為因應預測目的不同，有時需要各種預測水平{\rm (forecast horizon)}

，例如，月預測可供進料、生產、補貨及倉儲之參考，年預測則可作為產

能規畫、產品線規畫、投資決策等之準則。然而，預測結構卻會因水平的

不同而彼此相異，以致產生諸多預測值的矛盾。有鑑於此，本研究主要以

一簡單且具理論基礎的整合{\rm intergration)} 過程，解決預測值互相

矛盾的問題。由於年資料通常屬於連續性模式，月資料則多為季節性模式

，兩者透過的轉移函數形態截然不同，而且在解釋變數的選取上更是迥異

，因此，需要經由加權平均的整合，才能使月預測值的加總等於年預測值

。至於權數的決定則以離散程度為準則，由於年資料為月資料的加總，兩

者均值相差甚多，故以變異係數為測量離散情形的標準。本研究主要乃遵

循{\rm Box-Jenkins} 的模式建立法則，構建連續性轉移函數模式及季節

性、轉移函數模式，並加以整合調整。在實證分析中以台灣啤酒銷售量為

例說明預測流程，年銷量預測方面以國民所得為解釋變數; 月銷量預測方

面則以氣溫為解釋變數，最後以加權平均將兩者整合調整。

(1). 魏庭、澤(1982), 數理統計學
(2). 徐瑞玲(1988),『時間數列模型建立各種分析方法的比較與實證研
究』,政治大學統計研究所
(3). 顏月珠(1990), 商用統計學y 台北:三民。
(4). 周文賢及游文清(1990), 計量經濟與時間序列分析/SASETS之運
用, 台北, 教育部電算中心。
(5). 陳儀庭(1992), 『進出口貿易預測系統之建立運用-以個案為例』,
政治大學統計研究所

(1 )Abrahan1,B. , 'Seasonal time series and transfer fun ction modeling', J o'l/,rnal
of Business & Economic Statistics~3(1985) , No.4)56-361.
(2)Abrahanl,B. and Box,G.E.P.,'Bayesian analysis of SOllle outlier problems
in time series',Biom,ertrika,66(1979),229-236.
(3 )Akaike,H.' A llew look at statistical IDodel identification' ,LEEE Transactions
on Automatic Contral,AC-19(1974 ))16-723.
(4)Ansley,C.F.,'An algorithnl for the exact likelihood of a nlixed autoregressive
lll oving a'verage process' ,Biometrika, 66(1979) ,59-65.
(5)Ansley,C.F.and Newbold,P.,'On the bias in estinlates of forecast mean
squared error',Journal of American Statistz:cal Association, 76(1981),
569-578.
(6)Beguin,J .11. ,Gourieroux, C. and 110nfort,A., 'Identification of a Inixed autoregressive
moving average process:the corner method',in O.D.Anderson
(e.d. ), Journal of time series,Amsterdan1:N orth Holland, 1980.
(7)Bell,vV.R.,'An introduction to forecasting with time series 1110dels', Mathematics
and Econom,ics,3(1984),241-255.
(8)Bhansali,R.J.,'Autoregressive and \iVindo"\v Estimates of the Inverse Correlation
Function',B1:om.etrika, 67(1980),551-566.
(9)Binder,D.A.and Dick,J.P.,'l1nplications of survey designs for estilnating
seasonal ARl11A models',ASA Proceeding of Survey Resea.rch Methods
S ection,(1989) ,11-15 .
(10)Bowerman,B.L.and Jobe).11.,'Using Box-Jenkins ARII\1A Inodels to unify
the teaching of applied forecasting methodologies' ,ASA Proceeding of
Sta.tistical Ed'u.cation S ection,( 1987) ,131-134
(11)Box,G.E.P. and Jenkins,G.I\/I.,' Ti'me Series Analusis Forecasting and control',
San Francisco: Holden-Day (1976).
(12)Box,G.E.P. and Tiao,G.C.,'Intervention analysis with applications to economic
and environmental proble111', J o 'U.rnal of American Statistical
79.
(13)Box,G.E.P.and Pierce ,D.A., 'Distribution of residual autocorrelation in
autoregressive integrated moving average time series 1110dels' , Journal of
Am.erican Statistical Association,65(1970))509-1526.
(14)Burtschy,B.,'Variable seasonal transfer function',ASA proceedings of Busi-,
ness and Econom,ic st(J.tisi1:cs,6 (1985) ,614-618.
(15 )Chatfield, C., 'Inverse autocorrelations" J ouranl of Royal Statistical S oci-
ety, Ser.A,142(1979),363-377.
(16) Cleveland, vV.S., 'The inverse autocorrelations of a tilne series and thier
applications', Technometrics,14( 1972) ,277-298.
(17)Cleveland,\~T.P.and Tiao G.C.,'110deling seasonal tilne series', Economic
Appliq1tee,32(1979),107-129.
(18)Davies,N.and Petruccelli,J.D.'On the use of the general partial autocorrelation
frnction for order determintion in AR11A(p,q) processes' , Journal
American Sta.tistical Association, 79(1984),No.386,374-377.
(19)Davies,N. ,Triggs, C.11.and N ewbold,P. ,'Significance levels of the Box-Pierce'
potmanteau statistic in finite samples' ,Biometrika,64( 1977) ,517 -522.
(20)Durbin,J.,' The Fitting of Tim.e Series Models' ,Rev.Int.Inst.Stat.28(1960),
(21)Engle,Granger and Hallnlan,' 1/Ierging short- and long-Run forecasts an
application of seasonal cointegration to nlonthly electricity sales forecasting',
Journal of Econom.etrics,40(1989), 45-62.
(22 )Franke,J.,' A Levinson-Durbin recursion for autoregressive-moving average
processes' ,Biometn:/;;(J., 72( 1985) ,3,5 71-58l.
(23)Godfrey,L. G.) 'Testing the adequacy of a time series model' ,Bi07netrika,66
7')
(24)Gray,H.L.)(elley,G.D.and 11Icintire,D.D.,'A new approach to AR11A 1110delling',
Commv,nications in Sta.tistics ,B7(1978),l-77.
(25)Guerrero, V,lvl., 'Optimal conditional ARIj\/IA forecasts', J ourna.l of Foreca.
sting,8 (1989) ,215- 229.
(26)Guerrero,V,,'ARI11A forecasts with restrictions derived frOl11 a structural
change',Internation Journa.l of Forecasting,7(1991), 339-347.
(27)Hamil ton,D . C.and \Vatts,D. G., 'Interpreting patial autocorrelation func-
Tion of seasonal time series models’,Biometrika,65(1978), 135-140.
(28)Hannan,E.J.and Kavalieris,L.,’A method for autoregressive-moving average estimation’, Biometrika, 72(1984),2, 273-280.
(29)Hannan,E.J. and Rissanen,J.’Recursive estimation of mixed autoregressive moving average order’, Biometrika, 69(1982),1,81-94.
(30)Hillmer, S.C.and Tiao,G.C. Likelihood function of stationary multiple autoregressive moving average models’ , Journal of American Statistical 660
(31)Hosking,J.R.M.,’Lagrange multiplier test of time series models’ , Journal of Royal Statistical Society B, 42(1980), 170-181.
(32)Jenkins,G.M. and Alavi,A.S.,’Some aspects of modeling and ‘, Journal of Time Series Analysis, 2(1981), 1-47.
(33)Kashyap,R.L.’Inconsistency od the AIC rule for estimating the order if autoregressibe models’ ,LEEE Transactions on Automatic Control ,AC-025,(1980), 996-998.
(34)Klein,A.K.and Mélard, G.,’Fisher’s Information. Matrix for Seasonal Autoregressive Moving Average Models’, Journal of Time Series Analysis, 237.
(35)Latour,A.and Roy,R., ‘ Some exact results on the sample autocovariances of a saeaonal ARIMA model’, The Canadian Journal of Statistics, 15(1987), 283-291.
(36)Latour,A. and Roy,R.,’On the behavior of the sample autocovariances and autocorrelations of a seasonal ARIMA model’, Statistics & Probability Letters, 8(1989), 339-345.
(37)Ledolter,J.and Box, G.E.P., ’Conditions for the optimality of exponential
smoothing forecast procedures' ,Biometrika,25( 1 978),77-93.
(38)Ledolter,J. 'The effect of additive outliers on the forecasts from ARlTvIA
models',Internation Journal of Forecasting,5(1989),231-240.
(39)Lin,\iV.T.,'IVIodeling and forecasting hospital patient movements :univariate
and lnultiple tilne serie approaches' ,Internation Journal of Forecasting,
5(1989) ,195- 208.
(40)Liu,L.M.,'Sales forecasting using multi-equation transfer function 1110dels',
6(1987),223-238.
(41)Ljung,G.N.and Box,G.E.P.,'On a measure of lack of fit III tinle senes
models' ,Biometrika,65 (1978),297-303.
(42)Ljung,G.IVLand Box,G.E.P., (The likelihood function of stationary autoregressive
nloving average 1110clels',Biometrika,66 (1979),265-270.
(43)Lusk,E.J. and Neves,J.S.: A Comparative ARIT\1A of the 111 Series of
the Makriakis Con1petition' 1 J ov.rnal of Forccast1:ng ,3 (1984) ,329-332.
(44 )Makridakis,S. ,Andersen,A., Carbone,R. :Hibon,rv1. ,Filcles,R., Levlandowski,R.,
ton)., Parzen,E.and Winkler,R. , , The accuracy of extrapolation(tilne
series) mothods: results of a forecasting cOlnpetition', J ov.rnal of Forecasting
,1(1982),111-153.
(45 )Makridakis,S.ancl Hi bon,M.,' Accuracy of forecasting:an elnpirical investigation',
Journal of Royal Sta.stical Society A, 142(1979),97-145 .
(46)MIckenzin,' A note on using the interated form of ARlI\1A forecasts', International
Jov:rno.l of Forecasting,4(1988), 117-124.
(47)Mcleod ,A.I.,' On the distribution and applications of resid ual autocorrelations
in Box-J enkins 1110dels' , Journal of Roya.l Statistical Society B,40( 1978)302.
(48)Ivliller,M.,Erden,J.V. and \Vilus,R.,' An analysis of the accuracy of ARllvlA
nlodels in forecasting unemployment insurance workloads', ASA Statistical
Association,( 1984) ,547-552.
(49)Newbold,P.,'The equivalence of t,vo tests of tillle series nlodel adequacy
',Biometrika,67 (1980),463-465.
(50)Neftce,S.N.'Specification of economic time series lllodeis using Akaike's
cri terion' , Journal American Sta.tistical Association, 77(1982),N 0.375,540.
(51)Neter,J.,\Vasserman,\V. and Kutner)v.I.H.Applied liner statistical m.odels,華泰，台北,2nd,(1985).
(52)Newbold,P.,'The exact likelihood function for a lllixed autoregressivenloving
average process\Biometrika.,61(1974),423-426.
(53)Newbold,P.and Granger,C.W.J.,'Experience with forecasting univariate
time series and the combination of forecasts' , Journal of Royal Statistical
Society A,bf 137(1974),131-165.
(54)Newbold,P.,'ARllVIA model building and the tinle senes analysis approach
to forecasting ' ,Journal of foreca.sting, bf 2( 1983) )3-35.
(55)Nicholls,D.F. and Hall,A.D.,'The exact likelihood funtion of 111ultivariate
autoregressive-lVlmving average models', Biometrika,66(1979),259-264.
(56)Pena,D., 'The autocorrelation function of seasonal Afu\IIA 1110dels', 5 (1984),272.
(57)Phanl,D. T.,' Estilnation of autoregrcssiye paranleter and order selection
for ARlVIA models ',Jo7J.rnal of Time Series Analysis,bf 9(1988) ,3,265-279
(58)Poskitt,D.S.Clnd Tremayne,A.R., :Testing the specification of a fitted ARNIA
model' ,Biometrika,6 7(1980) ,359-363.
(59)Poskitt,D.S.and Tremayne,A.R.,'An approach to testing linear time series
models',Annals of Statistics,9(1981),976-986.
(60)Poskitt,D.S.and Tremayna,A.R.,'Diagnostic tests for multiple time series
lnodels',Annals of Statistics ,10(1982), 114-120.
(61 )Parzen,E.,' ARARlv1A lnodels for time series analysis and forecasting',
lournal of Forecasting, 1(1982),67-82.
(62)Pukkila,T. ,Koreisha,A. and Kallinen,A.,'The identification of AR}/IA
models',Biom,etrika, 77( 1990) :3,537-548.
(63)SAS/ETS User)s Gv,ide,Cary:SAS Institute Inc.,(1988).
(64)SAS/IML User-'s Gl1,ide,Cary:SAS Institute Inc.,(1988).
(65 )Schwarz, G.' Estimatinf the dimension of a n10del" ,Annals of Statistics
,6( 1978),461-464.
(66)Tasy:R.S.and Tiao,G.C.'Consistent estimates of autoregressive pal'ametel'S
and extended sample autocorrelation functions for stationary and
nonstationary ARj\lIA models', lo'U:rn.o.l A merican Statistical Association,
79 (1984 ),84-96.
(67)Tasy,R.S .and Tiao,G.C.'Use of cononical analusis in time series model
identification' ,Biom,etr-ika, 72(1985 ),2,299-315.
(68)Thompson,P.A.and Ivliller:R.B.,'Sampling the future:A Bayesian approacb
to forecasting fron1 univariate time series', lourno,l of Bv,siness and Econom,
ic Statistics,bf 4(1986),427-436.
(69)Tiao,G.C. and Box,G.E.P.,'IVlocleling multiple tilne series \vith applications',
lour-nal of Amer-ican Statistical, 7 6(1981), 802-816.
(70)Vandaele,\V.,Applied Tim,e Ser-ies o,nd Box-Jenkins Models,雙葉，台北
,(1983).
(71)Yajima,Y.,’Asymptotic properties of the sample autocorrelations and partial autocorrelation of a multiplicative ARIMA process’, 6(1985),No.3, 187-201.