Page 257 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Unit 20: Test of Consistency: Unit Test, Time Reversal Test, Factor Reversal Test and Circular Test

Notes
∑ p q
11
Here P 01 × Q 01 ≠ ∑ p q
00
Paasche’s Method

∑ p q ∑qp
11
11
P 01 = ∑ p q and Q 01 = ∑qp
01
01
∑ p q
P × Q ≠ 11
Here also 01 01 ∑ p q
00
This test is also satisfied by Fisher’s Method only.
∑ p q ∑ p q
11
10
P 01 = ∑ p q × ∑ p q
0 1
00
∑ 10 ∑qp qp
11
Q 01 = ∑ 00 × ∑qp qp
01
∑ 10 ∑ p q 11 ∑ p q 1 0 ∑q p q p
1 1
P × Q 01 = ∑ 00 × ∑ p q 01 × ∑ p q 00 × ∑ p q q p
01
0 1
⎛ pq ⎞ ∑ 2 ∑ p q
11
⎜ 11 ⎟ = = Value Index
00 ⎠
⎝ ∑ pq ∑ p q
00
Hence, Fisher’s Formula satisfies this test.
20.4 Circular Test

Another test of adequacy is circular test. This test is an extension of the time reversal test. It requires
that if an index is constructed for the year ‘b’ on year ’a’ and for the year ‘c’ on base year ‘b’ we should
get the same result as if we calculate directly an Index for ‘c’ on base year ‘a’ without going through
b as an intermediary.
Mathematically,
Let year a, b and c are denoted by 0, 1 and 2 respectively.
Condition of the test:
P × 01 P × 12 P = 1
20
For example, suppose prices have doubled in year 1 as compared to year 0, and again prices have
doubled in year 2 as compared to year 1, in such a case if we correlated the prices of year 2 and 0 then
1
we will find that prices of year 0 were of the prices of year 2.
4
Mathematically,
1
P 01 = 2; 12 = 2; 20 = 4
P
P
1
Here, P × 01 P × 12 P 20 = 22×× 4 = 1

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