## The Annals of Statistics

### Efficiency improvements in inference on stationary and nonstationary fractional time series

P. M. Robinson

#### Abstract

We consider a time series model involving a fractional stochastic component, whose integration order can lie in the stationary/invertible or nonstationary regions and be unknown, and an additive deterministic component consisting of a generalized polynomial. The model can thus incorporate competing descriptions of trending behavior. The stationary input to the stochastic component has parametric autocorrelation, but innovation with distribution of unknown form. The model is thus semiparametric, and we develop estimates of the parametric component which are asymptotically normal and achieve an M-estimation efficiency bound, equal to that found in work using an adaptive LAM/LAN approach. A major technical feature which we treat is the effect of truncating the autoregressive representation in order to form innovation proxies. This is relevant also when the innovation density is parameterized, and we provide a result for that case also. Our semiparametric estimates employ nonparametric series estimation, which avoids some complications and conditions in kernel approaches featured in much work on adaptive estimation of time series models; our work thus also contributes to methods and theory for nonfractional time series models, such as autoregressive moving averages. A Monte Carlo study of finite sample performance of the semiparametric estimates is included.

#### Article information

Source
Ann. Statist., Volume 33, Number 4 (2005), 1800-1842.

Dates
First available in Project Euclid: 5 August 2005

https://projecteuclid.org/euclid.aos/1123250230

Digital Object Identifier
doi:10.1214/009053605000000354

Mathematical Reviews number (MathSciNet)
MR2166563

Zentralblatt MATH identifier
1078.62096

#### Citation

Robinson, P. M. Efficiency improvements in inference on stationary and nonstationary fractional time series. Ann. Statist. 33 (2005), no. 4, 1800--1842. doi:10.1214/009053605000000354. https://projecteuclid.org/euclid.aos/1123250230

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