**Statistical Index Number**

**Statistical Index Number Terminologies**

In this part, we discuss about Index Number and related topics like:

Construction of Index Numbers Simple Aggregate Method Weighted Aggregate Method Laspeyres Method Paasche’s Method Fisher’s Method Marshall-Edge worth method | Chain Index Numbers Shifting of Index Numbers Splicing of Index Numbers Time Reversal Test Factor Reversal Test Circular Test |

**Statistical Index Numbers Terminologies**

**Statistical Index Number**

* Index Number* is a

*statistical*value that measures the change in a variable with respect to time.

In simple terms, Index Number may be expressed as a numerical figure which shows the level of a certain phenomenon at a certain period of time in comparison with the level of same phenomenon at another period of time.

**Construction of Index Numbers**

The following tables shows various methods of construction of Index Numbers

**Simple Aggregate method**

**Simple Aggregate Index** for a given period is calculated by dividing the aggregate of different values of specified period by the aggregate of values of the base period and multiplying the quotient by 100.

P_{01} = (åp_{1} / åp_{0}) ´ 100, where P_{01} = index of the current year with reference to the base period, åp_{1 }= total of the values of the current period, åp_{0 }= total of the value of the base period

Computation table of price index for 2009 on the basis of 2003 price (base year).

Items | P _{0}=Price(2003) | P _{1}=Price(2009) |

Sugar | 30 | 60 |

Rice | 15 | 20 |

Chicken | 15 | 20 |

Dal | 40 | 50 |

Fish | 30 | 40 |

Mustard Oil | 25 | 35 |

Egg | 20 | 25 |

Total (å=) | 175 | 250 |

P_{01} = [(åp_{1}) / (åp_{0}) ´ 100]= (250 / 175)´100= 143 The Index value indicates that there has been a rise of 43% in the price of 2009 in comparison to the price level of 2003.

**General Weighted Aggregate Method**

**General Weighted Aggregate Index** for any given period is calculated after assigning the suitable weight to the different items included in the index number.

P_{01} = (åp_{1}W / åp_{0}W) ´ 100, where P_{01} = index of the current year with reference to the base period, åp_{1 }=price of current year, åp_{0 }= price of base year, W=Weight

Table showing the computation of index number using general Weighted Aggregate Method.

Items | Weight | P _{0}=Pricein 2000 | P _{1}=Price in 2009 | P_{1}W | P_{0}W |

Dal | 4 | 8 | 10 | 40 | 32 |

Rice | 6 | 4 | 6 | 36 | 24 |

Oil | 4 | 3 | 5 | 20 | 12 |

Wheat | 3 | 10 | 8 | 24 | 30 |

Total | ¾ | ¾ | ¾ | 120 | 98 |

P_{01} = (åp_{1}W / åp_{0}W) ´ 100

= (120/98) ´ 100= 122.45 appx

The Index value indicates that there has been a rise of 22.45% in the price of 2009 in comparison to the price level of 2000.

**Weights Assigning for weighted aggregate Index**

In a weighted aggregates index, weights are assigned according to their significance and as a result the weighted index improves the accuracy of the general price level estimate based on the calculated index.

Some common methods of assigning weights are : 1.Laspeyres Method, 2.Paasche Method

Fixed Weight Aggregates Method, 3.Fisher’s Ideal Price Method, 4.Marshall-Edgeworth Method.

**Laspeyres Index**

**Laspeyres Index** method uses the quantities consumed during the base period in computing the index number. This method requires quantity measures for only one period.

Laspeyres price index calculates the changes in the aggregate value of the base year’s list of goods when valued at current year prices

Laspeyres index is simpler in calculation and can be computed once the current year prices are known as the weights are base year quantities in a price index. This also enables easy comparability of one index with another.

Laspeyres Price Index_{ }= [{(åP_{1} Q_{0}) / (åP_{0 }Q_{0})} / 100],

Laspeyres Quantity Index_{ }= [{(åP_{0} Q_{1}) / (åP_{0 }Q_{0})} / 100

Where, P_{0} = Prices in the base year. P_{1} = Prices in the current year, Q_{0} = Quantities in the base year, Q_{1} = Quantities in the current year

Laspeyres Index Computation Table

Commodity | Q_{0} (2004) | Q_{1}(2009) | P_{0} (2004) | P_{1}(2009) | P_{0} x Q_{0} | P_{1} x Q_{0} | P_{0} x Q_{1} |

Rice | 46.60 | 58.00 | 700 | 910 | 32,620.00 | 42,406.00 | 40,600.00 |

Jute | 14.57 | 17.92 | 620 | 950 | 9,033.40 | 13,841.50 | 11,110.40 |

Cotton | 69.46 | 85.10 | 205 | 300 | 14,239.30 | 20,838.00 | 17,445.50 |

Tea | 33.84 | 40.30 | 330 | 470 | 11,167.20 | 15,904.80 | 13,299.00 |

Total (å=) | | | | | 67,059.90 | 92,990.30 | 82,454.90 |

So, Laspeyres Price index = [{(åP_{1} Q_{0}) / (åP_{0 }Q_{0})} / 100] =

[{(92,990.30) / (67,059.90)}´100]=138.67

So, Laspeyres Quantity index = [{(åP_{0} Q_{1}) / (åP_{0 }Q_{0})} / 100] =

[{(82,454.90) / (67,059.90)}´100]=122.96

**Paasche’s Index**

**Paasche Index** method is similar to that of computing a Laspeyres Index. However, it uses quantity measures for the current period rather than for the base period.

Paasche Price Index_{ }= [{(åP_{1} Q_{1}) / (åP_{0} Q_{1}))} / 100],

Paasche Quantity Index_{ }= [{(åP_{1} Q_{1}) / (åP_{1}Q_{0}))} / 100],

Where, P_{0} = Prices in the base year. P_{1} = Prices in the current year, Q_{0} = Quantities in the base year, Q_{1} = Quantities in the current year

Laspeyres and Paasche methods tend to produce opposite extremes in index values calculated from the same data.

The use of Paasche index requires the continuous use of new quantity weights for each period considered.

Paasche index generally tends to underestimate the prices or has a downward bias.

Paasche index Computation Table (Base Year P_{0}, Q_{0} (2015), (Current Year P_{1}, Q_{1} (2016),

Commodity | P_{0} | Q_{0} | P_{1} | Q_{1} | P_{0}Q_{0} | P_{0}Q_{1} | P_{1}Q_{0} | P_{1}Q_{1} |

X | 3 | 18 | 4 | 15 | 54 | 45 | 72 | 60 |

Y | 5 | 6 | 5 | 9 | 30 | 45 | 30 | 45 |

U | 4 | 20 | 6 | 26 | 80 | 104 | 120 | 156 |

V | 2 | 14 | 4 | 15 | 28 | 30 | 56 | 60 |

| | | | | 192 | 224 | 278 | 321 |

Paasche Price Index

_{ }= [{(åP_{1} Q_{1}) / (åP_{0} Q_{1}))} / 100]

=321/224=143.30

Paasche Quantity Index=

[{(åP_{1} Q_{1}) / (åP_{1}Q_{0}))} / 100]

=321/278 = 115.47

**Fisher’s Ideal Index**

**Fisher’s Ideal Index** is calculated by taking the geometric mean of the Laspeyres and Paasche indices.

Fisher’s Ideal Index = Ö [ {(å P_{1}Q_{0} ) / (å P_{0}Q_{0} )} ´ {(å P_{1}Q_{1} ) / (å P_{0}Q_{1} )}] ´ 100

**Advantages of Fisher’s Ideal Index: **Fisher’s index uses geometric mean which is considered as better average for the construction of index numbers. 2. Laspeyres index and Paasche index indicate opposing characteristics and Fisher’s index reduces their respective bias demonstrated by the time reversal and factor reversal tests. 3. Both the current year and base prices and quantities are taken into account by this index.

However, Fisher Index is not widely used owing to the practical limitations of collecting data.

Fisher’s Ideal Index Computation Table for 2009 (P_{1}, Q_{1}) in respect of Base Year (P_{0}, Q_{0})

Ex

Commodities | P_{0} | Q _{0} | P_{1} | Q_{1} | P_{0}Q_{0} | P_{1}Q_{0} | P_{0}Q_{1} | P_{1}Q_{1} |

Tea, Grade I | 78 | 7 | 85 | 10 | 546 | 595 | 780 | 850 |

Tea, Grade Ii | 69 | 5 | 80 | 5 | 345 | 400 | 345 | 400 |

Tea, Grade III | 62 | 4 | 72 | 6 | 248 | 288 | 372 | 432 |

Total (å=) | | | | | 1139 | 1283 | 1497 | 1682 |

Fisher’s Index = Ö [ {(å P_{1}Q_{0} ) / (å P_{0}Q_{0} )} ´ {(å P_{1}Q_{1} ) / (å P_{0}Q_{1} )}] ´ 100

= [Ö{(1283/1139) ´ (1682/1497))}´ 100] = [Ö (1.1264 ´ 1.1235)] ´ 100 = 1.1249 ´ 100= 112.49

**Marshall-Edgeworth Index**

**Marshall Edgeworth Index** is the arithmetic average of the quantities of the base year and the current year as the weights of the items are taken into consideration.

The weight of an item = (q_{0}+q_{1})/2

P_{01}(ME) = [{(åp_{1}q_{0} + åp_{1}q_{1}) / (åp_{0}q_{0} + åp_{0}q_{1})} ´ 100]

**Marshall****–****Edge worth Weighted aggregative method Computation Table**

Example

Items | p_{0} | q_{0} | p_{1} | q_{1} | p_{1}q_{0} | p_{0}q_{0} | p_{1}q_{1} | p_{0}q_{1} |

Rice | 5 | 74 | 8 | 82 | 592 | 370 | 656 | 410 |

Wheat | 10 | 125 | 8 | 140 | 1000 | 1250 | 1120 | 1400 |

Jute | 14 | 40 | 12 | 33 | 408 | 396 | 396 | 462 |

Total(å=) | ¾ | ¾ | ¾ | ¾ | 2072 | 2180 | 2172 | 2272 |

P_{01}(ME) = [{(åp_{1}q_{0} + åp_{1}q_{1})/(åp_{0}q_{0} + åp_{0}q_{1})}´100]

= [{(2072+2172)/(2180+2272)}´100]

= (4244/4452)´100= 95.33

**Chain Index Numbers**

**Chain Index Number** is used in changing condition (like changing base).

The relatives of each year are first related to the preceding year called the link relatives and then they are chained together by successive multiplication to from a chain index.

Chain Index = [{(Link relative of current year x Chain Index of the previous year)} / 100]

Link relatives reveal annual change with reference to the previous year. But when they are chained, they change over to a fixed base from which they are chained

The chain index is useful where data for the whole period are not available or where commodity basket or the weights have to be changed. The link relatives of the current year and chain index from a given base will give a fixed base index with the given base year (as shown in last column of the Table)

Example

Year | Price | Link Relative | Chain Indices | |

2000 | 100 | 100 | 100 | |

2001 | 120 | (120/100) X 100 = 120.0 | (120/100)X 100 = 120.0 | |

2002 | 124 | (124/120) X 100 = 103.0 | (103/100)X 120 = 124.0 | |

2003 | 130 | (130/124) X 100 = 104.8 | (104.8/100) X 124 = 129.9 | |

2004 | 140 | (140/130) X 100 = 107.7 | (107.7/100) X 129.9 = 139.9 | |

2005 | 156 | (156/140) X 100 = 111.4 | (111.4/100) X 139.9 = 155.8 | |

2006 | 164 | (164/156) X 100 = 105.1 | (105.1/100) X 155.8 = 163.7 | |

**Shifting of Index Number**

**Shifting of Index** refers to change in reference base of an index number series from one time period to another. This is done without coming back to the original raw data and recomputing the entire series.

**Reasons for shifting the base : **When the previous base becomes too old and almost useless for comparisons. By shifting the base, it is possible to state the series in terms of a more recent time period. To show several series on the graph, it may be necessary to have the same base period.

Shifting price index = (Original Price Index) / (Price index of the year on which it has to be shifted)

Example : **Computation of Shifting Index numbers from 2005 to 2014**

Year | Index No. (2005=100) | Index No. (2011=100) | Year | Index No. (2005=100) | Index No. (2011=100) | |

2005 | 200 | (200/800) x 100 = 25.0 | 2010 | 820 | (820/800) x 100 = 102.5 | |

2006 | 220 | (220/800) x 100 = 27.5 | 2011 | 800 | (800/800) x 100 = 100.5 | |

2007 | 240 | (240/800) x 100 = 30.0 | 2012 | 760 | (760/800) x 100 = 95.5 | |

2008 | 400 | (400/800) x 100 = 50.0 | 2013 | 740 | (2740/800) x 100 = 92.5 | |

2009 | 800 | (800/800) x 100 = 100.0 | 2014 | 680 | (680/800) x 100 = 85.5 | |

**Splicing of Index Number**

**Splicing of Index** Numbers is a method of combining two or more series of overlapping index numbers to obtain a single index number on a common base. Splicing can be done only when index numbers are constructed with the same items for overlapping year.

Splicing like base shifting will give accurate result of the index number, It uses simple geometric mean and satisfy the circular test.

Spliced Index Number = (Index no. of current year x odd index of new base year) /100

**Ex. **A price index number series was revised by inclusion of certain new products, exclusion of some old products and change in the definition of some other products on reference base of 2004 and a revised series on a base of 2006.

Splicing Index Number Computation Table

Year | Old price index | Revised price index | Spliced price index |

(2004 = 100) | (2006 = 100) | (2006 = 100) | |

2003 | 96 | 87.27 | |

2004 | 100 | 90.91 | |

2005 | 105 | 95.45 | |

2006 | 110 | 100 | 100.00 |

2007 | 104 | 104.00 | |

2008 | 106 | 104.00 | |

2009 | 112 | 112.00 |

Spliced Index = (96/110) x 100 = 87.27

**Index Number Consistency Tests**

**Index Number Consistency Tests **are used put to verify the**consistency**, or adequacy **of**an**indexnumber**formula from different points**of**view.

**Time Reversal Test**

Time reversal test (developed by Prof. Irving Fisher) indicates that for a price/quantity index, if the time periods are reversed, the resulting index should be the reciprocal of the original price/quantity index.

I_{0},_{1} = 1/ (I_{1},_{0}), where I_{0},_{1} denotes the original index for the current year with a given base year, I_{1},_{0} denotes the resulting index, with time periods reversed, for the base year with the current year as a base year

Fisher’s Ideal Index, Marshall-Edgeworth Index and Fixed Weights Aggregate Index satisfy this test, but Laspeyres Index and Paasche Index do not.

**Factor Reversal Test**

As per this test (suggested by Prof. Irving Fisher), just as each formula should permit the interchange of the two time periods without giving inconsistent results, it ought to permit interchanging of the prices and should give the true value ratio.

The product of change in prices in the current year and the change in quantities in the current year should be equal to the value ratio, i.e = (åP_{1}Q_{1}) / (åP_{0}Q_{0}), where, P_{1} = Prices in the current year, P_{0} = Prices in the base year, Q_{0} = Quantities in the base year

The test can be applied to the index numbers by interchanging P to Q and Q to P.

Except Fischer’s Ideal Index, all other elementary indices, simple as well as weighted, fail to satisfy this test.

**Circular Test**

Circular test is an extension of the time reversal test. It is used while measuring price changes over a number of years with the shifting of base occurring frequently.

Index constructed for ‘X’ on base year ‘Y’ and for year ‘Y’ on base year ‘Z’ should yield the same result as an index constructed for ‘X’ on base year ‘Z’.

I_{0},_{1} x I_{1},_{2} x I_{2},_{0} = 1.

Laspeyres, Paasche and Fisher’s indices fail to satisfy this test. Only the fixed weight aggregates method and simple aggregates method satisfy this test.

The indices satisfying the circular test would be amenable to change from year to year without referring to the base year. The indices have the advantage of reduced computational work in the event of a change in the base year.

**Index Number Consistency Test – Problem and Solution**

**Index Number Consistency Test – Problems and Solutions**

Ex : Calculate Fisher’s ideal index number and prove that it satisfies Time Reversal and Factor Reversal tests:

**Computation Table**

p_{0} | q_{0} | p_{1} | q_{1} | p_{0}q_{0} | p_{0}q_{1} | p_{1}q_{0} | p_{1}q_{1} |

5.50 | 20 | 10.50 | 22 | 90 | 99 | 210 | 231.00 |

7.00 | 40 | 13.00 | 45 | 280 | 315 | 520 | 585.00 |

14.00 | 04 | 32.00 | 05 | 56 | 70 | 128 | 160.00 |

16.50 | 03 | 28.00 | 02 | 49.50 | 33 | 84 | 56.00 |

5.00 | 03 | 09.00 | 1.5 | 10 | 7.50 | 18 | 13.50 |

S = | 485.50 | 524.50 | 960 | 1045.50 |

Fisher’s ideal index number p_{01} = Ö [ {(å P_{1}Q_{0} ) / (å P_{0}Q_{0} )} ´ {(å P_{1}Q_{1} ) / (å P_{0}Q_{1} )}] ´ 100

[Ö{(960/485.5) x 1045.5 / 524.50)} x 100] = [Ö(1.9773) x (1.9933)] x 100= 1.9853×100=198.53%

Time reversal test is satisfied =

p_{01} x p_{10 }= [Ö {(å P_{1}Q_{0} ) / (å P_{0}Q_{0} )} ´ {(å P_{1}Q_{1} ) / (å P_{0}Q_{1})} ´ 100] x

[Ö {(å P_{0}Q_{1} ) / (å P_{1}Q_{1} )} ´ {(å P_{0}Q_{0} ) / (å P_{1}Q_{0} )} ´ 100]

=[Ö{(960/485.5) x (1045.5 / 524.50)} x 100] x [Ö{(1045.5 / 524.50)} x (485.5 / 960) x 100] = 1

So, Fisher’s ideal index number satisfies the Time Reversal Test

p_{01} x q_{01 }= [Ö {(å P_{1}Q_{0} ) / (å P_{0}Q_{0} )} ´ {(å P_{1}Q_{1} ) / (å P_{0}Q_{1} )} ´ 100] x [Ö {(å P_{0}Q_{1} ) / (å P_{0}Q_{0} )} ´ {(å P_{1}Q_{1} ) / (å P_{1}Q_{0} )} ´ 100]

= (960/485.5) x (1045.5 / 524.50)} x 100] x [{(524.5/485.5)x (1045.5 / 960) x 100]

= (åp_{1}q_{1}) / (åp_{0}q_{0})

So, Fisher’s ideal index number satisfies the Factor Reversal Test

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