We propose a sieve bootstrap procedure for time series with a deterministic trend. The sieve for constructing the bootstrap is based on nonparametric trend estimation and autoregressive approximation for some noise process. The bootstrap scheme itself does i.i.d. resampling of estimated innovations from fitted autoregressive models.
We show the validity and indicate second-order correctness of such sieve bootstrap approximations for the limiting distribution of nonparametric linear smoothers. The resampling can then be used to construct nonparametric confidence intervals for the underlying trend. In particular, we show asymptotic validity for constructing confidence bands which are simultaneously within a neighborhood of size in the order of the smoothing bandwidth.
Our resampling procedure yields satisfactory results in a simulation study for finite sample sizes. We also apply it to the longest series of total ozone measurements from Arosa (Switzerland) and find a significant decreasing trend.
"Sieve bootstrap for smoothing in nonstationary time series." Ann. Statist. 26 (1) 48 - 83, February 1998. https://doi.org/10.1214/aos/1030563978