Activity30

1. Core Concept: Time Series Anatomy with ETS

Why ETS? The ETS framework explicitly models three core components—level, trend, and seasonality—using a recursive structure. This differs from ARIMA, which typically uses differencing to remove trend or seasonality.

  1. Level (\(l_t\)): Baseline value
  2. Trend (\(b_t\)): Persistent upward/downward movement
  3. Season (\(s_t\)): Regular repeating pattern

1.1 Additive ETS Equations

For an Additive error, trend, and seasonality model, often denoted ETS(A,A,A):

\[ \begin{align} y_t &= l_{t-1} + b_{t-1} + s_{t-m} + \epsilon_t, \\ l_t &= l_{t-1} + b_{t-1} + \alpha \epsilon_t, \\ b_t &= b_{t-1} + \beta \epsilon_t, \\ s_t &= s_{t-m} + \gamma \epsilon_t, \end{align} \]

where \(m\) is the seasonal period (e.g., \(m=12\) for monthly data, \(m=4\) for quarterly), and \(\alpha, \beta, \gamma\) are smoothing parameters. A small \(\beta\) indicates a very slow‐changing trend, while a small \(\gamma\) indicates very stable seasonality.

Key Differences from ARIMA

  • ETS: Trend/seasonality are explicitly updated.
  • ARIMA: Trend/seasonality are removed by differencing.
  • ETS: Weighted averages via smoothing.
  • ARIMA: Linear combinations of past values and errors.

2. Extended Example: Tourism Demand in Sydney

Below is a quick demonstration of fitting an ETS model to tourism data in Sydney.

3. Activity: US GDP Forecasting

Here, we illustrate how to scale GDP by dividing by billions and compare ETS vs. ARIMA approaches in a fair manner. We also consider more sophisticated ETS variants (e.g., damped trend).

3.1 Data Preparation & Scaling

We now have a GDP_billions column that is easier to interpret than raw GDP (which can be in the trillions).

3.2 Simple ETS vs. ARIMA

Model A: Simple Exponential Smoothing (SES)

This model handles level only (no trend, no seasonality). In ETS notation: ETS(A,N,N).

Model B: ARIMA

We compare it with a simple differenced ARIMA(0,1,1). That is:

\[ y_t \;=\; y_{t-1} \;+\; \epsilon_t \;+\; \theta\,\epsilon_{t-1}. \]

3.3 More Sophisticated ETS Models

To capture trend, we might consider a damped trend approach, e.g. ETS(A,Ad,N), where:

\[ b_t = \phi\,b_{t-1} + \beta \epsilon_t, \quad 0 < \phi < 1. \]

This damping factor \(\phi\) shrinks the trend over time, preventing runaway forecasts.

Lab Activities

Activity 1: Simple Exponential Smoothing (SES)

  1. Fit an ETS(A,N,N) model to US GDP (in billions).
  2. Extract the smoothing parameter \(\alpha\).
  3. Interpret what \(\alpha\) implies about how quickly the model reacts to new data.
Answer

Activity 2: ARIMA with Automatic Selection

  1. Use ARIMA(GDP_billions) with default auto‐selection.
  2. Compare the chosen order \((p,d,q)\) with a manually specified ARIMA(0,2,2).
  3. Plot the forecasts to see which better captures the data trend.
Answer

Activity 3: Damped Trend ETS

  1. Fit an ETS(A,Ad,N) model.
  2. Inspect the damping parameter \(\phi\).
  3. If \(\phi\) is close to 1, interpret how that affects the forecast horizon.
Answer

Activity 4: Forecast Accuracy Comparison

  1. Generate 8‐year forecasts with each model (SES, ARIMA, Damped ETS).
  2. Compare the accuracy metrics (RMSE, MAE, MAPE).
  3. Conclude which model best fits US GDP data in billions.
Answer