The Annals of Statistics

Adaptive estimation in time-series models

Feike C. Drost, Chris A. J. Klaassen, and Bas J. M. Werker

Full-text: Open access


In a framework particularly suited for many time-series models we obtain a LAN result under quite natural and economical conditions. This enables us to construct adaptive estimators for (part of) the Euclidean parameter in these semiparametric models. Special attention is directed to group models in time series with the important subclass of models with time varying location and scale. Our set-up is confronted with the existing literature and, as examples, we reconsider linear regression and ARMA, TAR and ARCH models.

Article information

Ann. Statist., Volume 25, Number 2 (1997), 786-817.

First available in Project Euclid: 12 September 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62F12: Asymptotic properties of estimators 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

LAN in time series adaptive estimation semiparametrics


Drost, Feike C.; Klaassen, Chris A. J.; Werker, Bas J. M. Adaptive estimation in time-series models. Ann. Statist. 25 (1997), no. 2, 786--817. doi:10.1214/aos/1031833674.

Export citation


  • BICKEL, P. J. 1982. On adaptive estimation. Ann. Statist. 10 647 671. Z.
  • BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. 1993. Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press. Z.
  • BOLLERSLEV, T. 1986. Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31 307 327. Z.
  • CHAN, K. S., PETRUCCELLI, J. D., TONG, H. and WOOLFORD, S. W. 1985. A multiple-threshold Z. AR 1 model. J. Appl. Probab. 22 267 279. Z.
  • DROST, F. C. and KLAASSEN, C. A. J. 1997. Efficient estimation in semiparametric GARCH models. J. Econometrics. To appear. Z.
  • DROST, F. C., KLAASSEN, C. A. J. and WERKER, B. J. M. 1994. Adaptiveness in time-series Z. models. Asy mptotic Statistics P. Mandl and M. Huskova, eds. 203 211. physica, New ´ York. Z.
  • ENGLE, R. F. 1982. Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50 987 1008. Z.
  • ENGLE, R. F. and GONZALEZ-RIVERA, G. 1991. Semiparametric ARCH models. Journal of ´ Business and Economic Statistics 9 345 359. Z.
  • HAJEK, J. 1970. A characterization of limiting distributions of regular estimates. Z. Wahrsch. ´ Verw. Gebiete 14 323 330.
  • HAJEK, J. and SIDAK, Z. 1967. Theory of Rank Tests. Academia, Prague. ´ ´ Z.
  • HALL, P. and HEy DE, C. C. 1980. Martingale Limit Theory and Its Application. Academic Press, New York. Z.
  • JEGANATHAN, P. 1995. Some aspects of asy mptotic theory with applications to time series models. Econometric Theory 11 818 887. Z.
  • KLAASSEN, C. A. J. 1987. Consistent estimation of the influence function of locally asy mptotically linear estimates. Ann. Statist. 15 1548 1562. Z.
  • KOUL, H. L. and SCHICK, A. 1995. Efficient estimation in nonlinear time series models. Working paper, Michigan State Univ. and State Univ. New York, Binghamton. Z.
  • KREISS, J. P. 1987a. On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112 133. Z.
  • KREISS, J. P. 1987b. On adaptive estimation in autoregressive models when there are nuisance functions. Statist. Decisions 5 112 133. Z.
  • LINTON, O. 1993. Adaptive estimation in ARCH models. Econometric Theory 9 539 569. Z. Z.
  • NELSON, D. B. 1990. Stationarity and persistence in the GARCH 1, 1 model. Econometric Theory 6 318 334. Z.
  • NEWEY, W. K. 1990. Semiparametric efficiency bounds. J. Applied Econometrics 5 99 135. Z.
  • POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, New York. Z.
  • POTSCHER, B. M. 1995. Comment on ``Adaptive estimation in time series regression models'' by ¨ D. G. Steigerwald. J. Econometrics 66 123 129. Z.
  • ROBINSON, P. M. 1988. Semiparametric econometrics: a survey. J. Applied Econometrics 3 35 51. Z.
  • SCHICK, A. 1986. On asy mptotically efficient estimation in semiparametric models. Ann. Statist. 14 1139 1151. Z.
  • SCHICK, A. 1987. A note on the construction of asy mptotically linear estimates. J. Statist. Plann. Inference 16 89 105. Z.
  • STOKER, T. M. 1991. Lectures on Semiparametric Econometrics. CORE Foundation, Louvain la Neuve, Belgium. Z.
  • STEIGERWALD, D. G. 1992. Adaptive estimation in time-series regression models. J. Econometrics 54 251 275. Z.
  • STEIGERWALD, D. G. 1995. Reply to B. M. Potscher's comment on ``Adaptive estimation in time ¨ series regression models.'' J. Econometrics 66 131 132. Z.
  • TONG, H. and LIM, K. S. 1980. Threshold autoregression, limit cy cles and cy clical data. J. Roy. Statist. Soc. Ser. B 42 242 292.
  • WEFELMEy ER, W. 1994. Improving maximum quasi-likelihood estimators. Asy mptotic Statistics Z. P. Mandl and M. Huskova, eds. 467 474. physica, New York. ´ Z.
  • WEFELMEy ER, W. 1996. Quasi-likelihood models and optimal inference. Ann. Statist. 24 405 422. Z.
  • WEISS, A. A. 1986. Asy mptotic theory for ARCH models: estimation and testing. Econometric Theory 2 107 131.