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I am interested in fitting an ARMA-GARCH model to my data. After reading a few pages online I did so sequentially by first applying ARMA and then feeding the residuals into GARCH. I then took the estimated mean from ARMA and variance from GARCH to construct a forecast. However, I have not achieved too great of a result and after some further reading it seems this method is acceptable though not ideal. Others suggest the ARMA-GARCH parameters should be instead fit simultaneously.

Will doing so result in a different output or is it a matter of computational efficiency?

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Simultaneous estimation will surely produce different output than stepwise estimation. The conditional mean model is estimated assuming a GARCH-type conditional variance in the simultaneous case but a constant variance in the stepwise case, yielding different optima. The conditional variance model is estimated in line with the first optimum in the simultaneous case but in line with the second optimum in the stepwise case, again yielding different results in the two cases. There is no guarantee you will get "nicer" results (more significant coefficients and/or better behaved standardized errors) form simultaneous estimation, but conceptually this is more sensible approach as it avoids making conflicting assumptions at the different stages (unlike the two-step approach). See some related threads where this has been discussed before.

In terms of computational efficiency, I think stepwise estimation could be preferred. In stepwise estimation, fewer parameters are being optimized at a time, so the optimization task is easier and convergence is achieve with less trouble. Of course, there are two stages, so we have two easy tasks instead of a single, difficult one, and thus it could probably go either way.

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