Hopwood, William S.

McKeown, James C.

English

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UNlVtRSlTV OFILLINOIS LIBRARYAT URANA^HAMPAIGNBOOKSTACKSDigitized by the Internet Archivein 2011 with funding fromUniversity of Illinois Urbana-Champaignhttp://www.archive.org/details/ontreatmentofnon567hopwFaculty Working Papers^MHON THE TREATMENT OF NONSIGNIFICANT PARAMETERSIN ARIMA IDENTIFIED MODELS WHEN THE GENERATINGPROCESS IS FIRST ORDER AUTOREGRESSIVEWilliam S. Hdpwood, Assistant Professor,Department of AccountancyJames C. McKeown, Professor, Department ofAccountancyCollege of Commerce and Business AdministrationUniversity of Illinois at Urbona-Champalgnisfield, E., Microeconomics , Norton, New York, 1972.. Department of Justice, Expenditure and Employment Data forjCriminal Justice System , Washington, D.C., 1974.FACULTY WORKING PAPERSCollege of Commerce and Business AdministrationUniversity of Illinois at Urbana-ChampaignMay 4, 1979ON THE TREATMENT OF NONSIGNIFICANT PARAMETERSIN ARIMA IDENTIFIED MODELS WHEN THE GENERATINGPROCESS IS FIRST ORDER AUTOREGRESSIVEWilliam S. Hopwood, Assistant Professor,Department of AccountancyJames C. McKeown, Professor, Department ofAccountancy#567Summary;This study investigates the treatment of insignificant parameters in an identifiedautoregressive moving average model when the underlying process is generated by afirst order autoregressive process. The findings indicated that insignificantparameters should be dropped, and that the parameters in the reduced model shouldbe reestimated. Such a procedure, as compared to using insignificant parametersin the forecast model, produces a marked increase in forecast accuracy.INTRODUCTIONIn recent years there has been an increased focus on the use ofBox-Jenkins modeling procedures for purposes of forecasting univariatetime series. One problem that users of this methodology must face isthe proper treatment of insignificant parameters in the identified auto-regressive integrated moving average (ARIMA) model. In particular thedecision maker is faced with at least three alternatives including:(A) retain and use the model with insignificant parameters for fore-casting, (B) drop insignificant parameters and use the resulting reducedmodel for forecasting, (C) drop insignificant parameters, reestlmate theparameters in the reduced model and then forecast. The present studyinvestigates the use of these alternatives in the case where the gener-ating process is first order autoregressive. It is demonstrated thatthe highest degree of accuracy is achieved by alternative (C) and theleast degree of accuracy is achieved by alternative (A) . Furthermoreit is shown that there is little difference between alternatives (B) and(C). The following section discusses the problem in detail. Subsequentsections develop the methodology, findings and conclusions.THE PROBLEM OF DEALING WITH INSIGNIFICANT PARAMETERS IN ARIMA MODELSWhen an ARIMA model is identified containing insignificant parame-ters, it is not clear which of the above mentioned three alternatives isbest for the decision maker. If alternative (A) is selected there is areasonable possibility that a parameter will be used in the forecastmodel which is actually zero in the population model. On the other handif (B) is selected, and insignificant parameters are dropped before~2-forecasting, then the resulting model will not in general be one thatminimizes the sum of squares for the residuals. In addition the resid-uals for the reduced model are likely to be autocorrelated, since anyinsignificant parameters were probably included in the model for purposesof removing autocorrelation in the residuals. Finally if alternative(C) is selected, the problem of obtaining a minimum sum of squares isresolved. However it is very probable that the resultant residual serieswill be autocorrelated, and possibly more so than in case B.METHODOLOGYSimulation of the Time SeriesSimulation was used to obtain time series for study. The procedurewas to generate series from a first order autoregressive (AR1) populationand select those for analysis which contained at least one insignificantparameter after modeling. The AR1 process was selected because it often2occurs in business decision making contexts.3Several types of AR1 processes were generated via simulation. Inparticular, series of length 50 and 300 were generated. For each ofthese two lengths, first order parameter values of .2 and .7 were used.Both series length and p-3ra.meter size were considered as factors becausethey have an effect on the identifiability of the model. "Long" seriesare easier to identify than "short" series. Similarly series with "large"parameters are easier to identify than series with "small" parameters.The latter is true because the expectation of the first autocorrelationcoefficient is equal to the first order autoregressive parameter inthe population process. If the parameter is small, then the first-3-autocorrelation is likely to be buried in noise (i.e., found to beinsignificant). The former has been empirically demonstrated to betrue in the cases studied [5].Modeling of the Time Series and Computation of the Forecast ErrorsThe simulated series were modeled until 100 models for each of the4 series types were obtained such that each model contained at least 1insignificant parameter (with a = .05). These models were then usedto forecast from 1 to 30 steps ahead.The next step was to generate the theoretical forecast for eachmodel. This was done by applying the theoretical generating model tothe individual time series and then forecasting 1 to 30 steps ahead.Note that another alternative for obtaining the theoretical forecastswould be to extend the original simulated series an additional 30 stepsvia simulation. However the use of the former method is preferredbecause it produces conditional expectation forecasts given the knownmodel and observed series.Finally absolute and quadratic forecast errors were computed forall forecasts. Since the empirical findings were essentially the samefor both metrics, only the absolute error findings are presented.EMPIRICAL FINDINGSNotationally, let the 4 series types be represented by,S., S2 > S-and S, such that the following definitions hold:-4-Parameter Size inNotation Popula tion Model Series LengthSl.2 50S2 •.2 300S3.7 50S4.7 300Mean error profiles were plotted for S- , S„, S, and S, (Tables 1,2, 3 and 4 respectively). These contain the mean forecast error, for agiven series type, plotted for each of the thirty steps on the forecasthorizon.Tables 1, 2, 3 and 4About HereNote that in all cases alternative (A) consistently produces thelargest forecast errors. In addition, for all cases except S_, alterna-tive (C) produces the smallest forecast errors. In the case of S„alternatives (B) and (C) are virtually identical. Note also that in allcases the difference between alternatives decreases as the number ofsteps increases.Since the graphical profile analysis does not provide formal hypoth-esis testing, multivariate analysis of variance (MANOVA) was used to4assess differences in mean vectors. The design involved the use of ortho-gonal polynomial one sample tests for S- , S„, S_ and S, . Since the testswere one sample, no homogeneity assumptions for pooling of covarlancematrices were needed. This left the assumption of multivariate normalitywhich has been proven to hold for large samples via the multivariatecentral limit theorem [3].-5-Letting E.. equal the mean absolute forecast error for step 1(1 = 1, 15) under modeling alternative j (j = A,B,C) the following nullhypothesis was tested for S- , S-, S_ and S,:Hr 1,AE2,AJ15,A1,BJ2,B15, B= i E,1,C2,C15, CAlso, in the event that H. was rejected, the following tests were made:H2:V1,AJ2,AJ15,A•i. BJ2,BJ15,C1,BS2,B15,B_J-i,c*2,C*L3*<H3:*2,A15,AThe results of the tests are presented in Table 5,Table 5 About HereS2,C*u.«Rote that in general the MANOVA test statistics confirm the graphicalprofile analysis. This is demonstrated by the significant difference the-6-overall tests for S-, S„ and S„. Also note that for these 3 seriestypes that alternative (A) is significantly different from alternatives(B) and (C). The nonsignificance of the S, overall test and of theindividual (B)-(C) difference tests must be viewed in light of the factthat, for all 4 series types, the profiles demonstrated a consistent (C),(B), (A) ranking. However where the null is not rejected, the betweenalternative differences are at a minimum. This tends to Imply that thenonrejection is a result of the population differences being too smallfor the. multivariate test to measure.CONCLUSIONThe results indicate that insignificant parameters in identifiedARIMA models should be dropped, and that the remaining parameters inthe reduced model should be reestimated. Such a procedure produces amarked increase in forecast accuracy over the procedure of retaininginsignificant parameters. Furthermore most of this increased accuracycan be achieved by simply dropping the insignificant parameter beforeforecasting. The question of which of these two procedures should beused in practice must be addressed in terms of the required accuracy ofthe decision maker. However it seems that in no case would it be desir-able to retain nonsignificant parameters in the forecast model.The results are limited to the case where the underlying populationprocess is first order autoregressive. However since in practice thetrue generating process is typically unknown, the findings should be ofinterest to the decision maker who suspects that the population processis first order autoregressive.M/B/138JcxtoUd19V>6.3*'.ft.-?B.26\n,22.19.16.1*.12.06.04,0.0AAA~A~—AAA—5"-9A•-B Sn aS AR AT3 JTCB AC« AC A—C5CB ACB ACB A— x— .v—C AC AC« 4—cir—iCB AC AC A AC A AJCB 1 A ACB A A AC A A A-5 J -5C A A 4CB AAA A AC AA >A-z IXC Atr AAAAAr AC BSBb I-CB 33 AC C88 4C CC C. S8 AC C B }C b AC BBB A *—C 1.LCBS3 A ^CC CC CCB A A AC C3 A A ACB A AA A SA_"t ATS A S ITSCB A A B AC B A ABC?. 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CC CC C CCC P6C CC CC BB"»»«» CC 888888fl80BBPB68Bfl8 ,)B'»eRB!18PBB6BeBBBeEB8nRr\|:RP.SAAACC AAAAAAAAAAAAAAAAAAAAAAAAAAAAAACCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCa aA HB EPHP All A rR CCACA"*A HAFCA «C~n:ASCABC—r~BC-A CA CA CAPCA3C5 9CA C'IT -CBCac~2"c—cccecBCPCCXTTTTBCRCc5T—tt.0*.021C20IZ2* 2"31F3KEC*S1 xi-tl^NTable 5Summary of the Manova Tests for theEffect of Different Treatments of Insignificant Parameterssi1S2S3S4Hl FSignificanced.f.2.26.002830,702.64.000530,703.08.000130,701.1898.271730,705l FSignificanced.f.2.029.023115,853.4671.000215,853.40.000215,85NotTestedHt311FSignificanced.f.2.838.001415,852.74.001815,853.067.000615,85NotTestedi |SH4 ; fSignificanced.f.|1.02.4415,851.1989.288715,851.4337.150615,85NotTestedFOOTNOTESThe Box-Jenkins methodology refers to a method (summarized by Box andJenkins [2]) of model identification (selection), estimation, diagnosis,and forecasting for univariate time series. Recently this methodologyhas been used extensively in applications for a wide range of decisionmaking contexts [2], [5], [6], [7], [9], [10]. The results of the abovestudies have consistently demonstrated the powerful nature of Box-Jenkinsmodels.2For examples of the use of AR1 processes in business decision makingcontexts see [3].3The simulation involved superimposing the AR1 model on a normallydistributed white noise (not autocorrelated) series with a mean of zeroand variance of one. Since the models were of the form zt = <j>z , + u,the random numbers were generated for the residual series u . In orderto insure that the generated residuals were normal and not autocorrelated,the following procedure was used:(A) 10,000 normal (with mean zero and unity variance) random numberswere generated to constitute a normal "population". Let x. denotethe individual element number in this population (i = 1,10000).(B) The "population" was tested for normality. The null hypothesis ofnormality was not rejected.(C) The individual series (of length 50 or 300) were sampled from thepopulation. This was done, for a given series, by letting u.equal x where {&} is a sequence of uniformly distributed randomnumbers. The latter procedure effectively shuffles x. to insureindependence.(D) The simulated series were generated to contain an extra 30 unneededobservations. Then the first 50 observations were discarded. Thiswas done, in order to produce series that were reasonably indepen-dent of the starting value for z (a starting value of was usedsince it is the unconditional expectation value for a given obser-vation) .4For purposes of hypothesis testing, only the first 15 steps ahead wereused. This was necessary because inclusion of steps 16 to 30 resulted insevere numerical problems due to a high degree of multicollinearity.In addition examination of the profiles revealed that often 15 stepsahead the three alternatives tended to converge together, indicatingthat the excluded data contained little, if any, Information on dif-ferences between alternatives.The use of orthogonal polynomials for multivariate analysis is dis-cussed in Bock [1] and McCall and Applebaum [7].REFERENCES[1] Bock, R. Danell. Multivariate Statistical Methods in BehavioralResearch. New York, Nev7 York: McGraw-Hill, Inc., 1975.[2] Box, G. E. P. and G. M. Jenkins. Time Series Analysis: Fore-casting and Control . San Francisco, Calif.: Holden-Day, 1970.[3] Davis, Charles. A Case Study Investigation Into Certain Quanti-tative Aspects of Budget Forecasting , Unpublished Ph.D. Thesis:University of Illinois, 1979,[4] Harris, Richard J. A Primer of Multivariate Statistics . NewYork, New York: Academic Press, Inc., 1975.[5] Lorek, Kenneth S» and James C. McKeown. "The Effect on PredictiveAbility of Reducing the Number of Observations on a Time-SeriesAnalysis of Quarterly Earning Data." Journal of Accounting Research,Vol. 16, No. 1 (Spring, 1978), pp. 204-214.[6] Lorek, Kenneth S.; Charles L. McDonald and Dennis H. Patz. "AComparative Examination of Management Forecasts and Box-JenkinsForecasts of Earnings." The Accounting Review , Vol. 51, No. 2(April, 1976), pp. 321-330.[7] Mabert, Vincent A. and R. C. Radcliffe. "A Forecasting Methodologyas Applied to Financial Time Series." The Accounting Review,Vol. 49, No. 1 (January, 1974), pp. 61-75.[8] McCall, Robert B. and Mark I. Appelbaum. "Bias in the Analysis ofRepeated-Measures Designs: Some Alternative Approaches." ChildDevelopment,Vol. 44, No. 4 (December, 1973), pp. 401-415.[9] Nelson, Charles R. Applied Time Series Analysis for ManagerialForecasting. San Francisco, Calif.: Holden-Day, 1973.[10 J Rogalski, Richard J. "Bond Yields: Trends or Random Walks?"Decision Sciences, Vol. 6, No. 4 (October, 1975), pp. 688-699.am ^v a).-y

On the treatment of nonsignificant parameters in ARIMA identified models when the generating process is first order autoregressive / BEBR No. 567

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