Tests of Adequacy (TRT and FRT)

To ensure the reliability and accuracy of an index number, it must satisfy certain mathematical tests of consistency, known as Tests of Adequacy. The two most important tests are:

Time Reversal Test (TRT):

Time Reversal Test checks the consistency of an index number when time periods are reversed. In other words, if we calculate an index number from year 0 to year 1, and then from year 1 back to year 0, the product of the two indices should be equal to 1 (or 10000 when expressed as percentages).

Mathematical Condition:

P01 × P10 = 1

or

P01 × P10 = 10000

Where:

• P01​ = Price index from base year 0 to current year 1

• P10​ = Price index from current year 1 to base year 0

Interpretation:

This test ensures that the index number gives symmetrical results when the time order of comparison is reversed.

Which Formula Satisfies TRT?

• Fisher’s Ideal Index satisfies the Time Reversal Test.

• Laspeyres’ and Paasche’s indices do not satisfy this test.

Factor Reversal Test (FRT):

Factor Reversal Test checks whether the product of the Price Index and the Quantity Index equals the value ratio (i.e., the ratio of total expenditure in the current year to that in the base year).

Mathematical Condition:

P01 × Q01 = ∑P1Q1 / ∑P0Q0​​

Where:

• P01​ = Price index from base year to current year

• Q01​ = Quantity index from base year to current year

• ∑P1Q1​ = Total value in the current year

• ∑P0Q0​ = Total value in the base year

Interpretation:

This test checks whether the index number captures the combined effect of both price and quantity changes on total value.

Which Formula Satisfies FRT?

• Fisher’s Ideal Index satisfies the Factor Reversal Test.

• Laspeyres’ and Paasche’s indices do not satisfy this test.