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I have selected the best ARIMA(p,d,q) model via maximisation of AIC and got the estimates of its AR & MA coefficients.

However, after having subsequently modelled the conditional volatility with a GARCH(1,1) via rugarch package in $R$ (and having previously specified to consider ARMA(5,5) for conditional mean) the estimates for conditional mean comes out to be different!

Questions:

  • Is it normal? Why just adding a model for conditional variance I'm affecting the conditional mean model?
  • Should I force rugarch to use the the same estimates I got before by modelling just the residuals of the ARMA(5,5) model with GARCH? Would I get "better estimates" or not?
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  • $\begingroup$ In addition to the answer linked above, here is an intuitive argument. GARCH implies the observations have different conditional variances at different time points. Answer to Q1: Efficient estimation of a conditional mean model requires weighting the observations inversely proportionally to the conditional standard deviation. Answer to Q2: You should not force stepwise estimation as it is inefficient and moreover can be inconsistent. $\endgroup$ – Richard Hardy Apr 29 '18 at 12:15
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As discussed in the thread "ARMA/GARCH estimation in sequence", joint estimation of the conditional mean and variance models is the preferred choice due to its greater efficiency and possibly some consistency issues in stepwise estimation.

Answer to Q1:
GARCH implies the observations have different conditional variances at different time points. Efficient estimation of a conditional mean model requires weighting the observations inversely proportionally to the conditional standard deviation.

Answer to Q2:
You should not force stepwise estimation as it is inefficient and may be inconsistent. The estimates from stepwise estimation would thus be inferior to those from joint estimation.

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