Contents
1 Preface 7
2 What is a time series? 8
3 ARMAmodels 10
3.1 White noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Basic ARMAmodels . . . . . . . . . . . . . . . . . . . . . . . 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 ARMAmodels. . . . . . . . . . . . . . . . . . . . 17
3.5 Problems and Tricks . . . . . . . . . . . . . . . . . . . . . . . 19
4 The autocorrelation and autocovariance functions. 21
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Autocovariance and autocorrelation of ARMA processes. . . . 22
4.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 25
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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 ARMAmodels . . . . . . . . . . . . . . . . . . . . 32
5.2 State space representation . . . . . . . . . . . . . . . . . . . . 34
5.2.1 ARMAs in vector AR(1) representation . . . . . . . . 35
5.2.2 Forecasts fromvector 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 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2 Conditions for stationary ARMA’s . . . . . . . . . . . . . . . 41
6.3 Wold Decomposition theorem . . . . . . . . . . . . . . . . . . 43
6.3.1 What theWold theoremdoes 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
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7.4 Tricks and problems: . . . . . . . . . . . . . . . . . . . . . . . 55
7.5 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.5.1 Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.5.2 Definition, autoregressive representation . . . . . . . . 58
7.5.3 Moving average representation . . . . . . . . . . . . . . 59
7.5.4 Univariate representations . . . . . . . . . . . . . . . . 60
7.5.5 Effect 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 SpectralRepresentation 67
8.1 Facts about complex numbers and trigonometry . . . . . . . . 67
8.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 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 densitymatrix, cross spectral density . . . . . 75
8.2.3 Spectral density of a sum. . . . . . . . . . . . . . . . . 77
8.3 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.3.1 Spectrum of filtered series . . . . . . . . . . . . . . . . 78
8.3.2 Multivariate filtering formula . . . . . . . . . . . . . . 79
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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 filters, and an interpretation
of spectral density . . . . . . . . . . . . . . . . . 82
8.3.7 Constructing filters . . . . . . . . . . . . . . . . . . . . 84
8.3.8 Sims approximation formula . . . . . . . . . . . . . . . 86
8.4 Relation between Spectral, Wold, and Autocovariance representations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
9 Spectral analysis infinite samples 89
9.1 Finite Fourier transforms . . . . . . . . . . . . . . . . . . . . . 89
9.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 89
9.2 Band spectrumregression . . . . . . . . . . . . . . . . . . . . 90
9.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 90
9.2.2 Band spectrumprocedure . . . . . . . . . . . . . . . . 93
9.3 Cram´er or Spectral representation . . . . . . . . . . . . . . . . 96
9.4 Estimating spectral densities . . . . . . . . . . . . . . . . . . . 98
9.4.1 Fourier transformsample covariances . . . . . . . . . . 98
9.4.2 Sample spectral density . . . . . . . . . . . . . . . . . 98
9.4.3 Relation between transformed autocovariances and sample
density . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.4.4 Asymptotic distribution of sample spectral density . . 101
9.4.5 Smoothed periodogramestimates . . . . . . . . . . . . 101
9.4.6 Weighted covariance estimates . . . . . . . . . . . . . . 102
9.4.7 Relation between weighted covariance and smoothed
periodogramestimates . . . . . . . . . . . . . . . . . . 103
9.4.8 Variance of filtered data estimates . . . . . . . . . . . . 104
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9.4.9 Spectral density implied by ARMAmodels . . . . . . . 105
9.4.10 Asymptotic distribution of spectral estimates . . . . . . 105
10 Unit Roots 106
10.1 RandomWalks . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Randomwalk 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 stationary
processes in finite samples . . . . . . . . . . . . 119
10.4.2 Empirical work on unit roots/persistence . . . . . . . . 121
11 Cointegration 122
11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
11.2 Cointegrating regressions . . . . . . . . . . . . . . . . . . . . . 123
11.3 Representation of cointegrated system. . . . . . . . . . . . . . 124
11.3.1 Definition of cointegration . . . . . . . . . . . . . . . . 124
11.3.2 Multivariate Beveridge-Nelson decomposition . . . . . 125
11.3.3 Rank condition on A(1) . . . . . . . . . . . . . . . . . 125
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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 . . . . . . . . . . . . . .