Buch, Englisch, 307 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 487 g
A Concise Introduction
Buch, Englisch, 307 Seiten, Paperback, Format (B × H): 155 mm x 235 mm, Gewicht: 487 g
ISBN: 978-3-030-08975-7
Verlag: Springer International Publishing
Zielgruppe
Research
Autoren/Hrsg.
Fachgebiete
- Mathematik | Informatik Mathematik Stochastik Stochastische Prozesse
- Wirtschaftswissenschaften Volkswirtschaftslehre Volkswirtschaftslehre Allgemein Ökonometrie
- Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
- Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
- Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik
Weitere Infos & Material
1 Introduction. 1
1.1 What is a time series?. 1
1.2 Time series versus iid data. 2
2 Typical assumptions. 5
2.1 Fundamental properties. 5
2.1.1 Ergodic property with a constant limit. 5
2.1.2 Strict Stationarity. 7
2.1.3 Weak Stationarity. 7
2.1.4 Weak stationarity and Hilbert spaces. 9
2.1.5 Ergodic processes. 25
2.1.6 Sufficient conditions for the a.s. ergodic property with a constant limit. 26
2.1.7 Sufficient conditions for the L2-ergodic property with a constant limit. 27
2.2 Specific assumptions. 30
2.2.1 Gaussian processes. 30
2.2.2 Linear processes in L2(O). 31
2.2.3 Linear processes with E(X2t ) = 8. 34
2.2.4 Multivariate linear processes. 37
2.2.5 Invertibility. 38
2.2.6 Restrictions on the dependence structure. 49
3 Defining probability measures for time series. 55
3.1 Finite dimensional distributions. 55
3.2 Transformations and equations. 56
3.3 Conditions on the expected value. 57
3.4 Conditions on the autocovariance function. 58
3.4.1 Positive semidefinite functions. 59
3.4.2 Spectral distribution. 61
3.4.3 Calculation and properties of F and f.
4 Spectral representation of univariate time series. 81
4.1 Motivation. 81
4.2 Harmonic processes. 81
4.3 Extension to general processes. 84
4.3.1 Stochastic integrals with respect to Z. 84
4.3.2 Existence and definition of Z. 89
4.3.3 Interpretation of the spectral representation. 97
4.4 Further properties. 98
4.4.1 Relationship between ReZ and ImZ. 98
4.4.2 Frequency. 99
4.4.3 Overtones. 99
4.4.4 Why are frequencies restricted to the range [-p,p]?. 100
4.5 Linear filters and the spectral representation. 103
4.5.1 Effect on the spectral representation. 103
4.5.2 Elimination of Frequency Bands. 107
5 Spectral representation of real valued vector time series. 109
5.1 Cross-spectrum and spectral representation. 109
5.2 Coherence and phase. 116
6 Univariate ARMA processes. 127
6.1 Definition. 127
6.2 Stationary solution. 127
6.3 Causal stationary solution. 131
6.4 Causal invertible stationary solution. 133
6.5 Autocovariances of ARMA processes. 134
6.5.1 Calculation by integration. 134
6.5.2 Calculation using the autocovariance generating function. 135
6.5.3 Calculation using the Wold representation. 138
6.5.4 Recursive calculation. 139
6.5.5 Asymptotic decay. 140
6.6 Integrated, seasonal and fractional ARMA and ARIMA processes. 147
6.6.1 Integrated processes. 147
6.6.2 Seasonal ARMA processes.