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Regular version of the site
Bachelor 2020/2021

## Time Series and Stochastic Processes

Category 'Best Course for New Knowledge and Skills'
Area of studies: Applied Mathematics and Information Science
When: 4 year, 3 module
Mode of studies: distance learning
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 processes
Definitions 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 processes
Strictly 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 series
Linear 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.
• Forecasting
An 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) models
Sample 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 processes
Models 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 chains
Markov 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 Chains
A 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)

#### 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)

#### 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