STA 364 Spring 2025

Exam 2 Outline

This in-class exam is worth 100 points. The take-home is worth 100 points.

Outline

  1. Content in Exam 1 Outline HERE.

  2. ETS Models

  • Smoothing models with trend
  • Smoothing models with seasonality
  • Identifying when to use multiplicative models vs additive models
  • Comparing models
  • How ETS() auto selection function works.
  • Given the forecast variance formula, \(\hat{\sigma}_h\), calculate the prediction interval.
  1. ARIMA Models
  • Stationarity: Identify if a series is or is not stationary.
  • Differencing: What is it? What kinds? Why?
  • Backshift notation: What is it, and what does it do?
  • Autoregressive Models (AR): Conditions, how to fit, model equation.
  • Moving Average Models (MA): Conditions, how to fit, model equation.
  • ARIMA Models (see below): Write out the specific equation using the general equation and a 𝑝,𝑑, and 𝑞 (small) identifying the orders (\(p\) and \(q\)) by looking at a stationary series’s correlogram and PACF plot.
  • SARIMA Models: Same as ARIMA but with a season piece.
  • ARIMA vs ETA
  • ARIMA() function default algorithm.
  1. Overall
  • Fitting and comparing models on a test set
  • Forecasting with all models

ARIMA Process

  1. Plot the data. Identify any unusual observations.

  2. If necessary, transform the data (using a Box-Cox transformation) to stabilize the variance.

  3. If the data are non-stationary: take first differences of the data until the data are stationary.

  4. Examine the ACF/PACF (of the stationary series): Is an AR(\(p\)) or MA(\(q\)) model appropriate?

  1. AR(\(p\)): We have an AR(\(p\)) model when
  • the ACF is exponentially decaying or sinusoidal
  • there is a significant spike at lag \(p\) in PACF, but none beyond \(p\)
  1. MA(\(q\)): We have an MA(\(q\)) model when
  • the PACF is exponentially decaying or sinusoidal
  • there is a significant spike at lag \(q\) in ACF, but none beyond \(q\)

  1. Try your chosen model(s), and use the AICc to search for a better model.

  2. Check the residuals from your chosen model by plotting the ACF of the residuals, and doing a portmanteau test of the residuals. If they do not look like white noise, try a modified model.

  3. Once the residuals look like white noise, calculate forecasts.

Code Bank

You can also find the take-home exam code bank by clicking HERE.