statistics for business and economics: bab 18

7/24/2019 Statistics for Business and Economics: bab 18

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Chapter 18Forecasting

Time Series and Time Series MethodsComponents o a Time SeriesSmoothing Methods

Trend !ro"ection

Trend and Seasonal Components#egression $nal%sis&ualitati'e $pproaches to Forecasting


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Seasonal Component t represents an% repeating pattern+ less

than one %ear in duration+ in the time series The pattern duration can ,e as short as an

hour+ or e'en lessrregular Component

t is the 3catch-all4 actor that accounts orthe de'iation o the actual time series 'aluerom *hat *e *ould e.pect ,ased on theother components

t is caused ,% the short-term+unanticipated+ and nonrecurring actors thata5ect the time series

The Components o a Time Series


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Forecast $ccurac%

Mean S uared 7rror MS79 t is the a'erage o the sum o all the

s uared orecast errorsMean $,solute :e'iation M$:9

t is the a'erage o the a,solute 'alues o allthe orecast errors

;ne ma"or di5erence ,et*een MS7 and M$: isthat

the MS7 measure is in


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Mo'ing $'erages We use the a'erage o the most recent n

data 'alues in the time series as theorecast or the ne.t period

The a'erage changes+ or mo'es+ as ne*o,ser'ations ,ecome a'aila,le

The mo'ing a'erage calculation is

Mo'ing $'erage > most recent n data'alues9/ n

?sing Smoothing Methods in Forecasting


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Weighted Mo'ing $'erages This method in'ol'es selecting *eights or

each o the data 'alues and then computinga *eighted mean as the orecast

For e.ample+ a (-period *eighted mo'inga'erage *ould ,e computed as ollo*s

F t @ 1 > w 1 Y t - 2 9 @ w 2 Y t - 1 9 @ w ( Y t 9

*here the sum o the *eights w 'alues9is 1



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7.ponential Smoothing t is a special case o the *eighted mo'ing

a'erages method in *hich *e select onl%the *eight or the most recent o,ser'ation

The *eight placed on the most recento,ser'ation is the 'alue o the smoothingconstant+

The *eights or the other data 'alues arecomputed automaticall% and ,ecome

smaller at an e.ponential rate as theo,ser'ations ,ecome older


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7.ponential Smoothing

F t @ 1 > Y t @ 1 - 9F t

*here F t @ 1 > orecast 'alue or period t @

1 Y t > actual 'alue or period t @ 1

F t > orecast 'alue or period t

> smoothing constant 0 B B19


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7.ample 7.ecuti'e Seminars+ nc

7.ecuti'e Seminars specialiDes in conductingmanagement de'elopment seminars n order to

,etterplan uture re'enues and costs+ management

*ould liEeto de'elop a orecasting model or their 3TimeMana gement4 seminar

7nrollments or the past ten 3TM4 seminarsare

oldest9 ne*est9 Seminar 1 2 ( 6 = 8 A 10 Enroll. ( 0( (A 1(6( ( (8 ( 0


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7.ample 7.ecuti'e Seminars+ nc

7.ponential SmoothingLet > 2+ F 1 > Y 1 > (

F 2 > Y 1 @ 1 - 9F 1 > 2 ( 9 @ 8 ( 9 > ( F ( > Y 2 @ 1 - 9F 2 > 2 09 @ 8 ( 9 > ( 20 F > Y ( @ 1 - 9F ( > 2 ( 9 @ 8 ( 209 > ( 16

and so on


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7 uation or Linear Trend

T t > b 0 @ b 1 t

*here

T t > trend 'alue in period t b 0 > intercept o the trend line

b 1 > slope o the trend line

t > time ote t is the independent 'aria,le

?sing Trend !ro"ection in Forecasting


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Computing the Slope b 19 and ntercept b 09

b 1 > tY t - t Y t 9/n

t 2 - t 92 /n

b 0 > Y t /n 9 - b 1 t /n > Y - b 1 t

*hereY t > actual 'alue in period t n = num,er o periods in time series

?sing Trend !ro"ection in Forecasting


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7.ample Sail,oat Sales+ nc

Sail,oat Sales is a ma"or marine dealer inChicago The Grm has e.periencedtremendous sales gro*th in the past se'eral%ears Management *ould liEe to de'elop aorecasting method that *ould ena,le them to

,etter control in'entories The annual sales+ in num,er o ,oats+ orone particular sail,oat model or the past G'e%ears are

Year 1 2 ( Sales 11 1 20 26 (


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Linear Trend 7 uation

t Y t tY t t 2

1 11 11 1

2 1 2 8 ( 2 0 6 0 A 26 10 16 ( 1=0 2

Total 1 10 (=(



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Trend !ro"ection

b 1 > (=( - 1 9 10 9/ > 8

- 1 9 2 /

b 0 > 10 / - 8 1 / 9 > ( 6

T t > ( 6 @ 8 t

T 6 > ( 6 @ 8 69 > (8



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Trend and Seasonal Componentsin Forecasting

Multiplicati'e ModelCalculating the Seasonal nde.es:eseasonaliDing the Time Series?sing the :eseasonaliDing Time Series to denti % TrendSeasonal $d"ustmentsC%clical Component


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Multiplicati'e Model

?sing T t +S t + and It to identi % the trend+seasonal+ and irregular components at time t +*e descri,e the time series 'alue Y t ,% theollo*ing multiplicati'e time series model

Y t > T t x S t x It

T t is measured in units o the item ,eing

orecastS t and It are measured in relati'e terms+ *ith'alues a,o'e 1 00 indicating e5ects a,o'e thetrend and 'alues ,elo* 1 00 indicating e5ects

,elo* the trend


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:eseasonaliDing the Time Series

The purpose o Gnding seasonal inde.es is toremo'e the seasonal e5ects rom the timeseries

This process is called deseasonaliDing the timeseries

)% di'iding each time series o,ser'ation ,%the corresponding seasonal inde.+ the result isa deseasonaliDed time seriesWith deseasonaliDed data+ rele'ant

comparisons can ,e made ,et*eeno,ser'ations in successi'e periods


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?sing the :eseasonaliDing Time Seriesto denti % Trend

To identi % the linear trend+ *e use the linearregression procedure co'ered earlierH in thiscase+ the data are the deseasonaliDed timeseries 'aluesn other *ords+ Y t no* re ers to the

deseasonaliDed time series 'alue at time t andnot to the actual 'alue o the time series

The resulting line e uation is used to maEetrend pro"ections+ as it *as earlier


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Seasonal $d"ustments

The Gnal step in de'eloping the orecast is touse the seasonal inde. to ad"ust the trendpro"ection

The orecast or period t + season s + is o,tained,% multipl%ing the trend pro"ection or period t

,% the seasonal inde. or season s

Y t,s > Is Ib 0 @ b 1 t 9J


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7.ample 7astern $thletic Supplies

Management o 7$S *ould liEe to de'elopa

uarterl% sales orecast or one o theirtennis racEets

Sales o tennis racEets is highl% seasonal andhence an

accurate uarterl% orecast could aidsu,stantiall% in

ordering ra* material used in manu acturing The uarterl% sales data 000 units9 or thepre'ious

three %ears is sho*n on the ne.t slide


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2= 2=

Year QuarterSales 4-CMA 2-CMA

1 1 (2 A

00

( 62

1(

2=

0

2 16 2

6 00

2 11 6 0 6 (8

( 86 =

6 6(

( = = = 2



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Year Quarter Sales 2-CMA Seas-

Irreg1 1 (2 A( 6 1( 1 1=

2 0 0 (62 1 6 00 0 6=2 11 6 (8 1 =2( 8 6 6( 1 21

( = 2 0 1( 1 8 1( 0 62

2 1 8 0 1 =6( 11

(



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QuarterSeas-Irreg Values Seas. Index1 0 6=+ 0 62 0 62 1 =2+ 1 =6 1 =( 1 1=+ 1 21 1 1A

0 (6+ 0 1 0 (A

Total > ( A=

Seas.Index Ad . FactorAd .Seas.Index

0 6 /( A= 61 = /( A= 1 = (1 1A /( A= 1 1AA0 (A /( A= (A(

Total > 000



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(0 (0

Year QuarterSales Seas.Index

!eseas.Sales1 1 ( 6 82 A 1 = ( 1(( 6 1 1AA 00 2 (A(

0A2 1 6 6 11

2 11 1 = ( 6 2=( 8 1 1AA 6 6=

( (A( = 6(( 1 6 = 6(

2 1 1 = ( 8 6( 11 1 1AA A 1=

( (A( = 6(



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Seasonal $d"ustments

"eriod #rend Seasonal Quarterl$ t Forec. Index Forecast

1( A+1=A 6 6+012 1 A+ =2 1 = ( 16+=80 1 A+A66 1 1AA 11+A A16 10+( A (A( +0=1


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(( ((

Models )ased on Monthl% :ata

Man% ,usinesses use monthl% rather thanuarterl% orecasts

The preceding procedures can ,e applied *ithminor modiGcations

$ 12-month mo'ing a'erage replaces the -uarter mo'ing a'erage

12 monthl%+ rather than uarterl%+seasonal inde.es must ,e computed

;ther*ise+ the procedures are identical


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The multiplicati'e model can ,e e.panded toinclude a c%clical component that is e.pressedas a percentage o trend

Ko*e'er+ there are di culties in including ac%clical component

$ c%cle can span se'eral man%9 %ears andenough data must ,e o,tained to estimate

the c%clical component C%cles usuall% 'ar% in length

C%clical Component

t t t t t Y T C S I=


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#egression $nal%sis

;ne or more independent 'aria,les can ,eused to predict the 'alue o a single dependent'aria,le

The time series 'alue that *e *ant to orecastis the dependent 'aria,le

The independent 'aria,le s9 might include an%com,ination o the ollo*ing

!re'ious 'alues o the time series 'aria,leitsel

7conomic/demographic 'aria,les Time 'aria,les


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&ualitati'e $pproaches to Forecasting

:elphi Method t is an attempt to de'elop orecasts through

3group consensus 4 The goal is to produce a relati'el% narro*

spread o opinions *ithin *hich the ma"orit%

o the panel o e.perts concur7.pert Oudgment

7.perts indi'iduall% consider in ormationthat the% ,elie'e *ill in


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statistics for business and economics: bab 18

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