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We consider a time series model involving a fractional stochastic component,whose integration order can lie in the stationary/invertible or nonstationaryregions and be unknown, and an additive deterministic component consisting of ageneralized polynomial. The model can thus incorporate competing descriptionsof trending behavior. The stationary input to the stochastic component hasparametric autocorrelation, but innovation with distribution of unknown form.The model is thus semiparametric, and we develop estimates of the parametriccomponent which are asymptotically normal and achieve an M-estimationefficiency bound, equal to that found in work using an adaptive LAM/LANapproach. A major technical feature which we treat is the effect of truncatingthe autoregressive representation in order to form innovation proxies. This isrelevant also when the innovation density is parameterized, and we provide aresult for that case also. Our semiparametric estimates employ nonparametricseries estimation, which avoids some complications and conditions in kernelapproaches featured in much work on adaptive estimation of time series models;our work thus also contributes to methods and theory for nonfractional timeseries models, such as autoregressive moving averages. A Monte Carlo study offinite sample performance of the semiparametric estimates is included.

Efficiency improvements in inference on stationary and nonstationary fractional time series