Generalized Time Series Models and Spectral Density Based Parameters Estimation

Abstract

This thesis endeavors to investigate and propose some generalized time series models to extend the work available on classical and long memory models. The exploration encompasses various aspects, including the study of stationarity, invertibility, spectral densities, autocovariance functions, parameter estimation, and asymptotic properties of estimators for the introduced models. In this thesis work, we extend some classical and long memory models existing in literature to several directions. Initial part of the study focuses on developing and exploring the TAR(1) model by assuming tempered stable marginals for the AR(1) process, with a specific emphasis on its behavior under stationarity assumptions. In this context, the marginal probability density function of the error term is derived and it is shown that the distribution of error term is infinitely divisibility. The TAR(1) process serves as a generalization of well-established inverse Gaussian and one-sided stable autoregressive models. Furthermore, we study an autoregressive model of order one assuming tempered stable innovations. The subsequent step involves parameter estimation for both processes, a crucial aspect of model validation and applicability. Two distinct methodologies, namely conditional least squares and the method of moments, are employed in this estimation process. These techniques are then rigorously assessed and validated through simulated data, providing insights into the model’s performance under various conditions. The performance of the model is not only theoretically evaluated but also practically demonstrated through its application to both real and simulated datasets. Next, we introduce the Gegenbauer autoregressive tempered fractionally integrated moving average (GARTFIMA) process, aiming to generalize the existing GARMA and ARTFIMA models. A key motivation behind this extension is to tackle the unbounded spectral density, observed in the GARMA process. The analysis begins by comprehensively exploring the spectral density of the GARTFIMA process. Understanding the frequency components and their strengths within the time series is crucial for evaluating the model’s efficacy in capturing various patterns and behaviors. Subsequently, the autocovariance function is obtained using the spectral density of the process, providing insights into the temporal dependencies and relationships inherent in the data. To estimate the parameters of the GARTFIMA process, two distinct methodologies are employed. Firstly, a non-linear least square (NLS) based approach is utilized, which establishes a least square regression between empirical and theoretical spectral densities. Secondly, the Whittle likelihood estimation method is applied, emphasizing the statistical measure of discrepancy between the theoretical and observed spectral densities. The asymptotic properties of the Whittle likelihood estimators are obtained. The performance of these techniques is assessed on simulated data, providing their effectiveness. Additionally, the relevance and practical applicability of the GARTFIMA process are demonstrated through its application to real-world data. A comparative analysis against other time series models is conducted, highlighting the slightly better performance of the introduced model. Moreover, we extend the existing seasonal fractional ARUMA process by introducing a tempered fractional ARUMA process. This extension involves the incorporation of exponential tempering into the traditional seasonal fractional ARUMA model. The initial focus lies in establishing the fundamental characteristics of the introduced tempered fractional ARUMA process. This encompasses the conditions ensuring the stationarity and invertibility of the model. The analysis then delves into the spectral properties of the tempered fractional ARUMA model. To estimate the parameters of the tempered fractional ARUMA model, we again employ the Whittle likelihood estimation approach, which involves minimizing the contrast between the theoretical and observed spectral densities, providing a robust framework for parameter estimation. Additionally, the asymptotic properties of the estimators are investigated, offering valuable insights into their reliability and consistency as the sample size increases. Practical validation of the proposed estimation technique is conducted through a systematic assessment of its performance on simulated data. Lastly, the study extends to generalized ARMA processes, characterized by the type 2 Humbert polynomials and called Horadam ARMA and Horadam-Pethe ARMA processes. We examine the autocovariance function and its inherent properties for these models. By leveraging the minimum contrast Whittle likelihood estimation, we estimate the parameters of the Horadam ARMA and Horadam-Pethe ARMA processes. In addition to the conventional minimum contrast Whittle likelihood estimation, we also use the debiased Whittle likelihood estimation. This computationally efficient technique is designed to reduce biases inherent in the standard Whittle likelihood method. The incorporation of debiasing mechanisms enhances the robustness and accuracy of parameter estimates, particularly in scenarios where biases might distort the results. The assessment of the proposed parameter estimation methods is conducted through the use of simulated data for the Horadam ARMA process. This empirical evaluation serves as a crucial benchmark to gauge the effectiveness and reliability of the Whittle likelihood and debiased Whittle likelihood techniques.