Circular Test ⭕
The Circular Test is an extension of the Time Reversal Test for more than two years.
The Concept 💡
If we shift the base year in a circular manner, the product of the indices should bring us back to unity (1).
Imagine 3 years: Year 0, Year 1, Year 2.
- Calc Index of Year 1 on Year 0 (
P_01) - Calc Index of Year 2 on Year 1 (
P_12) - Calc Index of Year 0 on Year 2 (
P_20)
The product of all three should be 1.
Formula
P_01 * P_12 * P_20 = 1
Who Passes? ✅
This is a very strict test.
- Simple Geometric Mean of Price Relatives: Passes ✅
- Fisher’s Index: Does NOT pass ❌
- Laspeyres & Paasche: Do NOT pass ❌
This implies that while Fisher's is "Ideal" for 2 years, it's not logically consistent for a chain of years in a circle. However, in practice, the error is usually small.
Use Case
This test is important when we want to change the base year (Base Shifting) without changing the relationship between the other years.
Summary of All Tests 📋
| Test | Formula | Passed By |
|---|---|---|
| Unit | Independent of units | All except Simple Aggregative |
| Time Reversal | P_01 * P_10 = 1 | Fisher, Marshall-Edgeworth, Simple GM |
| Factor Reversal | P_01 * Q_01 = V_01 | Fisher |
| Circular | P_01 * P_12 * P_20 = 1 | Simple GM of Price Relatives |
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