ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat

Linear Time Series


FRANCQ Christian

Department: Finance


The first part of this course is devoted to studying univariate time series: first we present the principal statistical concepts, then estimation methods and tests; we examine the non-stationarity problem by studying the main unit root tests from a practical angle. The course is illustrated with practical examples. The second part of the course is devoted to studying stationary VAR models: we briefly present the general framework of multivariate stationary series before developing the specific case of VAR models. Finally, we take a quick look at the principles of cointegration.



  1. General remarks on second-order univariate stationary processes- Auto-covariances, partial and inverse auto-correlations, spectral density, innovation process. Statement of Wold's theorem. Statement of the asymptotic properties of empirical moments.
  2. AR, MA, ARMA, ARIMA processes- Definitions, canonical representation, properties, concept of initial condition, unit root tests. Identification, estimation and tests, forecasting.
  3. Stationary vector processes -Formal framework for studying these models, canonical representation and Wold representation, innovation process, stationary VAR models. Estimation, forecasting, causality tests, impulse-response function.
  4. Non-stationary vector processes and definition of cointegration -Cointegrated non-stationary VAR models and error-correction models. Estimation in a cointegrated VAR model. Cointegration tests: practical use.



Brockwell P.J., R.A. Davis : Time Series : Theory and Methods, Springer Verlag [24 BRO 00 A]
Gouriéroux C., A. Monfort Séries Temporelles et Modèles Dynamiques, Economica [24 GOU 00 B]
Hamilton J.D. Time Series Analysis, Princeton Univ. Press [24 HAM 00 A]
Lutkepohl H. Introduction to Multiple Time Series Analysis, Springer Verlag. [24 LUT 00 B]
Johansen S. Likelihood Based Inference in Cointegrated Vector Auto-Regression Models, Oxford University Press. [28 JOH 02 A]