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## Exact Maximum Likelihood Estimation of Structured or Unit Root Multivariate Time Series Models ∗ (2004)

Citations: | 7 - 1 self |

### Citations

2505 |
Co-integration and error correction: representation, estimation and testing
- Engle, Granger
- 1987
(Show Context)
Citation Context ... dimensional vectors. See also Söderström et al. (1998). The case of a VARMA process with unit roots (also called a co-integrated VARMA process) is generally handled through an error correction form (=-=Engle and Granger, 1987-=-). It is supposed that the determinant of the autoregressive polynomial has one or several roots equal to 1 and the other roots lying outside of the unit circle. It is equivalent to saying that the au... |

1194 | Time Series: Theory and Methods - Brockwell, Davis - 1991 |

452 | Testing for common trends
- STOCK, WATSON
- 1988
(Show Context)
Citation Context ...ta, the flour price data, the quarterly AAA corporate bonds and commercial paper series (Haugh and Box, 1977), the U.S. monthly housing data (Hillmer and Tiao, 1979), the U.S. monthly interest rates (=-=Stock and Watson, 1988-=-) and a simulated series (Nsiri and Roy, 1996). Second, limited Monte Carlo comparisons between our procedure and the conditional procedure described by Reinsel (1997), based on the accuracy of the es... |

218 |
The statistical Theory of Linear Systems
- E, Deistler
- 1988
(Show Context)
Citation Context ...Yt = Yt−h, the autoregressive and moving average polynomials by Φ(z) = Ik − Φ1z − ... − Φpz p , and Θ(z) = Ik − Θ1z − ... − Θqz q . We suppose that the VARMA model is identifiable (e.g. Hannan, 1969; =-=Hannan and Deistler, 1988-=-, Chap. 2) and invertible, that is the roots of det{Θ(z)} are greater than one in modulus. In the stationary case, the roots of det{Φ(z)} must also be outside the unit disk. In the sequel, we assume n... |

110 |
Elements of Multivariate Time Series Analysis
- Reinsel
- 1993
(Show Context)
Citation Context ... are generally estimated using least squares (in the pure autoregressive case, Ahn and Reinsel, 1990), approximated least squares (e.g. Poskitt and Lütkepohl, 1995) or conditional maximum likelihood (=-=Reinsel, 1997-=-, Yap and Reinsel, 1995). However, the small sample properties of these methods are not good for the moving average coefficients, at least. Yap and Reinsel (1995) discuss full-rank and reduced-rank es... |

106 | Evaluation of likelihood functions for Gaussian signals - Schweppe - 1965 |

82 |
Modeling Multiple Time Series With Applications
- Tiao, Box
- 1981
(Show Context)
Citation Context ...innovations are assumed to be known, instead of being determined conditionally on the observed series. Some other slightly improved procedures have been suggested by Hillmer and Tiao (1979) (see also =-=Tiao and Box, 1981-=-), and Nicholls and Hall (1979). More recently, that approach has been transformed by Mauricio (1995, 1997) into an exact and computationally efficient algorithm. Asymptotically, all these estimation ... |

68 |
Estimation for Partially Nonstationary Multivariate Autoregressive Models
- Ahn, Reinsel
- 1990
(Show Context)
Citation Context ...ay be some normalization in order to obtain a unique representation) and the innovation covariance matrix. The parameters are generally estimated using least squares (in the pure autoregressive case, =-=Ahn and Reinsel, 1990-=-), approximated least squares (e.g. Poskitt and Lütkepohl, 1995) or conditional maximum likelihood (Reinsel, 1997, Yap and Reinsel, 1995). However, the small sample properties of these methods are not... |

68 | Maximum likelihood fitting of ARMA models to time series with missing observations - Jones - 1980 |

61 | An innovations approach to least-squares estimation Part II: Linear smoothing in additive white noise [J - Kailath, Frost |

39 | Model Specification in Multivariate Time Series - Tiao, Tsay |

33 | An algorithm for the exact likelihood of a mixed autoregressive-moving average process - ANSLEY - 1979 |

27 | Algorithm for triangular decomposition of block Hankel and Toeplitz matrices with application to factoring positive matrix polynomials - RISSANEN - 1973 |

25 |
Finite Sample Properties of Estimators for Autoregressive Moving Average Models
- Ansley, Newbold
- 1980
(Show Context)
Citation Context ...to an exact and computationally efficient algorithm. Asymptotically, all these estimation methods are equivalent but their small sample properties known by Monte Carlo studies in the univariate case (=-=Ansley and Newbold, 1980-=-) are quite different. Although the autoregressive (AR) coefficients are estimated roughly with the same success, the moving average (MA) parameters suffer from a higher bias when the exact likelihood... |

19 | A New Algorithm for Optimal Filtering of Discrete-time Stationary Processes
- Lindquist
- 1974
(Show Context)
Citation Context ...rom a higher bias when the exact likelihood is not used. Some of the exact algorithms make use of a state space form and the fast Chandrasekhartype equation recursions of Morf et al. (1974) (see also =-=Lindquist, 1974-=-, and Rissanen, 1973). The latter recursions were first advocated for that purpose in the case of univariate time series by Caines and Rissanen (1974) and Pearlman (1980), and implemented in that case... |

17 |
Identification of Dynamic Regression (Distributed Lag) Models connecting IILvo Time Series
- Haugh, Box
- 1977
(Show Context)
Citation Context ...1989), Reinsel (1997), Yap and Reinsel (1995) have been fitted using the new procedure: the mink and muskrat data, the flour price data, the quarterly AAA corporate bonds and commercial paper series (=-=Haugh and Box, 1977-=-), the U.S. monthly housing data (Hillmer and Tiao, 1979), the U.S. monthly interest rates (Stock and Watson, 1988) and a simulated series (Nsiri and Roy, 1996). Second, limited Monte Carlo comparison... |

17 |
Likelihood Function of Stationary Multiple Autoregressive Moving Average Models
- Hillmer, Tiac
- 1979
(Show Context)
Citation Context ...en fitted using the new procedure: the mink and muskrat data, the flour price data, the quarterly AAA corporate bonds and commercial paper series (Haugh and Box, 1977), the U.S. monthly housing data (=-=Hillmer and Tiao, 1979-=-), the U.S. monthly interest rates (Stock and Watson, 1988) and a simulated series (Nsiri and Roy, 1996). Second, limited Monte Carlo comparisons between our procedure and the conditional procedure de... |

17 | On the specification of cointegrated autoregressive moving-average forecasting systems
- Poskitt
- 2003
(Show Context)
Citation Context ...esentation) and the innovation covariance matrix. The parameters are generally estimated using least squares (in the pure autoregressive case, Ahn and Reinsel, 1990), approximated least squares (e.g. =-=Poskitt and Lütkepohl, 1995-=-) or conditional maximum likelihood (Reinsel, 1997, Yap and Reinsel, 1995). However, the small sample properties of these methods are not good for the moving average coefficients, at least. Yap and Re... |

16 |
Exact Likelihood of Vector Autoregressive Moving-Average Process with Missing or Aggregated Data
- Ansley, Kohn
- 1983
(Show Context)
Citation Context ...state space form. However, we start with the Kalman filter recursions which are more general, and can be used for instance when there are missing data (although we shall not cover that case here, see =-=Ansley and Kohn, 1983-=-). We shall follow the presentation given by Shea (1987), using a state vector αt of dimension rk × 1, where r = max(p, q). We suppose that � Φ0 is invertible. When we have the echelon form, � Φ0 is a... |

15 |
Identi¯cation of Echelon Canonical Forms for Vector Linear ProcessesUsingLeastSquares,TheAnnalsofStatistics,20,195-215
- Poskitt
- 1992
(Show Context)
Citation Context ... scalar component model structure (SCM) of Tiao and Tsay (1989). In addition, several other specification procedures have been proposed recently, based upon a canonical ARMA echelon form (Tsay, 1991, =-=Poskitt, 1992-=-, Nsiri and Roy, 1992, 1996). It will be shown in Section 2 that the echelon form and the SCM share the same general appearance, despite the fact that the two specification procedures are not equivale... |

14 | The Euclid algorithm and the fast computation of cross–covariance and autocovariance sequences - Demeure, Mullis - 1989 |

11 | The Exact Likelihood of a Vector Autoregressive Moving Average Model - Shea - 1989 |

10 |
The identification of vector mixed autoregressive moving-average systems
- Hannan
- 1969
(Show Context)
Citation Context ...such that B h Yt = Yt−h, the autoregressive and moving average polynomials by Φ(z) = Ik − Φ1z − ... − Φpz p , and Θ(z) = Ik − Θ1z − ... − Θqz q . We suppose that the VARMA model is identifiable (e.g. =-=Hannan, 1969-=-; Hannan and Deistler, 1988, Chap. 2) and invertible, that is the roots of det{Θ(z)} are greater than one in modulus. In the stationary case, the roots of det{Φ(z)} must also be outside the unit disk.... |

10 | An algorithm for the exact likelihood of a stationary vector autoregressivemoving average model - Mauricio - 2002 |

10 | Derivation of the Theoretical Autocovariance Function of Autoregressi ve-Moving Average Time Series", Appl . Statist - McLeod - 1975 |

10 |
On the identification of ARMA echelon-form models
- Nsiri, Roy
- 1992
(Show Context)
Citation Context ...nt model structure (SCM) of Tiao and Tsay (1989). In addition, several other specification procedures have been proposed recently, based upon a canonical ARMA echelon form (Tsay, 1991, Poskitt, 1992, =-=Nsiri and Roy, 1992-=-, 1996). It will be shown in Section 2 that the echelon form and the SCM share the same general appearance, despite the fact that the two specification procedures are not equivalent and generally do n... |

10 |
Identifying multivariate time series models
- Tsay
- 1989
(Show Context)
Citation Context ...ion and from (2.8), we write p� (Ik − �Φ j=1 −1 0 � ΦjB j q� )Yt = (Ik − j=1 �Φ −1 0 � ΘjB j )εt. (4.1) For the scalar component model of Section 2.2, � Φ0 is obtained using a canonical analysis (see =-=Tsay 1989-=-) and is not considered as a parameter of the model. Besides that, the treatment is similar. 7sLet us denote Φj = � Φ −1 0 � Φj, j = 1, . . . , p, Φj = 0, j = p + 1, . . . , r, Θj = � Φ −1 0 � Θj, j =... |

9 | Estimating the Kronecker indices of cointegrated echelonform VARMA models - Bartel, Lütkepohl - 1998 |

9 |
Two canonical forms for vector ARMA processes
- Tsay
- 1991
(Show Context)
Citation Context ... and (b) the scalar component model structure (SCM) of Tiao and Tsay (1989). In addition, several other specification procedures have been proposed recently, based upon a canonical ARMA echelon form (=-=Tsay, 1991-=-, Poskitt, 1992, Nsiri and Roy, 1992, 1996). It will be shown in Section 2 that the echelon form and the SCM share the same general appearance, despite the fact that the two specification procedures a... |

8 | Exact Maximum Likelihood Estimation of Stationary Vector ARMA Models - Mauricio - 1995 |

8 |
Identification of refined ARMA echelon form models for multivariate time series
- Nsiri, Roy
- 1996
(Show Context)
Citation Context ...orate bonds and commercial paper series (Haugh and Box, 1977), the U.S. monthly housing data (Hillmer and Tiao, 1979), the U.S. monthly interest rates (Stock and Watson, 1988) and a simulated series (=-=Nsiri and Roy, 1996-=-). Second, limited Monte Carlo comparisons between our procedure and the conditional procedure described by Reinsel (1997), based on the accuracy of the estimators are presented in two cases: the eche... |

8 | An efficient and versatile algorithm for computing the covariance function of an ARMA process - Söderström, Ježek - 1998 |

8 | The estimation of parameters in multivariate time series models - Wilson - 1973 |

7 | Covariance matrix computations of the state variable of a stationary Gaussian process - AKAIKE - 1978 |

7 | Computation of the theoretical autocovariance function for a vector ARMA process - ANSLEY - 1980 |

6 | Asymptotic distribution of a simple linear estimator for VARMA models in echelon form - Dufour, Jouini - 2004 |

6 | An algorithm for the exact likelihood of highorder autoregressive - moving average process - Pearlman - 1980 |

6 | Tests for non-correlation of two cointegrated ARMA time series - Pham, Roy, et al. - 2003 |

6 | Properties of Infinite Covariance Matrices and Stability of Otimum Predictors," nformation Sciences 1 - Rissanen, Barbosa - 1969 |

5 | An algorithm for exact maximum likelihood estimation by means of (Caiman filtering - GARDNER, HARVEY, et al. - 1980 |

5 |
Algorithm AS197: A fast algorithm for the exact likelihood of autoregressive-moving average models
- Mélard
- 1984
(Show Context)
Citation Context ... the the procedure mentioned below. For univariate processes, the computation of the initial conditions of the Chadrasekhar recursions is reasonably fast, due to the existence of fast algorithms (see =-=Mélard, 1984-=-, Kohn and Ansley, 1985), which need approximately O(p 2 ) operations instead of O(p 3 ) for the direct method described by McLeod (1975). This is not true for multivariate processes. The method of An... |

4 | Maximum likelihood estimation of parameters in multivariate Gaussian stochastic processes - Caines, Rissanen - 1974 |

4 |
General structure and parametrization of ARMA and state-space systems and its relation to statistical problems
- Deistler
- 1985
(Show Context)
Citation Context ...cess {Yt } to admit an ARMA representation is that its dynamic dimension be finite. Also when dim Pt < ∞, for any basis of Pt we can find a corresponding ARMA representation of {Yt } (see for example =-=Deistler 1985-=- or Gouriéroux and Monfort 1990, chap. 8). A natural way to choose such a basis, when dim Pt = n, is to consider the one formed by the first n linearly independent components of the vector of predicto... |

4 |
Computing the likelihood and its derivatives for a Gaussian ARMA model
- Kohn, Ansley
- 1985
(Show Context)
Citation Context ...dure mentioned below. For univariate processes, the computation of the initial conditions of the Chadrasekhar recursions is reasonably fast, due to the existence of fast algorithms (see Mélard, 1984, =-=Kohn and Ansley, 1985-=-), which need approximately O(p 2 ) operations instead of O(p 3 ) for the direct method described by McLeod (1975). This is not true for multivariate processes. The method of Ansley (1980), which is u... |

3 | FIML estimation of the dynamic simultaneous equations model with ARIMA disturbances - Reinsel - 1979 |

3 | Estimation of multivariate time series - Shea - 1987 |

3 |
Parsimonious parametrization of vector autoregressive moving average models
- Tsay
- 1989
(Show Context)
Citation Context ...ion and from (2.8), we write p� (Ik − �Φ j=1 −1 0 � ΦjB j q� )Yt = (Ik − j=1 �Φ −1 0 � ΘjB j )εt. (4.1) For the scalar component model of Section 2.2, � Φ0 is obtained using a canonical analysis (see =-=Tsay 1989-=-) and is not considered as a parameter of the model. Besides that, the treatment is similar. 7sLet us denote Φj = � Φ −1 0 � Φj, j = 1, . . . , p, Φj = 0, j = p + 1, . . . , r, Θj = � Φ −1 0 � Θj, j =... |

3 | Some efficient computational procedures for high order ARMA models - Wilson, G - 1979 |

2 |
A method for generating independent realization of a multivariate normal stationary and invertible ARMA(p, q) process
- BARONE
- 1987
(Show Context)
Citation Context ...iments and for each of the two series lengths, the G05HDF subroutine of the NAG library (NAG, 1995) was used to generate Gaussian series from the corresponding standard VARMA data generating process (=-=Barone, 1987-=-, and Shea, 1988). Then, suitable transformations were applied to obtain the series from the non standard VARMA processes, For each replication, the two estimation methods (exact and conditional) were... |

2 | Exact likelihood for stationary vector autoregressive moving average processes - Dugré, Scharf, et al. - 1986 |

2 | Algorithmes pour l’estimation par pseudo-maximum de vraisemblance exacte pour des modèles VARMA sous forme classique et sous forme structurée. Unpublished Ph.D. Thesis. Institut de Statistique et de Recherche Opérationnelle, Université Libre de Bruxelles - Harti - 1996 |

2 | Computing covariances for scalar and vector ARMA processes - Harti, Melard, et al. - 2004 |

2 | A parsimonious representation and VARMA estimator for cointegration. Working paper - Hunter, Dislis - 1994 |

2 | A note on obtaining the theoretical autocovariances of an ARMA process - KOHN, ANSLEY - 1982 |

2 |
The SCA Statistical System Analysis: Reference Manual for Forecasting and Time series Analysis. Chicago: Scientific Computing Associates
- Lui, Hudak
- 1986
(Show Context)
Citation Context ... −0.51 −0.60 ⎦ ɛt − ⎣ 0.825 0 0 0.075 0 0 0.374 ⎤ 0 ⎥ 0 ⎦ ɛt−1. 0.215 0.55 0.83 −0.06 1.421 −1.256 −0.062 The results are close but not identical because the so-called exact estimation method of SCA (=-=Lui and Hudak, 1986-=-) computes, in fact, the exact likelihood for an autoregressive model, not for an ARMA model. Example 4 We consider the bivariate time series of seasonally adjusted monthly U.S. housing data consistin... |

2 | The exact likelihood function of stationary vector autoregressive moving average model - Mauricio - 1997 |

2 | Analyse de données chronologiques, Séminaire de mathématiques supérieures, Séminaire scientifique - Mélard - 1985 |

2 | Computation of theoretical autocovariance matrices of multivariate autoregressive moving average time series - Mittnik - 1990 |

2 | Computing theoretical autocovariances of multivariate autoregressive moving average models by using a block Levinson method - Mittnik - 1993 |

2 | The exact likelihood function of multivariate autoregressive moving average models - Nicholls, Hall - 1979 |

2 | Maximum likelihood estimation of multivariate ARMA processes via the Kalman filter - Shea - 1984 |

2 |
A note on the generation of independent realisations of a vector autoregressive moving average process, Journal of Time Series Analysis 9
- SHEA
- 1988
(Show Context)
Citation Context ...h of the two series lengths, the G05HDF subroutine of the NAG library (NAG, 1995) was used to generate Gaussian series from the corresponding standard VARMA data generating process (Barone, 1987, and =-=Shea, 1988-=-). Then, suitable transformations were applied to obtain the series from the non standard VARMA processes, For each replication, the two estimation methods (exact and conditional) were performed using... |

2 | The exact likelihood for a multivariate ARMA model - Solo - 1984 |

2 | Developments in multivariate covariance generation and factorization - Wilson, G - 1993 |

1 |
Nonlinear Methods in Econometrics. Amesterdam: NorthHolland Gouriéroux
- Goldfeld, Quandt
- 1972
(Show Context)
Citation Context ...ng programs. Both are based on the original Shea’s (1989) program. Implementation A was initiated by Harti (1996) and is home made except the optimization routine which is based on the library GQOPT (=-=Goldfeld and Quandt, 1972-=-). With respect to the Shea (1989) program, the following changes have been implemented in A: 1. the procedure described in Harti, Mélard and Pham (2004) has been used for computing the autocovariance... |

1 | Derivation of the unconditional state-covariance matrix for exact maximum-likelihood estimation of ARMA models - Mittnik - 1991 |

1 |
Algorithm 423: Linear equation solver
- Moler
- 1972
(Show Context)
Citation Context ...elements of the innovation covariance matrix, a k × k lower triangular matrix U has been used as a parameter, so that Σ = U ′ U; 2. some modifications have been operated in routines DECOMP and SOLVE (=-=Moler, 1972-=-) in order to avoid comparisons of real numbers with 0; 3. the original conditional maximum likelihood procedure included in Shea’s (1989) program does not work and has been corrected. 9sThe empirical... |

1 | Some new alorithms for recursive estimation on constant, linear, discrete-time systems - Morf, Sidhu, et al. - 1974 |

1 | Méthodes d’estimation de paramètres de modèles autorégressifs multivariés”. Unpublished Ph.D. Thesis. Université Joseph Fourier - Tong - 1991 |