Time Reversal Test is satisfied, when:

Question

Accepted Answer

The correct option is 2:

The Time Reversal Test (TRT) is a crucial test of consistency and reliability for an index number formula. It determines whether the index number gives the same result when the base period and the current period are interchanged. If an index number is calculated from base period '0' to current period '1' (denoted as $P_{01}$), and then the periods are reversed (from period '1' to period '0', denoted as $P_{10}$), the two results should be reciprocals of each other. In other words, if you calculate the price change from 2000 to 2010, and then calculate the price change from 2010 back to 2000, the two indices should logically multiply to 1, effectively neutralizing the time difference. The mathematical condition for an index number to satisfy the Time Reversal Test is therefore $P_{01} 	imes P_{10} = 1$. The only index number that satisfies this test perfectly is Fisher's Ideal Index, which is the geometric mean of the Laspeyres and Paasche indices.

The incorrect option is 1:
The condition $P\_{01} \times P\_{10} = 0$ implies that either the index from 0 to 1 or the index from 1 to 0 is zero, which is economically and mathematically impossible for a price index unless all prices or quantities in one of the periods were zero. Since price indices are typically positive measures of price change, this condition would invalidate the index formula entirely, indicating a massive failure of the test, not its satisfaction.

The incorrect option is 3:
The condition $P\_{01} \times P\_{10} \> 1$ means that when you multiply the index of change from period 0 to 1 and the index of change back from period 1 to 0, the result is greater than the unity. This indicates a definite inconsistency; the index formula systematically overestimates the true proportional price change when the base and current periods are reversed, thus failing the Time Reversal Test.

The incorrect option is 4:
The condition $P\_{01} \times P\_{10} \< 1$ means that the product of the two indices is less than one. This suggests that the index formula systematically underestimates the true proportional price change when the periods are reversed. Like the previous option, this inequality confirms that the index formula is inconsistent and does not satisfy the critical requirement of the Time Reversal Test.