1. John G. Zhang, Ph.D. Harper College jzhang@harpercollege.edu
Looking Ahead of the Curve: an ARIMA Modeling Approach to Enrollment Forecasting
John G. Zhang, Ph.D.
Harper College

2. Topics Why forecast How to forecast Why ARIMA What is ARIMA
How to ARIMA
How ARIMA did
Discussion
47th AIR Annual Forum

3. Why Forecast Queries and Reports: what was Dashboard: what is
Forecasts: what will be
Forecast for enrollment: more valuable for resources planning
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4. How to forecast Naïve forecast: random walk, moving average
Exponential smoothing
Markov chain
Regression
ARIMA
Others
Combining methods
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5. Why ARIMA Naïve forecast: best guess if no patterns
Exponential Smoothing: usually designed for one-step ahead forecast
Markov chain: see reference
Regression: frequently violates the assumption of uncorrelated errors
ARIMA: worked well, more later
Others: see reference
Combining Methods: non-directional
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6. What is ARIMA AutoRegressive Integrated Moving Average
Generally, the model is given by
47th AIR Annual Forum

7. where Xt is a time series value at time t, 0 is a constant,
B is a backshift or lag operator,
i is a number of lags or spans,
 is an error term at time t,
 and θ are AR and MA parameters, and
p, d, and q are the orders of AR, I, MA
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8. If p = 1, 1 = 1, d = 0, θ1= 0, random walk: (1 - B)(Xt – θ0) = t
if p = 1, d = 0, q = 1, ARMA(1, 1):
(1 - 1B)(Xt – θ0) = (1 - θ1B) t
If p = 1, d = 0, θ1 = 0, AR(1) model:
(1 - 1B)(Xt – θ0) = t
If p = 1, 1 = 1, d = 0, θ1= 0, random walk:
(1 - B)(Xt – θ0) = t
If 1 = 0, d = 0, θ1 = 0, constant:
(Xt – θ0) = t
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9. How to ARIMA Box and Jenkins (1976) notation: (p d q)(p d q)s
Four stages:
Identification
Estimation
Validation
Forecasting
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10. How to ARIMA SPSS Trends module: version 12 worked well
version 13 and 14: algorithms changed same data, same program, different forecast
SAS ETS module:
ARIMA procedure more flexible
forecast consistant
automation possible thanks to macros
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11. Identification Series Plot Autocorrelation plot
Dickey-Fuller test of unit root hypothesis
AR models to compare the log likelihood values for a series and its transformed series
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12. Identification Degree of differencing Order of AR Order of MA
Seasonality if any
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13. Estimation Q statistics Goodness-of-fit criteria: variance estimate
Akaike information criterion
Schwartz Bayesian criterion
Significance of parameters
Residuals analysis
Mean Absolute Percent Error
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14. Data
Time series data
Date variable: year, quarter, month, week, day, hour, minute, second
Enrollment data: FTE, headcount, seatcount
Data points
Nature of the series determines the forecast
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15. Patterns of Data
Trend: steady increase or decrease in the values of a times series
Cycle: long-term patterns of rising and falling data
Seasonality: regular change in the data values that occurs at the same time in a given period
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16. FTE
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17. FTE Pattern
Trendy: FTE increasing from 1998 to 2006, suggesting non-stationary and differencing necessary
Seasonal: higher in the Fall and Spring and lower in the Summer each and every year, implying a seasonal factor present as part of the model building process
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18. Autocorrelations and Partial Autocorrelations (ACF and PACF)
Lag Correlation
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PACF
Lag Correlation
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19. Q Statistics
Autocorrelation Check of Residuals
To Chi Pr >
Lag Square DF ChiSq Autocorrelations
<
<
<
<
Q Statistics show autocorrelations among various lags highly statistically significant
Autocorrelations were very high
Further actions needed
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20. FTE Forecast
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21. How ARIMA Did Accuracy: what matters most
2-period ahead: 0.74% (FTE) 0.50% (HC)
6-period ahead: 1.43% (FTE) 1.65% (HC)
10-period ahead: 1.40% (FTE) 2.52%(HC)
Forecast error bigger into distant future
Eleanor S. Fox (2005) 1.2% (4) 4.1% (8)
NCES (2003) 1.9% (2) 3.6% (6)
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22. Discussion
Theoretically factors includable along with the time series itself like in regression
Unemployment rate
Consumer Price Index (CPI)
High school student population
District population
Tuition
Forecasts used for forecasting?
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23. Discussion Stationarity and homogeneity Scarcity and spuriousness
Seasonality and outliers
Raw or cooked data
Data mining and stepwise
Fit and accuracy
Additive or multiplicative (subset/factored)
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24. Discussion Science and art Objective and Subjective
Quantitative and qualitative
Over-differencing and over-fitting
Parsimony and uncertainty
Simple or complex
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