From the literature I gather that exponential smoothing models can be recast as special cases of state space models. I haven't seen similar references w/r to ARIMA being considered state space models, yet the Statsmodels Python library includes SARIMAX in it's state space methods.

So are both ARIMA and Exponential Smoothing special cases of state space models? Is there work being done towards a unified state space approach that combines both families of models?

  • $\begingroup$ You may find the Basic Structural Model interesting; it's a fairly intuitive state space model family that includes the model corresponding to exponential smoothing but is considerably more general. It doesn't incorporate all of ARIMA but there's a lot of overlap. Andrew Harvey's 1989 book Forecasting. Structural Time Series Models and the Kalman Filter is a nice introduction. There are a number of other time-series model families that are state space models. $\endgroup$ – Glen_b Jan 22 '18 at 0:12

Yes indeed: both exponential smoothing and ARIMA are special cases of state space models. For ARIMA, see this talk by Rob Hyndman, and for Exponential Smoothing, see Forecasting with Exponential Smoothing - the State Space Approach. This underlies the fact that specific Exponential Smoothing methods can be shown to yield MSE-optimal point forecasts for certain ARIMA data generating processes, and vice versa.

Rob Hyndman works in general in a state space framework, as do other forecasters. The ARIMA and Exponential Smoothing special cases will not die out, because they are well established forecasting paradigms that are more easily explained than the general state space formulation. However, state space formulations allow "natural" extensions of the "classical" approaches, e.g., to including causal effects or complex seasonality.

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