Time Series and Index Numbers
What the exam tests
Time series analysis identifies trend, seasonal, cyclical and irregular components and uses moving averages or least squares to estimate trend. Index numbers measure relative change over time, especially price and cost-of-living indices.
Core rules
- Components: long-term trend (T), seasonal (S), cyclical (C) and irregular (I). Multiplicative model: Y = T × S × C × I; additive model: Y = T + S + C + I.
- Moving average: average consecutive k values; for even k, centre the averages by averaging adjacent pairs. Choose k equal to the seasonal period to remove seasonality.
- Least-squares trend: ŷ = a + bt with t coded so Σt = 0; b = Σty/Σt² and a = Σy/n.
- Price relative = (p1/p0) × 100. Simple aggregative index = (Σp1/Σp0) × 100.
- Laspeyres: L = (Σp1q0/Σp0q0) × 100. Paasche: P = (Σp1q1/Σp0q1) × 100.
- Fisher's ideal index: F = √(L × P), the geometric mean of Laspeyres and Paasche.
- Index numbers have no units; the base-year value is 100.
Worked example
Base: p0 = (2, 5), q0 = (10, 4). Current: p1 = (3, 6), q1 = (8, 5). Laspeyres = [(3×10 + 6×4)/(2×10 + 5×4)] × 100 = (54/40) × 100 = 135. Paasche = [(3×8 + 6×5)/(2×8 + 5×5)] × 100 = (54/41) × 100 = 131.7. Fisher = √(135 × 131.7) ≈ 133.3.
Traps
Swapping base and current quantities in Laspeyres or Paasche; using an arithmetic mean instead of a geometric mean for Fisher; failing to centre an even-order moving average; comparing index numbers with different base years; mistaking a seasonal rise for trend.