This textbook provides a self-contained presentation of the theory and models of time series analysis. Putting an emphasis on weakly stationary processes and linear dynamic models, it describes the basic concepts, ideas, methods and results in a mathematically well-founded form and includes numerous examples and exercises. The first part presents the theory of weakly stationary processes in time and frequency domain, including prediction and filtering. The second part deals with multivariate AR, ARMA and state space models, which are the most important model classes for stationary processes, and addresses the structure of AR, ARMA and state space systems, Yule-Walker equations, factorization of rational spectral densities and Kalman filtering. Finally, there is a discussion of Granger causality, linear dynamic factor models and (G)ARCH models. The book provides a solid basis for advanced mathematics students and researchers in fields such as data-driven modeling, forecasting and filtering, which are important in statistics, control engineering, financial mathematics, econometrics and signal processing, among other subjects.
Author(s): Manfred Deistler, Wolfgang Scherrer
Series: Lecture Notes in Statistics, 224
Publisher: Springer
Year: 2022
Language: English
Pages: 215
City: Cham
Preface
References
Contents
1 Time Series and Stationary Processes
1.1 Data Structure: Time Series
1.2 Stationary Processes and Covariance Function
1.3 The Time Domain of Stationary Processes
1.4 Examples of Stationary Processes
1.5 Examples of Non-stationary Processes
2 Prediction
2.1 Prediction from a Finite Past
2.2 Prediction from Infinite Past
2.3 Regular and Singular Processes and Wold Decomposition
3 Spectral Representation
3.1 The Fourier Representation of the Covariance Function
3.2 The Frequency Domain of Stationary Processes
3.3 Spectral Representation of Stationary Processes
4 Linear, Time-Invariant, Dynamic Filters and Difference Equations
4.1 Linear, Time-Invariant Transformations of Stationary Processes …
4.2 l1-Filter
4.3 Interpretation of Filters in the Frequency Domain
4.4 The Wiener Filter
4.5 Rational Filters
4.6 Difference Equations
5 Autoregressive Processes
5.1 The Stability Condition
5.2 Prediction
5.3 Spectral Density
5.4 The Yule–Walker Equations
5.5 The Unstable and Non-stationary Case
6 ARMA Systems and ARMA Processes
6.1 ARMA Systems and Their Solutions
6.2 The Factorization of Rational Spectra
6.3 From the Wold Representation to ARMA Parameters: Observational Equivalence and Identifiability
7 State-Space Systems
7.1 Linear State-Space Systems in Innovations Form
7.2 Controllability, Observability and Minimality of State-Space Systems
7.3 From the Wold Representation to a State-Space System
7.4 The Kalman Filter
8 Models with Exogenous Variables
8.1 General Structure
8.2 Structural Identifiability of ARX and ARMAX Models
8.3 Structural (V)AR Models
9 Granger Causality
9.1 Granger Causality
10 Dynamic Factor Models
10.1 Dynamic Factor Models—General Structure
10.2 Principal Components in the Frequency Domain
10.3 Generalized Dynamic Factor Models (GDFM)
11 ARCH and GARCH Models
11.1 Models for the Conditional Variance
Index
