AMS 586: Time Series

Linear time series models, moving average (MA), autoregressive (AR), ARMA and ARIMA models, estimation and forecasting, interval predictions, forecast errors, stationary processes in the frequency domain, state-space models.

Required Textbooks:

1. Analysis of Financial Time Series by Ruey S. Tsay, 3rd edition, 2010, Wiley
2. The Analysis of Time Series: An Introduction with R by Chris Chatfield and Haipeng Xing, 7th edition, 2019, Chapman and Hall/CRC

Supplementary Textbook: Time Series Analysis and Its Applications: With R Examples by Robert H. Shumway and David S. Stoffer, 4th edition, Springer, 2017

Learning Outcomes

1. Master the concepts of stationary time series:
   • Decomposition of time series into trend component, seasonal component, and stationary process;
   • Strictly stationary process versus weakly stationary process;
   • White noise and Gaussian white noise;
   • Autocovariance, mean and variance;
   • Autocorrelation function (ACF);
   • Partial autocorrelation function (PACF);
   • Autoregressive process (AR) – model introduction, condition for stationarity;
   • Moving average process (MA) – model introduction, condition for invertibility;
   • Autoregressive Moving Average Process (ARMA) – model introduction, conditions for stationarity and invertibility, three representations of ARMA;
   • Linear time series models.
2. Master statistical inference related to the stationary time series processes (AR, MA and ARMA):
   • Computation of the population & sample autocorrelations;
   • Computation of the population & sample partial autocorrelations;
   • Determination of the order of the AR processes based on PACF;
   • Determination of the order of the MA processes based on ACF;
   • Identification of the ARMA processes;
   • Estimation of AR, MA and ARMA;
   • Goodness-of-fit indices: AIC, AICC, BIC;
   • Normality test for the residuals;
   • Forecast with ARMA models;
   • Linear regression with ARMA errors;
   • Stationary processes in the frequency domain.
3. Master statistical concepts and inference related to the autoregressive integrated moving average processes (ARIMA) & Unit-Root Nonstationarity:
   • Random walk;
   • Random walk with drift;
   • Trend stationary time series;
   • ARIMA model and its reduction to ARMA through differencing;
   • Unit-root test;
   • Seasonal models and seasonal differencing;
   • Mastery of related statistical programs using R.
4. Demonstrate skills for statistical concepts and inference related to the conditional heteroscedastic models:
   • Volatility;
   • AutoRegressive Conditional Heteroskedasticity (ARCH) models;
   • Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) models;
   • ARMA-GARCH Models – identification, estimation and forecast;
   • Mastery of related statistical programs using R.
5. Demonstrate mastery of basic statistical concepts related to nonlinear models:
   • Bilinear models;
   • Threshold autoregressive (TAR) models;
   • Smooth transition AR (STAR) models;
   • Markov switching models;
   • Nonparametric methods.
6. Demonstrate skills for statistical concepts and inference related to the state-space models:
   • Local Trend Model;
   • Kalman Filter;
   • Linear state-space models;
   • Model transformation;
   • Structural equation modeling;
   • Mastery of related statistical programs using R or SAS.