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[下载]Time Series for Macroeconomics and Finance
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Time Series for Macroeconomics and Finance
John H. Cochrane1
Graduate School of Business
University of Chicago
Spring 1997; Pictures added Jan 2005
Contents
1 Preface
7
2 What is a time series?
8
3 ARMA models
10
3.1 White noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Basic ARMA models . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Lag operators and polynomials . . . . . . . . . . . . . . . . . 11
3.3.1
Manipulating ARMAs with lag operators. . . . . . . . 12
3.3.2
AR(1) to MA(∞) by recursive substitution . . . . . . . 13
3.3.3
AR(1) to MA(∞) with lag operators. . . . . . . . . . . 13
3.3.4
AR(p) to MA(∞), MA(q) to AR(∞), factoring lag
polynomials, and partial fractions . . . . . . . . . . . . 14
3.3.5
Summary of allowed lag polynomial manipulations . . 16
3.4 Multivariate ARMA models. . . . . . . . . . . . . . . . . . . . 17
3.5 Problems and Tricks . . . . . . . . . . . . . . . . . . . . . . . 19
4 The autocorrelation and autocovariance functions.
21
4.1 De?nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Autocovariance and autocorrelation of ARMA processes. . . . 22
4.2.1
Summary . . . . . . . . . . . . . . . . . . . . . . . . . 25
1
4.3 A fundamental representation . . . . . . . . . . . . . . . . . . 26
4.4 Admissible autocorrelation functions . . . . . . . . . . . . . . 27
4.5 Multivariate auto- and cross correlations. . . . . . . . . . . . . 30
5 Prediction and Impulse-Response Functions
31
5.1 Predicting ARMA models . . . . . . . . . . . . . . . . . . . . 32
5.2
State space representation . . . . . . . . . . . . . . . . . . . . 34
5.2.1
ARMAs in vector AR(1) representation
. . . . . . . . 35
5.2.2
Forecasts from vector AR(1) representation . . . . . . . 35
5.2.3
VARs in vector AR(1) representation. . . . . . . . . . . 36
5.3 Impulse-response function . . . . . . . . . . . . . . . . . . . . 37
5.3.1
Facts about impulse-responses . . . . . . . . . . . . . . 38
6 Stationarity and Wold representation
40
6.1 De?nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2 Conditions for stationary ARMA’s . . . . . . . . . . . . . . . 41
6.3 Wold Decomposition theorem . . . . . . . . . . . . . . . . . . 43
6.3.1
What the Wold theorem does not say . . . . . . . . . . 45
6.4 The Wold MA(∞) as another fundamental representation . . . 46
7 VARs: orthogonalization, variance decomposition, Granger
causality
48
7.1 Orthogonalizing VARs . . . . . . . . . . . . . . . . . . . . . . 48
7.1.1
Ambiguity of impulse-response functions . . . . . . . . 48
7.1.2
Orthogonal shocks . . . . . . . . . . . . . . . . . . . . 49
7.1.3
Sims orthogonalization–Specifying C(0) . . . . . . . . 50
7.1.4
Blanchard-Quah orthogonalization—restrictions on C(1). 52
7.2 Variance decompositions . . . . . . . . . . . . . . . . . . . . . 53
7.3 VAR’s in state space notation . . . . . . . . . . . . . . . . . . 54
2
7.4 Tricks and problems: . . . . . . . . . . . . . . . . . . . . . . . 55
7.5 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.5.1
Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.5.2
De?nition, autoregressive representation . . . . . . . . 58
7.5.3
Moving average representation . . . . . . . . . . . . . . 59
7.5.4
Univariate representations . . . . . . . . . . . . . . . . 60
7.5.5
E?ect on projections . . . . . . . . . . . . . . . . . . . 61
7.5.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.5.7
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.5.8
A warning: why “Granger causality” is not “Causality” 64
7.5.9
Contemporaneous correlation . . . . . . . . . . . . . . 65
8 Spectral Representation
67
8.1 Facts about complex numbers and trigonometry . . . . . . . . 67
8.1.1
De?nitions . . . . . . . . . . . . . . . . . . . . . . . . . 67
8.1.2
Addition, multiplication, and conjugation . . . . . . . . 68
8.1.3
Trigonometric identities . . . . . . . . . . . . . . . . . 69
8.1.4
Frequency, period and phase . . . . . . . . . . . . . . . 69
8.1.5
Fourier transforms . . . . . . . . . . . . . . . . . . . . 70
8.1.6
Why complex numbers? . . . . . . . . . . . . . . . . . 72
8.2 Spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.2.1
Spectral densities of some processes . . . . . . . . . . . 75
8.2.2
Spectral density matrix, cross spectral density . . . . . 75
8.2.3
Spectral density of a sum . . . . . . . . . . . . . . . . . 77
8.3 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.3.1
Spectrum of ?ltered series . . . . . . . . . . . . . . . . 78
8.3.2
Multivariate ?ltering formula . . . . . . . . . . . . . . 79
3
8.3.3
Spectral density of arbitrary MA(∞) . . . . . . . . . . 80
8.3.4
Filtering and OLS . . . . . . . . . . . . . . . . . . . . 80
8.3.5
A cosine example . . . . . . . . . . . . . . . . . . . . . 82
8.3.6
Cross spectral density of two ?lters, and an interpre-
tation of spectral density . . . . . . . . . . . . . . . . . 82
8.3.7
Constructing ?lters . . . . . . . . . . . . . . . . . . . . 84
8.3.8
Sims approximation formula . . . . . . . . . . . . . . . 86
8.4 Relation between Spectral, Wold, and Autocovariance repre-
sentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
9 Spectral analysis in ?nite samples
89
9.1 Finite Fourier transforms . . . . . . . . . . . . . . . . . . . . . 89
9.1.1
De?nitions . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.2 Band spectrum regression . . . . . . . . . . . . . . . . . . . . 90
9.2.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 90
9.2.2
Band spectrum procedure . . . . . . . . . . . . . . . . 93
9.3 Cram′er or Spectral representation . . . . . . . . . . . . . . . . 96
9.4 Estimating spectral densities . . . . . . . . . . . . . . . . . . . 98
9.4.1
Fourier transform sample covariances . . . . . . . . . . 98
9.4.2
Sample spectral density . . . . . . . . . . . . . . . . . 98
9.4.3
Relation between transformed autocovariances and sam-
ple density . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.4.4
Asymptotic distribution of sample spectral density . . 101
9.4.5
Smoothed periodogram estimates . . . . . . . . . . . . 101
9.4.6
Weighted covariance estimates . . . . . . . . . . . . . . 102
9.4.7
Relation between weighted covariance and smoothed
periodogram estimates . . . . . . . . . . . . . . . . . . 103
9.4.8
Variance of ?ltered data estimates . . . . . . . . . . . . 104
4
9.4.9
Spectral density implied by ARMA models . . . . . . . 105
9.4.10 Asymptotic distribution of spectral estimates . . . . . . 105
10 Unit Roots
106
10.1 Random Walks . . . . . . . . . . . . . . . . . . . . . . . . . . 106
10.2 Motivations for unit roots . . . . . . . . . . . . . . . . . . . . 107
10.2.1 Stochastic trends . . . . . . . . . . . . . . . . . . . . . 107
10.2.2 Permanence of shocks . . . . . . . . . . . . . . . . . . . 108
10.2.3 Statistical issues . . . . . . . . . . . . . . . . . . . . . . 108
10.3 Unit root and stationary processes . . . . . . . . . . . . . . . 110
10.3.1 Response to shocks . . . . . . . . . . . . . . . . . . . . 111
10.3.2 Spectral density . . . . . . . . . . . . . . . . . . . . . . 113
10.3.3 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 114
10.3.4 Random walk components and stochastic trends . . . . 115
10.3.5 Forecast error variances . . . . . . . . . . . . . . . . . 118
10.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 119
10.4 Summary of a(1) estimates and tests. . . . . . . . . . . . . . . 119
10.4.1 Near- observational equivalence of unit roots and sta-
tionary processes in ?nite samples . . . . . . . . . . . . 119
10.4.2 Empirical work on unit roots/persistence . . . . . . . . 121
11 Cointegration
122
11.1 De?nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
11.2 Cointegrating regressions . . . . . . . . . . . . . . . . . . . . . 123
11.3 Representation of cointegrated system. . . . . . . . . . . . . . 124
11.3.1 De?nition of cointegration . . . . . . . . . . . . . . . . 124
11.3.2 Multivariate Beveridge-Nelson decomposition . . . . . 125
11.3.3 Rank condition on A(1) . . . . . . . . . . . . . . . . . 125
5
11.3.4 Spectral density at zero . . . . . . . . . . . . . . . . . 126
11.3.5 Common trends representation . . . . . . . . . . . . . 126
11.3.6 Impulse-response function. . . . . . . . . . . . . . . . . 128
11.4 Useful representations for running cointegrated VAR’s . . . . . 129
11.4.1 Autoregressive Representations . . . . . . . . . . . . . 129
11.4.2 Error Correction representation . . . . . . . . . . . . . 130
11.4.3 Running VAR’s . . . . . . . . . . . . . . . . . . . . . . 131
11.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11.6 Cointegration with drifts and trends . . . . . . . . . . . . . . . 134

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