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1 FORECASTING: When esimaes of fuure condiions are made on a sysemaic basis, he process is referred o as forecasing and he figure and he saemen obained is known as a forecas. Forecasing is a service whose purpose is o offer he bes available basis for managemen expecaions of he fuure and o help managemen undersand he implicaions for he firm s fuure of he courses of acions o hem a presen. Forecasing is concerned wih wo main asks: firs, he deerminaion of he bes basis available for he formaion of inelligen managerial expecaions: and second, he handling of uncerainy abou he fuure, so ha implicaions of decisions become explici. Main Funcions of Forecasing: The following are main funcions of forecasing: (i) The creaion of plans of acion of acion. I is impossible o evolve a worhwhile sysem of business conrol wihou one accepable sysem of forecasing (ii) The second general use of forecasing is o found in monioring he coninuing progress of plans based on forecass. (iii) The forecas provides a warning sysem of he criical facors o be moniored regularly because hey migh drasically affec he performance of he plan. Seps in forecasing: The forecasing of business flucuaions consiss of he following seps. (i) Undersanding why changes in he pas have occurred: Forecas should use he daa on pas performance o ge a speedomeer reading of he curren rae and how far he rae is increasing and decreasing. (ii) Deermining which phases of business aciviy mus be measured: I is necessary o measure cerain phases of business aciviy in order o predic wha changes will probably follow he presen level of aciviy. (iii) Selecing and compiling daa o be used as measuring devices: There is an independen relaionship beween he selecion of saisical daa and deerminaion of why business flucuaions occur. (iv) Analysis of daa: Daa are analyzed in he ligh of one s undersanding of he reason why changes occur. Mehods of Forecasing: The following are some of he imporan mehods of forecasing: 1. Hisorical Analogy Mehod; 2. Field survey and opinion poll; 3. Exrapolaion 4. Regression Analysis 5. Economeric models 6. Lead Lag Analysis 7. Exponenial smoohing 8. Inpu-Oupu Analysis 9. Time series Analysis TIME SERIES ANALYSIS Time series: Arrangemen of Saisical daa in accordance wih occurrence of ime is known as ime series. A ime series may be mahemaically expressed by he funcional relaionship Y =f() where Y is he value of he variable under consideraion a ime. There are wo main goals of ime series analysis: (a) Idenifying he naure of he phenomenon represened by he sequence of observaions, and (b) Forecasing (predicing fuure values of he ime series variable). Boh of hese goals require ha he paern of observed ime series daa is idenified and more or less formally described. Once he paern Page# 1

2 is esablished, we can inerpre and inegrae i wih oher daa (i.e., use i in our heory of he invesigaed phenomenon, e.g., seasonal commodiy prices). Regardless of he deph of our undersanding and he validiy of our inerpreaion (heory) of he phenomenon, we can exrapolae he idenified paern o predic fuure. Role of Time series analysis: Time series analysis is of grea significance in decision-making for he following reasons. (i) I helps in he undersanding of pas behavior: By observing daa over a period of ime, one can easily undersand wha changes have aken place in he pas. Such analysis will be exremely helpful in predicing he fuure behavior. (ii) I helps in planning fuure operaions: If he regulariy of occurrence of any feaure over a sufficien long period could be clearly esablished hen, wihin limis predicion of probable fuure variaions would become possible. (iii) I helps in evaluaing curren accomplishmens: The acual performance can be compared wih he expeced performance and he cause of variaion analyzed. For example, if expeced sales for were colored TV ses and he acual sales were only one can invesigae he cause for he shorfall in achievemen. (iv) I faciliaes comparison: Differen ime series are ofen compared and imporan conclusions drawn here from. Componens of Time Series: Changes of daa wih change of ime depend on a number of causes; hese causes are known as he componens of ime series. The common componens of ime series are: 1. Trend or long erm movemen or Secular Trend is he long run direcion of he ime series. 2. Seasonal Variaion is he paern in a ime series wihin a year. These paerns end o repea hemselves from year o year. 3. Cyclical variaion is he flucuaion above and below he rend line. 4. Irregular or Random variaion is divided ino wo componens. [Episodic variaions are unpredicable, bu can usually be idenified, such as a flood of hurricane. Residual variaions refer o random in naure and canno be idenified.] Two General Aspecs of Time Series Paerns Mos ime series paerns can be described in erms of wo basic classes of componens: rend and seasonaliy. The former represens a general sysemaic linear or (mos ofen) nonlinear componen ha changes over ime and does no repea or a leas does no repea wihin he ime range capured by our daa (e.g., a plaeau followed by a period of exponenial growh). The laer may have a formally similar naure (e.g., a plaeau followed by a period of exponenial growh), however, i repeas iself in sysemaic inervals over ime. Those wo general classes of ime series componens may coexis in real-life daa. For example, sales of a company can rapidly grow over years bu hey sill follow consisen seasonal paerns (e.g., as much as 25% of yearly sales each year are made in December, whereas only 4% in Augus). Page# 2

3 This general paern is well illusraed in a "classic" Series G daa se (Box and Jenkins, 1976, p. 531) represening monhly inernaional airline passenger oals (measured in housands) in welve consecuive years from 1949 o 1960 (see example daa file G.sa and graph above). If you plo he successive observaions (monhs) of airline passenger oals, a clear, almos linear rend emerges, indicaing ha he airline indusry enjoyed a seady growh over he years (approximaely 4 imes more passengers raveled in 1960 han in 1949). A he same ime, he monhly figures will follow an almos idenical paern each year (e.g., more people ravel during holidays han during any oher ime of he year). This example daa file also illusraes a very common general ype of paern in ime series daa, where he ampliude of he seasonal changes increases wih he overall rend (i.e., he variance is correlaed wih he mean over he segmens of he series). This paern which is called muliplicaive seasonaliy indicaes ha he relaive ampliude of seasonal changes is consan over ime, hus i is relaed o he rend. Trend Analysis There are no proven "auomaic" echniques o idenify rend componens in he ime series daa; however, as long as he rend is monoonous (consisenly increasing or decreasing) ha par of daa analysis is ypically no very difficul. If he ime series daa conain considerable error, hen he firs sep in he process of rend idenificaion is smoohing. Smoohing always involves some form of local averaging of daa such ha he nonsysemaic componens of individual observaions cancel each oher ou. The mos common echnique is moving average smoohing which replaces each elemen of he series by eiher he simple or weighed average of n surrounding elemens, where n is he widh of he smoohing "window". Medians can be used insead of means. The main advanage of median as compared o moving average smoohing is ha is resuls are less biased by ouliers (wihin he smoohing window). Thus, if here are ouliers in he daa (e.g., due o measuremen errors), median smoohing ypically produces smooher or a leas more "reliable" curves han moving average based on he same window widh. The main disadvanage of median smoohing is ha in he absence of clear ouliers i may produce more "jagged" curves han moving average and i does no allow for weighing. Fiing a funcion. Many monoonous ime series daa can be adequaely approximaed by a linear funcion; if here is a clear monoonous nonlinear componen, he daa firs need o be ransformed o remove he nonlineariy. Usually a logarihmic, exponenial, or (less ofen) polynomial funcion can be used. Page# 3

4 Analysis of Seasonaliy Seasonal dependency (seasonaliy) is anoher general componen of he ime series paern. The concep was illusraed in he example of he airline passengers daa above. I is formally defined as correlaional dependency of order k beween each i'h elemen of he series and he (i-k)'h elemen and measured by auocorrelaion (i.e., a correlaion beween he wo erms); k is usually called he lag. If he measuremen error is no oo large, seasonaliy can be visually idenified in he series as a paern ha repeas every k elemens. Auocorrelaion correlogram. Seasonal paerns of ime series can be examined via correlograms. The correlogram (auocorrelogram) displays graphically and numerically he auocorrelaion funcion (ACF), ha is, serial correlaion coefficiens (and heir sandard errors) for consecuive lags in a specified range of lags (e.g., 1 hrough 30). Ranges of wo sandard errors for each lag are usually marked in correlograms bu ypically he size of auo correlaion is of more ineres han is reliabiliy (see Elemenary Conceps) because we are usually ineresed only in very srong (and hus highly significan) auocorrelaions. Examining correlograms. While examining correlograms, you should keep in mind ha auocorrelaions for consecuive lags are formally dependen. Consider he following example. If he firs elemen is closely relaed o he second, and he second o he hird, hen he firs elemen mus also be somewha relaed o he hird one, ec. This implies ha he paern of serial dependencies can change considerably afer removing he firs order auo correlaion (i.e., afer differencing he series wih a lag of 1). Parial auocorrelaions. Anoher useful mehod o examine serial dependencies is o examine he parial auocorrelaion funcion (PACF) - an exension of auocorrelaion, where he dependence on he inermediae elemens (hose wihin he lag) is removed. In oher words he parial auocorrelaion is similar o auocorrelaion, excep ha when calculaing i, he (auo) correlaions wih all he elemens wihin he lag are parially ou. If a lag of 1 is specified (i.e., here are no inermediae elemens wihin he lag), hen he parial auocorrelaion is equivalen o auo correlaion. In a sense, he parial auocorrelaion provides a "cleaner" picure of serial dependencies for individual lags (no confounded by oher serial dependencies). Page# 4

5 Removing serial dependency. Serial dependency for a paricular lag of k can be removed by differencing he series, ha is convering each i'h elemen of he series ino is difference from he (i-k)''h elemen. There are wo major reasons for such ransformaions. Firs, we can idenify he hidden naure of seasonal dependencies in he series. Remember ha, as menioned in he previous paragraph, auocorrelaions for consecuive lags are inerdependen. Therefore, removing some of he auocorrelaions will change oher auo correlaions, ha is, i may eliminae hem or i may make some oher seasonaliies more apparen. The oher reason for removing seasonal dependencies is o make he series saionary which is necessary for ARIMA and oher echniques. Sysemaic Paern and Random Noise As in mos oher analyses, in ime series analysis i is assumed ha he daa consis of a sysemaic paern (usually a se of idenifiable componens) and random noise (error) which usually makes he paern difficul o idenify. Mos ime series analysis echniques involve some form of filering ou noise in order o make he paern more salien. Models of Time Series: Time series may be affeced by one or more componens simulaneously. Two differen models are assumed in ime series. A. The addiive model: According o he addiive model, a ime series can be expressed as Y =T +S +C +I Where Y = Time series value a ime T = Trend values a ime S = Seasonal variaion a ime C = Cyclical variaion a ime I = Irregular variaion a ime B. The muliplicaive Model: In classical or radiional approach, i is assumed ha here is a muliplicaive relaionship among four componens. Any Paricular value Y is considered o be he produc of Trend (T ), Seasonal variaion (S ), Cyclical variaion (C ) and Irregular variaion (I ). Thus Y = T S C I 1. Trend (T ): By rend we mean he general endency of he daa o increase or decrease during a long period of ime. This is rue of mos of series of Business and Economic Saisics. Fore example an upward endency would be seen in daa peraining o populaion, agriculural producion, currency in circulaion ec., while, a downward endency will be noiced in daa of birh rae, deah rae ec. 2. Seasonal Variaion (S ): Seasonal variaions are he periodic and regular movemen in a ime series wih period less han one year. Fore example demand of umbrella in he rainy season, demand of worm clohe in he winer, demand of cold drinks in he summer ec. The facor ha causes seasonal variaions is (i) Climae and weaher condiions (ii) Cusoms, radiions and habis ec. Page# 5

6 3. Cyclical variaions (C ): The oscillaory movemens in a ime series wih period of oscillaion more han one year are ermed as cyclic flucuaions. One complee period is called a cycle. The cyclical movemens in a ime series are generally aribued o he socalled business cycle. There are four well-defined periods or phase in he business cycle namely prosperiy, recession (decline), depression and recovery and normally lass from seven o eleven years. 4. Irregular variaion (I ) : Besides rend, seasonal variaions and cyclical variaions, here are oher facors, which cause variaions in ime series. These variaions are purely random, unpredicable and are due o some irregular circumsances, which are beyond conrol of human hand. These irregular bu powerful flucuaions are due o floods, famines, revelaions, poliical unres, draugh ec. Mehod of Measuring Trend Trend can be measured by he following mehods: 1. The free hand or graphic mehod; 2. The semi-average mehod; 3. The mehod of moving average; 4. The leas squares mehod; 1. The graphic mehod: A free hand smooh curve obained on ploing. The value Y agains enables us o form an idea abou he general rend of he series. This mehod is simple and easier and does no require mahemaical skill. Bu in his mehod differen researcher may ge differen rend line for he same se of daa. Forecasing in his mehod is risky if he researcher is no efficien and experienced. 2. Mehod of semi average: In his mehod he whole daa is divided ino wo pars wih respec o ime. In case of odd number he wo pars are obaining by omiing he value corresponding o he middle of he series. Nex we compue he arihmeic mean for each par and plo hese wo averages agains he mid values of he respecive periods covered by each par. The line obained on joining hese wo poins is he required rend line This mehod is simple o undersand compared o he moving average mehod and he mehod of leas squares. This mehod assumes sraigh-line relaionship beween he ploed poins regardless of he fac wheher he relaionship exiss or no. 3. Mehod of moving averages: In his mehod 3, 4, or 5 years moving averages of he variable values are firs obained. Arihmeic mean of he firs hree years values are compued and placed agains he middle of hose years. Then excluding he firs year value, arihmeic mean of he 2 nd, 3 rd, and 4 h year values are calculaed and placed agains heir middle year. In his way 4 year, 5 year moving averages can be compued. The graph obained on ploing he moving average agains ime gives rend. Meris: *Long erm rend deerminaion is easy by he moving average mehod ** If an appropriae moving average can be aken, he irregular movemen is reduced o a grea exen. Limiaions:* Trend values for all he imes of ime series can no be esimaed by he mehod of moving average; some values a he saing and some values a he end may no found. ** Moving average are affeced by exreme values Page# 6

7 *** This mehod canno be used for forecasing fuure rend, which is he main objecive of he ime series analysis. 5. Leas squares mehod: This mehod is widely used in pracice. When his mehod is applied, a rend line is fied o he daa in such a manner ha he following wo condiions are saisfied: (i) Y Y ) 0 (ii) ( c ( Y Yc 2 ) is he leas. The sraigh line is represened by he equaion Y c =a+bx Where Y c denoe he rend values; Y acual values; a is he inercep; b is he slope of he line or amoun of change in Y variable ha is associaed wih a change of one uni in X variable. The long erm rend equaion (linear) esimaed by he leas squares equaion for ime is: Y' a b b a Y ( Y)( ) / n 2 ( ) 2 / n Y n b n The esimaed rend line becomes Yˆ aˆ bˆ. On he basis of his rend line, values of Y can be obained for differen values of X and predicion of fuure values can be done. Example: The owner of Srong Homes would like a forecas for he nex couple of years of new homes ha will be consruced in he Pisburgh area. Lised below are he sales of new homes consruced in he area for he las 5 years. Year Sales Toal 36.6 Year Sales Sales* Toal Develop a rend equaion using he leas squares mehod by leing 1997 be he ime period 1. b a Y Y n 2 b Y n 2 / n / n (15) / 5 (15) / Page# 7

8 The ime series equaion is: Y = The forecas for he year 2003 is: Y = (7) = If he rend is no linear bu raher he increases end o be a consan percen, he Y values are convered o logarihms, and a leas squares equaion is deermined using he logs. log( Y') [log( a)] [log( b)] Mehod of Moving Average: I consiss of measuremen of rend by smoohing ou he flucuaions of he daa by means of a moving average. Moving average of exen (or period) m is a series of successive averages (A.M.) of m erms a a ime, saring wih 1s, 2nd, 3rd erm ec. Thus he firs average is he mean of he 1s, m erms, he 2nd is he mean of he m erms from 2nd o (m+1)h erm and so on. Moving average is placed agains he middle value of he ime inerval i covers. When m is even he moving average does no coincide wih an original ime period and an aemp is made o synchronize he moving averages and he original daa by cenering he moving averages which consiss in he aking a moving average of exen wo, of hese moving averages and puing of hese values agains he middle ime period. The graph obained on ploing he moving averages agains ime gives rend. Example: The daa on he rice producion during in a cerain region are given below: Year: Producion (Ton) Deermine he rend by mehod of moving average. Soluion: I is clear from he daa ha a 4-year cycle is presen here. So 4-year moving averages are compued. Year Producion (Ton) 4-year moving 4-year moving 4-year moving oal average average (Cenered) Page# 8

9 The rend line is esimaed by ploing he 4-year moving averages along he y-axis agains he corresponding year ploed along he x-axis. The moving-average mehod is used o smooh ou a ime series. This is accomplished by moving he arihmeic mean hrough he ime series. The moving-average is he basic mehod used in measuring he seasonal flucuaion. To apply he moving-average mehod o a ime series, he daa should follow a fairly linear rend and have a definie rhyhmic paern of flucuaions. The mehod mos commonly used o compue he ypical seasonal paern is called he raioo-moving-average mehod. I eliminaes he rend, cyclical, and irregular componens from he original daa (Y). The numbers ha resul are called he ypical seasonal indexes. Sep 1: Deermine he moving oal for he ime series. Sep 2: Deermine he moving average for he ime series. Sep 3: The moving averages are hen cenered. Sep 4: The specific seasonal for each period is hen compued by dividing he Y values wih he cenered moving averages. Sep 5: Organize he specific seasonals in a able. Sep 6: Apply he correcion facor. The resuling series (sales) is called deseasonalized sales or seasonally adjused sales. The reason for deseasonalizing a series (sales) is o remove he seasonal flucuaions so ha he rend and cycle can be sudied. A se of ypical indexes is very useful in adjusing a series (sales, for example) Example : The daa on rice producion during in a large agriculural area. Year : Producion: (Tones) (a) fi a rend line by he mehod of 3-yearly moving average; (b) Fi a rend line by he mehod of leas squares and commen (c) Esimae he producion for he year Below are given he figures of food requiremen (in million ons) for a counry: Year Food grain Requiremen (Million ons) (i) (ii) Fi a sraigh Line by he Leas Squares Mehods and abulae he rend value Wha is he monhly increase of food requiremen for his counry? Page# 9

10 (iii) Esimae he food requiremen (in million ons) for Bangladesh in he year Soluion: (i) Compuaion of Trend value Year Food grain Requiremen (Y) Compuaional Table = Year Y Trend Values (in million ons) Eliminaion of Trend Le he Trend Equaion or Time series Equaion be Y=a + b b= Y ( ( Y).( ) )/ n / n = /11 = (66) /11 a = Y b. = = So he Trend Equaion or Time series Equaion is Y= (ii) Yearly increase of Food Demand as provided by linear rend is million ons or housand ons. So he monhly increase of Food Demand is /12= housand ons. (iii) The esimaed food requiremen (in million ons) in he year 2015 (= 20). Y= = Esimaed food requiremen is million ons 2. Below are given he figures of Mid-Year Populaion (in million ) for of a counry: Year Mid-Year Populaion (Million) Page# 10

11 (iv) Fi a sraigh Line by he Leas Squares Mehods and abulae he rend value (v) Wha is he monhly increase of populaion for his counry? (vi) Esimae he populaion for Bangladesh in he year Soluion: (i) Compuaion of Trend value Year Populaion in million (Y) Compuaional Table = Year Y 2 Trend Values Eliminaion (in million) of Trend Toal Le he Trend Equaion or Time series Equaion be Y=a + b b= Y ( ( Y).( ) )/ n / n = /11 = (66) / a = Y b. = = So he Trend Equaion or Time series Equaion is Y= (ii) Yearly increase of populaion as provided by linear rend is million or. So he monhly increasing number of people is million /12= (iii) The esimaed populaion for he counry in he year 2015 (= 20). Y= = million Page# 11

12 Exponenial Smoohing General Inroducion Exponenial smoohing has become very popular as a forecasing mehod for a wide variey of ime series daa. Hisorically, he mehod was independenly developed by Brown and Hol. Brown worked for he US Navy during World War II, where his assignmen was o design a racking sysem for fire-conrol informaion o compue he locaion of submarines. Laer, he applied his echnique o he forecasing of demand for spare pars (an invenory conrol problem). He described hose ideas in his 1959 book on invenory conrol. Hol's research was sponsored by he Office of Naval Research; independenly, he developed exponenial smoohing models for consan processes, processes wih linear rends, and for seasonal daa. Simple Exponenial Smoohing A simple and pragmaic model for a ime series would be o consider each observaion as consising of a consan (b) and an error componen (epsilon), ha is: X = b +. The consan b is relaively sable in each segmen of he series, bu may change slowly over ime. If appropriae, hen one way o isolae he rue value of b, and hus he sysemaic or predicable par of he series, is o compue a kind of moving average, where he curren and immediaely preceding ("younger") observaions are assigned greaer weigh han he respecive older observaions. Simple exponenial smoohing accomplishes exacly such weighing, where exponenially smaller weighs are assigned o older observaions. The specific formula for simple exponenial smoohing is: S = *X + (1- )*S -1 When applied recursively o each successive observaion in he series, each new smoohed value (forecas) is compued as he weighed average of he curren observaion and he previous smoohed observaion; he previous smoohed observaion was compued in urn from he previous observed value and he smoohed value before he previous observaion, and so on. Thus, in effec, each smoohed value is he weighed average of he previous observaions, where he weighs decrease exponenially depending on he value of parameer (alpha). If is equal o 1 (one) hen he previous observaions are ignored enirely; if is equal o 0 (zero), hen he curren observaion is ignored enirely, and he smoohed value consiss enirely of he previous smoohed value (which in urn is compued from he smoohed observaion before i, and so on; hus all smoohed values will be equal o he iniial smoohed value S 0 ). Values of in-beween will produce inermediae resuls. Even hough significan work has been done o sudy he heoreical properies of (simple and complex) exponenial smoohing he mehod has gained populariy mosly because of is usefulness as a forecasing ool. Thus, regardless of he heoreical model for he process underlying he observed ime series, simple exponenial smoohing will ofen produce quie accurae forecass. Choosing he Bes Value for Parameer (alpha) Gardner (1985) discusses various heoreical and empirical argumens for selecing an appropriae smoohing parameer. Obviously, should fall ino he inerval beween 0 (zero) and 1. Among praciioners, smaller han 0.30 is usually recommended. However, in he sudy by Makridakis (1982), values above.30 frequenly yielded he bes forecass. Page# 12

13 Esimaing he bes value from he daa. In pracice, he smoohing parameer is ofen chosen by a grid search of he parameer space; ha is, differen soluions for are ried saring, for example, wih = 0.1 o = 0.9, wih incremens of 0.1. Then is chosen so as o produce he smalles sums of squares (or mean squares) for he residuals (i.e., observed values minus one-sep-ahead forecass; his mean squared error is also referred o as ex pos mean squared error, ex pos MSE for shor). Indices of Lack of Fi (Error) The mos sraighforward way of evaluaing he accuracy of he forecass based on a paricular value is o simply plo he observed values and he one-sep-ahead forecass. This plo can also include he residuals (scaled agains he righ Y-axis), so ha regions of beer or wors fi can also easily be idenified. This visual check of he accuracy of forecass is ofen he mos powerful mehod for deermining wheher or no he curren exponenial smoohing model fis he daa. In addiion, besides he ex pos MSE crierion (see previous paragraph), here are oher saisical measures of error ha can be used o deermine he opimum parameer (see Makridakis, Wheelwrigh, and McGee, 1983): Mean error: The mean error (ME) value is simply compued as he average error value (average of observed minus one-sep-ahead forecas). Obviously, a drawback of his measure is ha posiive and negaive error values can cancel each oher ou, so his measure is no a very good indicaor of overall fi. Mean absolue error: The mean absolue error (MAE) value is compued as he average absolue error value. If his value is 0 (zero), he fi (forecas) is perfec. As compared o he mean squared error value, his measure of fi will "de-emphasize" ouliers, ha is, unique or rare large error values will affec he MAE less han he MSE value. Sum of squared error (SSE), Mean squared error. These values are compued as he sum (or average) of he squared error values. This is he mos commonly used lack-of-fi indicaor in saisical fiing procedures. Percenage error (PE). All he above measures rely on he acual error value. I may seem reasonable o raher express he lack of fi in erms of he relaive deviaion of he one-sepahead forecass from he observed values, ha is, relaive o he magniude of he observed values. For example, when rying o predic monhly sales ha may flucuae widely (e.g., seasonally) from monh o monh, we may be saisfied if our predicion "his he arge" wih abou ±10% accuracy. In oher words, he absolue errors may be no so much of ineres as Page# 13

14 are he relaive errors in he forecass. To assess he relaive error, various indices have been proposed (see Makridakis, Wheelwrigh, and McGee, 1983). The firs one, he percenage error value, is compued as: PE = 100*(X - F )/X where X is he observed value a ime, and F is he forecass (smoohed values). Mean percenage error (MPE). This value is compued as he average of he PE values. Mean absolue percenage error (MAPE). As is he case wih he mean error value (ME, see above), a mean percenage error near 0 (zero) can be produced by large posiive and negaive percenage errors ha cancel each oher ou. Thus, a beer measure of relaive overall fi is he mean absolue percenage error. Also, his measure is usually more meaningful han he mean squared error. For example, knowing ha he average forecas is "off" by ±5% is a useful resul in and of iself, whereas a mean squared error of 30.8 is no immediaely inerpreable. Auomaic search for bes parameer. A quasi-newon funcion minimizaion procedure (he same as in ARIMA is used o minimize eiher he mean squared error, mean absolue error, or mean absolue percenage error. In mos cases, his procedure is more efficien han he grid search (paricularly when more han one parameer mus be deermined), and he opimum parameer can quickly be idenified. The firs smoohed value S 0. A final issue ha we have negleced up o his poin is he problem of he iniial value, or how o sar he smoohing process. If you look back a he formula above, i is eviden ha you need an S 0 value in order o compue he smoohed value (forecas) for he firs observaion in he series. Depending on he choice of he parameer (i.e., when is close o zero), he iniial value for he smoohing process can affec he qualiy of he forecass for many observaions. As wih mos oher aspecs of exponenial smoohing i is recommended o choose he iniial value ha produces he bes forecass. On he oher hand, in pracice, when here are many leading observaions prior o a crucial acual forecas, he iniial value will no affec ha forecas by much, since is effec will have long "faded" from he smoohed series (due o he exponenially decreasing weighs, he older an observaion he less i will influence he forecas). Seasonal and Non-Seasonal Models Wih or Wihou Trend The discussion above in he conex of simple exponenial smoohing inroduced he basic procedure for idenifying a smoohing parameer, and for evaluaing he goodness-of-fi of a model. In addiion o simple exponenial smoohing, more complex models have been developed o accommodae ime series wih seasonal and rend componens. The general idea here is ha forecass are no only compued from consecuive previous observaions (as in simple exponenial smoohing), bu an independen (smoohed) rend and seasonal componen can be added. Gardner (1985) discusses he differen models in erms of seasonaliy (none, addiive, or muliplicaive) and rend (none, linear, exponenial, or damped). Addiive and muliplicaive seasonaliy. Many ime series daa follow recurring seasonal paerns. For example, annual sales of oys will probably peak in he monhs of November and December, and perhaps during he summer (wih a much smaller peak) when children are on heir summer break. This paern will likely repea every year, however, he relaive Page# 14

15 amoun of increase in sales during December may slowly change from year o year. Thus, i may be useful o smooh he seasonal componen independenly wih an exra parameer, usually denoed as (dela). Seasonal componens can be addiive in naure or muliplicaive. For example, during he monh of December he sales for a paricular oy may increase by 1 million dollars every year. Thus, we could add o our forecass for every December he amoun of 1 million dollars (over he respecive annual average) o accoun for his seasonal flucuaion. In his case, he seasonaliy is addiive. Alernaively, during he monh of December he sales for a paricular oy may increase by 40%, ha is, increase by a facor of 1.4. Thus, when he sales for he oy are generally weak, han he absolue (dollar) increase in sales during December will be relaively weak (bu he percenage will be consan); if he sales of he oy are srong, han he absolue (dollar) increase in sales will be proporionaely greaer. Again, in his case he sales increase by a cerain facor, and he seasonal componen is hus muliplicaive in naure (i.e., he muliplicaive seasonal componen in his case would be 1.4). In plos of he series, he disinguishing characerisic beween hese wo ypes of seasonal componens is ha in he addiive case, he series shows seady seasonal flucuaions, regardless of he overall level of he series; in he muliplicaive case, he size of he seasonal flucuaions vary, depending on he overall level of he series. The seasonal smoohing parameer. In general he one-sep-ahead forecass are compued as (for no rend models, for linear and exponenial rend models a rend componen is added o he model; see below): Addiive model: Forecas = S + I -p Muliplicaive model: Forecas = S *I -p In his formula, S sands for he (simple) exponenially smoohed value of he series a ime, and I -p sands for he smoohed seasonal facor a ime minus p (he lengh of he season). Thus, compared o simple exponenial smoohing, he forecas is "enhanced" by adding or muliplying he simple smoohed value by he prediced seasonal componen. This seasonal componen is derived analogous o he S value from simple exponenial smoohing as: Addiive model: I = I -p + *(1- )*e Muliplicaive model: I = I -p + *(1- )*e /S Pu ino words, he prediced seasonal componen a ime is compued as he respecive seasonal componen in he las seasonal cycle plus a porion of he error (e ; he observed Page# 15

16 minus he forecas value a ime ). Considering he formulas above, i is clear ha parameer can assume values beween 0 and 1. If i is zero, hen he seasonal componen for a paricular poin in ime is prediced o be idenical o he prediced seasonal componen for he respecive ime during he previous seasonal cycle, which in urn is prediced o be idenical o ha from he previous cycle, and so on. Thus, if is zero, a consan unchanging seasonal componen is used o generae he one-sep-ahead forecass. If he parameer is equal o 1, hen he seasonal componen is modified "maximally" a every sep by he respecive forecas error (imes (1- ), which we will ignore for he purpose of his brief inroducion). In mos cases, when seasonaliy is presen in he ime series, he opimum parameer will fall somewhere beween 0 (zero) and 1(one). Linear, exponenial, and damped rend. To remain wih he oy example above, he sales for a oy can show a linear upward rend (e.g., each year, sales increase by 1 million dollars), exponenial growh (e.g., each year, sales increase by a facor of 1.3), or a damped rend (during he firs year sales increase by 1 million dollars; during he second year he increase is only 80% over he previous year, i.e., $800,000; during he nex year i is again 80% less han he previous year, i.e., $800,000 *.8 = $640,000; ec.). Each ype of rend leaves a clear "signaure" ha can usually be idenified in he series; shown below in he brief discussion of he differen models are icons ha illusrae he general paerns. In general, he rend facor may change slowly over ime, and, again, i may make sense o smooh he rend componen wih a separae parameer (denoed [gamma] for linear and exponenial rend models, and [phi] for damped rend models). The rend smoohing parameers (linear and exponenial rend) and (damped rend). Analogous o he seasonal componen, when a rend componen is included in he exponenial smoohing process, an independen rend componen is compued for each ime, and modified as a funcion of he forecas error and he respecive parameer. If he parameer is 0 (zero), han he rend componen is consan across all values of he ime series (and for all forecass). If he parameer is 1, hen he rend componen is modified "maximally" from observaion o observaion by he respecive forecas error. Parameer values ha fall in-beween represen mixures of hose wo exremes. Parameer is a rend modificaion parameer, and affecs how srongly changes in he rend will affec esimaes of he rend for subsequen forecass, ha is, how quickly he rend will be "damped" or increased. Page# 16

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