1 Introduction 1
. 1.1 VectorAutoregressions...................... 1
. 1.2 LogLikelihoodFunction ..................... 2
. 1.3 ChoosingLagLength....................... 3
. 1.4 SYSTEMdefinitionandESTIMATE................6
. 1.5 VariablesandResiduals...................... 8
. 1.6 AlternativeEstimationMethods.................. 9
. 1.1 LagSelectionbyAIC .......................... 11 1.2 EstimationTechniques.........................12 1.3 LongLagVAR..............................13
2 Impulse Response Functions 14
. 2.1 MovingAverageRepresentation.................. 14
. 2.2 ComputingImpulseResponses .................. 16
. 2.3 Orthogonalization . . . . . . . . . . . . . . . . . . . . . . . . 21
. 2.4 VarianceDecomposition...................... 23
. 2.5 RATSTipsandTricks ...................... 26 2.1 IRFwithinputshocks......................... 28 2.2 IRFwithCholeskishocks....................... 29
3 Error Bands 30
. 3.1 Deltamethod .......................... 30
. 3.2 Bootstrapping.......................... 33 3.2.1Kilianbootstrap-after-bootstrap. . . . . . . . . . . . . . . . 34
. 3.3 MonteCarloIntegration ..................... 35 3.3.1 Creating%%RESPONSES...................38 3.3.2 The@MCPROCESSIRFprocedure.............. 39
3.4RATSTipsandTricks ...................... 42 3.4.1The%MODELfunctionfamily.................. 42 3.4.2 BlockSampling......................... 43
. 3.1 ErrorBandsbyDeltaMethod ..................... 45
. 3.2 ErrorBandsbyBootstrapping..................... 46
. 3.3 ErrorBandsbyKilianbootstrap ................... 47
. 3.4 ErrorBandsbyMonteCarlo...................... 48
. 3.5 Error Bands by Bootstrapping with Random Initial Values .. . . 49
4 Historical Decomposition and Counterfactual Simulations 51
. 4.1 HistoricalDecomposition..................... 51
. 4.2 CounterfactualSimulations.................... 53
. 4.3 ErrorBands........................... 54 4.1 HistoricalDecomposition........................54
5 Structural VAR’s 56
. 5.1 EigenFactorizations....................... 56
. 5.2 GeneralizedImpulseResponses.................. 57
. 5.3 ParametricStructuralVAR’s ................... 58
. 5.4 Identification........................... 61
. 5.5 Estimation............................62
. 5.6 StructuralResiduals....................... 67
. 5.7 ErrorBands........................... 67
. 5.7.1 Bootstrapping.......................... 68
. 5.7.2 Monte Carlo: Deriving the PosteriorDistribution . . . . . 68
. 5.7.3 MonteCarlowithImportanceSampling . . .. . . . . . . . 71
. 5.7.4 Monte Carlo with Random Walk Metropolis. . . . . . . . . 76
. 5.7.5 Monte Carlo with the Waggoner-ZhaSampler . . . . . . . 81
. 5.8 TipsandTricks ......................... 84 5.8.1 UsingPARMSETS........................ 84 5.8.2 Waggoner-Zhasampler..................... 85
. 5.1 Eigenfactorization........................... 87
. 5.2 SVAR:A-styleModel.......................... 88
Contents iii
5.3 SVAR:A-Bstylemodel ......................... 90 5.4 SVAR:ImportanceSamplingforErrorBands. . . . . . . . . . . . 92 5.5 SVAR:RandomWalkMetropolisforErrorBands.. . . . . . . . . 96 5.6 SVAR:Waggoner-ZhaSampler.................... 99
6 Semi-Structural VAR’s 101
. 6.1 ForcedFactorProcedure......................102
. 6.2 Short-andLong-RunRestrictions.................103
. 6.3 Multi-stepRestrictions......................105
. 6.4 ErrorBands...........................107 6.1 Blanchard-QuahDecomposition....................110 6.2 Multi-StepRestrictions.........................112 6.3 Short- and Long-RunRestrictions with Error Bands . . . . . . . . 114
7 Sign Restrictions 116
. 7.1 GeneratingImpulseVectors....................116
. 7.2 Penaltyfunctions.........................122
. 7.3 MultipleShocks.........................124 7.3.1 FORCEDFACTOR and the QR Decomposition . . . .. . . 125
. 7.4 ZeroConstraints.........................126
. 7.5 Fry-PaganCritique........................127
. 7.6 HistoricalDecomposition.....................130
. 7.7 Convenience functions for sign restrictions . . . . . . . . . . . . . 131 7.1 SignRestrictions:PartI ........................ 134 7.2 SignRestrictions:PartII........................136 7.3 SignRestrictions:PartIII....................... 139 7.4 SignRestrictions:MedianTargetMethod. . . . . . . . . . . . . . 141
A Probability Distributions 145
. A.1 Multivariate Normal . . . . . . . . . . . . . . . . . . . . . . . 145
. A.2 WishartDistribution . . . . . . . . . . . . . . . . . . . . . . . 146
. A.3 GammaDistribution . . . . . . . . . . . . . . . . . . . . . . . 147
. A.4 InverseGammaDistribution . . . . . . . . . . . . . . . . . . . 148
Contents iv
. B VAR Likelihood Function 149
. C Properties of Multivariate Normals 152
. D Deriving the Schwarz criterion 155
. E Delta method 157
. F Gibbs Sampling and Markov Chain Monte Carlo 158
Bibliography 162 Index 164
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