Time Series for Macroeconomics and Finance

by John H. Cochrane
Graduate School of Business
University of Chicago

Contents
1 Preface 7
2Whatisatimeseries? 8
3ARMAmodels 10
3.1 Whitenoise ............................ 10
3.2 BasicARMAmodels ....................... 11
3.3 Lagoperatorsandpolynomials ................. 11
3.3.1 ManipulatingARMAswithlagoperators. ....... 12
3.3.2 AR(1) to MA(∞)byrecursivesubstitution....... 13
3.3.3 AR(1) to MA(∞)withlagoperators........... 13
3.3.4 AR(p) to MA(∞), MA(q) to AR(∞), factoring lag
polynomials,andpartialfractions............ 14
3.3.5 Summary of allowed lag polynomial manipulations . . 16
3.4 MultivariateARMAmodels.................... 17
3.5 ProblemsandTricks ....................... 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

4.3 Afundamentalrepresentation .................. 26
4.4 Admissibleautocorrelationfunctions .............. 27
4.5 Multivariateauto-andcrosscorrelations............. 30
5 Prediction and Impulse-Response Functions 31
5.1 PredictingARMAmodels .................... 32
5.2 Statespacerepresentation.................... 34
5.2.1 ARMAsinvectorAR(1)representation ........ 35
5.2.2 ForecastsfromvectorAR(1)representation....... 35
5.2.3 VARsinvectorAR(1)representation........... 36
5.3 Impulse-responsefunction .................... 37
5.3.1 Factsaboutimpulse-responses.............. 38
6 Stationarity and Wold representation 40
6.1 Definitions....... ...................... 40
6.2 ConditionsforstationaryARMA’s ............... 41
6.3 WoldDecompositiontheorem .................. 43
6.3.1 WhattheWoldtheoremdoesnotsay.......... 45
6.4 The Wold MA(∞) as another fundamental representation . . . 46
7 VARs: orthogonalization, variance decomposition, Granger
causality 48
7.1 OrthogonalizingVARs ...................... 48
7.1.1 Ambiguityofimpulse-responsefunctions ........ 48
7.1.2 Orthogonalshocks .................... 49
7.1.3 Sims orthogonalization–Specifying C(0)........ 50
7.1.4 Blanchard-Quah orthogonalization—restrictions on C(1). 52
7.2 Variancedecompositions ..................... 53
7.3 VAR’sinstatespacenotation .................. 54

7.4 Tricksandproblems: ....................... 55
7.5 GrangerCausality......................... 57
7.5.1 Basicidea ......................... 57
7.5.2 Definition,autoregressiverepresentation ........ 58
7.5.3 Movingaveragerepresentation.............. 59
7.5.4 Univariaterepresentations ................ 60
7.5.5 Effectonprojections ................... 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 Contemporaneouscorrelation .............. 65
8SpectralRepresentation 67
8.1 Factsaboutcomplexnumbersandtrigonometry........ 67
8.1.1 Definitions......................... 67
8.1.2 Addition,multiplication,andconjugation........ 68
8.1.3 Trigonometricidentities ................. 69
8.1.4 Frequency,periodandphase............... 69
8.1.5 Fouriertransforms .................... 70
8.1.6 Whycomplexnumbers? ................. 72
8.2 Spectraldensity.......................... 73
8.2.1 Spectraldensitiesofsomeprocesses........... 75
8.2.2 Spectraldensitymatrix,crossspectraldensity..... 75
8.2.3 Spectraldensityofasum................. 77
8.3 Filtering... ........................... 78
8.3.1 Spectrum of filteredseries ................ 78
8.3.2 Multivariate filteringformula .............. 798.3.3 Spectral density of arbitrary MA(∞) .......... 80
8.3.4 FilteringandOLS .................... 80
8.3.5 Acosineexample..................... 82
8.3.6 Cross spectral density of two filters,andaninterpre-
tationofspectraldensity................. 82
8.3.7 Constructing filters.................... 84
8.3.8 Simsapproximationformula............... 86
8.4 Relation between Spectral, Wold, and Autocovariance repre-
sentations .. ........................... 87
9Spectralanalysisin finite samples 89
9.1 FiniteFouriertransforms..................... 89
9.1.1 Definitions......................... 89
9.2 Bandspectrumregression .................... 90
9.2.1 Motivation......................... 90
9.2.2 Bandspectrumprocedure ................ 93
9.3 Cram´ erorSpectralrepresentation................ 96
9.4 Estimatingspectraldensities................... 98
9.4.1 Fouriertransformsamplecovariances .......... 98
9.4.2 Samplespectraldensity ................. 98
9.4.3 Relation between transformed autocovariances and sam-
pledensity......................... 99
9.4.4 Asymptotic distribution of sample spectral density . . 101
9.4.5 Smoothedperiodogramestimates ............101
9.4.6 Weightedcovarianceestimates..............102
9.4.7 Relation between weighted covariance and smoothed
periodogramestimates ..................103
9.4.8 Variance of filtereddataestimates............1049.4.9 SpectraldensityimpliedbyARMAmodels.......105
9.4.10Asymptoticdistributionofspectralestimates......105
10 Unit Roots 106
10.1RandomWalks ..........................106
10.2Motivationsforunitroots ....................107
10.2.1Stochastictrends .....................107
10.2.2Permanenceofshocks...................108
10.2.3Statisticalissues......................108
10.3Unitrootandstationaryprocesses ...............110
10.3.1Responsetoshocks....................111
10.3.2Spectraldensity......................113
10.3.3Autocorrelation......................114
10.3.4Randomwalkcomponentsandstochastictrends....115
10.3.5Forecasterrorvariances .................118
10.3.6Summary .........................119
10.4 Summary of a(1)estimatesandtests...............119
10.4.1 Near- observational equivalence of unit roots and sta-
tionary processes in finitesamples............119
10.4.2Empiricalworkonunitroots/persistence........121
11 Cointegration 122
11.1 Definition ............. ................122
11.2Cointegratingregressions.....................123
11.3Representationofcointegratedsystem. .............124
11.3.1 Definitionofcointegration ................124
11.3.2MultivariateBeveridge-Nelsondecomposition .....125
11.3.3RankconditiononA(1) .................12511.3.4Spectraldensityatzero .................126
11.3.5Commontrendsrepresentation .............126
11.3.6 Impulse-responsefunction.................128
11.4UsefulrepresentationsforrunningcointegratedVAR’s.....129
11.4.1AutoregressiveRepresentations .............129
11.4.2ErrorCorrectionrepresentation .............130
11.4.3RunningVAR’s......................131
11.5AnExample............................132
11.6Cointegrationwithdriftsandtrends...............134