Time series can exhibit variety of patterns, which can be split into components representing different patterns

Extracting patterns & components yields two benefits:

• Understand series better
• Improve forecasts

Three types of patterns:

1. Trend: long-term increase or decrease in data
   • May change direction
2. Seasonal: influenced by seasonal factors
   • Always of fixed, known period
3. Cyclic: rises and falls not of fixed period

A series \(y_t\) has 3 components:

• seasonal component \(S_t\) (fixed-period variation)
• trend-cycle component \(T_t\) (long-term trend & non-fixed cycle)
• remainder component \(E_t\) (anything else in time series)

The data with the seasonal component removed (\(y_t - S_t\) or \(y_t / S_t\)) is the seasonally adjusted data.

• Useful if seasonal variation is not pattern of interest
• Comprises remainder and trend-cycle components
• Commonly used for economic data to study non-seasonal variation

Moving average of order \(m\) (\(m\)-MA): \[\hat{T}_t = \frac{1}{m} \sum_{j = -k}^k y_{t+j}\]

• Averages values \(y_j\) within \(k\) periods of \(t\)
• Eliminates some randomness, estimating trend-cycle
• Captures trend without minor fluctuation
• Increasing order increases smoothness

• The moving average of a moving average gives a smoother trend estimate
• Even-order MA of even-order MA will return a symmetric MA
  • i.e. takes same number of points before and after \(t\)
• Notation \(m_2 \times m_1\)-MA

Estimating Trend with Seasonal Data

Taking \(2 \times m\)-MA yields weighted MA of period \(m+1\) with first and last observations having half-weight. This can be used to view the trend absent seasonality

• Trend estimate unavailable for first & last \(m/2\) periods
  • Therefore no remainder \(\rightarrow\) all seasonality
• Assume seasonal component repeats from year to year
  • Unable to capture changes in seasonality over time
• Unable to detect unusual patterns

X-12-ARIMA

Method created to address shortcomings — 16-step process detailed in section 6/4 (pages 162-163). Requires proprietary software; no R package for performing this decomposition.

Seasonal and Trend decomposition using Loess (method for estimating nonlinear relationships) presents strengths over other methods:

• Handle any type of seasonality
• Seasonal component allowed to change over time
• User-controlled trend-cycle smoothness
• Robust to unusual observations

STL can only be used with additive decomposition – must log-transform multiplicative models

stl(x, s.window, t.window, ...)

A decomposed time series is written as:

• \(y_t = \hat{S}_t + \hat{A}_t\) with \(\hat{A}_t = \hat{T}_t + \hat{E}_t\) (additive)
• \(y_t = \hat{S}_t \hat{A}_t\) with \(\hat{A}_t = \hat{T}_t \hat{E}_t\) (multiplicative)

The two components \(\hat{S}_t\) & \(\hat{A}_t\) are forecast separately:

• \(\hat{S}_t\) assumed unchanging
  • Prior year's value taken (seasonal naive)
• \(\hat{A}_t\) forecast using non-seasonal method
  • Naive
  • Random walk with drift
  • ARIMA