Structural Breaks and ARMA process

[fchang]

Hi everyone,

I have been trying to play with some enrollment data from my postsecondary institution. I wanted to see if it was possible to use a simple ARMA process to forecast short term enrollment since what my institution does currently is just assume that enrollment will increase by 150 students every year, which has led them to budget deficits year after year. I have data from the 1999-2000 to 2023-2024 academic year.

Enrolllment GR.png (38.81 KiB) Viewed 16958 times

I applied growth rates and was doing some DF tests for stationarity and they were coming back as significant. However, the fact that it was choosing 0 lags for both AIC and BIC criteria had me double thinking:

Code: Select all

CALENDAR(A) 1999:1
DATA(FORMAT=XLSX,ORG=COLUMNS) 1999:01 2024:01 DATE ENROLLMENTUND CHANGEEU GREU ENROLLMENTTOT CHANGEET GRET

***stationarity test***
*significant at 1%, stationary 
@dfunit(det=constant) greu
*gives me zero lags from 6 AIC
@dfunit(det=constant,maxlags=6,method=aic) greu
*gives me zero lags from 6 BIC
@dfunit(det=constant,maxlags=6,method=bic) greu 

Also, when looking at the data, I was afraid there could be structural breaks. When doing a Perron test, and after testing for multiple breaks, it identified breaks during 2004, 2017, 2019, and 2022.

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@PERRONBREAKS(breaks=4,io=crash,lags=2,method=AIC) greu 
@PERRONBREAKS(breaks=4,io=crash,lags=2,method=BIC) greu 

*it identifies breaks on 2004, 2017, 2019, 2022 

I ended up going forward by looking at the ACF and PACF graphs to see which ARMA model could be used just for curiosity but I was still aware that the test identified breaks in the data.

After analyzing the graphs and trying a few ARMA models I ended up sticking to an ARMA (2,2) to forecast enrollment for the 2024-2025 academic year, which also ended up being 10 times more accurate than the "forecast" (even though it is more of an assumption) of 150 students that my postsecondary institution did.

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BOXJENK(CONSTANT,DEFINE=GREU_ARMA22,AR=2,MA=2) greu
UFORECAST(stderrs=stderrs4,EQUATION=GREU_ARMA22,PRINT) FORE_ARMA22GREU 2024:01 2026:01

Forecast ARMA22.png (24.81 KiB) Viewed 16958 times

When doing this, I wasn't trying to create the perfect model at first, I just wanted to see if Time Series Analysis could be used to improve accuracy in forecasting enrollment at my institution. However, if I was to go ahead and do this analysis more properly and potentially present this, what should I take into consideration to make my results more robust now that I know there were structural breaks in the data.

I am still new to Time Series so any help or guidance would be greatly appreciated!

[TomDoan]

If you're doing a unit root test on the difference (or rate of change) then you would certainly expect to reject unit roots. And the unit root tests for breaks are *not* tests for breaks, but test for unit roots allowing for breaks. (Breaks in the process can produce false acceptance of unit roots). See https://estima.com/webhelp/topics/breaks-unitroots.html.

To the eye, the only obvious break in this is in the volatility in the middle of the sample. You might want to find out if there is an explanation for that. However, when I look at this, I would certainly not expect that a forecast of a steady positive growth would be realistic.

Note that time series forecasts (in non-seasonal series) are generally quite boring shall we say. With the typical mean square error criterion, the worst type of forecast would be one which zigs when the data zags, so right down the middle will minimize the errors. That's what you're seeing---if you extend that out ten years, you will likely see an almost straight line trending slightly downwards.

[fchang]

If you're doing a unit root test on the difference (or rate of change) then you would certainly expect to reject unit roots. And the unit root tests for breaks are *not* tests for breaks, but test for unit roots allowing for breaks. (Breaks in the process can produce false acceptance of unit roots). See https://estima.com/webhelp/topics/breaks-unitroots.html.

To the eye, the only obvious break in this is in the volatility in the middle of the sample. You might want to find out if there is an explanation for that. However, when I look at this, I would certainly not expect that a forecast of a steady positive growth would be realistic.

Note that time series forecasts (in non-seasonal series) are generally quite boring shall we say. With the typical mean square error criterion, the worst type of forecast would be one which zigs when the data zags, so right down the middle will minimize the errors. That's what you're seeing---if you extend that out ten years, you will likely see an almost straight line trending slightly downwards.

Hi Tom,

Thanks for the insight! Do you think it is worthwhile looking at other time series models? The idea is see if I can suggest a model than forecasts enrollment in the short term, at most a year or two ahead.

[TomDoan]

I assume that any good "prediction" model for enrollment would take into account tuition relative to similar schools, reputation, ... What a time series model can do (at best) is to show whether a prediction is unrealistic (needless to say, usually unduly optimistic).

You might want to take a look at the West and Harrison book. https://estima.com/webhelp/topics/textb ... stHarrison. They use small state-space models to look at practical examples of time series forecasting. (All ARIMA models are state-space models, but not vice versa).