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As far as I know estimates of parameters of ARMA(1,1) are asymptotically optimal when fitted to data from ARMA(1,1)-GARCH(1,1) process, and only their variance increase, so when we assume large dataset, what kind of other issues could arise with estimators of ARMA(1,1) parameters ? If estimators are asymptotically optimal then does fitting first ARMA(1,1) and then GARCH(1,1) to obtained residuals of ARMA(1,1) is valid procedure ? What about variance of this whole procedure ?

I ask this question because, I often meet examples, when models are stacked one onto another and first one is called "filter" like the real estimation is conducted only in second step, but for me stacking models seems to increase overall variance, and is used because its more flexible (like, we don't have implemented procedure for estimation of SARIMAX-GARCH so first first we estimate SARIMAX then GARCH). For convenience we could assume AR(1) instead of ARMA(1,1).

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    $\begingroup$ Could you provide a reference for your statement estimates of parameters of ARMA(1,1) are asymptotically optimal when fitted to data from ARMA(1,1)-GARCH(1,1) process? I am not sure this is correct. While the coefficients of an autoregressive process AR($p$) can be estimated consistently under conditionally heteroskedastic errors (the basic argument being that OLS estimates are still consistent under heteroskedastic errors), this might not hold for an autoregressive moving-average process ARMA($p,q$). $\endgroup$ – Richard Hardy Feb 26 '16 at 17:16
  • $\begingroup$ @Richard Hardy I'm not sure so I ask here $\endgroup$ – Qbik Apr 20 '16 at 19:36

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