Time Series.

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

IRREGULAR VARIATIONS

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 TREND 1.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).

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,recove ry:During prosperityOptimisim, 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.

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,Epi demics 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

MULTIPLICATIVE RELATION SHIP O=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.

MEASUREMENT OF TREND :-

TREND LINE

GRAPHIC(FRE E HAND CURVE FITTING METHOD)

METHODS OF MOVING AVERAGES

METHOD OF SEMI 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.

POINTS TO CONSIDER:2. Curve should be smooth. 3. Number of Points above the Trend curve should be more or less equal to the number of points below it. 4. 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. 5. 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.

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.

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)

Year 1999 2000 2001 2002 2003 2004 2005 2006

X 0 1 2 3 4 5 6 7

Y 10 12 18 20 20 25 23 32

AVERAGE

Average=60/4=15

Average= 100/4=25

• 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.

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

2000 2001 2002 2003 2004 2005 2006

Sales 51 (000)

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.

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 53.5 54 (000)

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.

• 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.

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:-

Year 1989

Sales Year (million Rs) 10 1995

Sales (million Rs) 15

1990

15

1996

24

1991

20

1997

15

1992

25

1998

21

1993

15

1999

15

1994

12

2000

24

Year

3 year moving average

1989

Sales 3 year (million Rs) moving total 10

1990

15

45

15

1991

20

60

20

1992

25

60

20

1993

15

52

17

1994

12

42

14

Year

1995

Sales 3 year (million Rs) moving total 15 51

3 year moving average 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.

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:2. THIS METHOD IS QUITE SIMPLE. 3. THERE IS NO ELEMENT OF SUBJECTIVITY LIKE WE HAVE IN THE FREE HAND CURVE METHOD. 4. 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. 5. THIS METHOD IS VERY EFFECTIVE IF THE TREND OF SERIES IS VERY IRREGULAR.

DEMERITS:2. WE CAN’T COMPUTE TREND VALUES FOR ALL THE YEARS. 3. 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 METHOD FOR THIS 2 CONDITIONS ARE SATISFIED i.e 2. ∑(Y-Yc)=0 i.e the sum of deviations of the actual values of Y and the computed values of Y is zero. 3. ∑(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:3. The year selected as origin. 4. Unit of time represented by X i.e one,two or five years. 5. 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).

∑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 MIDWAY BETWEEN TWO MIDDLE YEARS.

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

X 2002 2003 2004

Y 1 1.8 3.3

2005 2006 2007

4.5 6.3 10

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

SEASONAL VARIATIONS SEASONAL

SIMPLE AVERAGES

RATIO TO TREND METHOD

RATIO TO M.A METHOD

LINK RELATIVE METHOD

METHOD OF SIMPLE AVERAGES:EASIEST METHOD OF CALCULATING SEASONAL INDEX. STEPS:4. ARRANGE THE DATA BY YEARS,MONTHS OR QUARTER AS THE CASE MAY BE. 5. FIND TOTAL OF EACH MONTH OR QUARTER FOR ALL THE YEARS. 6. FIND AVERAGE FOR EACH MONTH OR QUARTER FOR ALL THE YEARS. 7. FIND OVERALL AVERAGE OF THESES AVERAGES. 8. 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 YEAR

03

1st 2nd 3rd 4th QUARTE QUARTE QUARTE QUARTE R R R R 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

TOTA L AVER AGE SEAS ONAL

376

352

416

384

75.2

70.4

83.2

76.8

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

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:4. COMPUTE THE TREND VALUES BY APPLYING THE METHOD OF LEAST SQUARES. 5. 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. 6. 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. 7. 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 70

DEVIATI FROM MID YR -2

03

280

04

-140

360

90

-1

-90

05

400

100

0

0

06

520

130

1

130

07

680

170

2

280

N=5

∑=560

XY

∑=240

X2 4

TREND VALUES(Yc) 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

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 03

I

II

III

IV

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

Year/qu I II at 03 109.09 131.15

III

IV

04 05

86.08 77.67

122.35 106.42

109.89 90.72 93.91 79.34

06 07 Total Average Adjusted

85.04 105.96 463 92.77 92.05

114.29 117.20 591.42 118.28 117.36

97.84 105.52 514.62 102.92 102.12

TOTA L

107.46 93.15

85.52 97.04 445.77 89.1 84.47

403.12 400

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 METHOD Also known as the percentage of moving average method. Most widely used method for studying seasonal variations. STEPS:4. 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.

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

QUART GIVEN ER FIG

1

2

3

2001

I II III IV I II III IV

2 3 2 4 5 7 6 8

2002

4 FIG 2 FIG MOV MOV TOTAL TOTAL 4 11 14 18 22 26 27

5

25 32 40 48 53 56

4 FIG MOV AV5/8

GIVEN FIG% OF MOV

6 2*100/3 =67 3(appro 4*100/4 4 =100 5 100 6 117 6.5 app 92 7 114

YEAR

QUAR GIVEN TER FIG

1

2

3

2003

I II III IV

6 9 9 10

4 FIG 2 FIG 4 FIG GIVEN MOV MOV MOV FIG% OF TOTAL TOTAL AV5+8 MOV 4 5 6 29 31 33 -

60 64 -

7.5 8 -

80 113

Year

I

II

III

IV

TOTAL

2001 2002 2003 TOTAL

100 80 180

117 113 230

67 92 159

100 114 214

-

115

79.5

107

391.5

AVERA 90 GE

ADJUS 90*400 115*40 79.5*4 107*40 400 TED /391.5= 0/391.5 00/391. 0/391.5 =117.5 5=81.2 =109.3 91.5

LINK RELATIVE METHOD:ALSO CALLED PEARSON’S METHOD STEPS:4. CONVERT THE ORIGINAL DATA INTO LINK RELATIVES BY FORMULA:-LINK RELATIVE FOR ANY QUARTER=(CURRENT QUARTER VALUE)/PREVIOUS QUARTER VALUE*100. 5. AVERAGE THESE LINK RELATIVES FOR EACH QUARTER,THE AVERAGE BEING TAKEN ONE FOR THE GIVEN NO OF YEARS. 6. 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.

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

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: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

ADJUST ED CHAIN RELATIV ES

100

97.25*10 97.25*98. 95.97*11 0/100=97. 68/100= 4.56/100= 25 95.97 109.94 97.25+ .83= 98.08

95.97+1.6 109.94+ 6= 2.49= 97.63 112.43

97.63*10 112.43*1 100*100/ 97.63*10 SEASON 0/102.035 00/102.03 102.035= 0/102.035 AL = = 5= 98.10 INDICES 95.60 95.6 110.20

CHAIN RELATIVE OF 1 QUARTER ON BASIS OF IV QUARTER =87.94*109.94/100=96.68 DIFFERENCE 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.

• 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 VARIATIONS WE 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.

Prob:-

Year( 95’ X)

96’

97’

98’

99’

00’

Y

14

18

20

17

24

15

Soln:Yc=18+1.6X(Applying the method of least square). X 95’ 96’ 97’ 98’ 99’ 00’

Y 15 14 18 20 17 24

Yc 14 16 17 19 20 22

Y*100/Yc 107.1 87.5 105.9 105.3 85 109.1

Values of Y and Yc are then plotted on graph. Another method of measuring cyclical variation. (YHere,related cyclical residual= 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.

IRREGULAR VARIATION IT MAY BE REITERATED THAT VARIATIONS ARE NOT MERELY CYCLICAL BUT COMPRISE BOTH CYCLICAL AND IRREGULAR VARIATIONS. I=CI/C I =IRREGULAR VARIATION CI=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