1 Preface 7
2 What is a time series? 8
3 ARMA models 10
4 The autocorrelation and autocovariance functions. 21
5 Prediction and Impulse-Response Functions 31
6 Stationarity and Wold representation 40
7 VARs: orthogonalization, variance decomposition, Granger
causality 48
8 Spectral Representation 67
9 Spectral analysis in finite samples 89
10 Unit Roots 106
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 . . . . . . . . . . . . . . . 134