Copula Markov Models for Count Series with Excess Zeros

Abstract

Count time series data are observed in several applied disciplines such as environmental science, biostatistics, economics, public health, and finance. In some cases, a specific count, say zero, may occur more often than usual. Additionally, serial dependence might be found among these counts if they are recorded over time. Overlooking the frequent occurrence of zeros and the serial dependence could lead to false inference. In this chapter, Markov zero-inflated count time series models based on a joint distribution of consecutive observations are proposed. The joint distribution function of the consecutive observations is constructed through copula functions. First- or second-order Markov chains are considered with the univariate margins of zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), or zero-inflated Conway–Maxwell–Poisson (ZICMP) distributions. Under the Markov models, bivariate copula functions such as the bivariate Gaussian, Frank, and Gumbel are chosen to construct a bivariate distribution of two consecutive observations. Moreover, the trivariate Gaussian and max-infinitely divisible copula functions are considered to build the joint distribution of three consecutive observations. Likelihood-based inference is performed and asymptotic properties are studied. The proposed class of models is applied to arson counts example, which suggests that the proposed models are superior to some of the models in the literature.

Appendices

Appendix A: Trivariate Max-Id Copula Function with Positive Stable LT and Bivariate Gumbel

The following is a derivation of the trivariate max-id copula function with the positive stable LT and the bivariate Gumbel, which results in the trivariate extreme value copula function. The positive stable function is given by

$$\begin{aligned} \psi (s) = \exp {\{ -s^{1/\rho _1}\}},~~~~ \rho _1 \ge 1, \end{aligned}$$

with the corresponding functional inverse given by

$$\begin{aligned} \psi ^{-1}(t) = (-\log {t})^{\rho _1}. \end{aligned}$$

Hence, the trivariate max-id copula given in (7.3.7) becomes

$$\begin{aligned} F_{123}(y_t,y_{t-1},y_{t-2})= & {} \exp \bigg \{- \bigg (\sum _{j \in \{t,t-2\}} \\\\\Big [- & {} \log {H(e^{-0.5(- \log F_j)^{\rho _1}},e^{-0.5 (-\log F_{t-1})^{\rho _1}};{\rho _2})} \\\\+ & {} \frac{1}{2} (- \log F_j)^{\rho _1} \Big ] \bigg )^{1/\rho _1} \bigg \}. \end{aligned}$$

(7.6.1)

Now, if $H(.;\rho _2)$ is chosen to be the Gumbel copula, i.e.,

$$\begin{aligned} C(u_1,u_2; \rho )= \exp {\{ -([-\log {u_1}]^\rho +[-\log {u_2}]^\rho )^{1/\rho } \}}, \end{aligned}$$

then (7.6.1) becomes

$$\begin{aligned} F_{123}(y_t,y_{t-1},y_{t-2})= & {} \exp \bigg \{- \bigg (\sum _{j \in \{t,t-2\}} \Big [- \log \exp \Big \{ - \Big (\left[ - \log e^{-.5(-\log F_j)^{\rho _1}}\right] ^{\rho _2} \\\\+ & {} \left[ - \log e^{-.5(-\log F_{t-1})^{\rho _1}}\right] ^{\rho _2} \Big )^{1/\rho _2} \Big \} +\frac{1}{2} (- \log F_j)^{\rho _1} \Big ] \bigg )^{1/\rho _1} \bigg \} \\\\= & {} \exp \bigg \{- \bigg (\sum _{j \in \{t,t-2\}} \Big [\Big ( \Big [{\frac{1}{2^{\rho _2}}(-\log F_j)^{\rho _1 \rho _2}} \\\\+ & {} {\frac{1}{2^{\rho _2}}(-\log F_{t-1})^{\rho _1 \rho _2}} \Big )^{1/\rho _2} +\frac{1}{2} (- \log F_j)^{\rho _1} \Big ] \bigg )^{1/\rho _1} \bigg \} \\\\= & {} \exp \bigg \{- \bigg (\Big [ {\frac{1}{2^{\rho _2}}(-\log F_t)^{\rho _1 \rho _2}} + {\frac{1}{2^{\rho _2}}(-\log F_{t-1})^{\rho _1 \rho _2}} \Big ]^{1/\rho _2} \\\\+ & {} \Big [ {\frac{1}{2^{\rho _2}}(-\log F_{t-2})^{\rho _1 \rho _2}} + {\frac{1}{2^{\rho _2}}(-\log F_{t-1})^{\rho _1 \rho _2}} \Big ]^{1/\rho _2} \\\\+ & {} \frac{1}{2} (- \log F_t)^{\rho _1}+\frac{1}{2} (- \log F_{t-2})^{\rho _1} \bigg )^{1/\rho _1} \bigg \}. \end{aligned}$$

(7.6.2)

The bivariate margins of (7.6.2) are then given by

$$\begin{aligned} F_{j2}(y_j,y_{t-1})= & {} \psi \bigg (-\log {H(e^{-0.5\psi ^{-1}(F_j;\rho _1)},e^{-0.5\psi ^{-1}(F_{t-1};\rho _1)};{\rho _2})} \\\\+ & {} \frac{1}{2} \psi ^{-1}(F_j;\rho _1) + \frac{1}{2} \psi ^{-1}(F_{t-1};\rho _1); \rho _1 \bigg ) \\\\= & {} \exp \bigg \{- \bigg ( \log {H(e^{-0.5(-\log F_j)^{\rho _1}},e^{-0.5(-\log F_{t-1})^{\rho _1}};{\rho _2})} \\\\+ & {} \frac{1}{2} (-\log F_j)^{\rho _1} + \frac{1}{2} (- \log F_{t-1})^{\rho _1} \bigg )^{1/\rho _1} \bigg \} \\\\= & {} \exp \bigg \{- \bigg ( \Big ({\frac{1}{2^{\rho _2}}(-\log F_j)^{\rho _1\rho _2}} +{\frac{1}{2^{\rho _2}}(-\log F_{t-1})^{\rho _1\rho _2}}\Big )^{1/\rho _2} \\\\+ & {} \frac{1}{2} (-\log F_j)^{\rho _1} + \frac{1}{2} (- \log F_{t-1})^{\rho _1} \bigg )^{1/\rho _1} \bigg \}, \end{aligned}$$

for $j = t, t-2$, and

$$ F_{13}(y_t,y_{t-2}) = \exp \Big \{-\Big [(-\log F_t)^{\rho _1} + (- \log F_{t-2})^{\rho _1}\Big ]^{1/\rho _1}\Big \}.$$

Appendix B: R Codes for Data Analysis

The following R functions produce some of the results shown in Tables 7.1 and 7.2. They are negative log-likelihoods of the first-order Markov ZIP model and the negative log-likelihood of the second-order Markov ZIP model with the Gaussian Copula. The functions are constructed in a similar way to the ones in Joe (2014). However, zero-inflation covariates and their parameters are accounted for here. Also, the trivariate normal functions are approximated using the R package “mvtnorm” (Hothorn et al. 2001). First, we define the functions and then run the Arson example on both models.