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So far we have discussed various formulae for construction of weighted & unweighted index numbers.

However the problem still remains of selecting an appropriate method for the construction of an index number in a given situation. The following tests can be applied to find out the adequacy of an index number.

(1) Unit Test

(2) Time Reversal Test

(3) Factor Reversal Test

(4) Circular Test

**1. Unit Test -** This test requires that the index number formulae should be independent of the units in which prices or quantities of various commodities are quoted. For example in a group of commodities, while the price of wheat might be in kgs., that of vegetable oil may be quoted in per liter & toilet soap may be per unit.

Except for the simple (unweighted) aggregative index, all other formulae discussed above satisfy this test.

**2. Time Reversal Test -** The time reversal test is used to test whether a given method will work both backwards & forwards with respect to time. The test is that the formula should give the same ratio between one point of comparison & another no matter which of the two is taken as base.

The time reversal test may be stated more precisely as follows— If the time subscripts of a price (or quantity) index number formula be interchanged, the resulting price (or quantity) formula should be reciprocal of the original formula. i.e. if p_{0 }represents price of year 2011 and p_{1} represents price at year 2012 i.e.

should be equal to

symbolically, the following relation should be satisfied p_{01} x p_{10} = 1, Omitting the factor 100 from both the indices.

Where P_{01} is index for current year ‘1’ based on base year ‘0’ p_{l0} is index for year ‘0’ based on year ‘1’.

The methods which satisfy the following test are:-

(1) Simple aggregate index

(2) Simple geometric mean of price relative

(3) Weighted geometric mean of price relative with fixed weights

(4) Kelly’s fixed weight formula

(5) Fisher’s ideal formula

(6) Marshall-Edgeworth formula

This test is not satisfied by Laspeyres’ method & the Paasche’s method as can be seen from below—

1 (Laspeyres'Method)

Similarly when Paasche method is used—

On other hand applying Fisher’s formula

(Omitting the factor 100)

(Omitting the factor 100)

Hence the test is satisfied.

**3. Factor Reversal Test **- An Index number formula satiesfies this test if the product of the Price Index and the Quantity Index gives the True value ratio, omitting the factor 100 from each index. This test is satisfied if the change in the price multiplied by the change in quantity is equal to the change in the value.

Speaking precisely if p and q factors in a price (or quantity) index formula be interchanged, so that a quantity (or price) index formula is obtained the product of the two indices should give the true value ratio.

Symbolically,

= The True Value Ratio = TVR

Consider the Laspeyres formula of price index

Consider the quantity index by interchange p with q & q with p

Now

This test is not met.

This test is only met by Fisher’s ideal index. No other index number satisfies this test:

**Proof :**

Changing p to q and q to p, we get

= True Value Ratio

**4. Circular Test -** Circular test is an extension of time reversal test for more than two periods & is based on shiftability of the base period. For example, if an index is constructed for the year 2012 with the base of 2011 & another index for 2011 with the base of 2010. Then it should be possible for us to directly get an index for the year 2012 with the base of 2010. If the index calculated directly does not give an inconsistent value, the circular test is said to be satisfied.

This test is satisfied if— P_{01 }x P_{12 }x P_{20}= 1.

This test is satisfied only by the following index Nos. formulas—

(1) Simple aggregative index

(2) Simple geometric mean of price relatives

(3) Kelly’s fixed base method When the test is applied to simple aggregative method—

Hence, the simple aggregative formula satisfies circular test Similarly when it is applied to fixed weight Kelly’s method

This test is not satisfied by Fishers ideal index.

**Example 7 : **Compute Fisher’s Ideal Index and show that it satisfies time Reversal Test

**Solution :**

As in this problem value of each item is given we have to find out quantity by dividing the value by the price. Symbolically : Value = Price x Quantity

**Table : Calculation of Fisher’s Ideal Index**

Time reversal test is satisfied when

Since P_{01} x P_{10} = 1 hence Fisher’s ideal index satisfies time Reversal Test.

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