Understanding the Factor Reversal Test for Index Numbers

Index numbers are statistical tools used to measure changes in variables like price or quantity over time. A price index measures the average change in the prices of goods and services, while a quantity index measures the average change in their quantities. Different formulas exist for constructing these index numbers, such as Laspeyre's index, Paasche's index, and Fisher's ideal index.

To evaluate the quality and reliability of an index number formula, statisticians use various tests. One important test is the Factor Reversal Test.

What is the Factor Reversal Test?

The factor reversal test requires that the product of a price index and the corresponding quantity index for the same period should be equal to the value index for that period. The underlying idea is that the change in value from a base period to a current period is due to the combined changes in price and quantity.

Mathematically, the test is satisfied if:

\( P_{01} \times Q_{01} = V_{01} \)

Where:

• \( P_{01} \) is the price index number for the current period (1) relative to the base period (0).
• \( Q_{01} \) is the quantity index number for the current period (1) relative to the base period (0). The quantity index \( Q_{01} \) must be constructed using the same logic or weighting system as \( P_{01} \), or sometimes defined as the index obtained by swapping prices and quantities in the price index formula.
• \( V_{01} \) is the value index number for the current period (1) relative to the base period (0). The value index is simply the ratio of the total value of commodities in the current period to the total value in the base period.

The formula for the value index is:

\( V_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0} \times 100 \)

Where:

• \( p_0 \) and \( q_0 \) are the price and quantity of commodities in the base period.
• \( p_1 \) and \( q_1 \) are the price and quantity of commodities in the current period.
• \( \sum \) denotes the summation over all commodities.

Testing Different Index Number Formulas

Let's examine how some common index number formulas fare with the factor reversal test.

Laspeyre's Index

Laspeyre's price index uses base period quantities as weights:

\( P_L = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100 \)

Laspeyre's quantity index uses base period prices as weights:

\( Q_L = \frac{\sum q_1 p_0}{\sum q_0 p_0} \times 100 \)

Product of Laspeyre's indices:

\( P_L \times Q_L = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum q_1 p_0}{\sum q_0 p_0} \times (100)^2 \)

This product is generally not equal to \( \frac{\sum p_1 q_1}{\sum p_0 q_0} \times (100)^2 \). Therefore, Laspeyre's index does not satisfy the factor reversal test.

Paasche's Index

Paasche's price index uses current period quantities as weights:

\( P_P = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100 \)

Paasche's quantity index uses current period prices as weights:

\( Q_P = \frac{\sum q_1 p_1}{\sum q_0 q_1} \times 100 \)

Product of Paasche's indices:

\( P_P \times Q_P = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times \frac{\sum q_1 p_1}{\sum q_0 q_1} \times (100)^2 \)

This product is generally not equal to \( \frac{\sum p_1 q_1}{\sum p_0 q_0} \times (100)^2 \). Therefore, Paasche's index does not satisfy the factor reversal test.

Fisher's Ideal Index

Fisher's ideal index is the geometric mean of Laspeyre's and Paasche's indices. It is considered 'ideal' because it satisfies several tests, including the factor reversal test.

Fisher's price index:

\( P_F = \sqrt{P_L \times P_P} = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}} \times 100 \)

The corresponding Fisher's quantity index (obtained by swapping p and q in the price index formula, or calculated as the geometric mean of \(Q_L\) and \(Q_P\)):

\( Q_F = \sqrt{Q_L \times Q_P} = \sqrt{\frac{\sum q_1 p_0}{\sum q_0 p_0} \times \frac{\sum q_1 p_1}{\sum q_0 p_1}} \times 100 \)

Let's examine the product \( P_F \times Q_F \):

\( P_F \times Q_F = \left(\sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}} \times 100\right) \times \left(\sqrt{\frac{\sum q_1 p_0}{\sum q_0 p_0} \times \frac{\sum q_1 q_1}{\sum q_0 q_1}} \times 100\right) \)

Excluding the \( (100)^2 \) factor for simplicity during the test:

\( P_F \times Q_F = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1} \times \frac{\sum q_1 p_0}{\sum q_0 p_0} \times \frac{\sum q_1 q_1}{\sum q_0 q_1}} \)

Let's use simple notation for the summations:

• \( \sum p_1 q_0 = A \)
• \( \sum p_0 q_0 = B \)
• \( \sum p_1 q_1 = C \)
• \( \sum p_0 q_1 = D \)

When swapping p and q for the quantity index derivation:

• \( \sum q_1 p_0 = \sum p_0 q_1 = D \)
• \( \sum q_0 p_0 = \sum p_0 q_0 = B \)
• \( \sum q_1 p_1 = \sum p_1 q_1 = C \)
• \( \sum q_0 p_1 = \sum p_1 q_0 = A \)

So, the product becomes:

\( P_F \times Q_F = \sqrt{\frac{A}{B} \times \frac{C}{D} \times \frac{D}{B} \times \frac{C}{A}} \)

\( P_F \times Q_F = \sqrt{\frac{A \times C \times D \times C}{B \times D \times B \times A}} = \sqrt{\frac{A \times C^2 \times D}{A \times B^2 \times D}} \)

Assuming \(A, B, D \neq 0\), we can cancel terms:

\( P_F \times Q_F = \sqrt{\frac{C^2}{B^2}} = \frac{C}{B} \)

Substituting back the summations:

\( P_F \times Q_F = \frac{\sum p_1 q_1}{\sum p_0 q_0} \)

This result is the value index \( V_{01} \) (excluding the \( \times 100 \) factor, which would be consistent on both sides). Thus, Fisher's ideal index satisfies the factor reversal test.

Walsh Price Index

The Walsh price index uses a geometric mean of base and current period quantities as weights: \( P_W = \frac{\sum p_1 \sqrt{q_0 q_1}}{\sum p_0 \sqrt{q_0 q_1}} \times 100 \). The Walsh index also satisfies the factor reversal test, making it another valid index in this regard.

Conclusion

Among the commonly discussed index numbers in introductory statistics, Fisher's ideal index is known for satisfying both the time reversal test and the factor reversal test. While the Walsh index also satisfies the factor reversal test, Fisher's is often presented as the primary example of an index satisfying this property due to its widespread use and its definition as the geometric mean of Laspeyre's and Paasche's indices.

Therefore, Fisher's ideal index satisfies the factor reversal test.

Revision Table: Comparing Index Numbers and Factor Reversal Test

Additional Information on Index Number Tests

Besides the factor reversal test, other important tests for index numbers include:

• Time Reversal Test: This test requires that if the base period and current period are interchanged, the resulting index number should be the reciprocal of the original index number. Mathematically, \( P_{01} \times P_{10} = 1 \) (or \( P_{01} \times P_{10} = 10000 \) if the index is expressed as a percentage). Laspeyre's and Paasche's indices do not satisfy this test, but Fisher's ideal index does. Walsh index also satisfies the time reversal test.
• Circular Test: This test is an extension of the time reversal test for more than two periods. It requires that \( P_{01} \times P_{12} \times P_{20} = 1 \). None of Laspeyre's, Paasche's, or Fisher's indices satisfy the circular test.
• Identity Test: This test requires that if the current period is the same as the base period (i.e., \( p_1 = p_0 \) and \( q_1 = q_0 \)), the index number should be 1 (or 100%). All standard index number formulas satisfy this test.

Fisher's index is called 'ideal' because it satisfies both the time reversal and factor reversal tests, which are considered crucial properties for a good index number formula.