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Time Series
| Credits |
8 credit points |
| Instructors |
Vaart, A.W. van der (Vrije Universiteit) |
| E-mail |
avdvaart@math.leidenuniv.nl |
| Aim |
Making students familiar with the basic models for stochastic processes indexed by discrete time, especially for financial processes. |
| Description |
A time series is a sequence of random variables ordered according to an integer index, which is usually referred to as "time". This course is an introduction 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. Within the context of nonparametric estimation we discuss the ergodic theorem and extend the central limit theorem to dependent ("mixing") random variables. Spectral theory includes the definition, interpretation and properties of spectral measures, and their estimation from observed time series, using the (smoothed) periodogram. Methods for parameter estimation include least squares and maximum likelihood.
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. We spend time on the existence, stationary and stability of solutions to ARMA and GARCH equations, and formulate theorems on estimation methods, typically in the asymptotic setting of the number of observations tending to infinity. Interested students may experiment on simulated or real time series, for instance using the R package. However, the main focus of the course is on mathematical theory. |
| Organization |
Lectures. |
| Examination |
Written exam. |
| Literature |
Lecture notes by A.W. van der Vaart (freely downloadable from the web page). An expanded list of literature is given in the lecture notes. |
| Prerequisites |
Probability theory, some knowledge of statistics.
Measure-theoretic probability is strongly recommended. |
| Remarks |
http://www.math.vu.nl/~aad/timeseries |
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