BCOM - I

Business Statistics
Paper Code: BCOM-22-102
Topic – Index Number and Time Series

Meaning, Types and Uses, Methods of constructing

Price Index Number, Fixed – Base Method, Chain-
Base Method, Base conversion, Base shifting
Index Number :- deflating and splicing. Consumer Price Index
Number, Fisher’s Ideal Index Number,
Reversibility Test- Time and Factor;
Meaning, Importance and Components of a Time
Analysis of
Series. Decomposition of Time Series: - Moving
Time Series: -
Average Method and Method of Least square.

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 Unit IV

MEANING OF INDEX NUMBER (OR SIMPLY AN INDEX)

An index number is a specialised average designed to measure the changes in a
variable or a group of related variables with respect to time, geographic location
or other characteristics such as income, profession etc. Thus, when one says that
the wholesale price Index is 130 for Aug. 2005 compared to Aug. 2004, it means
the wholesale prices of commodities have increased by 30% during the year. An
index may be univariate index or composite index.
 Univariable Index – It is computed from a single variable.
 Composite Index – It is computed from a group of variables.
INDEX NUMBERS: MEANING AND CHARACTERISTICS
(A) Definition of index numbers
● According to Croxton and Cowden, index numbers are devices for measuring
differences in the magnitude of a group of related variables.
● According to Spiegal, an index number is a statistical measure designed to
show changes in a variable or a group of related variables with respect to time,
geographical locations, or other characteristics.
(B) Following are the important characteristics of index numbers.
(1) Expressed in percentage
● A change in terms of the absolute values may not be comparable.
● Index numbers are expressed in percentage, so they remove this barrier.
Although, we do not use the percentage sign.
● It is possible to compare the agricultural production and industrial production
and at the same time being expressed in percentage, we can also compare the
change in prices of different commodities.
(2) Relative measures or measures of net changes
● Index numbers measure a net or relative change in a variable or a group of
variables.

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● For example, if the price of a certain commodity rises from ₹10 in the year
2007 to ₹15 in the year 2017, the price index number will be 150 showing that
there is a 50% increase in the prices over this period.
(3) Measure change over a period of time or in two or more places
● Index numbers measure the net change among the related variables over a
period of time or at two or more places.
● For example, change in prices, production, and more, over the two periods or
at two places.
(4) Specialised average
● Simple averages like, mean, median, mode, and more can be used to compare
the variables having similar units.
● Index numbers are specialised average, expressed in percentage, and help in
measuring and comparing the change in those variables that are expressed in
different units.
● For example, we can compare the change in the production of industrial goods
and agricultural goods.
(5) Measuring changes that are not directly measurable
● Cost of living, business activity, and more are complex things that are not
directly measurable.
● With the help of index numbers, it is possible to study the relative changes in
such phenomena.
USES OF INDEX NUMBERS
Index numbers are one of the most widely used statistical tools. Some of the
advantages or uses of index numbers are as follows:
(1) Help in formulating policies
● Most of the economic and business decisions and policies are guided by the
index numbers. Example:
● To increase DA, the government refers to the cost-of-living index.
● To make any policy related to the industrial or agricultural production, the
government refers to their respective index numbers.

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(2) Help in study of trends
● Index numbers help in the study of trends in variables like, export-import,
industrial and agricultural production, share prices, and more.
(3) Helpful in forecasting
● Index numbers not only help in the study of past and present behaviour, they
are also used for forecasting economic and business activities.
(4) Facilitates comparative study
● To make comparisons with respect to time and place especially where units are
different, index numbers prove to be very useful.
● For example, change in ‘industrial production’ can be compared with change
in ‘agricultural production’ with the help of index numbers.
(5) Measurement of purchasing power of money to maintain standard of
living
● Index numbers, such as cost inflation index help in measuring the purchasing
power of money at different times between different regions.
● Such analysis helps the government to frame suitable policies for maintaining
or raising the standard of living of the people.
(6) Act as economic barometer
● Index numbers are very useful in knowing the level of economic and business
activities of a country. So, these are rightly known as economic barometers.
Conclusion-
1. Index Numbers reveal Trends and Tendencies. Index numbers reveal
tendencies of the phenomenon under study. By examining the index
numbers of wholesale prices, agricultural indestrial production, sales,
wagsn etc., one can draw a conclusion as to whether there is an upward
tendency or downward tendency.
2. Index Numbers held in framing suitable policies. The index numbers
provide some guidepost that one can use in making decision. Some of the
indices and their usefulness in policies are summarised below:

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 3. Index Numbers are useful in deflating. Deflating is the process of
eliminating the effect of changes in price level. Index numbers are useful
in converting the nominal wages into real wages, nominal income into real
income.
4. Index Numbers are useful for forecasting future economic activity.
5. Index Numbers are useful for studying long-term trend, seasonal and
cyclical variations.
Following are some of the problems involved in the construction of index
numbers:
(1) Purpose of index numbers
● Many different types of index numbers are constructed with different
objectives.
● Example: Price index, quantity index, consumer price index, wholesale price
index, and more
● So, the first important issue/problem is to define the objective for which the
index number is to be constructed.
(2) Selection of base period
● Base period is the period against which the comparisons are made.
● Selection of a suitable base period is a very crucial step.
● It should be of reasonable length and normal one, i.e., it should not be affected
by any abnormalities like, natural calamities, war, extreme business cycle
situations.
● It should neither be too close nor too far.
(3) Selection of commodities
● All the items cannot be included in the construction of an index number.
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● Nature and number of items to be included in an index number depends upon
the type of index to be constructed.
● For example, to construct a ‘consumer price index’ those commodities should
be considered that are generally consumed and the number should be neither too
small nor too big.
(4) Selection of sources of data
● Depending upon the type of index numbers, the correct source should be
selected for data.
● Like, to construct CPI, we need retail prices and to construct the wholesale
price index, we need wholesale prices. Accordingly, the right and reliable source
should be selected.
(5) Selection of weights
● The term ‘weight’ refers to the relative importance of different items in the
construction of index numbers.
● All the items do not have the same importance.
● So, it is necessary to adopt some suitable measures to assign weight.
(6) Selection of an appropriate formula
●There are various formulas for construction of index numbers like Laspeyres’
method, Paasche’s method, Fisher’s method, and more.
● No single formula is appropriate for all types of index numbers.
● The choice of formula depends upon the purpose of the available data.
LIMITATIONS OF INDEX NUMBERS
1. Index Numbers are based on only few items and not all items.
2. Index Numbers suffer from the limitations of Random sampling used in
selection of items.
3. Index Numbers generally do not take into account changes in the quality
of products.
4. Index Numbers suffer from the limitations of the method used for their
construction. Each method has its own merits and limitations. No particular
method is suitable for all circumstances.
5. Index numbers suffer from the selection of abnormal base period.

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 6. Index Numbers suffer from the problems of comparability and reliability.
7. International comparisons of index numbers are not possible since the
items of index differ from country to country.
Precautions to be taken while constructing an index number
 The first step is to receive a price quotation for the goods that have been
chosen. We all know that the cost of various goods differs from one
location to the next and even from one store to the next within the same
market. It’s also necessary to decide if wholesale or retail pricing is
required. The decision would be based on the index number’s intended use.
 The selection of an appropriate base year is another precaution in the
production of an index number. The base year serves as a point of reference
for comparisons. A regular year should be used as the basis year. It should
be free of anomalies such as wars, earthquakes, and other natural disasters.
 The next care to be considered while building the index number is to
consider the purpose of the index number. When a value index is required,
the calculation of a volume index is appropriate. The goods are chosen, and
their prices are determined with the help of index numbers.
 These are the precautions to be taken when dealing with the difficulties
faced in constructing index numbers.

Types of Index Numbers

There are various types of index numbers that have particular usage. We will
study the types of Index numbers to know the same. This section which is related
to the types of Index numbers will help the students to understand the importance
of each type in regard to the task which is practiced for.
1. Price Index
A price index number is used to measure how price alters across a period.
It will indicate the relative value and not the absolute value. The Consumer
Price Index (CPI) and Wholesale Price Index (WPI) are major examples of
a price index.

2. Quantity Index
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 A quantity index number is used to measure changes in the volume or
quantity of goods that are produced, consumed, and sold within a stipulated
period. It shows the relative change across a period for particular quantities
of goods. Index of Industrial Production (IIP) is an example of Quantity
Index.

3. Value Index
A value index number is formed from the ratio of the aggregate value for
a particular period with that of the aggregate value that is found in the base
period. The value index is utilized for inventories, sales, and foreign trade,
among others.

METHODS OF CONSTRUCTING INDEX NUMBERS

The various methods of constructing index numbers can be classified as follows:

UNWEIGHTED INDEX NUMBERS

In the unweighted index numbers, weights are not assigned to the various items.
Two unweighted index number are discussed below:
1. SIMPLE AGGREGATIVE METHOD
Simple Aggregative method is based on the assumption that the various items and
their prices are quoted in the same unit. Equal importance is given to all the items.
Under this method, the total of current year prices for the various commodities in

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question is divided by the total of base year prices and the quotient is multiplied
by 100.

Limitations of this index:

(i) It does not meet the units test. The units used in the price or quantity quotations
can affect the value of the index.
(ii) Relative importance of the item is ignored since equal importance is given to
all the items. Thus the weights are being implicit.
WEIGHTED INDEX NUMBERS
In Weighted Index Numbers, weights are assigned to various items. The various
methods of constructing weighted index numbers can be classified as follows:

Weighted Aggregative Indices

In the Weighted Aggregative Indices, weights are assigned to the various items.
In weighted aggregative price indices, the prices of the different items are
weighted by their quantities. The weights may be assigned in different methods.
Depending upon the method of assigning weights, a large number of
formulae have been devised. Some of them are as follows:
1. Laspeyres Method,

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2. Paasche Method,
3. Dorbish and Bowley’s Method,
4. Fisher’s ideal Method,
5. Marshall-Edgeworth Method, and
6. Kelly’s Method.
1. LASPEYRES METHOD
Meaning: The Laspeyres Price Index is a weighted aggregate price index where
the weights are the base period quantities. In general, Laspeyres index answers
the question, “What would be the valueof the base period list of goods when
valued at given period prices?”

Merit: The indices for different years can be compared with each other since
weights are the same base year weights.
Limitation: It does not take into consideration the consumption pattern. In case
of rising prices, it over- estimates the rise in prices.
ILLUSTRATION 1
From the following data, compute Laspeyre’s price index number for the current
year:

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Solution
Computation Table

2. PAASCHE METHOD
Meaning: The Paasche Price Index is a weighted aggregate price index where the
weights are the given period quantities. In general, Paasche price index answers
the question, “What would be the value of the given period list of goods when
valued at base period prices?”

Merit: It takes into consideration the consumption patterns.

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Limitation: The indices for different years cannot be compared with each other
since weights are not the same.
ILLUSTRATION 2
From the following data, compute Paasche’s price index number for the current
year:

Solution
Computation Table

3. DORBISH AND BOWLEY’S METHOD

Dorbish and Bowley Price Index is a weighted price index which is the arithmetic
mean of the Laspeyres and Paasche price indices. It is given by the formula:

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ILLUSTRATION 3
From the following data, compute Bowley’s price index number for the current
year:

Solution
Computation Table

4. Fisher’s Ideal Index Method

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Fisher’s Ideal Price Index is a weighted price index which is the geometric mean
of the Laspeyres and Paasche Price Indices. The Fisher’s ideal index is given by
the formula:

Why Ideal Index—Fisher’s formula is known as ‘Ideal’ because of the

following reasons:
1. It takes into account both current year as well as base year prices and quantities.
2. It satisfies both the time reversal test as well as the factor reversal test as
suggested by Fisher.
3. It is based on the geometric mean which is theoretically considered to be the
best average for constructing index numbers.
4. It is free from bias since there is complete cancellation of biases of the kinds
revealed by time reversal and factor reversal tests.
ILLUSTRATION 4
From the following data, compute Fisher’s price index number for the current
year:

Solution
Computation Table

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5. MARSHALL-EDGEWORTH METHOD
Marshall-Edgeworth Price Index is a weighted aggregative price index which also
takes into consideration both the current year as well as base year prices and
quantities. It is given by the formula:

ILLUSTRATION 5
From the following data, compute Marshall Edgeworth’s price index number for
the current year:

Solution
Computation Table
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6. KELLY’S METHOD
Kelly Price Index is a weighted aggregative price index which uses fixed weights.
Weights are the quantities which may relate to any period (not necessarily the
base year or current year) or which may be an average of the quantities of two or
three or more years. It is given by the formula:

Merits:
(i) It does not require yearly changes in the weights.
(ii) The base period can be changed without necessitating corresponding change
to the weights.
ILLUSTRATION 6
From the following data, compute Kelly’s price index number for the current
year:

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Solution
Computation Table

WEIGHTED AVERAGE OF RELATIVES METHOD

Under this method price index is constructed on the basis of price relatives and
not on the basis of absolute prices. A price index is obtained by taking the average
of all weighted price relatives. Such average can be obtained using arithmetic
mean or geometric mean.

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ILLUSTRATION 7 [Computation of Index Numbers by Weighted Average
of Price Relatives Method when there are two or more commodities]
From the following data, compute an index for the year 2002 taking 2001 as base
by weighted average of price relatives method using (a) arithmetic mean

Solution
 Index Numbers taking 2001 as base by Weighted Average of Price
Relatives Method using arithmetic mean

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QUANTITY OR VOLUME INDEX NUMBERS
Quantity Index Numbers measure change in quantities which may represent
the physical volume of production, employment etc. Prices are used as
weights. Quantity indices are obtained by changing p to q and q to p in the
various formulae discussed earlier. Some of the important Quantity Index
Formulae are given below:

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ILLUSTRATION 8 [Computation of Quantity Index Numbers]
From the following data, compute quantity index number for the current year
by applying:
1. Laspeyre’s Method
2. Paasche’s Method
3. Bowley’s Method
4. Fisher’s Ideal Method
5. Marshall Edgeworth’s Method

Solution
Computation Table

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VALUE INDEX NUMBERS (V)
A Value Index measures the change in actual values between the base and the
given period. It is obtained by dividing the sum of the value of a given year
by the sum of the values of the base year. In the form of a formula, it may be
expressed as follows:

ILLUSTRATION 9 [Computation of Value Index Number]

From the following data, compute value index number for the current year:

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Solution
Computation Table

TESTS OF ADEQUACY OF INDEX NUMBER FORMULAE

The following tests are suggested for choosing an appropriate index:
1. Unit Test,
2. Time Reversal Test,
3. Factor Reversal Test, and
4. Circular Test.
1. UNIT TEST
The unit test requires that the formula for constructing an index should be
independent of the units in which, or for which, prices and quantities are
quoted. All formulae except the simple (unweighted) aggregative index
formula satisfy this test.
2. TIME REVERSAL TEST

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Time reversal test is a test to determine whether a given method will work
both ways in time, forward and backward. According to Fisher, “The test is
that the formula for calculating the index number should be such that it will
give the same ratio between one point of comparison and the other, no matter
which of the two is taken as base.” In other words, the test requires that if an
index is constructed for the year 1 on the base year 0 and for the year 0 on the
base year 1, both the index numbers should be reciprocals of each other so that
their product is unity.

Methods which satisfy the time reversal test are:

1. The Fisher’s ideal formula,
2. Simple geometric mean of price relatives,
3. Aggregates with fixed weights,
4. The weighted geometric mean of price relatives if we use fixed weights,
and
5. Marshall-Edgeworth Method.
Let us now see how Fisher’s Ideal formula satisfies the test.

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3. FACTOR REVERSAL TEST
Factor Reversal test requires that the product of a price index and the quantity
index should be equal to the corresponding value index. According to Fisher,
“Just as each formula should permit the interchange of the two times without
giving inconsistent results, so it ought to permit interchanging the prices and
quantities without giving inconsistent result, i.e., the two results multiplied
together should give the true value ratio.” In other words, the test is that the
change in price multiplied by the change in quantity should be equal to the
total change in value. The total value of a given commodity in a given year is
the product of the quantity and the price per unit (total value = p × q). The
ratio of the total valuein one year to the total value in the preceding year is

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4. CIRCULAR TEST
Circular test is just an extension of the time reversal test. It requires that if an
index is constructed for the year a on base year b, and for the year b on base
year c, we should get the same result as if we calculated direct an index for a
on base year c without going through b as an intermediary.

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 ILLUSTRATION 10 [TIME REVERSAL AND FACTOR REVERSAL
TESTS]
From the following data, compute Fisher’s price and quantity index number
for the current year and check whether Fisher’s ideal index satisfies the time
reversal and factor reversal tests.

Solution
Computation Table

MEANING OF TIME SERIES

According to Spiegel, “A time series is a set of observations taken at specified times, usually
at ‘equal intervals’. Mathematically, a time series is defined by the values Y1, Y2 ... or variable
Y (temperature, closing price of a share, etc.) at times t1, t2.... Thus Y is a function of t,
symbolized by Y = F (t).”

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Examples of Time Series include the following:
1. Yearly National Income Data for the last 5, 6, 7 years or some other time period.
2. Yearly Production of Steel Data for the last 5, 6, 7 years or some other time period.
3. Yearly Population Data for the last 5, 6, 7 years or some other time period.
4. Yearly Sales Data for the last 5, 6, 7 years or some other time period.
5. Number of Marriages taking place during a certain period.
6. Number of Divorces taking place during a certain period.
7. Number of Accidents taking place during a certain period.
8. Number of Deaths taking place during a certain period.
9. Number of Births taking place during a certain period.
10. Number of Criminals entering into politics during a certain period.

ESSENTIAL REQUIREMENTS OF A TIME SERIES

1. Time series must consist of a homogeneous set of values.
2. It must consist of data for a sufficiently long period.
3. Time elapsing between various observations must as far as possible be equal.
4. The gaps if any in the data should be made up by interpolation.

MAIN OBJECTIVE OF ANALYSING TIME SERIES

The main objective of analysing time series is to understand, interpret and measure all types of
variations (i.e. changes) which characterise the time series and to use them for anticipating the
course of future events. Technically speaking the objective of analysing time series is to
ascertain the forces which are at work that pull or push the straight line and to use the results
for anticipating the future operations.
In nutshell, the main objectives of analysing time series are–
1. To study the past behaviour of the data.
2. To predict the future on the basis of past experience.
3. To segregate the effects of various factors which might affect time series.

UTILITY OF TIME SERIES ANALYSIS

1. Time series analysis helps in understanding past behaviour.
2. It facilitates the prediction of future on the basis of past experience.

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3. It facilitates the evaluation of current performance.
4. It facilitates intra and inter comparison. For example, comparison of production of a firm or
industry or economy of current year with that of previous year. Comparison of current year’s
production of two firms or economies.

COMPONENTS OF A TIME SERIES

The components of a time series are the various elements which can be segregated from the
observed data. The following are the broad classes of these components:

In short, a time series consists of four components as follows:

1. Secular Trend (T)
2. Seasonal Variations (S)
3. Cyclical Variations (C)
4. Irregular Movements (I)

1. SECULAR TRENDS (T)

Meaning of Secular Trend — Secular trend is the basic tendency of steady movements in a set
of observations to move in an upward or downward or constant direction over a fairly long
period of time. It does not include short term variations like seasonality, irregularity etc. Secular
trend is the long-term tendency of the time series to move in an upward or downward direction.
It indicates how, on the whole, it has behaved over the entire period under reference. These are
the result of long-term forces that gradually operate on the time series variable. A few examples
of these long term forces which make a time series to move in any direction over long period
of time are the long-term changes per capita income, technological improvements, growth of
population, changes in social norms etc. Trend of a time series characterizes the gradual and
consistent pattern of its changes. Some series of data recorded over a given period of time show
an upward trend, some show a downward trend, while others remain more or less at constant
level. Also, some series increase slowly and some increase fast, others decrease at varying

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rates. Again some series after a period of growth or decline reverse themselves and enter a
period of decline or growth. The secular trend is caused by basic factors underlying the series
in question.

Methods of Measuring Trend — Various methods that can be used for measurement of trend
are

(1) Freehand or Graphic method, (2) Semi-average method, (3) Moving average method, (4)
Least
squares method.

Types — Trend can be of three types as follows:

2. SEASONAL VARIATIONS (S)

Seasonal variations are the regular periodic changes which take place within a period of less
than a year and may take place daily, weekly, monthly or quarterly. It may be due to climatic
changes or due to the changes in the pattern of consumption or production, etc. For example,
sale of rain coats may go up during the rainy season and go down during the dry season;
increase of coal consumption may take place during winter; higher production of rice during
the harvesting season and lower production during cultivating season, higher sales of woollen
garments during winter season, and so on. After a careful study of seasonal behaviour we have
to collect suitable data; monthly, weekly, or quarterly, as the case may be. A study of pattern
is extremely useful for the management of business and industry.

3. CYCLICAL VARIATIONS (C)

Cyclical variations, which are also generally termed as business cycles, are the periodic
movements in the time series around the trend line. These are upswings and downswings in the
time series that are observable over extended periods of time. The amplitude or the frequency
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of occurrence of these cycles is not uniform.Such variations happen during intervals of time of
more than one year, say, once in 3 to 7 years. All economic activities get a boost during the
upward swings of business cycles and dip during the down word swings. The study of business
cycles helps us to take anti cyclical measures. Business cycles representing intervation. of (i)
prosperity (ii) recession (iii) depression (iv) recovery come under cyclical variation.

4. IRREGULARITIES (I)
These are irregular variations which occur on account of random external events. These
variations either go very deep downward or too high upward to attain peaks abruptly. These
variations may occur due to strikes, lockouts, floods, wars, elections etc. Although it is
ordinarily assumed that such events produce variations lasting only a short time, it is
conceivable that they may be so intense as to result in new cyclical or other movements.

MODELS OF TIME SERIES

The relationship among the different components of a time series may be expressed using either
the addition multiplication models.
1. Additive Model —

Additive model assumes that all the components of the time series are independent of one
another and describes all the components as absolute values. The original data (Y) is expressed
as a sum of four components as follows:

Y=T+C+S+I
where, Y = Observed value in a given time series, T = Trend, C = Cyclical Variations, S =
Seasonal Variations and I = Irregular Variations
2. Multiplicative Model —

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Multiplicative model assumes that all the four components are due to different causes but they
are not necessarily independent and they can affect one another. It describes only the trend (T)
as an absolute value while other components (i.e., C, S & I) are expressed as rate or percentage.
Thus, a seasonal index of 89% would indicate that the actual value is expected to be 11% lower
than it would be without the seasonal influence. The original data (Y) is expressed as a product
of four components as follows:

Y=T×C×S×I
METHODS OF MEASURING TREND
There are four methods of measuring trend in time series:

METHOD OF LEAST SQUARES

Meaning—Method of Least Squares is a mathematical technique employed for finding
equation of a specified type of curve which best fits a given observation relating to two
variables. This is based on the principle that for the “best-fitting” curve the sum of the squares
of difference between the observed and the corresponding estimated values obtained from the
equation, should be the minimum possible.
Features — The main features of the least square method are as follows:
1. The sum of deviations of the actual values of Y from the computed values of Y is zero i.e.,
(Y – Yc) = 0.
2. The sum of the squares of the deviations of the actual values of Y and computed values of
Y is least. i.e., (Y – Yc)2 is least. That is why it is called the method of least squares and the
line obtained by this method is called as ‘the line of best fit’. The least squares method may be
used to fit a straight line trend or a parabolic trend.
Straight Line Trend — Straight line trend is represented by the equation:
Yc = a + bX
where, Yc = Trend values.
a = Y intercept or trend value at origin i.e., when X = 0.

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b = Slope of the trend line or the amount of change in trend value per unit of X. Slope
may be positive, negative or zero.
X = Time unit which may be one year, half year, one quarter or one month or one week
or one day.
Along with the trend equation, the following three things should also be specified.
1. Origin — (Specify the date/year selected as origin)
2. X Unit — (Specify the time unit whether one year or half year or one quarter or one month)
3. Y Unit — (Specify the unit in which Y is being measured e.g. whether sales in rupees, sales
in
units, production in tonnes)
How to determine the value of ‘a’ and ‘b’ — The values of ‘a’ and ‘b’ can be found by solving
the
following normal equations:
Y = Na + bX ... Equation I
XY = aX + bX2 ... Equation II
where, Y = Sum of actual values of Y variable.
N = No. of years or months or any other period.
X = Sum of deviations from the origin.
XY = Sum of the products of deviations from the origin and actual values.
X2 = Sum of squares of deviations from the origin.
Notes:
(i) Equation II can be obtained by multiplying Equation I by X.
(ii) If middle period is taken as origin and deviations are taken from the middle period, X
would also be zero provided there is no gap in the data given. When X = 0, the above
normal equation would be reduced to:
Y = Na ... Equation I
XY = bX2 ... Equation II

Business Statistics
Dr. Sofia Khan
MERITS AND LIMITATIONS OF METHOD OF LEAST SQUARES

HOW TO CALCULATE TREND VALUES (YC)

Business Statistics
Dr. Sofia Khan
Trend values (Yc) can be obtained by putting the value of X in the straight line trend equation
of the form Yc = a + bX.
For example, if straight line trend equation is Yc = 30 + 9X and the value of X = –2, – 1, 0, 1,
2, the trend values will be obtained as follows:

ILLUSTRATION 1 (a) [Calculation of Straight Line Trend Equation when odd number of
years are given.] The following data relate to sales of TUSHAR Ltd.

(a) Fit a Straight Line Trend by the method of least squares and tabulate the trend values.
(b) Eliminate the trend using additive model. What components of the time series are thus left
over?
(c) Estimate the likely sales for the year 2006.
(d) By what year the company’s expected sales would be 84 lakhs?
Solution
(a) & (b) Fitting a Straight Line Trend by the Method of Least Squares

Business Statistics
Dr. Sofia Khan
Hence the annual trend equation is given by:
Y = 30 + 9X
[Origin = 2002, X units = One year, Y units = Annual sales in lakhs of Rs.]
(b) After eliminating the trend only cyclical and irregular variations are left since seasonal
variations are absent as the annual data is given.
(c) Likely sales for the year 2006: For 2006, X = 4
Y2006 = 30 + 9 × 4 = 66
(d) 84 = 30 + 9X or X = (84 – 30)/9 = 6 years frorm origin (i.e. 2002)
Hence, in the year 2008 (i.e. 2002 + 6 years the company’s expected sales would be Rs. 84
lakhs.

Business Statistics
Dr. Sofia Khan