged responses as covariates to resolve the temporal correlation. The proposed model can be viewed as an extension of the binomial time series model presented by Kedem and Fokianos [20].

Let yt denote the binomial count response. Define the information set

$$\mathcal{F}\_{t-1} = \sigma\{y\_{t-1}, y\_{t-2}, \dots, \mathbf{x}\_{t}\} \tag{4}$$

so as to represent all that is known to the observer at time t about the response and any relevant covariate processes. Thus, the vector x<sup>t</sup> represents a collection of past and possibly present time-dependent covariates that are observed at time t � 1. In the present setting, x<sup>t</sup> may be viewed as either fixed or random. Conditioning on the information F<sup>t</sup>�1, the response is assumed to follow a ZIB distribution with probability mass function defined as follows:

$$f\_t\left(y\_t|\mathcal{F}\_{t-1};\pi\_t,\omega\_t\right) = \begin{cases} \omega\_t + (1-\omega\_t)(1-\pi\_t)^{n\_t} & \text{if } y\_t = 0, \\\\ (1-\omega\_t)\binom{n\_t}{y\_t}\pi\_t^{y\_t}(1-\pi\_t)^{n\_t-y\_t} & \text{if } y\_t > 0. \end{cases} \tag{5}$$

Similarly, ω<sup>t</sup> and π<sup>t</sup> represent the zero-inflation parameter and the intensity parameter, respectively. Both parameters are modeled via logit link functions. Specifically, we assume that

$$\mathbf{1}\operatorname{logit}(w\_t) = \mathbf{x}\_{\mathbf{1},t}^T \boldsymbol{\gamma}\_t \tag{6}$$

$$\mathbf{1}\text{logit}(\pi\_t) = \mathbf{x}\_{2,t}^T \boldsymbol{\beta} + \sum\_{j=1}^p \phi\_j \mathbf{y}\_{t-j^\prime} \tag{7}$$

where x1,<sup>t</sup> and x2,<sup>t</sup> are sets of time-dependent explanatory variables for the corresponding vectors of regression coefficients γ and β, and φ = [φ1,…,φp] <sup>Τ</sup> is a vector of autoregressive coefficients corresponding to the past responses [yt� 1,…, yt� <sup>p</sup>] Τ . For simplicity, we treat the zero-inflation parameter ω<sup>t</sup> as a constant that does not vary over time. In the observation-driven ZIB model, serial correlation is accommodated by introducing lagged values of the response to the linear predictor.

The partial data likelihood of the observed series is

$$\text{PL}\left(\boldsymbol{\theta}\right) = \prod\_{t=1}^{n} f\_t\left(y\_t|\mathcal{F}\_{t-1}\right) \tag{8}$$

where θ = [β, φ, γ] <sup>Τ</sup> is the vector of unknown parameters. The partial likelihood does not require the derivation of the joint distribution of the response and the covariates, and is largely simplified relative to the full likelihood. This approach facilitates conditional inference for a fairly large class of transitional processes where the response depends on its past values.

The log-likelihood for the observation-driven ZIB model is

$$\log \text{PL}(\boldsymbol{\theta}) = \sum\_{t=1}^{n} \log \left\{ \omega\_{l} I\_{\left(y\_{t} = 0\right)} + (1 - \omega\_{l}) \binom{n\_{t}}{y\_{t}} \pi\_{t}^{y\_{t}} (1 - \pi\_{t})^{n\_{t} - y\_{t}} \right\}.\tag{9}$$

The vector θ^ obtained by maximizing the partial likelihood is called the maximum partial likelihood estimator (MPLE).

Similar to Section 2.1, we can apply the EM algorithm or the Newton–Raphson method to obtain the MPLE. This estimation process can be conveniently conducted in practice using standard software tools available for fitting classical ZIB models. In SAS, we can use the finite mixture models (FMM) procedure to fit the observation-driven ZIB model, while we can use function gamlss in the package generalized additive models for location scale and shape (GAMLSS) for model fitting in R. Hypothesis testing for θ is carried out through the partial likelihood method. The common tests are based on Wald statistics, score statistics, and partial likelihood ratio statistics. All of these tests are conducted based on the framework for classical maximum likelihood inference.
