This book brings together a variety of non-Gaussian autoregressive-type models to analyze time-series data. This book collects and collates most of the available models in the field and provide their probabilistic and inferential properties. This book classifies the stationary time-series models into different groups such as linear stationary models with non-Gaussian innovations, linear stationary models with non-Gaussian marginal distributions, product autoregressive models and minification models. Even though several non-Gaussian time-series models are available in the literature, most of them are focusing on the model structure and the probabilistic properties.
Author(s): N. Balakrishna
Publisher: Springer
Year: 2022
Language: English
Pages: 243
City: Singapore
Preface
Contents
About the Author
Acronyms
1 Basics of Time Series
1.1 Useful Characteristics of Time Series
1.2 Linear Time Series Models
1.2.1 Autoregressive Models
1.2.2 Moving Average Models
1.2.3 Autoregressive Moving Average Models
1.3 Random Coefficient AR Models
1.3.1 Random Lag Autoregressive Model of Order p (RLAR(p))
1.4 Other Non-linear Time Series Models
1.5 Non-Gaussian Time Series
1.6 Model Specifications
1.6.1 Marginal Specific Models
1.6.2 Error Specific Models
1.6.3 Conditionally Specified Models
References
2 Statistical Inference for Stationary Linear Time Series
2.1 Methods of Estimation
2.2 Yule-Walker Method of Estimation
2.3 Maximum Likelihood Methods
2.3.1 ML Method for ARMA Models with Non-Gaussian Innovations
2.3.2 MLE for Stationary AR(p) Model
2.3.3 Modified Method of Maximum Likelihood Estimation (MMLE)
2.3.4 Maximum Probability Estimation
2.4 Quasi-Maximum Likelihood Method
2.5 Method of Conditional Least Squares
2.5.1 Two-Stage Conditional Least Squares Method
2.6 Generalized Method of Moments
2.7 Godambe Estimating Functions and Quasi-likelihood Methods
2.7.1 Estimating Functions for Stochastic Processes
2.7.2 Quasi-likelihood Scores Based on Conditional Mean and Variance
2.7.3 Asymptotic Theory of Estimating Functions
2.8 Other Methods of Estimation
2.9 Methods of Model Identification, Diagnosis and Forecasting
References
3 AR Models with Stationary Non-Gaussian Positive Marginals
3.1 Constant Coefficient Exponential Autoregressive Models
3.1.1 First-Order Exponential Autoregressive Models
3.1.2 Higher Order Exponential Autoregressive Models
3.1.3 ACF of EAR(p) Processes
3.2 Estimation for Stationary Exponential AR Models
3.2.1 Estimation in the Presence of Zero-Defects
3.2.2 Conditional Least Square Method for EAR(p) Models
3.3 Random Coefficient Exponential AR Models
3.3.1 Transposed EAR (TEAR) Models
3.3.2 New Exponential AR(1) (NEAR(1)) Model
3.3.3 Generalized Exponential AR(1) (GEAR(1)) Model
3.3.4 New Exponential AR(2) (NEAR(2)) Model
3.3.5 NEAR(p) Models
3.4 Estimation for Random Coefficient Exponential AR Models
3.4.1 Estimation for NEAR(1) Model
3.4.2 Estimation for NEAR(2) Model
3.4.3 Estimation in NEAR(p) Model
3.4.4 Quasi-likelihood Estimates for NEAR(p) Model
3.5 Gamma Autoregressive Models
3.5.1 Constant Coefficient Gamma AR(1) Models
3.5.2 Random Coefficient GAR(1) Models
3.5.3 Beta-Gamma ARMA Models
3.5.4 Gamma Models by Random Thinning
3.5.5 GAR(1) Models with Conditional Specifications
3.6 Estimation for Gamma Time Series
3.7 Other Non-negative Stationary AR(1) Models
3.7.1 Mixed Exponential AR(1) Model
3.7.2 Birnbaum-Saunders AR Model
3.7.3 Inverse Gaussian Time Series Models
3.7.4 Mittag-Leffler Processes
References
4 AR Models with Stationary Non-Gaussian Real-Valued Marginals
4.1 Constant Coefficient Laplace AR Models
4.2 Random Coefficient Laplace AR Models
4.3 Estimation for Symmetric Laplace AR Models
4.3.1 Parameter Estimation in NLAR(p) Model
4.3.2 Quasi-likelihood Estimates for NLAR(p) Model
4.4 AR Models with Generalized Laplace Marginals
4.5 Gumbel's Extreme Value AR Model
4.5.1 Properties of the Model
4.5.2 Properties of the Innovation
4.5.3 Maximum Likelihood Estimation
4.6 ARMA Models with Stationary Stable Distributions
4.7 Stationary Time Series with Absolutely Continuous Innovations
4.7.1 Stationary Cauchy AR Models
4.7.2 Logistic AR Process
4.7.3 Hyperbolic Secant Autoregressive Models
4.8 Other Real-Valued Non-Gaussian AR Models
References
5 Some Non-linear AR-type Models for Non-Gaussian Time Series
5.1 Minification Models
5.1.1 Existence and Stationarity
5.1.2 General Properties of the Minification Process
5.2 Minification Processes with Specified Marginals
5.2.1 Exponential and Weibull Minification Processes
5.2.2 Link Between AR(1) and Minification Models
5.2.3 Pareto and Semi-Pareto Minification Processes
5.2.4 Parameter Estimation
5.2.5 Generalization of Minification Models
5.3 Product Autoregressive Models
5.3.1 General Properties of PAR Models
5.4 PAR(1) Models with Specified Marginals
5.4.1 Lognormal PAR(1) Model
5.4.2 Exponential PAR(1) Model
5.4.3 Weibull PAR(1) Model
5.4.4 Gamma PAR(1) Model
5.4.5 Generalized Gamma PAR(1) Models
5.5 Simulation of PAR(1) Sequences
5.6 Inference for PAR(1) Models
5.6.1 Method of Maximum Likelihood
5.6.2 Parameter Estimation by Conditional Least Squares
5.6.3 Method of Estimating Functions
References
6 Linear Time Series Models with Non-Gaussian Innovations
6.1 Introduction
6.2 Stationary Marginal Distributions for Models with Real-Valued Innovations
6.2.1 ARMA Models with Stable Innovations
6.2.2 AR Models with Symmetric Alpha Stable Innovations
6.2.3 AR Models with Cauchy Innovations
6.3 AR Models with Other Non-Gaussian Symmetric Innovations
6.3.1 ARMA Models with Laplace Innovations
6.4 Estimation for AR Models with Symmetric Innovations
6.4.1 Estimating Function Method for AR(1) Model with Laplace Innovations
6.4.2 Estimation for ARMA Models with GED Innovations
6.4.3 Estimation for AR Models with Student's-t Innovations
6.5 ARMA Models with Asymmetric Real-Valued Innovations
6.5.1 Asymmetric Laplace Innovations
6.5.2 AR Models with Asymmetric Skew Normal Innovations
6.5.3 AR Models with Skew Exponential Power Innovations
6.6 AR Models with Mixture-Type Innovations
6.6.1 Generalized Hyperbolic Innovations
6.6.2 Normal Inverse Gaussian Innovations
6.6.3 Finite Mixture Autoregressive Models
6.7 Autoregressive Models with Positive Innovations
6.7.1 Autoregressive Models with Exponential Innovations
6.7.2 Autoregressive Models With Slowly Varying Innovations
6.7.3 Models with Other Positive Innovations
References
7 Autoregressive-Type Time Series of Counts
7.1 Introduction
7.2 Early Models for Discrete Time Series
7.3 Models Based on Random Thinning
7.3.1 Integer-Valued AR(1)-type Models
7.3.2 Integer-Valued AR(p)-type Models
7.4 INAR Models with Specific Marginal Distributions
7.4.1 Poisson INAR Models
7.4.2 Geometric INAR Models
7.4.3 Negative Binomial INAR Models
7.4.4 INAR Models on Finite State Space
7.4.5 INAR Models with Specified Innovations
7.5 Observation-Driven Models for Time Series of Counts
7.5.1 Observation-Driven Autoregressive-Type Models
7.5.2 Observation-Driven Autoregressive Models with Specific Conditional Distributions
7.6 Estimation
7.6.1 Yule-Walker Method of Estimation
7.6.2 Method of Conditional Least Squares
7.6.3 Method of Maximum Likelihood for INAR Models
7.6.4 Conditional Maximum Likelihood Method for Observation-Driven Autoregressive Models
References
