Circular Test

Another test of the adequacy of index number formula is what is known as ‘circular test’. If in the use of index numbers interest attaches not merely to a comparison of two years, but to the measurement of price changes over a period of years. It is frequently desirable to shift the base. Clearly, the desirability of this property is that it enables us to adjust the index values from period to period without referring each time to the original base. A test of this shift ability of base. A test of this shift ability of base is called to the circular test.

This test is just an extension of the time reversal test. The test requires that if an index is constructed for the year a on base year b, and for the year b on base year c, we ought to get the same result as if we calculated direct an index for a on base year c without going through b as an intermediary.

Symbolically, if there are three indices P01, P12 and P20 the Circular test will be satisfied if
        P₀₁ X P₁₂ X P₂₀ = 1

The Laspeyres index does not satisfy the test as can be seen from the following:
If the three years are 0, 1, 2, the index by Laspeyres method will be


The product of all these is not equal to 1. Hence the test is not satisfied. Similarly, it can be shown that the Paasche’s index and Fisher’s index do not satisfy the test. However, the simple aggregative method and the fixed weight aggregative method satisfy the test as can be seen from the following:

When the test is applied to the simple aggregative method, we will get:


Similarly, when applied to fix weight aggregative method, we will get:


An index which satisfies this test has the advantages of reducing the computations every time a change in the base year has to be made. Such index numbers can be adjusted from year to year without referring each time to the original bases.

The circular test is not met by the ideal index or by any of weighted aggregative with changing weights. This test is met by simple geometric mean of price relatives and the weighted aggregative fixed weights.

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