Single exponential smoothing

Question

I think my question is quite simple and stupid:
What do we forecast using single exponential smoothing model: the next value of the observed time series or the next value of the level which lies in its basis?
If I got it correctly, an observed value of the time series with no trend is comprised of a level component and a stochastic one (which is impossible to predict).

I think it's quite possible using least squares to determine not only the optimal value of $\alpha$ (smoothing parameter) but the initial value $y_0$ for single smooing model as well. I've easily done that for some simulated data following formulae (3.10a), (3.10b) from Hyndman et al. Is it of any use for this model or just a nonsence?

Answer 1

1) The goal of exponential smoothing is to estimate the level. Since the level changes randomly from one period to the next, but the change has zero mean, this means that estimating the current value of the level, predicting the next value of the level, and predicting all future values of the level, all result in the same estimate / prediction - although of course the standard errors of those three estimations / predictions will be different.

"Estimating the level" is equivalent to "predicting the next value of $Y$", as the difference between the level and the next value of $Y$ is a zero-mean random number, so you can't make a better prediction than just using the estimate of the level. (Well, if you don't use squared-error loss, and your error distribution isn't normal, you often can, but that's not on this question's topic - and typically it's assumed that you are using squared-error loss and do have normal distributions.)

The difference between your two equations is essentially notational; when using exponential smoothing, you'll always use your most recent data point, and depending upon how you think about the problem, that can be denoted $Y_t$ or $Y_{t-1}$; in either case, though, it's the most recent observation, which is what really matters.

2) Your approach of jointly determining $y_0$ and $\alpha$ via least squares is just fine - much better than ignoring $y_0$ if you don't have much data.

Answer 2

I'm not quite sure if I understood right what you ask about. But I've seen two different implementations of exponential smoothing model. The first one can be called "forcasting manner". The current point $Y_t$ does not participate in determining its smoothed value $S_t$, which is determined by the preceeding point and its smoothed value:

$S_t=\alpha*Y_{t-1}+(1-\alpha)*S_{t-1}$

Majority of packages (including SPSS, STATISTICA) and textbooks adopt this manner. In the second, "predicting manner", the current point participates in determining its smoothed value:

$S_t=\alpha*Y_{t}+(1-\alpha)*S_{t-1}$

I saw this implementation in packages Statit and Stadia. You can transform the smoothed series left after one approach into the series left after the other approach by lagging or leading by one observation.