### Using Time Series Modeling to Forecast Enrollments

```Using Time Series Modeling to
Forecast Enrollments
Robert J. Marsh, Ph.D.
Associate Dean of Research and Assessment
North Central Michigan College
MI/AIR, November 8, 2013
Why forecast enrollment?
 Declining enrollment since 2010
 Declining property tax base
 Declining state appropriation
 Uncertain budgeting process due to lack of data
 ACCURATE FORECASTS ARE MORE IMPORTANT
THAN EVER
Previous forecasts
 Intuitive, sense of potential population
 Conversations with HS counselors
 Conversations with workforce organizations and advisory boards
 Assume flat enrollment for budget; be pleased with an increase
 Assume a 2-3% year-over-year increase
 Held steady for many years
 Extrapolate from recent terms’ trend
 Monitor statewide trends among CCs
 No data-informed methodology
An enrollment forecast model
 Possible factors

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Area population
Tuition
Area unemployment
Past enrollments
Competition, especially online
Tuition differential, four-year vs. community college
Attitude towards education in general
 Community colleges in particular
Regression approach
 Identify factors (xi), find coefficients (bi)
F = b1x1 + b2 x2 +... + bn xn
 Identifying relevant or interesting (or pertinent) factors
 Enrollments can be more dependent on past
enrollments than outside factors
Area population
 Sometimes highly correlated, but not always.
Time series forecasting
 Relies on past demand to forecast future
 Stores only one past period’s demand (originally
necessary)
 Typically forecasts one period ahead
Time series forecasting
 Simple: forecast = immediate past period
Ft = At-1
 Moving average: average of n past periods
Ft =
A t-1 +At-2 + At-3
3
n
åA
t-i
Ft =
i=1
n
( n = 3)
Exponential Smoothing
 Originated in the production/operations management field, 50+ years
ago (management science)
 Primary formula
Ft = a At-1 + (1- a ) Ft-1
0.0 £ a £1.0
where a is the “smoothing coefficient”
 Relies on the past period’s actual (At-1) and forecast (Ft-1) values.
 An a close to 1.0 results in a very reactive model, one more
responsive to very recent actual data.
 All that needed to be stored were the immediate past actual and
forecast values.
(Holt, 1957)
Exponential Smoothing
 Each period’s forecast builds on all past periods,
although only immediate past period’s needs to be
stored
F0 = A0
F1 = a A0 + (1- a )F0
F2 = a A1 + (1- a )F1
= a A1 + (1- a ) [a A0 + (1- a )F0 ]
F3 = a A2 + (1- a )F2
= a A2 + (1- a )éëa A1 + (1- a ) (a A0 + (1- a )F0 )ùû
“Exponential” Smoothing
 Reliance on progressively older data drops off exponentially
F0 = A0
F1 = a A0 + (1- a )F0
F2 = a A1 + (1- a )F1
= a A1 + (1- a ) [a A0 + (1- a )F0 ]
F2 = a A1 + a (1- a ) A0 + (1- a ) F0
2
F3 = a A2 + (1- a )F2
= a A2 + (1- a )éëa A1 + (1- a ) (a A0 + (1- a )F0 )ùû
F3 = a A2 + a (1- a ) A1 + a (1- a ) A0 + (1- a ) F0
2
3
Exponential Smoothing with trend
 Trends within our actual enrollment patterns.
Ft = a At-1 + (1- a ) ( Ft-1 + Tt-1 )
where
Tt = b ( Ft - Ft-1 ) + (1- b ) Tt-1
ft = Ft + Tt
0.0 £ b £1.0
where b is the “trend factor”
(Holt & Winters, 1960; Hopp & Spearman, 2011)
Seasonality
 Enrollment is seasonal, with fall typically being higher
than winter
Fall
Winter
 Add a seasonality factor by superimposing a sinusoidal
function
 Exclude summers (much lower)
Seasonality
st = H sin éën (t -1) p + f ùû
where
n = frequency H = amplitude
f = phase
t = numberof period
f £p
 Sine function is periodic over 2p, every two periods
(every fall) the forecast tends to be higher.
ft = Ft +Tt + st
(Middleton, 2010)
Methodology
 Obtained 30 years of enrollment data (credit hours)
 60 periods (no summers)
 Built model in Excel® 2011
 Used Solver® to “optimize” the model
 Minimized sum of squared differences (the objective
function, Z)
n
SSD = Z = å( Fi - Ai )
i=3
2
by varying a, b, H, n and f
subject to
0.0 £ a £ 1.0
0.0 £ b £ 1.0
f £p
Year
1982
1982
1983
1983
1984
YrTm
198230
198250
198330
198350
198430
Actual
14649
14778
14028
12113
13419
Smooth F
14649.00
14778.00
14028.00
12113.50
13419.00
Smooth T
Seasonality
FORECAST
129.00
129.00
129.00
129.00
1303.54
-1295.51
1287.47
-1279.44
16210.54
12861.49
13529.97
Z=
Parameters
a
b
H
n
f
1.0
0.0
11084.685
1.000233
3.02226
Squared Diff
4763482.0
559493.0
12314.0
5335291.26
Excel®-Solver®
Z
a, b, H, n, f
a
b
f
Results
 Initially used actual data through Winter 2011-12 to forecast
Fall 2012-13
 Discovered strong “starting point dependency” within the
non-linear model in Solver
 Not optimal; best described as heuristic
Results through Winter 2011-12, showing “best”:
a start
0.5
0.0
1.0
b start
0.5
0.0
1.0
a
0.856
1.0
0.736
b
0.654
0.0
0.987
H
n
10422.91 1.0002481
10067.420 1.0001813
10290.092 1.0002618
f
2.997
3.0150
2.9990
SSD
3.17 E+07 *
3.23 E+07
3.22 E+07
MAPE = 3.3%. Using model would have produced a 1.7% error for W12
Results
 Used “best” from W12 to forecast Fall 2012
 F12 forecast =
 F12 actual =
 Difference =
 Model MAPE =
21,980
21,878
0.5%
3.3%
 Incorporated actual F12 data into model to forecast
Winter 2013
a start
0.5
0.0
1.0
b start
0.5
0.0
1.0
a
0.848
1.0
0.742
b
0.670
0.0
0.987
H
SSD
n
F
10186.25 1.0002578 2.993902 3.17 E+07 *
10186.26 1.0002604 3.011174 3.44 E+07
10392.14 1.000247 3.000360 3.22 E+07
Results
 Used “best” from F12 to forecast Winter 2013
 W13 forecast =
 W13 actual =
 Difference =
 Model MAPE =
19,783
21,081
– 6.2%
3.4%
 Resulted in an under-forecast
 Assumes past practices continue
 More focused recruitment effort made
0
201330
201230
201130
201030
200930
200830
200730
200630
200530
200430
200330
200230
200130
200030
199930
199830
199730
199630
199530
199430
199330
199230
199130
199030
198930
198830
198730
198630
198530
198430
198330
198230
Model fit
Credit hours by semester
30000
25000
20000
15000
Actual
Forecast
10000
5000
Discussion
 Model is not optimal
 Non linear objective function; heuristic methodology
 Does not include any outside variables (as written)
 Solver® is blunt instrument
 Can be built on simple software platform
 Good visualization with graphs
 Relies on known data
 Very quick to update and do what-if analysis
 Pretty accurate
I’m happy to share
Bob Marsh
North Central Michigan College
231.439.6353
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Discussion
?
```