ANALYSIS OF TIME SERIES

Time series is an arrangement of statistical data in accordance with it’s time of

occurance.It shows the dynamic pace of

movement of a phenomenon over a period

of time.

e.g.

1. Annual production of wheat/rice in India over a Number of Yrs.

2. Daily sales of Ansal Plaza.

3. The daily closing of price of a

share in the stock market

4.Profit earned by a company for each of the past 10 yrs.

5.Number of MBA students enrolled in last 5yrs etc…

In today’s changing Economic phases all

variables relating to Buz.,commerce and

Economics...Production,costs,profits,sales,

N.Y.,etc are directly affected by the course

of time.

THE STATISTICAL TECHNIQUE APPLIED

TO MEASURE THE TIME BASED DATA

OVER A PERIOD OF TIME IS KNOWN

AS TIME SERIES ANALYSIS.

MATHEMATICALLY :-

y=f(t)

y= value of the variable at time “t”

If the values of a variable over time period

t1,t2,t3…..tn are

y1 y2 y3….yn respectively then

t: t1 t2 t3…………..tn.

y: y1 y2 y3…………..yn.

Thus,time series has a bivariate Distribution,

One of the two variable will be (I.V.)

and the other(D.V.).

The value of “t” may be yearly,monthly,weekly,daily,hourly.

TIME SERIES CONSISTS OF DATA

ARRANGED CHRONOLOGICALLY.

BASIC REQUIRMENTS OF TIME SERIES:-

1.HOMOGENITY OF DATA:-i.e. all the terms/data in the time series must be related to the same phenomenon.2.LONG PERIOD:-3.TIME GAPS:-Should be of equal intervals.4.APPLICATION OF INTERPOLATION:- In case there are gaps b/w the data these should be made up by proper interpolation to arrive at proper results.

ELEMENTS/COMPONENTS OF TIME SERIES:-

It’s generally seen that the values of a time series show various types of fluctuations over a period of time which are caused by multiple forces and reasons.These force are also called as variations,the pattern,the movements,the elements,components of time series.

COMPONENTS OF TIME SERIES

SECULAR TREND/LONG

TERM VARIATIONS

SEASONAL VARIATIONS

CYCLICAL VARIATIONS

IRREGULARVARIATIONS

SECULAR TREND

Refers to the general tendency of the time

series data to raise, reduce or remain

constant over a period of time.The secular

trend can be either upwards or

downwards.(Diag).

TYPES OF TREND1.LINEAR TREND:-when the rate of gradual growth/decline of a time series remains constant in the long run.Mathematically:- Y(t)= a+bx Y(t)=Trend value of a variable

X= time a,b= constants(a=intercept of y axis,b=slope

of the line).(Diag)

2.NON LINEAR TREND:- When the long run

gradual growth/decline in a series is not at a constant rate.

SEASONAL VARIATIONS These periodic Movements in Buz. Activity

occurs at regular intervals i.e. every year. As these occur during a period of 12 months they can be predicted fairly accurately. These will be there if the data are recorded quarterly, monthly, weekly, daily, hourly, etc.

The amplitude will be different but the time period will be same.

CAUSES:-

1.Natural factors:-these are weather, seasons,climatic changes.

2.Man made conventions:-The changes in habits, customs,fashion,conventions

(Diwali,dushera,Christmas,baishakhi).

CAUSES:-

1.Natural factors:-these are weather, seasons,climatic changes.

2.Man made conventions:-The changes in habits, customs,fashion,conventions

(Diwali,dushera,Christmas,baishakhi).

The study of seasonal variations is of utmost importance to Buz.,producers,sales

managers etc in framing future policies pertaining to purchase, production, sales, advertising program.

The study of seasonal variations is of utmost importance to Buz.,producers,sales

managers etc in framing future policies pertaining to purchase, production, sales, advertising program.

CYCLICAL VARIATIONS These cycles are generally repeated at

intervals ranging from 3-10 yrs.These are caused by complex combinations of forces affecting the equilibrium b/w Demand and supply. These are also called Buz.

Cycles as these are of longer durations than 12 months.(Diag)

Prosperity,recession,depression,recovery:-

During prosperityOptimisim, high profits, expansion, development,money demand increase,”r” increases, transport facilities decreases,Prices decreases, unemployment increases…..Pessimism increases.

Prosperity,recession,depression,recovery:-

During prosperityOptimisim, high profits, expansion, development,money demand increase,”r” increases, transport facilities decreases,Prices decreases, unemployment increases…..Pessimism increases.

IMPORTANCE OF CYCLICAL VARIATIONS

1.Helps in formation of Buz. Policies.

2.Estimation of future behaviour which results in arranging suitable safeguards.

3.Idea of Irregular fluctutations can be easily studied.

DISTINGUISH BETWEEN CYCLICAL AND SEASONAL VARIATIONS:-

1.Time period:-The duration of S.V. is one year whereas that of C.V. it is greater than one year.

2. Degree of accuracy:-C.V. can’t be estimated accuarately,whereas S.V. can be estimated with high degree of accuracy.

DISTINGUISH BETWEEN CYCLICAL AND SEASONAL VARIATIONS:-

1.Time period:-The duration of S.V. is one year whereas that of C.V. it is greater than one year.

2. Degree of accuracy:-C.V. can’t be estimated accuarately,whereas S.V. can be estimated with high degree of accuracy.

3. Causes:-The main cause of S.V. are Weather,customs,traditions,whereas the C.V. are caused due to complex economic activities.

4.Calculations and measurement:-Both have different methods form them.

5.Effect:-The shadow effects of C.V. influence the Economy as well as Buz. Activity as a whole where as the shadow effects of S.V. differ from occupation to occupation.

6.Activities of Preceding period:-C.V. depends upon the activities of preceding period whereas the S.V. have origin the the year itself.It is independent of the activities of preceding year.

3. Causes:-The main cause of S.V. are Weather,customs,traditions,whereas the C.V. are caused due to complex economic activities.

4.Calculations and measurement:-Both have different methods form them.

5.Effect:-The shadow effects of C.V. influence the Economy as well as Buz. Activity as a whole where as the shadow effects of S.V. differ from occupation to occupation.

6.Activities of Preceding period:-C.V. depends upon the activities of preceding period whereas the S.V. have origin the the year itself.It is independent of the activities of preceding year.

IRREGULAR OR RANDOM VARIATIONS

Also known as Residual, Episode,Erratic or Accidental predictions. These don’t have any pattern and are not repetitive. These results from Non Recurring circumstances like…Wars, Strikes,Lockout,Droughts,Tremors,Storms,Epidemics etc…

The best can be done is to get rough estimates of these variations and accordingly make provisions for such abnormalities during normal times in Buz.

ANALYSIS OF TIME SERIES:- ANALYSIS OF TIME SERIES:-

ANALYSIS OF TIME SERIES

MULTIPLICATIVE RELATION

SHIPO=T*S*C*I`

ADDITIVE RELATIONSHIP

O=T+S+C+I

Where O=Original data

T= Trend.

S=seasonal component.

C=Cyclical variations. I =Irregular variations.

Where O=Original data

T= Trend.

S=seasonal component.

C=Cyclical variations. I =Irregular variations.

MEASUREMENT OF TREND :-MEASUREMENT OF TREND :-

TREND LINE

GRAPHIC(FREE HAND CURVE FITTING

METHOD)

METHOD OF SEMI

AVERAGES

METHODS OF MOVING

AVERAGES

LEAST SQUARES METHOD

GRAPHIC METHOD Simplest and most flexible method of

Estimating the trend.

Consist of first obtaining a HISTOGRAM by plotting the time series value on a graph paper and then drawing a free hand curve through the points so that it correctly shows the long term tendency of the data. Shows a rising trend or a declining trend.(Diag)

STEPS:-

• Plot the time series data on a graph.

• Examine the direction from the information.

• According to the best personnel judgments, draw a straight line which will best fit to the data. This line will show the direction of the trend.

STEPS:-

• Plot the time series data on a graph.

• Examine the direction from the information.

• According to the best personnel judgments, draw a straight line which will best fit to the data. This line will show the direction of the trend.

POINTS TO CONSIDER:-1. Curve should be smooth.2. Number of Points above the Trend curve

should be more or less equal to the number of points below it.

3. Sum of vertical deviations of the given points above the trend line should be roughly equal to the sum of vertical deviations of the points below the trend line so that they may balance each other.

4. Sum of the Squares of the vertical deviations of the given points from the trend line is Minimum as possible.

POINTS TO CONSIDER:-1. Curve should be smooth.2. Number of Points above the Trend curve

should be more or less equal to the number of points below it.

3. Sum of vertical deviations of the given points above the trend line should be roughly equal to the sum of vertical deviations of the points below the trend line so that they may balance each other.

4. Sum of the Squares of the vertical deviations of the given points from the trend line is Minimum as possible.

Prob:-

Fit a trend line to the following data by the free hand curve method.

Prob:-

Fit a trend line to the following data by the free hand curve method.

YEAR Production(Tonnes)

1988 40

1989 44

1990 48

1991 42

1992 46

1993 50

1994 46

1995 52

1996 59

Soln:- Diag.

Merits:-

• Simple and time saving

• No mathematical calculations.

• Very flexible method.Tells if the trend is linear or non linear.

Demerits:-

• Highly subjective in nature.

• Dangerous to use for predictions.

Soln:- Diag.

Merits:-

• Simple and time saving

• No mathematical calculations.

• Very flexible method.Tells if the trend is linear or non linear.

Demerits:-

• Highly subjective in nature.

• Dangerous to use for predictions.

METHODS OF SEMI AVERAGES

Here the whole data is divide into two equal parts preferably with same number of years. The averages of each part is calculated and then a trend line through these averages is calculated.

a) Even Number of years:-

Prob:-Production from the 1999-2006 is given. Fit a trend line.

Year 99 00 01 02 03 04 05 06

Prod 10 12 18 20 20 25 23 32

(Tonnes)

a) Even Number of years:-

Prob:-Production from the 1999-2006 is given. Fit a trend line.

Year 99 00 01 02 03 04 05 06

Prod 10 12 18 20 20 25 23 32

(Tonnes)

Year X Y AVERAGE

1999 0 10

2000 1 12

2001 2 18 Average=60/4=15

2002 3 20

2003 4 20

2004 5 25 Average=

2005 6 23 100/4=25

2006 7 32

Year X Y AVERAGE

1999 0 10

2000 1 12

2001 2 18 Average=60/4=15

2002 3 20

2003 4 20

2004 5 25 Average=

2005 6 23 100/4=25

2006 7 32

• The Average of 1999-2002 is 15 (b/w 2000-2001)

• The Average of 2003-2006=25. (b/w 2004-2005)

• These points are joined by a straight line , which is semi variable trend line(Diag.)

• The Average of 1999-2002 is 15 (b/w 2000-2001)

• The Average of 2003-2006=25. (b/w 2004-2005)

• These points are joined by a straight line , which is semi variable trend line(Diag.)

b)Odd Numbers of years:-Here either the middle year is excluded or the series may be split into unequal parts. If in the series one particular year has been abnormal year, it is advisable to omit that year to make trend line more realistic.

b)Odd Numbers of years:-Here either the middle year is excluded or the series may be split into unequal parts. If in the series one particular year has been abnormal year, it is advisable to omit that year to make trend line more realistic.

Prob:-Fit a trend line by Semi Avg. method. Also find trend values.

Yr 2000 2001 2002 2003 2004 2005 2006

Sales

(000)

51 54 57 55 54 56 58

Sol:-Here middle year shall be left out and the Av. Of 1st 3yrs and last 3yrs shall be obtained.i.e.

2000-2002 =51+54+57/3 =54

2004-2006 =54+56+58/3 =56

Thus we get 2 pts 54 and 56. By joining these 2pts we shall obtain the required trend line.This can be used for Prediction.

Sol:-Here middle year shall be left out and the Av. Of 1st 3yrs and last 3yrs shall be obtained.i.e.

2000-2002 =51+54+57/3 =54

2004-2006 =54+56+58/3 =56

Thus we get 2 pts 54 and 56. By joining these 2pts we shall obtain the required trend line.This can be used for Prediction.

COMPUTATION OF TREND VALUES:- The trend values (Yc) can be computed from the annual change.

Annual change = Diff. in Semi-Av values/Diff. in 2yrs to which S.A.V. belongs

= 56-54/2005-2001 = 2/4 =0.5

On this basis we can compute the trend line of Sales.The trend values are

COMPUTATION OF TREND VALUES:- The trend values (Yc) can be computed from the annual change.

Annual change = Diff. in Semi-Av values/Diff. in 2yrs to which S.A.V. belongs

= 56-54/2005-2001 = 2/4 =0.5

On this basis we can compute the trend line of Sales.The trend values are

Yr 2000 2001 2002 2003 2004 2005 2006

Sales

(000)

53.5 54 54.5 55 55.5 56 56.5

MERITS:-

• EASY TO UNDERSTAND AND TO APPLY.

• LINE CAN BE EXTENDED TO OBTAIN FUTURE ESTIMATES.

DEMERITS:-

• ASSUMES STRAIGHT LINE RELATIONSHIP WHICH MAY NO EXIST IN REAL WORLD.

MERITS:-

• EASY TO UNDERSTAND AND TO APPLY.

• LINE CAN BE EXTENDED TO OBTAIN FUTURE ESTIMATES.

DEMERITS:-

• ASSUMES STRAIGHT LINE RELATIONSHIP WHICH MAY NO EXIST IN REAL WORLD.

• USE OF ARITHMETIC MEAN CAN ALSO BE QUESTIONED DUE TO ITS LIMITATIONS.

• USE OF ARITHMETIC MEAN CAN ALSO BE QUESTIONED DUE TO ITS LIMITATIONS.

METHOD OF MOVING AVERAGES

Here number of items taken for averaging will be the number required to cover period over which the fluctuations occur.

This average is taken as the Normal/Trend value for the unit of time falling at the middle of period covered in calculation of average.The series may be given in odd or even number of years.

METHOD OF MOVING AVERAGES

Here number of items taken for averaging will be the number required to cover period over which the fluctuations occur.

This average is taken as the Normal/Trend value for the unit of time falling at the middle of period covered in calculation of average.The series may be given in odd or even number of years.

ODD PERIOD:

3 YEARS:- (a+b+c)/3,

(b+c+d)/3,

(c+d+e)/3,………………

5 YEARS:- (a+b+c+d+e)/5,

(b+c+d+e+f)/5,

(c+d+e+f+g)/5,………………

ODD PERIOD:

3 YEARS:- (a+b+c)/3,

(b+c+d)/3,

(c+d+e)/3,………………

5 YEARS:- (a+b+c+d+e)/5,

(b+c+d+e+f)/5,

(c+d+e+f+g)/5,………………

Suppose we are given a time series for 12 years-1989 to 2000 relating to sales of a certain business firm.these data are given below.Find out three year moving averages,starting from 1989:-

Suppose we are given a time series for 12 years-1989 to 2000 relating to sales of a certain business firm.these data are given below.Find out three year moving averages,starting from 1989:-

Year Sales

(million Rs)

Year Sales

(million Rs)

1989 10 1995 15

1990 15 1996 24

1991 20 1997 15

1992 25 1998 21

1993 15 1999 15

1994 12 2000 24

Year Sales

(million Rs)

3 year moving total

3 year moving average

1989 10

1990 15 45 15

1991 20 60 20

1992 25 60 20

1993 15 52 17

1994 12 42 14

Year Sales

(million Rs)

3 year moving total

3 year moving average

1995 15 51 17

1996 24 54 18

1997 15 60 20

1998 21 54 18

1999 15 63 21

2000 24

THE MOVING AVERAGE IS THEN PLOTTED ON GRAPH.

THE MOVING AVERAGE IS THEN PLOTTED ON GRAPH.

EVEN PERIOD:-

IF THE MOVING AVERAGE IS AN EVEN PERIOD SAY 2,4,6 YEARLY,THE MOVING TOTAL AND MOVING AVERAGE ARE PLACED AT THE CENTRE OF THE TIME SPAN FROM WHICH THEY ARE CALCULATED FALL BETWEEN TWO TIME PERIODS.

EVEN PERIOD:-

IF THE MOVING AVERAGE IS AN EVEN PERIOD SAY 2,4,6 YEARLY,THE MOVING TOTAL AND MOVING AVERAGE ARE PLACED AT THE CENTRE OF THE TIME SPAN FROM WHICH THEY ARE CALCULATED FALL BETWEEN TWO TIME PERIODS.

MERITS:-1. THIS METHOD IS QUITE SIMPLE.2. THERE IS NO ELEMENT OF SUBJECTIVITY

LIKE WE HAVE IN THE FREE HAND CURVE METHOD.

3. IT’S QUITE FLEXIBLE.IT IMPLIES IF WE ADD SOME VALUES,WE WILL GET SOME MOVE TREND VALUES.THAT MEANS A FEW MORE OBSERVATIONS MAY BE ADDED TO GIVEN DATA WITHOUT AFFECTING THE TREND VALUES ALREADY CALCULATED.

4. THIS METHOD IS VERY EFFECTIVE IF THE TREND OF SERIES IS VERY IRREGULAR.

MERITS:-1. THIS METHOD IS QUITE SIMPLE.2. THERE IS NO ELEMENT OF SUBJECTIVITY

LIKE WE HAVE IN THE FREE HAND CURVE METHOD.

3. IT’S QUITE FLEXIBLE.IT IMPLIES IF WE ADD SOME VALUES,WE WILL GET SOME MOVE TREND VALUES.THAT MEANS A FEW MORE OBSERVATIONS MAY BE ADDED TO GIVEN DATA WITHOUT AFFECTING THE TREND VALUES ALREADY CALCULATED.

4. THIS METHOD IS VERY EFFECTIVE IF THE TREND OF SERIES IS VERY IRREGULAR.

DEMERITS:-

1. WE CAN’T COMPUTE TREND VALUES FOR ALL THE YEARS.

2. WE REQUIRE A CAUTION IN SELECTING THE PERIOD OF MOVING AVERAGE.THERE ARE NO HARD AND FAST RULES AS FAR AS THE CHOICE OF PERIOD IS CONCERNED.IT DEPENDS EXCLUSIVELY ON THE PERSONAL JUDGEMENT.

DEMERITS:-

1. WE CAN’T COMPUTE TREND VALUES FOR ALL THE YEARS.

2. WE REQUIRE A CAUTION IN SELECTING THE PERIOD OF MOVING AVERAGE.THERE ARE NO HARD AND FAST RULES AS FAR AS THE CHOICE OF PERIOD IS CONCERNED.IT DEPENDS EXCLUSIVELY ON THE PERSONAL JUDGEMENT.

LEAST SQUARE METHODFOR THIS 2 CONDITIONS ARE

SATISFIED i.e

1. ∑(Y-Yc)=0 i.e the sum of deviations of the actual values of Y and the computed values of Y is zero.

2. ∑(Y-Yc)2=minimum i.e sum of the squares of deviations of the actual values of Y and the computed values is minimum from this line.

Equation :-Yc=a+bX

3 things to consider:-

1. The year selected as origin.

2. Unit of time represented by X i.e one,two or five years.

3. Unit in which Y is measured i.e in rupees,metres,tonnes etc.

Equation :-Yc=a+bX

3 things to consider:-

1. The year selected as origin.

2. Unit of time represented by X i.e one,two or five years.

3. Unit in which Y is measured i.e in rupees,metres,tonnes etc.

HOW TO DETERMINE ‘a’ AND ‘b’.

Normal equation:-

∑Y=Na+b∑X

∑XY=a∑X+b∑X2

If ∑X=0

(calculation becomes simple when midpoint in time is taken as origin because -ve values in first half of series will balance out +ve values of second half so ∑X=0).

HOW TO DETERMINE ‘a’ AND ‘b’.

Normal equation:-

∑Y=Na+b∑X

∑XY=a∑X+b∑X2

If ∑X=0

(calculation becomes simple when midpoint in time is taken as origin because -ve values in first half of series will balance out +ve values of second half so ∑X=0).

∑X=0, a=∑Y/N and b=∑XY/∑X2.

ODD NUMBER OF YEARS:-WHEN DEVIATIONS ARE TAKEN FROM MIDDLE YEAR, ∑X WOULD ALWAYS BE ZERO.

EVEN NUMBER OF YEARS:- ∑X WOULD BE ZERO IF X ORIGIN IS KEPT MID-WAY BETWEEN TWO MIDDLE YEARS.

∑X=0, a=∑Y/N and b=∑XY/∑X2.

ODD NUMBER OF YEARS:-WHEN DEVIATIONS ARE TAKEN FROM MIDDLE YEAR, ∑X WOULD ALWAYS BE ZERO.

EVEN NUMBER OF YEARS:- ∑X WOULD BE ZERO IF X ORIGIN IS KEPT MID-WAY BETWEEN TWO MIDDLE YEARS.

PROB:-FIT A STRAIGHT LINE TAKING X AS INDEPENDENT VARIABLE.

PROB:-FIT A STRAIGHT LINE TAKING X AS INDEPENDENT VARIABLE.

X Y

2002 1

2003 1.8

2004 3.3

2005 4.5

2006 6.3

2007 10

SOLN:-SOLN:-

X Y XY X2 Yc

2002 1 0 0 0.23

2003 1.8 1.8 1 1.93

2004 3.3 6.6 4 3.63

2005 4.5 13.5 9 5.33

2006 6.3 25.2 16 7.03

2007 10 50 25 8.73

∑=97.1 ∑=55

26.9=6a+15b------------(i)

97.1=15a+55b------------(ii)

Multiply (i) by 5 and eqn (ii) by 2

134.5=30a+75b

194.2=30a+110b

b=1.7

a=0.233

Yc=0.23+1.7X

26.9=6a+15b------------(i)

97.1=15a+55b------------(ii)

Multiply (i) by 5 and eqn (ii) by 2

134.5=30a+75b

194.2=30a+110b

b=1.7

a=0.233

Yc=0.23+1.7X

SEASONAL VARIATIONS

SEASONAL

SIMPLE AVERAGES

RATIO TO TREND METHOD

RATIO TO M.A METHOD

LINK RELATIVEMETHOD

METHOD OF SIMPLE AVERAGES:-EASIEST METHOD OF CALCULATING SEASONAL

INDEX.STEPS:-1. ARRANGE THE DATA BY YEARS,MONTHS OR

QUARTER AS THE CASE MAY BE.2. FIND TOTAL OF EACH MONTH OR QUARTER FOR

ALL THE YEARS.3. FIND AVERAGE FOR EACH MONTH OR QUARTER

FOR ALL THE YEARS.4. FIND OVERALL AVERAGE OF THESES AVERAGES.5. SEASONAL INDEX IS EXPRESSED AS EACH

MONTHLY OR QUARTERLY AVERAGE AS A PERCENTAGE OF OVERALL AVERAGE.

METHOD OF SIMPLE AVERAGES:-EASIEST METHOD OF CALCULATING SEASONAL

INDEX.STEPS:-1. ARRANGE THE DATA BY YEARS,MONTHS OR

QUARTER AS THE CASE MAY BE.2. FIND TOTAL OF EACH MONTH OR QUARTER FOR

ALL THE YEARS.3. FIND AVERAGE FOR EACH MONTH OR QUARTER

FOR ALL THE YEARS.4. FIND OVERALL AVERAGE OF THESES AVERAGES.5. SEASONAL INDEX IS EXPRESSED AS EACH

MONTHLY OR QUARTERLY AVERAGE AS A PERCENTAGE OF OVERALL AVERAGE.

EXAMPLE:-ASSUMING TREND IS ABSENT,DETERMINE IF THERE IS ANY SEASONALITY IN DATA GIVEN BELOW

EXAMPLE:-ASSUMING TREND IS ABSENT,DETERMINE IF THERE IS ANY SEASONALITY IN DATA GIVEN BELOW

YEAR 1st

QUARTER

2nd

QUARTER

3rd

QUARTER

4th

QUARTER

03 72 68 80 70

04 76 70 82 74

05 74 66 84 80

06 76 74 84 78

07 78 74 86 82

03 72 68 80 70

04 76 70 82 74

05 74 66 84 80

06 76 74 84 78

07 78 74 86 82

TOTAL

376 352 416 384

AVERAGE

75.2 70.4 83.2 76.8

SEASONAL

98.4 92.1 108.9 100.5

AVERAGE QUARTERLY AVERAGE=

(75.2+70.4+83.2+76.8)/4=76.4

SEASONAL INDEX FOR Ist QUARTER=75.2*100/76.4=98.4

SEASONAL INDEX FOR 2nd QUARTER=

70.4*100/76.4=92.1

SEASONAL INDEX FOR 3rd QUARTER=83.2*100/76.4=108.9

SEASONAL INDEX FOR 4th QUARTER=76.8*100/76.4=100.5

AVERAGE QUARTERLY AVERAGE=

(75.2+70.4+83.2+76.8)/4=76.4

SEASONAL INDEX FOR Ist QUARTER=75.2*100/76.4=98.4

SEASONAL INDEX FOR 2nd QUARTER=

70.4*100/76.4=92.1

SEASONAL INDEX FOR 3rd QUARTER=83.2*100/76.4=108.9

SEASONAL INDEX FOR 4th QUARTER=76.8*100/76.4=100.5

RATIO TO TREND METHOD:-IMPROVEMENT OVER SIMPLE AVERAGE METHOD.BASED ON

ASSUMPTION THAT SEASONAL FLUCTUATIONS FOR ANY SEASON ARE A CONSTANT FACTOR OF THE TREND.

STEPS:-1. COMPUTE THE TREND VALUES BY APPLYING THE METHOD

OF LEAST SQUARES.2. DIVIDE THE ORIGINAL DATA BY TREND VALUES AND

MULTIPLY THESE RATIOS BY 100.THESE VALUES ARE FREE FROM TREND BUT CONTAIN SEASONAL,CYCLICAL AND IRREGULAR COMPONENTS OF TIME SERIES.

3. TO REMOVE EFFECT OF CYCLICAL AND IRREGULAR COMPONENTS THE PROCESS OF AVERAGING THE PERCENTAGES FOR EACH QUARTER IS ADOPTED SO THAT THE SEASONAL VARIATIONS ARE LEFT.EITHER MEAN OR MEDIAN CAN BE USED FOR THIS PURPOSE.

4. THIS SEASONAL INDICES OBTAINED IN STEP 3 ARE ADJUSTED TO TOTAL OF 400 FOR QUARTERLY DATA AND 1200 FOR MONTHLY DATA BY MULTIPLYING EACH INDEX BY A SUITABLE FACTOR IN ORDER TO GET FINAL SEASONAL INDICES.

RATIO TO TREND METHOD:-IMPROVEMENT OVER SIMPLE AVERAGE METHOD.BASED ON

ASSUMPTION THAT SEASONAL FLUCTUATIONS FOR ANY SEASON ARE A CONSTANT FACTOR OF THE TREND.

STEPS:-1. COMPUTE THE TREND VALUES BY APPLYING THE METHOD

OF LEAST SQUARES.2. DIVIDE THE ORIGINAL DATA BY TREND VALUES AND

MULTIPLY THESE RATIOS BY 100.THESE VALUES ARE FREE FROM TREND BUT CONTAIN SEASONAL,CYCLICAL AND IRREGULAR COMPONENTS OF TIME SERIES.

3. TO REMOVE EFFECT OF CYCLICAL AND IRREGULAR COMPONENTS THE PROCESS OF AVERAGING THE PERCENTAGES FOR EACH QUARTER IS ADOPTED SO THAT THE SEASONAL VARIATIONS ARE LEFT.EITHER MEAN OR MEDIAN CAN BE USED FOR THIS PURPOSE.

4. THIS SEASONAL INDICES OBTAINED IN STEP 3 ARE ADJUSTED TO TOTAL OF 400 FOR QUARTERLY DATA AND 1200 FOR MONTHLY DATA BY MULTIPLYING EACH INDEX BY A SUITABLE FACTOR IN ORDER TO GET FINAL SEASONAL INDICES.

YEAR I II III IV

03 60 80 72 68

04 68 104 100 88

05 80 116 108 96

06 108 152 136 124

07 160 184 172 164

YEAR YEARLY

TOTAL

QUA T.AVEG

DEVIATIFROM

MID YR

XY

03 280 70 -2 -140

04 360 90 -1 -90

05 400 100 0 0

06 520 130 1 130

07 680 170 2 280

N=5 ∑=560 ∑=240

X2 TREND VALUES(Yc)

4 64

1 88

0 112

1 136

4 160

∑X2=10

Y=a+bX

a= ∑Y/N=560/5=112

b= ∑XY/ ∑X2=240/10=24

Y=112+24X

Yearly increment in trend value is b=24.Thus Per quarter it’s 24/4=6

Y=a+bX

a= ∑Y/N=560/5=112

b= ∑XY/ ∑X2=240/10=24

Y=112+24X

Yearly increment in trend value is b=24.Thus Per quarter it’s 24/4=6

Calculation of Quarterly Trend Values:-

For 1999,the trend value for the mid year,i.e half of 2nd quarter and half of 3rd quarter is 64.quarterly increment=6.

So trend value of 2nd quarter=64-6/2=61.

3rd quarter=64+6/2=67.

1st quarter=64-6-6/2=55.

4th quarter=64+6+6/2=73.

Calculation of Quarterly Trend Values:-

For 1999,the trend value for the mid year,i.e half of 2nd quarter and half of 3rd quarter is 64.quarterly increment=6.

So trend value of 2nd quarter=64-6/2=61.

3rd quarter=64+6/2=67.

1st quarter=64-6-6/2=55.

4th quarter=64+6+6/2=73.

Year/quat

I II III IV

03 55 61 67 73

04 79 85 91 97

05 103 109 115 121

06 127 133 139 145

07 151 157 163 169

1st quarter of 2003,%ge=60*100/55=109.09

2nd quarter of 2003,%ge=80*100/61=131.15

1st quarter of 2003,%ge=60*100/55=109.09

2nd quarter of 2003,%ge=80*100/61=131.15

Year/quat

I II III IV TOTAL

03 109.09 131.15 107.46 93.15

04 86.08 122.35 109.89 90.72

05 77.67 106.42 93.91 79.34

06 85.04 114.29 97.84 85.52

07 105.96 117.20 105.52 97.04

Total 463 591.42 514.62 445.77

Average 92.77 118.28 102.92 89.1 403.12

Adjusted 92.05 117.36 102.12 84.47 400

Total of average=

92.77+118.28+102.92+89.15=403.12

400/403.12=0.992

Final indices are obtained.

Total of average=

92.77+118.28+102.92+89.15=403.12

400/403.12=0.992

Final indices are obtained.

RATIO TO MOVING AVERAGE METHODAlso known as the percentage of moving average

method. Most widely used method for studying seasonal variations.

STEPS:-1. Eliminate seasonality from the data by ironing

it out of the original data. Obtain centered 4 quarters(12 months) moving average values for the given series. Since the variations recur after 4 quarters for quarterly data, a 4 quarterly moving average will wipe out the seasonal variations provided they are of constant pattern and intensity. Thus the centered 4 quarter moving average approximates trend and cyclical components.

RATIO TO MOVING AVERAGE METHODAlso known as the percentage of moving average

method. Most widely used method for studying seasonal variations.

STEPS:-1. Eliminate seasonality from the data by ironing

it out of the original data. Obtain centered 4 quarters(12 months) moving average values for the given series. Since the variations recur after 4 quarters for quarterly data, a 4 quarterly moving average will wipe out the seasonal variations provided they are of constant pattern and intensity. Thus the centered 4 quarter moving average approximates trend and cyclical components.

2. Express the original data for each quarter as a percentage of the centred 4 quarter moving average corresponding to it.

3.Arrange these percentage acc. to years and quarters.

4.By averaging these percentages for each quarter, seasonal indices are obtained. For averaging mean or median may be used.

5.Sum of these indices should be 400(or 1200)for quarterly(monthly)data.if it is not so, then an adjustment is made to eliminate this discrepency.seasonal indices obtained in step 4 are adjusted to total of 400(or 1200) by multiplying each index by a suitable factor in order to get final seasonal indices.

2. Express the original data for each quarter as a percentage of the centred 4 quarter moving average corresponding to it.

3.Arrange these percentage acc. to years and quarters.

4.By averaging these percentages for each quarter, seasonal indices are obtained. For averaging mean or median may be used.

5.Sum of these indices should be 400(or 1200)for quarterly(monthly)data.if it is not so, then an adjustment is made to eliminate this discrepency.seasonal indices obtained in step 4 are adjusted to total of 400(or 1200) by multiplying each index by a suitable factor in order to get final seasonal indices.

PROBLEM:-calculate seasonal indices by the ratio to moving average method from the following data.

PROBLEM:-calculate seasonal indices by the ratio to moving average method from the following data.

YEAR I II III IV

2001 2 3 2 4

2002 5 7 6 8

2003 6 9 9 10

YEAR QUARTER

GIVEN

FIG

4 FIG

MOV

TOTAL

2 FIG

MOV

TOTAL

4 FIG

MOV

AV5/8

GIVEN FIG%

OF MOV

1 2 3 4 5 6

2001 I

II

III

IV

2

3

2

4

11

14

18

25

32

3(appro

4

2*100/3

=67

4*100/4

=100

2002 I

II

III

IV

5

7

6

8

22

26

27

40

48

53

56

5

6

6.5 app

7

100

117

92

114

YEAR QUARTER

GIVEN

FIG

4 FIG

MOV

TOTAL

2 FIG

MOV

TOTAL

4 FIG

MOV

AV5+8

GIVEN FIG%

OF MOV

1 2 3 4 5 6

2003 I

II

III

IV

6

9

9

10

29

31

33

-

60

64

-

-

7.5

8

-

80

113

Year I II III IV TOTAL

2001

2002

2003

-

100

80

-

117

113

67

92

-

100

114

-

-

-

-

TOTAL 180 230 159 214

AVERAGE

90 115 79.5 107 391.5

ADJUSTED

90*400/391.5=

91.5

115*400/391.5=117.5

79.5*400/391.5=81.2

107*400/391.5

=109.3

400

LINK RELATIVE METHOD:-ALSO CALLED PEARSON’S METHODSTEPS:-1. CONVERT THE ORIGINAL DATA INTO LINK

RELATIVES BY FORMULA:-LINK RELATIVE FOR ANY QUARTER=(CURRENT QUARTER VALUE)/PREVIOUS QUARTER VALUE*100.

2. AVERAGE THESE LINK RELATIVES FOR EACH QUARTER,THE AVERAGE BEING TAKEN ONE FOR THE GIVEN NO OF YEARS.

3. CONVERT THESE LINKS INTO CHAIN RELATIVES ON THE BASE OF THE FIRST SEASON BY FORMULA:-

LINK RELATIVE METHOD:-ALSO CALLED PEARSON’S METHODSTEPS:-1. CONVERT THE ORIGINAL DATA INTO LINK

RELATIVES BY FORMULA:-LINK RELATIVE FOR ANY QUARTER=(CURRENT QUARTER VALUE)/PREVIOUS QUARTER VALUE*100.

2. AVERAGE THESE LINK RELATIVES FOR EACH QUARTER,THE AVERAGE BEING TAKEN ONE FOR THE GIVEN NO OF YEARS.

3. CONVERT THESE LINKS INTO CHAIN RELATIVES ON THE BASE OF THE FIRST SEASON BY FORMULA:-

=LINK RELATIVE OF THAT QUARTER*CHAIN RELATIVE OF PREVIOUS QUARTER/100.

4.THE LAST CHAIN RELATIVE SHOULD ALSO BE 100.BUT DUE TO EFFECT OF LONG TERM CHANGES,THIS IS NOT USUALY SO.THEREFORE,IT IS NECESSARY TO ADJUST THE CHAIN RELATIVE FOR THE EFFECT OF THE TREND.IF THE LAST CHAIN RELATIVE IS GREATER THAN 100,THE CORRECTION FACTOR IS SUBTRACTED;IF IT IS LESS THAN 100,THE CORRECTION FACTOR IS TO BE ADDED.

=LINK RELATIVE OF THAT QUARTER*CHAIN RELATIVE OF PREVIOUS QUARTER/100.

4.THE LAST CHAIN RELATIVE SHOULD ALSO BE 100.BUT DUE TO EFFECT OF LONG TERM CHANGES,THIS IS NOT USUALY SO.THEREFORE,IT IS NECESSARY TO ADJUST THE CHAIN RELATIVE FOR THE EFFECT OF THE TREND.IF THE LAST CHAIN RELATIVE IS GREATER THAN 100,THE CORRECTION FACTOR IS SUBTRACTED;IF IT IS LESS THAN 100,THE CORRECTION FACTOR IS TO BE ADDED.

5.FINALLY EXPRESS THE CORRECT CHAIN RELATIVES AS PERCENTAGES OF THEIR AVERAGES.THE RESULT FIGURES ARE THE REQUIRED SEASONAL INDICES.

5.FINALLY EXPRESS THE CORRECT CHAIN RELATIVES AS PERCENTAGES OF THEIR AVERAGES.THE RESULT FIGURES ARE THE REQUIRED SEASONAL INDICES.

PROBLEM:-PROBLEM:-

YEARS I II III IV

1997

1998

1999

2000

2001

2002

2003

283

210

194

159

184

179

200

258

208

168

162

179

182

204

244

204

159

168

176

182

207

260

241

183

189

197

219

243

SOLN:-SOLN:-

YEARS I II III IV

1997

1998

1999

2000

2001

2002

2003

TOTAL

AVERAGE

-

80.76

80.49

86.88

97.35

90.86

91.32

527.66

87.94

91.17

99.05

86.59

101.88

97.28

102.79

102.00

680.76

97.25

94.57

98.07

94.64

103.7

98.32

100.00

101.47

690.77

98.68

106.56

118.14

115.09

112.50

111.93

120.33

117.39

801.94

114.56

CHAIN RELATIV

ES

100 97.25*100/100=97.

25

97.25*98.68/100=

95.97

95.97*114.56/100=

109.94

ADJUSTED

CHAIN RELATIV

ES

SEASONAL

INDICES

100

100*100/102.035=

98.10

97.25+

.83=

98.08

97.63*100/102.035

=

95.6

95.97+1.66=

97.63

97.63*100/102.035

=

95.60

109.94+

2.49=

112.43

112.43*100/102.03

5=

110.20

CHAIN RELATIVE OF 1 QUARTER ON BASIS OF IV QUARTER

=87.94*109.94/100=96.68DIFFERENCE BETWEEN CHAIN

RELATIVE OF 1 QUARTER :96.68-100= -3.32 -3.32/4= -.83 AVERAGE OF ADJUSTED CHAIN

RELATIVES=(109+98.08+97.63+112.43)/ 4

=102.035.

CHAIN RELATIVE OF 1 QUARTER ON BASIS OF IV QUARTER

=87.94*109.94/100=96.68DIFFERENCE BETWEEN CHAIN

RELATIVE OF 1 QUARTER :96.68-100= -3.32 -3.32/4= -.83 AVERAGE OF ADJUSTED CHAIN

RELATIVES=(109+98.08+97.63+112.43)/ 4

=102.035.

MERITS:-• HELPFUL IN SHORTTERM BUSINESS

PLANNING,ECONOMIC FORECASTING AND MANAGERIAL CONCEPT.

• GREAT USE TO A BUSINESS CONCERN IN SCHEDULING ITS SEASONAL FINANCING,LABOUR INTAKE,PERSONNEL REQUIREMENT,ADVERTISING PROGRAMMES AND PURCHASES.

MERITS:-• HELPFUL IN SHORTTERM BUSINESS

PLANNING,ECONOMIC FORECASTING AND MANAGERIAL CONCEPT.

• GREAT USE TO A BUSINESS CONCERN IN SCHEDULING ITS SEASONAL FINANCING,LABOUR INTAKE,PERSONNEL REQUIREMENT,ADVERTISING PROGRAMMES AND PURCHASES.

• HELPS TO UNDERSTAND CURRENT MONTHLY VARIATIONS BY COMPARING THEM WITH CORRESPONDING SEASONAL INDICES OF A BUSINESS CONCERN AND THUS THEY ARE USEFUL TO CONTROL THE OPERATIONS IN BUSINESS CONCERN.

LIMITATIONS:-EFFECT OF RANDOM FLUCTUATIONS ON A

SEASONAL INDICES CANNOT BE COMPLETELY RULED OUT.

• REPRESENT AN AVERAGE PATTERN FOR THE YEARS UNDER STUDY AND THUS CANNOT BE EXPECTED THAT THE PATTERN WILL BE ACCURATELY REPEATED IN A PARTICULAR YEAR.

• HELPS TO UNDERSTAND CURRENT MONTHLY VARIATIONS BY COMPARING THEM WITH CORRESPONDING SEASONAL INDICES OF A BUSINESS CONCERN AND THUS THEY ARE USEFUL TO CONTROL THE OPERATIONS IN BUSINESS CONCERN.

LIMITATIONS:-EFFECT OF RANDOM FLUCTUATIONS ON A

SEASONAL INDICES CANNOT BE COMPLETELY RULED OUT.

• REPRESENT AN AVERAGE PATTERN FOR THE YEARS UNDER STUDY AND THUS CANNOT BE EXPECTED THAT THE PATTERN WILL BE ACCURATELY REPEATED IN A PARTICULAR YEAR.

CYCLICAL VARIATIONSWE MAY CALCULATE THE SEASONAL

INDEX NUMBER• IF TIME COSISTS OF ONLY ANNUAL

DATA,THEN THE SEASONAL VARIATION IS NON EXISTANT i.e WE CONSIDER ONLY SECULAR,CYCLICAL AND IRREGULAR COMPONENTS.

• SECULAR TREND BY TREND LINE.SO,WE CAN ISOLATE TWO COMPONENTS.

CYCLICAL VARIATIONSWE MAY CALCULATE THE SEASONAL

INDEX NUMBER• IF TIME COSISTS OF ONLY ANNUAL

DATA,THEN THE SEASONAL VARIATION IS NON EXISTANT i.e WE CONSIDER ONLY SECULAR,CYCLICAL AND IRREGULAR COMPONENTS.

• SECULAR TREND BY TREND LINE.SO,WE CAN ISOLATE TWO COMPONENTS.

FIND THE CYCLICAL VARIATION BY DIVIDING THE ACTUAL VALUE(Y) BY CORRESPONDING VALUE Yc FOR EACH ITEM.RESULTANT IS MULTIPLIED BY 100.

PERCENT OF TREND,C=Y*100/Yc.

C=CYCLICAL VARIATION.

Y=ACTUAL VALUE.

Yc=ESTIMATED VALUE.

Yc=a+bX.

FIND THE CYCLICAL VARIATION BY DIVIDING THE ACTUAL VALUE(Y) BY CORRESPONDING VALUE Yc FOR EACH ITEM.RESULTANT IS MULTIPLIED BY 100.

PERCENT OF TREND,C=Y*100/Yc.

C=CYCLICAL VARIATION.

Y=ACTUAL VALUE.

Yc=ESTIMATED VALUE.

Yc=a+bX.

Prob:-Prob:-

Year(X)

95’ 96’ 97’ 98’ 99’ 00’

Y 15 14 18 20 17 24

Soln:-

Yc=18+1.6X(Applying the method of least square).

Soln:-

Yc=18+1.6X(Applying the method of least square).

X Y Yc Y*100/Yc

95’ 15 14 107.1

96’ 14 16 87.5

97’ 18 17 105.9

98’ 20 19 105.3

99’ 17 20 85

00’ 24 22 109.1

Values of Y and Yc are then plotted on graph.

Another method of measuring cyclical variation.

Here,related cyclical residual=

(Y-Yc)*100/Yc.

Values of Y and Yc are then plotted on graph.

Another method of measuring cyclical variation.

Here,related cyclical residual=

(Y-Yc)*100/Yc.

(Y-Yc)*100/Yc

7.1

-12.5

5.9

5.3

-15

9.1

Cyclical data ca be used only for past data and not for forecasting cyclical variation in future.

Cyclical data ca be used only for past data and not for forecasting cyclical variation in future.

IRREGULAR VARIATIONIT MAY BE REITERATED THAT

VARIATIONS ARE NOT MERELY CYCLICAL BUT COMPRISE BOTH CYCLICAL AND IRREGULAR VARIATIONS.

I=CI/CI =IRREGULAR VARIATIONCI=IRRECULAR AND CYCLICAL

VARIATION.C=CYCLICAL VARIATION.

IRREGULAR VARIATIONIT MAY BE REITERATED THAT

VARIATIONS ARE NOT MERELY CYCLICAL BUT COMPRISE BOTH CYCLICAL AND IRREGULAR VARIATIONS.

I=CI/CI =IRREGULAR VARIATIONCI=IRRECULAR AND CYCLICAL

VARIATION.C=CYCLICAL VARIATION.

ALTHOUGH IT IS EXTREMELY DIFFICULT TO MEASURE IRREGULAR VARIATION,WE CAN IDENTIFY THE CAUSE FOR IRREGULAR VARIATION.E.g-STRIKES AND LOCKOUTS

ALTHOUGH IT IS EXTREMELY DIFFICULT TO MEASURE IRREGULAR VARIATION,WE CAN IDENTIFY THE CAUSE FOR IRREGULAR VARIATION.E.g-STRIKES AND LOCKOUTS