Confidence Regions for Trends in Time Series: a Simultaneous Approach with a Sieve Bootstrap

We study a sieve bootstrap procedure for time series with a deterministic trend. The sieve for constructing the bootstrap is based on autoregressive a p p r o ximation. Given time series data, one would rst use a preliminary estimate of the trend of the underlying time series and then approximate the noise process by a large autoregressive model of increasing order as the sample size grows. The bootstrap scheme is based on resampling estimated innovations of tted autoregressive models. We show t h e v alidity of such s i e v e bootstrap approximations for the limiting distribution of linear trend estimators, such as general regression predictors or kernel smoothers. This bootstrap scheme can then be used to construct simultaneous conn-dence intervals for the trend, where the simultaneity c a n b e a c hieved over a range of points which can be chosen by the user. The time series context is substantially diierent from the independent setup: methods from the independent, adapted to the dependent case, seem to loose much of their accuracy. Our resampling procedure yields satisfactory results in a simulation study for nite sample sizes.