Time Series Analysis
Time series questions ask you to name and interpret the components, to smooth a series by moving averages, and to fit a linear trend by least squares for forecasting. The arithmetic is light but the placement of moving averages and the coding of time are easy to get wrong.
Components
A time series has four components: secular trend T (long-term direction), seasonal variation S (pattern within a year), cyclical variation C (multi-year waves) and irregular or random movement I. Additive model: Y = T + S + C + I. Multiplicative model: Y = T × S × C × I, where the seasonal index is expressed as a percentage.
Measuring trend
Moving averages: for odd order m = 2k + 1, average the k values on each side, which centres the average on a period. For even order such as 4, a single moving average falls between periods, so a centred 2 × 4 moving average is required. Least squares: fit y = a + bx; coding x as deviations from the middle period gives Σx = 0, hence a = Σy/n and b = Σxy/Σx².
Worked example
Yearly sales for five years are 10, 12, 13, 15, 16. With x = −2, −1, 0, 1, 2: Σy = 66, so a = 13.2. Σxy = −20 − 12 + 0 + 15 + 32 = 15 and Σx² = 10, so b = 1.5. The trend line is y = 13.2 + 1.5x with x = t − 3. Forecast for year 6 (x = 3): 13.2 + 4.5 = 17.7.
Traps
- Writing an even-order moving average against a period instead of centring it between periods.
- Using the additive model when the seasonal swing grows with the level of the series, where multiplicative fits.
- Forgetting to subtract the middle value when coding x, so Σx ≠ 0.
- Reading a seasonal peak as part of the trend.