1. Chapter 4 Test for index numbers Time-reversal test
Dr. A. PHILIP AROKIADOSS
Assistant Professor
Department of Statistics
St. Joseph’s College (Autonomous)
Tiruchirappalli

2. What is an index number?
The problem of how to construct an index number is as much of economic theory as of statistical technique. Frisch (1936)

3. What is an index number? Definition 1:
An index number of prices shows the average percentage change of prices from one point of time to another.
The percentage change in the price of a single product from one time to another is found by dividing its price at time t by its price in time 0.
Pt/P0: price relative of a commodity in relation to these two points in time.
An index number of the prices of a collection of products is the average of their price relatives.
Most people understand what a high cost of living is or what low prices mean but little idea of how the height of the high cost of living or the lowness of the low cost of living is to be measured.
It is to measure these magnitudes that index numbers were invented.
If all prices moved up or down in perfect unison there would be no need for index numbers. But since in reality the prices of different articles move very differently then some sort of average or compromise must be used.
This average is the index number.
This definition is based on prices but in a like manner an index number can be calculated for wages, for quantities of goods imported or exported, for quantities produced in the economy.
Here the definition is expressed in terms of time, but can apply the same definition of comparisons between two places. But usually, index numbers are used to indicate price movements in time.

4. What is an index number? Definition 2:
An index number is limited to the measure of changes in a magnitude between one situation to another. The two situations compared are in no way restricted; they may be two time periods (e.g. CPI), or two situations in a spatial sense (e.g. two cities or two or more countries - PPPs), or two groups of individuals (e.g. one and two-person pensioner families).
Since index numbers measure changes, they are expressed with one selected situation as 100. This is called the “time” reference base of the series of index numbers.
From Allen, p. 2.

5. What is an index number? Definition 2 (cont’d):
In an annual series for example, the reference base is the year taken with the level of 100 for comparison. In another year, the index number may be 126. This shows an increase of 26% over these two years.
From Allen, p. 2.

6. Example (changing a time base)
2001
2002
2003
2004
Number of employees, 000’s
8741
8727
8432
8062
Series with 2001 as 100
100
99.8
96.5
92.2
The math…
100/96.5 x 100
99.8/96.5 x 100
96.5/96.5 x 100
92.2/96.5 x 100
Series with 2003 as 100
103.7
103.5
95.6
Example of index number
Explain table
Show that changing base is simply an arithmetic switch
Cannot compare employment index of 100 for 2001 with that of Not a meaningful comparison.
Can show that switching the base does not change the rates of change: employment fell 3.5% from 2001 to 2003 and fell 4.4% from 2003 to 2004 regardless of the base year used.

7. The index number problem
How exactly should the microeconomic information involving possibly millions of prices and quantities be aggregated into a smaller number of price and quantity variables?
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The problem that arises is how to combine the relative changes in the prices of various commodities into a single index number that can meaningfully be interpreted as a measure of the relative change in the general price level.
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The number of physically distinct goods and unique types of services that consumers can purchase is in the millions. On the business or production side of the economy, there are even more products that are actively traded. The reason is that firms not only produce products for final consumption, they also produce exports and intermediate products that are demanded by other producers. Firms collectively also use millions of imported goods and services, thousands of different types of labor services, and hundreds of thousands of specific types of capital. If we further distinguish physical products by their geographic location or by the season or time of day that they are produced or consumed, then there are billions of products that are traded within each year in any advanced economy. For many purposes, it is necessary to summarize this vast amount of price and quantity information into a much smaller set of numbers. The question that this chapter addresses is the fol- lowing:

8. The index number problem
An index number reduces all the distinct prices for the class of goods in question to a single number.
Cuts through the noise thus helps in seeing the big picture.
Hides potentially important details, some prices may be increasing but many others can be decreasing.
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Different index numbers, i.e., different forms of averages, tend to lead to different results, some of them will register positive changes and others negative, so that different indices may be moving in different directions !!!!!
The number of physically distinct goods and unique types of services that consumers can purchase is in the millions. On the business or production side of the economy, there are even more products that are actively traded. The reason is that firms not only produce products for final consumption, they also produce exports and intermediate products that are demanded by other producers. Firms collectively also use millions of imported goods and services, thousands of different types of labor services, and hundreds of thousands of specific types of capital. If we further distinguish physical products by their geographic location or by the season or time of day that they are produced or consumed, then there are billions of products that are traded within each year in any advanced economy. For many purposes, it is necessary to summarize this vast amount of price and quantity information into a much smaller set of numbers. The question that this chapter addresses is the fol- lowing:

9. Uses of index numbers Broad indexes Narrow indexes
It measures the economy’s price level.
Changes in the Cost-of-living
Help in measuring GDP
Narrow indexes
Tracks changes in the price of certain products
Help in guiding investments
Stumpage fees and other contracts.

10. Some notable price indexes
CPI
IPPI
PCE deflator
GDP deflator
Productivity indexes

11. Some basic index number formulas: Historical overview
1738 Dutot compared prices for 24 items from the time of Louis XII and Louis XIV using the following formula:

12. Some basic index number formulas: Historical overview
1747 and 1780: Tabular format by the Colony of Massachusetts to counter the effects of the depreciation of money. 5 bushels of corn, sixty-eight Pounds and four-seventh parts of a Pound of beef, 10 Pounds of sheep’s wool, and sixteen pounds of shoe leather shall then cost, more or less than one hundred and thirty pounds current money, at the then current prices of the said articles.

13. Some basic index number formulas: Historical overview
1764: Carli in Italy looked at the change in the prices for grain, wine, and oil between 1500 and 1750 to show the effect of the discovery of America on the purchasing power of money.

14. Some basic index number formulas: Historical overview
1822: Lowe index or fixed basket index.

15. Some basic index number formulas: Historical overview
1863: Jevons (father of index numbers) worked out index numbers for for English prices back to He was interested in showing the fall in the value of gold as a result of the outpouring of gold mines beginning in 1849.
Father because he seems to have inspired others to explore this field.

16. Some basic index number formulas
Laspeyres and Paasche price indexes in response to the need for more precision with regards to the basket.

17. Laspeyres price index

18. Paasche price index

19. Paasche quantity index

20. Laspeyres quantity index

21. Fisher indexes

22. Marshall-Edgeworth price indexes

23. Value aggregates

24. Example: Simple index
June
Granny Smith: $0.72/each
Red Delicious: $0.75/each
Fuji: $0.50/each
Gala: $0.75/each
Russets: $0.90/each

25. Example: Simple index (cont’d)
July
Granny Smith: $0.80/each
Red Delicious: $0.85/each
Fuji: $0.55/each
Gala: $0.77/each
Russets: $0.90/each

26. Example: Simple index (cont’d)
0.72 = 5
0.774 = 5
Dutot = 0.774/0.72 = or 7.5%
Carli = 3.87/3.62 = or 6.9%

27. Basic index number theory
Choosing an index number formula: two approaches
“AXIOMATIC” approach: the theoretical underpinnings of index numbers are built upon certain postulates (or axioms). Irving Fisher, 1922; Eichhorn and Voeller (EV), 1983
“ECONOMIC THEORETIC” approach: seeks to define the price or volume indices with reference to underlying utility or production functions.
Two basic approaches to index number theory
Which is the best criteria?
The axiomatic approach starts from the actual prices observed in two periods or situations being compared. The Ps and Qs are treated as independent variables.This is in contrast to the economic approach where the Qs are a function of P.

28. Axiomatic approach
The axiomatic approach to the selection of an appropriate index formulation specifies a number of desirable properties an index formulation should possess.
These properties are imbedded in axioms.
Potential indexes are then evaluated against the specified properties and the index that passes the most tests is the preferred one.
EV define an index as a function of the observed prices and quantities which satisfies four basic axioms.

29. Before we start, some simple notation
Pij: price index number for year j compared with year i.
Qij: quantity index number for year j compared with year i.
Vij: value index number for year j compared with year i.
Pji: price index number for year i compared with year j.

30. Axiomatic (or test) approach
Four basic axioms:
Monotonicity: a price index is increased whenever any of the prices in the current period are increased or any of the prices in the base period are lowered.
Proportionality: when all prices in the current period are uniformly greater or lower than those in the base period by some fixed proportion, the index should equal that proportion.

31. Axiomatic (or test) approach (cont’d)
Price dimensionality: the same proportional change in the unit of currency in both periods (e.g., from dollars to euros) does not change the index.
Commensurability: a change in the unit of quantity for any commodity in both periods (e.g., from pounds to kilos) does not change the index.

32. Axiomatic (or test) approach (cont’d)
These axioms are described as “basic properties which are desirable for every price (or quantity) index”.
Indexes that have these properties automatically satisfy various tests of the type which Irving Fisher proposed.
Identity test
Weak proportionality test
Mean value test
Here are some others:

33. Fisher’s proposed tests
Time-reversal: requires that Pij x Pji = 1
Factor-reversal: requires that Pij x Qij = Vij
Determinateness: requires that Pij and Qij shall be positive, finite and determinate regardless of the price and quantity value assumed by an individual commodity.
Commensurability: requires that Pij and Qij be independent of the scale of measurement of the prices and quantities of the individual commodities.
Proportionality: requires that Pij and Qij are linear homogeneous functions of the observed prices and quantities, respectively.
Some are the same suggested from EV but are shown here for completeness.

34. Eichhorn and Voeller tests
The number of indices satisfying the four basic properties are too broad.
Eichhorn and Voeller (EV) suggest adding the following conditions:
Product test: The product of a price and a quantity index should equal the expenditure ratio where the price and quantity indices do not necessarily have to have the same form, but must satisfy the previous four basic axioms of a price index.
Time-reversal test
Factor-reversal test

35. Eichhorn and Voeller tests
More on the product test
Weak version of Fisher’s “factor reversal” test
It is extremely important for economic analysis whenever time-series data are available in current values.
The product test requires that when the change in the current values is divided by a price index, we should come out with a recognisable and acceptable quantity index, even if it has a different form or properties from the price index.
E.G.: Laspeyres Quantity Index with Paasche Price Index satisfy the product test but not the factor-reversal test.

36. Eichhorn and Voeller tests
By adding the product test to the four basic axioms, the number of acceptable indexes is till quite (too) broad.
But if Fisher’s circular test (P1,2 x P2,3 = P1,3) is added, then the set of possible indices is “empty”.
Example of an “inconsistency theorem” or “non-existence theorem”.
Circular test = circularity test = transitivity.
Circularity means that a direct comparison between periods 1 and 3, should give the same result as and indirect comparison via period 2.

37. Eichhorn and Voeller tests
It can and will be shown that the and index cannot satisfy the proportionality test and at the same time satisfy the transitivity condition.
Which of the conditions should be relaxed in order to be able to define an index?
More on this later.

38. Economic theoretic approach
Remember that with the axiomatic approach, prices and quantities were treated as separate independent variables.
Two price vectors and two quantity vectors
With the economic approach, the quantities are a function of the prices.
Two price vectors plus a functional relationship connecting the prices to the quantities in periods 1 and 2 respectively.
Parameters of the function are unknown hence economic theoretic indices cannot be estimated.
Two price vectors and two quantity vectors in periods 1 and 2 respectively.

39. Economic theoretic approach
Two main functions relating Qs to Ps:
Utility functions
Production functions
Or the generic: “aggregator function”
The classic example of an economic theoretic index is the Cost-of-living-index (COLI).
COLI: “the ratio of the minimum expenditures required to attain a particular indifference curve (level of welfare) under two price regimes”.
Two price vectors and two quantity vectors in periods 1 and 2 respectively.

40. END