Fundamentals of Business Mathematics and Statistics · Index Numbers and Time Series

Time Reversal and Factor Reversal Tests for Index Numbers

Updated 10 October 2026 · Fact-checked

Tests of adequacy check whether an index number formula behaves consistently. The time reversal test needs P01 × P10 = 1. The factor reversal test needs P01 × Q01 = Σp1q1 ÷ Σp0q0. Fisher's index satisfies both, so it is called ideal. To solve, compute the formula forwards and backwards, multiply, and compare.

Understand Tests of Adequacy: Time and Factor Reversal Tests

An index number can be built in many ways. Laspeyres, Paasche, Marshall-Edgeworth and Fisher all give different answers for the same data. So how do you judge whether a formula is any good? You apply tests. These are called tests of adequacy or tests of consistency.

The time reversal test asks: if you swap the base year and the current year, does the new index come out as the reciprocal of the old one? If prices rose 50% from 2023 to 2024, going back from 2024 to 2023 should show a fall to 1 ÷ 1.5 of the price level. So the two indices multiplied together should equal 1 (or 10,000 if both are written with the ×100).

The factor reversal test asks: if you swap the roles of price and quantity in the formula, does the product of the price index and the quantity index equal the true change in total value? The true value ratio is Σp1q1 ÷ Σp0q0, which is total spend in the current year over total spend in the base year. A good formula should split this value change cleanly into a price part and a quantity part.

The circular test extends the time reversal test to three or more years. It needs P01 × P12 × P20 = 1. It matters when you want to shift base without recomputing everything.

Fisher's index is the geometric mean of Laspeyres and Paasche. It passes both the time reversal and factor reversal tests. This is why it is called the ideal index. It does not pass the circular test in general.

Laspeyres and Paasche on their own generally fail both the time reversal and factor reversal tests. The Marshall-Edgeworth index satisfies the time reversal test but does not satisfy the factor reversal test.

Key formulas to remember

Time reversal test: P01 × P10 = 1; P10 is the same formula with base and current year swapped. If each index is written with ×100, the product is 10,000.
Factor reversal test: P01 × Q01 = Σp1q1 ÷ Σp0q0; Q01 is the quantity index formed by swapping p and q in the price index formula. The right side is the value ratio.
Circular test: P01 × P12 × P20 = 1; Extension of time reversal to three periods. Fisher's index does not satisfy it in general.
Fisher's price index: P01 = √[(Σp1q0 ÷ Σp0q0) × (Σp1q1 ÷ Σp0q1)] × 100; Geometric mean of Laspeyres and Paasche. Satisfies time reversal and factor reversal.
Fisher's quantity index: Q01 = √[(Σq1p0 ÷ Σq0p0) × (Σq1p1 ÷ Σq0p1)] × 100; Obtained from the price index by interchanging p and q.
Fisher's reversed index: P10 = √[(Σp0q1 ÷ Σp1q1) × (Σp0q0 ÷ Σp1q0)]; Swap the subscripts 0 and 1 everywhere in P01.
Marshall-Edgeworth price index: P01 = [Σp1(q0 + q1) ÷ Σp0(q0 + q1)] × 100; Satisfies the time reversal test but does not satisfy the factor reversal test.

How to solve Tests of Adequacy: Time and Factor Reversal Tests questions

Use this order for any numerical question that asks you to verify or test an index formula.

1Write the data in a table with p0, q0, p1, q1 and add columns for p0q0, p1q0, p0q1 and p1q1.
2Find the four totals: Σp0q0, Σp1q0, Σp0q1 and Σp1q1.
3Write the index formula you are testing (usually Fisher) and compute P01 as a ratio, without ×100 for now.
4For the time reversal test, write P10 by swapping 0 and 1 everywhere, compute it, and check that P01 × P10 = 1.
5For the factor reversal test, write Q01 by swapping p and q in P01, compute it, and find P01 × Q01.
6Compute the value ratio Σp1q1 ÷ Σp0q0 and compare it with P01 × Q01.
7State the conclusion clearly: the test is satisfied if both sides are equal, otherwise it is not.

Quickest way: Use the four totals only

When to use it: Use this when the question gives a full table and four options with numerical values.

Compute only Σp0q0, Σp1q0, Σp0q1 and Σp1q1. Every test uses just these four numbers.
Note that for Fisher, P10 is the reciprocal of P01 by structure, so the time reversal product is exactly 1. Do not waste time if you only need to name the test result.
For factor reversal, remember that P01 × Q01 simplifies to Σp1q1 ÷ Σp0q0 for Fisher. Compute just that ratio to match an option.
If the question asks which formula satisfies both tests, pick Fisher. If it asks which fails both, think of Laspeyres or Paasche.
Keep fractions unsimplified until the end so square roots come out as perfect squares.

Common mistakes in Tests of Adequacy: Time and Factor Reversal Tests

Swapping only the prices in P10 and leaving the quantities as they were.
Students think reversing time means reversing only the price columns.
Fix: In P10, swap every 0 and 1 in the whole formula. Base year becomes current year for both p and q.
Getting Q01 by just reading off a different column without swapping p and q.
The quantity formula looks similar to the price formula and gets mixed up.
Fix: Take the price formula and replace every p with q and every q with p. Write Q01 in full before computing.
Checking time reversal as P01 × P10 = 100 when the indices are in ratio form, or 1 when they include ×100.
The ×100 is applied in some steps but not others.
Fix: Choose one form. In ratio form the product is 1. With ×100 on both, the product is 10,000.
Saying Laspeyres or Paasche satisfies the factor reversal test, or that Marshall-Edgeworth satisfies it.
They are familiar formulas, so students assume they pass every test.
Fix: Remember that Laspeyres and Paasche generally fail both the time and factor reversal tests. Marshall-Edgeworth passes time reversal but fails factor reversal. Fisher is the one that passes both.
Claiming Fisher's index satisfies the circular test.
Students over-generalise the word ideal.
Fix: Fisher passes time reversal and factor reversal only. It generally fails the circular test.
Rounding the square root early and then finding the product is not exactly 1.
Decimal approximations of P01 and P10 do not multiply to exactly 1.
Fix: Keep values as fractions inside the root and simplify at the end, or multiply the two expressions under one root.

Worked examples

Example 1

For two items, the data are: Item A: p0 = ₹2, q0 = 10, p1 = ₹3, q1 = 5. Item B: p0 = ₹4, q0 = 5, p1 = ₹6, q1 = 10. Verify the time reversal test for Fisher's index.

Show the solution

Σp0q0 = 2×10 + 4×5 = 20 + 20 = 40.
Σp1q0 = 3×10 + 6×5 = 30 + 30 = 60.
Σp0q1 = 2×5 + 4×10 = 10 + 40 = 50.
Σp1q1 = 3×5 + 6×10 = 15 + 60 = 75.
P01 = √[(60 ÷ 40) × (75 ÷ 50)] = √(1.5 × 1.5) = 1.5.
P10 is found by swapping 0 and 1: P10 = √[(Σp0q1 ÷ Σp1q1) × (Σp0q0 ÷ Σp1q0)] = √[(50 ÷ 75) × (40 ÷ 60)] = √(2/3 × 2/3) = 2/3.
P01 × P10 = 1.5 × 2/3 = 1.

Answer: P01 × P10 = 1, so Fisher's index satisfies the time reversal test. With ×100 on each index, the indices are 150 and 200/3 (about 66.67). Their product is 150 × 200/3 = 10,000, so the product ÷ 10,000 = 1.

Example 2

Using the same data (Item A: p0 = ₹2, q0 = 10, p1 = ₹3, q1 = 5; Item B: p0 = ₹4, q0 = 5, p1 = ₹6, q1 = 10), verify the factor reversal test for Fisher's index.

Show the solution

From the totals: Σp0q0 = 40, Σp1q0 = 60, Σp0q1 = 50, Σp1q1 = 75.
Fisher's price index in ratio form: P01 = √[(Σp1q0 ÷ Σp0q0) × (Σp1q1 ÷ Σp0q1)] = √[(60 ÷ 40) × (75 ÷ 50)] = 1.5.
Swap p and q to get the quantity index: Q01 = √[(Σq1p0 ÷ Σq0p0) × (Σq1p1 ÷ Σq0p1)].
Here Σq1p0 = 50, Σq0p0 = 40, Σq1p1 = 75 and Σq0p1 = 60.
Q01 = √[(50 ÷ 40) × (75 ÷ 60)] = √(1.25 × 1.25) = 1.25.
P01 × Q01 = 1.5 × 1.25 = 1.875.
Value ratio = Σp1q1 ÷ Σp0q0 = 75 ÷ 40 = 1.875.

Answer: P01 × Q01 = 1.875 = Σp1q1 ÷ Σp0q0, so Fisher's index satisfies the factor reversal test.

Exam tips

Most questions are MCQs that ask which formula satisfies a given test. Memorise: Fisher passes time reversal and factor reversal, and generally fails the circular test.
For a numerical MCQ, compute only the four totals and use them. Do not build full index tables unless required.
Check whether the options show the index with ×100 or in ratio form before you choose the answer.
For a circular test question with P01 and P12 given, the missing index is P20 = 1 ÷ (P01 × P12). Work it as a simple reciprocal.
There is no negative marking, so always mark an answer. Eliminate options that break the rule, such as a time reversal product that is not 1.

Practice questions from Index Numbers and Time Series

Tests of Adequacy: Time and Factor Reversal Tests: frequently asked questions

Why is Fisher's index called the ideal index?

Fisher's index is the geometric mean of Laspeyres and Paasche. It satisfies both the time reversal test and the factor reversal test, which most other formulas do not. That is why it is called ideal. It still does not satisfy the circular test in general.

What does the time reversal test check?

It checks that swapping the base year and the current year gives the reciprocal index. In symbols, P01 × P10 = 1 when indices are in ratio form. If each index has ×100, the product should be 10,000.

How do I verify the factor reversal test in a numerical?

Compute the price index P01 and the quantity index Q01 by swapping p and q in the same formula. Multiply them and compare with Σp1q1 ÷ Σp0q0. If they are equal, the test is satisfied.

Do Laspeyres and Paasche pass these tests?

In general, no. Laspeyres and Paasche generally fail both the time reversal test and the factor reversal test. Taking their geometric mean gives Fisher's index, which passes both. Marshall-Edgeworth passes time reversal but not factor reversal.