Time series

This course is an introduction to the theory of statistical time series with special attention for financial time series.

A/ statistical time series/ is a sequence of random variables $X_t$, the index $t\in Z$ being referred to as ``time''. Thus a time series is a "discrete time stochastic process". Typically the variables are dependent and one aim is to predict the ``future'' given observations $X_1,\ldots, X_n$ on the ``past''. Although the basic statistical concepts apply (such as likelihood, mean square errors, etc.) the dependence gives time series analysis a distinctive flavour. The models are concerned with specifying the time relations, and the probabilistic tools (e.g. the central limit theorem) must go beyond results for independent random variables.

The course is an introduction for mathematics students to the theory of time series, including prediction theory, spectral (=Fourier) theory, and parameter estimation.

Among the special time series models we discuss are the classical ARMA processes, and also the GARCH and stochastic volatility processes, which have become popular models for financial time series. We study the existence of stationary versions of these processes, and, if time allows, also the unit-root problem and co-integration. Time permitting we also discuss state space models, which include Markov processes and hidden Markov processes,
and derive the famous Kalman filter, which is a recursive algorithm to compute predictions.

Within the context of nonparametric estimation we may discuss the ergodic theorem and extend the central limit theorem to dependent ("mixing") random variables. Thus the course is a mixture of probability and statistics, with some Hilbert space theory coming in to develop the spectral theory and the prediction problem.