On unified model selection for stationary and nonstationary short- and long-memory autoregressive processes

SUMMARY The question of model choice for the class of stationary and nonstationary, fractional and nonfractional autoregressive processes is considered. This class is defined by the property that the dth difference, for -2 < d < oo, is a stationary autoregressive process of order p0 < 00. A version of the Akaike information criterion, AIC, for determining an appropriate autoregressive order when d and the autoregressive parameters are estimated simultaneously by a maximum likelihood procedure (Beran, 1995) is derived and shown to be of the same general form as for a stationary autoregressive process, but with d treated as an additional estimated parameter. Moreover, as in the stationary case, this criterion is shown not to provide a consistent estimator of p0. The corresponding versions of the BIC of Schwarz (1978) and the HIC of Hannan & Quinn (1979) are shown to yield consistent estimators of po. The results provide a unified treatment of fractional and nonfractional, stationary and integrated nonstationary autoregressive models.

[1]  R. Bhansali,et al.  Convergence of Moments of Least Squares Estimators for the Coefficients of an Autoregressive Process of Unknown Order , 1991 .

[2]  J. Beran.  Fitting long-memory models by generalized linear regression , 1993 .

[3]  S. Koreisha,et al.  A Comparison among Identification Procedures for Autoregressive Moving Average Models , 1991 .

[4]  M. Taqqu,et al.  Large-Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series , 1986 .

[5]  Yoshihiro Yajima,et al.  ON ESTIMATION OF LONG-MEMORY TIME SERIES MODELS , 1985 .

[6]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1972 .

[7]  Sergio G. Koreisha,et al.  The identification of ARMA models , 1990 .

[8]  R. Bhansali,et al.  Some properties of the order of an autoregressive model selected by a generalization of Akaike∘s EPF criterion , 1977 .

[9]  R. J. Bhansali,et al.  Consistent Recursive Estimation of the Order of an Autoregressive Moving Average Process , 1991 .

[10]  A. Raftery,et al.  Space-time modeling with long-memory dependence: assessing Ireland's wind-power resource. Technical report , 1987 .

[11]  H. Akaike.  A Bayesian extension of the minimum AIC procedure of autoregressive model fitting , 1979 .

[12]  Herman J. Bierens,et al.  Testing stationarity and trend stationarity against the unit root hypothesis , 1993 .

[13]  R. Dahlhaus.  Efficient parameter estimation for self-similar processes , 1989, math/0607078.

[14]  R. Shibata.  Selection of the order of an autoregressive model by Akaike's information criterion , 1976 .

[15]  C. Granger,et al.  AN INTRODUCTION TO LONG‐MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING , 1980 .

[16]  S. Koreisha,et al.  New approaches for determining the degree of differencing necessary to induce stationarity in ARIMA models , 1993 .

[17]  Ruey S. Tsay,et al.  Order Selection in Nonstationary Autoregressive Models , 1984 .

[18]  B. G. Quinn,et al.  The determination of the order of an autoregression , 1979 .

[19]  Jan Beran,et al.  Maximum Likelihood Estimation of the Differencing Parameter for Invertible Short and Long Memory Autoregressive Integrated Moving Average Models , 1995 .

[20]  D. Surgailis,et al.  A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate , 1990 .

[21]  J. Stock.  Deciding between I(1) and I(0) , 1992 .

[22]  R. J. Bhansali,et al.  A derivation of the information criteria for selecting autoregressive models , 1986, Advances in Applied Probability.

[23]  George E. P. Box,et al.  Time Series Analysis: Forecasting and Control , 1977 .

[24]  E. Hannan.  The Estimation of the Order of an ARMA Process , 1980 .

[25]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[26]  R. Bhansali.  Recursive Order Selection for an Arma Process , 1994 .

[27]  Peter C. B. Phillips,et al.  Posterior Odds Testing for a Unit Root with Data-Based Model Selection , 1994, Econometric Theory.

[28]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .