On Adaptive Estimation in Nonstationary Arma Models with Garch Errors

This paper considers adaptive estimation in nonstationary autoregressive moving average models with the noise sequence satisfyinga generalized autoregressive conditional heteroscedastic process. The locally asymptotic quadratic form of the log-likelihood ratio for the model is obtained. It is shown that the limit experiment is neither LAN nor LAMN, but is instead LABF. For the model with symmetric density of the rescaled error, a new efficiency criterion is established for a class of defined $M_{\nu}$-estimators. It is shown that such efficient estimators can be constructed when the density is known. Using the kernel estimator for the score function, adaptive estimators are constructed when the density of the rescaled error is symmetric, and it is shown that the adaptive procedure for the parameters in the conditional mean part uses the full sample without splitting. These estimators are demonstrated to be asymptotically efficient in the class of $M_{\nu}$-estimators. The paper includes the results that the stationary ARMA-GARCH model is LAN, and that the parameters in the model with symmetric density of the rescaled error are adaptively estimable after a reparameterization of the GARCH process. This paper also establishes the locally asymptotic quadratic form of the log-likelihood ratio for nonlinear time series models with ARCH-type errors.

[1]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[2]  P. Protter,et al.  Weak Limit Theorems for Stochastic Integrals and Stochastic Differential Equations , 1991 .

[3]  Anton Schick,et al.  A note on the construction of asymptotically linear estimators , 1987 .

[4]  Oliver Linton,et al.  Adaptive Estimation in ARCH Models , 1993, Econometric Theory.

[5]  Jushan Bai,et al.  ON THE PARTIAL SUMS OF RESIDUALS IN AUTOREGRESSIVE AND MOVING AVERAGE MODELS , 1993 .

[6]  Kreiss Jens-Peter,et al.  On Adaptive Estimation In Autoregressive Models When There Are Nuisance Functions , 1987 .

[7]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[8]  Bas J. M. Werker,et al.  Adaptive estimation in time-series models , 1997 .

[9]  P. Phillips.  Partially Identified Econometric Models , 1988, Econometric Theory.

[10]  P. Robinson.  Semiparametric econometrics: A survey , 1988 .

[11]  D. McLeish.  Dependent Central Limit Theorems and Invariance Principles , 1974 .

[12]  Jens-Peter Kreiss.  On Adaptive Estimation in Stationary ARMA Processes , 1987 .

[13]  Daniel B. Nelson.  Stationarity and Persistence in the GARCH(1,1) Model , 1990, Econometric Theory.

[14]  Weakly Adaptive Estimators in Explosive Autoregression , 1990 .

[15]  P. Jeganathan,et al.  On Asymptotic Inference in Linear Cointegrated Time Series Systems , 1997, Econometric Theory.

[16]  J. Hannan,et al.  On estimation and adaptive estimation for locally asymptotically normal families , 1982 .

[17]  P. Jeganathan.  Some Aspects of Asymptotic Theory with Applications to Time Series Models , 1995, Econometric Theory.

[18]  P. Bickel.  Efficient and Adaptive Estimation for Semiparametric Models , 1993 .

[19]  Andrew A. Weiss,et al.  Asymptotic Theory for ARCH Models: Estimation and Testing , 1986, Econometric Theory.

[20]  P. Hall,et al.  Martingale Limit Theory and its Application. , 1984 .

[21]  Chris Chatfield,et al.  Introduction to Statistical Time Series. , 1976 .

[22]  Shiqing Ling.  On the Probabilistic Properties of a Double Threshold ARMA Conditional Heteroskedastic Model , 1999 .

[23]  K. W.,et al.  LIMITING DISTRIBUTIONS OF MAXIMUM LIKELIHOOD ESTIMATORS FOR UNSTABLE AUTOREGRESSIVE MOVING-AVERAGE TIME SERIES WITH GENERAL AUTOREGRESSIVE HETEROSCEDASTIC ERRORS , 2002 .

[24]  Chris A. J. Klaassen,et al.  Efficient estimation in semiparametric GARCH models , 1996 .

[25]  P. Bickel.  On Adaptive Estimation , 1982 .

[26]  A. Schick.  On Asymptotically Efficient Estimation in Semiparametric Models , 1986 .

[27]  Peter C. B. Phillips,et al.  Optimal Inference in Cointegrated Systems , 1991 .

[28]  Michael McAleer,et al.  A Survey of Recent Theoretical Results for Time Series Models with GARCH Errors , 2001 .

[29]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[30]  Bruce E. Hansen,et al.  Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase Power , 1995, Econometric Theory.

[31]  Le Cam,et al.  Locally asymptotically normal families of distributions : certain approximations to families of distributions & thier use in the theory of estimation & testing hypotheses , 1960 .

[32]  Wai Keung Li,et al.  On Fractionally Integrated Autoregressive Moving-Average Time Series Models with Conditional Heteroscedasticity , 1997 .

[33]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[34]  Anton Schick,et al.  Efficient estimation in nonlinear autoregressive time-series models , 1997 .

[35]  Lucien Le Cam,et al.  Locally Asymptotically Normal Families , 1990 .

[36]  L. Lecam.  On the Assumptions Used to Prove Asymptotic Normality of Maximum Likelihood Estimates , 1970 .

[37]  Anton Schick,et al.  Adaptive estimation in a random coefficient autoregressive model , 1996 .

[38]  J. Hájek.  Local asymptotic minimax and admissibility in estimation , 1972 .

[39]  Michael McAleer,et al.  NECESSARY AND SUFFICIENT MOMENT CONDITIONS FOR THE GARCH(r,s) AND ASYMMETRIC POWER GARCH(r,s) MODELS , 2002, Econometric Theory.

[40]  Michael McAleer,et al.  Estimation and Testing for Unit Root Processes with GARCH (1, 1) Errors: Theory and Monte Carlo Evidence , 2003 .

[41]  K. Do,et al.  Efficient and Adaptive Estimation for Semiparametric Models. , 1994 .

[42]  D. Shin,et al.  UNIT ROOT TESTS BASED ON ADAPTIVE MAXIMUM LIKELIHOOD ESTIMATION , 1999, Econometric Theory.

[43]  R. Engle,et al.  Semiparametric ARCH Models , 1991 .

[44]  G. Kallianpur.  Stochastic Filtering Theory , 1980 .

[45]  Efficient estimation in some missing data problems , 1988 .