Tests of Adequacy for Index Numbers

Tests of Adequacy: Importance

As we have seen, various formulas exist for constructing index numbers, particularly weighted price and quantity indices (such as Laspeyres, Paasche, Fisher, Marshall-Edgeworth, etc.). Each formula uses different weighting schemes and aggregation methods, leading to potentially different index values for the same set of data. This raises a fundamental question: how do we evaluate and compare these different formulas to determine which is more appropriate or "better" for a given purpose?

To address this, statisticians and economists have developed a set of criteria or standards, often referred to as tests of adequacy or tests of consistency. These are mathematical tests designed to check whether an index number formula possesses certain logical and desirable properties. These properties ensure that the index number behaves predictably and consistently under different scenarios, such as changing the base period or swapping the roles of price and quantity.

Passing these tests suggests that an index number formula is mathematically sound and less likely to produce misleading results due to its internal structure. Conversely, failing a test highlights potential weaknesses or biases in the formula. It's important to note that no single index number formula perfectly satisfies all desirable tests under all circumstances. However, these tests provide a framework for understanding the strengths and weaknesses of different formulas and help in choosing the most suitable one for a specific application.

The main tests commonly considered in the theory of index numbers include:

• Unit Test: Checks if the index number is independent of the units in which prices or quantities are quoted (e.g., $\textsf{₹}$/kg vs $\textsf{₹}$/quintal). Most standard index number formulas (including weighted ones) satisfy this test because they use ratios where units cancel out. The Simple Aggregate Method is a notable exception as discussed previously.
• Time Reversal Test: Checks if the index remains consistent when the base and current periods are interchanged.
• Factor Reversal Test: Checks if the product of a price index and the corresponding quantity index (calculated using the same formula) equals the value index.
• Circular Test: An extension of the Time Reversal Test, it checks consistency across three or more periods (e.g., index from 0 to 1, 1 to 2, and 2 to 0 should be consistent).

Evaluating index number formulas based on these tests provides valuable insights into their properties and helps in the selection of an appropriate formula for practical applications, such as calculating inflation rates, production indices, or trade volume changes.

Time Reversal Test

Concept

The Time Reversal Test is a fundamental test of consistency for index number formulas. It examines whether the index number formula maintains its consistency when the roles of the base period (period 0) and the current period (period 1) are swapped. If an index number formula satisfies this test, it means that the relative change measured from period 0 to period 1 is the reciprocal of the relative change measured from period 1 to period 0, using the same formula.

In simple terms, if a price index from period 0 to period 1 shows a $20\%$ increase (i.e., $P_{01} = 120$), then the price index from period 1 to period 0 should show a decrease such that it returns the value to its original level. If we use index values as percentages (e.g., 120), calculating the index from 1 to 0 should give a result ($P_{10}$) such that $P_{01} \times P_{10} / 100 = 100$. If we use index values as ratios (e.g., 1.20), then $P_{01} \times P_{10} = 1$.

Mathematical Condition

An index number formula $I_{01}$ for measuring change from period 0 to period 1 is said to satisfy the Time Reversal Test if:

where $I_{01}$ is the index formula for period 1 with base period 0 (calculated as a ratio, i.e., without multiplying by 100), and $I_{10}$ is the index formula for period 0 with base period 1. The formula for $I_{10}$ is obtained by simply interchanging the subscripts 0 and 1 in the formula for $I_{01}$.

If the index number is calculated as a percentage (i.e., formula $\times 100$), the test condition becomes:

where $P_{01} = I_{01} \times 100$ and $P_{10} = I_{10} \times 100$. Let's verify using the ratio form $I_{01} \times I_{10} = 1$. If $P_{01} = I_{01} \times 100$ and $P_{10} = I_{10} \times 100$, then $\frac{P_{01}}{100} \times \frac{P_{10}}{100} = 1$, which simplifies to $P_{01} \times P_{10} = 10000$. Both conditions are equivalent, but working with the ratio form $I_{01} \times I_{10} = 1$ is usually simpler for proving satisfaction of the test.

Checking Common Price Indices for Time Reversal Test

Let's check whether the prominent weighted aggregate price index formulas satisfy the Time Reversal Test:

1. Laspeyres Price Index ($P^L$):

   The ratio form of the Laspeyres formula from period 0 to 1 is: $$I_{01}^L = \frac{\sum p_1 q_0}{\sum p_0 q_0}$$

   To get the formula for $I_{10}^L$ (index from period 1 to 0), we interchange the subscripts 0 and 1:

   $$I_{10}^L = \frac{\sum p_0 q_1}{\sum p_1 q_1}$$

   Now, check the product $I_{01}^L \times I_{10}^L$:

   $$I_{01}^L \times I_{10}^L = \left(\frac{\sum p_1 q_0}{\sum p_0 q_0}\right) \times \left(\frac{\sum p_0 q_1}{\sum p_1 q_1}\right)$$

   (Product of the ratios)

   In general, $\left(\frac{\sum p_1 q_0}{\sum p_0 q_0}\right) \times \left(\frac{\sum p_0 q_1}{\sum p_1 q_1}\right)$ is not equal to 1. Therefore, the Laspeyres Price Index does not satisfy the Time Reversal Test.

2. Paasche Price Index ($P^P$):

   The ratio form of the Paasche formula from period 0 to 1 is: $$I_{01}^P = \frac{\sum p_1 q_1}{\sum p_0 q_1}$$

   To get the formula for $I_{10}^P$, we interchange the subscripts 0 and 1:

   $$I_{10}^P = \frac{\sum p_0 q_0}{\sum p_1 q_0}$$

   Now, check the product $I_{01}^P \times I_{10}^P$:

   $$I_{01}^P \times I_{10}^P = \left(\frac{\sum p_1 q_1}{\sum p_0 q_1}\right) \times \left(\frac{\sum p_0 q_0}{\sum p_1 q_0}\right)$$

   (Product of the ratios)

   In general, $\left(\frac{\sum p_1 q_1}{\sum p_0 q_1}\right) \times \left(\frac{\sum p_0 q_0}{\sum p_1 q_0}\right)$ is not equal to 1. Therefore, the Paasche Price Index does not satisfy the Time Reversal Test.

3. Fisher's Ideal Price Index ($P^F$):

   The ratio form of Fisher's formula from period 0 to 1 is: $$I_{01}^F = \sqrt{I_{01}^L \times I_{01}^P} = \sqrt{\left(\frac{\sum p_1 q_0}{\sum p_0 q_0}\right) \left(\frac{\sum p_1 q_1}{\sum p_0 q_1}\right)}$$

   To get the formula for $I_{10}^F$, we interchange the subscripts 0 and 1 in the formula for $I_{01}^F$ (which is equivalent to finding $I_{10}^L$ and $I_{10}^P$ and taking their geometric mean):

   $$I_{10}^F = \sqrt{I_{10}^L \times I_{10}^P} = \sqrt{\left(\frac{\sum p_0 q_1}{\sum p_1 q_1}\right) \left(\frac{\sum p_0 q_0}{\sum p_1 q_0}\right)}$$

   Now, check the product $I_{01}^F \times I_{10}^F$:

   $$I_{01}^F \times I_{10}^F = \sqrt{\left(\frac{\sum p_1 q_0}{\sum p_0 q_0}\right) \left(\frac{\sum p_1 q_1}{\sum p_0 q_1}\right)} \times \sqrt{\left(\frac{\sum p_0 q_1}{\sum p_1 q_1}\right) \left(\frac{\sum p_0 q_0}{\sum p_1 q_0}\right)}$$

   (Product of the Fisher ratios)

   Combine the terms under a single square root:

   $$I_{01}^F \times I_{10}^F = \sqrt{\left(\frac{\sum p_1 q_0}{\sum p_0 q_0}\right) \times \left(\frac{\sum p_1 q_1}{\sum p_0 q_1}\right) \times \left(\frac{\sum p_0 q_1}{\sum p_1 q_1}\right) \times \left(\frac{\sum p_0 q_0}{\sum p_1 q_0}\right)}$$

   (Combining terms under root)

   Rearrange the terms inside the square root to see cancellations:

   $$I_{01}^F \times I_{10}^F = \sqrt{\left(\frac{\cancel{\sum p_1 q_0}}{\cancel{\sum p_1 q_0}}\right) \times \left(\frac{\cancel{\sum p_1 q_1}}{\cancel{\sum p_1 q_1}}\right) \times \left(\frac{\cancel{\sum p_0 q_1}}{\cancel{\sum p_0 q_1}}\right) \times \left(\frac{\cancel{\sum p_0 q_0}}{\cancel{\sum p_0 q_0}}\right)}$$

   (Cancelling terms)

   $$I_{01}^F \times I_{10}^F = \sqrt{1 \times 1 \times 1 \times 1} = \sqrt{1} = 1$$

   [Condition satisfied]

   Therefore, Fisher's Ideal Price Index satisfies the Time Reversal Test.

4. Marshall-Edgeworth Price Index ($P^{ME}$):

   The ratio form of the Marshall-Edgeworth formula from period 0 to 1 is: $$I_{01}^{ME} = \frac{\sum p_1 (q_0 + q_1)}{\sum p_0 (q_0 + q_1)}$$

   To get the formula for $I_{10}^{ME}$, we interchange the subscripts 0 and 1:

   $$I_{10}^{ME} = \frac{\sum p_0 (q_1 + q_0)}{\sum p_1 (q_1 + q_0)}$$

   Now, check the product $I_{01}^{ME} \times I_{10}^{ME}$:

   $$I_{01}^{ME} \times I_{10}^{ME} = \left(\frac{\sum p_1 (q_0 + q_1)}{\sum p_0 (q_0 + q_1)}\right) \times \left(\frac{\sum p_0 (q_1 + q_0)}{\sum p_1 (q_1 + q_0)}\right)$$

   (Product of the ratios)

   Since $(q_0 + q_1) = (q_1 + q_0)$, the terms involving sums cancel out:

   $$I_{01}^{ME} \times I_{10}^{ME} = \frac{\cancel{\sum p_1 (q_0 + q_1)}}{\cancel{\sum p_0 (q_0 + q_1)}} \times \frac{\cancel{\sum p_0 (q_0 + q_1)}}{\cancel{\sum p_1 (q_0 + q_1)}}$$

   (Cancelling terms)

   $$I_{01}^{ME} \times I_{10}^{ME} = 1$$

   [Condition satisfied]

   Therefore, the Marshall-Edgeworth Price Index satisfies the Time Reversal Test.

In summary, among the weighted aggregate indices discussed, only Fisher's Ideal Index and the Marshall-Edgeworth Index satisfy the Time Reversal Test. This indicates a mathematical consistency in measuring relative change regardless of the direction of time (which period is chosen as the base).

Factor Reversal Test

Concept

The Factor Reversal Test is another crucial consistency test for index number formulas. It focuses on the relationship between price index numbers and quantity index numbers for the same set of commodities over the same two periods. The test posits that if a price index ($P_{01}$) measures how much prices have changed from period 0 to period 1, and a corresponding quantity index ($Q_{01}$) measures how much quantities have changed over the same period, then the product of these two indices should equal the index that measures the total change in the value of the commodities over the same period.

The total value of a set of commodities in a period is the sum of the products of the price and quantity for each commodity in that period ($\text{Value} = \sum p \times q$). The Value Index ($V_{01}$) measures the relative change in this total value from period 0 to period 1.

Total Value in Base Period ($V_0$) = $\sum p_0 q_0$

Total Value in Current Period ($V_1$) = $\sum p_1 q_1$

The Value Index from period 0 to 1, as a ratio, is: $$V_{01} = \frac{V_1}{V_0} = \frac{\sum p_1 q_1}{\sum p_0 q_0}$$ (Often multiplied by 100 to express as a percentage).

The essence of the Factor Reversal Test is that the overall change in the value of a basket of goods should be fully accounted for by the changes in the prices of those goods and the changes in the quantities of those goods. The test ensures that the formula treats price and quantity factors symmetrically.

Mathematical Condition

An index number formula satisfies the Factor Reversal Test if the product of the price index ($I_{01}^P$) and the corresponding quantity index ($I_{01}^Q$), both calculated using the same formula structure, equals the Value Index ($V_{01}$). The test condition, using index ratios (before multiplying by 100), is:

where $I_{01}^P$ is the price index formula ratio from period 0 to 1, and $I_{01}^Q$ is the quantity index formula ratio from period 0 to 1. The key to constructing $I_{01}^Q$ from $I_{01}^P$ is to interchange the symbols $p$ and $q$ in the formula for $I_{01}^P$, while keeping the subscripts (0 and 1) in their respective positions.

If the indices are expressed as percentages ($P_{01} = I_{01}^P \times 100$, $Q_{01} = I_{01}^Q \times 100$, $V_{01} = \frac{\sum p_1 q_1}{\sum p_0 q_0} \times 100$), the test condition becomes:

Using the ratio form (Equation i) is usually simpler for proving satisfaction of the test.

Constructing Corresponding Quantity Indices

To check the Factor Reversal Test, we first need to derive the quantity index formula corresponding to each price index formula by swapping $p$ and $q$ in the formula structure.

• Laspeyres Quantity Index ($Q^L$): The Laspeyres Price Index uses base quantities ($q_0$) as weights for prices. Interchanging $p$ and $q$, the Laspeyres Quantity Index uses base prices ($p_0$) as weights for quantities.

  $$I_{01}^L (\text{Price}) = \frac{\sum p_1 q_0}{\sum p_0 q_0}$$

  Swapping $p$ and $q$ gives the Laspeyres Quantity Index:

  $$\mathbf{I_{01}^L (\text{Quantity}) = Q_{01}^L = \frac{\sum q_1 p_0}{\sum q_0 p_0}}$$

• Paasche Quantity Index ($Q^P$): The Paasche Price Index uses current quantities ($q_1$) as weights for prices. Interchanging $p$ and $q$, the Paasche Quantity Index uses current prices ($p_1$) as weights for quantities.

  $$I_{01}^P (\text{Price}) = \frac{\sum p_1 q_1}{\sum p_0 q_1}$$

  Swapping $p$ and $q$ gives the Paasche Quantity Index:

  $$\mathbf{I_{01}^P (\text{Quantity}) = Q_{01}^P = \frac{\sum q_1 p_1}{\sum q_0 p_1}}$$

• Fisher's Ideal Quantity Index ($Q^F$): Since Fisher's Ideal Index is the geometric mean of Laspeyres and Paasche, the Fisher's Ideal Quantity Index is the geometric mean of the Laspeyres Quantity Index and the Paasche Quantity Index.

  $$I_{01}^F (\text{Price}) = \sqrt{I_{01}^L (\text{Price}) \times I_{01}^P (\text{Price})} = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1}}$$

  Swapping $p$ and $q$ in the overall structure or within the L and P components gives:

  $$\mathbf{I_{01}^F (\text{Quantity}) = Q_{01}^F = \sqrt{\left( \frac{\sum q_1 p_0}{\sum q_0 p_0} \right) \times \left( \frac{\sum q_1 p_1}{\sum q_0 p_1} \right)}}$$

• Marshall-Edgeworth Quantity Index ($Q^{ME}$): The Marshall-Edgeworth Price Index uses $(q_0 + q_1)$ as weights for prices. Swapping $p$ and $q$, the Quantity Index uses $(p_0 + p_1)$ as weights for quantities.

  $$I_{01}^{ME} (\text{Price}) = \frac{\sum p_1 (q_0 + q_1)}{\sum p_0 (q_0 + q_1)}$$

  Swapping $p$ and $q$ gives the Marshall-Edgeworth Quantity Index:

  $$\mathbf{I_{01}^{ME} (\text{Quantity}) = Q_{01}^{ME} = \frac{\sum q_1 (p_0 + p_1)}{\sum q_0 (p_0 + p_1)}}$$

Checking Common Indices for Factor Reversal Test

Let's check if $I_{01}^P \times I_{01}^Q = \frac{\sum p_1 q_1}{\sum p_0 q_0}$ for each method:

1. Laspeyres:

   $I_{01}^L (\text{Price}) \times I_{01}^L (\text{Quantity}) = \left( \frac{\sum p_1 q_0}{\sum p_0 q_0} \right) \times \left( \frac{\sum q_1 p_0}{\sum q_0 p_0} \right)$

   This product $\frac{(\sum p_1 q_0) \times (\sum q_1 p_0)}{(\sum p_0 q_0) \times (\sum q_0 p_0)}$ is generally not equal to $\frac{\sum p_1 q_1}{\sum p_0 q_0}$. Therefore, the Laspeyres Index fails the Factor Reversal Test.

2. Paasche:

   $I_{01}^P (\text{Price}) \times I_{01}^P (\text{Quantity}) = \left( \frac{\sum p_1 q_1}{\sum p_0 q_1} \right) \times \left( \frac{\sum q_1 p_1}{\sum q_0 p_1} \right)$

   This product $\frac{(\sum p_1 q_1) \times (\sum q_1 p_1)}{(\sum p_0 q_1) \times (\sum q_0 p_1)}$ is generally not equal to $\frac{\sum p_1 q_1}{\sum p_0 q_0}$. Therefore, the Paasche Index fails the Factor Reversal Test.

3. Fisher:

   $I_{01}^F (\text{Price}) \times I_{01}^F (\text{Quantity}) = \sqrt{\left( \frac{\sum p_1 q_0}{\sum p_0 q_0} \right) \left( \frac{\sum p_1 q_1}{\sum p_0 q_1} \right)} \times \sqrt{\left( \frac{\sum q_1 p_0}{\sum q_0 p_0} \right) \left( \frac{\sum q_1 p_1}{\sum q_0 p_1} \right)}$

   Combine under a single square root:

   $$ = \sqrt{\left( \frac{\sum p_1 q_0}{\sum p_0 q_0} \right) \left( \frac{\sum p_1 q_1}{\sum p_0 q_1} \right) \left( \frac{\sum q_1 p_0}{\sum q_0 p_0} \right) \left( \frac{\sum q_1 p_1}{\sum q_0 p_1} \right)}$$

   Rearrange the terms inside the square root:

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

   Wait, let's re-arrange carefully. The correct algebraic proof shows that the terms simplify nicely.

   $$I_{01}^F (\text{Price}) \times I_{01}^F (\text{Quantity}) = \sqrt{\frac{\sum p_1 q_0 \cdot \sum p_1 q_1 \cdot \sum q_1 p_0 \cdot \sum q_1 p_1}{\sum p_0 q_0 \cdot \sum p_0 q_1 \cdot \sum q_0 p_0 \cdot \sum q_0 p_1}}$$

   Recognize that $q_1 p_0 = p_0 q_1$ and $q_0 p_1 = p_1 q_0$. So, $\sum q_1 p_0 = \sum p_0 q_1$ and $\sum q_0 p_1 = \sum p_1 q_0$. Similarly $q_1 p_1 = p_1 q_1$ and $q_0 p_0 = p_0 q_0$.

   The expression becomes:

   $$= \sqrt{\frac{\sum p_1 q_0 \cdot \sum p_1 q_1 \cdot \sum p_0 q_1 \cdot \sum p_1 q_1}{\sum p_0 q_0 \cdot \sum p_0 q_1 \cdot \sum p_0 q_0 \cdot \sum p_0 q_1}}$$

   $$= \sqrt{\frac{(\sum p_1 q_1)^2 \times (\sum p_1 q_0) \times (\sum p_0 q_1)}{(\sum p_0 q_0)^2 \times (\sum p_0 q_1) \times (\sum p_1 q_0)}}$$

   $$= \sqrt{\frac{(\sum p_1 q_1)^2}{(\sum p_0 q_0)^2}} = \frac{\sum p_1 q_1}{\sum p_0 q_0}$$

   [Condition satisfied]

   Therefore, Fisher's Ideal Index satisfies the Factor Reversal Test.

4. Marshall-Edgeworth:

   $I_{01}^{ME} (\text{Price}) \times I_{01}^{ME} (\text{Quantity}) = \left( \frac{\sum p_1 (q_0 + q_1)}{\sum p_0 (q_0 + q_1)} \right) \times \left( \frac{\sum q_1 (p_0 + p_1)}{\sum q_0 (p_0 + p_1)} \right)$

   This product $\frac{\sum (p_1 q_0 + p_1 q_1) \times \sum (q_1 p_0 + q_1 p_1)}{\sum (p_0 q_0 + p_0 q_1) \times \sum (q_0 p_0 + q_0 p_1)}$ is generally not equal to $\frac{\sum p_1 q_1}{\sum p_0 q_0}$. Therefore, the Marshall-Edgeworth Index fails the Factor Reversal Test.

The Factor Reversal Test highlights that Laspeyres and Paasche indices are not consistent in how they treat price and quantity changes. Fisher's Ideal Index is unique among these common weighted aggregate indices in satisfying both the Time Reversal Test and the Factor Reversal Test, contributing to its theoretical significance as an "ideal" measure that symmetrically accounts for both price and quantity changes.

Circular Test (Implicit)

Concept

The Circular Test is an extension of the Time Reversal Test. While the Time Reversal Test deals with consistency between two periods (0 and 1), the Circular Test extends this concept to three or more periods (say, 0, 1, and 2). It checks whether the index number formula provides consistent results when the base period is shifted across multiple periods. Specifically, it requires that the index number calculated from period 0 to period 1 ($I_{01}$), when multiplied by the index number from period 1 to period 2 ($I_{12}$), should equal the index number calculated directly from period 0 to period 2 ($I_{02}$).

In other words, if prices increase by $10\%$ from period 0 to 1, and then by $15\%$ from period 1 to 2, the overall price change from period 0 to 2 should ideally be $(1.10 \times 1.15 - 1) \times 100 = 26.5\%$. The Circular Test checks if the formula results in this consistency across linked periods and direct calculation.

Mathematical Condition

An index number formula $I_{ij}$ (as a ratio) for measuring change from period $i$ to period $j$ is said to satisfy the Circular Test if, for any three periods 0, 1, and 2:

Or, equivalently, for any sequence of periods $t_0, t_1, \dots, t_k, t_{k+1}, \dots, t_n$:

$$I_{t_0 t_1} \times I_{t_1 t_2} \times \dots \times I_{t_{n-1} t_n} = I_{t_0 t_n}$$

A common way to express this is $I_{01} \times I_{12} \times I_{20} = 1$, which follows directly from $I_{02} = I_{01} \times I_{12}$ and the Time Reversal Test applied to $I_{02}$, where $I_{20} = 1/I_{02}$.

If the indices are expressed as percentages ($P_{ij} = I_{ij} \times 100$), the condition becomes:

Or $P_{01} \times P_{12} \times P_{20} = 100 \times 100 \times 100 = 1000000$ (using percentage indices directly is confusing for this test, sticking to ratio form is clearer).

Checking Common Indices for Circular Test

Let's consider whether the weighted aggregate price index formulas generally satisfy the Circular Test for three periods 0, 1, and 2:

• Laspeyres Price Index ($P^L$):

  $$I_{01}^L = \frac{\sum p_1 q_0}{\sum p_0 q_0}$$

  $$I_{12}^L = \frac{\sum p_2 q_1}{\sum p_1 q_1}$$

  $$I_{02}^L = \frac{\sum p_2 q_0}{\sum p_0 q_0}$$

  Checking $I_{01}^L \times I_{12}^L = I_{02}^L$: $\left(\frac{\sum p_1 q_0}{\sum p_0 q_0}\right) \times \left(\frac{\sum p_2 q_1}{\sum p_1 q_1}\right)$ is generally not equal to $\frac{\sum p_2 q_0}{\sum p_0 q_0}$. Laspeyres fails the Circular Test. (It fails because the base quantity $q_0$ in $I_{01}^L$ and $I_{02}^L$ is constant, but $I_{12}^L$ uses $q_1$ as weight for period 1 to 2 change, which is inconsistent across the chain).

• Paasche Price Index ($P^P$):

  $$I_{01}^P = \frac{\sum p_1 q_1}{\sum p_0 q_1}$$

  $$I_{12}^P = \frac{\sum p_2 q_2}{\sum p_1 q_2}$$

  $$I_{02}^P = \frac{\sum p_2 q_2}{\sum p_0 q_2}$$

  Checking $I_{01}^P \times I_{12}^P = I_{02}^P$: $\left(\frac{\sum p_1 q_1}{\sum p_0 q_1}\right) \times \left(\frac{\sum p_2 q_2}{\sum p_1 q_2}\right)$ is generally not equal to $\frac{\sum p_2 q_2}{\sum p_0 q_2}$. Paasche fails the Circular Test. (It fails because the current quantity weights $q_1$ and $q_2$ keep changing).

• Fisher's Ideal Price Index ($P^F$):

  Since Fisher's Index is the geometric mean of Laspeyres and Paasche, and neither Laspeyres nor Paasche satisfy the Circular Test, Fisher's Ideal Index also generally does not satisfy the Circular Test.

• Marshall-Edgeworth Price Index ($P^{ME}$):

  $$I_{01}^{ME} = \frac{\sum p_1 (q_0 + q_1)}{\sum p_0 (q_0 + q_1)}$$

  $$I_{12}^{ME} = \frac{\sum p_2 (q_1 + q_2)}{\sum p_1 (q_1 + q_2)}$$

  $$I_{02}^{ME} = \frac{\sum p_2 (q_0 + q_2)}{\sum p_0 (q_0 + q_2)}$$

  Checking $I_{01}^{ME} \times I_{12}^{ME} = I_{02}^{ME}$: $\left( \frac{\sum p_1 (q_0 + q_1)}{\sum p_0 (q_0 + q_1)} \right) \times \left( \frac{\sum p_2 (q_1 + q_2)}{\sum p_1 (q_1 + q_2)} \right)$ is generally not equal to $\frac{\sum p_2 (q_0 + q_2)}{\sum p_0 (q_0 + q_2)}$. Marshall-Edgeworth fails the Circular Test.

So, the common weighted aggregate indices (Laspeyres, Paasche, Fisher, Marshall-Edgeworth) do not satisfy the Circular Test. This is because their weighting schemes (fixed base, fixed current, or average) applied across multiple periods lead to inconsistencies when linking indices.

Which indices *do* satisfy the Circular Test?

• Simple Geometric Mean of Price Relatives: The formula $I_{01} = \left( \prod_{i=1}^{N} \frac{p_{1i}}{p_{0i}} \right)^{1/N}$.

  $I_{01} \times I_{12} = \left( \prod \frac{p_1}{p_0} \right)^{1/N} \times \left( \prod \frac{p_2}{p_1} \right)^{1/N} = \left( \prod \left( \frac{p_1}{p_0} \times \frac{p_2}{p_1} \right) \right)^{1/N} = \left( \prod \frac{p_2}{p_0} \right)^{1/N} = I_{02}$.

  This index satisfies the Circular Test. However, it is an unweighted index and suffers from that major limitation.
• Kelly's Fixed Weight Index: An aggregate index using arbitrarily fixed weights $w_i$ (not necessarily $q_0$ or $q_1$), $I_{01} = \frac{\sum p_1 w}{\sum p_0 w}$.

  $I_{01} \times I_{12} = \left(\frac{\sum p_1 w}{\sum p_0 w}\right) \times \left(\frac{\sum p_2 w}{\sum p_1 w}\right) = \frac{\cancel{\sum p_1 w}}{\sum p_0 w} \times \frac{\sum p_2 w}{\cancel{\sum p_1 w}} = \frac{\sum p_2 w}{\sum p_0 w} = I_{02}$.

  This index satisfies the Circular Test. The challenge here is choosing appropriate fixed weights $w_i$ that remain relevant over extended periods.

While the Circular Test is desirable for chaining index numbers, its failure by indices like Laspeyres and Paasche implies that chaining them (e.g., calculating $P_{01} \times P_{12}$ to get a measure of $P_{02}$) can lead to drift or inconsistency over long periods compared to a direct calculation of $P_{02}$ using the same formula. Official statistical agencies often use chained indices (like chained Laspeyres or chained Fisher) as a compromise, accepting the failure of the strict Circular Test in favour of using more relevant quantities from more recent periods, while acknowledging that different base periods will yield slightly different results for distant periods.