<p><br/></p><p></p><p>Time Series for Macroeconomics and Finance<br/>John H. Cochrane1<br/>Graduate School of Business<br/>University of Chicago<br/>5807 S. Woodlawn.<br/>Chicago IL 60637<br/>(773) 702-3059<br/><a href="mailto:john.cochrane@gsb.uchicago.edu">john.cochrane@gsb.uchicago.edu</a><br/>Spring 1997; Pictures added Jan 2005</p><p></p><p></p><p>Contents<br/>1 Preface 7<br/>2 What is a time series? 8<br/>3 ARMAmodels 10<br/>3.1 White noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10<br/>3.2 Basic ARMAmodels . . . . . . . . . . . . . . . . . . . . . . . 11<br/>3.3 Lag operators and polynomials . . . . . . . . . . . . . . . . . 11<br/>3.3.1 Manipulating ARMAs with lag operators. . . . . . . . 12<br/>3.3.2 AR(1) to MA(∞) by recursive substitution . . . . . . . 13<br/>3.3.3 AR(1) to MA(∞) with lag operators. . . . . . . . . . . 13<br/>3.3.4 AR(p) to MA(∞), MA(q) to AR(∞), factoring lag<br/>polynomials, and partial fractions . . . . . . . . . . . . 14<br/>3.3.5 Summary of allowed lag polynomial manipulations . . 16<br/>3.4 Multivariate ARMAmodels. . . . . . . . . . . . . . . . . . . . 17<br/>3.5 Problems and Tricks . . . . . . . . . . . . . . . . . . . . . . . 19<br/>4 The autocorrelation and autocovariance functions. 21<br/>4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21<br/>4.2 Autocovariance and autocorrelation of ARMA processes. . . . 22<br/>4.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 25<br/>1<br/>4.3 A fundamental representation . . . . . . . . . . . . . . . . . . 26<br/>4.4 Admissible autocorrelation functions . . . . . . . . . . . . . . 27<br/>4.5 Multivariate auto- and cross correlations. . . . . . . . . . . . . 30<br/>5 Prediction and Impulse-Response Functions 31<br/>5.1 Predicting ARMAmodels . . . . . . . . . . . . . . . . . . . . 32<br/>5.2 State space representation . . . . . . . . . . . . . . . . . . . . 34<br/>5.2.1 ARMAs in vector AR(1) representation . . . . . . . . 35<br/>5.2.2 Forecasts fromvector AR(1) representation. . . . . . . 35<br/>5.2.3 VARs in vector AR(1) representation. . . . . . . . . . . 36<br/>5.3 Impulse-response function . . . . . . . . . . . . . . . . . . . . 37<br/>5.3.1 Facts about impulse-responses . . . . . . . . . . . . . . 38<br/>6 Stationarity and Wold representation 40<br/>6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40<br/>6.2 Conditions for stationary ARMA’s . . . . . . . . . . . . . . . 41<br/>6.3 Wold Decomposition theorem . . . . . . . . . . . . . . . . . . 43<br/>6.3.1 What theWold theoremdoes not say . . . . . . . . . . 45<br/>6.4 The Wold MA(∞) as another fundamental representation . . . 46<br/>7 VARs: orthogonalization, variance decomposition, Granger<br/>causality 48<br/>7.1 Orthogonalizing VARs . . . . . . . . . . . . . . . . . . . . . . 48<br/>7.1.1 Ambiguity of impulse-response functions . . . . . . . . 48<br/>7.1.2 Orthogonal shocks . . . . . . . . . . . . . . . . . . . . 49<br/>7.1.3 Sims orthogonalization–Specifying C(0) . . . . . . . . 50<br/>7.1.4 Blanchard-Quah orthogonalization—restrictions on C(1). 52<br/>7.2 Variance decompositions . . . . . . . . . . . . . . . . . . . . . 53<br/>7.3 VAR’s in state space notation . . . . . . . . . . . . . . . . . . 54<br/>2<br/>7.4 Tricks and problems: . . . . . . . . . . . . . . . . . . . . . . . 55<br/>7.5 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . 57<br/>7.5.1 Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . 57<br/>7.5.2 Definition, autoregressive representation . . . . . . . . 58<br/>7.5.3 Moving average representation . . . . . . . . . . . . . . 59<br/>7.5.4 Univariate representations . . . . . . . . . . . . . . . . 60<br/>7.5.5 Effect on projections . . . . . . . . . . . . . . . . . . . 61<br/>7.5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 62<br/>7.5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 63<br/>7.5.8 A warning: why “Granger causality” is not “Causality” 64<br/>7.5.9 Contemporaneous correlation . . . . . . . . . . . . . . 65<br/>8 SpectralRepresentation 67<br/>8.1 Facts about complex numbers and trigonometry . . . . . . . . 67<br/>8.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 67<br/>8.1.2 Addition,multiplication, and conjugation . . . . . . . . 68<br/>8.1.3 Trigonometric identities . . . . . . . . . . . . . . . . . 69<br/>8.1.4 Frequency, period and phase . . . . . . . . . . . . . . . 69<br/>8.1.5 Fourier transforms . . . . . . . . . . . . . . . . . . . . 70<br/>8.1.6 Why complex numbers? . . . . . . . . . . . . . . . . . 72<br/>8.2 Spectral density . . . . . . . . . . . . . . . . . . . . . . . . . . 73<br/>8.2.1 Spectral densities of some processes . . . . . . . . . . . 75<br/>8.2.2 Spectral densitymatrix, cross spectral density . . . . . 75<br/>8.2.3 Spectral density of a sum. . . . . . . . . . . . . . . . . 77<br/>8.3 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78<br/>8.3.1 Spectrum of filtered series . . . . . . . . . . . . . . . . 78<br/>8.3.2 Multivariate filtering formula . . . . . . . . . . . . . . 79<br/>3<br/>8.3.3 Spectral density of arbitrary MA(∞) . . . . . . . . . . 80<br/>8.3.4 Filtering and OLS . . . . . . . . . . . . . . . . . . . . 80<br/>8.3.5 A cosine example . . . . . . . . . . . . . . . . . . . . . 82<br/>8.3.6 Cross spectral density of two filters, and an interpretation<br/>of spectral density . . . . . . . . . . . . . . . . . 82<br/>8.3.7 Constructing filters . . . . . . . . . . . . . . . . . . . . 84<br/>8.3.8 Sims approximation formula . . . . . . . . . . . . . . . 86<br/>8.4 Relation between Spectral, Wold, and Autocovariance representations<br/>. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87<br/>9 Spectral analysis infinite samples 89<br/>9.1 Finite Fourier transforms . . . . . . . . . . . . . . . . . . . . . 89<br/>9.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 89<br/>9.2 Band spectrumregression . . . . . . . . . . . . . . . . . . . . 90<br/>9.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 90<br/>9.2.2 Band spectrumprocedure . . . . . . . . . . . . . . . . 93<br/>9.3 Cram&acute;er or Spectral representation . . . . . . . . . . . . . . . . 96<br/>9.4 Estimating spectral densities . . . . . . . . . . . . . . . . . . . 98<br/>9.4.1 Fourier transformsample covariances . . . . . . . . . . 98<br/>9.4.2 Sample spectral density . . . . . . . . . . . . . . . . . 98<br/>9.4.3 Relation between transformed autocovariances and sample<br/>density . . . . . . . . . . . . . . . . . . . . . . . . . 99<br/>9.4.4 Asymptotic distribution of sample spectral density . . 101<br/>9.4.5 Smoothed periodogramestimates . . . . . . . . . . . . 101<br/>9.4.6 Weighted covariance estimates . . . . . . . . . . . . . . 102<br/>9.4.7 Relation between weighted covariance and smoothed<br/>periodogramestimates . . . . . . . . . . . . . . . . . . 103<br/>9.4.8 Variance of filtered data estimates . . . . . . . . . . . . 104<br/>4<br/>9.4.9 Spectral density implied by ARMAmodels . . . . . . . 105<br/>9.4.10 Asymptotic distribution of spectral estimates . . . . . . 105<br/>10 Unit Roots 106<br/>10.1 RandomWalks . . . . . . . . . . . . . . . . . . . . . . . . . . 106<br/>10.2 Motivations for unit roots . . . . . . . . . . . . . . . . . . . . 107<br/>10.2.1 Stochastic trends . . . . . . . . . . . . . . . . . . . . . 107<br/>10.2.2 Permanence of shocks . . . . . . . . . . . . . . . . . . . 108<br/>10.2.3 Statistical issues . . . . . . . . . . . . . . . . . . . . . . 108<br/>10.3 Unit root and stationary processes . . . . . . . . . . . . . . . 110<br/>10.3.1 Response to shocks . . . . . . . . . . . . . . . . . . . . 111<br/>10.3.2 Spectral density . . . . . . . . . . . . . . . . . . . . . . 113<br/>10.3.3 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . 114<br/>10.3.4 Randomwalk components and stochastic trends . . . . 115<br/>10.3.5 Forecast error variances . . . . . . . . . . . . . . . . . 118<br/>10.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 119<br/>10.4 Summary of a(1) estimates and tests. . . . . . . . . . . . . . . 119<br/>10.4.1 Near- observational equivalence of unit roots and stationary<br/>processes in finite samples . . . . . . . . . . . . 119<br/>10.4.2 Empirical work on unit roots/persistence . . . . . . . . 121<br/>11 Cointegration 122<br/>11.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122<br/>11.2 Cointegrating regressions . . . . . . . . . . . . . . . . . . . . . 123<br/>11.3 Representation of cointegrated system. . . . . . . . . . . . . . 124<br/>11.3.1 Definition of cointegration . . . . . . . . . . . . . . . . 124<br/>11.3.2 Multivariate Beveridge-Nelson decomposition . . . . . 125<br/>11.3.3 Rank condition on A(1) . . . . . . . . . . . . . . . . . 125<br/>5<br/>11.3.4 Spectral density at zero . . . . . . . . . . . . . . . . . 126<br/>11.3.5 Common trends representation . . . . . . . . . . . . . 126<br/>11.3.6 Impulse-response function. . . . . . . . . . . . . . . . . 128<br/>11.4 Useful representations for running cointegrated VAR’s . . . . . 129<br/>11.4.1 Autoregressive Representations . . . . . . . . . . . . . 129<br/>11.4.2 Error Correction representation . . . . . . . . . . . . . 130<br/>11.4.3 Running VAR’s . . . . . . . . . . . . . . . . . . . . . . 131<br/>11.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132<br/>11.6 Cointegration with drifts and trends . . . . . . . . . . . . . . . 134</p>