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Statistics for stochastic processes


Statistics for stochastic processes


Academic year 2017/2018

Course ID
Teaching staff
Prof. Elvira Di Nardo
Joseph Rinott
1st year
Teaching period
Second semester
D.M. 270 TAF B - Distinctive
Course disciplinary sector (SSD)
MAT/06 - probabilita' e statistica matematica
Formal authority
Type of examination
Good knowledge of probability theory and the basics of stochastic processes. In more details you will need
- laws of large numbers and central limit theorems
- measure theory
- conditional expectations
- L^p spaces with respect to a probability measure
- Hilbert spaces (some introductory material on this topic is present in the text books)

Sommario del corso


Course objectives

The goal of lectures is to introduce statistical inference for time series taking into account both the theoretical/mathematical aspects and their practical application to data analysis.

Time series are considered, aiming to characterize properties, asymptotic behavior, estimations and forecasting,  spectral analysis as well as decomposition in trend and seasonal components. Such concepts are applied to the analysis of simulated data or existing databases in order to infer and validate a model supporting the data.


Results of learning outcomes

At the end of the course, students will have understood how to model time series with focus on forecasting and estimation of the moments, of the spectrum and of the parameters of time series models. 

 Moreover they will know which are the main steps of the analysis of a dataset, and which tools are available to this aim:
- descriptive statistics, moment and spectrum estimation
- formulation of models, parameter estimation, model selection, model verification
- forecasting



Course delivery

We will mainly deliver frontal lectures, but a computer lab is also included. During the lectures we will alternate a formal presentation of some topics, including proofs and technical details, with a more informal part where we will introduce some concepts that are useful for the analysis of data sets. In the computer lab we will use R to simulate and analyse datasets from ARMA processes or existing databases. We refer to some particular packages useful to deal with simulations, decompositions and forecasting.


Learning assessment methods

Who wants to be examined on the syllabus of 

a.a.<2015/16: send an e-mail to Elvira Di Nardo, one week before the practical session, to organize the methods

a.a.=2015/16: a practical session on the analysis of a dataset in the computer lab is followed by writing a short essay on one of the arguments introduced  by Prof. Sirovich. The final evaluation with a regular oral examination will be after the correction of this essay and the analysis in the computer lab a couple of days later.

a.a.=2016/17: a practical session on the analysis of a dataset in the computer lab is followed by writing a short essay on one of the arguments introduced  by Prof. Rinott. The final evaluation with a regular oral examination will be after the correction of this essay and the analysis in the computer lab a couple of days later.


Support activities

Computer lab.



Time series: weak and strong stationarity. Autocovariance and autocorrelation functions: characterizations. Linear processes. Time invariant linear filters: estimation of trend and seasonal component. Backward shift and difference operators. Not uniqueness of modeling. Invertibility. Q-dependence and q-correlation: Moving Average models of order q. Causality: AutoRegressive models of order p and their multivariate representation as AR(1) model. ARMA models. How to construct ARMA models whose solution is causal, invertible and not redundant. Autocovariance function of ARMA models and homogeneous linear difference equations. Yule-Walker equations for AR(p) model. Yule-Walker estimators. Partial autocorrelation function. Forecasting: conditional mean and best linear predictor. The n-step ahead predictor when the covariance matrix is non-singular and singular. Perfect predictable time series and the Wold decomposition. Choosing p and q from data:  Akaike's criterion. The ARIMA procedure in R: forecasting and estimation by suing innovations. Spectral representation of simple processes. Herglotz Theorem. Spectral density, the relation to characteristic functions and their inversion in probability. Computing the spectral density for ARMA models. Applying the spectral density to obtain causal invertible models. Stochastic integrals: definition, existence, examples, properties, relation to spectral distributions. Spectral representation of stationary processes by stochastic integrals and applications to prediction in ARMA. Estimation of the mean, the covariance, the partial autocorrelation. Estimation of the parameters and model selection. Diagnostic tools. Asymptotic theory: m-dependent variables and the associated CLT. 

Computer lab: simulation and statistical analysis of time series with R.


Suggested readings and bibliography


Lectures will not adhere to the material of any single text, but the students can find material on the topics we teach on different books. References for each topic will be made available during the course.

- Brockwell and Davis, Introduction to Time Series and Forecasting, Second Edition. Springer texts in statistics. 2002

- Brockwell and Davis, Time Series, theory and methods, Springer (collana SSS), New York, 1991

- Shumway and Stoffer, Time series Analysis and Its Applications, Springer, 2011.

- William W. S. Wei Time series analysis, Addison-Wesley Publishing Company, 1990

For the Lab, refer to


Class schedule

Last update: 24/04/2018 12:17
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