# 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 Xt, the index t in ZZ 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 X1,..., Xn 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 the GARCH, which have become popular models for financial time series. We study the existence of stationary versions of these processes. If time allows we also treat the unit-root problem and co-integration. State space models include Markov processes and hidden Markov processes, with stochastic volatility processes as a special case, popular in finance. Filtering theory, in particular the famous Kalman filter, is an important topic for this processes. The extent of coverage of these topics changes from year to year.

Within the context of nonparametric estimation we extend the central limit theorem to dependent ("mixing") random variables. To treat maximum likelihood we shall develop the martingale central limit theorem.

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 R, 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 R is provided.

We assume that the audience is familiar with measure theory, and basic concepts of statistics. Knowledge of measure-theoretic probability and stochastic convergence concepts (convergence in distribution and probability, Slutsky, Delta-method, CLT) is highly recommended. Knowledge of Hilbert spaces is convenient. 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-Castelle1984,  S\'eries d'Observations Irr\'eguli\`eres,  Masson, Paris.
• PJ Brockwell, RA Davis,  1991, Time Series: Theory and Methods,  Springer, New York.

These books are a bit dated. (For instance, they do not treat GARCH models.) An expanded list of literature is provided with the lecture notes.

## Course spring 2013

 Lecture hour Wednesdays 14.00-16.45, starting February 6, ending May 15 or 22. No Lecture on March 13. Lecture room M607, VU Science Building. EXCEPT on FEBRUARY 20: F647 and MARCH 27 : C121 Office hour On appointment, in 223 MI Leiden University, or KdV University of Amsterdam. Registration Via mastermath. Lecturer Prof.dr. A.W. van der Vaart. Exam Written, Wednesday June 5, 14.00-17.00, VU-Science building (WN), D107, VU Science Building. The exam is on the part of the lectures listed below. You are expected to know the general flow of the course, to be able to formulate and apply theorems (exact for non-technical ones), to be able to solve problems as indicated, to rework examples, and to know the proofs of: 1.27/1.29, 4.4, 4.5, 5.9, 6.11, 8.8+8.10, 8.30, 9.17, 12.1, 12.4. See the example exams below. Retake Exam Written, Wednesday June 26, 14.00-17.00, VU-Medical Faculty (MF) G513. Registration by email to avdvaart at math.leideuniv.nl at the latest one week before the exam is required for admission to this exam. Oral Exam An oral exam on appointment is possible only in exceptional cases, and typically only after first taking a written exam. Grades Grades will be communicated to the mastermath administration, and can be obtained on request by sending an email to the lecturer, approximately two weeks after the exam. Credits 8. Lecture Notes downloadable at beginning or during semester (last corrected and updated April 2.)

To gain better insight in time series we recommend that students perform  some computer simulations. One possibility is the use of R (or its precessor Splus). A short introduction to R/Splus functions that deal with time series can be found here. (This presumes basic knowledge of R/Splus).

## Preliminary course schedule (will be adapted during the semester)

 Week Subject Exercises Hand in 6 Feb Chapter 1. 1.4, 1.9, 1.11, 1.18, 1.29, 1.32, 1.33. 1.7, 1.14, 1.40 13 Feb Sections 2.1-2.6. 2.2, 2.11, 2.12, 2.15, 2.16, 2.27, 2.30. Read Sections 3.1-3.2. 2.9, 2.13, 2.32 20 Feb Sections 3.3, 4.1, 4.2, 4.3, 5.1. 4.6, 4.9 4.2, 4.10. 27 Feb Sections 5.1, (5.2), 3.5 5.5, (5.12) 5.2, (5.13) 6 March Sections 3.6, 5.2, 5.3, 5.4, part of 6.1. 5.14, 6.1, 6.4. 6.8. 13 March NO LECTURE 20 March Sections 6.1, 8.1. 6.13, 6.18, 8.7. 6.14, 6.17. 27 March Sections 8.2, 8.4, 8.5 (self study), 8.6. 8.7, 8.34. 8.12, 8.19. 3 April Sections 8.8, (part of) Section 9.1. 9.6, 9.8 (as in lectures notes of April) 9.1, 9.9. 10 April Sections 9.1, 9.2, 10.1 (not 10.5-10.9), 10.2. 9.18, 10.2. 9.20. 17 April NO LECTURE 24 April Sections 11.1, 11.2 (11.2.1 until Exercise 11.11), 11.3 11.1, 11.11, 11.15. 11.2, 11.21. 1 May Chapter 12 12.2, 12.3. 12.6. 8 May Sections 4.5, 13.1, 13.2. 13.6 13.5 15 May Sections 13.3, 13.4, 13.5. 13.8 none

The exercises are highly recommended. The ones under "hand in" you may hand in for correction, on paper at the beginning of next week's lectures. Doing so may raise your final mark by maximally 1 point: [%correct x 1 point]. All other exercises, even the ones not listed, are recommended as well!