NTE N A CASE-STUDY IN B-JENKINS SEASNAL FRECASTING F TIME SERIES BY STEFFEN L. LAURITZEN TECHNICAL REPRT N. 16 APRIL 1974 PREPARED UNDER CNTRACT N00014-67-A-0112-0030 (NR-042-034) FR THE FFICE F NAVAL RESEARCH THEDRE W. ANDERSN, PRJECT DIRECTR DEPARTMENT F STATISTICS STANFRD UNIVERSITY STANFRD, CALIFRNIA
NTE N A CASE-STUDY IN B-JENKINS SEASNAL FRECASTING F TIME SERIES STEFFEN L. LAURITZEN STANFRD UNIVERSITY TECHNICAL REPRT N. l6 APRIL 197^ PREPARED UNDER THE AUSPICES F FFICE F NAVAL RESEARCH CNTRACT N0001U-6T-A-0112-0030 T.. ANDERSN, PRJECT DIRECTR DEPARTMENT F STATISTICS STANFRD UNIVERSITY STANFRD, CALIFRNIA
_1_ Recently, the Jurnal f the Ryal Statistical Sciety brught a lively discussin f the Bx-Jenkins prcedure fr frecasting seasnal time series. This tk place in tw papers, ne by Chatfield and Prther (l973) and ne by Bx and Jenkins (1973). The discussin was centered arund a particular case study f sme sales figures. Bx and Jenkins refer t several ther case-studies in their reply, including a paper n telephne installatins and remvals by Thmpsn and Tia (l97l) The riginal data in this analysis are nt in the paper, but can be fund in the preceding technical reprt frm University f Wiscnsin [Thmpsn and Tia (1970)]. The present authr had sme time ag an pprtunity t lk at these data and the findings might be f sme general interest in view f the recent discussin. The data cnsist f the mnthly ttals f telephne installatins and remvals made by the Wiscnsin Telephne Cmpany frm January 1951 thrugh ctber 1966. The bjective f the analysis is t btain gd frecasts fr the tw series. The authrs cnsider the lgarithms f the tw series, respectively dented by and Y. Via the prcedure described in the bk by Bx and Jenkins (1970) ARIMA mdels are btained fr Z =, - Y and Y. The parameters are estimated and frecasts made accrding t the estimated mdels. The frecasts f bth series frm Nvember 1966 thrugh ctber 1968 are cmpared t actual bservatins with quite gd results. Fllwing the standard ntatin the mdels fr the Z and Y series were as fllws: (1 - B 12 ) (1 - ^B) Z t = (1 - e^b 12 ) (1 - e^b) a t, (l) where a, are uncrrelated randm disturbances and
(1 - (J) 3 B 3 ) (1 - * 12 B 12 ) Y t = (1-6 9 B 9-6 12 B 12-6 13 B 13 )b t, (2) The mdel (2) seems t be quite cmplicated, and (als suggested by Thmpsn and Tia) calendar irregularities might be the reasn. If ne cnsiders the number f installatins r remvals in any given mnth t be rughly prprtinal t the number f weekdays in that mnth, the effect f the calendar irregularities wuld be present in the Y -series but nt in the Z -series as this invlves nly the rati between the number f installatins and remvals. 3D as a simple methd f frecasting the number f remvals in any given mnth ne might suggest that the average daily number f remvals made in that mnth wuld increase by a fixed percentage ver the crrespnding number in the same mnth f the preceding year. That is, if we dente the number f weekdays in mnth t by d, rfe culd let the frecast f Y, 1 <_ j <^ 12, frm Y,,Y be Wk + j = Y T + j-12 " lq S Vj-12 + l0s WfcfJ + (k+l) (.3) Averaging ver (Y - lg d ) = (Y,, - lg d _) suggests that 0 = 0.0^9 wuld be a gd value fr 9, crrespnding t a 5% yearly increase. The authr tried t d exactly as in the paper by Thmpsn and Tia apart frm substituting the "naive" methd (3) fr the frecasts btained frm the mdel (2). That is Z was frecasted frm the mdel (l) and Y by (3) and then = Y + Z frecasted as "0 "C ~G Tables 1 and 2 shw the percentage errr f frecasts fr the ttal number f installatins and remvals, respectively, btained by the tw
-3- methds fr the 2k mnths frm Nvember 1966 thrugh ctber 1968. These data are pltted n Figures 1 and 2. Table 1 Percentage Errr in Frecasts f the Number f Telephne Remvals. Nv. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. ct. Perid 1966 196T % errr by riginal methd +6 +6-5 +3 +15 +12 -k +1 +3 +7 +^ -7 # errr by naive methd 0-1 0+1 +6 +5 0 +1 +2 +8 0-2 Nv. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. ct. Perid 1968 % errr by riginal methd +8 +6-10 -5 +lh +lk -12 +7-12 +9 +7 0 $ errr by naive methd +8-10 -5-7 +2 +17-8 +8 -h +5 +3 +9
-h- Table 2 Percentage Errr in Frecasts f the Number f Telephne Installatins. Nv. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. ct. Fend 1966 1967 % errr by riginal methd +9 +5 +2 +5 +9 +10 +2-2 +7 +7 +6-6 % errr by naive methd +3-2 +7 +3 0 +3 +6-2 +6 +8 +2-1. n Nv. Dec. Jan. Feb. Mar. Apr. May June July Aug. Sept. ct. Perid 196T 1968 % errr by riginal methd +8 +1-3 -2 +lk +26-10 +lk -11 0 +6-6 $ errr by naive methd +8-15 +2 -U +2 +29-6 +15-3 - 1 * +2 +3 It is surprising but bvius that the "naive" methd seems t give much better results fr the frecasts f the remvals and it des nt d wrse fr the frecasts f the number f installatins. Nne f the methds culd frecast the strike in April 1968.
-5- +20 - +15 - +10 - +5 - % errr Jan. 67 0 CD x <p * t * & -10.. i 0 riginal methd naive methd Jan. 68 <8> * Q 1 1 1 I» 1 f < ( 1 t * ) 1 V time -15 +20.. +15 +10.. +5 Fig. 1. $ Frecast Errr in Frecasts f Number f Remvals Jan. 67 Jan. 68 x x 0 -I,!_ x - riginal methd 0 naive methd «s D & * 9 ^ I,,, f t +-, 1+ { I -, «I I 1 I I I 1 -p. time. b 0-10: -ir F ig. 2 # Frecast Errr in Frecasts f Number f Installat ins
REFERENCES Bx, G. E. P., and G. M. Jenkins (1970). Time Series Analysis, Frecasting and Cntrl, Hlden-Day, San Francisc. Bx, G. E. P., and G. M. Jenkins (1973). Sme Cmments n a paper by Chatfield and Prther and n a Review by Kendall. J.R. Statist. Sc. A, 136, part 3, 337-352. Chatfield, C, and D. L. Prther (1973). Bx-Jenkins seasnal frecasting: prblems in a case study (with Discussin). J.R. Statist. Sc. A., 136, 295-336. Thmpsn, H.E., and G. C. Tia (1971). Analysis f Telephne Data. Bell J. Ecn. Manag. Sei., 2_, 51^-5^1 Thmpsn, H. E., and G. C. Tia (1970). Analysis f Telephne Data. Tech. Reprt W. 222, Department f Statistics, University f Wiscnsin.
-7- TECHNICAL REPRTS FFICE F NAVAL RESEARCH CNTRACT N0001U-67-A-0112-0030 (NR-l+2-03 1^) 1. "Cnfidence Limits fr the Expected Value f an Arbitrary. Bunded Randm Variable with a Cntinuus Distributin Functin," T.. Andersn, ctber 1, 1969. 2. "Efficient Estimatin f Regressin Cefficients in Time Series," T. W. Andersn, ctber 1, 1970. 3. "Determining the Apprpriate Sample Size fr Cnfidence Limits fr a Prprtin," T. W. Andersn and H. Burstein, ctber 15, 1970. k. "Sme General Results n Time-rdered Classificatin," D. V. Hinkley, July 30, 1971. 5. "Tests fr Randmness f Directins against Equatrial and Bimdal Alternatives," T. W. Andersn and M. A. Stephens, August 30, 1971. 6. "Estimatin f Cvariance Matrices with Linear Structure and Mving Average Prcesses f Finite rder," T. W. Andersn, ctber 29, 1971. 7. "The Statinarity f an Estimated Autregressive Prcess," T. W. Andersn, Nvember 15, 1971-8. "n the Inverse f Sme Cvariance Matrices f Teplitz Type," Raul Pedr Mentz, July 12, 1972. 9. "An Asympttic Expansin f the Distributin f "Studentized" Classificatin Statistics," T.. Andersn, September 10, 1972. 10. "Asympttic Evaluatin f the Prbabilities f Misclassificatin by Linear Discriminant Functins," T. W. Andersn, September 28, 1972. 11. "Ppulatin Mixing Mdels and Clustering Algrithms," Stanley L. Sclve, February 1, 1973. 12. "Asympttic Prperties and Cmputatin f Maximum Likelihd Estimates in the Mixed Mdel f the Analysis f Variance," Jhn James Miller, Nvember 21, 1973. 13. "Maximum Likelihd Estimatin in the Birth-and-Death Prcess," Niels Keiding, Nvember 28, 1973. li+. "Randm rthgnal Set Functins and Stchastic Mdels fr the Gravity Ptential f the Earth," Steffen L. Lauritzen, December 27, 1973. 15. "Maximum Likelihd Estimatin f Parameters f an Autregressive Prcess with Mving Average Residuals- and ther Cvariance' Matrices with Linear Structure," T. W. Andersn, December, 1973. 16. "Nte n a Case-Study in Bx-Jenkins Seasnal Frecasting f Time Series," Steffen L. Lauritzen, April, 197^.
UNCLASSIFIED SECURITY CLASSIFICATIN F THIS PAGE (When Data Entered) REPRT DCUMENTATIN PAGE 1. REPRT NUMBER Technical Reprt N. l6 4. TITLE fand Subtitle) NTE N A CASE-STUDY IN B-JENKINS SEASNAL FRECASTING F TIME SERIES READ INSTRUCTINS BEFRE CMPLETING FRM 2. GVT ACCESSIN N. 3. RECIPIENT'S CATALG NUMBER S. TYPE F REPRT a PERID CVERED TECHNICAL REPRT * PERFRMING RG. REPRT NUMBER 7. AUTHRf Steffen L. Lauritzen CNTRACT R GRANT NUMBERf») NIU-67-A-II2-3 9. PERFRMING RGANIZATIN NAME AND ARESS Department f Statistics Stanfrd University Stanfrd, Califrnia 9^305 II. CNTRLLING FFICE NAME AND ADDRESS ffice f Naval Research Statistics & Prbability Prgram Cde ij-36 Arlingtn, Virginia 22217 10. PRGRAM ELEMENT, PRJECT, TASK AREA a WRK UNIT NUMBERS NR-0U2-03^ 12. REPRT DATE APRIL 11, 197U 13. NUMBER F PAGES JL 14. MNITRING AGENCY NAME & ADDRESSf/f dllterent frm Cntrlling ttlce) IS. SECURITY CLASS, (l thle reprt) 16. DISTRIBUTIN STATEMENT (l this Reprt) Unclassified ISa. DECLASSIFI CATIN/DWNGRADING SCHEDULE Reprductin in whle r in part is permitted fr any purpse f the United States Gvernment. Distributin is unlimited. 17. DISTRIBUTIN STATEMENT (l the abstract entered In Blck 20, It dlllerent frm Reprt) t8. SUPPLEMENTARY NTES 19. KEY WRDS (Cnttnu n reverse aide II neceeeary and Identify by blck number) 1. time series 2. autregressive-mving average mdels 3. frecasting 20. ABSTRACT (Cntinue n reverse tide It necessary and Identity by blck number) It is shwn that in a particular example a straight frward frecasting prcedure gives better results than frecasts btained frm a fitted ARIMA mdel. DD,^73 1473 EDITIN F I NV 8S IS BSLETE S/N 0102-014- 6601! Unclassified SECURITY CLASSIFICATIN F THlS>AGE (When Dmf Entered)