Advanced Time Series Models

The approach to time series analysis presented so far relied on a frequentist notion of probability. The aim thereby is to estimate the “true” parameters θ $$\theta $$ of a model from the data (sample). While the data are treated as stochastic, the parameters are considered to be fixed non-stochastic quantities. While probability statements bear on the estimator of θ $$\theta $$ for repeated samples, the Bayesian approach models the uncertainty over the unobserved parameters via an prior distribution. This allows to model explicitly the information, respectively, the perception the researcher has on the parameter. Thereby, the concept of a probability is a subjective one, which does not rely on the idea of a repeated sample. The emphasis in the analysis is then how the observed data modify the prior subjective perception of the researcher. Berger (Statistical decision theory and Bayesian analysis. Springer Series in Statistics, (2nd ed.). Springer Science+Business Media, 1985), Cox (Principles of statistical inference. Cambridge University Press, 2006), Gorroochurn (Classic problems in probability. Wiley, 2012, Kapitel 14), or Diaconis and Skyrms (Ten great ideas about chance. Princeton University Press, Princeton, 2018, Kapitel 6) provide good intelligible introductions to both views, which go beyond its pure statistical significance.