Schimek, Michael G. (2003), Semiparametric Regression Smoothing and Feature Detection, Computing Science and Statistics, 35, I2003Proceedings/SchimekMichael/SchimekMichael.paper.pdf ,
Let us assume a regularly spaced time series with white noise or weak stationary autoregressive errors of known order. Correct nonparametric estimation of a smooth trend in a series of dependent observations asks for a regression technique which takes care of the specific error structure. Such an approach has been given in Schimek and Schmaranz (1994). Here we extend it to a semiparametric concept in which one or more artificial dummy input series are introduced to analyze certain features in the time series which are of interest beyond long-term trend. Unbiased partial spline fitting (Schimek, 2002) is a recent approach to evaluate such a model. An open problem since Schimek and Schmaranz (1994) is the choice of the degree of smoothing, more so in the semiparametric context where dummy (feature) testing is involved. Whilst the trend is fitted nonparametrically by a smoothing spline, features are tested parametrically via a simple approximately N(0,1) or F-distributed test statistic.
Criteria are derived and estimates given to calculate the significance trace (introduced in a nonparametric setting by Azzalini and Bowman, 1993). The p-values are plotted as a function of the smoothing parameter across a typical range of values. Doing so we can study test significance as a function of the degree of nonparametric smoothing, thus having a systematic handle for feature detection. The main complication here is computational costs. Finally we illustrate the approach on a real example.