STAT 520 Final Exam Review Sheet

I.  What Are Time Series Data?
  A. Key difference between time series data and cross-sectional data
  B. Scatterplots of time series values vs. lagged values to see (lag-1) autocorrelation
  C. Concept of Seasonality
II. Fundamental Mathematical Concepts
  A. Moments of a Stochastic Process
    1. Mean function E(Y_t)
    2. Variance function var(Y_t)
    3. Autocovariance function cov(Y_t, Y_s), also can be written cov(Y_t, Y_{t-k}) where k is the lag
    4. Autocorrelation function and formula relating autocorrelation to autocovariance and variance
    5. Interpretations of Autocovariance and Autocorrelation and advantages of interpreting Autocorrelation
  B. Calculating Variances, Covariance, Correlations of Linear Combinations of Random Variables
    1. "FOIL" approach
    2. How to deal with constants
  C. Writing autocovariance and autocorrelation functions as piecewise functions of the lag
  D. Concept of Stationarity
    1. Implications of being stationary of the mean function, variance function, autocovariance function
    2. Showing a process is (or is not) weakly stationary
    3. White noise process
III. Regression Trend Models for Time Series
  A. Decomposing a time series into a deterministic trend and a random noise process
  B.  Simplest Trend Model:  A constant mean
  C. Other trend models
    1. Linear time trend
    2. Estimating coefficient parameters with least squares
    3. Quadratic time trend
    4. Seasonal means model
    5. Harmonic regression model
  D. Regression Output from Software
    1. Finding estimated coefficients from software output
    2. Writing estimated trend model equation
    3. Residual standard deviation s
    4. Adjusted R^2
    5. AIC and BIC
    6. Using these criteria to selection the best trend model
  E. Residual Analyses and How to Interpret them
    1. Plots of (Standardized) Residuals against Time
    2. Plots of Residuals against Fitted Values
    3. Normal Q-Q plot of residuals
    4. Shapiro-Wilk test on residuals
    5. Runs Test on residuals
    6. Sample ACF of residuals
  F. Investigating Relationships with Lagged Values of Time Series
  G. Dealing with Nonstationary Time Series
    1. Detrending
    2. Differencing
IV. Stationary Models
  A. Moving Average (MA) Processes
    1. Model equation for an MA(q) process
    2. MA(1), MA(2), ... models --- what does the value of q mean?
    3. True autocorrelations of an MA process are zero for which lags?
  B. Autoregressive (AR) Processes
    1. Model equation for an AR(p) process
    2. Formula for lag-k autocorrelation for an AR(1) process
    3. How the autocorrelations of an AR process behave as k increases
  C. ARMA processes and their model equation
V. Nonstationary Models
  A. Nonstationarity due to nonconstant mean function
  B. Differencing
    1. Definition of first differences
    2. Definition of second differences
    3. Effect of differencing on stationarity
  C. ARIMA(p,d,q) model
  D. Role of constant term in ARIMA model
  E. Log transformation to achieve stationary (when is it useful?)
  F:  More general:  Box-Cox approach
VI. Model Specification
  A. Use of the Sample ACF
    1.  Using the ACF to help specify an MA model
  B. Use of the Sample PACF
    1.  Using the PACF to help specify an AR model
  C. EACF to help specify ARMA models
  D. Specifying actual time series using the ACF, PACF, EACF
    1. Using ACF (and plot of the original time series) to assess nonstationarity
    2. Avoiding overdifferencing
    3. Augmented Dickey-Fuller Test for Nonstationarity
  E. Using AIC and BIC to compare candidate models
    1. When can models be compared with AIC/BIC and when can they not be?
VII. Parameter Estimation
  A. Method of Moments
    1. MOM estimator of phi in an AR(1) model
    2. What type of model is MOM not good for?
  B. Least Squares
  C. Maximum Likelihood
    1. Need to specify noise distribution
    2. Similar to LS estimators for large sample sizes
  D. Large-sample CIs for the model coefficients
    1. Formula for CI for phi in an AR(1) model
VIII. Model Diagnostics
  A. Residual Analysis to Check Model Fit
    1. (Standardized) residuals plotted against time
    2. Normal Q-Q plots of residuals
    3. ACF plots of residuals
    4. Ljung-Box and Runs Tests
  B. Overfitting approach
IX. Forecasting
  A. Forecasting with Simple Deterministic Trend Models
  B. Formula for forecasts with an AR(1) model
  C. Effect of presence of constant term on the forecasts with a nonstationary model
  D. Getting/Interpreting Forecasts using R
    1. Idea of Lead time
    2. Behavior of forecasts with stationary models when the lead time increases
    3. Subtleties related to forecasting with nonstationary models
  E. Prediction Intervals and interpreting R output
X. Seasonal Models
  A. Concept of the Seasonal Period
  B. Seasonal MA(Q) models and Seasonal AR(P) models and their model equations
    1. For which lags are the autocorrelations zero?
    2. Using the ACF and PACF to identify AR(P) or MA(Q) models
  C. Seasonal ARMA models
  D. Concept of Seasonal Differencing and why it is useful
  E. Definition of a SARIMA(p,d,q)x(P,D,Q)_s model
XI. Relationships between Time Series; Outliers
  A. Response series Y_t and Covariate series X_t
  B. Cross-Correlation function (lag-0 and lag-k)
  C. One series "leading" another
  D. Prewhitening to avoid spurious correlations
  E. Regression with Time Series with Autocorrelated Errors
    1. Specifying ARMA model for noise process
  F. Additive Outliers and Innovative Outliers and the Distinction between them