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 . . . . . . . 133.3.3 AR(1) to MA(
∞) with lag operators. . . . . . . . . . . 133.3.4 AR(p) to MA(
∞), MA(q) to AR(∞), factoring lagpolynomials, 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 De
finitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Autocovariance and autocorrelation of ARMA processes. . . . 22
4.2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 25
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 De
finitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.2 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 . . . 467 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) . . . . . . . . 507.1.4 Blanchard-Quah orthogonalization—restrictions on
C(1). 527.2 Variance decompositions . . . . . . . . . . . . . . . . . . . . . 53
7.3 VAR’s in state space notation . . . . . . . . . . . . . . . . . . 54
7.4 Tricks and problems: . . . . . . . . . . . . . . . . . . . . . . . 55
7.5 Granger Causality . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.5.1 Basic idea . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.5.2 De
finition, autoregressive representation . . . . . . . . 587.5.3 Moving average representation . . . . . . . . . . . . . . 59
7.5.4 Univariate representations . . . . . . . . . . . . . . . . 60
7.5.5 E
ffect on projections . . . . . . . . . . . . . . . . . . . 617.5.6 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
......
扫码加好友，拉您进群
全部版块
我的主页
收藏
