The goal of this course is to introduce and explain the statistical methods used for the analysis and forecasting of certain financial time series. This domain of applications have given rise to substantial modeling efforts in the last decades, which allow one to consider many types of financial time series (price returns, rates, transactions data) : linear time series, conditionally heteroscedastic time series, multivariate time series, discrete time series and so on. The main classes of linear and non-linear models will be introduced as well as the statistical methods associated to them. The main prerequisites to attend this course are the bases of linear algebra, Hilbert geometry, probability and statistics. It is assumed that the students have a good knowledge of linear models for univariate time series such as ARMA processes.
Topic: Stochastic autoregressive processes
- Introduction, the basic models (ARMA, random walk, Mart. diff., Markov chains).
- Financial time series and their characteristics.
- Conditional volatility, (G)ARCH models, Stoch. vol. models.
- Other non-linear models, time series of counts.
Topic: From univariate to multivariate time series.
- Covariance operator, coherence function of multivariate time series.
- Granger causality, VARMA models.
- Dynamic linear models, filtering, forecasting, smoothing, likelihood, examples.
Topic: From stationary to non-stationary models.
- Increment stationary models, Integration order, ARIMA, ARFIMA processes.
- Unit root and detection of a change in the mean.
 R. Douc, E. Moulines, and D. S. Stoffer. Nonlinear time series. Chapman & Hall/CRC Texts in Statistical Science Series. Chapman & Hall/CRC, Boca Raton, FL, 2014. Theory, methods, and applications with R examples.
 H. L¨utkepohl. New Introduction to Multiple Time Series Analysis. Cambridge University Press, 2005.
 R. Shumway and D. Stoffer. Time Series Analysis and Its Applications. New York : Springer, 3rd edition, 2011.
 R. Tsay. Analysis of Financial Time Series, volume 543. Wiley-Interscience, 2005.
 R. S. Tsay. Multivariate time series analysis. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2014. With R and financial applications.