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I've seen so far two definitions of the term "auto-regressive" model when it comes to time series modeling:

  • The first definition is just basic AR models and their relatives such as ARMA and ARIMA, where the relationship between the future values and the past value is linear and the AR term can be written as $Y_{t} = a_1Y_{t-1} + a_2Y_{t-2} +...+ a_nY_{t-n}$.

  • The second definition extends to the general non-linear case, for example as used in Bergmeir et al. "A Note on the Validity of Cross-Validation for Evaluating Autoregressive Time Series Prediction" or as used in describing Auto-Regressive neural networks. In this case any model that describes $Y_t = f(Y_{t-1},Y_{t-2},...,Y_{t-n})$ is auto-regressive.

But based on the second definition, why aren't models of the exponential smoothing family considered auto-regressive? When you expand the equation for exponential smoothing, you eventually end up with a non linear function of the form:

$Y_t = f(Y_{t-1},Y_{t-2},...,Y_{0})$ (or if you want to be nit-picky $Y_t = f(Y_{t-1},Y_{t-2},...,Y_{t-\infty})$)

So an exponential smoothing model is also auto-regressive, no?

In fact, isn't just about any univariate time series model that doesn't include exogenous variables auto-regressive?

What am I missing here?

My motivation here is to understand why the result in Bergmeir et al. applies to some models and not others.

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For an autoregressive model, non-linear or linear, the number of lags must be finite. An ETS(A,N,N) model can be written as an AR($\infty$) model, but not as an autoregressive model with finite lags. A few other exponential smoothing models can be written in AR($\infty$) form, but none can be written as an autoregressive model with finite lags. See https://otexts.org/fpp2/arima-ets.html for the details.

The results in my paper with Bergmeir and Koo require you to estimate the model using a finite set of autoregressive predictors, so the number of lags must be much less than the number of observations.

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