# Gaussian ARMA process with non-gaussian innovations

Can a gaussian ARMA process have non-gaussian innovations? (i.e., is there an ARMA process that is gaussian, but the corresponding innovations are not gaussian)?

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No............. – Rob Hyndman Mar 9 '12 at 1:30
many thanks for your comment. but could you provide a sort of explanation for your answer (or a hint to literature that provides a formal proof or something the like?) – s_2 Mar 10 '12 at 3:19
@mpiktas has provided the details in his answer. – Rob Hyndman Mar 10 '12 at 4:07

If a process is Gaussian, then any linear combination of its elements is Gaussian too. Take for example the AR(1) process:

$$X_t=\phi X_{t-1}+\varepsilon_t$$

Then

$$\varepsilon_t=X_t-\phi X_{t-1},$$

i.e. the innovation is a linear combination of the process elements, hence it should be Gaussian if $\{X_t\}$ is Gaussian.

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@Rob, thanks for edits. The articles in English will remain a mystery for me :) – mpiktas Mar 9 '12 at 7:11
thanks for your comment. i thought something along the same lines... but does it hold for a general ARMA(p,q) process as well? if so, how would you prove the statement formally (w/o assuming invertibility, causality etc.) – s_2 Mar 10 '12 at 3:21
what if there is an MA part too? – s_2 Mar 10 '12 at 14:10

Alternatively you could check out literatures on Generalized Autoregressive Moving Average Model (GARMA), which extend univariate ARMA models to non-Gaussian situations, which extends the Generalized Linear Modelds (exponential family distributions) to incoproate time dependence in the observations. I assume this is what you are interested. Gaussian ARMA is with Gaussian error process, but in GARMA you can assume non-Gaussian distributions in the error process.

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Sorry, I am not able to put comments for this post. If this is regarded as a comment, please someone helps to convert it. Also some time series data, like tourists or airline passage data, they traditionally were modeled using seasonal ARIMA models. But as they are also count data, you can model them using GARMA with poisson process too. – Fred Mar 9 '12 at 9:30
thanks for the comment! – s_2 Mar 10 '12 at 3:21