Bachelor
2019/2020

## Time Series and Stochastic Processes

Type:
Elective course (Applied Mathematics and Information Science)

Area of studies:
Applied Mathematics and Information Science

Delivered by:
Department of Mathematics

When:
4 year, 3 module

Mode of studies:
offline

Instructors:
Elena R. Goryainova

Language:
English

ECTS credits:
4

### Course Syllabus

#### Abstract

This course presents an introduction to time series analysis and stochastic processes and their applications in operations research and management science. Time series includes the description of the following models: white noise, Moving average models MA(q), Autoregressive models AR(p), Autoregressive-moving average ARMA(p,q) models, Nonlinear Autoregressive Conditional Heteroskedasticity (ARCH(p)) and Generalised Autoregressive Conditional Heteroskedasticity (GARCH(p;q)) models and VAR models. Also, the solution of the problem of identification of the ARMA process, including the model selection, estimation of the model parameters and verification of the adequacy of the selected model, is given. Methods for reducing some non-stationary time series to stationary ones by removing trend and seasonal components are described. Then, the Dolado-Jenkinson-Sosvilla-Rivero procedure is presented to distinguish non-stationary time series such as Trend-stationarity (TSP) and Difference-stationarity (DSP). The procedure for diagnosing the presence of spurious regression is also considered. Stochastic processes are discussed on a basic process Brownian motion and Poisson process. The method for constructing optimal forecasts for Gaussian stochastic processes and stationary time series is given. At the end of the course Markov chains and continuous-time Markov chains are considered. For these models, the conditions for the existence of a stationary distribution are established. In particular, are found the final distribution for the processes of «birth and death» and for the queueing system M/M/n/r.

#### Learning Objectives

- To familiarize students with the concepts, models and statements of the theory of time series analysis and stochastic processes.

#### Expected Learning Outcomes

- Know basics of time series analysis and stochastic processes
- Be able to choose adequate models in practical socio-economic problems
- Have skills in model construction and solving problems of time series analysis and stochastic processes

#### Course Contents

- Basic concepts of the theory of stochastic processesDefinitions of a stochastic process (SP), Time series, realizations (or sample-paths) of the process, finite-dimensional distribution functions of stochastic process. Kolmogorov consistency theorem (without prove). Main characteristics of time series (expectations, variance, moments, covariance function, correlation function. Properties of a covariance function. Examples.
- Some types of stochastic processesStrictly stationary stochastic process, weakly stationary stochastic process, relationship between weak and strict stationarity. Stochastic process with independent increments. stochastic process with orthogonal increments. Poisson stochastic process. Gaussian stochastic process, finite-dimensional density function of a Gaussian stochastic process. The Wiener process (Brownian Motion). Relationship between random walk and Brownian motion. Filtration problem.
- Main models of stationary time seriesLinear stochastic process. Lag (or back shift) operator. Discrete white noise. Moving average models MA(q). Condition of invertibility MA(q). Autoregressive models AR(p). Condition of stationarity AR(p). Autoregressive-moving average ARMA(p; q) models. Conditions of stationarity ARMA(p; q). Time-varying volatility. The notion of conditional volatility. Nonlinear Autoregressive Conditional Heteroskedasticity (ARCH(p)) and Generalised Autoregressive Conditional Heteroskedasticity (GARCH(p;q)) models.
- ForecastingAn optimal in the mean square sense predictor. A mean square error of the predictor. The theorem on the best (in the mean square sense) predictor (with prove). Forecasting of Gaussian processes. Theorem on Normal Correlation. Forecasting of stationary time series.
- Identification, estimation and testing of ARMA(p,q) modelsSample autocorrelation function (ACF), sample partial autocorrelation function (PACF), correlograms. Statistical properties of sample ACF and sample PACF. Goodness of fit in time series models. Yule-Walker’s method for estimating the parameters of AR(p) models. Backcasting procedure for estimating the parameters of MA(q) models. Recursive least squares (LS) method for estimating the parameters of ARMA(p,q) models. Distribution of LS estimates in ARMA(p,q) models. Check residuals for white noise. Akaike information criterion (AIC). Schwarz information criterion (SIC). Ljung-Box and Box-Pierce Q-tests. Jarque-Bera test for checking the normality of residuals.
- Identification of nonstationary stochastic processesModels with Trend and Seasonality. Box-Jenkins methodology. Difference operator. ARIMA models. Trend-stationarity stochastic process (TSP), Difference-stationarity stochastic process (DSP). Spurious regressions. Problem of the unit root. Dickey-Fuller test. Augmented Dickey-Fuller tests. Dolado-Jenkinson-Sosvilla-Rivero procedure.
- Vector autoregressive models.Causality.Vector autoregressive models. ADL models. Cointegrated series. The notion of causality. Granger causality.
- Markov chainsMarkov processes as generalizations of IID variables and of deterministic dynamical systems. The Markov property and the strong Markov property. Classifications of States of Markov chain. Ergodic Markov chain. Limiting distribution of Markov chain. The Classical Ruin Problem.
- Continuous-Time Markov ChainsA series of events. Chapman-Kolmogorov Equations. Ergodic properties of homogeneous Markov chains. Birth and Death Processes. Queuing theory.

#### Assessment Elements

- SG (Participation in the statistical game)
- HW (Homework)
- MEX (mid-term exam)
- EX (final exam)Exam form: The exam is conducted in writing using asynchronous proctoring. Asynchronous proctoring means that all the student's actions during the exam will be “watched” by the computer. The exam process is recorded and analyzed by artificial intelligence and a human (Proctor). Please be careful and follow the instructions clearly! Platform for conducting: The exam is conducted on the Moodle platform, an online platform for conducting test tasks of various levels of complexity. Proctoring by using the system Eczemas. The link to the completion of the exam task will be placed in the LMS. You must sign up for the exam 15 minutes before it starts. Technical requirements and rules of the exam: https://elearning.hse.ru/student_steps To participate in the exam, the student must: Prepare an identity document (passport, spread with name and photo) for identification before starting the examination task; Check the operation of the video camera and microphone, the speed of the Internet (for best results, we recommend connecting your computer to the network via a cable); Prepare the tools necessary for completing the examination tasks. Disable applications other than the browser that will be used to log in to the StartExam platform in the computer's task Manager. If one of the necessary conditions for participation in the exam cannot be met, it is necessary to inform the teacher or an employee of the training office about this 7 days before the date of the exam in order to make a decision about the student's participation in the exams. During the exam, students are not allowed to: Turn off the video camera or microphone; Leave the place where the exam task is performed (go beyond the camera's viewing angle); Look away from your computer screen or desktop; Use smart gadgets (smartphone, tablet, etc.); To attract outsiders to assist in the examination, speak with outsiders during the execution of the tasks; Read tasks out loud. Breaking the link: A short-term communication failure during an exam is considered to be the loss of a student's network connection with the Examus platform for no more than 1 minute. A long-term communication failure during an exam is considered to be the loss of a student's network connection to the Examus platform for more than 1 minute. Long-term communication failure during the exam is the basis for the decision to terminate the exam and the rating “unsatisfactory” (0 on a ten-point scale. In case of a long-term violation of communication with the Examus platform during the examination task, the student must notify the teacher, record the fact of loss of communication with the platform (screenshot, response from the Internet provider) and contact the training office with an explanatory note about the incident to make a decision on retaking the exam.

#### Interim Assessment

- Interim assessment (3 module)0.4 * EX (final exam) + 0.2 * HW (Homework) + 0.3 * MEX (mid-term exam) + 0.1 * SG (Participation in the statistical game)

#### Bibliography

#### Recommended Core Bibliography

- Brockwell, P. J., & Davis, R. A. (2002). Introduction to Time Series and Forecasting (Vol. 2nd ed). New York: Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=108031
- Dolado, J. J., Jenkinson, T., & Sosvilla-Rivero, S. (1990). Cointegration and Unit Roots. https://doi.org/10.1111/j.1467-6419.1990.tb00088.x
- 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
- Gebhard Kirchgässner, Jürgen Wolters, & Uwe Hassler. (2013). Introduction to Modern Time Series Analysis. Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.b.spr.sptbec.978.3.642.33436.8
- Maddala, G. S., & Kim,In-Moo. (1999). Unit Roots, Cointegration, and Structural Change. Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.b.cup.cbooks.9780521587822
- Shir︠i︡aev, A. N. (1999). Essentials Of Stochastic Finance: Facts, Models, Theory. Singapore: World Scientific. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=91430

#### Recommended Additional Bibliography

- Banerjee, A., Dolado, J. J., Galbraith, J. W., & Hendry, D. (1993). Co-integration, Error Correction, and the Econometric Analysis of Non-Stationary Data. Oxford University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsrep&AN=edsrep.b.oxp.obooks.9780198288107
- Box, G. E. P., Reinsel, G. C., & Jenkins, G. M. (2008). Time Series Analysis : Forecasting and Control (Vol. 4th ed). Hoboken, N.J.: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=588017
- Hamilton, J. D. . (DE-588)122825950, (DE-576)271889950. (1994). Time series analysis / James D. Hamilton. Princeton, NJ: Princeton Univ. Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edswao&AN=edswao.038453134
- 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
- Lütkepohl, H., & Krätzig, M. (2004). Applied Time Series Econometrics. Cambridge, UK: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=164387
- Maronna, R. A. (2018). Robust Statistics : Theory and Methods (with R) (Vol. Second edition). [Place of publication not identified]: Wiley. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1921437
- Williams, R. J. (2006). Introduction to the Mathematics of Finance. Providence: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971271