# Forecasting with exponential smoothing - impact of regression to the mean? Assumptions for forecasting?

I have a question regarding the impact of regression to the mean when using forecasting with exponential smoothing. Unfortunately, I do not have access to the book on which this analysis was based [@Rob Hyndman, Forecasting with Exponential Smoothing: The State Space Approach (New York: Springer, 2008)], nor am I familiar with forecasting with exponential smoothing (FWES).

A report was submitted to me which uses FWES to evaluate the success of a school truancy program. Numbers of absences per week were collected for each week 10 weeks prior to the intervention and again for the following 5 weeks. There is no control group, and all data are from people who received an intervention for a given number of unexcused absences. The number of unexcused absences in the 10 weeks prior to the intervention were used to forecast the expected weekly number of absences for the next 5 weeks. The slope of this line was then compared to the actual number of weekly absences.

My problem with this analysis is that as far as I can tell, by definition, the intervention is going to be given at the peak, that is, the point at which the student's truancy has reached a high enough level to warrant an intervention. The weeks prior to this there would be fewer absences. As you might expect, the line in the initial 10 weeks shows a steady (jagged) increase and following the intervention, the forecasted line shows a continued increase (albeit, a slightly slower increase), while the actual number of absences plummets.

The student who conducted these analyses assures me that forecasting with exponential smoothing accounts for regression to the mean, but I don't see how it can. Also, I am wondering what the assumptions are for FWES - for example, is random selection necessary, and is that a valid explanation for why these 10 weeks of data can't create a valid forecast? Or is this in fact a valid analysis?

## 1 Answer

Exponential moving averages overweight the most recent observations by a factor $\alpha$ to forecast the next value in a time-series, thus relying mainly on the innovation process found in the recent observations. In plain english, we can say that the larger the difference in the most recent observations (compared to their previous), the larger the forecast will diverge from the long term trend.

This is quite different than optimizing for $\mathbb{E}[X_{t+1}|X_{t...}]$, where the regressor optimization is based on the correlation between $X$ and the previous values, rather than the innovation process. Specifically, an exponential smoothing forecast is equivalent to an ARIMA(0,1,1) with the MA $\theta$ parameter matching $1 - \alpha$, while a regression model is equivalent to an ARIMA(1,0,0) model.

You can introduce an IV that quantifies the amount of intervention during the program in a regression model. The coefficient should have an inverse sign to the student's truancy value at $t-1$. An exponential moving average can only include data for the same series.

• So if I understand your answer correctly, not only will forecasting not account for regression to the mean, it will actually exacerbate the problem by weighting the most recent cases more heavily, when we know that the design assures that the most recent cases will be at their highest point? Jun 5, 2012 at 18:39
• Yes. I don't see how a simple exponential moving average can forecast the intervention, or even a cyclical pattern. It would be helpful if you can ask your student what does she mean by regression to the mean in exp. forecasting, and add her point to the question. Jun 5, 2012 at 19:02
• Thanks. I suspect that the student doesn't understand that choosing the correct method of analysis determines whether or not the results are valid. Instead, he seems to think that you plug the numbers into your formula and if the method is a valid method, then the numbers that come out will be valid. That's the reason I was wondering about the assumptions of this method. If there are contraindications for using FWES, that would help me to explain why his use is problematic, in a way that explaining regression to the mean has not been able to. Jun 5, 2012 at 19:38