Forecasting Pre-K Enrollment In Georgia Counties
Forecasting Pre-K Enrollment in Georgia CountiesTable of ContentsExecutive Summary . iiiI.Introduction . 1II.Data . 2Population of Four Year Olds. 2CDC Data on Births . 2Geolytics Forecast . 3Pre-K Enrollment. 4III.Methodology . 5Forecasting Population of Four Year Olds . 5Estimating Enrollment Ratio . 11Pre-K Enrollment. 12IV.“How To” / Manual . 14Re-Estimating Four Year Old Population with Additional Data . 14Adjustment of Number of Lags in ARMA(p,q) Model . 15Reproducing the Forecast of Population of Four Year Olds . 20Adjusting Enrollment Ratios . 20Appendices . 22ii
Forecasting Pre-K Enrollment in Georgia CountiesExecutive SummaryTo forecast pre-kindergarten (Pre-K) enrollment in Georgia by county for2007 through 2011 we use data on actual Pre-K enrollment and data on thepopulation of four year olds. Due to data limitations, we rely on forecasting thepopulation of four year olds from 2005 forward and we forecast Pre-K enrollmentbased on past relationships between population and Pre-K enrollment.Data for the four year old population come from the SurveillanceEpidemiology and End Results (SEER) database at the National Cancer Institute. Weperform a number of consistency checks on the data base by comparing our forecastwith the one obtained from the Centers for Disease Control (CDC) data on births aswell as from Geolytics population forecasts. Geolytics is a private company thatdevelops population forecasts. Data for Pre-K enrollment comes from Bright fromthe Start (BFTS).We use a county specific time-series autoregressive moving averageeconometric model, an ARMA(p,q) model, to forecast population. Statistical testsand in sample forecasts show that the model is able to explain the data very well andthat we are able to predict trends that have been occurring so far. Although there arevariations across counties, aggregate forecasts for Georgia are nearly identical toobserved numbers for the in sample forecasts. Nevertheless, one needs to check forerrors in purely statistical forecasts and assess them in relation to externalinformation.This document provides a manual on the forecasting methodology to provideguidance to someone who is interested in replicating results or reestimating the modelwhen the new data becomes available.iii
Forecasting Pre-K Enrollment in Georgia CountiesI.IntroductionBright from the Start (BFTS), Georgia Department of Early Care andLearning, administers Georgia’s pre-kindergarten program (Pre-K) in addition toother administrative and policy oversight related to early learning and child care. Aspart of its annual planning and budget activities, BFTS forecasts the Pre-K populationby county for the state. The Pre-K forecasts by county can be used for programbudgeting, for analysis of the coverage of Pre-K, and for long-term planning relatedto expansion of the Pre-K program.The Andrew Young School of Policy Studies, Georgia State University wascontracted to provide a methodology for forecasting the Pre-K population. Thisreport provides a manual that documents the methodology and provides the actualforecast by county for 2007-2011. This methodology is to be “handed off” to BFTSalong with training so that future forecasting can be done in-house at BFTS.1
Forecasting Pre-K Enrollment in Georgia CountiesII.DataTwo key components of data used to forecast Pre-K enrollment are data onactual Pre-K enrollment and data on population of four year olds. As data on actualPre-K enrollment are available only from 2001 to 2006, it is nearly impossible to usea statistical model to accurately forecast 5 periods ahead given we have only 6 actualvalues of enrollment. Therefore, we rely on forecasting the population of all fouryear olds. As our forecasting capabilities are much greater given that population datafor four year olds are available from 1969 to 2004, we first forecast the population offour year olds, and then we estimate Pre-K enrollment based on some pastrelationship between population and Pre-K enrollment. We apply this relationship tothe forecast of four year olds to derive the forecast of Pre-K enrollment.The following section describes the data used to forecast the population offour year olds. It describes the data on actual Pre-K enrollment that is used todetermine the relationship between population and enrollment.Population of Four Year OldsTo forecast the population of four year olds our model relies on the previouspopulation of four year olds. Data we use come from the Surveillance Epidemiologyand End Results (SEER) database at the National Cancer Institute. The database hasestimates of four year olds by county for each year from 1969 to 2004. This relativelylong time series allows us to use a purely statistical approach in forecastingpopulation from 2007 to 2011. Nevertheless, we have used several different datasources to forecast the population of four year olds as a robustness check. Thesesources are briefly described below.CDC Data on BirthsCenter for Disease Control and Prevention (CDC) provides data on births bycounty by year from 1968 to 2004. The unit of observation is birth, not county, andtherefore one first needs to aggregate births by county. These data can be very usefulin estimating population of four year olds as the data contain a lot of explanatory2
Forecasting Pre-K Enrollment in Georgia Countiesvariables for each newborn. However, using these data to forecast population of fouryear olds requires making two strong assumptions:1. Newborns did not move from their county of birth until they were overfour years old (or that movement is purely random so that on averagepopulation stays the same);2. The county where the birth certificate was issued is same as the county ofresidence.Making such strong assumptions that can be easily violated may lead to inconsistentestimates of population of four year olds at the county level and in turn toinconsistent estimates of Pre-K enrollment at the county level.Geolytics ForecastA previous AYSPS forecast used population estimates and projections fromGeolytics for 2006 and 2011 for the “less than 5” age group as the base of theforecast of four year olds. Then, several calculations were done in order to derive thepopulation of four year olds from the base group. These calculations involve usingCensus data on “3-4 year old” and data on first grade enrollment to derive a finalpopulation forecast. However, by using these data we are making the followingassumptions:1. There will be no migration between counties in the period from 2006 to2011;2. All children enrolled in public schools (i.e. there were no childrenenrolled in home schooling programs and there were no children enrolledin private schools);3. All children enrolled in public schools in the county where they attendedPre-K;4. In Census population estimate of “3-4 year old,” exactly half are four yearolds and half are three year olds.Again, making such strong assumptions may lead to inconsistent estimates ofpopulation of four year olds at the county level and in turn to inconsistent estimatesof Pre-K enrollment at the county level. We therefore rely on SEER data for our final3
Forecasting Pre-K Enrollment in Georgia Countiesforecast. However, we compare our final forecast with the forecast based onGeolytics data, and we describe this in more detail in the next section.Pre-K EnrollmentData on actual Pre-K enrollment are collected from Bright from the Start.These data are available by county by year from 2001 to 2006. We do not use thesedata to forecast future Pre-K enrollment alone. Nevertheless, we use these data toderive relationship between four year olds and Pre-K enrollment in each county. Inturn our final forecast of Pre-K enrollment will be influenced by these data as theydetermine relationship between population of four year olds and Pre-K enrollment.In almost every county in Georgia we observe an increase in enrollment overtime. This increase in enrollment is much greater (in relative terms) than the increasein population. Therefore, it appears that the enrollment rate has been increasing at afaster rate when compared to the increase in the population of four year olds.However, given only 6 observations, we cannot estimate nor impose any relationshipthat would suggest how this enrollment is to increase in years 2007 through 2011.In addition, there may have been exogenous factors that caused the increasein enrollment. One such factor is a change in policy or targeted focus, and we knowthat such changes have occurred. Therefore, given policy changes and relatively fewobservations of actual enrollment, one would need to rely on “expert opinion” inorder to forecast future Pre-K enrollment, as previous realizations are not very goodpredictors of future ones. In the next two sections, we will discuss in more detailhow such “expert opinion” can be incorporated to adjust estimates of enrollmentratios and forecasts of Pre-K enrollment.4
Forecasting Pre-K Enrollment in Georgia CountiesIII.MethodologyThis section briefly describes the methodology used to forecast Pre-Kenrollment by county. Our forecasting has two components: forecasting population offour year olds and calculating enrollment from the forecasted population. Thepopulation forecasts are based on a purely statistical time-series model. To derive ourfinal Pre-K enrollment forecast, we multiply our population forecast of four year oldsby the estimated enrollment ratio. Depending on the estimated enrollment ratio, wecan have more than one forecasts of Pre-K enrollment, despite having only a singleforecast of population of four year olds.First we describe the methodology used to forecast population of four yearolds. Next, we describe how we calculate the enrollment ratio, and finally wedescribe how these two are combined to obtain forecast of Pre-K enrollment.Forecasting Population of Four Year OldsTo forecast the population of four year olds by county we can rely on purelystatistical models (where we use only previous population to forecast the futurepopulation) or on structural models (“more economic” models where we impose astructure and use other variables such as per capita income and mother’s laborparticipation, as well as population itself to forecast future population). Althoughstructural models have many advantages, they require us to impose a relationshipbetween four year old population and some other variables (such as income, femaleunemployment, etc.). Therefore, these models are very data intensive. In addition,using structural model raises the following issues:1. What variables should be included in the model? Although somevariables, such as births four years ago, have a strong statisticalrelationship to population of four year olds, one can argue that populationof four year olds is influenced by various other factors and there is noclear-cut point at which one can say what variables should and whatshould not be included in the model.2. Asides from selecting relevant variables, one also needs forecasts of suchvariables in order to derive a forecast of four year old population. Someforecasts, such as per capita income, are relatively easy to obtain.However, other variables for which forecasts are not available would5
Forecasting Pre-K Enrollment in Georgia Countiesrequire doing additional forecast and they would thus increase forecasterror.Therefore, we rely on a purely statistical models to forecast the population offour year olds. Although these models are less intuitive, they are very powerful if thetrends are stable as they are in this case.The population forecast literature has some doubts about using standardforecast models in the very long run. However, forecasting seven periods aheadshould not raise any serious concerns under the assumption that there would not beany drastic scenarios. Nevertheless, as Lee and Tuljapurkar point out, “one should notrely on mechanical time series forecasts in any case; they should be assessed inrelation to external information.” 1We have used several methods, and our final forecasts are based on anautoregressive moving average model, ARMA(p,q), where p denotes maximumautoregressive order included in the model and q denotes maximum moving averageorder included in the model. Our model is outlined in the following equation:(1)popi ,t ϕi1 popi ,t 1 . ϕip popi ,t p ε i ,t θi1ε i ,t 1 . θiqε i ,t qwhere pop denotes population, i denotes county, and t denotes year. We let p and qvary by county, and the choice of p and q is data driven (i.e. we let the data tell uswhat lags are significant and should be included). Therefore, we do not impose amodel where we assume each county is same; rather, we allow each county to followits own structure so it best fits the data.The county-specific models outlined in Equation (1) fit the data well. Inaddition, they are relatively easy to recreate and reanalyze. 2 However, these modelsare not always intuitive, and they can be very weak in forecasting in cases where theydo not fit the data very well. Nevertheless, there are several techniques available fordetermining how well a model fits the data.1Lee, Ronald and Shripad Tuljapurkar (1998). "Population Forecasting for Fiscal Planning: Issuesand Innovations." Burch Center Working Paper Series, University of California, Berkeley.2Section IV focuses on recreating and reanalyzing these models when new four year oldpopulation data become available.6
Forecasting Pre-K Enrollment in Georgia CountiesOne approach for determining how well the model fits the data would be toperform a Portmanteau Q-test for white noise in the residuals after the data have beenfitted. If fitted residuals appear to be random (i.e. if Q-statistic is significant at thecommon confidence levels), then we can be confident that model fits the data well.Although less orthodox, another approach in determining how well the modelfits the data would be to perform in sample forecasts and then compare those withactual realizations. We did this exercise assuming that the last observation availablewas in year 2000, and we started forecasting at 2001. When we aggregate our insample forecast over all Georgia counties, we see that forecast was able to predictpopulation with only small error. Table 1 outlines our results.TABLE 1. ACTUAL VS. FORECASTEDFOUR YEAR OLD 1,640121,4222003122,879123,917Nevertheless, there are variations from county to county on how well the model fitsthe data. In addition to looking at how well we forecasted point estimates of thepopulation of four year olds, it is also interesting to see how confident we can be inour point estimate. The following two figures present in sample point forecasts andconfidence intervals of forecasts for selected counties.7
Forecasting Pre-K Enrollment in Georgia CountiesFIGURE 1. IN-SAMPLE FORECAST OF POPULATION OF FOUR YEAR OLDS 90199119921993199419951996Confidence Intervals1997199819992000Actual Values2001200220032004Forecasted ValuesFIGURE 2. IN-SAMPLE FORECAST OF POPULATION OF FOUR YEAR OLDS 951996Confidence Intervals8199719981999Actual Values2000200120022003Forecasted Values2004
Forecasting Pre-K Enrollment in Georgia CountiesFrom the figures wee see that forecast is able to capture trends in both cases –in Gwinnett County where there is a steady increase in population, as well as inRabun County where population is much more volatile. From the figures we also seethat confidence intervals are much different in these two counties. Although bothconfidence intervals are based on two standard deviations (little above 95 percentconfidence level), we see that confidence interval for Gwinnett County is much wider(in relative and absolute terms) than the confidence interval for Rabun County. Whilewider absolute deviation can just reflect the fact that Gwinnett County has a muchlarger population, wider relative deviation also reflects the fact that the model has abetter fit for the Rabun County data than it does for the Gwinnett County data.Forecasts of four year olds by county for years 2007 through 2011 areincluded in Table A1 in Appendix 1. There we see that forecasts in almost allcounties, with several exceptions, predict an increase in population of four year oldsfrom 2007 to 2011. Although the population growth varies from county to county,when the population for all of Georgia is considered, we see a growth in populationof four year old of 2.7 percent from 2007 to 2011. The following two figures presentpoint forecasts and confidence intervals of forecasts for selected counties.9
Forecasting Pre-K Enrollment in Georgia CountiesFIGURE 3. FORECAST OF POPULATION OF FOUR YEAR OLDS (GWINNETT 12002200320042005Confidence Intervals200620072008Actual Values200920102011Forecasted ValuesFIGURE 4. FORECAST OF POPULATION OF FOUR YEAR OLDS (RICHMOND 12002200320042005Confidence Intervals1020062007Actual Values200820092010Forecasted Values2011
Forecasting Pre-K Enrollment in Georgia CountiesFrom the figures we see that confidence intervals are much different in thesetwo counties. Although both confidence intervals are based on two standarddeviations (little above 95 percent confidence level), we see that confidence intervalfor Gwinnett County is much wider (in relative and absolute terms) than theconfidence interval for Richmond County. While wider absolute deviation can againjust reflect the fact that Gwinnett County has higher population, wider relativedeviation reflects the fact that model fits the Richmond County data better than it fitsthe Gwinnett County data. 3When compared 4 to the previous AYSPS previous that were based onGeolytics’ estimates of 0-5 years old and adjusted for estimated fraction of 4 yearolds in that group, the new estimates are very close for most years. However, the newforecast underestimates population compared to old forecast. This deviation isespecially noticeable in the final forecast years.In most cases the two forecasts differ by very little in relative terms, but thereare some outliers where the two forecasts differ by a lot. Average absolute deviationfor all counties and years is 17 percent. Median absolute deviation for all countiesand years is 12.9 percent. However, absolute deviations for all counties and yearsrange from 0 percent to 161.1 percent.This comes as no surprise when we take into account that the new forecast ispurely statistical, and that even when the predictions for a given cou
Learning, administers Georgia’s pre-kindergarten program (Pre-K) in addition to other administrative and policy oversight related to early learning and child care. As part of its annual planning and budget activities, BFTS forecasts the Pre-K population by county for the state. The Pre-K for
Forecasting with R Nikolaos Kourentzesa,c, Fotios Petropoulosb,c aLancaster Centre for Forecasting, LUMS, Lancaster University, UK bCardi Business School, Cardi University, UK cForecasting Society, www.forsoc.net This document is supplementary material for the \Forecasting with R" workshop delivered at the International Symposium on Forecasting 2016 (ISF2016).
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