Time-series methods: Marketing Engineering Technical Note 1

Transcription

1 Time-series mehods: Markeing Engineering Technical Noe 1 Table of Conen Inroducion Time Series Techniques Smoohing echniques Moving average Double moving average Exponenial smoohing Double exponenial smoohing Adapive filering Box-Jenkins models Decomposiional mehods Summary References Inroducion The logic behind ime series mehods is ha pas daa incorporae enduring paerns ha will carry forward ino he fuure and ha can be uncovered hrough quaniaive analysis. Thus he forecasing ask becomes, in essence, a careful analysis of he pas plus an assumpion ha he same paerns and relaionships will hold in he fuure. There are a number of ime-series analysis and forecasing mehods, differing mainly in he way pas observaions are relaed o he forecass. Many of hese mehods, such as moving averages, and exponenial smoohing are available in Excel, as add-ins. Time Series Techniques Smoohing echniques: The noion underlying smoohing mehods is ha 1 This echnical noe is a supplemen o Chaper 5 of Principles of Markeing Engineering, by Gary L. Lilien, Arvind Rangaswamy, and Arnaud De Bruyn (2007). (All righs reserved) Gary L. Lilien, Arvind Rangaswamy, and Arnaud De Bruyn. No o be re-produced wihou permission. Visi for addiional informaion.

2 here is some specific paern in he values of he variables o be forecas, which is represened in pas observaions, along wih random flucuaions or noise. Using smoohing mehods, he analys ries o disinguish he underlying paern from he random flucuaions by eliminaing he laer. For example, by averaging ou shor-erm flucuaions in a sales daa series could reveal he longer-erm paerns or cycles in sales. Formally, for simple moving averages le S = forecas a ime, X = acual value a ime, and N = number of values included in average. Then forecasing wih moving averages can be represened as 1 X X N S 1 X + = i = + S, (1) N N i= N + 1 Thus, he moving average is simply he unweighed mean of he previous N observaions. Eq. (1) makes i clear ha he new forecas S +1 is a funcion of he preceding moving-average forecas S. Furhermore, if X corresponds o a change (e.g., sep change) in he basic paern of variable X, i is difficul for he mehod o accoun for ha change. Noe also ha he larger N is, he smaller (X -X -N )/N will be and he greaer he smoohing effec will be. In he double moving average, one sars by compuing a se of single moving averages and hen compues anoher moving average based on he values of he firs. Wih a rend, a single or double moving average lags he acual series. Also, he double moving average is always below he simple moving average. Thus i is possible o forecas by aking he difference beween he single moving average and he double moving average and adding i back o he single moving average. This forecasing echnique is called he double moving averages wih rend adjusmens. The exponenial-smoohing approach is very similar o he movingaverage mehod, differing in ha he weighs given o pas observaions are no consan hey decline exponenially so ha more recen observaions ge more weigh han earlier values. Choice of he smoohing facor is lef o he analys. Mos ofen he analys selecs a value experimenally from a se of wo 2

3 or hree differen rial values. Wih he foregoing noaion, he procedure can be represened by S + 1 = αx + (1 α) S, (2) where 0 α 1 is seleced empirically by he analys. A high value of α gives pas forecass and pas daa (included in S ) lile weigh, whereas a low value of α weighs he mos recen period very lighly compared wih all oher pas observaions. The mehod of double exponenial smoohing is analogous o ha of double moving averages, and easily adaps o changes in paerns, such as sep changes. Adapive filering (i.e., removing noise from signal) is anoher approach for deermining he mos appropriae se of weighs, where he weighs change o adjus o he changes in he ime series being filered. Noice ha all he mehods oulined so far are based on he idea ha a forecas can be made as a weighed sum of pas observaions: S W X, (3) = + 1 i i= N + 1 i where S +1 = forecas for period +1; W i = weigh assigned o observaion i; X i N = observed value a i, as before; and = number of observaions used in compuing S +1 (and so is equal o he number of weighs required). The weighs are deermined by an ieraive process ha minimizes he average mean-squared forecasing error. Box-Jenkins: This refers o a class of mehods and a philosophy for approaching forecasing problems. Using i an analys can develop an adequae model for almos any paern of daa. However, i is sufficienly complex ha is 3

4 users mus have a cerain amoun of experise. Box and Jenkins propose hree general classes of models for describing any ype of saionary process (processes ha remain in equilibrium abou a consan mean level): (1) auoregressive (AR), (2) moving average (MA), and (3) mixed auoregressive and moving average (ARMA). If a series is increasing or decreasing wih ime, we can remove his (rend) by aking differences, ΔY = Y Y (4). 1 and hen developing an ARMA model for ΔY. The original series Y can be recovered by successively adding in he ΔY, saring a Y 0. If he rend is nonlinear, several successive differences (d) may be required o produce a saionary ARMA series. (Recall ha if you differeniae Y=X 2 wice d 2 Y/dX 2 you ge a consan, 2. The differencing operaion here is analogous and produces he same resul.) Again, he original series can be recovered by summing d imes. Such a series is called an inegraed ARMA series, denoed as ARIMA (p, d, q), where p is he order (number of periods used) of he AR par, q is he order of he MA par, and d is he level of difference used o produce saionariy. There are also mulivariae exensions of he ARMA models, known as mulivariae ARMA, or MARMA. They combine powerful ime-series forecasing echniques wih explanaory variables and causal models (Hanssens, Parsons, and Schulz 1990). Applying he ARMA and MARMA mehods requires more echnical experise and experience han many of he oher mehods we describe. EXAMPLE Exhibi 1 shows how some of hese forecasing mehods perform on daa drawn from he Naional Bureau of Economic Research. Using he meanabsolue-percen error (MAPE) as he measure of forecasing abiliy, Box-Jenkins does bes in his case. However, he naive mehod is he hird bes ou of he six mehods, suggesing ha more sophisicaed mehods do no always perform beer han simple ones. 4

5 EXHIBIT 1 A comparison of he forecasing accuracy of six forecasing mehods; (a) gives acual daa for fabricaed meal producs while (b), columns (2) hrough (7), gives he forecasing accuracy of six mehods. Source: Naional Bureau of Economic Research Series MDCSMS. Decomposiional mehods: The forecasing mehods described hus far are based on he idea ha we can disinguish an underlying paern in a daa series 5

6 from noise by smoohing (averaging) pas values. The smoohing eliminaes noise so ha we can projec he paern ino he fuure and use i as a forecas. These mehods make no aemp o idenify individual componens of he basic underlying paern. However, in many cases we can break he paern down (decompose i) ino sub-paerns ha idenify each componen of he series separaely. Wih such a breakdown we can frequenly improve accuracy in forecasing and beer undersand he series. Decomposiional mehods assume ha all series are made up of paerns plus error. The objecive is o decompose he paern of he series ino rend, cycle, and seasonaliy: where X = f I, T, C, E ). (5) ( X = ime series a ime ; I = seasonal componen (or index) a ; T = rend componen a ; C = cyclical componen a ; and E = error or random componen a. The exac funcional form of Eq. (5) depends on he decomposiional mehod used. The mos common form is a muliplicaive model: X = I T C E. (6) An addiive form is used ofen, as well. An example of addiive decomposiion is given in Exhibi 2. 6

7 Sales (Unis) Variable Acual Fis Trend Forecass Time Series Decomposiion Plo Time (Monhs) EXHIBIT 2: This char how he acual daa for25 periods are decomposed ino wo addiive componens, namely rend and seasonaliy o creae he fied daa. We also show how he forecass would be generaed from his series for periods 25 o 30. Alhough here are a number of decomposiional mehods, hey all seem o follow he same basic process: 1. For he series X compue a moving average of lengh N, where N is he lengh of he seasonaliy (e.g., N=12 wih monhly daa). This averaging will eliminae seasonaliy by averaging seasonally high periods wih seasonally low periods; and because random errors have no sysemaic paern, i reduces randomness as well. 2. Separae he oucome of he N-period moving average from he original daa period o obain rend and cyclicaliy. If he model is muliplicaive, you do his by dividing he original series by he smoohed series, leaving seasonaliy and error: X1 T + C = moving average) = I E. ( (7) 3. Isolae he seasonal facors by averaging hem for each daa poin in a season over he complee lengh of he series. 4. Specify he appropriae form of he rend (linear, quadraic, exponenial) 7

8 and calculae is value a each period T. You can do his by using regression analysis or moving averages wih rend adjusmens. 5. Use he resuls o separae ou he cycle from he rend+cycle (i.e., he moving average). 6. When you have separaed he seasonaliy, rend, and cyclicaliy from he original daa series, you can idenify he remaining randomness, E. Decomposiional mehods are widely used and have been developed empirically and esed on housands of series. Alhough hey do no have a sound saisical base, he mehods are inuiive and geared o he praciioner and, herefore, he opposie of such procedures as he Box-Jenkins approach, which is derived from heory. Decomposiional mehods appear o be mos appropriae for shor- or medium-erm forecasing and are mainly suied o macroeconomic series. Summary Time series mehods help markeers o generae forecass as a funcion of ime, i.e., wha will happen o a paricular daa series (e.g., sales, rial rae) a ime in he fuure? Unlike naïve mehods ha simply projec he pas ono he fuure, he mehods described in his noe aemp o isolae he enduring paerns hidden in a daa series by removing noise from he signal. Thus, ime series mehods are mos useful in siuaions when enduring paerns repea hemselves in he fuure. They are also mos useful when our ineres ceners mainly on forecasing, and no on explaining or diagnosing a daa paern. We have described some simple echniques ha can be implemened in spreadshees. More complex mehods such as Kalman filers, Bayesian filers, and Fuzzy filers, can also be used in more sophisicaed sysems for analysis of ime series daa. References Hanssens, Dominique M.; Parsons, Leonard J.; and Schulz, Randall L., 1990, Marke Response Models: Economeric and Time Series Analysis, Kluwer Academic Publishers, Boson. 8