CLASSICAL TIME SERIES DECOMPOSITION

Transcription

1 Time Series Lecure Noes, MSc in Operaional Research Lecure CLASSICAL TIME SERIES DECOMPOSITION Inroducion We menioned in lecure ha afer we calculaed he rend, everyhing else ha remained (according o ha calculaion approach) was called he residuals. We also saed ha here are oher mehods ha give paricular imporance o hese residuals'. These mehods rea his remainder as a valuable componen of he ime series, and he basis for using hese mehods is o find he componens ha make up he series. Le us develop furher our iniial formula according o which: Time series = Paern + Residuals If we give a name o he paern and call i a rend, hen he residuals could be called variaions around he rend. The variaions could be of a differen naure and we can differeniae he following variaions: Cyclical variaions Seasonal variaions Irregular variaions In he pas, when his mehod was developed for census purposes, rend was called a `secular variaion' or a `secular endency'. Almos a hundred years ago, saisicians developed a mehod ha we, call oday a classical decomposiion mehod, and he basic assumpion of his mehod was ha each ime series consised of he following componens: T = rend C = cyclical componen, S = seasonal componen, I = irregular variaions Decomposiion echniques are among he oldes of he forecasing mehods. Economiss have used hese echniques since he beginning of he cenury o idenify he business cycle. Decomposiion mehods are among he easies o undersand and use, especially for shor-erm forecasing. In addiion, unforunaely, he decomposiion mehod is basically inuiive and here are number of heoreical weaknesses in is approach. However, hese do no preven he posiive resuls obained in he pracical applicaion of he mehod. Addiive and Muliplicaive Models Time-series models can basically be classified ino wo ypes: addiive model and muliplicaive models. For an addiive model, we assume ha he daa is he sum of he ime-series componens, ha is,. X = T + C + S + I + e. X = T C S I + e If he daa does no conain one of he componens, he value for ha componen is equal o zero. In an addiive model he seasonal or cyclical componen is independen of he rend, and hus he magniude of he seasonal swing (movemen) is consan over he ime, as illusraed in Fig. (a). Figure (a): addiive model-he magniude of he seasonal swing is consan over ime.

2 Time Series Lecure Noes, Figure (b) muliplicaive model he magniude of he seasonal swing is proporional o he rend. In a muliplicaive model, he daa is he produc of he various componens, ha is, X = T S C e, If rend, seasonal variaion, or cycle is missing, hen is value is assumed o be. As shown in Fig. b, he seasonal (or cyclical) facor of a muliplicaive model is proporional (a raio) o he rend, and hus he magniude of he seasonal swing increases or decreases according o he behavior of he rend. Alhough mos daa ha possess seasonal (cyclical) variaions canno be precisely classified as addiive or muliplicaive in naure, we usually look a he forecass obained using boh models and choose he model ha yields he smalles SSE and ha seems appropriae for he daa in quesion. The Seasonal and Cyclical Componens Daa ha is repored quarerly, monhly, weekly, ec. and ha demonsraes a yearly periodic paern is said o conain a seasonal componen or facor. A seasonal series may be rended or unrended. I may or may no possess a cyclical componen. However, in mos cases seasonaliy is easier o model han rend or cycle because i has a clearly repeiive -monh or -quarer paern. Trend may be linear or curvilinear, cycles can be any lengh and may repea a irregular inervals, bu seasonaliy is usually well defined. In he decomposiion mehod, he seasonal componen is he firs componen ha is modeled in he ime series. Addiive Decomposiion The addiive decomposiion mehod is appropriae for modeling ime-series daa conaining rend, seasonal, and error componens, if we can assume he following: We have an addiive model ( X = T + S + C + e ), he error erms are random, and he seasonal componen for any one season is he same in each year. Seps in he Decomposiion Mehod Figure Quarerly sales figures for he years 985 hrough 988 Figure displays quarerly sales figures for he years 985 hrough 988. Clearly, sales of he company is seasonal. There is a definable drop in sales during he firs quarer and a rise o a peak during he hird quarer. There also appears o be an upward rend in he daa. We assume ha we have an addiive model and use his daa o explain and demonsrae he following seps in he addiive decomposiion mehod. To accomplish his, cenered moving averages (wo-period moving averages of he iniial moving averages) are compued:

3 Time Series Lecure Noes, cenered moving average (CMA,) = rend + cycle, Table Obaining he esimaes for seasonaliy and in an addiive decomposiion model of consrucion sales X ( ) Year Quarer T X Moving CMA S + e S d Ave; T C As shown in he Table,. firs moving average = ( ) / = 7. 6 second moving average = ( ) / = hird moving average = ( ) / = 6. ec. Thus, CMA = ( ) / = 5. CMA = ( ) / = 58. CMA 5 = ( ) / = 6.9. Subrac he CMA (T + C ) from he daa. The difference is equal o S + e In he example, S + e = = 0.56 S + e = = 7.56 S5 + e5 = = 8.9 ec.. Remove he error ( e ) componen from S + e by compuing he average for each of he seasons. Tha is, Table Seasonal values of daa series is given in he Table. Quarer Quarer Quarer Quarer S =-0.5 S =5.0 S =.5 S =5. These are he four esimaes for he seasonal componens.. These averaged seasonal esimaes should add up o zero. If hey do no, we mus adjus hem (normalize hem) so ha hey will be final adjusmen (normalizaion) consiss of subracing a consan ( Sn / L) from each esimae. Using he daa from Table, ( Sn / L) = ( ) / = 0. he final seasonal esimaes are = = 0.86 S = =. 96 S

4 Time Series Lecure Noes, S =.5 0. =. S = = Deseasonlize he daa by subracing from i heir proper seasonal esimaes: di = x i Sn For example, d = 6 ( 0.86) = 6.86 d = 6.8 (.96) =.8 d = 6.9 (.) = d = 65.7 (.79) = 60.9 d 6 = 50.8 (.79) = Perform he proper regression analysis on he deseasonlized daa o obain he appropriae model for he rend. The appropriae model for he daa in Table is a linear model ( - es for he slope =.). The equaion o he model he rend is T = An esimae or forecas for any ime period can be found by adding ogeher he esimaes for he various componens. For our example, he forecas for he sevenh ime period would be Xˆ = T7 + S7 + C7, T 7 = (7) = S7 = S =. C 7 = 0 (we are assuming here is no cycle) Xˆ = = 79.5 Figure Graph of acual and esimaed values of daa in he Table... Forecas and Confidence Inervals we discussed he mehods of forecasing ime series daa an addiive model. The procedure was o add he appropriae esimaes for he various componens ogeher. Thus, he forecas conrucing for he firs quarer of 989 should be Xˆ 7 = T7 + S7 T 7 = (7) = S 7 = S = X 7 = (-0.86) = 90.5 Since he decomposiion mehod is basically inuiive, wihou any sound saisical heory behind i, here is no "saisically correc" confidence inerval for x. However, here is an inuiive mehod for consrucing confidence inervals for decomposiion forecass. The mehod is simply o use he inerval error for he rend model as he measure of he inerval error for x. This can be compued from he resuls of he regression analysis on he deseasonalized daa. Therefore, he confidence inerval for xˆ, is compued as Xˆ ± α / Se (correcion facor) where S e = he sandard error of esimae ( MSE ) from he appropriae rend regression analysis, ( p ) and where correcion facor = + + where p is he ph poin. n ( ) n which is obained from he resuls of he regression analysis of he deseasonalized daa.

5 Time Series Lecure Noes, 5 In our example, he rend sandard error for he deseasonalized daa is equal o 5.86, and he correcion facor for he linear model is equal o.9. Thus, an approximae 95 percen confidence inerval for X 7 is 90.5 ±.5 x (5.86) x(.9), or (75.9 o 50.) We should sress ha his echnique yields only an approximae (alhough fairly accurae) confidence inerval and does no have a sound heoreical basis. Muliplicaive Decomposiion The muliplicaive decomposiion mehod (someimes called he raio o rend or he raio o moving average) is very similar o ha of he addiive decomposiion mehod. For a muliplicaive decomposiion model, we assume he following: X, is a produc of he various componens, including error ( X = TSC e ) erms are random and he seasonal facor for any one season is he same for each year. Seps in he Decomposiion Mehod To illusrae he echniques used in he muliplicaive decomposiion mehod, we will use U.S. quarerly reail sales daa for he years As seen in Fig., and he daa is in he Table, here is a regular seasonal paern in he series. Pronounced peaks during he fourh quarer and hrough during he firs quarer of each year are apparen. There also appears o be an upward rend in he daa. Thus, we proceed o isolae hese componens by applying he following seps o he daa: Sales(millions of dollars) Acvual values of U.S reail sales(98-987) Qurer Figure : Acual values are shown in he Table for U.S reail sales (98-987). For he acual ime series, compue (as in he addiive model) a cenered moving average of lengh L. The moving average and cenered moving average for he firs hree quarers of he reail sales daa (Table ) are Quarerly daa (in millions of dollars) was obained by summing monhly daa. 00 for compuaional convenience divided he new series. Linear regression resuls a = b = 8.79 es for slope = 7.97, r = compued in he following manner: firs moving average = (,88 +,9 +,80 +,505)/ =,0.75 second moving average = (,9 +,80 +,505 +,00)/ =,8.5 hird moving average = (,80 +,505 +,00 +,9)/ =,88.5 ec. and CMA = (,0.75 +,8.5)/ =,.5 CMA = (,8.5 +,88.5)/ =,6.5 CMA 5 = (, ,6.5) / =,5 (see Table.) ec.

6 Time Series Lecure Noes, 6 Table : Obaining he esimaes for seasonaliy and rend in a muliplicaive decomposiion model of reail sales in (99-997) Year Quarer T x Sales Moving Ave; CMA T C S e S Divide he CMA, ( T C ) ino he daa. The quoien is equal o S ne : X = T S C e ( TSC e ) (TC ) = Se In our example S e = 80.5 = e = S = S5 e5 = 00 5 = ec: d Remove he error ( e ) componen from S ne by compuing he average for each of he seasons: Table Seasonal values of daa series is given in he Table Quarer Quarer Quarer Quarer S =0.905 S =.0 S =.00 S =.07 These are he four esimaes for he seasonal componens.. These averaged seasonal esimaes should, for a muliplicaive model, add up L (he number of seasons in a year). If hey do no, we mus normalize hem so ha hey will. The final normalizaion consiss of muliplying each esimae by he consan L ( Sn ). Using he daa fin he Table, L ( Sn ) = /( ) =.005 The final seasonal esimaes are S = = 0.9 S = =. 05 S = =.005 S = = Deseasonalize he daa by dividing i by he proper seasonal esimaes: d = X S, For example, d, =,88/ 0.90 =,86.97 d =,9 /.05 =, d =,80 /.005 =,6.79 d =,505 /.075 =, d 6 =,59 /.075 =,868.8

7 Time Series Lecure Noes, 7 6. Perform he proper regression analysis on he deseasonalized daa o obain he appropriae rend model (linear, quadraic, exponenial, ec.). The appropriae rend model for he deseasonalized daa in our example is a linear regression model. The F-es for he coefficien of deerminaion (0.979) and he -es for he slope are boh significan a he 0.05 level (see Table ). The equaion o model he rend is T = An esimae of forecas for any ime period consiss of he produc of he individual componen esimaes a : X = TSC e, Thus, he forecas for he daa in he Table sales in he fourh quarer of he second year ( =8) is X 8 = T8S8C 8 T 7 =, x8 = 75., S 8 = S =.075 C 8 = I (we are assuming here is no cycle) X 8 = 75. x.075 x = Figure : Graph of acual and esimaed values of sales in Table Forecass and Confidence Inervals The esimaes for rend, seasonal variaion, and cycle obained by he muliplicaive decomposiion mehod are used o describe he ime series or o forecas fuure values of he daa. As discussed in an earlier secion, he forecas for ime in a muliplicaive model is he produc of he individual esimaes for ime period. Using our example as shown in Table., sales, we can obain he poin esimae for he second quarer in 988 by he following mehod: X = T S 8 8 8C8 T 8 = x (8) = 96. S 8 =S =.05 C 8 = (we are assuming here is no cycle) X 8 = 0.69 As in he addiive decomposiion mehod, here is no "saisically correc" way o compue a confidence inerval for he poin esimae. This mehod uses he inerval error for he rend componen as he measure of he inerval error of X. Thus, he formula for he confidence inerval can be wrien as xˆ ± α / Se (correcion facor) S where e = he sandard error of esimae ( MSE ) from he appropriae rend regression analysis, and where correcion facor = + ( p ) + n ( ) n where p is he ph ime poin which we concern for confidence inerval, which is obained from he resuls of he regression analysis of he deseasonalized daa. In our example, he rend sandard error for he deseasonalized reail sales is 5.56 and he correcion facor for he 8h ime period is.9. An approximae 95 percen confidence inerval for xˆ 8 would be 0.69 ±.5(5.56)(.9), or (96.57 o 08.80) We can be 95 percen confiden ha he reail sales for he second quarer of 998 will be somewhere beween and (in housands).

8 Time Series Lecure Noes, 8 Tes For Seasonaliy In he preceding examples, we have confirmed he presence of a seasonal componen (before isolaing i) by inspecing he graph of he daa and by prior knowledge of he behavior of he series. There are imes, however, when he presence of a significan seasonal componen is quesionable. In hese insances, somehing more han a visual inspecion of he graph is needed. One such mehod is o apply he Kruskal-Wallis one-way analysis of variance es o he oucomes ha were obained by subracing or dividing he CMAs ino he daa. These oucomes supposed o be conained jus he seasonal and error componens. If here is no specific seasonal componen, he oucomes should consis of nohing bu random error and hus heir disribuion should be he same for all seasons. This means ha if hese oucomes are ranked and he ranks are grouped by seasons, hen he average rank for each season should be saisically equal o he average rank of any oher season. The Kruskal-Wallis es, a nonparameric es analogous o he parameric one-way analysis of variance es, will deermine wheher or no he sums of he rankings (and hus he means) are differen (or he same) beween he various groups (seasons). Table 5: Tesing for seasonaliy: compuaion of he Kruskal-Wallis Saisic for he daa in he Table T Quarer S e Rank Sum of ranks by quarer Hypoheses: : S = S = S = S (here is no seasonaliy) H 0 = H a : Sn for some seasons (here is a seasonaliy in he daa) α = 0.05 This can be accomplished by compuing he saisic R i H = (N + ) N(N + ) n i where N = he oal number of rankings R i = he sum of he rankings in a specific season n i = he number of rankings in a specific season. In he muliplicaive decomposiion example, four seasonal facors (0.90,.05,.005, and.075) were firs isolaed and hen used o deseasonalize he daa. However, if hese four seasonal facors were saisically equal o, hese procedures would no be necessary. By ranking he S e, values and applying he Kruskal-Wallis es o he sums of he seasonal ranks, he presence of he seasonal facors can be saisically confirmed. As seen in Table 5 when he ranks for each quarer are summed and he H saisic is calculaed, he compued value (9.67) is greaer han he abled (criical) value ( χ = 7. 8, df = ). This resul would lead us o conclude ha here is seasonaliy in he reail sales daa in he Table. Tabled (criical) value, χ wih df = L - : χ = 7. 8, df = Compued value: R i H = (N + ) N(N + ) n i 6 7 H = () = 9.67 ()

9 Time Series Lecure Noes, 9 Decision rule: 9.67 > 7.8, herefore rejec H o. Noe, he procedures are he same for esing for seasonaliy in he addiive model- H0 : S = S = S = S = 0. For furher discussion of he Kruskal-Wallis es and oher ess for seasonaliy. Advanages and Disadvanages of he Decomposiion Mehod The main advanages of he decomposiion mehod are he relaive simpliciy of he procedure (i can be accomplished wih a hand calculaor) and he minimal sar-up ime. The disadvanages include no having sound saisical heory behind he mehod, he enire procedure mus be repeaed each ime a new daa poin is acquired, and, as in some oher ime-series echniques, no ouside variables are considered. However, he decomposiion mehod is widely used wih much success and accuracy, especially for shor-erm forecasing. Moving Averages Forecasing If we hen eliminae he firs daa poin from his inerval and add he fourh one, heir moving average is again a forecas for he fifh period. As he mahemaicians would say: x + x + x x N+ M = N where M, is a moving average a he poin, x, is a daa poin in he ime series, and N is he number of daa in he period for which he moving average is calculaed. Therefore, he forecas for he fuure period could be expressed as: x + x + x x N+ F = + N If we agree ha his is a very easy and elegan way of forecasing he fuure, we have o say also ha here are quie a few limiaions o his forecasing mehod. In he firs place his mehod canno forecas he non-saionary series a well enough. Imagine ha we have consanly increasing numbers in he series. Now, having described he rend, we can use moving averages o consruc an equaion ha will simulae our upwards or downwards series. This could be, of course, a simple sraigh-line equaion ha would handle he non saionary ime series. Wha has been menioned above abou every daa poin consising of single and double moving averages, could be convered ino an inercep (or parameer a) of our projeced sraigh line. Therefore: a = M M where M is a single moving average and M is a double moving average. To find he parameer b we have o use he following formula: b = (M M ) N We can now say ha every daa on our ime series could be approximaed by: F = a + b + I is very simple o calculae he forecass of our ime series by saying ha: F + = a + mb Where m is a number of periods ahead, which we are forecasing. m By using he example from Table 6, we can see he mechanics of calculaing forecass. If we compare single and double moving average forecass, hen graphically hey appear as shown in Fig. 5. We can see ha double moving average forecass are following he paern of he original ime series in a no lively fashion, which is exacly wha we waned o achieve in he case of he nonsaionary series.

10 Time Series Lecure Noes, 0 Original, Single and Double Moving Average Forecsa Value Period Figure 5 Single and double moving average forecass Table 6 Single moving averages and double moving average models Period Series M M a b M Forecas = a + b

11 Time Series Lecure Noes, Case Sudy Quesion. (a) Is i possible o apply he decomposiion mehod o Quarerly elecriciy demand in Sri Lanka daa? If so sugges a mehodology. (b) Using he muliplicaive mehod, generae he forecas for he daa as in he Table 7. Table 7: Quarerly elecriciy demand in Sri Lanka for he period of 977/Q o 997/Q 977/ / / / / / / / / / / / / / / / / / / / / Soluion: Table 8 for Calculaed Trend and Seasonal Values under Muliplicaive Model Year Observe Trend Seasonal Year Observer Trend Seasonal &Qr r Values Values &Qr Demand Values Values Demand 977/ / / / / / / / / / /

12 Time Series Lecure Noes, / / / / / / / / / / / demand(mn was) Acual and rend curve of he elecriciy demand in Sri Lanka Quarer Figure 6 Acual and rend values of elecriciy demand daa, Table 9 Calculaed Seasonal Indices for elecriciy demand daa Year Q Q Q Q

13 Time Series Lecure Noes, Toal S-Indices Regression Analysis The regression equaion is T = Predicor Coef SDev T P Consan Elecriciy demand (Mn Was) Acual values versus forecas values of elecriciy demand daa Quarer Figure 7 Acual and esimaed values of elecriciy demand in Sri Lanka Forecass The esimaes for rend, seasonal variaion, and cycle obained by he muliplicaive decomposiion mehod are used o describe he ime series or o forecas fuure values of he daa. As discussed in an earlier secion, he forecas for ime in a muliplicaive model is he produc of he individual esimaes for ime period. Using our example as shown in Table 8 and 9, elecriciy demand in Sri Lanka, we can obain he poin esimae for he second quarer in 998 by he following mehod: X = T S C85 T 85 = x(85)+0.069x(85 ) = S 85 =S =.006 C 85 = (we are assuming here is no cycle) Forecas elecriciy demand a 998/Q is X 85 = Similarly, X 86 = T86S86C86 T 86 = x(86)+0.069x(86 ) = S 86 =S =.0 C 86 = (we are assuming here is no cycle) Forecas elecriciy demand a 998/Q is X 85 =

14 Time Series Lecure Noes, Case sudy Quesion Table 0, shows monhly empirical daa se having seasonaliy. Use appropriae mehod o find adjused seasonal indices. (a) Use muliplicaive decomposing mehod o generae forecass. (b) Explain ineresing feaures in he analysis. Table 0 Monhly daa series for he period of hree years Monh Value Monh Value Monh Value Monh Value Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Value Emprical monhy(original) daa series Monh Figure8 Original daa Series as in he Table 0 Regression Analysis: The regression equaion is T = Predicor Coef SDev T P Consan C We have used a ime series ha has monhly values, and ha is four years long. As in he previous example wih he cyclical componen, we have found he rend value of he ime series. If we make a graphic presenaion of wha we have done so far, hen i will appear as shown in Fig. 8.

15 Time Series Lecure Noes, 5 Table Compuaion of seasonal index, by using he muliplicaive decomposiion model of monhly daa series in Table.0 Monh Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec X T Inde x S e S e Monh Y Jan Feb March Apr May June July Aug Sep Oc Nov Dec Y /Jan Feb March Apr May June July Aug Sep Oc Nov Dec Table Seasonal Indices of daa in able Adjused monhly Monh Year Year Year Year indices Jan Feb Mar Apr May June July Aug Sep Oc Nov Dec Sum 00 X T Index S e S e

16 Time Series Lecure Noes, 6 Derended daa series as shown in Table 0 Values Monh Figure 9 Derended values of daa in he Table 0 Value Deseasonalised daa series of daa in Table Monh Figure0 Deseasonalised values of daa in he Table 0 Residual Series of daa in he Table 0 Value Monh Figure Residuals(or Irregularaies) of daa in he Table 0 Seasonal Indices of he daa in he Table 9 Seasonal Index Monh Figure Monhly Indices of daa in he Table 0 For he sake of his exercise, if we now divide each acual value wih a responding ypical seasonal index value (see Table 0), we can how he ime series would look if here were no seasonal influences a all. is now jus a deseasonalized value wih some irregular componens presen i (Fig. 0). If we eliminae he irregular componen by dividing he deasonalized values of he series wih he rend, wha we are lef wih is a measure of oher influences or an index of irregular flucuaions (see Table 0). Each value is elling us how much a paricular monh is affeced o he oher influences, and no only seasonal flucuaions. Graphically i appears as shown in Fig.. The adjused seasonal indices as in he Table wih he fuure linear rend values are used o

17 Time Series Lecure Noes, 7 esimae forecas for nex year. This recomposed series, which represens our forecas, is presened in he form of a graph (Fig. ). Acual and Forecased values of daa in he Table.9 values Monh Figure Acual and forecased values of he daa in he Table 0 Obviously, here are several ways of doing wha we have jus described. This migh auomaically imply ha he mehod of classical decomposiion is no paricularly accurae. Forunaely, his is no rue. I is a fairly accurae mehod, bu he problem wih i is ha i is very arbirary, as he resuls can vary depending on who is doing he forecasing. From he classical decomposiion mehod were developed several new modificaions during he 960s and 970s, noe ha hey are jus modernized and compuerized versions of he good old classical decomposiion mehod. Exercises. Consider he ime series daa in Table.Sales of a paricular company, for he period of (ons) Year Q Q Q Q Table.Sales of a paricular company (a) Draw a graph of x() agains and commen upon wheher or daa appears saionary. (b) Calculae a single and double moving average forecas of lengh superimpose boh forecass on o your graph. (c) Calculae he forecas for quarer, 980 using he double moving and compare his resul wih your resul from decomposiion mehod.. Year Q Q Q Q Table Sales of heaing oil. (a) Draw a graph of x() agains and commen upon wheher or no he daa is saionary. (b) Superimpose on o he above graph a moving average forecas of suiable lengh. Give reasons for your model choice. (c) Use his moving average o provide a forecas for he firs quarer of year.. Table 5 gives he index of average earnings of insurance, banking and finance employees (base: 976 = 00). Year March June Sep: Dec:

18 Time Series Lecure Noes, (a) Draw a graph of x() agains and superimpose on he graph moving averages of lengh in order o show he general rend. (b) Provide an index forecas for he four quarers of 98.. Year Q Q Q Q Table 6: Number of cans (in 00000s) of lager sold manufacurer in each quarer of hree successive years. (a) Draw a graph of x() agains and commen upon wheher or no he daa is saionary. (b) Superimpose ono he above graph a moving average forecas of suiable lengh. Give reasons for your model choice. (c) Use his moving average o provide a forecas for he firs quarer of Table 7 shows he number of cans (in 00000s) of lager sold by a manufacurer in each quarer of hree successive years. Table 6 Year Q Q Q Q (a) Draw a graph of x() agains. (b) Compue he rend equaion using linear regression analysis. (c) Wha are he differences beween an `addiive' and `muliplicaive models when calculaing he seasonal componens? (d) Esimae he number of cans sold in each quarer of 988 using boh he muliplicaive (A= T x S) and `addiive' (A = T + S) models. 6. Use he daa in he able., relae o he sales of heaing oil. (a) Draw a graph of x() agains. (b) Compue he rend equaion using linear regression analysis. (c) Use he Muliplicaive model, A = T x S, o provide a forecas for he firs quarer of year. 7. Table 8 below gives he index of average earnings (GB) of insurance, banking and finance employees (base: 976 = 00). Table 8 Year Q Q Q Q (a) Draw a graph of x() agains. ` (b) Compue he rend equaion using linear regression analysis. (c) Provide an index forecas for Q, Q and Q in he year. Give reasons for your model choice.