What is formula check whether it satisfy time reversal test?

There isn't a single, universal formula to check if something satisfies the time reversal test. The test itself depends on the context. The time reversal test is a property of an index number (like a price index or a quantity index), not a general mathematical formula.

Here's how the time reversal test works and how you'd check it in different contexts:

Understanding the Time Reversal Test

The time reversal test checks for symmetry in time. It asks: If we calculate an index from period 0 to period 1, and then calculate the index from period 1 back to period 0, should we get a result consistent with the original calculation? Ideally, the product of the two indices should be 1 (or very close to it, depending on the acceptable margin of error).

How to Check (depending on the context):

Let's say we have a price index formula. We'll use *P* for prices, *Q* for quantities, and subscripts 0 and 1 to represent the two periods. Different index number formulas exist (Laspeyres, Paasche, Fisher, etc.).

1. Laspeyres Price Index:

* Formula (0 to 1): `L<sub>01</sub> = Σ(P<sub>1</sub> * Q<sub>0</sub>) / Σ(P<sub>0</sub> * Q<sub>0</sub>)`

* Formula (1 to 0): `L<sub>10</sub> = Σ(P<sub>0</sub> * Q<sub>1</sub>) / Σ(P<sub>1</sub> * Q<sub>1</sub>)`

* Time Reversal Test: `L<sub>01</sub> * L<sub>10</sub> = ?` This generally *doesn't* equal 1. Laspeyres fails the time reversal test.

2. Paasche Price Index:

* Formula (0 to 1): `P<sub>01</sub> = Σ(P<sub>1</sub> * Q<sub>1</sub>) / Σ(P<sub>0</sub> * Q<sub>1</sub>)`

* Formula (1 to 0): `P<sub>10</sub> = Σ(P<sub>0</sub> * Q<sub>0</sub>) / Σ(P<sub>1</sub> * Q<sub>0</sub>)`

* Time Reversal Test: `P<sub>01</sub> * P<sub>10</sub> = ?` This also generally *doesn't* equal 1. Paasche fails the time reversal test.

3. Fisher Ideal Index: The Fisher index is designed to satisfy the time reversal test.

* Formula (0 to 1): `F<sub>01</sub> = √(L<sub>01</sub> * P<sub>01</sub>)`

* Formula (1 to 0): `F<sub>10</sub> = √(L<sub>10</sub> * P<sub>10</sub>)`

* Time Reversal Test: `F<sub>01</sub> * F<sub>10</sub> ≈ 1` (It will be approximately 1 due to rounding errors in calculations; ideally it equals 1).

In Summary:

There's no single formula. You apply the time reversal test by:

1. Identifying the index number formula you're using (Laspeyres, Paasche, Fisher, etc.).

2. Calculating the index going from period 0 to period 1.

3. Calculating the index going from period 1 to period 0.

4. Multiplying the two indices. If the result is approximately 1, the formula satisfies the time reversal test. If not, it fails. The acceptability of deviations from 1 depends on the context and required precision.