**Contents**

The course is an introduction to the theory of time series, including prediction theory, spectral (=Fourier) theory, and parameter estimation. A time series is a sequence of random variables ordered according to an integer index, which is usually referred to as “time”. 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, 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.

**Prerequisites**

Basic concepts of probability and measure theory, for instance at the level of the Measure Theoretic Probability course. Basic knowledge of statistics (completion of the Asymptotic Statistics course is useful, but is not assumed).

**Literature**

The course is based on the lecture notes written by prof. dr. A.W. van der Vaart.

**R**

The main focus of the course is on mathematical theory. However, interested students are encouraged to experiment on simulated or real time series using R (a free software environment for statistical computing and graphics). This gives additional insight that is hard to obtain from theory only.

Since the R interface is rather basic, you might also consider installing RStudio, a free and open source integrated development environment for R (runs under Windows, Mac, or Linux). A handy tool included in R, that enables embedding the R code within LaTeX documents to generate a pdf file with analyses, graphics and the results of computations is Sweave. Sweave is fully supported by RStudio.

**People**

Lectures by Shota Gugushvili, homework grading by Dong Yan.

**Office hour**

No fixed office hour. Send an email beforehand.

**Schedule**

Spring semester, Wednesdays 14:00 – 16:45. First lecture on February 4.

**Location**

VU, Mathematics and Physics (W&N) Building, De Boelelaan 1085, room C121; see a map of the VU campus and travel directions.

**Homework**

Homework assignments are due in one week, at the beginning of the lecture. It is allowed to work in pairs (in fact it is strongly encouraged to do so). Homework may be submitted electronically as a pdf file. In that case please email it to the assistant (address above).

**Exam**

3 June 2014, 14:00-17:00, Room Q112.

24 June 2014, 14:00-17:00, F612.

The exam is about all the material covered in the course. For a passing grade you need to know the basic definitions (e.g. of various types of time series, spectral theory etc.) and results and how to use them to solve the problems. The exam level corresponds approximately to the level of homework assignments, so that working through exercises in the lecture notes is recommended. You are expected to be able to reproduce the exact statements and detailed proofs of the following results (numbering refers to the lecture notes): 4.4, 4.5, 6.2, 8.10, 8.30.

**Grading**

The final grade is a combination of the results of the homework assignments (30%) and the written exam (70%).

**Useful links**

Webpage Time Series

Course homepage from the previous years

Mastermath website