**Lecture times:** Tuesday 16H00 (LT1)

**Tutorial times:** Thursday 16H00 (Lab)

The course provides an accessible introduction to the application of time series methods. Topics covered include an introduction to the dynamic properties of time series, stuctural breaks, univariate autoregressive moveing average models, forecast evaluation, state-space models, various unit root tests, univariate volatility models, autoregressive distriubted lag models, vector autoregression models, structural vector autoregression models, Bayesian vector autoregression models, dynamic factor models, cointegration and error-correction models, multivariate volatility models, regime-switching models, and deep learning models. The course will also emphasize recent developments in time series analysis and areas of ongoing research.

The main objective of the course is to develop the skills that are needed to conduct empirical research using time series data. Therefore, the course provides students with an understanding of the techniques that are required to select, estimate, and assess the quality of time series models. Special attention will be placed on limitations and pitfalls of different methods. On completing the course, students will be able to make use of a wide variety of time series methods, which may be applied to various areas of interest. Students will also be able to critically evaluate and summarize the results of time series models. As a substantial part of the course will be dedicated towards applying these research methods, students will also be introduced to the popular **R** computing language, which is freely available to all users.^{2}

In addition to detailed class notes and selected journal articles, we will also consult the following texts. Further details may be found on my website at: https://www.economodel.com/time-series-analysis.

**Main texts:**

Bjørnland, H. and L. Thorsrud, 2014, *Applied Time Series for Macroeconomics*, Gyldendal Akademisk: Oslo.

Enders, W., 2014, *Applied Econometric Time Series Analysis*, 4th Edition, John Willey & Sons: New York.

Hamilton, J., 1994, *Time Series Analysis*, Princeton University Press: Princeton.

Pesaran, M. H., 2015, *Time Series and Panel Data Econometrics*, Oxford University Press: Oxford.

Shumway, R. and D. Stoffer, 2010, *Time Series Analysis and its Applications with R Examples*, 3rd Edition, Springer-Verlag: New York.

**Supplementary texts:**

Box, G. E. P., G. M. Jenkins and G. C. Reinsel, 2008, *Time Series Analysis - Forecasting and Control*, 4th Edition, John Wiley & Sons: New York.

Campbell, J. Y., A. W. Lo and J. Y. MacKinlay, 1997, *The Econometrics of Financial Markets*, Princeton University Press: Princeton.

Commandeur, J. F. F. and S. J. Koopman, 2007, *An Introduction to State Space Time Series Analysis*, Oxford University Press: Oxford.

Durbin, J. and S. J. Koopman, 2012, *Time Series Analysis by State Space Methods*, 2nd Edition, Oxford University Press: Oxford.

Frances, P.H. and D. van Dijk, 2004, *Non-linear time series models in empirical finance*, Cambridge University Press: Cambridge.

Ghysels, E. and M. Marcellino, 2018, *Applied Economic Forecasting using Time Series Methods*, Oxford University Press: Oxford.

Gourieroux, C. and J. Jasiak, 2001, *Financial Econometrics*, Princeton University Press: Princeton.

Kilian, L. and H Lütkepohl, 2017, *Structural Vector Autoregressive Analysis*, Cambridge University Press: Cambridge.

Koop, G., 2003, *Bayesian Econometrics*, John Wiley & Sons: New York.

Koop, G. and D. Korobilis, 2010, *Bayesian Multivariate Time Series Methods for Empirical Macroeconomics*, Foundations and Trends in Econometrics, Now Publishers: New York.

Juselius, K., 2006, *The Cointegrated VAR Model: Methodology and Applications*, Oxford University Press: Oxford.

Lütkepohl, H., 2006, *New Introduction to Multiple Time Series Analysis*, Springer: New York.

Martin, V., S. Hurn and D. Harris, 2013, *Econometric Modelling with Time Series: Specification, Estimation and Testing*, Cambridge University Press: Cambridge

Nielsen, A., 2019, *Practical Time Series Analysis: Prediction with Statistics and Machine Learning*, O’Reilly Media: Sebastopol, California.

Petris, G., S. Petrone and P Campagnoli, 2009, *Dynamic Linear Models with R*, Springer: New York.

Pfaff, B., 2008, *Analysis of Integrated and Cointegrated Time Series with R*, Springer: New York.

Taylor, S. J., 2005, *Asset Price Dynamics, Volatility, and Predictability*, Princeton University Press: Princeton.

Taylor, S. J., 2008, *Modelling Financial Time Series*, 2nd Edition, World Scientific Publishing: London.

Tsay, R. S., 2010, *Analysis of Financial Time Series*, 3rd Edition, John Willey & Sons: New York.

Tsay, R. S., 2014, *Multivariate Time Series Analysis: With R and Financial Applications*, John Willey & Sons: New York.

Tsay, R. S. and R. Chen, 2018, *Nonlinear Time Series Analysis*, John Willey & Sons: New York.

Assessment consists of a final examination that counts 50% towards the final mark. Students will be required to complete a group project that will count a further 35%, and a test will be written midway through the semester for the remaining 15%. Most of the weekly tutorials will closely follow the course outline and where required, homework will need to be completed prior to moving onto new work.

Univariate methods

**Introduction:**

*Common features of economic and financial time series data, decomposing the data, autoregressive & moving average processes, autocorrelation and partial autocorrelation functions, stochastic linear difference equations, characteristic roots, eigenvalues, impact multipliers & calculation of IRFs*. Notes supplemented by: Enders ch 1; Gourieroux & Jasiak ch 1; Box, Jenkins & Reinsel ch 1; Hamilton ch 1 & 2; Shumway & Stoffer ch 1.**Structural breaks:**

*TBC*. Notes supplemented by:**Univariate autoregressive moving average models:**

*Box-Jenkins approach for model identification & evaluation, autocorrelation & partial autocorrelation functions, information criteria for model selection*. Notes supplemented by: Enders ch 2; Tsay ch 1 & 2; Gourieroux & Jasiak ch 2, Hamilton ch 3; Shumway & Stoffer ch 3.**Forecasting and out-of-sample evaluations:**

*Point forecasts, density forecasts, evaluation using bias, RMSE, DM, LS, and PIT, constructing a rolling forecast experiment, alternative forecasting models*. Notes supplemented by: Bjørnland ch 3; Hamilton ch 4; and various articles.**Univariate state-space models:**

*The methodology of state-space models & the use of the Kalman filter, model construction for stochastic levels, slopes, seasonals, time-varying parameters, and multivariate extensions*. Notes supplemented by: Commandeur ch 1-12; Durbin & Koopman ch 1-3; Hamilton ch 13; Koop ch 8; Petris, Petrone & Campagnoli.**Decompositions and spectral analysis:***The use of filters in the time domain with the Hodrick-Prescott and Beveridge-Nelson decompositions, spectral decompositions and Baxter-King filter, the use of periodograms, time-frequency decompositions and wavelet applications*. Notes supplemented by: Shumway & Stoffer ch 4; Enders ch 4; Hamilton ch 6.**Nonstationary and unit root tests:***Tests for random walks, trend stationary, non-stationarity in the presence of structural change, power of these tests*. Notes supplemented by: Pfaff ch 5 & 6; Enders ch 4; Bjornland ch 4; Martin ch 17.**Univariate volatility models:**

*Univariate ARCH, GARCH, GARCH-M, EGARCH, TGARCH, IGARCH, etc., univariate SV models*. Notes supplemented by: Tsay ch 3; Gourieroux & Jasiak ch 6; Enders ch 3; Campbell, Lo, & MacKinlay ch 12; Martin ch 8 & 20.

Multivariate methods

**Autoregressive distributed lag models:**

*TBC*. Notes supplemented by:**Vector autoregressive models:**

*Identification issues in VAR models, Choleski factorisation & SVAR decompositions (Sims ’86 & Blanchard-Quah), constructing IRFs, variance decompositions, historical decomposition*. Notes supplemented by: Lutkepohl ch 1-12; Enders ch 5; Pfaff ch 2; Bjornland ch 7 & 8; Gourieroux & Jasiak ch 3; Martin ch 14.**Structural vector autoregressive models:**

*Identification issues in VAR models, Choleski factorisation & SVAR decompositions (Sims ’86 & Blanchard-Quah), constructing IRFs, variance decompositions, historical decomposition*. Notes supplemented by: Lutkepohl ch 1-12; Enders ch 5; Pfaff ch 2; Bjornland ch 7 & 8; Gourieroux & Jasiak ch 3; Martin ch 14.**Bayesian vector autoregressive models:**

*Bayesian VARS, stationary and nonstationary priors, including time-varying parameters, endogenous structural breaks, stochastic variable selection*. Notes supplemented by: Lutkepohl ch 1-12; Koop & Korobilis; and various articles.**Cointegration and error correction models:**

*General principles of cointegration, the Engle-Granger technique, introduction to the Johansen procedure, calculating the determinant, rank and the importance of the characteristic roots / eigenvalues, testing restrictions on the Johansen model, alternative framework including the SVAR, ARDL & general-to-specific approach*. Notes supplemented by: Enders ch 6; Lutkepohl ch 14; Bjornland ch 9; Martin ch 18; Juselius ch5-8; Pfaff ch 7 & 8.**Dynamic factor models:**

*Principal components models, factor augmented VARs, incorporating data at different frequencies, nowcasting techniques*. Notes supplemented by articles including: J. Breitung (2005); J. Stock & M. Watson (2010); Giannone, Reichlin & Small (2008); Doz, Giannone & Reichlin (2011).**Multivariate volatility models:**

*Multivariate GARCH, including heteroskedastic features in a VAR model*. Notes supplemented by:**Nonlinear regime-switching models:***TAR, STAR, MSW, and methods that test for nonlinearity*. Notes supplemented by: Enders ch 7; Frances & van Dijk ch 3 & ch 5; Tsay ch 4; Hamilton ch 22; Campbell, Lo, & MacKinlay ch 12, Martin ch19.

Machine learning methods

**Regression trees and clustering models:***Decision trees, clustering*. Notes supplemented by:**Deep learning models:***Neural Networks*. Notes supplemented by:

Appendices

**Appendix A: Mathematics**

**Appendix B: Probability and Statistics**

email: kevin.kotze@uct.ac.za↩

While numerous examples and exercises will be provided in

**R**, students can use their preferred open source scripting language (eg.**Python**or**Julia**) to complete their assignments.↩