Why Simple Exponential Smoothing Constrains Alpha to Zero Through One

Summary

The document addresses why the smoothing coefficient in simple exponential smoothing is normally restricted to values between zero and one. In the stated forecasting recurrence, the next forecast combines the latest observation with the prior forecast, weighted by alpha and one minus alpha. Keeping alpha within the stated interval makes the update a convex combination, so the forecast lies between those two inputs. The response describes this restriction as part of the original model’s definition and says estimation software should enforce it.

The question arose after an ARIMA fit suggested a second difference, while a separately fitted exponential-smoothing model yielded alpha above one. The answer notes that a coefficient outside the usual range can indicate trend behavior and suggests double exponential smoothing as a more suitable model to consider. This is a brief conceptual response, not a derivation or empirical comparison; it does not establish that every alpha estimate above one proves a trend or provide a full model-selection procedure.

Key ideas

• Simple exponential smoothing defines alpha between zero and one to form a convex combination.
• Within that interval, each forecast lies between the latest observation and the previous forecast.
• An estimate above one falls outside the stated simple-smoothing definition.
• The response suggests considering double exponential smoothing when trend behavior is present.

Tags

# Exponential Smoothing - Alpha greater than 1


# Exponential Smoothing - Alpha greater than 1












Simple stats question.

I'm having trouble finding anything in the literature as to why the smoothing coefficient can never be greater than 1. This question was started by me doing time series ARIMA model. I estimated the model would be (0,1,1) or exponential smoothing, turned out it was (0,2,0). I decided to model it as exponential smoothing anyways and found that the alpha was about 1.4.

Where $Forecast(t+1)=\alpha Actual(t)+(1-\alpha)Forecast(t)$

Doing some rough googling I'm told alpha isn't supposed to be greater than 1 but no actual reasons are given. If someone can provide some insight or point me in the right direction I'd appreciate that.

## Answer by Dave Harris (score 1)

https://quant.stackexchange.com/a/46007

The answer is at least, in part, definitional. The original definition of the process constrains the model to $0<\alpha<1$ to assure a convex combination of the two terms. It assures that the prediction is between the two values at all times. Software to estimate the solution should be properly constrained so that a result of 1.4 cannot happen. The presence of an $|\alpha|>1$ implies the existence of a trend so that you should at least be using double exponential smoothing.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.