Tests of Consistency 🧪

How do we know if a formula is "Good"? Statisticians have developed certain mathematical tests that a perfect index number formula should satisfy.

There are 4 main tests:

1. Unit Test
2. Time Reversal Test
3. Factor Reversal Test
4. Circular Test

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1. Unit Test 📏

Requirement: The index number formula should be independent of the units in which prices and quantities are quoted.

• Whether we quote Milk in Litres or Millilitres, the Index Number answer should remain the same.

Who passes?

• All formulas EXCEPT the Simple Aggregative Method satisfy this test.
• This is why Simple Aggregative is considered poor.

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2. Time Reversal Test (TRT) ⏳

Proposed by: Irving Fisher.

Requirement: If we reverse the time periods (make Base Year = Current and Current Year = Base), the index number should be the reciprocal of the original.

P01 * P10 = 1

• P01: Index of year 1 on base 0.
• P10: Index of year 0 on base 1.

Who passes?

• Fisher’s Ideal Index ✅
• Marshall-Edgeworth Method ✅
• Simple Geometric Mean of Price Relatives ✅
• Laspeyres and Paasche DO NOT pass. (Because Laspeyres uses fixed q0, if you reverse time, it would use q1, changing the formula structure).

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Checking Fisher's TRT (Proof) 🧩

Formula for P01:

√(∑ p1 q0 / ∑ p0 q0) * (∑ p1 q1 / ∑ p0 q1)

Formula for P10 (Swap 0 and 1):

√(∑ p0 q1 / ∑ p1 q1) * (∑ p0 q0 / ∑ p1 q0)

Multiply P01 * P10:

√ [ (∑ p1 q0 / ∑ p0 q0) * (∑ p1 q1 / ∑ p0 q1) ] * [ (∑ p0 q1 / ∑ p1 q1) * (∑ p0 q0 / ∑ p1 q0) ]

• ∑ p1 q0 cancels with denominator.
• ∑ p0 q0 cancels with numerator.
• ∑ p1 q1 cancels with denominator.
• ∑ p0 q1 cancels with numerator.

Result = √1 = 1. Passed!

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Summary

Test	Condition	Who Passes?
Unit Test	Independent of units	All except Simple Aggregative
Time Reversal	P01 * P10 = 1	Fisher, Marshall-Edgeworth

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