Confidence Regions for Trends in Time Series:
We study a sieve bootstrap procedure for time series with a deterministic trend. The sieve for constructing the bootstrap is based on autoregressive approximation. Given time series data, one would first 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 fitted autoregressive models. We show the validity of such sieve 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 confidence intervals for the trend, where the simultaneity can be achieved over a range of points which can be chosen by the user. The time series context is substantially different from the independent set-up: 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 finite sample sizes.