Time Series and Index Numbers
This domain covers data observed over time and the construction of index numbers that track price or quantity change. Expect items on decomposition, smoothing, and the choice among index formulas and their known biases.
Components and smoothing
- A series is decomposed into trend, seasonal, cyclical and irregular components. The additive model is y = T + S + C + I; the multiplicative model is y = T times S times C times I.
- A moving average whose order equals the season length removes seasonality; an even order needs a second, centred averaging step because the first pass lands between periods.
- A seasonal index averages the detrended values for each season; adjust so additive indices sum to 0 and multiplicative indices sum to 400.
Index numbers
- Simple price relative: current price divided by base price times 100. A simple aggregate index sums current prices and divides by summed base prices, ignoring quantities entirely.
- Laspeyres index uses base-period quantities: sum of current price times base quantity divided by sum of base price times base quantity, times 100. It tends to overstate a cost-of-living increase because the basket is not updated.
- Paasche index uses current quantities and tends to understate it.
- Fisher's ideal index is the geometric mean of the Laspeyres and Paasche indices.
- To deflate, divide a nominal value by the price index and multiply by 100 to obtain a real value.
Worked example
Base prices 2 and 5, base quantities 10 and 4; current prices 3 and 6, current quantities 8 and 6. Laspeyres = (3 times 10 + 6 times 4) divided by (2 times 10 + 5 times 4) times 100 = 54/40 times 100 = 135. Paasche = (3 times 8 + 6 times 6) divided by (2 times 8 + 5 times 6) times 100 = 60/46 times 100 = 130.4. Fisher = root of 135 times 130.4 = 132.7. Laspeyres sits higher here because it keeps the base basket, which is weighted toward the item whose price rose most sharply.
Traps
- Averaging four quarterly seasonal indices and expecting 100.
- Using current quantities in a Laspeyres index or base quantities in a Paasche index.
- Splicing two index series without rescaling them to a common base.
- Comparing nominal values over time without deflating by a price index.