Time Series Analysis

Recommended Prerequisites

Knowledge in Probability, Statistics and Stochastic Processes; experience with statistical software.

Teaching Methods

Classes are expository and include examples and exercises for applying the acquired knowledge. Throughout the semester, students will do a computational assignment. This assignment is geared towards the treatment of  different type temporal data.

During the semester students may use tutorial time to clarify their difficulties in grasping the theory and in gaining practical knowledge, as well as in the development of the necessary skills for the computational assignment.

Learning Outcomes

The aim of this course is to provide mathematical methodologies for describing, analysing and forecasting time random features. We begin by studying the general class of linear models which describe different types of data. The study of non-linear models, particularly adjusted for volatile data, is also another purpose of this course. Thus, we develop the study of conditionally heteroscedastic and bilinear processes. Counting stochastic systems are also considered, studying recent integer-valued time series models. Fitting this kind of modeling to real data is another goal of this class.

This course allows developing the following skills: ability to calculate; using computational tools; knowledge of mathematical results; ability to generalize and abstract; formulating and solving problems; design and use of mathematical models for real situations. On a personal level, it allows to develop individual initiative, teamwork, research and independent learning.

Work Placement(s)

No

Syllabus

Time series.  ARMA modeling with conditional heteroskedastic errors: power GARCH and GTARCH processes (general settings, stationarity, ergodicity, moments). Bilinear processes (brief reference).

Integer-valued time series. Thinning operator, INARMA and INGARCH stochastic processes (general settings, stationarity, ergodicity, moments).

Statistical analysis of time series. Estimation, forecasting and testing in some of the models studied.

Assessment Methods

Assessment
There are 2 types of grading: during the semester requires taking 1 mid-term exam (75% of the final grade) and doing scientific or computational assignments (representing 25% of the final grade). Grading by final examination includes taking an exam (75% of the final grade) and doing scientific or computational assignments (25% of the final grade).: 100.0%

Bibliography

E. Gonçalves and N. Mendes-Lopes, Séries Temporais: Modelações lineares e não lineares,  2ª edição, Sociedade Portuguesa de Estatística, 2008.

E. Gonçalves, N. Mendes-Lopes and F. Silva, Infinitely divisible distributions in integer-valued GARCH models, Journal of Time Series Analysis 36, 503-527, 2015.

Ch. Gouriéroux and A. Monfort, Séries Temporelles et Modèles Dynamiques, Economica, 1990.

C. Martins,  Modelos Bilineares em Séries Temporais. Propriedades probabilistas e decisão estatística, Tese de doutoramento, Univ. Coimbra, 2000.

H. Tong, Non-linear Time Series: a Dynamical System Approach, Oxford University Press, 1990.

C.H. Weiβ, Thinning Operations for Modeling Time Series of Counts – a survey, Advances in Statistical Analysis 92, no. 3, 319-341, 2008.

P.J. Brockwell and R.A. Davis, Time Series: Theory and Methods, 2nd edition, Springer-Verlag, 2006.

J. Fan and Q. Yao, Nonlinear Time Series: Nonparametric and Parametric Methods, Springer-Verlag, 2003.

R. Ferland, A. Latour, and D. Oraichi, Integer-valued GARCH process, Journal of Time Series Analysis 27, no. 6, 923-942, 2006.

Ch. Francq and J.M. Zakoian, GARCH models, Wiley, 2010.

E. Gonçalves, J. Leite,  and N. Mendes-Lopes, On the probabilistic structure of power threshold generalized ARCH stochastic processes, Statistics and Probability Letters 82, 1597-1609, 2012