Abstract
The limiting distribution of the normalized periodogram ordinate is used to test for unit roots in the first-order autoregressive model Ζst=α Ζs-1,t+βΖs,t-1-αβ Ζs-1,t-1+ɛst. Moreover, for the sequence α n = e c/n, β n = e d/n of local Pitman-type alternatives, the limiting distribution of the normalized periodogram ordinate is shown to be a linear combination of two independent chi-square random variables whose coefficients depend on c and d. This result is used to tabulate the asymptotic power of a test for various values of c and d. A comparison is made between the periodogram test and a spatial domain test.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akidi, Y. (1995). Periodogram analysis for unit roots, Ph.D. Dissertation, Department of Statistics, North Carolina State University, Raleigh, North Carolina.
Basu, S. and Reinsel, G. C. (1994). Regression models with spatially correlated errors, J. Amer. Statist. Assoc., 89, 88–99.
Bhattacharyya, B. B., Khalil, T. M. and Richardson, G. D. (1996). Gauss-Newton estimation of parameters for a spatial autoregression model, Statist. Probab. Lett., 28, 173–179.
Bhattacharyya, B. B., Richardson, G. D. and Franklin, L. A. (1997). Asymptotic inference for near unit roots in spatial autoregression, Ann. Statist., 25, 1709–1724.
Bickel, P. J. and Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic processes and some applications, Ann. Math. Statist., 42, 1656–1670.
Billingsley, P. (1968). Convergence of Probability Measures, Wiley, New York.
Bobkoski, M. J. (1983). Hypothesis testing in nonstationary time series, Ph.D. Dissertation, Department of Statistics, University of Wisconsin, Madison, Wisconsin.
Breiman, L. (1968). Probability, Addison-Wesley, Reading, Massachusetts.
Chan, N. H. and Wei, C. Z. (1987). Asymptotic inference for nearly nonstationary AR(1) processes, Ann. Statist., 15, 1050–1063.
Cullis, B. R. and Gleeson, A. C. (1991). Spatial analysis of field experiments—an extension to two dimensions, Biometrics, 47, 1449–1460.
Dickey, D. A. and Fuller, W. A. (1979). Distribution of the estimators for autoregressive time series with a unit root, J. Amer. Statist. Assoc., 74, 427–431.
Dickey, D. A. and Fuller, W. A. (1981). Likelihood ratio statistics for autoregressive time series with a unit root, Econometrica, 49, 1057–1072.
Hobson, E. W. (1957). The Theory of Functions of a Real Variable and the Theory of Fourier Series, Dover Publications, New York.
Phillips, P. C. B. (1987). Towards a unified asymptotic theory for autoregression, Biometrika, 74, 535–547.
Yeh, J. (1963). Cameron-Martin translation theorems in the Wiener space of functions of two variables, Trans. Amer. Math. Soc., 107, 409–420.
Author information
Authors and Affiliations
About this article
Cite this article
Bhattacharyya, B.B., Li, X., Pensky, M. et al. Testing for Unit Roots in a Nearly Nonstationary Spatial Autoregressive Process. Annals of the Institute of Statistical Mathematics 52, 71–83 (2000). https://doi.org/10.1023/A:1004184932031
Issue Date:
DOI: https://doi.org/10.1023/A:1004184932031