Two crucial sources for time series analysis differ in a critical formula for equivalence between simple exponential smoothing (SES) and ARIMA(0, 1, 1).

From Hyndman's F:PP:

$\theta_1 = \alpha - 1$

and from R. Nau's Statistical Forecasting:

$\theta_1 = 1- \alpha$

Can someone help me understand the apparent discrepancy?

  • $\begingroup$ The ETS model formulation is not the same as simple exponential smoothing. SES relies on a $-$ sign in front of the term involving the error but ETS has a $+$ sign in front, hence the relationship between the two (different) $\theta_1$s. $\endgroup$
    – jbowman
    Feb 27, 2018 at 21:25

1 Answer 1


There are two common parameterizations of ARIMA models. My book follows the R convention, while Nau follows the Box-Jenkins convention. The formulations are both correct and equivalent.

  • 1
    $\begingroup$ Would you happen to know which parameterization is used by the python statsmodel statespace library? $\endgroup$
    – Brian R
    Feb 27, 2018 at 21:38
  • $\begingroup$ No idea, and the online documentation doesn't seem to clearly specify it. $\endgroup$ Feb 27, 2018 at 23:09

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