Abstract
A nonstationary simultaneous autoregressive model \({X^{(n)}_k=\alpha \Big(X^{(n)}_{k-1}+X^{(n)}_{k+1}\Big)+\varepsilon_k, k=1, 2, \ldots , n-1}\), is investigated, where \({X^{(n)}_0}\) and \({X^{(n)}_n}\) are given random variables. It is shown that in the unstable case α = 1/2 the least squares estimator of the autoregressive parameter converges to a functional of a standard Wiener process with a rate of convergence n 2, while in the stable situation |α| < 1/2 the estimator is biased but asymptotically normal with a rate n 1/2.
Similar content being viewed by others
References
Baran S, Pap G, Zuijlen MV (2007) Asymptotic inference for unit roots in spatial triangular autoregression. Acta Appl Math 96: 17–42
Besag J (1974) Spatial interaction and the statistical analysis of lattice systems. J R Stat Soc Ser B Stat Methodol 36: 192–236
Besag J (1975) Statistical analysis of non-lattice data. Statistician 24: 179–195
Billingsley P (1968) Convergence of probability measures. Wiley, New York
de Luna X, Genton MG (2002) Simulation-based inference for simultaneous processes on regular lattices. Stat Comput 12: 125–134
Guyon X (1995) Random fields on a network: modeling, statistics and applications. Springer, New York
Hasza DP, Fuller WA (1979) Estimation for autoregressive processes with unit roots. Ann Stat 7: 1106–1120
Jacod J, Shiryaev AN (1987) Limit theorems for stochastic processes. Springer, Berlin
Robinson PM, Vidal Sanz J (2006) Modified Whittle estimation of multilateral models on a lattice. J Multivar Anal 97: 1090–1120
Shiryayev AN (1984) Probability. Springer, New York
Whittle P (1954) On stationary processes in the plane. Biometrika 41: 434–449
Author information
Authors and Affiliations
Corresponding author
Additional information
This research has been supported by the Hungarian Scientific Research Fund under Grant No. OTKA-T079128/2009.
Rights and permissions
About this article
Cite this article
Baran, S., Pap, G. Asymptotic inference for a one-dimensional simultaneous autoregressive model. Metrika 74, 55–66 (2011). https://doi.org/10.1007/s00184-009-0289-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-009-0289-5