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Danish National Research Foundation Network in Mathematical Physics and Stochastics
Funded by The Danish National Research Foundation

MaPhySto Concentrated Advanced Course and

MaPhySto Workshop on

Non-Linear Time Series Modeling

Course: Monday, September 27, 2004 - Thursday, September 30, 2004

Workshop: Friday, October 1, 2004. Speakers only by invitation

at the University of Copenhagen

Main Lectures by

Richard A. Davis (Colorado State University, Ft. Collins)

The following distinguished researchers have agreed to give lecturers:

Peter Brockwell (Colorado State University, Ft. Collins), Alexander Lindner (TU Munich), Philippe Soulier (University Paris X), Catalin Starica (Chalmers University Gothenburg)

The Concentrated Advanced Course and the Workshop will be given at the Institute of Mathematical Sciences, University of Copenhagen, HC Ørsted Institute. See the Infomation below how to get to the Institute. There will be 6 hours of lectures per day, 4 given by Richard A. Davis.

Generous financial support for Richard A. Davis by the Villum Kann Rasmussen Foundation is greatfully acknowledged.

The course is jointly organized by MaPhySto and the Graduate School of Mathematics and its Applications at the Institute of Mathematical Sciences.

The course is organized by Søren Asmussen (University of Aarhus), Henrik Hult (University of Copenhagen), Thomas Mikosch (University of Copenhagen), and Michael Sørensen (University of Copenhagen)

Description of Richard A. Davis's course

Much of the recent interest in time series modeling has focused on data from financial markets, from communications channels, from engineering, and from environmental applications where the need for non-Gaussian, non-linear, and continuous-time models is clear. Another rapidly developing area is the analysis of time series of counts, which has very broad application in view of the host of integer-valued time series which cannot be satisfactorily handled within the classical framework of Gaussian-like series. The rapid advances in the practical application of both continuous-time and discrete-time non-Gaussian and non-linear models has raised a host of interesting theoretical questions as well as suggesting a great many future directions for the practical application of stochastic modeling.

In this course, nonlinear time series models will be developed to model a wide-range of phenomena. These include generalized state space models for modeling time series of counts, GARCH and stochastic volatility models for modeling financial data, and continuous-time models that incorporate long memory. There will also be a discussion on fitting models with structural breaks.

The lectures will be complemented with practical hands-on experience in modeling data sets provided by the presenters.

Structure of Richard A. Davis's course

  • 1. Introduction to linear and nonlinear time series models (4 lectures). Notes in pdf
  • a) Why nonlinear models?

    b) Classification of white noise c) Wold decomposition d) Prediction

    e) Different types of white noise

    f) Tests of white noise

    g) Reversibility

    h) All-pass models

  • 2. Time series models in finance (3 lectures) Notes in pdf
  • a) GARCH models

    i) basic properties

    ii) estimation

    iii) extreme value theory

    iv) properties of sample correlations

    b) Stochastic volatility models

    i) autocorrelation function

    ii) extreme value theory

    iii) estimation

  • 3. Nonlinear and nonGaussian state-space models (4 lectures)Notes in pdf
  • a) Parameter-driven models

    b) Observation-driven models

    c) Background on linear state-space models

    d) Kalman filtering

    e) Estimation for parameter-driven models

    i) importance sampling

    ii) approximate likelihood estimation

    f) Estimation for GLARMA models

  • 4. Piecewise AR models (2 lectures)Notes in pdf
  • a) Minimum description length

    b) Genetic algorithm

    The Concentrated Advanced Course aims at the graduate student in probability theory, statistics, finance, economics, insurance mathematics and the researcher who wants to get an overview of methods and techniques on modeling time series, as well as at the researcher and the practitioner who intend to apply non-linear time series models.

    The course will be accessible for Masters students with a background in statistics, econometrics or time series analysis.

    Abstracts of the lectures will frequently be updated

    Alexander Lindner GARCH processes - probabilistic properties

    The focus of these lectures will be on financial time series models, in particular on the GARCH family and their probabilistic properties. We will discuss

  • existence of a stationary solution
  • Markov properties
  • moment conditions
  • tail behaviour
  • extremal behaviour
  • As continuous-time counterparts of the (discrete-time) GARCH processes we present GARCH diffusion limits and the COGARCH process, the latter being a GARCH type process driven by a Levy process.

    Here do you find the transparencies of the 1st and 2nd Lecture .

    Philippe Soulier Modelling and estimating long memory in non linear time series Slides in pdf

    Gaussian and linear long memory processes have been quite exhaustively studied in the past twenty five years, and a full statistical theory is available for them but most applications require non-Gaussian, non-linear models. In the first lecture, after giving a working definition of long memory, we will present several classes of non-linear long memory processes and review their main probabilistic properties. The second lecture will address the issue of estimating long memory.

    Peter Brockwell Continuous-time ARMA processes

    We will discuss

  • their properties
  • applications
  • non-linear continuous-time ARMA processes
  • related inference problems
  • Catalin Starica 1. Is GARCH as good a model as the Nobel prize accolades would imply?

    We investigate the relevance of the stationary, non-linear, conditional, parametric modeling paradigm embodied by the Garch(1,1) process to describing and forecasting the dynamics of returns of the Standard & Poors 500 (S&P 500) stock market index.

    A detailed analysis of the series of S&P returns featured in the illustration of the use of the Garch(1,1) model in estimating and forecasting volatility given in Section 3.2 of the Advanced Information note on the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel reveals that the Garch(1,1) model severely over-estimated the unconditional variance of returns during the period under study.

    We test and reject the hypothesis that a Garch(1,1) process is the true data generating process of the longer sample of returns on the S&P 500 stock market index between March 4, 1957 and October 9, 2003. We investigate then the alternative use of the Garch(1,1) process as a local, stationary approximation of the data and find that the Garch(1,1) model fails during significantly long periods to provide a good local description to the time series of returns on the S&P 500 and Dow Jones Industrial Average indexes.

    Catalin Starica 2. When did the 2001 recession really start?

    We develop a non-parametric, non-stationary framework for business-cycle dating based on an innovative statistical methodology known as Adaptive Weights Smoothing (AWS). The methodology is used both for the study of the individual macroeconomic time series relevant to the dating of the business cycle as well as for the estimation of their joint dynamic.

    We find evidence of a change in the methodology of the NBER's Business-Cycle Dating Committee: an extended set of five monthly macroeconomic indicators replaced in the dating of the last recession the set of indicators emphasized by the NBER's Business-Cycle Dating Committee in recent decades. This change seems to seriously affect the continuity in the outcome of the dating of business cycle.

    We find that, independent of the set of coincident indicators monitored, the last economic contraction began in November 2000, four months before the date of the NBER's Business-Cycle Dating Committee.

    Programme of the Course

    The Course is followed by a One-Day Workshop on Non-Linear Time Series Modeling

    The lecturers of the MaPhySto Course and some other invited speakers will present results on their most recent research on modeling with non-linear time series.

    Other speakers will be announced later.

    Talks are only by invitation.

    Among others, the speakers will cover the following topics:

  • continuous-time models, continuous ARMA models
  • state-space models in economics, finance and the environment
  • finding structural break points
  • statistics of long-range dependent time series
  • time series of counts with applications in epidemiology and the environment
  • Programme of the Workshop


  • Javier Hidalgo (LSE) and Jens-Peter Kreiss (Braunschweig) Bootstrap specification tests for linear covariance stationary processes. Abstract
  • Philippe Soulier (Paris X) Long memory point processes. Abstract
  • Francois Roueff (TÚlÚcom Paris) Recursive estimation of smoothly time-varying autoregressive processes. Abstract
  • Richard A. Davis (Ft. Collins) and Thomas Mikosch (Copenhagen) Extreme value theory for space-time processes with heavy-tailed distributions. Abstract
  • Michael Neumann (Braunschweig) Doukhan's concept of weak dependence - examples and basic tools. Abstract
  • Peter Brockwell (Ft. Collins) Fractionally integrated ARMA processes with continuous time parameter Abstract
  • Alexander Lindner (Munich) Stationarity of generalised Ornstein-Uhlenbeck processes Abstract
  • Registration

    There will be a regular registration fee of 500 DKK (for Danes) or 70 Euros (for non-Danes) for all participants of the Course. The fee covers lunches from Monday to Friday, coffee and cake during the coffee breaks.

    Participation in the Workshop is free. It is possible to order lunch on Friday, 1 October, 2004, for a fee of 100 DKK (for Danes) or 15 Euros (for non-Danes). If you do not need lunch, please register anyway.

    The participants are expected to have their expenses covered by their home institutions or from other sources.

    We intend to have an excursion to some sights in the neighborhood of Copenhagen on Wednesday afternoon, followed by a dinner in the center of Copenhagen. We intend to have a bus excursion to North Sealand, the island on which Copenhagen is located. Among others, we will visit Louisiana, the world famous museum of modern arts, located in a charming park close to the sea. There will be extra charges for the excursion and the dinner to be paid on arrival at the conference site.

    Please register via the registration form at your earliest convenience but before August 31, 2004.

    The programmes of the Course and the Workshop will be on the web after September 1, 2004.

    The course will start on Monday, 9:15, and stop on Thursday, 16:30. Similar times apply to the workshop on Friday: 9:00-17:00.

    More Information

    We have a page with information on how to get to the HC Ørsted Institute, where the Course will be given. Here is a map with the HC Ørsted Institute.

    Do not hesitate to contact the local organizers (hult@math.ku.dk) for more information.

    This document was last modified September 24, 2004. Questions or comments to the contents of this document should be directed to hult@math.ku.dk.