Tests of Adequacy: Time Reversal and Factor Reversal

Not every formula that "looks reasonable" is actually consistent. Statisticians test an index number formula against two conditions.

(A) Time Reversal Test

An index formula passes this test if, when the base year and current year are interchanged, the two resulting indices (expressed as ratios, not multiplied by 100) are exact reciprocals of each other:

Here $P_{01}$ is the index of year 1 on year 0, and $P_{10}$ is the index of year 0 on year 1 (i.e. every $p_0\leftrightarrow p_1$ and $q_0\leftrightarrow q_1$ swapped together). Laspeyres and Paasche individually fail this test — swapping the years in Laspeyres' formula does not give the reciprocal of the original; it actually gives Paasche's formula computed for the reversed years. Fisher's Ideal Index satisfies this test exactly, because it is built symmetrically from both.

(B) Factor Reversal Test

An index formula passes this test if the product of its price index and the corresponding quantity index (obtained from the same formula with prices and quantities interchanged) equals the true value ratio: …