1. Deirdre Syms, George Xia Macomb Community College November 5, 2015
Using Multivariate Time Series Forecasting Models for Enrollment Predictions
Deirdre Syms,
George Xia
Macomb Community College
November 5, 2015

2. Bring the Case into Statistical Analysis and Forecasting
A very interesting Macomb Community degree credit side student enrollment pattern over decades matches the local economic situation:
When the local economy turns better, the enrollment figure went down
When the local economy turns worse, the enrollment figure went up
As a typical suburban two-year college, we provide quality education with lower costs to our students, thus we assume local economic data such as unemployment rates are affect our students’ enrollment behaviors.
We/I do a lot of student enrollment reports at MCC, including each semester’s internal enrollment report, each middle fall’s IPEDS enrollment report, etc. Almost all these reports focus on the data facts and descriptive statistics. They are not related to inferential statistical analysis.
Today, there are a lot of statistical analysis and forecasting tools for us to use in this case if we want to dig more useful information from our enrollment data, especially predicting the near future student enrollment. The tools may includes the recent versions of SAS/ETS and SPSS.

3. Pre-Data Analysis Considerations
Before every inferential statistical analysis with modeling building, we have to examine if our statistical model(s) chosen is appropriate to our case.
For example, in the most common analysis using ordinary two-sample t-test, each observation of data has to meet with three assumptions: independent each other, statistically identical variation, and normally distributed.
To qualify a series of data to use statistical software such as SAS/ETS, the time series data need assumptions such as:
Data points gathered sequentially through equal time periods
Future value depends upon the present value and the present value depends on the past observations, the series length is 30 or more
Stochastically stationary or can be mathematically converted into stationary

4. SAS/ETS Proc ARIMA Data Analysis –Univariate Time Series – Model 1
Proc ARIMA, what does ARIMA mean? It means:
A procedure of Auto Regressive Integrative Moving Average
There are three stages to proc ARIMA:
Identification stage: to determine the degree of difference, d; also can use “nlag” to set up number of lags for autocorrelations are shown
Estimation and diagnostic checking stage: to determine degree of auto regression, p; and degree of moving average, q;
Forecasting stage: set up number of periods need to be predicted
Thus ARIMA has three important parameters: ARIMA (p, d, q). In our case, after back and forth checking, we picked p = 1, nlag = 8
d = 2, and q = 0, i.e., we have ARIMA (1, 2, 0) model, and we
set up our forecast periods as 3.

5. SAS Code for ARIMA Procedure – Univariate Time Series – Model 1
title1 'Macomb Community College Enrollment Study'; title2 'Fall Fall 2015 Study'; title3 'ARIMA Method'; data a(keep=Year Fall UnempRt); set sasuser.enrlhsty2015; *if year > 2012 then delete; proc arima data=a; identify var=fall(2) nlag=8; run; estimate p=1; forecast lead=3 out=results id=year;

6. SAS Proc STATESPACE – Multi-Valuate Time Series – Model 2
We found that there is another time series that has similar behavior as our student enrollment series, that is the local unemployment rate series.
SAS/ETS Proc STATESPACE is appropriate for jointly forecasting two more related time series having dynamic interactives. Our case fits the model well.
From our ARIMA (1,2,0) model, we know that we chose difference d = 2, so we also set up the second series d = 2
The state space procedure approaches to modeling a multivariate stationary time series automatically (Akaike 1976). Therefore, we don’t have to set up p and q parameters for it.

7. SAS Code for STATESPACE – Multivariate Time Series – Model 2
title1 'Macomb Community College Enrollment Study'; title2 'Fall Fall 2015 Study'; title3 'Multivariate Time Series Method'; data a1; set a; *if year > 2012 then delete; proc statespace data=a1 out=out2 lead=3 noprint; var Fall(2) UnempRt(2); id Year; proc print data=out2 id Year; run;

8. Actual vs Forecasting by Model 2(1)

9. Actual vs Forecasting by Model 2(2)

10. Forecasting Test Result for Fall
Model Building Test -- Fall
Year
Fall MCC
Method I - ARIMA
Method II - MultiVTS
3-Year
Actual Enrolled
Forecasted
Difference
2012
24,160
24,702
542
24,443
283
2013
23,725
24,392
667
24,061
336
2014
23,370
24,881
1,511
24,240
870
2-Year
23,852
127
23,775 50
23,801
431
23,788
418
1-Year
23,041
(329)
23,637
267

11. Forecasting Result for Fall
Forecasting Result -- Fall
Year
As of Today
Method I - ARIMA
Method II - MultiVTS
2015
22,352
23,041
689
23,105
753
2016
23,356
22,902
2017
23,212
22,409

12. Forecasting Test Result for Winter
Model Building Test -- Winter
Year
Winter MCC
Method I - ARIMA
Method II - MultiVTS
3-Year
Actual Enrolled
Forecasted
Difference
2013
23,986
24,163
177
23,245
(741)
2014
23,690
23,671
(19)
23,290
(400)
2015
22,168
24,045
1,877
23,009
841
2-Year
23,353
(337)
22,814
(876)
23,356
1,188
22,902
734
1-Year
22,846
678
22,472
304

13. Forecasting Result of Winter
Forecasting Result -- Winter
Year
Actual Enrolled
Method I - ARIMA
Method II - MultiVTS
2016
21,643
21,003
2017
21,585
21,052
2018
21,303
20,363

14. Research Notes
Two types of time series models we mentioned, a univariate one and a multivariate one, we can compare and judge which one is better. The other model such as using SAS Proc PDL (polynomial distributed lag) also can be used.
Since enrollment forecasts are difficult to make in periods of irregular patterns when turning points are unexpected. Therefore, subjective forecasting should not be ruled out. Actually, if we integrate our administrative experience and quantitative analysis, the forecasting may more accurate.
Some factors are important, such as marketing effort, improve our program and curriculum set up, teaching reputation, etc., however, these factors are very hard to quantified and hard to bring in. Also, comparing with most of four-year colleges and universities, we charge lower tuition and fees, a small increase of tuition and fees will not affect enrollment much.

15. References
SAS Institute (1993), SAS/ETS User’s Guide V6 second Edition
SAS Inc.
Abraham, B. and Ledolter, J. (1983). Statistical Method for Forecasting
Chen, C. (2008) Meharry Medical College, An Integrated Enrollment Forecast Model, IR Applications V. 15
Xia, Z. 2001, Macomb Community College Student Enrollment Model Building with Trend Analysis and Forecasting