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Notational Conventions, Symbols and cronyms The following notational conventions will be used throughout the text: Bold lower case letters indicate vectors. Bold upper case letters indicate matrices. Both notations [ B] and [, B] will be used, depending on convenience, for column-wise partitioned matrices B B Both notations and will be used, depending on C D C D convenience, for block matrices Symbols and cronyms r () ρ g ind() D, Meaning generalized inverse of rank of reflexive generalized inverse transpose of Moore-Penrose inverse of index of Drazin inverse of Section 1.1 () 1.1 1.1 () 1.1 1.1 () 1.1 (Definition 5) 1.1 (Definition 6) r right inverse of 1.1 (Definition 7) l left inverse of 1.1 (Definition 8) det () determinant of 1.1 ',( ) orthogonal complement of specular directional inverses 1.2 () 1.2 (Remark 4) l s s left orthogonal complement of 1.2 () r right orthogonal complement of 1.2 (ibid)
216 Notational Conventions, Symbols and cronyms (z) matrix polynomial of z 1.6 () ( z), ( z), ( z) dot notation for derivatives 1.6,, short notation for 1.6 ( 1), (1), (1), (1) + adjoint of 1.6 tr trace of 1.6 vec staked form of 1.6 L lag operator 1.8 backward difference operator 1.8 () 1 antidifference operator 1.8 () Σ B u indefinite sum operator Kronecker product of and B vector of 1 s 1.8 (ibidem) 1.10 1.10 E expectation operator 2.1 Γ (h) autocovariance matrix of order h 2.1 I (d) integrated process of order d 2.1 (Definition 5) I (0) stationary process 2.1 WN (n) n-dimensional white noise 2.2 () δ v discrete unitary function 2.2 VM (q) vector moving average process 2.2 () of order q VR (p) vector autoregressive process of 2.2 (Definition 5) order p VRM (p, q) vector autoregressive moving 2.2 (Definition 7) average process of order (p, q) CI (d, b) cointegrated system of order 2.4 (Definition 6) PCI (d, b) B (d, b) polynomially cointegrated system of order (d, b) Hadamard product of and B 2.4 (Definition 7) 3.1
K List of Definitions Section 1.1 Generalized Inverse... 1 Reflexive Generalized Inverse... 2 Moore-Penrose Inverse...2 Definition 4 Nilpotent Matrix...3 Definition 5 Index of a Matrix...3 Definition 6 Drazin Inverse... 3 Definition 7 Right Inverse...4 Definition 8 Left Inverse...4 Section 1.2 Row ernel...7 Orthogonal Complement... 7 Left and Right Orthogonal Complements...17 Section 1.6 Matrix Polynomial...38 Zero of a Matrix Polynomial... 38 Nullity... 38 Definition 4 Pole... 39 Definition 5 Order of Poles and Zeros... 39 Definition 6 Characteristic Polynomial...43 Section 1.7 Order of a Pole in a Laurent Expansion... 52 Section 1.8 Backward Difference Operator...65 ntidifference Operator... 66
T 218 List of Definitions Section 2.1 Definition 4 Definition 5 Stationary Processes...128 Stationarity in Mean...128 Covariance Stationarity...129 Stationarity in the Wide Sense...129 Integrated Processes...129 Section 2.2 Definition 4 Definition 5 Definition 6 Definition 7 White Noise... 130 Vector Moving-verage Processes...131 First Difference of a White Noise...133 Second Difference of a White Noise...133 Vector utoregressive Processes...134 Invertible Processes...137 Vector utoregressive Moving-verage Processes...139 Section 2.4 Definition 4 Definition 5 Definition 6 Definition 7 Random Walk... 145 Random Walk With Drift...146 Cumulated Random Walk...147 Deterministic T rends... 147 Stochastic rends... 148 Cointegrated Systems...149 Polynomially Cointegrated Systems...150 Section 3.2 Basic VR Model...167 Error Correction Model...168