# Re: Residuals from AR1xAR1 model

From: Arthur <asremlforum_at_VSNI.CO.UK>
Date: Thu, 25 Sep 2008 20:43:03 +0100

Hi Kevin,

I'm glad to see someone adding a bit of interest to this forum.
Hopefully others will contribute to this discussion.

Haslett and Haslett (I say a draft paper, 2006, H\&H, hopefully it is published now) review residuals
and identify three basic kinds: marginal, model and conditional residuals.
The marginal residual is $\vec{\hat{e}}=\vec{y}-\vec{X}\vec{\hat{\tau}}$;
the model or innovation residuals ($\upsilon$) pertain to a particular structural variance model such as the innovation error in an AR process; the fully conditional residual ($\tilde{e}_{(i)}$) sometimes referred to as drop-one residuals and are the residuals obtained from predicting the observation ignoring the actual observation (estimated using the out() term in ASReml as a fixed effect.

In ASReml, I have used the term residual loosely to refer to the random term in the model that has variance structure R. Under a tighter definition,
the residual can be regarded as Zu+e (wth variance R+ZGZ')
which is the residual from the fixed model.

I am unaware of any rule that says residuals should be independent any more than that they should have equal variance. Nevertheless, most of us were first introduced to residuals as IID Normal variables in a simle fixed model setting.

So, yes under a typical AR1 x AR1 model, the 'residuals are correlated.
Now you could add to the model a random term 'units' which are assumed IID and are residuals in that sense though fitted in the G structure.
And of course you could fit the A1 x AR1 'residuals' as random effects and
the IID residuals as the deviations from the fitted model.

So, it is convenient to talk about residuals in a loose sense, but when looking at their atributes, we need to think about their correlation structure and exactly where they fit into the mixed model.

Like many things in fixed models, they do not always carry over simly to mixed models.

Regards to the family

Arthur

------------------------
Arthur Gilmour

Principal Research Scientist (Biometrics)
NSW Dept Primary Industries

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