Chapter 1 Overview of Time Series

Transcription

1 Chaper 1 Overview of Time Series 1.1 Inroducion Analysis Mehods and SAS/ETS Sofware Opions How SAS/ETS Sofware Procedures Inerrelae Simple Models: Regression Linear Regression Highly Regular Seasonaliy Regression wih Transformed Daa Inroducion This book deals wih daa colleced a equally spaced poins in ime. The discussion begins wih a single observaion a each poin. I coninues wih k series being observed a each poin and hen analyzed ogeher in erms of heir inerrelaionships. One of he main goals of univariae ime series analysis is o forecas fuure values of he series. For mulivariae series, relaionships among componen series, as well as forecass of hese componens, may be of ineres. Secondary goals are smoohing, inerpolaing, and modeling of he srucure. Three imporan characerisics of ime series are ofen encounered: seasonaliy, rend, and auocorrelaion. Seasonaliy occurs, for example, when daa are colleced monhly and he value of he series in any given monh is closely relaed o he value of he series in ha same monh in previous years. Seasonaliy can be very regular or can change slowly over a period of years. A rend is a regular, slowly evolving change in he series level. Changes ha can be modeled by loworder polynomials or low-frequency sinusoids fi ino his caegory. For example, if a plo of sales over ime shows a seady increase of $500 per monh, you may fi a linear rend o he sales daa. A rend is a long-erm movemen in he series. In conras, auocorrelaion is a local phenomenon. When deviaions from an overall rend end o be followed by deviaions of a like sign, he deviaions are posiively auocorrelaed. Auocorrelaion is he phenomenon ha disinguishes ime series from oher branches of saisical analysis. For example, consider a manufacuring plan ha produces compuer pars. Normal producion is 100 unis per day, alhough acual producion varies from his mean of 100. Variaion can be caused by machine failure, abseneeism, or incenives like bonuses or approaching deadlines. A machine may malfuncion for several days, resuling in a run of low produciviy. Similarly, an approaching deadline may increase producion over several days. This is an example of posiive auocorrelaion, wih daa falling and saying below 100 for a few days, hen rising above 100 and saying high for a while, hen falling again, and so on. Anoher example of posiive auocorrelaion is he flow rae of a river. Consider variaion around he seasonal level: you may see high flow raes for several days following rain and low flow raes for several days during dry periods.

2 2 SAS for Forecasing Time Series Negaive auocorrelaion occurs less ofen han posiive auocorrelaion. An example is a worker's aemp o conrol emperaure in a furnace. The auocorrelaion paern depends on he worker's habis, bu suppose he reads a low value of a furnace emperaure and urns up he hea oo far and similarly urns i down oo far when readings are high. If he reads and adjuss he emperaure each minue, you can expec a low emperaure reading o be followed by a high reading. As a second example, an ahlee may follow a long workou day wih a shor workou day and vice versa. The ime he spends exercising daily displays negaive auocorrelaion. 1.2 Analysis Mehods and SAS/ETS Sofware Opions When you perform univariae ime series analysis, you observe a single series over ime. The goal is o model he hisoric series and hen o use he model o forecas fuure values of he series. You can use some simple SAS/ETS sofware procedures o model low-order polynomial rends and auocorrelaion. PROC FORECAST auomaically fis an overall linear or quadraic rend wih auoregressive (AR) error srucure when you specify METHOD=STEPAR. As explained laer, AR errors are no he mos general ypes of errors ha analyss sudy. For seasonal daa you may wan o fi a Winers exponenially smoohed rend-seasonal model wih METHOD=WINTERS. If he rend is local, you may prefer METHOD=EXPO, which uses exponenial smoohing o fi a local linear or quadraic rend. For higher-order rends or for cases where he forecas variable Y is relaed o one or more explanaory variables X, PROC AUTOREG esimaes his relaionship and fis an AR series as an error erm. Polynomials in ime and seasonal indicaor variables (see Secion 1.3.2) can be compued as far ino he fuure as desired. If he explanaory variable is a nondeerminisic ime series, however, acual fuure values are no available. PROC AUTOREG reas fuure values of he explanaory variable as known, so user-supplied forecass of fuure values wih PROC AUTOREG may give incorrec sandard errors of forecas esimaes. More sophisicaed procedures like PROC STATESPACE, PROC VARMAX, or PROC ARIMA, wih heir ransfer funcion opions, are preferable when he explanaory variable's fuure values are unknown. One approach o modeling seasonaliy in ime series is he use of seasonal indicaor variables in PROC AUTOREG o model a highly regular seasonaliy. Also, he AR error series from PROC AUTOREG or from PROC FORECAST wih METHOD=STEPAR can include some correlaion a seasonal lags (ha is, i may relae he deviaion from rend a ime o he deviaion a ime 12 in monhly daa). The WINTERS mehod of PROC FORECAST uses updaing equaions similar o exponenial smoohing o fi a seasonal muliplicaive model. Anoher approach o seasonaliy is o remove i from he series and o forecas he seasonally adjused series wih oher seasonally adjused series used as inpus, if desired. The U.S. Census Bureau has adjused housands of series wih is X-11 seasonal adjusmen package. This package is he resul of years of work by census researchers and is he basis for he seasonally adjused figures ha he federal governmen repors. You can seasonally adjus your own daa using PROC X11, which is he census program se up as a SAS procedure. If you are using seasonally adjused figures as explanaory variables, his procedure is useful.

3 Chaper 1: Overview of Time Series 3 An alernaive o using X-11 is o model he seasonaliy as par of an ARIMA model or, if he seasonaliy is highly regular, o model i wih indicaor variables or rigonomeric funcions as explanaory variables. A final inroducory poin abou he PROC X11 program is ha i idenifies and adjuss for ouliers. * If you are unsure abou he presence of seasonaliy, you can use PROC SPECTRA o check for i; his procedure decomposes a series ino cyclical componens of various periodiciies. Monhly daa wih highly regular seasonaliy have a large ordinae a period 12 in he PROC SPECTRA oupu SAS daa se. Oher periodiciies, like muliyear business cycles, may appear in his analysis. PROC SPECTRA also provides a check on model residuals o see if hey exhibi cyclical paerns over ime. Ofen hese cyclical paerns are no found by oher procedures. Thus, i is good pracice o analyze residuals wih his procedure. Finally, PROC SPECTRA relaes an oupu ime series Y o one or more inpu or explanaory series X in erms of cycles. Specifically, cross-specral analysis esimaes he change in ampliude and phase when a cyclical componen of an inpu series is used o predic he corresponding componen of an oupu series. This enables he analys o separae long-erm movemens from shor-erm movemens. Wihou a doub, he mos powerful and sophisicaed mehodology for forecasing univariae series is he ARIMA modeling mehodology popularized by Box and Jenkins (1976). A flexible class of models is inroduced, and one member of he class is fi o he hisoric daa. Then he model is used o forecas he series. Seasonal daa can be accommodaed, and seasonaliy can be local; ha is, seasonaliy for monh may be closely relaed o seasonaliy for his same monh one or wo years previously bu less closely relaed o seasonaliy for his monh several years previously. Local rending and even long-erm upward or downward drifing in he daa can be accommodaed in ARIMA models hrough differencing. Explanaory ime series as inpus o a ransfer funcion model can also be accommodaed. Fuure values of nondeerminisic, independen inpu series can be forecas by PROC ARIMA, which, unlike he previously menioned procedures, accouns for he fac ha hese inpus are forecas when you compue predicion error variances and predicion limis for forecass. A relaively new procedure, PROC VARMAX, models vecor processes wih possible explanaory variables, he X in VARMAX. As in PROC STATESPACE, his approach assumes ha a each ime poin you observe a vecor of responses each enry of which depends on is own lagged values and lags of he oher vecor enries, bu unlike STATESPACE, VARMAX also allows explanaory variables X as well as coinegraion among he elemens of he response vecor. Coinegraion is an idea ha has become quie popular in recen economerics. The idea is ha each elemen of he response vecor migh be a nonsaionary process, one ha has no endency o reurn o a mean or deerminisic rend funcion, and ye one or more linear combinaions of he responses are saionary, remaining near some consan. An analogy is wo lifeboas adrif in a sormy sea bu ied ogeher by a rope. Their locaion migh be expressible mahemaically as a random walk wih no endency o reurn o a paricular poin. Over ime he boas drif arbirarily far from any paricular locaion. Neverheless, because hey are ied ogeher, he difference in heir posiions would never be oo far from 0. Prices of wo similar socks migh, over ime, vary according o a random walk wih no endency o reurn o a given mean, and ye if hey are indeed similar, heir price difference may no ge oo far from 0. * Recenly he Census Bureau has upgraded X-11, including an opion o exend he series using ARIMA models prior o applying he cenered filers used o deseasonalize he daa. The resuling X-12 is incorporaed as PROC X12 in SAS sofware.

4 4 SAS for Forecasing Time Series How SAS/ETS Sofware Procedures Inerrelae PROC ARIMA emulaes PROC AUTOREG if you choose no o model he inpus. ARIMA can also fi a richer error srucure. Specifically, he error srucure can be an auoregressive (AR), moving average (MA), or mixed-model srucure. PROC ARIMA can emulae PROC FORECAST wih METHOD=STEPAR if you use polynomial inpus and AR error specificaions. However, unlike FORECAST, ARIMA provides es saisics for he model parameers and checks model adequacy. PROC ARIMA can emulae PROC FORECAST wih METHOD=EXPO if you fi a moving average of order d o he dh difference of he daa. Insead of arbirarily choosing a smoohing consan, as necessary in PROC FORECAST METHOD=EXPO, he daa ell you wha smoohing consan o use when you invoke PROC ARIMA. Furhermore, PROC ARIMA produces more reasonable forecas inervals. In shor, PROC ARIMA does everyhing he simpler procedures do and does i beer. However, o benefi from his addiional flexibiliy and sophisicaion in sofware, you mus have enough experise and ime o analyze he series. You mus be able o idenify and specify he form of he ime series model using he auocorrelaions, parial auocorrelaions, inverse auocorrelaions, and cross-correlaions of he ime series. Laer chapers explain in deail wha hese erms mean and how o use hem. Once you idenify a model, fiing and forecasing are almos auomaic. The idenificaion process is more complicaed when you use inpu series. For proper idenificaion, he ARIMA mehodology requires ha inpus be independen of each oher and ha here be no feedback from he oupu series o he inpu series. For example, if he emperaure T in a room a ime is o be explained by curren and lagged furnace emperaures F, lack of feedback corresponds o here being no hermosa in he room. A hermosa causes he furnace emperaure o adjus o recen room emperaures. These ARIMA resricions may be unrealisic in many examples. You can use PROC STATESPACE and PROC VARMAX o model muliple ime series wihou hese resricions. Alhough PROC STATESPACE and PROC VARMAX are sophisicaed in heory, hey are easy o run in heir defaul mode. The heory allows you o model several ime series ogeher, accouning for relaionships of individual componen series wih curren and pas values of he oher series. Feedback and cross-correlaed inpu series are allowed. Unlike PROC ARIMA, PROC STATESPACE uses an informaion crierion o selec a model, hus eliminaing he difficul idenificaion process in PROC ARIMA. For example, you can pu daa on sales, adverising, unemploymen raes, and ineres raes ino he procedure and auomaically produce forecass of hese series. I is no necessary o inervene, bu you mus be cerain ha you have a propery known as saionariy in your series o obain heoreically valid resuls. The saionariy concep is discussed in Chaper 3, The General ARIMA Model, where you will learn how o make nonsaionary series saionary. Alhough he auomaic modeling in PROC STATESPACE sounds appealing, wo papers in he Proceedings of he Ninh Annual SAS Users Group Inernaional Conference (one by Bailey and he oher by Chavern) argue ha you should use such auomaed procedures cauiously. Chavern gives an example in which PROC STATESPACE, in is defaul mode, fails o give as accurae a forecas as a cerain vecor auoregression. (However, he saionariy of he daa is quesionable, and saionariy is required o use PROC STATESPACE appropriaely.) Bailey shows a PROC STATESPACE

5 Chaper 1: Overview of Time Series 5 forecas considerably beer han is compeiors in some ime inervals bu no in ohers. In SAS Views: SAS Applied Time Series Analysis and Forecasing, Brocklebank and Dickey generae daa from a simple MA model and feed hese daa ino PROC STATESPACE in he defaul mode. The dimension of he model is overesimaed when 50 observaions are used, bu he procedure is successful for samples of 100 and 500 observaions from his simple series. Thus, i is wise o consider inervening in he modeling procedure hrough PROC STATESPACE s conrol opions. If a ransfer funcion model is appropriae, PROC ARIMA is a viable alernaive. This chaper inroduces some echniques for analyzing and forecasing ime series and liss he SAS procedures for he appropriae compuaions. As you coninue reading he res of he book, you may wan o refer back o his chaper o clarify he relaionships among he various procedures. Figure 1.1 shows he inerrelaionships among he SAS/ETS sofware procedures menioned. Table 1.1 liss some common quesions and answers concerning he procedures. Figure 1.1 How SAS/ETS Sofware Procedures Inerrelae PROC VARMAX Mulivariae Models wih Random Inpus PROC STATESPACE Mulivariae Models wih Random Inpus PROC ARIMA Inervenion Models Transfer Funcion Models PROC FORECAST METHOD=EXPO Exponenial Smoohing Models PROC AUTOREG Auocorrelaed Residuals PROC FORECAST METHOD=STEPAR Time Series Errors

6 6 SAS for Forecasing Time Series Table 1.1 Seleced Quesions and Answers Concerning SAS/ETS Sofware Procedures Quesions 1. Is a frequency domain analysis (F) or ime domain analysis (T) conduced? 2. Are forecass auomaically generaed? 3. Do prediced values have 95% confidence limis? 4. Can you supply leading indicaor variables or explanaory variables? 5. Does he procedure run wih lile user inervenion? 6. Is minimal ime series background required for implemenaion? 7. Does he procedure handle series wih embedded missing values? Answers SAS/ETS Procedures FORECAST T Y Y N Y Y Y AUTOREG T Y* Y Y Y Y Y X11 T Y* N N Y Y N X12 T Y* Y Y Y N Y SPECTRA F N N N Y N N ARIMA T Y* Y Y N N N STATESPACE T Y Y* Y Y N N VARMAX T Y Y Y Y N N MODEL T Y* Y Y Y N Y Time Series Forecasing Sysem T Y Y Y Y Y Y * = requires user inervenion N = no = supplied by he program T = ime domain analysis F = frequency domain analysis Y = yes 1.3 Simple Models: Regression Linear Regression This secion inroduces linear regression, an elemenary bu common mehod of mahemaical modeling. Suppose ha a ime you observe Y. You also observe explanaory variables X 1, X 2, and so on. For example, Y could be sales in monh, X 1 could be adverising expendiure in monh, and X 2 could be compeiors' sales in monh. Oupu 1.1 shows a simple plo of monhly sales versus dae.

7 Chaper 1: Overview of Time Series 7 Oupu 1.1 Producing a Simple Plo of Monhly Daa A muliple linear regression model relaing he variables is Y = β + β X + β X +ε For his model, assume ha he errors ε have he same variance a all imes are uncorrelaed wih each oher ( ε and have a normal disribuion. ε are uncorrelaed for differen from s) s These assumpions allow you o use sandard regression mehodology, such as PROC REG or PROC GLM. For example, suppose you have 80 observaions and you issue he following saemens: TITLE PREDICTING SALES USING ADVERTISING ; TITLE2 EXPENDITURES AND COMPETITORS SALES ; PROC REG DATA=SALES; MODEL SALES=ADV COMP / DW; OUTPUT OUT=OUT1 P=P R=R; RUN;

8 8 SAS for Forecasing Time Series Oupu 1.2 shows he esimaes of β 0, β 1, and β 2❶. The sandard errors ❷ are incorrec if he assumpions on ε are no saisfied. You have creaed an oupu daa se called OUT1 and have called for he Durbin-Wason opion o check on hese error assumpions. Oupu 1.2 Performing a Muliple Regression PREDICTING SALES USING ADVERTISING EXPENDITURES AND COMPETITORS' SALES The REG Procedure Model: MODEL1 Dependen Variable: SALES Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model E E Error E C Toal E13 Roo MSE R-square Dep Mean Adj R-sq C.V Parameer Esimaes ❶ Parameer ❷ Sandard T for H0: Variable DF Esimae Error Parameer=0 Prob > T INTERCEP ADV COMP Durbin-Wason D ❸ (For Number of Obs.) 80 1s Order Auocorrelaion ❹ The es saisics produced by PROC REG are designed specifically o deec deparures from he null hypohesis ( H 0 : ε uncorrelaed) of he form H 1 : ε = ρε + 1 e where ρ < 1 and e is an uncorrelaed series. This ype of error erm, in which ε is relaed o ε 1, is called an AR (auoregressive) error of he firs order.

9 Chaper 1: Overview of Time Series 9 The Durbin-Wason opion in he MODEL saemen produces he Durbin-Wason es saisic ❸ where ( ) 2 n 2 ˆ ˆ / ˆ n d = Σ ε ε Σ ε = 2 1 = 1 ˆ ˆ ˆ ε ˆ = Y β 0 β 1X1 β 2X 2 2 If he acual errors ε are uncorrelaed, he numeraor of d has an expeced value of abou 2( n 1) σ 2 and he denominaor has an expeced value of approximaely n σ. Thus, if he errors ε are uncorrelaed, he raio d should be approximaely 2. Posiive auocorrelaion means ha ε is closer o ε 1 han in he independen case, so ε ε 1 should be smaller. I follows ha d should also be smaller. The smalles possible value for d is 0. If d is significanly less han 2, posiive auocorrelaion is presen. When is a Durbin-Wason saisic significan? The answer depends on he number of coefficiens in he regression and on he number of observaions. In his case, you have k=3 coefficiens ( β, β, and β for he inercep, ADV, and COMP) and n=80 observaions. In general, if you wan o es for posiive auocorrelaion a he 5% significance level, you mus compare d=2.046 o a criical value. Even wih k and n fixed, he criical value can vary depending on acual values of he independen variables. The resuls of Durbin and Wason imply ha if k=3 and n=80, he criical value mus be beween d L =1.59 and d U =1.69. If d is less han d L, hen you would rejec he null hypoheses of uncorrelaed errors in favor of he alernaive: posiive auocorrelaion. Since d>2, which is evidence of negaive auocorrelaion, compue d =4 d and compare he resuls o d L and d U. Specifically, because d (1.954) is greaer han 1.69, you are unable o rejec he null hypohesis of uncorrelaed errors. If d were less han 1.59 you would rejec he null hypohesis of uncorrelaed errors in favor of he alernaive: negaive auocorrelaion. Noe ha if 1.59 < d < 1.69 you canno be sure wheher d is o he lef or righ of he acual criical value c because you know only ha 1.59 < c < 1.69 Durbin and Wason have consruced ables of bounds for he criical values. Mos ables use k =k 1, which equals he number of explanaory variables, excluding he inercep and n (number of observaions) o obain he bounds d L and d U for any given regression (Draper and Smih 1998). * Three warnings apply o he Durbin-Wason es. Firs, i is designed o deec firs-order AR errors. Alhough his ype of auocorrelaion is only one possibiliy, i seems o be he mos common. The es has some power agains oher ypes of auocorrelaion. Second, he Durbin-Wason bounds do no hold when lagged values of he dependen variable appear on he righ side of he regression. Thus, if he example had used las monh's sales o help explain his monh's sales, you would no know correc bounds for he criical value. Third, if you incorrecly specify he model, he Durbin- Wason saisic ofen lies in he criical region even hough no real auocorrelaion is presen. Suppose an imporan variable, such as X 3 =produc availabiliy, had been omied in he sales example. This omission could produce a significan d. Some praciioners use d as a lack-of-fi saisic, which is jusified only if you assume a priori ha a correcly specified model canno have auocorrelaed errors and, hus, ha significance of d mus be due o lack of fi. * Exac p-values for d are now available in PROC AUTOREG as will be seen in Oupu 1.2A laer in his secion.

10 10 SAS for Forecasing Time Series The oupu also produced a firs-order auocorrelaion, ❹ denoed as ρ ˆ = When n is large and he errors are uncorrelaed, ( ) 1/2 ˆ 1/2 ˆ / 1 2 n ρ ρ is approximaely disribued as a sandard normal variae. Thus, a value ( ) 1/2 ˆ 1/2 ˆ / 1 2 n ρ ρ exceeding is significan evidence of posiive auocorrelaion a he 5% significance level. This is especially helpful when he number of observaions exceeds he larges in he Durbin-Wason able for example, 80 (.283)/ = You should use his es only for large n values. I is subjec o he hree warnings given for he Durbin-Wason es. Because of he approximae naure of he 1/2 ( 2 ) 1/2 ˆ / 1 ˆ Wason es is preferable. In general, d is approximaely ( 1 ρˆ ) This is easily seen by noing ha 2 ρ ˆ = εˆ εˆ / ˆ ε 1 2. n ρ ρ es, he Durbin- and d = ( εˆ εˆ ) / εˆ Durbin and Wason also gave a compuer-inensive way o compue exac p-values for heir es saisic d. This has been incorporaed in PROC AUTOREG. For he sales daa, you issue his code o fi a model for sales as a funcion of his-period and las-period adverising. PROC AUTOREG DATA=NCSALES; MODEL SALES=ADV ADV1 / DWPROB; RUN; The resuling Oupu 1.2A shows a significan d=.5427 (p-value.0001 <.05). Could his be because of an omied variable? Try he model wih compeior s sales included. PROC AUTOREG DATA=NCSALES; MODEL SALES=ADV ADV1 COMP / DWPROB; RUN; Now, in Oupu 1.2B, d = is insignifican (p-value.2239 >.05). Noe also he increase in R-square (he proporion of variaion explained by he model) from 39% o 82%. Wha is he effec of an increase of $1 in adverising expendiure? I gives a sales increase esimaed a $6.04 his period bu a decrease of $5.18 nex period. You wonder if he rue coefficiens on ADV and ADV1 are he same wih opposie signs; ha is, you wonder if hese coefficiens add o 0. If hey do, hen he increase we ge his period from adverising is followed by a decrease of equal magniude nex

11 Chaper 1: Overview of Time Series 11 period. This means our adverising dollar simply shifs he iming of sales raher han increasing he level of sales. Having no auocorrelaion eviden, you fi he model in PROC REG asking for a es ha he coefficiens of ADV and ADV1 add o 0. PROC REG DATA = SALES; MODEL SALES = ADV ADV1 COMP; TEMPR: TEST ADV+ADV1=0; RUN; Oupu 1.2C gives he resuls. Noice ha he regression is exacly ha given by PROC AUTOREG wih no NLAG= specified. The p-value (.077>.05) is no small enough o rejec he hypohesis ha he coefficiens are of equal magniude, and hus i is possible ha adverising jus shifs he iming, a emporary effec. Noe he label TEMPR on he es. Noe also ha, alhough we may have informaion on our company s plans o adverise, we would likely no know wha our compeior s sales will be in fuure monhs, so a bes we would have o subsiue esimaes of hese fuure values in forecasing our sales. I appears ha an increase of $1.00 in our compeior s sales is associaed wih a $0.56 decrease in our sales. From Oupu 1.2C he forecasing equaion is seen o be PREDICTED SALES = COMP ADV ADV1 Oupu 1.2A Predicing Sales from Adverising AUTOREG Procedure Dependen Variable = SALES Ordinary Leas Squares Esimaes SSE E9 DFE 77 MSE Roo MSE SBC AIC Reg Rsq Toal Rsq Durbin-Wason PROB<DW Variable DF B Value Sd Error Raio Approx Prob Inercep ADV ADV

12 12 SAS for Forecasing Time Series Oupu 1.2B Predicing Sales from Adverising and Compeior s Sales PREDICTING SALES USING ADVERTISING EXPENDITURES AND COMPETITOR'S SALES AUTOREG Procedure Dependen Variable = SALES Ordinary Leas Squares Esimaes SSE E9 DFE 76 MSE Roo MSE SBC AIC Reg Rsq Toal Rsq Durbin-Wason PROB<DW Variable DF B Value Sd Error Raio Approx Prob Inercep COMP ADV ADV Oupu 1.2C Predicing Sales from Adverising and Compeior s Sales PREDICTING SALES USING ADVERTISING EXPENDITURES AND COMPETITOR'S SALES Dependen Variable: SALES Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Toal Roo MSE R-square Dep Mean Adj R-sq C.V Parameer Esimaes Parameer Sandard T for H0: Variable DF Esimae Error Parameer=0 Prob > T INTERCEP COMP ADV ADV Durbin-Wason D (For Number of Obs.) 80 1s Order Auocorrelaion PREDICTING SALES USING ADVERTISING EXPENDITURES AND COMPETITOR'S SALES Dependen Variable: SALES Tes: TEMPR Numeraor: DF: 1 F value: Denominaor: DF: 76 Prob>F:

13 Chaper 1: Overview of Time Series Highly Regular Seasonaliy Occasionally, a very regular seasonaliy occurs in a series, such as an average monhly emperaure a a given locaion. In his case, you can model seasonaliy by compuing means. Specifically, he mean of all he January observaions esimaes he seasonal level for January. Similar means are used for oher monhs hroughou he year. An alernaive o compuing he welve means is o run a regression on monhly indicaor variables. An indicaor variable akes on values of 0 or 1. For he January indicaor, he 1s occur only for observaions made in January. You can compue an indicaor variable for each monh and regress Y on he welve indicaors wih no inercep. You can also regress Y on a column of 1s and eleven of he indicaor variables. The inercep now esimaes he level for he monh associaed wih he omied indicaor, and he coefficien of any indicaor column is added o he inercep o compue he seasonal level for ha monh. For furher illusraion, Oupu 1.3 shows a series of quarerly increases in Norh Carolina reail sales; ha is, each poin is he sales for ha quarer minus he sales for he previous quarer. Oupu 1.4 shows a plo of he monhly sales hrough ime. Quarerly sales were compued as averages of hree consecuive monhs and are used here o make he presenaion brief. A model for he monhly daa will be shown in Chaper 4. Noe ha here is a srong seasonal paern here and perhaps a mild rend over ime. The change daa are ploed in Oupu 1.6. To model he seasonaliy, use S1, S2, and S3, and for he rend, use ime, T1, and is square T2. The S variables are ofen referred o as indicaor variables, being indicaors of he season, or dummy variables. The firs CHANGE value is missing because he sales daa sar in quarer 1 of 1983 so no increase can be compued for ha quarer. Oupu 1.3 Displaying Norh Carolina Reail Sales Daa Se OBS DATE CHANGE S1 S2 S3 T1 T2 1 83Q Q Q Q Q (More Oupu Lines) 47 94Q Q

14 14 SAS for Forecasing Time Series Oupu 1.4 Ploing Norh Carolina Monhly Sales Now issue hese commands: PROC AUTOREG DATA=ALL; MODEL CHANGE = T1 T2 S1 S2 S3 / DWPROB; RUN;

15 Chaper 1: Overview of Time Series 15 This gives Oupu 1.5. Oupu 1.5 Using PROC AUTOREG o Ge he Durbin- Wason Tes Saisic Dependen Variable = CHANGE AUTOREG Procedure Ordinary Leas Squares Esimaes SSE DFE 41 MSE Roo MSE SBC AIC Reg Rsq Toal Rsq Durbin-Wason PROB<DW Variable DF B Value Sd Error Raio Approx Prob Inercep T T S S S PROC AUTOREG is inended for regression models wih auoregressive errors. An example of a model wih auoregressive errors is where Y = β 0 + β 1 X 1 + β2 X 2 + Z Z = ρ Z 1 + g Noe how he error erm Z is relaed o a lagged value of iself in an equaion ha resembles a regression equaion; hence he erm auoregressive. The erm g represens he porion of Z ha could no have been prediced from previous Z values and is ofen called an unanicipaed shock or whie noise. I is assumed ha he e series is independen and idenically disribued. This one lag error model is fi using he /NAG=1 opion in he MODEL saemen. Alernaively, he opions /NLAG=5 BACKSTEP can be used o ry 5 lags of Z, auomaically deleing hose deemed saisically insignifican. Our reail sales change daa require no auocorrelaion adjusmen. The Durbin-Wason es has a p-value >0.05; so here is no evidence of auocorrelaion in he errors. The fiing of he model is he same as in PROC REG because no NLAG specificaion was issued in he MODEL saemen. The parameer esimaes are inerpreed jus as hey would be in PROC REG; ha is, he prediced change PC in quarer 4 (where S1=S2=S3=0) is given by PC =

16 16 SAS for Forecasing Time Series and in quarer 1 (where S1=1, S2=S3=0) is given by PC = ec. Thus he coefficiens of S1, S2, and S3 represen shifs in he quadraic polynomial associaed wih he firs hrough hird quarers and he remaining coefficiens calibrae he quadraic funcion o he fourh quarer level. In Oupu 1.6 he daa are dos, and he fourh quarer quadraic predicing funcion is he smooh curve. Verical lines exend from he quadraic, indicaing he seasonal shifs required for he oher hree quarers. The broken line gives he predicions. The las daa poin for 1994Q4 is indicaed wih an exended verical line. Noice ha he shif for any quarer is he same every year. This is a propery of he dummy variable model and may no be reasonable for some daa; for example, someimes seasonaliy is slowly changing over a period of years. Oupu 1.6 Ploing Quarerly Sales Increase wih Quadraic Predicing Funcion To forecas ino he fuure, exrapolae he linear and quadraic erms and he seasonal dummy variables he requisie number of periods. The daa se exra lised in Oupu 1.7 conains such values. Noice ha here is no quesion abou he fuure values of hese, unlike he case of compeior s sales ha was considered in an earlier example. The PROC AUTOREG echnology assumes perfecly known fuure values of he explanaory variables. Se he response variable, CHANGE, o missing.

17 Chaper 1: Overview of Time Series 17 Oupu 1.7 Daa Appended for Forecasing OBS DATE CHANGE S1 S2 S3 T1 T2 1 95Q Q Q Q Q Q Q Q Combine he original daa se call i NCSALES wih he daa se EXTRA as follows: DATA ALL; SET NCSALES EXTRA; RUN; Now run PROC AUTOREG on he combined daa, noing ha he exra daa canno conribue o he esimaion of he model parameers since CHANGE is missing. The exra daa have full informaion on he explanaory variables and so prediced values (forecass) will be produced. The prediced values P are oupu ino a daa se OUT1 using his saemen in PROC AUTOREG: OUTPUT OUT=OUT1 PM=P; Using PM= requess ha he prediced values be compued only from he regression funcion wihou forecasing he error erm Z. If NLAG= is specified, a model is fi o he regression residuals and his model can be used o forecas residuals ino he fuure. Replacing PM= wih P= adds forecass of fuure Z values o he forecas of he regression funcion. The wo ypes of forecas, wih and wihou forecasing he residuals, poin ou he fac ha par of he predicabiliy comes from he explanaory variables, and par comes from he auocorrelaion ha is, from he momenum of he series. Thus, as seen in Oupu 1.5, here is a oal R-square and a regression R-square, he laer measuring he predicabiliy associaed wih he explanaory variables apar from conribuions due o auocorrelaion. Of course in he curren example, wih no auoregressive lags specified, hese are he same and P= and PM= creae he same variable. The prediced values from PROC AUTOREG using daa se ALL are displayed in Oupu 1.8.

18 18 SAS for Forecasing Time Series Oupu 1.8 Ploing Quarerly Sales Increase wih Predicion Because his example shows no residual auocorrelaion, analysis in PROC REG would be appropriae. Using he daa se wih he exended explanaory variables, add P and CLI o produce prediced values and associaed predicion inervals. PROC REG; MODEL CHANGE = T T2 S1 S2 S3 / P CLI; TITLE QUARTERLY SALES INCREASE ; RUN;

19 Chaper 1: Overview of Time Series 19 Oupu 1.9 Producing Forecass and Predicion Inervals wih he P and CLI Opions in he Model Saemen QUARTERLY SALES INCREASE Dependen Variable: CHANGE Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Toal Roo MSE R-square Dep Mean Adj R-sq C.V Parameer Esimaes Parameer Sandard T for H0: Variable DF Esimae Error Parameer=0 Prob > T INTERCEP T T S S S Quarerly Sales Increase Dep Var Predic Sd Err Lower95% Upper95% Obs CHANGE Value Predic Predic Predic Residual (more oupu lines) Sum of Residuals 0 Sum of Squared Residuals Prediced Resid SS (Press) For observaion 49 an increase in sales of (i.e., a decrease) is prediced for he nex quarer wih confidence inerval exending from o This is he ypical afer-chrismas sales slump.

20 20 SAS for Forecasing Time Series Wha does his sales change model say abou he level of sales, and why were he levels of sales no used in he analysis? Firs, noice ha a cubic erm in ime, b 3, when differenced becomes a quadraic erm: b 3 b( 1) 3 = b( ). Thus a quadraic plus seasonal model in he differences is associaed wih a cubic plus seasonal model in he levels. However if he error erm in he differences saisfies he usual regression assumpions, which i seems o do for hese daa, hen he error erm in he original levels can possibly saisfy hem he levels appear o have a nonsaionary error erm. Ordinary regression saisics are invalid on he original level series. If you ignore his, he usual (incorrec here) regression saisics indicae ha a degree 8 polynomial is required o ge a good fi. A plo of sales and he forecass from polynomials of varying degree is shown in Oupu The firs hing o noe is ha he degree 8 polynomial, arrived a by inappropriae use of ordinary regression, gives a ridiculous forecas ha exends verically beyond he range of our graph jus a few quarers ino he fuure. The degree 3 polynomial seems o give a reasonable increase while he inermediae degree 6 polynomial acually forecass a decrease. I is dangerous o forecas oo far ino he fuure using polynomials, especially hose of high degree. Time series models specifically designed for nonsaionary daa will be discussed laer. In summary, he differenced daa seem o saisfy assumpions needed o jusify regression. Oupu 1.10 Ploing Sales and Forecass of Polynomials of Varying Degree

21 Chaper 1: Overview of Time Series Regression wih Transformed Daa Ofen, you analyze some ransformed version of he daa raher han he original daa. The logarihmic ransformaion is probably he mos common and is he only ransformaion discussed in his book. Box and Cox (1964) sugges a family of ransformaions and a mehod of using he daa o selec one of hem. This is discussed in he ime series conex in Box and Jenkins (1976, 1994). Consider he following model: Y X ( ) = β β 0 1 ε Taking logarihms on boh sides, you obain Now if ( ) = ( β ) + ( β ) + ( ε ) log Y log log X log η = log( ε ) 0 1 and if η saisfies he sandard regression assumpions, he regression of log(y ) on 1 and X produces he bes esimaes of log( β ) and log( β 0 1 ). As before, if he daa consis of (X 1, Y 1 ), (X 2, Y 2 ),..., (X n, Y n ), you can append fuure known values X n+1, X n+2,..., X n+s o he daa if hey are available. Se Y n+1 hrough Y n+s o missing values (.). Now use he MODEL saemen in PROC REG: MODEL LY=X / P CLI; where LY=LOG(Y); is specified in he DATA sep. This produces predicions of fuure LY values and predicion limis for hem. If, for example, you obain an inerval 1.13 < log(y n+s ) < 2.7 you can compue exp( 1.13) =.323 and exp(2.7) = o conclude.323 < Y n+s < Noe ha he original predicion inerval had o be compued on he log scale, he only scale on which you can jusify a disribuion or normal disribuion. When should you use logarihms? A quick check is o plo Y agains X. When Y X ( ) = β β 0 1 ε he overall shape of he plo resembles ha of X ( ) Y = β β 0 1

22 22 SAS for Forecasing Time Series See Oupu 1.11 for several examples of his ype of plo. Noe ha he curvaure in he plo becomes more dramaic as β moves away from 1 in eiher direcion; he acual poins are scaered 1 X around he appropriae curve. Because he error erm ε is muliplied by β0( β 1), he variaion around he curve is greaer a he higher poins and lesser a he lower poins on he curve. Oupu 1.11 Ploing Exponenial Curves Oupu 1.12 shows a plo of U.S. Treasury bill raes agains ime. The curvaure and especially he variabiliy displayed are similar o hose jus described. In his case, you simply have X =. A plo of he logarihm of he raes appears in Oupu Because his plo is sraigher wih more uniform variabiliy, you decide o analyze he logarihms.

23 Chaper 1: Overview of Time Series 23 Oupu 1.12 Ploing Niney- Day Treasury Bill Raes Oupu 1.13 Ploing Niney- Day Logged Treasury Bill Raes

24 24 SAS for Forecasing Time Series To analyze and forecas he series wih simple regression, you firs creae a daa se wih fuure values of ime: DATA TBILLS2; SET TBILLS END=EOF; TIME+1; OUTPUT; IF EOF THEN DO I=1 TO 24; LFYGM3=.; TIME+1; DATE=INTNX('MONTH',DATE,1); OUTPUT; END; DROP I; RUN; Oupu 1.14 shows he las 24 observaions of he daa se TBILLS2. You hen regress he log T-bill rae, LFYGM3, on TIME o esimae log( β 0) and log( β 1) in he following model: ( ) ( ) ( ) LFYGM3 log log *TIME log = β + β + ε 0 1 You also produce prediced values and check for auocorrelaion by using hese SAS saemens: PROC REG DATA=TBILLS2; MODEL LFYGM3=TIME / DW P CLI; ID DATE; TITLE 'CITIBASE/CITIBANK ECONOMIC DATABASE'; TITLE2 'REGRESSION WITH TRANSFORMED DATA'; RUN; The resul is shown in Oupu Oupu 1.14 Displaying Fuure Dae Values for U.S. Treasury Bill Daa CITIBASE/CITIBANK ECONOMIC DATABASE OBS DATE LFYGM3 TIME 1 NOV DEC JAN FEB MAR (More Oupu Lines) 20 JUN JUL AUG SEP OCT

25 Chaper 1: Overview of Time Series 25 Oupu 1.15 Producing Prediced Values and Checking Auocorrelaion wih he P, CLI, and DW Opions in he MODEL Saemen CITIBASE/CITIBANK ECONOMIC DATABASE REGRESSION WITH TRANSFORMED DATA Dependen Variable: LFYGM3 Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Toal Roo MSE R-square Dep Mean Adj R-sq C.V Parameer Esimaes Parameer Sandard T for H0: Variable DF Esimae Error Parameer=0 Prob > T INTERCEP TIME REGRESSION WITH TRANSFORMED DATA Dep Var Predic Sd Err Lower95% Upper95% Obs DATE LFYGM3 Value Predic Predic Predic Residual 1 JAN FEB MAR APR MAY (More Oupu Lines) 251 NOV (More Oupu Lines) 270 JUN JUL AUG SEP OCT Sum of Residuals 0 Sum of Squared Residuals Prediced Resid SS (Press) DURBIN-WATSON D (FOR NUMBER OF OBS.) 1ST ORDER AUTOCORRELATION ❶ 250 ❷ ❸

26 26 SAS for Forecasing Time Series Now, for example, you compue: Thus, ( )( ) ( ) ( )( ) < log β < < β < is a 95% confidence inerval for β 0. Similarly, you obain < β < which is a 95% confidence inerval for β 1. The growh rae of Treasury bills is esimaed from his model o be beween 0.46% and 0.54% per ime period. Your forecas for November 1982 can be obained from < < so ha 6.61 < FYGM3 251 < is a 95% predicion inerval for he November 1982 yield and exp(2.377) = is he prediced value. Because he disribuion on he original levels is highly skewed, he predicion does no lie midway beween 6.61 and 17.55, nor would you wan i o do so. Noe ha he Durbin-Wason saisic is d= However, because n=250 is beyond he range of he Durbin-Wason ables, you use ρ ˆ = o compue ( ) 1/ 2 ˆ ρ ρ = n 1/ 2 ˆ / 1 2 which is greaer han A he 5% level, you can conclude ha posiive auocorrelaion is presen (or ha your model is misspecified in some oher way). This is also eviden in he plo, in Oupu 1.13, in which he daa flucuae around he overall rend in a clearly dependen fashion. Therefore, you should recompue your forecass and confidence inervals using some of he mehods in his book ha consider auocorrelaion. 2 Suppose X=log(y) and X is normal wih mean M x and variance σ. Then y = exp(x) and y has x median exp(m x ) and mean exp(m x + ½σ 2 ) For his reason, some auhors sugges adding half he x error variances o a log scale forecas prior o exponeniaion. We prefer o simply exponeniae and hink of he resul, for example, exp(2.377) = 10.77, as an esimae of he median, reasoning ha his is a more credible cenral esimae for such a highly skewed disribuion.