A time series is a sequence of random variables ordered according to an integer index, which is usually referred to as "time". The course aims at making students familiar with mathematical fundamentals of times series analysis and in this context covers topics such as prediction theory, spectral theory, and parameter estimation. Among the time series models we discuss are the classical ARMA processes, as well as the GARCH processes, which have become popular models for financial time series. Within the context of nonparametric estimation we extend the central limit theorem from the i.i.d. setting to dependent 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 existence, stationarity 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.
Basic concepts of probability and measure theory, for instance at the level of the Measure Theoretic Probability (http://www.math.leidenuniv.nl/~gugushvilis/mtp.html) course. A summary of useful results is provided in the lecture notes.
Basic knowledge of statistics, for instance at the level of L. Wasserman, All of Statistics, Chapters 6, 9-10, 13 (http://www.springer.com/gp/book/9780387402727). Completion of the Asymptotic Statistics (http://www.mastermath.nl/program/Fall_2013/Asymptotic_Statistics/) course is useful, but is not assumed.
- Docent: Shota Gugushvili