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AMS 586, Time Series

Analysis in the frequency domain. Periodograms, approximate tests, relation to regression theory. Pre-whitening and digital fibers. Common data windows. Fast Fourier transforms. Complex demodulation, GibbsÕ phenomenon issues. Time-domain analysis.

Prerequisites: AMS 570 or AMS 572

3 credits, ABCF grading


May be offered in either Spring or Fall semester

Actuarial Exam: A student receiving a B- or better in this course and in  AMS 578 satisfies the Actuarial Exam test in Applied Statistics, through the  Society of Actuaries Validation by Educational Experience program. For more details about actuarial preparation at Stony Brook see  Actuarial Program 

Fall 2022 Semester:
Text: "Analysis of Financial Time Series (CourseSmart)" by Ruey S. Tszy, 3rd edition (August 30, 2010), published by Wiley, ISBN #978-0-470414354 (recommended/optional)

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.