Time Series Analysis

Syllabus

1. Overview of methods of the course Time series analysis, conditions for completing the course. Objectives of the analysis of socio-economic time series, as realization of stochastic process, its properties, stationarity (weak, strong, Gaussian). Graphic methods of analysis of time series components (methods of extracting stochastic or analytical trend from time series; classical decomposition - mechanical smoothing and extracting from the components). 2. Random process and its moments. Stationary random process and its properties. Stationary time series and their extrapolations (naive forecasts, constant trend, terminal moving averages as forecast techniques). Stationarity tests. ACF and PACF of a random processes. Transformations of nonstationary series to stationary (differentiation, notation using the backward operator; Box – Cox transformation). 3. Trends in the time series (linear, quadratic, exponential, hyperbolic, Gomperz curve) and t-tests of their parameters in a statistical software application. Estimation of the random component and its variance. Average errors of model residuals, definition, interpretation (MSE, RMSE, ME, MAE, MAPE, MPE) and comparison of systematic bias of models according to software outputs. 4. Analysis of residuals. White noise and its properties (independence, homoscedasticity, normality). Results of graphical and numerical tests (nonparametric tests of independence in the time series application). Histogram, Box-Plot, normal probability plot in time series application – interpretation of the results. 5. Tests of non-correlation. Sample ACF and sample PACF (Bartlett test, empirical rule). Postmenteau tests (Box-Pierce, Ljung-Box). Their applications for time series and for the random component of models. 6. Partition into estimation and validation/verification period. Ex-post and ex-ante extrapolations. Comparison of model quality (interpolation and extrapolation period). Evaluation of the errors of extrapolation. Assessment of model suitability, including of information criteria for model selectin (AIC, BIC, Theil's U, Adjusted coefficient of determination). 7. Exponential smoothing and forecasting - Brown models. Holt model of exponential smoothing. 8. Autoregressive models of stationary process AR (p). Models of moving averages of stationary process MA (q). Properties of ACF and PACF of these processes. Random walk process - AR (1) process with unit root. 9. Models of stationary ARMA processes (p,q) and properties of their ACF and PACF. Integrated ARIMA models (p,d,q) and their properties. Preliminary determination of the number of parameters, verification of the initial assumptions of the models. Forecasting of non-seasonal time series by ARIMA models - applications. 10. Time series with seasonal component. Seasonal decomposition. Seasonal indices and seasonally adjusted time series. Extrapolation of seasonally adjusted series. Extrapolation of time series with seasonality. A combination of classical and adaptive forecasting methods. Holt-Winter model of exponential smoothing. 11. ARIMA models with seasonal component. 12. Practical advice and summary of the steps of the Box-Jenkins methodology in the phases of identification, estimation and verification. Verification of the accuracy of short-term forecasts. 13. Presentation of the results of an illustrative application of the studied curriculum - a case study on a real financial time series. Repetition and discussion of the subject problems