Time Series and Index Numbers
Time series and index numbers analyse change over time. The exam tests the components of a series, moving-average trend, seasonal variation, and the construction and interpretation of Laspeyres, Paasche and Fisher index numbers.
Core rules
- A time series has trend (long-term direction), seasonal variation (repeating within a year), cyclical variation (longer swings) and irregular variation.
- A moving average smooths short-term fluctuations to reveal trend; the period should match the seasonal cycle, e.g. a 4-quarter moving average for quarterly data.
- Simple price index = (p₁/p₀) × 100. A base year should be normal, not abnormal.
- Laspeyres index uses base-year quantities: (Σp₁q₀/Σp₀q₀) × 100. Paasche uses current quantities: (Σp₁q₁/Σp₀q₁) × 100.
- Fisher’s ideal index is the geometric mean of Laspeyres and Paasche: √(L × P). It is often preferred because it uses both quantity sets.
Worked example
Take two commodities with base prices p₀ = 2 and 4, base quantities q₀ = 10 and 5, and current prices p₁ = 3 and 5. Base expenditure is Σp₀q₀ = 2(10) + 4(5) = 40. At current prices with base quantities, Σp₁q₀ = 3(10) + 5(5) = 55. The Laspeyres price index is (55/40) × 100 = 137.5, meaning a 37.5% rise in the base-basket cost. If current quantities were q₁ = 8 and 6, the Paasche denominator would be Σp₀q₁ = 2(8) + 4(6) = 40 and the numerator Σp₁q₁ = 3(8) + 5(6) = 54, giving 135. Fisher’s index would then be √(137.5 × 135) ≈ 136.2.
Traps
- Weighting a price index by current quantities when the question asks for Laspeyres.
- Using a strike-affected or shortage year as base.
- Confusing seasonal with cyclical movement.
- Using a moving average of the wrong period and leaving endpoints without trend values.