Time Series Analysis and Forecasting
This domain tests decomposition of a series into trend, seasonal, cyclical and irregular components, and simple forecasting. The exam favours moving averages, seasonal indices and least-squares trend lines.
Core rules
- Additive model: Y = T + S + C + I; multiplicative model: Y = T × S × C × I.
- A moving average smooths a series to estimate trend. For an even period such as 4 quarters, centre the average by averaging two consecutive moving averages.
- Seasonal index: in a multiplicative model the indices average 1 (or 100%); in an additive model they sum to 0.
- Deseasonalise by dividing by the seasonal index (multiplicative) or subtracting it (additive).
- Forecast = trend estimate × seasonal index (multiplicative).
- Least-squares trend: b = (nΣty − ΣtΣy)/(nΣt² − (Σt)²), a = ȳ − b t̄, with t coded consecutively.
Worked example
A quarterly sales series has a seasonal index of 1.15 for Q1. Actual Q1 sales are 230 units. Deseasonalised sales = 230/1.15 = 200. If the trend line predicts 210 for the next Q1, the seasonal forecast is 210 × 1.15 = 241.5 units. The reasoning is that the observed figure contains both trend and seasonal effects; dividing removes the seasonal effect before forecasting.
Traps
- Using an uncentred 4-quarter moving average, which misaligns the trend with the original periods.
- Forgetting to rescale seasonal indices so they average 1.
- Adding a multiplicative seasonal index instead of multiplying.
- Extrapolating a trend far beyond the data range.