There are certain tests which are put to verify the consistency, or adequacy of an index number formula from different points of view. The most popular among these are the following tests:
- Order reversal test.
- Time reversal test.
- Factor reversal test.
- Circular test.
- Unit test.
At the outset, it should be noted that it is neither possible nor necessary for an index-number formula to satisfy all the tests mentioned above. But, an ideal formula should be such that it satisfies the maximum possible tests which are relevant to the matter under study. However, the various tests cited above are explained here as under:
1. Order reversal test
This test requires that a formula of Index number should be such that the value of the index number remains the same, even if, the order of arrangement of the items is reversed, or altered. As a matter of fact, this test is satisfied by all the twelve methods of index number explained above.
2. Time reversal test
This test has been put forth by Prof. Irving Fisher, who proposes that a formula of index number should be such that it turns the value of the index number to its reciprocal when the time subscripts of the formula are reversed i.e. 0 is made 1,and 1 is made 0. According to this proposition, if the index number of the current period on the basis of the current period i.e.
P01 is 200, the index number of the base period on the basis of the current period i.e. P10 would be 50. Thus, when the value of is 2 times the base year price, the value of P10 is. As such, an index number formula, in order to satisfy this test must prove the following equation:
P01 x P10 = 1
As a matter of fact, this test is satisfied by most of the formula of index number except those of Laspeyre and Paasche which is shown in the table as on the next page.
Besides the seven methods shown in the table, both the simple and weighted geometric mean of piece relatives, also, satisfy this time reversal test.
3. Factor reversal test
This test has also been purforth by Prof. Irving Fisher, who proposes that a formula of index number should be such that it permits the interchange of the price, and the quantity factors without giving inconsistent result i.e. the two results multiplied together should give the true ratio in as much as th4e product of price and quantity is the value of a thing.
Thus, for the Factor Reversal test, a formula of index number should satisfy the following equation:
Price index × Quantity Index = Value Index
∴ P01 x Q01 = V01 , ∑P1 Q1/ ∑P0 Q0
Most of the formulae of index number discussed above fail to satisfy this acid test of consistency except that of Prof. Irving Fisher. This is the reason for which Prof. Fisher claims his formula to be an ideal one. The table displayed on page 631 bears a testimony to the test mentioned above.
4. Circular test
This test has been purforts by Westergaard and recommended by C.M. Walsch in extension of the times reversal test purforth by Prof. Fisher. This test requires that an index number formula should be such that an index number formula should be such that it works in a circular fashion. This means that if an index is computed for the period 1 on the base period 0, another index is computed for the period 2 on the base period 0 on the base period 2, the product of all these indices should be equal to 1. Thus, a formula to satisfy the test should comply with the following equation
P01 × P12 × P20 = 1.
An index formula which satisfies this test enjoys the advantage of reducing the computation work every time a change in the base year is made. As it will be seen from the table exhibited on page 632, this test is not satisfied by most of the important index formula viz. Fisher’s, Laspeyre’s, Paasche’s, Marshall and Edge worth’s, Drobish and Bowley’s etc.
However, the following three methods satisfy the test:
(i) Simple aggregative method
(ii) Weighted aggregative method
(iii) Kelley’s method.
5. Unit test
This is a common test which requires that an index number formula should be such that it does not affect the value of the index number, even if, the units of the price quotations are altered viz. price per kg, converted into price per quintal or vice versa. This test is satisfied by all the index formula except the simple aggregative method under which the value of the index number changes radically, if the units of price quotations of any of the items included in the index number are changed.