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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.

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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.

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    $\begingroup$ Hi: Statistically speaking, it's a only a myth that the smoothing parameter in SES has to be between zero and one. ( what david said about convexity is true but not requited ). Using an $\alpha$ of 1.4 is not a problem. But, if you are going to use 1.4, then (1 - \alpha ) = -0.4 which you might or might not want. It sounds like you might be interested in using a koyck distributed lag because that will introduce another parameter into the SES model, namely $\beta$, that adds flexibility. Google for "koyck distributed lag" if you're interested. $\endgroup$ – mark leeds Jun 10 at 1:00
  • $\begingroup$ Note that another reason for the "myth" is because, if the smoothing parameter is not between zero and one, then the equivalent ARIMA(0,1,1) is not invertible. But, invertibility is a theoretical construct that doesn't necessarily have to be adhered to in practice. $\endgroup$ – mark leeds Jun 10 at 13:03
  • $\begingroup$ @markleeds when I originally planned the post, I considered adding the $|\alpha|>1$ case, but there is a non-trivial problem which you refer to regarding invertibility. I don't think that is a trivial issue. I would have to find the article, but Fisher in 1934 uses a similar, non-time series, problem to show how dangerous the MVUE can be in a similar case in that the estimators are unbiased but the sampling distribution is so wide as to be dangerous. The language he used was "accurate but not correct." It concerns me more because of the local optimization used. $\endgroup$ – Dave Harris Jun 10 at 20:34
  • $\begingroup$ @markleeds do you have an article you could cite on why I shouldn't be concerned. $\endgroup$ – Dave Harris Jun 10 at 20:35
  • $\begingroup$ Hi David: lack of invertibility just means that the model can't be written as AR model. But if you don't consider the equivalence of ES and ARIMA(0,1,1) and just think of the ES model as a model itself, then the question of invertibility doesn't arise. In fact, even the notion that the two weights have to sum to one is just an adage due to the fact that generally speaking, one intuitively wants weights to add to 1.0. One can put any weight on the current value and the previous forecast, they don't need to sum to 1. $\endgroup$ – mark leeds Jun 11 at 12:03

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