Factorization of a Spectral Density with Smooth Eigenvalues of a Multidimensional Stationary Time Series
The aim of this paper to give a multidimensional version of the classical one-dimensional case of smooth spectral density. A spectral density with smooth eigenvalues and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Econometrics |
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| Online Access: | https://www.mdpi.com/2225-1146/11/2/14 |
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| author | Tamás Szabados |
| author_facet | Tamás Szabados |
| author_sort | Tamás Szabados |
| collection | DOAJ |
| description | The aim of this paper to give a multidimensional version of the classical one-dimensional case of smooth spectral density. A spectral density with smooth eigenvalues and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mo>∞</mo></msup></semantics></math></inline-formula> eigenvectors gives an explicit method to factorize the spectral density and compute the Wold representation of a weakly stationary time series. A formula, similar to the Kolmogorov–Szeg<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi mathvariant="normal">o</mi><mo>”</mo></mover></mrow></semantics></math></inline-formula> formula, is given for the covariance matrix of the innovations. These results are important to give the best linear predictions of the time series. The results are applicable when the rank of the process is smaller than the dimension of the process, which occurs frequently in many current applications, including econometrics. |
| first_indexed | 2024-03-11T02:34:08Z |
| format | Article |
| id | doaj.art-dfe0b42353424c04be36a0812e2fba70 |
| institution | Directory Open Access Journal |
| issn | 2225-1146 |
| language | English |
| last_indexed | 2024-03-11T02:34:08Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Econometrics |
| spelling | doaj.art-dfe0b42353424c04be36a0812e2fba702023-11-18T10:05:17ZengMDPI AGEconometrics2225-11462023-05-011121410.3390/econometrics11020014Factorization of a Spectral Density with Smooth Eigenvalues of a Multidimensional Stationary Time SeriesTamás Szabados0Department of Mathematics, Budapest University of Technology and Economics, 1111 Budapest, HungaryThe aim of this paper to give a multidimensional version of the classical one-dimensional case of smooth spectral density. A spectral density with smooth eigenvalues and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>H</mi><mo>∞</mo></msup></semantics></math></inline-formula> eigenvectors gives an explicit method to factorize the spectral density and compute the Wold representation of a weakly stationary time series. A formula, similar to the Kolmogorov–Szeg<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mover><mi mathvariant="normal">o</mi><mo>”</mo></mover></mrow></semantics></math></inline-formula> formula, is given for the covariance matrix of the innovations. These results are important to give the best linear predictions of the time series. The results are applicable when the rank of the process is smaller than the dimension of the process, which occurs frequently in many current applications, including econometrics.https://www.mdpi.com/2225-1146/11/2/14multidimensional stationary time seriessmooth spectral densityspectral factorbest linear prediction |
| spellingShingle | Tamás Szabados Factorization of a Spectral Density with Smooth Eigenvalues of a Multidimensional Stationary Time Series Econometrics multidimensional stationary time series smooth spectral density spectral factor best linear prediction |
| title | Factorization of a Spectral Density with Smooth Eigenvalues of a Multidimensional Stationary Time Series |
| title_full | Factorization of a Spectral Density with Smooth Eigenvalues of a Multidimensional Stationary Time Series |
| title_fullStr | Factorization of a Spectral Density with Smooth Eigenvalues of a Multidimensional Stationary Time Series |
| title_full_unstemmed | Factorization of a Spectral Density with Smooth Eigenvalues of a Multidimensional Stationary Time Series |
| title_short | Factorization of a Spectral Density with Smooth Eigenvalues of a Multidimensional Stationary Time Series |
| title_sort | factorization of a spectral density with smooth eigenvalues of a multidimensional stationary time series |
| topic | multidimensional stationary time series smooth spectral density spectral factor best linear prediction |
| url | https://www.mdpi.com/2225-1146/11/2/14 |
| work_keys_str_mv | AT tamasszabados factorizationofaspectraldensitywithsmootheigenvaluesofamultidimensionalstationarytimeseries |