Newbold Et Al 94 Adventures With Arima Software

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514 P. Newbold et al. I I nternat ional Journal of Forecasti ng 10 (1994) 573-581

In this paper we report analyses of five timeseries through every package we could find onthe Urbana-Champaign campus of the Universityof Illinois. Some packages allow more than oneestimation procedure, in which case all wereused. We tried to put ourselves in the position ofthe novice, or unsophisticated, user. For exam-ple, although many programs allow the user arange of optional modifications, we generally ranthem in default mode.

Our sample of time series is very small, andwas certainly not randomly chosen. Indeed, welooked for cases where we would expect to find,on a priori grounds, differences in results amongprograms. However, we did not have to lookvery hard: three of our five series are taken fromthe same secondary source. We would not viewany of our examples as remotely pathological.None of our series is short by the standardsdeemed adequate by the textbooks. Perhaps oneof our models might be regarded as being gener-ously parameterized, but by no means ridiculous-ly so. In short, we believe that novice users willinevitably encounter many similar cases.

The set of programs we have employed doesnot include all of those currently available, but ismore than adequate to illustrate our point. Wemust emphasize that it is not our intention tocritically review these programs or to comparetheir merits. This is not a competition at the endof which a winner will be announced. For thatreason, the names of the packages used havebeen suppressed. Our aim is to illustrate whatcan and, we believe, often will happen when thesame model is fitted to the same data by differ-ent programs.

2. Some sources of differences in estimatesAssuming that the error terms generating anARIMA model are normally distributed, theexact likelihood function can be computed, al-lowing full maximum likelihood estimation(ML). Simulation evidence reported by Ansleyand Newbold (1980) suggests a preference forML over alternatives that are approximations to

full maximum likelihood in moderate-sized sam-ples. However, many commercially availablesoftware packages implement one or both of thecomputationally convenient approximations toML proposed by Box and Jenkins (1970). Theseare conditional least squares (CLS) and thebackcasting method (BC). (The latter is some-times called backforecasting.) Two of our pack-ages implemented ML, six implemented CLS,and seven implemented BC. In addition, twopackages implemented other different approxi-mations to full maximum likelihood. (These aredenoted Other 1 and Other 2 in our tabulatedresults.)Since the objective functions maximized differamong the packages, we would naturally expectsome differences among the point estimates ofthe ARIMA model parameters. The resultspresented in Section 3 illustrate just how largethese differences can be in practice, and conse-quently how much divergence among forecastsmight result when the same data and model areanalyzed through different programs.

Whatever procedure is employed, point esti-mation can be viewed as the minimization of thesum of squares of quantities that are non-linearfunctions of the parameters. Therefore, softwaremust incorporate a non-linear regression algo-rithm. This is another potential source of differ-ences among estimates. The algorithms requireinitial estimates of the parameters, and optimi-zation proceeds iteratively until convergence isachieved. Termination may occur after somespecified number of iterations or when relativechange in either the objective function or theestimates is less than some specified amount.Many programs allow the user to specify initialvalues or termination criteria. However, puttingourselves in the position of the novice user, weran all programs in default mode, with oneexception. When it was indicated that the defaultmaximum number of iterations had beenexhausted, we increased that number. We sus-pect this to be common practice among users.The example given in Section 4 illustrates howdifferently structured optimization algorithmscan yield very different parameter estimates.


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P. Newbold et al. I I nternational Journ al of Forecasting 10 1994) 573-581 575

3 Four seasonal series

In this section we report the analyses of fourmonthly seasonal time series. These are:

Series A (168 observations): Naturallogarithms of index of kilowatt hours of electrici-ty used.Series B (120 observations): Housing starts.Series C (120 observations): Housing sales.

Series D (77 observations): Common logarith-ms of sales of a company.The first three series were taken from Pan-kratz (1991, pp. 133-134, 219-220, 366, 370-371). To all of these we fitted the airline model:(1 - B) l - B12)X, = (1 - W) l - 0B12)Er,where B is the back-shift operator and E, is aninnovation error presumed to be white noise.Series D was taken from Chatfield and Prothero(1973), and to this we fitted the model:(1 - +B)(l - B) l - B*)X, = (1 - @B~*)E, .

1)For all series, the models fitted and the trans-

formations employed were those of the originalsource.Simulations reported by Ansley and Newbold(1980) suggest that, in the aggregate, the largestdifferences in the performance of different es-timators occur when the true values of seasonalmoving average parameters are close to theboundary of the invertibility region 1O < 1, inthe case of the models here). Accordingly, inchoosing our series we looked for reportedestimates in this area. In addition, since theChatfield/Prothero paper occasioned such livelydebate on its publication, we were intrigued tospeculate on what these authors would havefound given access to software available today.

Table 1 shows coefficient estimates for thefour series, together with their reported esti-mated standard errors, from 17 programs. Towhat extent our novice user would be perplexedby differences of these magnitudes in the esti-mates is a matter of speculation. These differ-

ences are least for series A and greatest forseries D, but since the former has as many as 168observations (or 155 after differencing), it wouldnot be unreasonable to expect more consistency.As predicted, the largest differences in the pointestimates are for the seasonal moving averageparameter, where the ranges are 0.803-1.000 forseries A, 0.754-1.000 for series B, 0.731-1.000for series C, and 0.447-1.000 for series D.

Rather than discuss further our results seriesby series, it is more appropriate to do so in termsof estimation method. This is so since it appearsthat each program has achieved its objective: themajor differences arise because the sums ofsquares functions being minimized are not thesame.

Maximum likelihood. The two maximumlikelihood programs produced very similar pointestimates. For series B and C, ML 2 gaveestimates of precisely one for the seasonal mov-ing average parameter, while the correspondingML 1 estimates were close to one. Such esti-mates might suggest over-differencing. On theother hand, they could reflect the pile-up effectdiscussed by Cryer and Ledolter (1981) andShephard and Harvey (1990). These authorsshow that even when the true parameter is wellinside the invertibility region, the likelihoodfunction can quite often have a global maximumon the boundary. The user might respond tothese estimates by fitting an alternative modelinvolving seasonal dummy variables. In thiscontext, it is interesting to note that the CLS andBC estimates of these parameters are signifi-cantly less than one.

Condit i onal l east squares. The six CLS pro-grams produced very similar results for all fourseries. Simulation results of Ansley and Newbold(1980) suggest that, when the true value of theseasonal moving average parameter is close tothe boundary of the invertibility region, con-ditional least squares estimators can be seriouslybiased towards zero. Certainly the conditionalleast squares estimates are substantially lowerthan the maximum likelihood estimates for seriesB, C, and D. However, this outcome is notinevitable, since for series A the maximum


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576 P. Newbold et al. I International Journal of Forecasting 10 (1994) 573-. ITable 1Parameter estimates for some seasonal models

Series APackage e &

Series B Series C Series De 6 e & & a

ML1ML2

CLS 1CLS 2CLS 3CLS 4

CLS 5CLS 6

BClBC2BC3BC4BC5BC6BC7

Other 1Other 2

0.693 0.803(0.057) (0.076)0.694 0.804

(0.056) (0.075)0.697

(0.058)0.696

(0.059)0.698

(0.058)0.697

(0.058)0.697(0.067)0.697

(0.058)

0.859(0.052)0.859

(0.053)0.859

(0.052)0.859

(0.052)0.859(0.053)0.859

(0.053)0.704(0.057)0.705(0.057)0.700(0.057)0.700(0.059)0.700(0.058)0.708(0.066)0.704(0.057)

0.893(0.024)0.893(0.025)0.876(0.026)0.943(0.015)0.887(0.059)0.946(0.025)0.893(0.026)

0.691(0.059)0.693(0.059)

1.000(0.161)0.803(0.048)

0.270(0.087)0.269

(0.085)0.290

(0.093)0.289

(0.094)0.290

(0.094)0.290

(0.094)0.290(0.088)0.290

(0.094)0.286(0.093)0.285(0.094)0.284(0.090)0.280(0.094)0.286(0.094)0.309(0.084)0.286(0.094)0.259(0.094)0.271(0.093)

0.967(0.601)1 ooo(153.6)0.755

(0.071)0.754

(0.074)0.755

(0.072)0.755

(0.071)0.755(0.072)0.755

(0.072)0.843(0.035)0.843(0.035)0.876(0.032)0.936(0.019)0.844(0.081)0.924(0.057)0.844(0.037)1.000(0.193)1 ooo(0.001)

0.200(0.086)0.216

(0.083)0.245

(0.094)0.244

(0.095)0.245

(0.095)0.247

(0.096)0.245(0.088)0.245

(0.095)0.191(0.095)0.189(0.096)0.205(0.091)0.203(0.095)0.189(0.096)0.198(0.108)0.189(0.096)0.231(0.094)0.201(0.095)

0.967(0.724)1000(57.3)0.733

(0.072)0.731

(0.074)0.733

(0.072)0.731

(0.074)0.733(0.073)0.733

(0.073)0.891(0.034)0.890(0.035)0.881(0.030)0.937(0.018)0.891(0.071)0.950(0.049)0.891(0.036)1.000

(0.193)1.000(0.007)

-0.454(0.114)

-0.453(0.113)

-0.526(0.116)

-0.528(0.117)

-0.526(O.llY)

-0.528(0.118)

-0.526(0.122)-0.526(0.119)

-0.458(0.113)

-0.457(0.115)

-0.448(0.102)

-0.430(0.114)

-0.458(0.114)-0.425(0.119)

-0.458(0.115)

-0.430(0.116)

-0.468(0.111)

0.725(0.195)0.727

(0.193)0.462

(0.125)0.458

(0.128)0.452

(0.128)0.447

(0.130)0.462(0.147)0.452

(0.129)0.796

(0.050)0.794

(0.053)0.814

(0.048)0.908

(0.029)0.795

(0.117)0.911(0.050)0.795

(0.052)1.000

(0.250)0.696

(0.091)Note: Figures in parentheses are the reported standard errors

likelihood estimates of 0 are somewhat lowerthan the conditional least squares estimates.

Backcasting. There is considerably more vari-ability among the point estimates from thisapproach than among those based on eithermaximum lik,elihood or conditional least squares.Presumably, this is because the backcasting ap-proach is imprecisely defined, since a truncationrule must be specified. The BC 1, BC 2, BC 5,

and BC 7 point estimates are very close indeed,while those from BC 3 do not differ much fromthese four. (Oddly, the standard errors reportedby BC 5 can differ substantially from those ofother programs in this group.) On the otherhand, the BC 4 and BC 6 estimates can be quitedifferent from those of the other five programs,but are very close to each other (though theassociated standard errors need not be).


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Other The point estimates from Other 2were very close to the full maximum likelihoodestimates, though the reported standard errorsassociated with the seasonal moving averageparameter estimates were quite different. Un-iquely among our packages, Other 1 producedseasonal moving average parameter estimates ofone (to three decimal places) for series A and D,as well as for series B and C.

We believe the novice user is likely to beperplexed by findings similar to those of Table 1,whatever the purpose of the analysis. It couldpossibly be argued that some users are indiffer-ent to anything but the forecasts produced by thefitted models. These models might differ in otherrespects, however. In Table 2, we have attempt-ed to produce a measure of divergence amongthe forecasts. For forecasts up to 12 monthsahead, we have computed the range (highest-lowest) for our 17 sets of forecasts. To stan-dardize units, these ranges have been divided bythe forecast standard errors provided by the ML2 program. (In fact, these standard errors differlittle from one program to another.) Again, thedifferences are generally smallest for series Aand largest for series D. We are not sure howlarge a difference should be to cause concern,but find it difficult to be sanguine about the factthat two researchers on our campus could quiteeasily produce forecasts differing by as much as

Table 2Range of forecasts, divided by forecast standard error (asreported by ML 2)

SeriesHorizon A B C D1 0.16 0.58 0.26 0.932 0.26 0.70 0.28 1.113 0.11 0.37 0.38 1.544 0.14 0.31 0.34 0.985 0.15 0.43 0.31 0.946 0.44 0.34 0.38 0.897 0.13 0.21 0.38 0.848 0.20 0.23 0.46 0.679 0.10 0.25 0.39 0.81

10 0.11 0.17 0.42 0.5611 0.14 0.20 0.47 0.3312 0.14 0.27 0.41 0.19

four-tenths of a standard error, after fitting thesame model to the same data.

By any standards, the differences for series Dreported in Table 2 look rather alarming. Havingfitted the model (1) to these data and computedforecasts, Chatfield and Prothero (1973) werealso alarmed. They regarded their point forecastsof sales at the peak of the following year (6-months ahead) as much higher than canreasonably be expected. Indeed, it is this ob-servation that prompted their paper on problemsin applying the ARIMA methodology. Thepaper provoked a lively discussion, includingBox and Jenkins (1973). Much of this discussioncentered on the appropriateness or otherwise ofthe logarithmic transformation. It appears that,had the class of power transformations of Boxand Cox (1964) been considered, the logarithmictransformation would have been rejected infavor of something close to the cube root.Nevertheless, the incorporation of power trans-formations into ARIMA analysis has not becomestandard practice in the intervening years. In-deed, we believe that the typical novice user-and many sophisticated users-would follow theoriginal lead of Chatfield and Prothero, takinglogarithms after a casual inspection of the graphof the series.In Table 3, we examine the sales forecasts forthe peak month of the next year which could beachieved with the programs at our disposaltoday. We have listed the forecasts of log salestaken directly from the program output and havesimply taken antilogarithms to generate forecastsof sales. To keep our results comparable withthose of Chatfield and Prothero, we have notcorrected for bias, as for example in Grangerand Newbold (1976). To do so would reduce themagnitudes somewhat, but would not change therelativities greatly: our impression is that suchbias corrections are rarely applied in practice.The forecasts range from sales of 1061-1392units-that is, the highest forecast exceeds thelowest by over 31% The highest of our fore-casts, 1392 from BC 3, appears to be somethingof an outlier. In fact, it is almost identical to theinitial forecast (1387) reported by Chatfield andProthero. These authors noted that, while their


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578 P. Newbold et al. I Int ernat ional Journal of Forecasti ng 10 (1994) 573-581Table 3Six-months ahead forecasts for Series DPackage Log sales SalesML1 3.077 1194ML2 3.077 1195CLS 1 3.028 1066CLS 2 3.027 1063CLS 3 3.027 1063CLS 4 3.026 1061CLS 5 3.028 1066CLS 6 3.027 1063

Package Log sales SalesBCl 3.079 1199BC2 3.079 1199BC3 3.144 1392BC4 3.100 1258BC5 3.079 1199BC6 3.108 1282BC7 3.086 1219Other 1 3.112 1293Other 2 3.070 1176

program employed backcasting in estimation,initial values of the innovations were set to zeroin calculating forecasts. Subsequently, they re-computed forecasts through backcasting initialvalues, obtaining 1221 for predicted sales in thisparticular month. This they deemed morereasonable, but rather high: it is remarkablyclose to what we obtained from BC 7, and just alittle higher than results from BC 1, BC 2, andBC 5. Thus, for this data set, a user could haveachieved virtually the same parameter estimatesthrough the same estimation method, and yetradically different forecasts.

Full maximum likelihood estimation was notavailable to Chatfield and Prothero. Had it been,they would have obtained a peak forecast of1194 from ML 1 or 1195 from ML 2, just a littlelower than their second reported figure. Ironical-ly, conditional least squares certainly was avail-able to these authors. We would not have rec-ommended that they use it, but, had they doneso, they would have obtained a peak forecast inthe neighborhood of 1065-and presumablythere would have been no publication

4. A non seasonal seriesIn this section we discuss the analysis of aseries of 160 quarterly observations on thelogarithms of U.S. real gross national product(1950-89). We fitted to these data the ARIMA

(2, 1, 2) model

1 - 41B W2)[ l - B)X, -4= (1 - 8,B - lYZB2)&, 2)Although our series is quite long, the model

(2) could be regarded as being generously pa-rameterized. There are, however, sound reasonsfor interest in this model. The usual formal andinformal criteria for choosing a parsimoniousARIMA representation suggest that either anARIMA (1, 1, 0) or an ARIMA (0, 1, 2) modelmight be adequate. The ARIMA (2, 1, 2) modelnests both of the simpler models and, through asecond autoregressive term, allows the possibilityof cyclical behavior that is sometimes thought tobe appropriate for economic time series. A usermight well want to fit (2), either as a check onthe adequacy of the simpler models or as part ofa search for a parsimonious model through anorder selection criterion. Watson (1986), Clark(1987), and Campbell and Mankiw (1987) haveall considered the modelling of this time seriesover an earlier, but strongly overlapping period.Campbell and Mankiw fitted several ARIMAmodels, as tools for the estimation of the per-sistence of economic shocks. They singled outthe model (2), in addition to the two simplermodels, for discussion. Both Watson and Clarkfitted unobserved components structural models,of the type discussed in detail by Harvey (1989).Their models are constrained versions of a fullARlMA (2, 1, 2) model, which is the simplestmodel to contain ARIMA (1, 1, 0), ARIMA (0,1, 2)) and the Watson-Clark structural models asspecial cases. It therefore seems reasonable that


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an economist would be interested in the estima-tion of the full model (2), at least as a startingpoint for further analysis.

Table 4 shows the estimates we obtained forthis model. (Although the mean change, p in(2), was estimated with the other parameters,these estimates reveal nothing of great interest,and are excluded from the table.) The programsused here are the same as in the previoussection, with two exceptions. First, we have no

results for BC 4, which failed to execute. Sec-ond, one of our packages permits two automatedselections of initial parameter values. In thiscase, these led to different fitted models. Thesecond of these is denoted BC 8.It is quite clear from Table 4 that substantiallydifferent parameter estimates were possible,depending on which program was used, for thisdata set. In the final column of the table, wehave calculated the log likelihoods for each set of

Table 4Parameter estimates for an ARIMA(2, 1, 2) modelPackage 1 &,ML1 0.577 -0.299

(0.409) (0.274)ML2 0.664 -0.388

(0.338) (0.231)

6,0.217

(0.398)0.312

(0.327)

e:-0.397(0.169)

-0.459(0.152)

Log likelihood514.381514.481

CLS 1 0.685 -0.382(0.386) (0.250)

CLS 2 0.126 0.059(0.670) (0.439)

CLS 3 1.344 -0.532(0.163) (0.141)

CLS 4 0.131 0.329(0.687) (0.522)

CLS 5 0.685 -0.383(0.282) (0.209)CLS 6 0.569 -0.202

(0.374) (0.265)BCl 1.774

(0.003)BC2 0.140

(0.630)BC3 0.704

(0.393)BC5 0.663

(0.324)BC6 0.663

(0.227)BC7 0.663(0.315)

BC8 1.680(0.034)

-0.797(0.010)0.073

(0.423)-0.392(0.251)

-0.398(0.224)

-0.400(0.224)-0.400(0.217)

-0.749(0.036)

0.331(0.376)

-0.240(0.665)1.030

(0.189)-0.197(0.704)0.332(0.261)0.224

(0.376)1.461

(0.000)-0.230(0.625)0.351

(0.383)0.311

(0.311)0.311

(0.202)0.311(0.302)1.328

(0.074)

-0.424 514.447(0.161)

-0.161 513.639(0.246)

-0.239 512.280(0.164)0.138 512.065

(0.373)-0.425 514.448(0.220)-0.274 514.044(0.177)

-0.466 512.009(0.015)

-0.157 513.664(0.239)

-0.426 514.440(0.160)

-0.475 514.478(0.149)

-0.478 514.477(0.257)-0.477 514.478(0.146)

-0.392 513.049(0.098)

Other 1 0.663 -0.399 0.311 -0.476 514.478(0.319) (0.222) (0.306) (0.148)

Other 2 0.548 -0.191 0.203 -0.271 514.084(0.655) (0.407) (0.647) (0.219)Note: Figures in parentheses are the reported standard errors.


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580 P. Newbold et al. I I nternat ional Journal of Forecast ing IO 1994) S-581

parameter estimates. The highest of these is forML 2.

The substantial differences in Table 4 cannotbe attributed to differences in the objectivefunction being optimized. For example, theestimates produced by CLS 1, CLS 5, BC 3, BC5, BC 6, BC 7 and Other 1 are very close to theML 2 estimates, though the BC 6 standard errorsdiffer somewhat from the others. In fact, what-ever function is optimized for this particularmodel and data, that function is multi-modal,and rather flat around the optimum.

The ML 1 estimates are quite close to the ML2 estimates. The difference is due to earlytermination of the ML 1 iterations. When thedefault value for relative change in the objectivefunction was relaxed, forcing further iterations,ML 1 produced the same point estimates as ML2. It might be argued that the ML 1 and ML 2estimates reported in Table 4 differ only insub-stantially in relation to their standard errors.There is, however, a sense in which differentconclusions might be drawn from these two setsof estimates. One way in which a user mightassess whether a model is over-parameterized isthrough the r-ratios associated with the estimatesof the highest order autoregressive and movingaverage parameters in the model. The f-ratiosassociated with the ML 1 estimates of (&, 0,)given in Table 4 are ( - 1.09, - 2.35) while thosefor ML 2 are (- 1.68, - 3.01). Presumably, onthis basis, the ML 1 user would be more inclinedto drop the second autoregressive parameterfrom the model than would the ML 2 user.

The CLS 6 and Other 2 point estimates arealso fairly close to the ML 2 estimates-at leastin relation to the standard errors. However, thecause of the difference here is not early termina-tion of the optimization iterations. Indeed, wesupplied both programs with the ML 2 pointestimates as initial values. Both iterated awayfrom these initial values, again yielding theestimates reported in Table 4. It appears thenthat, in this case, the cause of different estimatesis differences in the functions being minimized.

Point estimates that are all very small com-pared with their standard errors were obtainedfrom CLS 2, CLS 4, and BC 2. Perhaps surpris-

ingly, in the case of CLS 2 and BC 2 the issue isagain early termination of the iterations, Whenfurther iterations were forced by relaxing thetermination criteria, estimates very close to thoseof ML 2 were obtained. In the case of CLS 4, weforced as many iterations as the program permit-ted. The estimates moved a little, but remainedfar from the ML 2 estimates.

Finally, Table 4 reports three sets of pointestimates-from CLS 3, BC 1, and BC &-thatare spectacularly different from the ML 2 esti-mates, and with standard errors that are smallenough to suggest the ML 2 estimates to beimplausible values for the parameters. The BC 1and BC 8 estimates are close to one another andto the boundaries of the stationarity and inver-tibility regions, which accounts for the smallstandard errors. The CLS 3 estimates differsomewhat from the other two, reflecting differ-ences between conditional least squares andbackcasting near the boundary of the invertibilityregion. Given the substantial differences be-tween these three estimates on the one hand andthe ML 2 estimates on the other, one wouldexpect to find that the corresponding values ofthe log likelihood differed by more than thefigures shown in Table 4. This suggests that thefunctions that are being minimized have multiplelocal minima. To check this possibility, we beganby supplying ML 2 with the CLS 3, BC 1, andBC 8 point estimates as initial values. In none ofthese cases was ML 2 able to achieve its maxi-mum likelihood estimates as reported in Table 4:indeed, ML 2 reported final estimates that didnot differ greatly from these initial values. Next,we supplied these three packages with the ML 2point estimates as initial values. BC 1 and BC 8then reported final estimates that were very closeto the ML 2 estimates. CLS 3 iterated awaysomewhat from the initial estimates, achievingfinal estimates almost identical to those reportedfor CLS 6 in Table 4. We conclude that, for thisparticular data set, the initial parameter esti-mates can be crucially important in determiningthe final estimates and that it is this factor thataccounts for the largest of the differences amongthe point estimates reported in Table 4.

Our example in this section has illustrated that


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substantial differences among ARIMA modelparameter estimates may not only be due todifferences in the functions that are minimized.The structure of the minimization algorithm,including specification of initial values and termi-nation criteria, can also substantially influencethe estimates achieved.

5. ConclusionsWe have been at some pains to emphasize thatthe results in this paper are no more thanillustrations of what a user could find. Neverthe-

less, it is tempting to say something about how auser should proceed. Our preference is to esti-mate ARIMA models through full maximumlikelihood. This preference is based on thesimulation evidence of Ansley and Newbold(1980), and the examples presented here havenot, and cannot have, added to that evidence,beyond illustrating differences that can arisewhen alternatives are employed. We know of noevidence arguing any of its approximations to bein any sense preferable to full maximum likeli-hood, beyond the question of computationalcost. The practical importance of this factor israpidly becoming negligible. Many packagescurrently available allow estimation by two ormore approaches. (Indeed, it is a littte disap-pointing that several incorporate just those twoapproaches discussed by Box and Jenkins in1970.) We do not know what the novice userwould make of options of this sort, and areuncomfortable trying to explain in apphedcourses why this situation prevails.

Our results have illustrated how the user maybe confronted with substantially different esti-mates and forecasts, depending on which pack-age is used, and consequently the difficulty inreplicating published results. This state of affairsis unfortunate, and could certainly discouragethe novice from attempting to implement theARIMA methodology.

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of estimates for autoregressive moving average models,Journal of Econometr ics, 13, 159-183.

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Campbell, J.Y. and N.G. Mankiw, 1987. Are output fluctua-tions transitory?, Quar terl y Journal of Economi cs, 102,X57-880.

Chatfield, C. and D.L. Prothero, 1973, Box-Jenkins seasonalforecasting: Problems in a case study, Journal of the RoyalStat isti cal Society, A. 136, 295-336.

Clark, P.K., 1987. The cyclical component of U.S. economicactivity, Qz~~~ter~y Journal of Economi cs, 102, 797-814.

Cryer, J.D. and J. Ledolter, 1981, Small sample properties ofthe maximum likelihood estimator in the first order movingaverage model, Biometr ik a. 68, 691-694.

Granger. C.W.J. and P. Newbold, 1976, Forecasting trans-formed series, Journal of the Royal Stati stical Society , B,38, 189-203.

Harvey, A.C. _ 1989, Forecast ing. St ructur al Tim e Seri esM odels and the Kalr nan Fil ter (Cambridge UniversityPress, Cambridge).

Pankratz, A., 1991, Forecast ing w it h Dy nami c RegressionM odels (Wiley, New York).

Shephard, N.G. and A.C. Harvey, 1990, On the probabilityof estimating a deterministic component in the local levelmodel, Journal of Time Seri es Anal ysis, 11, 339-347.

Watson, M.W.. 1986. Univariate detrending methods withstochastic trends, Journal of M onetar y Economi cs, 18,49-75.

Biographies: Paul NEWBOLD is Professor of Economics atthe University of Illinois, Urbana-Champaign. He haspublished extensively in the areas of time-series analysisand forecasting.

Christos AGIAKLOGLOU received his Ph.D. in economicsfrom the University of Illinois, Urbana-Champaign. He iscurrently serving in the Greek navy.

John MILLER received his Ph.D. in economics from theUniversity of Illinois, Urbana-Champaign. He is nowemployed as a senior analyst with Lehman Brothers, NewYork.

Newbold Et Al 94 Adventures With Arima Software

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