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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
7/24/2019 Statistics for Business and Economics: bab 18
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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
?sing Smoothing Methods in Forecasting
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?sing Smoothing Methods in Forecasting
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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?sing Smoothing Methods in Forecasting
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 (=(
7.ample Sail,oat Sales+ nc
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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
7.ample Sail,oat Sales+ nc
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1A 1A
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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2( 2(
?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
7.ample 7astern $thletic Supplies
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28 28
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
(
7.ample 7astern $thletic Supplies
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2A 2A
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
7.ample 7astern $thletic Supplies
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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(
7.ample 7astern $thletic Supplies
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(2 (2
7.ample 7astern $thletic Supplies
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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0 0
&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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