Some Generalized and Heavy-tailed Time Series Models

Abstract

The collection of random variables {Yt : t ∈ T} defined on the same sample space with time T as the index set is known as a time series. The most fundamental and easy-to-understand time series models in the literature are autoregressive (AR), moving average (MA), and the mixture of AR and MA model known as the autoregressive moving average (ARMA) model. These models are defined as the linear combination of previous terms and error terms or innovation terms, where these error terms are assumed to be normal with mean 0 and constant variance σ2. However, asymmetry, skewness, and non-Gaussian behavior are commonly observed in many real-life phenomena. For example, in financial data, stock prices exhibit non-Gaussian behavior with extreme values, meteorological data show asymmetry due to extreme weather events and long-term climate changes, in traffic f low, sudden congestion, accidents, or disruptions can lead to non-Gaussian behavior and asymmetry in data, and so on. To efficiently capture these events, different non-Gaussian models are considered, such as the AR model with exponential, Student’s t-distribution, Laplace, Cauchy and other distributions for innovation terms. In this thesis, we initiate our study of the AR model by considering non-Gaussian innovation terms, specifically focusing on the semi-heavy-tailed and heavy-tailed classes of distribution. First, we consider the AR(p) model with normal inverse Gaussian innovation terms, which has semi-heavy-tail behavior. We propose using the expectation-maximization (EM) algorithm for parameter estimation of the model. Further, we conduct an extensive simulation study to assess the method’s performance and compare the EM method with Yule-Walker and conditional least squares methods. We also apply the proposed model to three real datasets, namely, Google equity closing price, US gasoline price and NASDAQ historical data. In the next chapter, we consider the AR(p) model with Cauchy-distributed innovation terms. Again, this distribution is heavy-tailed with infinite mean and variance, effectively capturing extreme events. We make use of the mixture representation of the Cauchy distribution, and employ the EM algorithm for estimation. We also discuss another method based on the empirical characteristic function for parameter estimation. A simulation study is performed to compare the EMmethod with maximum likelihood estimation for the Cauchy distribution. Next, we delve into a class of geometric infinitely divisible random variables by examining their Laplace exponents, characterized by Bernstein functions. We introduce AR models with geometric infinitely divisible (gid) marginals, namely geometric tempered stable, geometric gamma, and geometric inverse Gaussian. We also provide some distributional properties and the limiting behavior of the probability densities of these random variables at 0+. Further, we present parameter estimation methods for the introduced AR(1) model, using both conditional least squares and the method of moments. The performance of estimation methods for the AR(1) model is assessed using simulated data. From empirical study on geometric tempered stable, geometric gamma, and geometric inverse Gaussian distributions, we conclude that these distributions belong to the class of semi-heavy-tailed distribution. Until now, the focus of the work has been on one of the fundamental time series models, namely the AR model, which is applied to stationary data. The autoregressive integrated moving average (ARIMA) models accommodate non-stationary time series data by employing integer order differencing. It involves lagged innovation terms along with differencing steps. An extension of this model is referred to as the autoregressive fractionally integrated moving average (ARFIMA) model, which has a fractional differencing operator. Using similar approach, we introduce a new model by considering two different types of Humbert polynomials and call these models as type 1 and type 2 Humbert fractionally differenced autoregressive moving average (HARMA) models. We also establish the stationarity and invertibility conditions of these introduced models. The focus is particularly directed towards Pincherle ARMA, Horadam ARMA, and Horadam-Pethe ARMA processes, which are particular cases of HARMA models. The Whittle quasi-likelihood method is employed for parameter estimation of the introduced processes. This method yields consistent and normally distributed estimators, and its effectiveness is further assessed through a simulation study for the Pincherle ARMA process. Finally, the Pincherle ARMA model is applied to Spain’s 10-year treasury bond yield data, demonstrating its effectiveness in capturing the dynamics of the market.