THE UNIVERSITY OF TEXAS AT AUSTIN McCombs School of Business

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1 THE UNIVERITY OF TEXA AT AUTIN McCombs chool of Business TA 7.5 Tom hively CLAICAL EAONAL DECOMPOITION - MULTIPLICATIVE MODEL Examples of easonaliy 8000 Quarerly sales for Wal-Mar for quarers a l e s Quarerly sales for The Gap for quarers

2 Quarerly daa for privae housing sars for 6 quarers Monhly U.. clohing sales for years Jan Jan Jan Jan Jan Jan Jan Jan Jan Jan Jan Jan

3 A ime series is ypically decomposed ino four componens: () easonal componen ( ); () Trend componen (T ); () Cyclical componen (C ); () Irregular componen (I ). The rend and cyclical componens are ofen difficul o separae excep in he long run. (ee he figures on he following page.) For his reason, he rend and cyclical componens are ofen combined ino one componen - he Trend/Cycle componen (TC ). Two ypes of models are used for seasonal decomposiion: () Addiive model: y = T C I or y = TC I () Muliplicaive model: y = T C I or y = TC I We will discuss muliplicaive models since hese are ypically he models ha are used in pracice. Muliplicaive models are used when here is increasing volailiy such as in he sales daa from The Gap (see page ). The reason for he increasing volailiy in ales for The Gap is ha ales grow abou 0% on average beween he hird and fourh quarer every year. A he beginning of he series, where ales are approximaely $00 million, a 0% change in ales from he hird o fourh quarer gives a change of (0.0)($00m) = $0m. A he end of he series, where ales are approximaely $,000 million (i.e. $ billion), a 0% change in ales gives a change of (0.0)($,000m) = $00m.

4 Copper prices (in consan 997 dollars) ploed for differen lengh periods Copper Prices - 8 Monhs Copper Prices - Years Jan, 956 Jan, 957 Jan, Copper Prices - Years Copper Prices - 98 Years

5 Calculaion of Cenered Moving Averages Excel preadshee for Compuing Cenered Moving Averages Row A B C D Y ThreeYearMovingAverage (y - y y )/ 75.0 = AVERAGE(A:A) = AVERAGE(A:A5) FiveYearMovingAverage (y - y - y y y )/ = AVERAGE(A:A6) = AVERAGE(A:A7) evenyearmovingaverage (y - y - y - y y y y )/ = AVERAGE(A:A8) 8.87 = AVERAGE(A:A9) Y and Three-Year Moving Average Y ThreeYearMovingAverage 5

6 Classical easonal Decomposiion for he Muliplicaive Model: y = TC I From here on, he Trend/Cycle componen will be denoed T (raher han TC ). Four eps in a Classical easonal Decomposiion Procedure for a Muliplicaive Model () Compue an iniial esimae of he Trend/Cycle componen T (he esimae is denoed Tˆ ) using he cenered moving average Tˆ = 0.5y - 0.5y - 0.5y 0.5y 0.5y. () Compue y ˆ. This is an esimae of he I componens. T () Compue an esimae of (denoed Ŝ ) by averaging he esimaes of he I componens across corresponding quarers o average ou he I componens. For example, for quarerly daa, he I erm is averaged across all he firs quarers, all he second quarers, ec. () easonally adjus he original daa using y ˆ. 6

7 Imporan Quesion Regarding ep : How good an esimae is Tˆ of T? To answer his quesion, firs consider annual (simulaed) daa wih no seasonaliy. Therefore, he model is y = T I. Row A B C D Year Trend T Irregular I Daa=Trend*Irregular y = T I

8 Trend Componen Irregular Componen Daa = Trend*Irregular

9 To ge an esimae of T we will compue a weighed moving average of he y s o average ou he I componens. In paricular, Tˆ = 0.5y - 0.5y - 0.5y 0.5y 0.5y. uppose for he momen here is no irregular componen (i.e. I = in each period ). As shown below, even wih no irregular componen I we canno recover T exacly, alhough we can come very close. Row A B C D E Year Trend T Irregular I Daa = Trend*Irregular y = T I = T () = T Trend Esimae Tˆ = 0.5y - 0.5y - 0.5y 0.5y 0.5y = 0.5*D 0.5*D 0.5*D 0.5*D5 0.5*D

10 Now consider he acual daa y = T I. As shown below, we canno recover T exacly because of he variabiliy in he I componen. Row A B C D E Year Trend T Irregular I Daa=Trend*Irregular y = T I Trend Esimae Tˆ = 0.5y - 0.5y - 0.5y 0.5y 0.5y 0.80 = 0.5*D 0.5*D 0.5*D 0.5*D5 0.5*D

11 Trend Daa=Trend*Irregular Trend There are wo sources of esimaion error when esimaing T : () Even wih no irregular componen he weighed moving average canno recover T exacly (his is ypically a very small source of error). The only ime T can be recovered exacly is when T = α β (i.e. he rend is linear). () The I componen canno be compleely averaged ou wih a shor weighed moving average. The size of he esimaion error due o I depends on how much variabiliy here is in I (i.e. i depends on wha he variance of I is).

12 Now consider esimaing T in a full seasonal model (using simulaed quarerly daa). The model is y = T I. Row A B C D E F Time Quarer Trend easonal Irregular Daa=Trend*easonal*Irregular T I y = T I Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr

13 Trend Componen easonal Componen Irregular Componen Daa = Trend*easonal*Irregular

14 To ge an esimae of he T componen we will compue a weighed moving average of y o average ou he and I componens: Tˆ = 0.5y - 0.5y - 0.5y 0.5y 0.5y. uppose for he momen here is no seasonal or irregular componen (i.e. = and I = in each period ). As shown below, even wih no irregular componen I we canno recover T exacly, alhough we can come very close. Row A B C D E F G Quarer Time Trend T easonal Irregular I Daa = Trend*easonal*Irregular y = T I Trend Esimae- No easonal Or Irregular Comp Tˆ = 0.5y - 0.5y Qr Qr y 0.5y 0.5y = 0.5*F 0.5*F 0.5*F 0.5*F5 0.5*F6 5 Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr

15 Now suppose here is a seasonal componen bu no irregular componen (i.e. I = in each period so y = T ). As shown below, we can obain an accurae (bu no perfec) esimae of T. Row A B C D E F G Quarer Time Trend T easonal Irregular I Daa = Trend*easonal*Irregular y = T I Trend Esimae- No Irregular Componen Tˆ = 0.5y - 0.5y Qr Qr y 0.5y 0.5y = 0.5*F 0.5*F 0.5*F 0.5*F5 0.5*F6 5 Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr

16 Now consider he full model y = T I. If he variabiliy in I is no large we can obain a fairly accurae esimae of T. Row A B C D E F G Quarer Time Trend T easonal Irregular I Daa = Trend*easonal*Irregular y = T I Qr Qr Trend Esimae Tˆ = 0.5y - 0.5y - 0.5y 0.5y 0.5y = 0.5*F 0.5*F 0.5*F 0.5*F5 0.5*F6 5 Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr

17 If he variabiliy in I is large, his makes i more difficul o obain an accurae esimae of T. Row A B C D E F G Quarer Time Trend T easonal Irregular I Daa = Trend*easonal*Irregular y = T I Qr Qr Trend Esimae Tˆ = 0.5y - 0.5y - 0.5y 0.5y 0.5y.86 = 0.5*F 0.5*F 0.5*F 0.5*F5 0.5*F6 5 Qr Qr Qr Qr Qr Qr Qr True Trend Esimae Large I Esimae mall I The full spreadshee o implemen classical decomposiion in a muliplicaive model is given on he nex page. 7

18 Row A B C D E F G H Quarer Daa y Qr 87.8 Qr Qr Qr T Esimae Tˆ = 0.5y - 0.5y - 0.5y 0.5y 0.5y = 0.5*B 0.5*(BBB5) 0.5*B6 08. = 0.5*B 0.5*(BB5B6) 0.5*B7.767 = 0.5*B 0.5*(B5B6B7) 0.5*B8 5.9 = 0.5*B5 0.5*(B6B7B8) 0.5*B9 xi Esimae y Tˆ.08 = B/C.9 = B5/C = B6/C6 0.8 = B7/C7 8 Qr Qr easonalum This column sums he I componens for corresponding quarers 7.59 = UMIF(A$:A$, "Qr ",D$:D$) 9. = UMIF(A$:A$, "Qr ",D$:D$) 5.55 = UMIF(A$:A$, "",D$:D$) = UMIF(A$:A$, "Qr ",D$:D$) easonalavg This column is an average of he I componens for corresponding quarers.05 = E/7.0 = E5/ = E6/ = E7/7 Esimae Ŝ This column forces he componens o average o one 0.79 Copied manually 0.85 Copied manually.05 = *F/UM(F$:F$7).0 = *F5/UM(F$:F$7) 0.79 = *F6/UM(F$:F$7) 0.85 = *F7/UM(F$:F$7).05 Copied manually.0 Copied manually easonaladjy y ˆ = B/G 0. = B/G 0.0 = B/G = B5/G5.07 = B6/G6.9 = B7/G7.60 = B8/G8.66 = B9/G Qr Qr Qr Qr Qr Qr

19 9 Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr Qr easonally Adjused Y

20 Excel preadshee for Classical easonal Decomposiion of ales for The Gap Row A B C D E F G H Quarer ales y 0575 Qr 06 Qr Qr Qr 6760 T Esimae Tˆ = 0.5y - 0.5y - 0.5y 0.5y 0.5y = 0.5*B 0.5*(BBB5) 0.5*B = 0.5*B 0.5*(BB5B6) 0.5*B = 0.5*B 0.5*(B5B6B7) 0.5*B = 0.5*B5 0.5*(B6B7B8) 0.5*B9 xi Esimae y Tˆ.077 = B/C.5 = B5/C5 0.8 = B6/C = B7/C7 8 Qr Qr easonalum This column sums he I componens for corresponding quarers 0.57 = UMIF(A$:A$, "Qr ",D$:D$).56 = UMIF(A$:A$, "Qr ",D$:D$) 8.58 = UMIF(A$:A$, "",D$:D$) 8.60 = UMIF(A$:A$, "Qr ",D$:D$) easonalavg This column is an average of he I componens for corresponding quarers.06 = E/0.5 = E5/ = E6/ = E7/0 Esimae Ŝ This column forces he componens o average o one 0.85 Copied manually 0.86 Copied manually.06 = *F/UM(F$:F$7).5 = *F5/UM(F$:F$ = *F6/UM(F$:F$ = *F7/UM(F$:F$7.06 Copied manually.5 Copied manually easonaladjales y ˆ 78.7 = B/G 97.7 = B/G = B/G 96. = B5/G = B6/G = B7/G7 9.8 = B8/G = B9/G Qr Qr Qr Qr Qr Qr

21 Qr Qr Qr Qr Qr Qr.E Qr Qr.6E Qr.5E easonally Adjused ales for The Gap

22 Algebra used o find he normalizing consan for he seasonal facors in Column G We wan o find a normalizing consan, denoed c, so ha when we divide each of he seasonal facors by c hey average o one, i.e. we wan o find c so ha = c c c c. Using some algebra, his gives = c ( ) = c ( ) c = Therefore, he normalized seasonal facors are ( ), ( ), ( ) and ( ), or equivalenly,,, and. This is he formula used in Column G in he above spreadshees.

23 easonally adjus sales for The Gap using atools atools insrucions o seasonally adjus daa To use atools in Excel, you mus firs open i ouside Excel by clicking on is icon. Then inside Excel, click on atools in he menu a he op of he Excel screen. (Please noe ha click will always refer o a lef click; if a righ click is needed, I ll wrie righ click ). To run an analysis using atools, you mus firs creae a atools daa se conaining he variable(s) you wan o analyze. To do his, click on Daa e Manager in he op lef hand corner of he atools screen. In he Daa e Manager dialog box, click on New, click on he elec he range icon immediaely o he righ of he Excel Range box, highligh he column in he Excel workshee conaining quarerly sales for The Gap, click OK, and hen click OK again. To seasonally adjus sales, click on Time eries and Forecasing a he op of he atools screen, and hen click on Forecas. In he atools-forecas dialog box, click he box nex o ales, click on Time cale (a his poin in he semeser don worry abou any of he oher opions in he Forecas dialog box), click on Quarerly in he new dialog box ha comes up, click on Deseasonalize, and hen click OK. The seasonally adjused sales daa will be pu ino a new workshee labelled Forecas (he seasonally adjused daa will be in a column labelled Deseason ales par way down he workshee). We will discuss he oher columns in his workshee as he semeser goes along. The columns in he workshee labelled Forecas relaed o seasonally adjused sales are: eason Deseason ales Index ales

24 easonally adjus sales for The Gap using R IMPORTANT: You are no responsible for running R scrips in his class. I have only included he R scrip for his analysis in case you are ineresed. R scrip o decompose sales for The Gap ino rend, seasonal and irregular (random) componens ##################################################################################################################### # # You mus se he working direcory properly # If he R package ggplo is no insalled hen i mus be insalled by yping he command (a he "> promp"): # insall.packages("ggplo") # You mus ype (a he "> promp"): library (ggplo) # To run, ype (a he "> promp"): source("classicaleasonaldecomposiion_ales_thegap.r") # (where ClassicaleasonalDecomposiion_ales_TheGap.R is he name of file conaining he R scrip given below) # ##################################################################################################################### # # Open file for oupu # sink ("C:/Users/shively/Box ync/courses/7_pring05/r_crips/classical_easonal_decomposiion/00pracice.x", append=fale, spli=true) # # Read daa # file <- "ales_thegap_ da" ales_able <- read.able(file, header = FALE, sep = "") colnames(ales_able) <- c("time", "ales") n_obs = nrow(ales_able) ca ("Number of observaions is:", n_obs, "\n", "\n") prin(head(ales_able)) # # Plo ales vs. Time # g = ggplo()

25 g <- g geom_line(daa=ales_able, aes(time,ales), color="black", ly=) ggsave('plo_ales.pdf', g) shell.exec(file.pah(gewd(), "plo_ales.pdf")) # # ave daa as a ime series objec # ales_ime_series <- s(ales_able[], frequency=) colnames(ales_ime_series) <- "ales" # # Decompose ales ino rend, seasonal and irregular componens, and plo and prin componens # ales_ime_series_componens <- decompose(ales_ime_series, ype="muliplicaive") pdf("plo_seasonal_componens.pdf") plo(ales_ime_series_componens) shell.exec(file.pah(gewd(), "plo_seasonal_componens.pdf")) dev.off () prin (ales_ime_series_componens) # # Close file for oupu # closeallconnecions() Oupu from R Number of observaions is: Time ales $x Qr Qr Qr Qr

26 $seasonal Qr Qr Qr Qr $rend Qr Qr Qr Qr NA NA NA NA 6

27 $random Qr Qr Qr Qr NA NA NA NA $figure [] $ype [] "muliplicaive" ar(,"class") [] "decomposed.s" 7

28 00000 ales Time

29 Decomposiion of muliplicaive ime series random seasonal rend observed e05 6e05 e Time