Copula-Based Markov Models for Time Series

This chapter briefly describes the main ideas of the book: time series data and copula-based Markov models for serial dependence. For illustration, we introduce five datasets, namely, the chemical process data, S&P 500 stock market index data, the batting average data in MLB, the stock price data of Dow Jones Industrial Average, and data on the number of arsons.

This chapter introduces the basic concepts on copulas and Markov models. We review the formal definition of copulas with its fundamental properties. We then introduce Kendall’s tau as a measure of dependence structure for a pair of random variables, and its relationship with a copula. Examples of copulas are reviewed, such as the Clayton copula, the Gaussian copula, the Frank copula, and the Joe copula. Finally, we introduce the copula-based Markov chain time series models and their fundamental properties.

We propose an estimation method under a copula-based Markov model for serially correlated data. Motivated by the fat-tailed distribution of financial assets, we select a normal mixture distribution for the marginal distribution. Based on the normal mixture distribution for the marginal distribution and the Clayton copula for serial dependence, we obtain the corresponding likelihood function. In order to obtain the maximum likelihood estimators, we apply the Newton–Raphson algorithm with appropriate transformations and initial values. In the empirical analysis, the stock price of Dow Jones Industrial Average is analyzed for illustration.

Kim et al. (Commun Stat: Simul Comput, 2019) proposed control charts of mean by applying the statistical process control (SPC) method of Emura et al. (Commun Stat: Simul Comput 46:3067–3087, 2017) under serial dependence after accounting for the directional dependence by diverse copula functions. To illustrate the method proposed by Kim et al. (Commun Stat: Simul Comput, 2019), we revisit the case study of Major League Baseball (MLB), where the SPC method is performed for the batting average (BA) and earned run average (ERA) data from 1998 to 2016 seasons to detect a large abnormal season of Minnesota Twins. The R codes in Appendix are helpful for users who implement the proposed method.

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.