In ARMA model we postulate that predictions of time series can be calculated as a linear function of $N$ previous observations (AR part) and $K$ differences between the previous observations and the corresponding predictions (MA part).

Are there extensions of this linear model in which we use a nonlinear function of previous observations and predictions?

Are there even more general models in which we also calculate some auxiliary parameters as nonlinear functions of previous observations, predictions and auxiliary parameters and then use these auxiliary parameters to calculate next predictions (and next values of the auxiliary parameters)?

If got it right, the sate space model goes in approximately this direction by introducing auxiliary (or hidden) parameters but in this model everything is linear. Are there nonlinear extensions of the state space model?


I just found here the following statement:

If the time series is Gaussian (i.e., normally distributed) then the best linear forecast is in fact the best of all possible forecasts: No nonlinear forecast can do better in terms of mean squared prediction error. Thus, as long as the series is Gaussian, we need look no further than the linear methods (e.g., ARMA forecasting) already presented.

Can somebody explain how Gaussianity of noise is related to linearity of the model. Why we cannot have a nonlinear function of previous observations as an estimation of mean and then have a Gaussian noise on top of that. For example: $y_i = \alpha \cdot y_{i-1}^2 + \beta \cdot y_{i-2}^3 + n(0, \sigma^2)$

  • $\begingroup$ You can certainly estimate a non-linear model like the one you have shown above. The question is, why would you want to? Normality and linearity are chosen for convenience, they are related because linear functions of independant normal variables equate to other normally distributed variables. The normal likelihood is also globally convex and relatively easy to estimate over. Many other distributions do not share these properties. There are extensions though. for example, stochastic volatility models are nonlinear state space and regime switching results in non-normal forecast distributions. $\endgroup$ – Zachary Blumenfeld Dec 2 '15 at 11:19
  • $\begingroup$ @ZacharyBlumenfeld, concerning "why do I want to use a nonlinear model". The only reason is that an assumption of linearity and normality in many realistic cases might be to strict. So, just to get a better predictions one need to go beyond these assumptions. $\endgroup$ – Roman Dec 2 '15 at 12:09
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    $\begingroup$ To the last sentence in the question: The statement starts with "If the time series is Gaussian" and is an assumption on the time series itself not on the residual process. $\endgroup$ – Josef Dec 2 '15 at 13:25
  • $\begingroup$ Consider linking your other post (corresponding to the ADDED part) to this one, and vice versa. $\endgroup$ – Richard Hardy Dec 2 '15 at 19:54

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