Review Outline for Time Series

1. Fundamental concepts (2.1-2.7)

1. Stationarity: covariance stationary
2. Autocovariance and autocorrelation functions (ACF)
3. Partial autocorrelation function (PACF)
4. Sample ACF and PACF
5. Stationary = has a moving average representation
6. Invertible = has an autoregressive representation
7. Linear difference equations: know how solution depends on roots

2. Stationary (ARMA) models (3.1-3.4)

1. AR(p): difference equation representation, stationarity conditions, characteristics of ACF and PACF
2. MA(q): difference equation representation, invertibility conditions, characteristics of ACF and PACF
3. ARMA(p,q): difference equation representation, stationarity and invertibility conditions, characteristics of ACF and PACF
4. MA representation of stationary ARMA
5. AR representation of invertible ARMA

3. Nonstationary (ARIMA) models (4.1-4.3)

1. Differencing: how does differencing reduce a nonstationary model to a stationary one?
2. Features of ACF and PACF that suggest nonstationarity
3. Random walk and IMA type models
4. Variance nonstationarity and transformations: log transformation

4. ARIMA forecasting (5.1-5.3)

1. Computing minimum mean square error forecasts by conditional expectations
2. ARMA forecast error analysis derived from MA representation (since it is stationary) (5.2.8-5.2.10)
3. Extension to ARIMA forecast error analysis: contrast forecast intervals for stationary and nonstationary processes

5. Model identification (6.1)

1. Time series graphs and differenced time series graphs
2. ACF and PACF: what to look for?

6. Model estimation (7.1-7.2, 7.5-7.7)

1. Yule-walker equations for AR(p)
2. R function arima
3. Diagnostics: standardized residuals, goodness-of-fit test
4. Model selection criteria: AIC(M)

1. Difference equation representation
2. Characteristics of seasonal processes, ACFs, PACFs

1. Spectrum of absolutely summable autocovariance function: Fourier transform
2. Relation between spectrum and autocovariance generating function
3. *General form of spectrum for ARMA models: study special cases, characteristics of the spectrum