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Asymptotic properties of conditional least-squares estimators for array time series

Abstract

The paper provides a kind of Klimko–Nelson’s theorems alternative in the case of conditional least-squares and M-estimators for array time series when the assumptions of almost sure convergence cannot be established. We do not assume stationarity nor even local stationarity. Besides, we provide sufficient conditions for two of the assumptions and a procedure for the evaluation of the information matrix in array time series. In addition to time-dependent models, illustrations to a threshold model and a count data model are given.

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Acknowledgements

Guy Mélard has benefited from a Belgian research grant F.R.S.-FNRS 1.5.261.09. We thank Prof. Nikolaos Kourogenis, University of Piraeus (Greece), for providing the reference of Brown and Eagleson (1971). We thank also Abdelkamel Alj for his comments on the proof of Theorem 2.1. The second author thanks Marcella Niglio (University of Salerno, Italy) and Luisa Bisaglia (University of Padua, Italy) for their invitations to their universities, Marcella Niglio for her remarks, and Konstantinos Fokianos for references and discussions on count data models. This paper was presented at several places by the second author who thanks Aboubacar Amiri, Marianne Clausel, Christian Francq, Marc Hallin, Yacouba Boubacar Mainassara, and Davy Paindaveine, among others. Finally, we are grateful to the Editor and to the two referees (in particular, we appreciated the insistence of one of them), whose suggestions lead to a much better paper.

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Correspondence to Guy Mélard.

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Azrak, R., Mélard, G. Asymptotic properties of conditional least-squares estimators for array time series. Stat Inference Stoch Process 24, 525–547 (2021). https://doi.org/10.1007/s11203-021-09242-8

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Keywords

  • Klimko–Nelson’s theorems
  • Non-stationary process
  • Multivariate time series
  • Time-varying models
  • Information matrix

Mathematics Subject Classification

  • Primary 62M10
  • 60K35
  • secondary 60G12