©2003 Thomson/South-Western 1 Chapter 16 – Time Series Analysis and Index Numbers Slides prepared...

©2003 Thomson/South-Western 1

Chapter 16 –Chapter 16 –

Time Series Time Series Analysis and Analysis and Index NumbersIndex Numbers

Slides prepared by Jeff Heyl, Lincoln UniversitySlides prepared by Jeff Heyl, Lincoln University©2003 South-Western/Thomson Learning™

Introduction toIntroduction to Business StatisticsBusiness Statistics, 6e, 6eKvanli, Pavur, KeelingKvanli, Pavur, Keeling


Times Series AnalysisTimes Series Analysis

Time series represents a variable Time series represents a variable observed across timeobserved across time

Components of a time seriesComponents of a time series

Trend (TR)Trend (TR) Seasonal variation (S)Seasonal variation (S) Cyclical variation (C)Cyclical variation (C) Irregular activity (I)Irregular activity (I)


Trend (TR)Trend (TR)

Linear TrendLinear Trend

TR = TR = bb00 + + bb11tt

Quadratic TrendQuadratic Trend

TR = TR = bb00 + + bb11tt + + bb22tt22

Decaying TrendDecaying Trend

TR = TR = bb00 + + bb11 oror TR = TR = bb00 + + bb11ee-1-111

tt


Power ExamplePower Example

Figure 16.1Figure 16.1

300 300 –

200 200 –

100 100 –Po

wer

co

nsu

mp

tio

n (

mil

lio

n k

wh

)P

ow

er c

on

sum

pti

on

(m

illi

on

kw

h)

|19941994

|19951995

|19961996

|19971997

|19981998

|19991999

|20002000

|20012001

t t (time)(time)|19921992

|19931993


Employees ExampleEmployees Example


11.0 11.0 –

10.0 10.0 –

9.0 9.0 –

8.0 8.0 –

7.0 7.0 –

6.0 6.0 –

5.0 5.0 –

4.0 4.0 –

3.0 3.0 –

2.0 2.0 –

1.0 1.0 –

Nu

mb

er o

f em

plo

yees

(th

ou

san

ds)

Nu

mb

er o

f em

plo

yees

(th

ou

san

ds)

|19941994

|19951995

|19961996

|19971997

|19981998

|19991999

|20002000

|20012001

tt

TrendTrend


Linear TrendsLinear Trends

YYtt

tt

(a) Increasing trend(a) Increasing trend

YYtt

tt

(b) decreasing trend(b) decreasing trend



Curvilinear ModelsCurvilinear Models


YYtt

ttbb22 < 0 < 0

(a)(a)

YYtt

ttbb22< 0< 0

(b)(b)


Curvilinear ModelsCurvilinear Models


YYtt

ttbb22 > 0 > 0

(c)(c)

YYtt

ttbb2 2 > 0> 0

(d)(d)


Seasonality (S)Seasonality (S)

Seasonal variation refers to periodic Seasonal variation refers to periodic increases or decreases that occur within increases or decreases that occur within a calendar year in a time series. The key a calendar year in a time series. The key is that these movements in the time is that these movements in the time series follow the same pattern each yearseries follow the same pattern each year


Seasonal VariationSeasonal Variation

40 40 –

35 35 –

30 30 –

25 25 –

20 20 –

15 15 –

10 10 –

Po

wer

co

nsu

mp

tio

n (

mil

lio

ns

kw

h)

Po

wer

co

nsu

mp

tio

n (

mil

lio

ns

kw

h)

| | | | | | | | |Jan Jul DecJan Jul Dec

19991999

Jan Jul DecJan Jul Dec

20002000

Jan Jul DecJan Jul Dec

20012001



Seasonal VariationSeasonal Variation


4 4 –

3 3 –

2 2 –

1 1 –Sa

les

of

Wil

dc

at

sa

ilb

oa

tsS

ale

s o

f W

ild

ca

t s

ail

bo

ats

(mil

lio

ns

of

do

lla

rs)

(mil

lio

ns

of

do

lla

rs)

|JulyJuly19981998

|JulyJuly19991999

|JulyJuly20002000

|JulyJuly20012001

Linear trendLinear trend

tt


Cyclical Variation (C)Cyclical Variation (C)

Cyclical variation describes a gradual Cyclical variation describes a gradual cyclical movement about the trend; it is cyclical movement about the trend; it is generally attributable to business and generally attributable to business and economic conditionseconomic conditions

The length of the cycle is the period of The length of the cycle is the period of that cycle and is measured from one that cycle and is measured from one peak to the nextpeak to the next


Cyclical VariationCyclical VariationC

yclic

al a

ctiv

ity

Cyc

lical

act

ivit

y

ZZ11

PP11

VV11

ZZ22PP22

VV22

tt



Textile ExampleTextile Example

4.0 4.0 –

3.5 3.5 –

3.0 3.0 –

2.5 2.5 –

2.0 2.0 –

1.5 1.5 –

1.0 1.0 –

Co

rpo

rate

tax

es

Co

rpo

rate

tax

es

(mil

lio

ns

of

do

llar

s)(m

illi

on

s o

f d

oll

ars)

11 22 33

|19751975

|19851985

|19951995

|20002000



Irregular ActivityIrregular Activity

Irregular activity consists of what is Irregular activity consists of what is “left over” after accounting for the “left over” after accounting for the effect of any trend, seasonality, or effect of any trend, seasonality, or cyclical activitycyclical activity


Combining ComponentsCombining Components

Additive StructureAdditive Structure

yytt = TR = TRtt + + SStt + + CCtt + + IItt

Multiplicative StructureMultiplicative Structure

yytt = TR = TRtt • • SStt • • CCtt • • IItt


Measuring TrendMeasuring TrendLinear TrendLinear Trend

∑∑tt = 1 + 2 + ... + = 1 + 2 + ... + TT = = TT((TT + 1) + 1)

22

∑∑tt22 = 1 + 4 + ... + = 1 + 4 + ... + TT22 = =TT((TT + 1)(2 + 1)(2TT + 1) + 1)

66

tt = = = =∑∑tt

TTTT + 1 + 1

22

bb11 = =1212BB - 6( - 6(TT + 1) + 1)AA

TT((TT22 - 1) - 1)bb00 = - = - bb11

AA

TTTT + 1 + 1

22


Trend Line using Coded DataTrend Line using Coded Data

12 12 –––

9 9 –––

6 6 –––

3 3 –––– | | | | | | | |

1994

1994

(( tt =

1)

= 1

)

1995

1995

(( tt =

2)

= 2

)

YearYear

2001

2001

(( tt =

8)

= 8

)

Nu

mb

er o

f em

plo

yees

Nu

mb

er o

f em

plo

yees

(th

ou

san

ds)

(th

ou

san

ds)

YYtt

tt



Trend Line using Coded DataTrend Line using Coded Data

12 12 –––

9 9 –––

6 6 –––

3 3 –––– | | | | | | | |

1994

1994

(( tt =

1)

= 1

)

1995

1995

(( tt =

2)

= 2

)

YearYear

2001

2001

(( tt =

8)

= 8

)

Nu

mb

er o

f em

plo

yees

Nu

mb

er o

f em

plo

yees

(th

ou

san

ds)

(th

ou

san

ds)

YYtt

tt


yt = TRt = b0 + b1t


Excel SolutionExcel Solution



Quadratic TrendQuadratic Trend



Illustration of Quadratic Illustration of Quadratic Trend LinesTrend Lines

YYtt

Time (Time (tt))

tt = - = -bb11

22bb22

YYtt

Time (Time (tt))

tt = - = -bb11

22bb22

AA BB



Measuring Cyclical ActivityMeasuring Cyclical Activity

yytt = TR = TRtt • • CCtt • • IItt

CCtt yytt

yytt

^̂


Complete CycleComplete Cycle

YYtt

1 complete cycle1 complete cycle

TrendTrend

CCtt > 1 > 1

CCtt < 1 < 1

TimeTime



Trend and Cyclical ActivityTrend and Cyclical Activity

tt yyt t yyt t CCtt

11 1.11.1 1.1251.125 .978.97822 2.42.4 2.5292.529 .949.94933 4.64.6 3.9333.933 1.1691.16944 5.45.4 5.3375.337 1.0121.01255 5.95.9 6.7416.741 .875.87566 8.08.0 8.1458.145 .982.98277 9.79.7 9.5499.549 1.0161.01688 11.211.2 10.95310.953 1.0221.022

yytt

yytt

^̂^̂

Table 16.1Table 16.1


Cyclical ActivityCyclical Activity

11.0 11.0 –

10.0 10.0 –

9.0 9.0 –

8.0 8.0 –

7.0 7.0 –

6.0 6.0 –

5.0 5.0 –

4.0 4.0 –

3.0 3.0 –

2.0 2.0 –

1.0 1.0 –

Nu

mb

er o

f em

plo

yees

(th

ou

san

ds)

Nu

mb

er o

f em

plo

yees

(th

ou

san

ds)

|19941994

|19951995

|19961996

|19971997

|19981998

|19991999

|20002000

|20012001

tt

yytt = = -.279 + 1.404-.279 + 1.404tt

(trend line)(trend line)

Actual Actual yytt

YYtt



Cyclical ComponentsCyclical Components

1.15 1.15 –

1.10 1.10 –

1.05 1.05 –

1.00 1.00 –

.95 .95 –

.90 .90 –

StartStart EndEnd

CCtt

tt|11

|22

|33

|44

|55

|66

|77

|88

19941994 19961996 19981998 20002000



Additive Seasonal VariationAdditive Seasonal Variation

100 units100 units

100 units100 units

100 units100 units

TrendTrend

Actual time seriesActual time series

|

WinterWinter19991999

|


|


tt

YYtt

2000 2000 –

1500 1500 –

1000 1000 –

500 500 –

Un

its

sold

Un

its

sold



Jetski SalesJetski Sales

700 700 –

600 600 –

500 500 –

400 400 –

300 300 –

200 200 –

100 100 –Sal

es (

ten

s o

f th

ou

san

ds

of

do

llar

s)S

ales

(te

ns

of

tho

usa

nd

s o

f d

oll

ars)

YYtt

TRTRtt = 100 + 20 = 100 + 20tt

Estimated sales Estimated sales using trend and using trend and seasonalityseasonality

tt|

11|

22|

33|

44|

55|

66|

77|

88|

99|

1010|

1111|

1212|

1313|

1414|

1515|

1616|

1717|

1818|

1919|

2020



Heat Pump SalesHeat Pump Sales


100 units100 units

250 units250 units

180 units180 units

TrendTrend

Actual time seriesActual time series

|


|


|


tt

YYtt

2000 2000 –

1500 1500 –

1000 1000 –

500 500 –

Un

its

sold

Un

its

sold


Jetski Sales - Jetski Sales - Multiplicative Season VariationMultiplicative Season Variation


700 700 –

600 600 –

500 500 –

400 400 –

300 300 –

200 200 –

100 100 –Sal

es (

ten

s o

f th

ou

san

ds

of

do

llar

s)S

ales

(te

ns

of

tho

usa

nd

s o

f d

oll

ars)

YYtt

TRTRtt = 100 + 20 = 100 + 20tt

Estimated sales Estimated sales using trend and using trend and seasonalityseasonality

tt|

11|

22|

33|

44|

55|

66|

77|

88|

99|

1010|

1111|

1212|

1313|

1414|

1515|

1616|

1717|

1818|

1919|

2020


Four Step Procedure for Four Step Procedure for DecompositionDecomposition

1.1. Determine a seasonal index, SDetermine a seasonal index, Stt, for each , for each

time periodtime period

2.2. Deseasonalize the data, dDeseasonalize the data, dtt = = TRTRtt • C • Ctt • I • Itt

3.3. Determine the trend component, Determine the trend component, TRTRtt

4.4. Determine the cyclical component, CDetermine the cyclical component, Ctt


Centered Moving AveragesCentered Moving Averages

TimeTime QuarterQuarter tt yytt Moving TotalsMoving Totals

19901990 11 11 8585 (1) 263(1) 26322 22 4141 (2) 268(2) 26833 33 9292 (3) 270(3) 27044 44 4545 and so onand so on

19911991 11 55 909022 66 434333 77 959544 88 4747

19921992 11 99 9292.. .. .... .. .... .. ..



Sales Data for Video-CompSales Data for Video-Comp

YearYear Quarter 1Quarter 1 Quarter 2Quarter 2 Quarter 3Quarter 3 Quarter 4Quarter 4

19981998 2020 1212 4747 606019991999 4040 3232 6565 767620002000 5656 5050 8585 10010020012001 7575 7070 101101 123123



Moving Averages for Video-CompMoving Averages for Video-CompCenteredCentered Ratio toRatio to

MovingMoving MovingMoving MovingMovingYearYear QuarterQuarter tt yytt TotalTotal AverageAverage AverageAverage

19981998 11 11 2020 —— ——22 22 1212 —— —— ——33 33 4747 139139 37.2537.25 1.261.2644 44 6060 159159 42.2542.25 1.421.42

19991999 11 55 4040 179179 47.0047.00 .85.8522 66 3232 197197 51.2551.25 .62.6233 77 6565 213213 55.2555.25 1.181.1844 88 7676 229229 59.2059.20 1.281.28

20002000 11 99 5656 247247 64.2564.25 .87.8722 1010 5050 267267 69.7569.75 .72.7233 1111 8585 291291 75.1375.13 1.131.1344 1212 100100 310310 80.0080.00 1.251.25

20012001 11 1313 7575 330330 84.5084.50 .89.8922 1414 7070 346346 89.3889.38 .78.7833 1515 101101 369369 —— ——44 1616 123123 —— —— ——



Smoothing a Time SeriesSmoothing a Time Series

120 120 –

100 100 –

80 80 –

60 60 –

40 40 –

20 20 –Sal

es (

nu

mb

er o

f u

nit

s)S

ales

(n

um

ber

of

un

its)

Moving averages Moving averages (no seasonality)(no seasonality)

|

11|

22|

33|

4419981998

|

11|

22|

33|

4419991999

|

11|

22|

33|

4420002000

|

11|

22|

33|

4420012001

tt

YYtt

Quarters by yearQuarters by year



Ratios for Each QuarterRatios for Each Quarter

Quarter 1Quarter 1 Quarter 2Quarter 2 Quarter 3Quarter 3 Quarter 4Quarter 4

1.261.26 1.421.42.85.85 .62.62 1.181.18 1.281.28.87.87 .72.72 1.131.13 1.251.25.89.89 .78.78

TotalTotal 2.612.61 2.122.12 3.573.57 3.953.95AverageAverage 0.8700.870 0.7070.707 1.1901.190 1.3171.317

—— ——

—— ——



DeseasonalizedDeseasonalizedValuesValues

SeasonalSeasonalYearYear tt YYtt Index (Index (SStt)) ddtt = — = —

YYtt

SStt

19981998 11 2020 .852.852 23.4723.4722 1212 .692.692 17.3417.3433 4747 1.1661.166 40.3140.3144 6060 1.2901.290 46.5146.51

19991999 11 4040 .852.852 46.9546.9522 3232 .692.692 46.2446.2433 6565 1.1661.166 55.7555.7544 7676 1.2901.290 58.9158.91

20002000 11 5656 .852.852 65.7365.7322 5050 .692.692 72.2572.2533 8585 1.1661.166 72.9072.9044 100100 1.2901.290 77.5277.52

20012001 11 7575 .852.852 88.0388.0322 7070 .692.692 101.16101.1633 101101 1.1661.166 86.6286.6244 123123 1.2901.290 95.3595.35

Deseasonalizing Deseasonalizing DataData



Total U.S. Retail TradeTotal U.S. Retail Trade

19971997 19981998 19991999 20002000JanJan 134.738134.738 139.935139.935 146.613146.613 158.691158.691FebFeb 130.255130.255 135.538135.538 145.121145.121 164.725164.725MarMar 148.497148.497 151.118151.118 165.736165.736 183.875183.875AprApr 145.703145.703 155.820155.820 166.011166.011 178.776178.776MayMay 156.603156.603 162.797162.797 173.496173.496 190.753190.753JunJun 150.915150.915 159.701159.701 171.286171.286 187.868187.868JulJul 153.200153.200 161.541161.541 172.364172.364 182.891182.891AugAug 156.782156.782 162.369162.369 174.788174.788 191.647191.647SepSep 149.407149.407 155.747155.747 169.809169.809 183.229183.229OctOct 157.523157.523 164.528164.528 174.740174.740 186.550186.550NovNov 161.925161.925 169.914169.914 185.347185.347 198.706198.706DecDec 203.117203.117 215.590215.590 238.452238.452 243.255243.255



Summary of RatiosSummary of Ratios

Month (Period)Month (Period)

JanJan FebFeb MarMar AprApr MayMay JunJun JulJul AugAug SepSep OctOct NovNov DecDec

19971997 0.9930.993 1.0131.013 0.9640.964 1.0131.013 1.0361.036 1.2951.29519981998 0.8880.888 0.8570.857 0.9520.952 0.9790.979 1.0181.018 0.9940.994 1.0001.000 1.0011.001 0.9540.954 1.0021.002 1.0291.029 1.2991.29919991999 0.8780.878 0.8640.864 0.9810.981 0.9760.976 1.0141.014 0.9920.992 0.9900.990 0.9960.996 0.9590.959 0.9800.980 1.0321.032 1.3171.31720002000 0.8710.871 0.8990.899 0.9960.996 0.9630.963 1.0221.022 1.0031.003AverageAverage 0.8790.879 0.8370.837 0.9770.977 0.9730.973 1.0181.018 0.9960.996 0.9940.994 1.0041.004 0.9590.959 0.9980.998 1.0331.033 1.3041.304



Deseasonalized DataDeseasonalized Data

••••••••••••

•••••••••• ••••••••••••••••••

••

•••• ••••••••••

••••••

•••••••• ••

••••••••••

•• ••••••••••

200 200 –

190 190 –

180 180 –

170 170 –

160 160 –

150 150 –

140 140 –|

00|

1010|

2020|

3030|

4040|

5050

Time (Time (tt))

Des

easo

nal

ized

Val

ues

(D

esea

son

aliz

ed V

alu

es (

ddtt))



Cyclical ComponentsCyclical Components

11 153.349153.349 146.470 - .934(1) = 146.979146.470 - .934(1) = 146.979 1.04331.0433 ——22 149.238149.238 146.470 - .934(2) = 147.913146.470 - .934(2) = 147.913 1.00901.0090 1.0251.02533 152.165152.165 146.470 + .934(3) = 148.847146.470 + .934(3) = 148.847 1.02231.0223 1.0111.01144 149.871149.871 146.470 + .934(4) = 149.782146.470 + .934(4) = 149.782 1.00061.0006 1.0151.01555 153.899153.899 146.470 + .934(5) = 150.716146.470 + .934(5) = 150.716 1.02111.0211 1.0071.00766 151.602151.602 146.470 + .934(6) = 151.650146.470 + .934(6) = 151.650 0.99970.9997 1.0101.010.. .. .. .. .... .. .. .. .... .. .. .. ..

tt ddtt ddtt —(= —(= CCtt • • IItt))ddtt

ddtt

^̂^̂

3-Month3-MonthMovingMoving

Average (Average (CCtt))



Plot of Cyclical ActivityPlot of Cyclical Activity


1.03 1.03 –

1.02 1.02 –

1.01 1.01 –

1.00 1.00 –

0.99 0.99 –

0.98 0.98 –

0.97 0.97 –|

00|

1010|

2020|

3030|

4040|

5050

Cyc

lical

Co

mp

on

ents

Cyc

lical

Co

mp

on

ents

MonthMonth FebFeb JanJan JanJan JanJanYearYear 19971997 19981998 19991999 20002000


Excel PlotsExcel Plots



Index NumbersIndex Numbers

19851985 19901990 19951995 20002000

WageWage $7.05$7.05 $8.50$8.50 $10.90$10.90 $12.50$12.50

Index (base = 1980)Index (base = 1980) 100100 120.6120.6 154.6154.6 177.3177.3



Price IndexesPrice Indexes

Simple Aggregate Price Index = • 100Simple Aggregate Price Index = • 100∑∑PP11

∑∑PP00

Weighted Aggregate Price Index = • 100Weighted Aggregate Price Index = • 100∑∑PP11QQ

∑∑PP00QQ

Laspeyres Index = • 100Laspeyres Index = • 100∑∑PP11QQ00

∑∑PP00QQ00

∑∑PP11QQ11

∑∑PP00QQ11

Paasche Index = • 100Paasche Index = • 100


Prices of Four ItemsPrices of Four Items

ItemItem 19901990 20002000

EggsEggs .75 (doz).75 (doz) 1.351.35

ChickenChicken .95 (lb).95 (lb) 1.791.79

CheeseCheese .89 (lb).89 (lb) 1.851.85

Auto batteryAuto battery $31.00 (each)$31.00 (each) $55.00 (each)$55.00 (each)


LongLifeLongLife

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