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. This course is an introduction for mathematics students to the theory of time series, including prediction theory, spectral (=Fourier) theory, and parameter estimation. Among the 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. State space models include Markov processes and hidden Markov processes. We do not go into much detail in the probabilistic properties of such processes, but methods of parameter estimation apply to such processes and we may discuss prediction through the famous Kalman filter. 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. Many of the procedures that we discuss are implemented in the statistical computer package Splus, and are easy to use. We recommend trying out these procedures, because they give additional insight that is hard to obtain from theory only. A hand-out on Splus is provided. We assume that the audience is familiar with measure theory, and basic concepts of statistics. Knowledge of measure-theoretic probability, stochastic convergence concepts, and Hilbert spaces is recommended. We presume no knowledge of time series analysis. We provide full lecture notes. Two books that cover a part of the course are: R Azencott, D Dacunha-Castelle, 1984, S\'eries d'Observations Irr\'eguli\`eres, Masson, Paris. PJ Brockwell, RA Davis, 1991, Time Series: Theory and Methods, Springer, New York. |