The advances in information technology and survey methods have increased the availability of intra-daily, daily and weekly time series. Seasonality is perhaps the most prominent features of time series that are observed at the sub-annual frequency. Modelling seasonality in high frequency time series serves a variety of analytical purposes, including seasonal adjustment and forecasting. It also poses several important challenges. First of all the seasonal period of the annual and the monthly cycles are neither constant nor integral. Secondly, in order to accommodate abrupt seasonal patterns, several harmonic cycles are needed to complement the fundamental one or several season-specific individual effects are required. This poses variable selection or regularization problems of the kind that are typical of high-dimensional inferential settings. Thirdly, the need for methods that are robust to outliers is reinforced by the fact that the effects of outlying observations are not smoothed by temporal aggregation and that they are relatively more frequent. The robustness can be enforced either by an outlier detection procedure or by robust filtering methods. The tutorial aims at reviewing the solutions that have been provided by the literature and at exposing some of the challenges open to further research. In particular, it focuses on parametric and semiparametric models of seasonality within an unobserved components framework, where the seasonal component is estimated along with other components.

Prediction has been traditionally approached via a model-based paradigm, i.e., (a) fit a model to the data at hand, and (b) use the fitted model in order to extrapolate/predict future data. Due to both mathematical and computational constraints, 20th century statistical practice focused mostly on parametric models. Fortunately, with the advent of widely accessible powerful computing in the late 1970s, computer-intensive methods such as the bootstrap and cross-validation have freed practitioners from the limitations of parametric models. Nevertheless, there is a further step one may take, i.e., going beyond even nonparametric models. The Model-Free Prediction Principle is based on the simple notion of transforming a complex dataset to one that is easier to work with, e.g., i.i.d. or Gaussian. Coupled with resampling, the Model-Free Prediction Principle further allows us to go beyond point prediction in order to construct frequentist prediction intervals without resort to unrealistic assumptions such as normality or linearity. The tutorial will focus on comparing Model-based to Model-Free prediction in the case of time series data.