Bachelor
2020/2021

# Time Series and Stochastic Processes

Category 'Best Course for New Knowledge and Skills'

Type:
Compulsory course (HSE University and University of London Double Degree Programme in Data Science and Business Analytics)

Area of studies:
Applied Mathematics and Information Science

Delivered by:
Big Data and Information Retrieval School

Where:
Faculty of Computer Science

When:
3 year, 1-4 module

Mode of studies:
offline

Language:
English

ECTS credits:
8

Contact hours:
198

### Course Syllabus

#### Abstract

This course is conducted at Data Science and Business Analytics program and is provided to 3rd-year undergraduates who have studied a course covering basic probability and statistical inference. A half of this course introduces concepts of Markov chains, random walks, martingales as well as of to the time series. The course requires basic knowledge in probability theory and linear algebra. It introduces students to the modeling, quantification and analysis of uncertainty. The main objective of this course is to develop the skills needed to do empirical research in fields operating with time series data sets. The course aims to provide students with techniques and receipts for estimation and assessment of quality of economic models with time series data. The course will also emphasize recent developments in Time Series Analysis and will present some open questions and areas of ongoing research.

#### Learning Objectives

- The main objective of this course is to develop the skills needed to do empirical research in fields operating with time-series data sets.
- The course will emphasize recent developments in Time Series Analysis and will present some open questions and areas of ongoing research.

#### Expected Learning Outcomes

- Students get an understanding of techniques and receipts for estimation and assessment of the quality of economic models with time-series data.

#### Course Contents

- Discrete-time martingale theory.
- Continuous-time stochastic processes.At this point, we shall focus on the most important stochastic processes in continuous time, and explore some of their remarkable properties. Topics: the Poisson process; Brownian motion and its emergence as a limit of random walks; Markov and martingale properties; quadratic variation; reflection principle and maxima of Brownian motion.
- Stochastic calculus and differential equations.Continuing the development, the theory of stochastic integration, the cornerstone of the mathematical finance theory, will be presented in this section. Topics: definition and properties of the stochastic integral; The Itˆo-Doeblin formula; Brownian market model and wealth-process dynamics; Girsanov’s theorem; martingale representation theorem; the Feynman-Kac formula
- Continuous-time financial models.Finally, applications of the previous theory will be given which will clearly illustrate the use of mathematical tools. Topics: fundamental theorem(s) of asset pricing; risk-neutral probability measures and valuation of contingent claims; the Black-Scholes PDE for valuation of European options.
- Autoregressive-moving average models ARMA (p,q) Moving average models МА(q). Condition of invertibility. Autoregressive models АR(р). Yull-Worker equations. Stationarity conditions. Autoregressive-moving average models ARMA (p,q). Coefficient estimation in ARMA (p,q) processes. Box-Jenkins’ approach Coefficients estimation in autoregressive models. Coefficient estimation in ARMA (p) processes.Quality of adjustment of time series models. AIC information criterion. BIC information criterion. “Portmonto”-statistics. Box-Jenkins methodology to the identification of stationary time series models.
- Forecasting in the framework of Box-Jenkins model. Forecasting, trend and seasonality in Box-Jenkins model.
- Non-stationary time series, TSP or DSP: methodology of research. Segmented trends and structure changes
- Regressive dynamic models. Autoregressive models with distributed lags (ADL).
- Vector autoregression model and co-integration.
- Time series co-integration. Co-integration regression. Testing of co-integration. Vector autoregression and co-integration. Co-integration and error correction model.

#### Assessment Elements

- FallMock
- Winter MockThe exam may be carried out online via distance learning platforms.
- Homework
- Spring Mock
- Final ExamThe exam may be carried out online via distance learning platforms.
- UoL Exam
- Quizzes

#### Interim Assessment

- Interim assessment (2 module)0.4 * FallMock + 0.2 * Homework + 0.4 * Winter Mock
- Interim assessment (4 module)0.1 * Final Exam + 0.1 * Homework + 0.05 * Quizzes + 0.1 * Spring Mock + 0.65 * UoL Exam

#### Bibliography

#### Recommended Core Bibliography

- Enders, W. (2015). Applied Econometric Time Series (Vol. Fourth edition). Hoboken, NJ: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1639192
- Harvey, A. C. (1993). Time Series Models (Vol. 2nd ed). Cambridge, Mass: MIT Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=11358
- Mills, T. C., & Markellos, R. N. (2008). The Econometric Modelling of Financial Time Series: Vol. 3rd ed. Cambridge University Press.

#### Recommended Additional Bibliography

- Bartoszyński, R., & Niewiadomska-Bugaj, M. (2008). Probability and Statistical Inference (Vol. 2nd ed). Hoboken, N.J.: Wiley-Interscience. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=219782
- Freund, J. E., Miller, I., & Miller, M. (2014). John E. Freund’s Mathematical Statistics with Applications: Pearson New International Edition (Vol. Eighth edition, Pearson new international edition). Essex, England: Pearson. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1418305
- Hogg, R. V., Zimmerman, D. L., & Tanis, E. A. (2015). Probability and Statistical Inference, Global Edition (Vol. Ninth edition. Global edition). Boston: Pearson. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1419274
- Larsen, R. J., & Marx, M. L. (2015). An introduction to mathematical statistics and its applications. Slovenia, Europe: Prentice Hall. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.19D77756
- Lindgren, B. W. (1993). Statistical Theory (Vol. Fourth edition). Boca Raton, Florida: Routledge. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1683924