References. Press, New York

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1 References Banerjee, Dolado J, Galbraith JW, Hendry DF (1993) Co-integration, errorcorrection and the econometric analysis of non stationary data. Oxford University Press, New York Bellman R (1970) Introduction to matrix analysis, 2nd edn. McGraw Hill, New York Beveridge S, Nelson CR (1981) new approach to the decomposition of economic time series into permanent and transient components with particular attention to measurement of the business cycle. J. Monet Econ 7: Blanc-Lapierre, Fortet R (1953) Théorie des fonctions aléatoires. Masson, Paris Campbell SL, Meyer CD (1979) Generalized inverse of linear transformations. Pitman, London Charenza WW, Deadman DF (1992) New directions in econometric practise. Edgar, ldershot, UK Chipman JS, Rao MM (1964) The treatment of linear restrictions in regression analysis. Econometrica 32: Cramer H, Leadbetter MR (1967) Stationary and related stochastic processes. Wiley, New York Dhrymes PJ (1971) Distributed lags: problems of estimation and formulation. Holden Day, San Francisco Elaydi SN (1996) n introduction to difference equations. Springer, New York Faliva M (1987) Econometria: principi e metodi. UTET, Torino Faliva M, Zoia MG (1999) Econometria delle serie storiche. Giappichelli, Torino Faliva M, Zoia MG (2002) On a partitioned inversion formula having useful applications in econometrics. Econom Theory 18: Faliva M, Zoia MG (2003) new proof of the representation theorem for I(2) processes. J Interdiscipl Math 6: Franchi M (2007) The integration order of vector autoregressive processes. Econom Theory 23: Fraser R, Duncan WJ, Collar R (1963) Elementary matrices and some applications to dynamics and differential equations. Cambridge University Press, Cambridge Gantmacher FR (1959) The theory of matrices. Chelsea, New York Goldberger S (1964) Econometric theory. Wiley, New York Greville TNE (1960) Some applications of the pseudoinverse of a matrix. Siam Rev 2:15 22 Hansen PR, Johansen S (1998) Workbook on cointegration. Oxford University Press, New York Hatanaka M (1996) Time-series-based econometrics. Oxford University Press, New York Jeffrey (1992) Complex analysis and applications. CRC, London

2 214 References Johansen S (1995) Likelihood-based inference in cointegrated vector autoregressive models. Oxford University Press, New York Lancaster P, Tismenetsky M (1985) The theory of matrices, 2nd edn. cademic Press, New York Lütkepohl H (1991) Introduction to multiple time series analysis. Springer, Berlin Magnus JR, Neudecker H (1999) Matrix differential calculus with applications to statistics and econometrics. Wiley, New York Markuscevich I (1965) Theory of a complex variable. Prentice Hall, London Marsaglia G, Styan GPH (1974) Equalities and inequalities for ranks of matrices. Linear Multilinear lgebra 2: Mickens RE (1990) Difference equations. Theory and applications, 2nd edn. Van Nostrand Reinhold, New York Neudecker H (1968) The Kronecker matrix product and some of its applications in econometrics. Statistica Neerlandica 22:69 82 Papoulis (1965) Probability, random variables and stochastic processes. McGraw Hill, New York Pringle RM, Rayner (1971) Generalized inverse matrices with applications to statistics. Griffin, London Rao CR (1973) Linear statistical inference and its applications, 2nd edn. Wiley, New York Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York Searle SR (1982) Matrix algebra useful for statistics. Wiley, New York Stock JH, Watson MW (1988) Variable trends in economic time series. J Econ Persp 2: Styan GPH (1973) Hadamard products and multivariate statistical analysis. Linear lgebra ppl 6: Theil H (1971) Principles of econometrics. North Holland, msterdam Vladimirov VS (1984) Equations of mathematical physics. MIR, Moscow Yaglom M (1962) Stationary random functions. Prentice Hall, Englewood Cliffs, NJ Wold H (1938) study in the analysis of stationary time series (2nd edn. 1954) lmqvist & Wiksells, Uppsala Zoia MG (2006) new algebraic approach to representation theorems for (co) integrated processes up to the second order. Cahiers du Département d économétrie, Université de Genève 6:1 20

3 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 (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)

4 216 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) 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

5 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 Nullity Definition 4 Pole Definition 5 Order of Poles and Zeros Definition 6 Characteristic Polynomial...43 Section 1.7 Order of a Pole in a Laurent Expansion Section 1.8 Backward Difference Operator...65 ntidifference Operator... 66

6 T 218 List of Definitions Section 2.1 Definition 4 Definition 5 Stationary Processes Stationarity in Mean Covariance Stationarity Stationarity in the Wide Sense Integrated Processes Section 2.2 Definition 4 Definition 5 Definition 6 Definition 7 White Noise Vector Moving-verage Processes First Difference of a White Noise Second Difference of a White Noise Vector utoregressive Processes Invertible Processes Vector utoregressive Moving-verage Processes Section 2.4 Definition 4 Definition 5 Definition 6 Definition 7 Random Walk Random Walk With Drift Cumulated Random Walk Deterministic T rends Stochastic rends Cointegrated Systems Polynomially Cointegrated Systems Section 3.2 Basic VR Model Error Correction Model...168