Page 256 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes words, Simple Aggregative Method does not satisfy this test. The index number reckoned forward
should be the reciprocal of that reckoned backward’. Thus, an ideal Index Number formula should
work both ways, i.e., forward as well as backward.
Mathematically, the following relation should be satisfied:
P 01 × P 10 = 1 or P ×
10
100 100 01 P = 1
Laspeyre’s Method

∑ p q ∑ p q ∑ pq ∑ p q
01
10
10
0 1
P ×
P
P 01 = ∑ p q and 10 = ∑ p q ; 01 P 10 = ∑ pq × ∑ p q ≠ 1
1 1
11
00
00
P × P ≠ 1 hence, test is not satisfied.
Q 01 10
Paasche’s Method
∑ p q ∑ p q
00
11
P 01 = ∑ p q and P = ∑ p q
10
01
10
∑ pq ∑ p q
P × P 11 × 0 0 ≠ 1 , hence, test is not satisfied.
Here, also 01 01 = ∑ pq ∑ p q
1 0
01
The test is not satisfied by Laspeyre’s and the Paasche’s method. However, Fisher’s Method satisfies
the test.
∑ p q ∑ p q ∑ p q ∑ p q
00
11
01
10
P 01 = ∑ p q × ∑ p q and P = ∑ p q × ∑ p q
10
00
11
10
0 1
∑ p q ∑ p q ∑ p q ∑ p q
0 1
11
10
0 0
P × P 10 = ∑ p q × ∑ p q × ∑ p q × ∑ p q
01
11
0 1
10
0 0
i.e., P × P 10 = 1
01
or P × P 10 = 1
01
P × P
Since, 01 10 = 1, the Fisher’s Ideal Index satisfies the test.
20.3 Factor Reversal Test
This is another test suggested by Fisher. According to Fisher Just as our formula should permit the
interchange of two factors without giving inconsistent results, so without to permit interchange of
prices and quantities without giving inconsistent results, i.e., the two results multiplied together
should give the true value ratio.
Mathematically,
P 01 × Q 01 ∑ p q P × ∑ p q
11
11
100 100 = ∑ p q or 01 Q 01 = ∑ p q
00
00
Laspeyre’s & Paasche’s method does not satisfy this test also like the above test.
∑ p q ∑qp
10
10
P 01 = ∑ p q and Q 01 = ∑qp
00
00
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