Abstract
This paper deals with inference in a class of stable but nearly-unstable processes. Autoregressive processes are considered, in which the bridge between stability and instability is expressed by a time-varying companion matrix \(A_{n}\) with spectral radius \(\rho (A_{n}) < 1\) satisfying \(\rho (A_{n}) \rightarrow 1\). This framework is particularly suitable to understand unit root issues by focusing on the inner boundary of the unit circle. Consistency is established for the empirical covariance and the OLS estimation together with asymptotic normality under appropriate hypotheses when A, the limit of \(A_n\), has a real spectrum, and a particular case is deduced when A also contains complex eigenvalues. The asymptotic process is integrated with either one unit root (located at 1 or \(-1\)), or even two unit roots located at 1 and \(-1\). Finally, a set of simulations illustrate the asymptotic behavior of the OLS. The results are essentially proved by \(L^2\) computations and the limit theory of triangular arrays of martingales.
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Notes
Double indexing is customary to such representations: \(X_{n,\, k}\) is the kth observation of a time series of size n (apart from the initial value). The triangular form of the process is \(\{ X_{1,\, 0} X_{1,\, 1} \}, \{ X_{2,\, 0}, X_{2,\, 1}, X_{2,\, 2} \}, \ldots , \{ X_{n,\, 0}, \ldots , X_{n,\, n} \}\).
To be rigorous, one should write \(\widehat{\theta }_{n,\, n}\) instead of \(\widehat{\theta }_{n}\) to emphasize that the OLS is a function of \(X_{n,\, 0}, \ldots , X_{n,\, n}\). Similarly, \(S_{n}\) will be used for \(S_{n,\, n}\) (and \(T_{n}\) for \(T_{n,\, n}\), etc.) to lighten the notation when no confusion can arise.
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Acknowledgements
The authors sincerely thank the anonymous reviewer and the associate editor for their comments and references which have clearly contributed to the improvement of the paper. This research benefited from the support of the ANR project ‘Efficient inference for large and high-frequency data’ (ANR-21-CE40-0021).
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Badreau, M., Proïa, F. Consistency and asymptotic normality in a class of nearly unstable processes. Stat Inference Stoch Process 26, 619–641 (2023). https://doi.org/10.1007/s11203-023-09290-2
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DOI: https://doi.org/10.1007/s11203-023-09290-2
Keywords
- Nearly unstable autoregressive process
- OLS estimation
- Asymptotic behavior
- Companion matrix
- Unit root
- Martingales