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Is there any justfification for producing a standard error of a single exponentially weighted coefficient?

If yes, how can we interpret the p-value?

Background

I use SAS ETS to estimate a single exponential smoothing model. When i select the optimize option for the estimation of the smoothing weight I get the estimated value (optimal), a standard error, a t-value and a p-value. How is the standard error calculated (e.g. what is the formula for it) and what is the menaing of the p-value. In the users guide it says that:

The standard errors associated with the smoothing weights are calculated from the Hessian matrix of the sum of squared, one-step-ahead prediction errors with respect to the smoothing weights used in the optimization process.

As far as I know the optimal weight is calcualted with a non-linear optimization procedure (heuristcs search with objective function to be minimzed the MSE or somthing similar). Based on this the statistical concepts are not applicable in this case (e.g. if we repeat the calculations in another sample then in the 95% of the cases etc.) So what is the meaning of the p-value and the standard error calcualtion? Do we have to check whether the smoothing coefficient is significant or not or we just disregard the standard error and the p-values?

Is there any use of the standard error, the t-value and the p-value?

Andy W
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Andreas Zaras
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1 Answers1

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There is an underlying statistical model where the SES weight is a parameter. So one use for the standard error is to get a confidence interval on the parameter. However, since the parameter is not really of direct interest, there is little point in doing this. The p-value on the test for whether the parameter is equal to zero may be of interest in testing whether the underlying process is a random walk. But there are better tests for that.

So while there is some meaning in these calculations, there is little value to be gained. Consequently, I don't think there is much point in providing the standard errors or p-values in software output.

Rob Hyndman
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  • Hi Rob, Thanks for your answer. Do you have in mind any material on the web or somewhere else on the method used to calculate the standard error (The standard errors associated with the smoothing weights are calculated from the Hessian matrix of the sum of squared, one-step-ahead prediction errors with respect to the smoothing weights used in the optimization process)? – Andreas Zaras Sep 13 '11 at 06:34
  • Most likely SAS is assuming an ARIMA(0,1,1) model with Gaussian errors for the SES forecasts. So look at a time series book on ARIMA models. Note that other models are possible. See Chatfield et al (JRSSD, 2001). – Rob Hyndman Sep 13 '11 at 07:35