Time Series and Index Numbers
Domain 8: Time Series and Index Numbers
This domain tests the decomposition of a time series into its components, the methods for isolating trend and seasonality, and the construction and testing of price index numbers.
Core rules
- Components: T (long-term trend), S (seasonal, period one year or less), C (cyclical, multi-year), I (irregular). Models: additive Y = T + S + C + I or multiplicative Y = T × S × C × I.
- Trend by moving averages: the period equals the seasonal length; an even period needs a centred two-stage average (four-quarter averages averaged in pairs).
- Trend by least squares: fit y = a + bx with x measured as deviations from the middle period, so Σx = 0, a = ȳ and b = Σxy/Σx².
- Seasonal indices by simple averages: index = (seasonal average / overall average) × 100, adjusted to total 400 for four quarters.
- Price relative = (p₁/p₀) × 100. Laspeyres L = Σp₁q₀/Σp₀q₀ × 100; Paasche P = Σp₁q₁/Σp₀q₁ × 100; Fisher's ideal F = √(L × P).
- Fisher satisfies the unit, time reversal and factor reversal tests; Laspeyres and Paasche fail time and factor reversal; only chain-base indices satisfy the circular test.
Worked example
Base year: p₀ = (2, 5), q₀ = (10, 4). Current year: p₁ = (3, 8), q₁ = (12, 3). Laspeyres = (3×10 + 8×4)/(2×10 + 5×4) × 100 = 62/40 × 100 = 155.0. Paasche = (3×12 + 8×3)/(2×12 + 5×3) × 100 = 60/39 × 100 = 153.8. Fisher = √(155.0 × 153.8) = 154.4. Reasoning: Laspeyres weights by base-year quantities and overstates the rise when consumers switch away from commodities whose prices rose most; Paasche weights by current quantities and understates it; Fisher's geometric mean balances the two biases.
Traps
- Using q₁ in the Laspeyres denominator or q₀ in the Paasche denominator.
- Failing to centre an even-period moving average, which misaligns the trend with the original data.
- Leaving seasonal indices that do not total 400 (or 1200 for months).
- Treating an index number as a rupee value; it is a pure, unit-free ratio.
- Choosing a base year distorted by drought, war or another abnormal event.