90481 - Advanced Time Series

Learning outcomes

By the end of the course the student is able to analyse data generated by GARCH, DCS, long memory processes and make inference on the moment estimators.

Course contents

The course introduces the main statistical methods for the analysis and forecasting of time series. Particular attention is devoted to the specification, interpretation and estimation of time series models, as well as to model selection, diagnostic checking and forecasting. The theoretical concepts are illustrated through examples and applications to real and simulated data.

Introduction

Basic objectives of time series analysis, graphical representation of time series, descriptive methods, trend and seasonal components, transformations, sample mean, sample autocovariance function and sample autocorrelation function.

Stationary Processes

Weak stationarity, autocovariance and autocorrelation functions, white noise, linear processes, moving average processes, autoregressive processes and their main properties.

ARMA Models

Autoregressive, moving average and ARMA models. Causality, invertibility, autocorrelation and partial autocorrelation functions, parameter estimation, model identification and diagnostic checking.

Spectral Analysis

Introduction to frequency-domain methods for stationary time series, spectral density functions and their main properties, periodogram and basic principles of spectral estimation.

Modeling and Forecasting with ARMA Processes

Main stages of time series modelling, model specification and selection, parameter estimation, residual analysis, forecasting, prediction intervals and assessment of forecasting accuracy.

Nonstationary and Seasonal Time Series Models

Trend-stationary and difference-stationary processes, differencing, ARIMA models, seasonal time series, seasonal differencing, seasonal ARIMA models, estimation, diagnostic checking and forecasting.

Multivariate Time Series

Introduction to multivariate stationary processes, cross-covariance and cross-correlation functions, vector autoregressive models, estimation and multivariate forecasting.

State-Space Models

State-space representation of time series models, linear state-space models, prediction and updating equations, the Kalman filter, parameter estimation and forecasting.

Forecasting Techniques

Introduction to alternative forecasting techniques, construction of prediction intervals and evaluation and comparison of forecasting performance.

Readings/Bibliography

The main textbook is: Brockwell, P.J. and Davis, R.A. (2016), Introduction to Time Series and Forecasting, 3rd edition, Springer.

The chapters and sections included in the course programme will be specified during the lectures. Additional lecture notes, exercises and supplementary materials will be made available through the Virtuale platform.

Teaching methods

The course consists of lectures and guided exercise sessions. The lectures introduce the theoretical foundations of the main time series models and the methods used for model specification, estimation, diagnostic checking and forecasting.

The theoretical concepts are illustrated through worked examples and applications to real and simulated time series. Exercise sessions are intended to support students in applying the methods introduced during the lectures and in interpreting the results of a time series analysis.

Assessment methods

Assessment is based on a written examination covering the topics presented during the course.

The examination includes theoretical questions and exercises aimed at assessing the student’s understanding of the main time series models, their statistical properties and the procedures used for estimation, model selection, diagnostic checking and forecasting. Students may also be asked to interpret graphical, numerical or software output related to a time series analysis.

Teaching tools

Lectures are supported by slides, lecture notes, exercises, datasets and software code. Empirical examples and applications are implemented in R.

Teaching materials, datasets, exercises, software code, and course-related communications are made available through the Virtuale platform.

Office hours

See the website of Silvia Bianconcini