Demand forecasting by time series analysis

What is Demand?
 Demand is a buyer's willingness and ability to pay a
price for a specific quantity of a good or service.
Demand refers to how much (quantity) of a product or
service is desired by buyers at various prices. The
quantity demanded is the amount of a product people
are willing to buy at a certain price; the relationship
between price and quantity demanded is known as the
demand.

What is demand forecasting?
 “The science and art of the converting the Qualitative
understanding of a market into Quantitative data.”
 Demand forecasting is the activity of estimating the
quantity of a product or service that consumers will
purchase. Demand forecasting involves techniques
including both informal methods, such as educated
guesses, and quantitative methods, such as the use of
historical sales data or current data from test markets.
Demand forecasting may be used in making pricing
decisions, in assessing future capacity requirements, or in
making decisions on whether to enter a new market.

Importance of demand forecasting
 Distribution of resources
 Helps in avoiding wastages of resources
 Serves as a direction to production
 Pricing
 Helps in devising sales policy
 Decrease of business risk
 Inventory management

Levels of Demand Forecasting
1. Micro level or firm level
 This refers to the demand forecasting by the firm for its product.
The management of a firm is really interested in such forecasting.
Generally speaking, demand forecasting refers to the forecasting of
demand of a firm.
2. Industry level
 Demand forecasting for the product of an industry as a whole is
generally undertaken by the trade associations and the results are
made available to the members. A member firm by using such data
and information may determine its market share.
3. Macro level
 Estimating industry demand for the economy as a whole will be
based on macroeconomic variables like national income, national
expenditure, consumption function, index of industrial production,
aggregate demand, aggregate supply etc; generally, it is undertaken
by national institutes, govt. Agencies etc. Such forecasts are helpful
to the Government in determining the volume of exports and
imports, control of prices etc.

Criteria for Good Demand
Forecasting
 Accuracy
 Plausibility
 Simplicity
 Durability
 Accuracy
 Plausibility
 Simplicity
 Durability

Meaning of Time Series:
 A sequence of numerical data points in successive
order, usually occurring in uniform intervals. In
general, a time series is simply a sequence of numbers
collected at regular intervals over a period of time.
 Time series analysis comprises methods for analyzing
time series data in order to extract meaningful
statistics and other characteristics of the data. Time
series forecasting is the use of a model to predict
future values based on previously observed values

Components of Time Series
 The change which are being in time series, They are
effected by Economic, Social, Natural, Industrial &
Political Reasons. These reasons are called components
of Time Series.
 Secular trend :-
 Seasonal variation :-
 Cyclical variation :-
 Irregular variation :-

Secular trend:
 Secular movements indicate the general conditions and
direction in which graph of a time series move in relatively
a long period of time.
 A time series data may show upward trend or downward
trend for a period of years and this may be due to factors
like:
 increase in population,
change in technological progress ,
large scale shift in consumers
demands,

Seasonal variation:
 Seasonal variation are short-term fluctuation in a time
series which occur periodically in a year. This
continues to repeat year after year.
 Time series also undergo changes during seasonal sales
of a company. During festival season, sales clearance
season etc., we come across most unexpected changes.
 The major factors that are weather conditions and
customs of people.

Cyclical Variations:
 Cyclical variations are recurrent upward or downward
movements in a time series but the period of cycle is greater
than a year. Also these variations are not regular as seasonal
variation.
 A business cycle showing these oscillatory movements has to
pass through four phases-prosperity, recession, depression
and recovery. In a business, these four phases are completed
by passing one to another in this order.

Irregular variation:
 Irregular variations are fluctuations in time series that
are short in duration, erratic in nature and follow no
regularity in the occurrence pattern. These variations
are also referred to as residual variations since by
definition they represent what is left out in a time
series after trend, cyclical and seasonal variations.
 Irregular fluctuations results due to the occurrence of
unforeseen events like :
Floods,
Earthquakes,
Wars,
Famines

Time Series Model
 Addition Model:
Y = T + S + C + I
Where:- Y = Original Data
T = Trend Value
S = Seasonal Fluctuation
C = Cyclical Fluctuation
I =
I = Irregular Fluctuation
• Multiplication Model:
Y = T x S x C x I
or
Y = TSCI

Measurement of Secular trend:-
 The following methods are used for calculation of trend:
 Free Hand Curve Method:
 Semi – Average Method:
 Moving Average Method:
 Least Square Method:

Free hand Curve Method
 In this method the data is denoted on graph paper. We take
“Time” on ‘x’ axis and “Data” on the ‘y’ axis. On graph there
will be a point for every point of time. We make a smooth
hand curve with the help of this plotted points.
Example:
Draw a free hand curve on the basis of the
following data:
Years 1989 1990 1991 1992 1993 1994 1995 1996
Profit
(in
‘000)
148 149 149.5 149 150.5 152.2 153.7 153

145
146
147
148
149
150
151
152
153
154
155
1989 1990 1991 1992 1993 1994 1995 1996
Profit ('000)
Trend Line
Actual Data

Semi – Average Method:
 In this method the given data are divided in two parts,
preferable with the equal number of years.
 For example, if we are given data from 1991 to 2008,
i.e., over a period of 18 years, the two equal parts will
be first nine years, i.e.,1991 to 1999 and from 2000 to
2008. In case of odd number of years like, 9, 13, 17,
etc.., two equal parts can be made simply by ignoring
the middle year. For example, if data are given for 19
years from 1990 to 2007 the two equal parts would
be from 1990 to 1998 and from 2000 to 2008 - the
middle year 1999 will be ignored.

Example:
Find the trend line from the
following data by Semi – Average Method:-
Year 1989 1990 1991 1992 1993 1994 1995 1996
Production
(M.Ton.)
150 152 153 151 154 153 156 158
There are total 8 trends. Now we distributed it in equal part.
Now we calculated Average mean for every part.
First Part = 150 + 152 + 153 + 151 = 151.50
4
Second Part = 154 + 153 + 156 + 158 = 155.25
4

Year
(1)
Production
(2)
Arithmetic Mean
(3)
1989
1990
1991
1992
1993
1994
1995
1996
150
152
153
151
154
153
156
158
151.50
155.25

146
148
150
152
154
156
158
160
1989 1990 1991 1992 1993 1994 1995 1996
Production
Production
151.50
155.25

Moving Average Method:-
 It is one of the most popular method for calculating Long Term Trend. This
method is also used for ‘Seasonal fluctuation’, ‘cyclical fluctuation’ & ‘irregular
fluctuation’. In this method we calculate the ‘Moving Average for certain years.
 For example: If we calculating ‘Three year’s Moving Average’ then according to
this method:
=(1)+(2)+(3) , (2)+(3)+(4) , (3)+(4)+(5), ……………..
3 3 3
Where (1),(2),(3),………. are the various years of time series.
Example: Find out the five year’s moving Average:
Year 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996
Price 20 25 33 33 27 35 40 43 35 32 37 48 50 37 45

Year
(1)
Price of
sugar (Rs.)
(2)
Five year’s moving
Total
(3)
Five year’s moving
Average (Col 3/5)
(4)
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
20
25
33
30
27
35
40
43
35
32
37
48
50
37
45
-
-
135
150
165
175
180
185
187
195
202
204
217
-
-
-
-
27
30
33
35
36
27
37.4
39
40.4
40.8
43.4
-
-

 This method is most widely in practice. When this method is
applied, a trend line is fitted to data in such a manner that the
following two conditions are satisfied:-
 The sum of deviations of the actual values of y and computed values of y is zero.
 i.e., the sum of the squares of the deviation of the actual and computed values is
least from this line. That is why method is called the method of least squares.
The line obtained by this method is known as the line of `best fit`.
is least
  0 cYY
 2
  cYY
Least Square Method

 The Method of least square can be used either to fit a straight line
trend or a parabolic trend.
 The straight line trend is represented by the equation:-
= Yc = a + bx
Where, Y = Trend value to be computed
X = Unit of time (Independent Variable)
a = Constant to be Calculated
b = Constant to be calculated
Example:-
Draw a straight line trend and estimate trend value for 1996
Year 1991 1992 1993 1994 1995
Production 8 9 8 9 16

Year
(1)
Deviation From
1990
X
(2)
Y
(3)
XY
(4)
X2
(5)
Trend
Yc = a + bx
(6)
1991
1992
1993
1994
1995
1
2
3
4
5
8
9
8
9
16
8
18
24
36
80
1
4
9
16
25
5.2 + 1.6(1) = 6.8
5.2 + 1.6(2) = 8.4
5.2 + 1.6(3) = 10.0
5.2 + 1.6(4) = 11.6
5.2 + 1.6(5) = 13.2
N= 5
= 15 =50 = 166 = 55
X
Y XY 2
X
’
Now we calculate the value of two constant ‘a’ and ‘b’ with the help
of two equation:-
Solution:-

  
 


2
XbXaXY
XbNaY
Now we put the value of :-
50 = 5a + 15(b) ……………. (i)
166 = 15a + 55(b) ……………… (ii)
Or 5a + 15b = 50 ……………… (iii)
15a + 55b = 166 …………………. (iv)
Equation (iii) Multiply by 3 and subtracted by (iv)
-10b = -16
b = 1.6
Now we put the value of “b” in the equation (iii)
    NXXYYX ,&,,, 2

= 5a + 15(1.6) = 50
5a = 26
a = = 5.2
As according the value of ‘a’ and ‘b’ the trend line:-
Yc = a + bx
Y= 5.2 + 1.6X
Now we calculate the trend line for 1996:-
Y1996 = 5.2 + 1.6 (6) = 14.8
5
26

Shifting The Trend Origin:-
 In above Example the trend equation is:
Y = 5.2 + 1.6x
 Here the base year is 1993 that means actual base of
these year will 1st July 1993. Now we change the base
year in 1991. Now the base year is back 2 years unit
than previous base year.
 Now we will reduce the twice of the value of the ‘b’ from the
value of ‘a’.
 Then the new value of ‘a’ = 5.2 – 2(1.6)
 Now the trend equation on the basis of year 1991:
Y = 2.0+ 1.6x

Parabolic Curve:-
 Many times the line which draw by “Least Square
Method” is not prove ‘Line of best fit’ because it is not
present actual long term trend So we distributed Time
Series in sub- part and make following equation:-
Yc = a + bx + cx2
 If this equation is increase up to second degree then it is
“Parabola of second degree” and if it is increase up to third
degree then it “Parabola of third degree”. There are three
constant ‘a’, ‘b’ and ‘c’.
 Its are calculated by following three equation:-

 If we take the deviation from ‘Mean year’ then the
all three equation are presented like this:
  
 
 



422
2
2
XcXaYX
XbXY
XCNaY
  
  
  



4322
32
2
XcXbXaYX
XcXbXaXY
XcXbNaY
Parabola of second degree:-

Year Production Dev. From Middle
Year
(x)
xY x2 x2Y x3 x4 Trend Value
Y = a + bx + cx2
1992
1993
1994
1995
1996
5
7
4
9
10
-2
-1
0
1
2
-10
-7
0
9
20
4
1
0
1
4
20
7
0
9
40
-8
-1
0
1
8
16
1
0
1
16
5.7
5.6
6.3
8.0
10.5
= 35 = 0 =12 = 10 = 76 = 0 = 34
Y X XY  2
X YX 2
 3
X  4
X
Example:
Draw a parabola of second degree from the following data:-
Year 1992 1993 1994 1995 1996
Production (000) 5 7 4 9 10

  
 
 



422
2
2
XcXaYX
XbXY
XNaY
Now we put the value of
35 = 5a + 10c ………………………… (i)
12 = 10b ………………………… (ii)
76 = 10a + 34c ……………………….. (iii)
From equation (ii) we get b = = 1.2
    NXXXXYYX ,&,,,,, 432
10
12
We take deviation from middle year so the
equations are as below:

Equation (ii) is multiply by 2 and subtracted from (iii):
10a + 34c = 76 …………….. (iv)
10a + 20c = 70 …………….. (v)
14c = 6 or c = = 0.43
Now we put the value of c in equation (i)
5a + 10 (0.43) = 35
5a = 35-4.3 = 5a = 30.7
a = 6.14
Now after putting the value of ‘a’, ‘b’ and ‘c’, Parabola of second
degree is made that is:
Y=6.14+1.2x+0.43x2
14
6

Parabola of Third degree:-
 There are four constant ‘a’, ‘b’, ‘c’ and ‘d’ which are calculated
by following equation.
 The main equation is Yc = a + bx + cx2 + dx3.
 There are also four normal equation.
  
  
   
  




65433
54322
432
32
XdXcXbXaYX
XdXcXbXaYX
XdXcXbXaXY
XdXcXbNaY

Methods Of Seasonal Variation:-
 Seasonal Average Method
 Link Relative Method
 Ratio To Trend Method
 Ratio To Moving Average Method

Seasonal Average Method
 Seasonal Averages = Total of Seasonal Values
No. Of Years
 General Averages = Total of Seasonal Averages
No. Of Seasons
 Seasonal Index = Seasonal Average
General Average

EXAMPLE:-
 From the following data calculate quarterly seasonal indices
assuming the absence of any type of trend:
Year I II III IV
1989
1990
1991
1992
1993
-
130
120
126
127
-
122
120
116
118
127
122
118
121
-
134
132
128
130
-

Solution:-
Calculation of quarterly seasonal indices
Year I II III IV Total
1989
1990
1991
1992
1993
-
130
120
126
127
-
122
120
116
118
127
122
118
121
-
134
132
128
130
-
Total 503 476 488 524
Average 125.75 119 122 131 497.75
Quarterly
Turnover
seasonal
indices
124.44 = 100
101.05 95.6 98.04 105.03

 General Average = 497.75 = 124.44
4
Quarterly Seasonal variation index = 125.75 x 100
124.44
So as on we calculate the other seasonal indices

Link Relative Method:
 In this Method the following steps are taken for
calculating the seasonal variation indices
 We calculate the link relatives of seasonal figures.
Link Relative: Current Season’s Figure x 100
Previous Season’s Figure
 We calculate the average of link relative foe each
season.
 Convert These Averages in to chain relatives on the
basis of the first seasons.

 Calculate the chain relatives of the first season on the
base of the last seasons. There will be some difference
between the chain relatives of the first seasons and the
chain relatives calculated by the pervious Method.
 This difference will be due to effect of long term
changes.
 For correction the chain relatives of the first season
calculated by 1st method is deducted from the chain
relative calculated by the second method.
 Then Express the corrected chain relatives as
percentage of their averages.

Ratio To Moving Average Method:
 In this method seasonal variation indices are
calculated in following steps:
 We calculate the 12 monthly or 4 quarterly moving
average.
 We use following formula for calculating the moving
average Ratio:
Moving Average Ratio= Original Data x 100
Moving Average
Then we calculate the seasonal variation indices on the
basis of average of seasonal variation.

Ratio To Trend Method:-
 This method based on Multiple model of Time Series.
In It We use the following Steps:
 We calculate the trend value for various time duration
(Monthly or Quarterly) with the help of Least Square
method
 Then we express the all original data as the percentage
of trend on the basis of the following formula.
= Original Data x 100
Trend Value
Rest of Process are as same as moving Average Method

Methods Of Cyclical Variation:-
Residual Method
References cycle analysis
method
Direct Method
Harmonic Analysis Method

Residual Method:-
 Cyclical variations are calculated by Residual Method . This method is
based on the multiple model of the time Series. The process is as below:
 (a) When yearly data are given:
In class of yearly data there are not any seasonal variations so original data
are effect by three components:
• Trend Value
• Cyclical
• Irregular
 First we calculate the seasonal variation indices according to
moving average ratio method.
 At last we express the cyclical and irregular variation as the
Trend Ratio & Seasonal variation Indices
(b) When monthly or quarterly data are given:

Measurement of Irregular Variations
 The irregular components in a time series represent
the residue of fluctuations after trend cycle and
seasonal movements have been accounted for. Thus if
the original data is divided by T,S and C ; we get I i.e. .
In Practice the cycle itself is so erratic and is so
interwoven with irregular movement that is impossible
to separate them.

Demand forecasting by time series analysis

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