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A SHORT‐TERMFORECASTING PROCEDUREFOR INSTITUTIONENROLLMENTSCharles Barry Pfitzner aa Randolph‐Macon College , Ashland, VirginiaPublished online: 03 Aug 2006.

To cite this article: Charles Barry Pfitzner (1987) A SHORT‐TERM FORECASTINGPROCEDURE FOR INSTITUTION ENROLLMENTS, Community Junior CollegeResearch Quarterly of Research and Practice, 11:3, 141-152, DOI:10.1080/0361697870110301

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A SHORT-TERM FORECASTING PROCEDUREFOR INSTITUTION ENROLLMENTS

CHARLES BARRY PFITZNERRandolph-Macon College, Ashland, Virginia

Institutions of higher education are finding that forecastingenrollments is of critical importance in the current environ-ment of steady and declining student populations. Accurateshort-term enrollment forecasts provide valuable informationto administrators for budgeting, planning, and (in the caseof state supported institutions) negotiating with fundingagencies. This research suggests that Box-Jenkins (ARIMA)models may be used to produce accurate short-range forecastsof seasonal enrollment data. One-step ahead (one academicquarter ahead) forecast errors of Full Time Equivalent Students(FTES) for the Virginia Community College System are on theorder of 3.6% with errors increasing in magnitue to 5.6% forforecasts one year into the future. The methodology employedin this paper to forecast VCCS enrollments can be adaptedeasily and inexpensively to forecasting enrollments at otherinstitutions.

INTRODUCTION

Research on enrollment forecasting generally focuses (with notableexception; see Salley, 1979 and Weiler, 1981) on long-term projectionby curve fitting, structural (econometric) modeling, or some combina-tion of these two methods (Armstrong, 1981; Miller & McGill, 1984;Wing, 1974). These procedures, since they are generally based onannual data, are not intended to produce forecasts for the short-term,where the short-term is defined as a forecast horizon of one to foursubperiods (quarters, in the case of the VCSS) ahead. Short-termforecasts of this nature are the focus of this research.

This paper applies the now widely known Box-Jenkins time seriesmethodology (Box & Jenkins, 1976) to enrollment data for the VirginiaCommunity College System (VCCS). This methodology, a subset ofgeneral univariate statistical time-series methods, has been shown toproduce accurate forecasting results in a variety of applications.The methods and results of this research should be of general interest

Community/Junior College Quarterly, 11:141-152, 1987 141Copyright © 1987 by Hemisphere Publishing Corporation

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to those concerned with forecasting student enrollments, since Box-Jenkins methods have been successfully applied to many time seriesthat exhibit trend, seasonal variation, or both. Though the methodpresented here is univariate, that is, based purely on the past beha-vior of enrollment data, the procedure can be expanded to includeeconomic explanation as well.

The remainder of the paper is organized into five sections. The dataset is presented in the first section. In the second section, theBox-Jenkins methodology is briefly described, and the applicabilityof the method to the VCCS data is discussed. The third section de-scribes the modeling of the series and presents the estimated equa-tions. Next, the forecasts, along with forecast error statistics,are presented and analyzed. The fifth section presents the implica-tions for practice.

THE DATA SET

The official data set on VCSS enrollment begins in the Summer quarterof 1966 and ends in Fall quarter of 1986. The enrollment variableis "end of quarter" full time equivalent students (FTES). Twenty-three colleges, located throughout the state, are members of theVirginia Community College System.

The data on FTES for the Virginia Community College System are showngeographically in Figure 1. The data shows the obvious repetitiveseasonality that would be expected relative to quarter; the peaksoccur in the Fall, consecutively lower Winter and Spring enrollmentsfollow, and the seasonal trough is the Summer FTES. In addition, thedata clearly show a trend change approximately at the Summer sessionof 1975. The period from 1969 to 1975 corresponds roughly to theperiod when VCCS Colleges and campuses were being built and thus newgeographical areas were being served. After this building phasethere is little or no secular trend present in the data.

THE BOX-JENKINS APPROACH AND THE VCCS DATA

The Box-Jenkins approach, also called ARIMA (autoregressive integratedmoving average) modeling, consists of four steps or stages. The firststage is the specification stage. The characteristics of the timeseries are analyzed in order to specify alternative models for esti-mation and forecasting. The second stage is the estimation phase,wherein the parameters of the proposed model or models are computed.Diagnostic checks, the third step, indicate whether or not the esti-mated model is adequate, and suggest necessary modifications. Finally,the model is used to generate forecasts.

In univariate ARIMA modeling, the forecasts are based purely on thepast behavior of the time series under analysis. Thus, an obviouscost advantage of this approach over multivariate methods is thatdata collection and analysis is limited to a single variable. Apotential drawback is that no structural (economic) explanation isprovided. Concerning a data set such as VCCS enrollments, this means

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F68 F70 F72 F74 F76 F78 F80 F82

Quarters (Labels at Fall - AH. Years)

F84 F86

FIGURE 1. Quarterly FTES enrollments

that measures such as tuition rates, the state of the economy, andthe number of graduating high school seniors are ignored when fore-casts are made. Nonetheless, it will be shown that the procedureresults in accurate prediction over the forecast period, as judgedby forecast error statistics.

A series such as the VCCS enrollments is, on a priori grounds, alogical candidate for ARIMA modeling. First, as previously mentioned,the data follow a definite, repetitive seasonal pattern, (seeFigure 1.) ARIMA models are capable of accurately simulating thesepatterns. (ARIMA models do not handle changes in the seasonalitypatterns well.) Second, the VCCS, like many similar institutions,follows an essentially "open door" enrollment policy. It would beconsiderably more difficult to forecast enrollments based on the pastbehavior of the series if the institution changed entrance criteriato stimulate or discourage enrollment. Third, the forecast periodchosen (beginning in Fall 1980) follows the "building phase". Obvi-ously, it would be more difficult to forecast enrollments accuratelywhen new campuses and colleges are being built.

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SPECIFICATION AND ESTIMATION

The previously mentioned change in trend of the series at Summerquarter 1975 presents a problem in modeling the data. Short ofattempting all of the possible alternative procedures, the researchermust make choices in determing how to handle the trend change. Again,the period preceding Summer 1975 was the period when new collegesand campuses were entering the system. If a choice is made to in-corporate the trend change in the forecasting equation, severaloptions are available. Techniques such as intervention analysis(similar to the use of dummy variables in regression) could be intro-duced to account for this building phase. This would be a complicatedchoice since numerous campuses opened at irregular intervals duringthe period prior to 1975, and classes were often started using otherfacilities before college buildings were completed. Thus it would benecessary to enter a number of variables and it would be difficult todetermine the proper time frame for entering each. A second possi-bility is to fit a growth curve to the data, compute the residualsfrom that model, and model the residuals as an ARIMA process whichwould capture the seasonal effects. This procedure is also somewhatcomplicated, since it would require two sets of estimations and fore-casts. Still other methods could be applied. However, a much simplermodel can be produced by simply confining the estimation and forecast-ing period to the VCCS data after Spring term 1975. The latter dataset is of sufficient length to estimate the model, and the more recentpast is of greatest interest in forecasting this series. This choiceis not without cost, of course, since the earlier data might containinformation that is relevant to forecasting the future. The research-er (or administrator) must make a judgement as to whether the reducedmodeling and estimation costs of the simpler model outweigh the possi-ble loss of information. In this research, the forecasting accuracyof the model suggests that any information is minimal over theforecast period.

Certain data sets require complicated ARIMA models. Fortunately, forthe VCCS data set, a very simple model emerges. (See the Appendix tothis paper for a discussion of the technical aspects of the seriestransformation and the model specification.) The forecasting equationis a first-order autoregressive model at lag one. This means that theequation is obtained by regressing the value of the "transformed"series in time period "t" on its immediately preceding value. (The"transformed" series refers to the fact that the square root of theoriginal series has been computed, and that the resulting series hasbeen de-seasonalized.) If the current value is denoted as Yt and,therefore, the preceding value is Yj. _,, this process may be writtenas

Yt = (Dt.! + at,

where 0 is the parameter to be estimated, a t is an error term that isfree of autocorrelation, and the constant term is ignored. Thevariable is recast in its original form for forecasting. Table 1presents the estimated equations at the beginning and end of theforecast period.

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TABLE 1. Parameter estimates

ParameterPeriod Estimate (fl)) Q statistic P valueFall 1976 - .7095 Q(8) = 1.55 7992Summer 1980 (3.98)

Fall 1976 - .7579 Q(18) = 8.81 .964Spring 1986 (7.12)

The values below the parameter estimates are t statistics, the Q ,statistic is the Box-Pierce Q (the number of lags in parentheses),which assesses the presence of autocorrelation in the residuals ofthe estimated model. The P value relates to the Q statistic andrepresents the probability of the null hypothesis that autocorrela-tion is absent in the residuals. The high P values, therefore, indi-cate that the residuals are uncorrelated, and that the model estima-tion is proper in light of this criterion, since the objective ofARIMA modeling is to "use" the autocorrelation of the series in theestimation of the equation, reducing the residual autocorrelation tozero. The forecasting equation is re-estimated each quarter through-out the forecast period, a procedure known as sequential updating,meaning that new (but not very different) parameter estimates arecomputed each quarter. In each estimation the value of 0 is signifi-cant at the alpha = .01 level, and other standard criteria of estima-tion adequacy are met.

FORECASTING RESULTS

The forecasts are presented in table 2 for the forecast period, Fall1980 through Fall 1987. One, two, three, and four-step ahead(quarters ahead) forecasts are reported along with the actual FTESfor each quarter. It should be emphasized all forecasts are "realtime" forecasts, meaning that all of the information necessary toestimate the model and generate the forecasts was known at the fore-cast origin. The reader may examine the forecast and actual valuesfor each quarter and each forecast horizon. For example, the firstrow in the table contains the date and actual FTES for Fall quarterof 1980 in the first two columns. The number under "one" in thefirst row, 55353.8, is a one-step ahead forecast for Fall quarter of1980. This means that the equation is estimated using data throughthe Summer Quarter of 1980, and that estimation is used to produce aforecast for the Fall of 1980. The 55353.8 is thus a one-step (onequarter) ahead forecast; it is somewhat lower than the actual Fall

It should be noted that the Box-Pierce Q statistic should include aminimum of approximately 20 lags for strict reliability; the smallnumber of observations in the first equation precludes approachingthat number.

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1980 enrollment of 57968. The number under "two" in the first row,54907.9, is a two-step ahead forecast, also for Fall quarter 1980,meaning that the estimating equation for that forecast utilizes datathrough only Spring quarter of 1980, hence, it is two-steps (twoquarters) ahead. Accordingly, the forecasts under "three" and "four"are obtained by estimating the equation through Winter 1980 and Fall1979, respectively, and then generating forecasts for the Fall of 1980.The remainder of the table is produced by the same estimation andforecasting procedure as additional (later) actual data points areadded to the series.

TABLE 2. Enrollment forecasts

FORECAST STEPS (QUARTERS) AHEAD

QUARTER

1980-F1981-W1981-Sp1981-Su1981-F1981-W1982-Sp1982-Su1982-F1983-W1983-Sp1983-Su1983-F1984-W1984-Sp1984-Su1984-F1985-W1985-Sp1985-Su1985-F1986-W1986-Sp1986-Su1986-F1987-W1987-Sp1987-Su1987-F

ACTUALFTES

57968518994819420376598565368450389185315738453193487961932657294501944567117892523634610641244172625151144742406391852154851

ONE

55353.852394.848049.121368.059716.853664.49895.321734.258117.052345.250413.818023.558670.853486.047106.518021.955645.646591.342547.415579.851554.945510.240251.916965.953194.147223.7

TWO

54907.950045.048569.621275.561179.353560.749880.621478.162071.052952,49768,1874857044,54367,49305,18576.4558334921342988163364896345542.840826.716768.351122.945979.842544.2

THREE

54749.649689.246435.421638.761029.95477549800214706172255714502661844058001532354988219737568694935845118.816589.950203.543396.84085117063508494446841582.419574.1

FOUR

54278.449560.46141.20301.61657.54641.50841.21427.61712.55444.752307.718692.257575.053938.949080.019996.058099.50141,45231.17717.250610.44474.39058.17076.651262.344262.840441.719039.456340.9

Figures 2 and 3 are graphical representations of the forecast andactual values of the FTES; the one and three-step ahead forecastsare chosen for display. Clearly, the one-step ahead and the three-

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7 0

60 -

50 -

•a 40 -

-c 30 -

20 -

10 -

1980-F 1981-F 1982-F 1983-F 1984—F 1985-F 198S-F(Quarters)+ One—Step Forecasts

FIGURE 2. One-step ahead forecasts and actual values

1980-F 1981-F 1982-F 1983—F 1984--F 1985—F 1986-F

D Actual FTES(Quarters)

+ 3—Step Forecasts

FIGURE 3. Three-step ahead forecasts and actual values

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148 C. B. PFTTZNEE

step ahead forecasts "track" the actual data very well, though theone-step ahead forecasts are, as expected, more accurate. Carefulexamination of the three-step ahead forecasts reveals a tendency ofthe forecast values to persist above or below the actual FTES in con-secutive quarters. For example, the forecast values are lower thanthe actual FTES for Fall, Winter, and Spring of the 1980-81 academicyear, and higher than the actual values for Fall, Winter, and Springof the 1984-85 academic year. This pattern is not surprising. If,for example, Fall enrollments are lower than expected (and forecast),the lower trend in actual enrollments is likely to persist throughthe following quarters for the obvious reasons. However, Fall quarterenrollment information is not "known" to the model until the followingsummer, when forecasting three steps ahead. Thus the forecasts tendto persist above the actual FTES and, hence, the forecast errors arecorrelated for forecasts that are more than one step ahead. (Granger& Newbold, 1977, p. 281) In general, the longer the forecast horizon,the stronger the autocorrelation in the forecast error series.

Table 3 presents the traditional error statistics for each set offorecasts. Reported are the mean squared error (MSE), root meansquared error (RMSE), mean absolute error (MAE), with the latter twomeasures expressed also in percentage terms (%RMSE and %MAE). Inaddition, values of Theil's U statistic are reported in table 3.

TABLE 3. Enrollment Forecast Error Statistics

Perhaps the easiest error statistic to interpret is the mean absoluteerror (though the MSE or RMSE is generally preferred in comparingforecasting models; see Granger and Newbold, p. 280) and the percen-tage mean absolute error. The one-step ahead forecasts for the fore-cast period "miss" the actual value by an average of 1,200.1 FTES

Let e . = z^ - f^, where e . is the forecast error, z- is the actualvalue of the series at time period t, and f . is the forecast value ofthe time series for time t. Then the mean squared error is 2{e^) /n,where n is the number of forecasts, and the RMSE is simply the squareroot of the MSE. The mean absolute error is Slej-l/n. To compute the%RMSE, the squared error is divided by the actual value of the seriesin that period, these values are then averaged and the square root istaken. For the %MAE, the absolute error is divided by the actualvalue prior to averaging.

ERROR FORECAST STEPS (QUARTERS) AHEADSTATISTIC ONE TWO THREE FOUR

MSE 2,417,372.1 5,157,921.1 6,342,413.5 7,455,444.9RMSE 1554.8 2271.1 2518.4 2730.5WISE 5.27 5.85 6.51 6.76MAE 1200.1 1790.3 2155.8 2260.93SMAE 3.57 4.60 5.50 5.55THEIL'S U 0.60 0.88 0.97 1.06

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(MAE) = 1,200.1). This error increases, as expected, as the forecasthorizon is increased, to 2,260.9 for the four-step ahead (one yearahead) forecasts. On a relative basis, the one-step ahead forecasterrors average only 3.6%. In fact, with the exception of the Summer1982 quarter, all of the one quarter ahead forecast errors are smallerthan 9%. (Enrollment declined in Summer 1982 and was 17% smaller thanthe one-step ahead forecast; this FTES decline preceded a drop inenrollment over the following academic year.) The percentage meanabsolute errors increase to 4.6, 5.5, and 5.55 for the two, three andfour-step ahead forecasts respectively.

Theil's U statistic is intended to indicate the effectiveness of aforecasting procedure by comparison to a "naive" forecast of no changein the level of the series. Theil's U is the ratio of the RMSE of themodel to the RMSE of the no change forecast. (Doan & Litterman, 1984,p. 20-6) Clearly, if the value of U is less than one, the model fore-casts are more accurate than the naive forecasts. In this research theno change error is computed from season to season. That is, the naiveforecast for a given quarter is the actual value of the series fourquarters prior. This criterion is not stringent when there is a strongtrend present in the series, since any reasonable forecasting methodis likely to produce a value less than one in such a setting. However,in this case, the forecast period is one in which the data are virtual-ly trendless, and therefore Theil's U represents a more rigorous com-parison. The RMSE for the no change forecast is 2,588 FTES. The RMSEfor each forecast horizon is divided by this value (2,588) producingthe ratios at the bottom of table 3. The one-step ahead forecasts aresubstantially superior to the no change forecasts (since U < 1 ) ; thetwo-step ahead forecasts are superior, but less so; the three-stepahead forecasts are marginally superior and the four-step ahead fore-casts are marginally inferior. To summarize, the forecasting procedureperforms better than using last year's corresponding quarter as aforecast for all forecast horizons except four quarters ahead.

One particular forecast period of interest is the Fall quarter of 1984.The VCCS experienced a sharp decline in enrollment in this quarter andcorrespondingly lower enrollments in the following quarters (seeTable 3), causing great concern on the part of Virginia state govern-ment officials and VCCS administrators. According to published re>-ports, (The Richmond Times-Dispatch, October 3, 1984, p. 1) the VCCShad forecast 59,600 FTES for that quarter, producing an error of 7,237FTES. The model utilized in this paper produces forecasts that arealso higher than actual enrollments, Lut that reduce the forecastingerror to 5,736, 4,507, 3,471, and 3,283 FTES for four, three, two, andone steps ahead respectively. Importantly, these ARIMA forecasts con-tain no judgemental component that might account for economic influ-ences. The level of tuition was increased substantially (13%) for thefall of 1984, and the Virginia economy was expanding rapidly. A nega-tive influence of economic expansion on the level of educationalenrollments has been documented by other researchers (Miller & McGill,1984; Salley, 1979), and rapidly rising tuition rates would be expectedto depress enrollment levels as well. An administrator could effec-tively utilize such economic factors in modifying univariate forecasts.

It is worthwhile to question the value of one-quarter ahead forecasts

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for budgeting and management. The VCCS, as mentioned above, usesend of quarter enrollment figures as the official data series. Thefigures for the current quarter are known just prior to (or perhapseven later than) the start of the next quarter. Yet, in any case,the minimum lead time on the forecasts is still one quarter for theofficial series, the "end of quarter" FTES. In utilizing the one-step ahead forecases the administrator may make several adjustmentsin the forecasting procedure as well. FTES at the beginning of aquarter are likely to bear a predictable relationship to end ofquarter figures, so that the forecast lead could be increased byestimating end of quarter figures based on opening enrollment data.It may be possible to drop the Summer quarter from the data seriesaltogether, thereby increasing the forecast lead time for the impor-tant Fall quarter. Finally, the structure of ARIMA models fit toother data sets may dictate a less important role for the immediatelypreceding quarter. This research does not attempt to provide empiri-cal evidence on any of these issues.

IMPLICATIONS FOR PRACTICE

Short-term projections of the type presented in this paper may be ofvalue in a number of areas of budgeting and planning. The direct andobvious budget effects include the fact that tuition receipts repre-sent an important portion of current operating revenues and thatfunding agency support is generally based on some measure of enroll-ments. These revenues are, of course, the immediate constraint oncurrent expenditures (Salley, 1979, p. 323). Accurate short-termforecasts allow for added lead time for the administration of revenueshortfalls and excesses. Other planning related to the level ofenrollments include the hiring of adjunct faculty, classroom utili-zation (including, in many colleges and universities, the use of off-campus classroom space), and other regular budget decisions such asthe purchase of materials and equipment, and related inventory levels.In addition, since the number (and timing) of course offerings affectenrollment levels, those course offerings should be determined inlight of the expected (forecast) enrollment levels. That is, giventhe general level of anticipated enrollment, too many course offeringsdilute the numver of students per course (or section) causing classesto be cancelled if student-faculty productivity ratios are to be met.In contrast, too few course offerings discourage student registrationsand, therefore, reduce tuition revenues. A reliable forecasting methodcan be of value in choosing the appropriate number of course offerings.

As a practical matter, the formulation and use Box-Jenkins type fore-casts has been, in the past, a procedure that has required a greatdeal of expertise on the part of the user and access to mainframe com-puting software. While user expertise is still desirable and useful,there currently exist a number of software firms that are marketingmicroputer software that not only performs Box-Jenkins estimations andforecasts, but that is also very nearly "automatic" and inexpensive.Given the ubiquity of the microcomputer, the accuracy of the fore-casts, and the availability of such programs, it would seem thatinstitutions similar to the Virginia Community College System will beable to generate "in house" forecasts similar to those in this paper.

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REFERENCES

Armstrong, D. F. (May/June 1981). Enrollment Projection Within aa Decision-Making Framework. Journal of Higher Education, 52,295-309.

Box, G. E. D., and G. M. Jenkins. (1976). Time Series Analysis:Forecasting and Control. San Francisco: Holden Day.

Cox, C. (1984, October 3). Rolls off 7%, community colleges fear.The Richmond Times-Dispatch, p. 1.

Doan, T. A., and R. B. Litterman. (1984). RATS: User's Manual.Minneapolis: VAR Econometrics.

Granger, C. W. J., and P. Newbold. (1977). Forecasting EconomicTime Series. New York: Academic Press.

Hoenack, S. A., and W. C. Weiler. (January 1979). The Demand forHigher Education and Institutional Enrollment Forecasting.Economic Inquiry, 17, 89-113.

Miller, J. C , and P. A. McGill. (May, 1984). Forecasting StudentEnrollment. Community and Junior College Journal, 31-33.

Pindyck, R. S., and D. J. Rubinfeld. (1981). Econometric Modelsand Economic Forecasting. New York: McGraw-Hill.

Salley, C. D. (May/June 1979). Short-Term Enrollment Forecastingfor Accurate Budget Planning. Journal of Higher Education, 50,323-33.

Vandaele, W. (1983). Applied Time Series and Box-Jenkins Models.New York: Academic Press.

Weiler, W. C. (May/June 1981). A Model for Short-Term InstitutionalEnrollment Forecasting. Journal of Higher Education, 51, 314-27.

Wing, P. (1974). Higher Education Enrollment Forecasting - AManual for State-Level Agencies. Boulder, Colorado: NationalCenter for Higher Education Management Systems.

APPENDIX: A TECHNICAL DISCUSSION OF THE MODEL

The variance of the FTES series is stabilized by a square roottransformation. This is a first step in specification of ARIMAmodels; the series is transformed, if necessary, so that the varianceis constant throughout the series. The variance of original data inthis case increases slightly over time. When the variance eitherincreases or decreases secularly, logarithmic or square root trans-formations are commonly utilized to produce a stable variance; thelatter is chosen here. (A logarithmic transformation produces avariance that declines slightly over time.)

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Data are differenced in AIMA modeling if there exists a trend withrespect to time or a strong seasonal pattern to the data. The FTESdata have no secular trend over the estimation period (after 1975),so consecutive differencing is unnecessary. Seasonal differencing,however, is required to remove the strong seasonality exhibited bythe data set. First-order seasonal differences are sufficient toremove this seasonality. The seasonal differencing is accomplishedby subtracting the value of the series "s" periods prior to thecurrent value from the current value of the series, where "s" is theseasonal span. If Yt is the current value of a quarterly series,the transformation Yt - Yt_. is applied throughout the data seriessince the span is four forquarterly data.

After the above transformations of the data, the series has a con-stant variance, is stationary in the mean (no secular trend ispresent), and the seasonality has been removed. Analysis of theautocorrelation function (ACF) and the partial autocorrelationfunction (PACF) provide indications as to the proper specificationof the forecasting model to be estimated on the transformed dataseries. These diagnostics indicate that a first-order autoregressivemodel is appropriate.

The estimated parameter is significant throughout the estimationperiod. The probability of white noise in the residuals exceeds 95%in each case, and examination of the autocorrelation function of theresiduals reveals no troublesome spikes. The model estimation isthus judged adequate throughout the period.

For the reader familiar with the backshift notation common to ARIMAmodeling, the model for the end of the period may be written:(1 - .7579B) (1 - B4) (FTES)1/2 = at.

Received February 2, 1987Accepted February 23, 1987

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