Since its inception in 1964, the Projections of Education Statistics series has been providing projections of key education statistics to policy makers, educators, researchers, the press, and the general public. This edition of Projections of Education Statistics is the thirtyeighth in the series.
Appendix A contains this introduction, which provides a general overview of the projection methodology, as well as six additional sections, which discuss the specific methodology for the different statistics projected. Appendix A contains seven sections:
This introduction
Two major projection techniques were used to develop the projections presented in this publication:
Two different types of exponential smoothing, single exponential smoothing and double exponential smoothing, were used in producing the projections presented in this publication.
Single exponential smoothing was used when the historical data had a basically horizontal pattern. Single exponential smoothing produces a single forecast for all years in the forecast period. In developing projections of elementary and secondary enrollments, for example, the rate at which students progress from one particular grade to the next (e.g., from grade 2 to grade 3) was projected using single exponential smoothing. Thus, this percentage was assumed to be constant over the forecast period.
In general, exponential smoothing places more weight on recent observations than on earlier ones. The weights for observations decrease exponentially as one moves further into the past. As a result, the older data have less influence on the projections. The rate at which the weights of older observations decrease is determined by the smoothing constant.
When using single exponential smoothing for a time series, P_{t}, a smoothed series, , is computed recursively by evaluating
where 0< ∝ ≤1 is the smoothing constant.
By repeated substitution, we can rewrite the equation as
where time, s, goes from the first period in the time series, 0, to time period t1.
The forecasts are constant for all years in the forecast period. The constant equals
where T is the last year in the estimation sample and k > 0.
These equations illustrate that the projection is a weighted average based on exponentially decreasing weights. For higher smoothing constants, weights for earlier observations decrease more rapidly than for lower smoothing constants.
Double exponential smoothing with one smoothing constant was used when the time series was expected to change linearly with time. Double exponential smoothing produces different forecasts for different years in the forecast period, reflecting trend patterns. This technique was used to forecast the number of doctor’s degrees awarded to men and women.
Double exponential smoothing with one smoothing constant applies the single smoothing method twice (using the same parameter). Double smoothing of a series P_{t} is defined by the recursions:
where:
S_{t} = the single smoothed series;
D_{t} = the double smoothed series; and
0 < ∝ ≤ 1
Note that double smoothing is a single parameter smoothing method with a damping factor ∝. Forecasts from double smoothing are computed as
where T is the last year in the estimation sample and k > 0. The last expression shows that forecasts from double smoothing lie on a linear trend with intercept 2S_{T} – D_{T} and slope ∝(S_{T} – D_{T} )/(1 – ∝).
As with previous editions of the Projections of Education Statistics, a smoothing constant of 0.4 was used for both single and double exponential smoothing. For more information about exponential smoothing, see Diebold (2001).
Multiple linear regression was used in cases where a strong relationship exists between the variable being projected (the dependent variable) and independent variables. This technique can be used only when accurate data and reliable projections of the independent variables are available. Key independent variables for this publication include demographic and economic factors. For example, current expenditures for public elementary and secondary education are related to economic factors such as disposable income and education revenues from state sources. The sources of the demographic and economic projections used for this publication are discussed below, under “Assumptions.”
The equations in this appendix should be viewed as forecasting rather than structural equations. That is, the equations are intended only to project values for the dependent variables, not to reflect all elements of underlying social, political, and economic structures. Available data precluded the building of largescale structural models. The particular equations shown were selected on the basis of their statistical properties, such as coefficients of determination (R^{˛}s), the tstatistics of the coefficients, the DurbinWatson statistic, the BreuschGodfrey Serial Correlation LM test statistic, and residual plots.
The functional form primarily used is the multiplicative model. When used with two independent variables, this model takes the form:
This equation can easily be transformed into the linear form by taking the natural log (ln) of both sides of the equation:
One property of this model is that the coefficient of an independent variable shows how responsive in percentage terms the dependent variable is to a one percent change in that independent variable (also called the elasticity). For example, a 1 percent change in X_{1} in the above equation would lead to a b_{1} percent change in Y.
All projections are based on underlying assumptions, and these assumptions determine projection results to a large extent. It is important that users of projections understand the assumptions to determine the acceptability of projected time series for their purposes. All the projections in this publication are to some extent dependent on demographic and/or economic assumptions.
Many of the projections in this publication are demographically based on the U.S. Census Bureau’s 2008 National Population Projections (August 2008) and the Interim State Population Projections (April 2005).
The two sets of Census Bureau population projections are produced using cohortcomponent models. In order for the nationallevel population projections by age, sex, and race/ethnicity to be consistent with the most recent historical estimates released by the Census Bureau, the projections were ratioadjusted by applying the ratio of the last historical estimate to the corresponding projections year to the projections for each age, sex, and race/ethnicity combination. This allows for a consistent set of historical estimates and projections. For more information on the methodology used for Census Bureau population projections, see appendix C, Data Sources.
The enrollment projections in this publication depend on Census Bureau population projections for the various age groups that attend school. The future fertility rate assumption (along with corresponding projections of female populations) determines projections of the number of births, a key factor for population projections. The fertility rate assumption plays a major role in determining population projections for the age groups enrolled in nursery school, kindergarten, and elementary grades. The effects of the fertility rate assumption are more pronounced toward the end of the forecast period, while immigration assumptions affect all years. For enrollments in secondary grades and college, the fertility rate assumption is of no consequence, since all the population cohorts for these enrollment ranges have already been born.
Various economic variables are used in the forecasting models for numbers of elementary and secondary teachers, public elementary and secondary school expenditures, and postsecondary enrollment.
The source of these variables is the trend scenario of the “U.S. Monthly Model November 2009: ShortTerm Projections” developed by the economic consulting firm IHS Global Insight. The trend scenario depicts a mean of possible paths that the economy could take over the forecast period, barring major shocks. The economy, in this scenario, evolves smoothly, without major fluctuations.
For details about the primary assumptions used in this edition of Projections of Education Statistics, see table A1 on page 83.
Projections of time series usually differ from the final reported data due to errors from many sources. This is because of the inherent nature of the statistical universe from which the basic data are obtained and the properties of projection methodologies, which depend on the validity of many assumptions.
The mean absolute percentage error (MAPE) is one way to express the forecast accuracy of past projections. This measure expresses the average absolute value of errors over past projections in percentage terms. For example, an analysis of projection errors over the past 25 editions of Projections of Education Statistics indicates that the MAPEs for public school enrollment in grades K–12 for lead times of 1, 2, 5, and 10 years were 0.3, 0.6, 1.3, and 2.3 percent, respectively. For the 1yearout projection, this means that one would expect the projection to be within 0.3 percent of the actual value, on average.
For a list of MAPEs for selected national statistics in this publication, see table A2 on page 84. Sections A.1 through A.5 each contain a text table (exhibits A3 through A7) that presents the MAPEs for the key national statistics of that section. Each exhibit appears directly after the discussion of accuracy of that section’s national projections. For a list of MAPEs by state and region for public elementary and secondary enrollment, see tables A5 through A7 on pages 92–97 and for a list of MAPEs by state and region for the number of high school graduates in public elementary and secondary schools, see table A8 on pages 102–103.
Exhibits A1 and A2 present an example of how the MAPEs were constructed using actual values for national public elementary and secondary enrollment projections for schools years 2004 through 2007 and enrollment projections from the last four editions of the Projections of Education Statistics. The top panel of Exhibit A1 shows the actual values for school years 2004 through 2007 and enrollment projections for each year from the Projections of Education Statistics to 2015 with the number of projections decreasing by one for each subsequent edition. The bottom panel of Exhibit A1 shows the percentage differences between the actual values and the projected values. For example, the projected value for 2004 presented in the Projections of Education Statistics to 2005 was 0.5 lower than the actual value for that year.
The top panel of Exhibit A2 shows the absolute value of the percent differences from Exhibit A1 arranged by lead time rather than year. Hence, the 0.2 appearing in the column for lead times of 1 year and the row for projections from the Projections of Education Statistics to 2016 indicates that projection of the oneyearout forecast from the Projections of Education Statistics to 2016 differed by 0.2 in absolute terms from its actual value. The MAPE for each lead time show in the bottom panel of Exhibit A2 were calculated by computing the average of the absolute values of the percentage differences for that lead time. Unlike the MAPEs on exhibits A3 through A7 and appendix tables A2 and A5 through A8, the MAPEs on Exhibit A2 are presented for illustrative purposes only. They are different from the MAPEs for public elementary and secondary enrollment projections elsewhere in this report because the MAPEs in the example were calculated using only the last 4 editions of the Projections of Education Statistics.
The number of years used in the analysis of the projection error differs by statistics both because projections of additional education statistics have been added to the report over time and because, for some statistics, there have been such a substantial change in the methodology used to produce the projections that the projections produced using the earlier methodology were not included in the analysis of the projection error.
Source  Year  

2004  2005  2006  2007  
Enrollment in thousands  
Actual  48,795  49,113  49,316  49,293 
Projected enrollment in thousands  
Projections of Education Statistics 2015  48,560  48,710  48,948  49,091 
Projections of Education Statistics 2016  †  49,028  49,370  49,610 
Projections of Education Statistics 2017  †  †  49,464  49,644 
Projections of Education Statistics 2018  †  †  †  49,470 
Percentage difference between actual and projected values  
Projections of Education Statistics 2015  0.5  0.8  0.7  0.4 
Projections of Education Statistics 2016  †  0.2  0.1  0.6 
Projections of Education Statistics 2017  †  †  0.3  0.7 
Projections of Education Statistics 2018  †  †  †  0.4 
† Not applicable. SOURCE: U.S. Department of Education, National Center for Education Statistics, Common Core of Data (CCD), “State Nonfiscal Survey of Public Elementary/Secondary Education,” 2004–05 through 2007–08; and Projections of Education Statistics, various editions. (This exhibit was prepared September 2010.) 
Source  Lead time (years)  

1  2  3  4  
Absolute value of percentage difference between actual and projected values  
Projections of Education Statistics 2015  0.5  0.8  0.7  0.4 
Projections of Education Statistics 2016  0.2  0.1  0.6  † 
Projections of Education Statistics 2017  0.3  0.7  †  † 
Projections of Education Statistics 2018  0.4  †  †  † 
Mean absolute percentage error  
Example  0.3  0.5  0.7  0.4 
† Not applicable. NOTE: The mean absolute percentage errors presented on this table are for illustrative purpose only. SOURCE: U.S. Department of Education, National Center for Education Statistics, Common Core of Data (CCD), “State Nonfiscal Survey of Public Elementary/Secondary Education,” 2004–05 through 2007–08; and Projections of Education Statistics, various editions. (This exhibit was prepared September 2010.) 
Variable  Assumption  

Demographic variables  
Population  Projections are consistent with the Census Bureau middle series estimates ^{1}  
18 to 24yearold population  Census Bureau middle series projection: average annual growth rate of 0.1%  
25 to 29yearold population  Census Bureau middle series projection: average annual growth rate of 0.8%  
30 to 34yearold population  Census Bureau middle series projection: average annual growth rate of 1.3%  
35 to 44yearold population  Census Bureau middle series projection: average annual growth rate of 0.3%  
Economic variables  
Disposable income per capita in constant dollars 
Annual percent changes range between 0.7% and 2.6% with an annual growth rate of 1.5% 

Education revenue receipts from state sources per capita in constant dollars 
Annual percent changes range between 1.3% and 2.7% with an annual growth rate of 1.4% 

Inflation rate  Inflation rate ranges between 0.9% and 2.1%  
Unemployment rate (men)  
Ages 18 and 19  Remains between 20.1% and 30.3%  
Ages 20 to 24  Remains between 11.7% and 18.7%  
Age 25 and over  Remains between 5.0% and 8.3%  
Unemployment rate (women)  
Ages 18 and 19  Remains between 14.6% and 20.6%  
Ages 20 to 24  Remains between 8.9% and 12.9%  
Age 25 and over  Remains between 4.5% and 6.7%  
^{1} As the Census projections were not updated to reflect 2008 Census Bureau population estimates, the Census Bureau agespecific population projections for each year were adjusted by multiplying the ratio of the total Census Bureau estimate for 2008 to the total Census Bureau projection for 2008. SOURCE: U.S. Department of Commerce, Census Bureau, Population Estimates, retrieved October 13, 2009, from http://www.census.gov/popest/national/asrh/2008nataf.html; and Population Projections, retrieved November 2, 2009, from http://www.census.gov/popest/national/asrh/2008nataf.html; and IHS Global Insight, "U.S. Monthly Model November 2009 ShortTerm Projections." (This table was prepared March 2010.) 
Statistic  Lead time (years)  

1  2  3  4  5  6  7  8  9  10  
Public elementary and secondary schools  
Prekindergarten–12 enrollment ^{1}  0.3  0.6  0.8  1.1  1.3  1.5  1.8  1.9  2.1  2.3 
Prekindergarten–8 enrollment ^{1}  0.4  0.7  0.9  1.2  1.4  1.7  2.0  2.4  2.8  3.1 
9–12 enrollment ^{1}  0.4  0.7  0.9  1.1  1.3  1.6  2.0  2.3  2.3  2.2 
High school graduates ^{2}  1.0  1.0  1.6  1.7  1.7  2.2  2.9  3.7  4.0  3.8 
Elementary and secondary teachers ^{3}  1.0  1.4  1.7  2.4  3.0  3.6  3.9  4.4  5.1  6.3 
Total current expenditures ^{4 }  1.2  2.1  2.2  2.3  2.7  3.5  4.2  4.3  4.1  4.4 
Private elementary and secondary schools ^{5}  
Prekindergarten–12 enrollment  3.4  4.6  3.7  6.1  5.5  6.0  0.5  1.0  —  — 
Prekindergarten–8 enrollment  3.5  4.9  4.1  6.6  6.0  6.7  0.9  1.1  —  — 
9–12 enrollment  3.0  3.8  2.3  4.5  3.6  3.5  0.7  0.7  —  — 
High school graduates  0.9  0.9  2.0  2.8  5.9  5.6  2.2  2.2  —  — 
Degreegranting institutions ^{6}  
Total enrollment  1.4  2.4  2.9  3.4  4.6  6.1  8.0  9.8  10.1  10.4 
Men  1.5  2.9  3.5  4.1  5.2  6.6  8.1  9.8  10.2  10.3 
Women  1.5  2.4  3.0  3.2  4.2  5.8  7.9  9.8  9.9  10.4 
4year institutions  1.5  2.5  3.3  4.1  5.3  6.8  8.8  10.9  11.5  12.4 
2year institutions  2.0  3.4  3.8  4.0  4.8  5.0  6.6  7.8  7.6  6.9 
—Not available. ^{1} MAPEs for public PK–12 enrollments were calculated using the last 26 editions of Projections of Education Statistics. ^{2} MAPEs for public high school graduates were calculated from the past 19 editions of Projections of Education Statistics. ^{3} Data for teachers expressed in fulltime equivalents. MAPEs for teachers were calculated from the past 19 editions containing teacher projections. ^{4} In constant dollars based on the Consumer Price Index for all urban consumers, Bureau of Labor Statistics, U.S. Department of Labor. MAPEs for current expenditures were calculated using projections from the last 19 editions containing current expenditure projections. ^{5} MAPEs for private PK–12 enrollments and high school graduates were calculated from the past 8 editions of Projections of Education Statistics. ^{6} MAPEs for degreegranting institution enrollments were calculated using the last 12 editions of Projections of Education Statistics. NOTE: Mean absolute percentage error is the average value over past projections of the absolute values of errors expressed in percentage terms. No MAPEs are presented for degrees conferred as the current model used for producing their projections has been used for only one other edition of the Projections of Education Statistics. Calculations were made using unrounded numbers. Some data have been revised from previously published numbers. SOURCE: U.S. Department of Education, National Center for Education Statistics, Projections of Education Statistics, various issues. (This table was prepared February 2010.) 