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I'm trying to understand how ETS selects whether to use a damped model via information criteria (I'm not sure which of AIC, AICc or BIC are used).

I have a time series and I'm comparing two ETS models, one that sets damped = TRUE and the other that sets damped = FALSE. Visually, the model with damped = FALSE provides a better fit by capturing the trend.

Undamped:

undamped

Damped:

damped

Yet by AIC, AICc and BIC, damped provides a much better fit. Why do these penalized likelihoods prefer the damped model instead of the undamped model?

Undamped:

ETS(M,A,N) 

Call:
 ets(y = daily_max, damped = FALSE) 

  Smoothing parameters:
    alpha = 0.1162 
    beta  = 0.0025 

  Initial states:
    l = 3.703 
    b = 0.4918 

  sigma:  0.2977

     AIC     AICc      BIC 
7703.744 7703.829 7726.542 

Damped:

ETS(M,Ad,N) 

Call:
 ets(y = daily_max, damped = TRUE) 

  Smoothing parameters:
    alpha = 0.088 
    beta  = 1e-04 
    phi   = 0.8773 

  Initial states:
    l = 0.4713 
    b = 2.8166 

  sigma:  0.3013

     AIC     AICc      BIC 
7680.163 7680.283 7707.521 
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The AIC and AICc are optimal for one-step forecasting. Over the course of your historical data, the damped model would do reasonably well, with the local trend sometimes heading up and sometimes heading down. If you think the trend near the end of the series will continue into the forecast period, then just set damped=FALSE.

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    $\begingroup$ Can you explain why "The AIC and AICc are optimal for one-step forecasting?" Does this imply that some other criterion is more appropriate for a longer horizon? $\endgroup$ – Rylan Schaeffer Dec 10 '18 at 22:12
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    $\begingroup$ Asymptotically, minimizing the AIC is equivalent to minimizing the cross-validated one-step forecast MSE (Stone JRSSB 1977 jstor.org/stable/2984877). It is this property that makes the AIC so useful in model selection when the purpose is prediction. $\endgroup$ – Rob Hyndman Dec 11 '18 at 0:49

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