Two-Stage ARMA and GARCH Estimation Under Conditional Heteroskedasticity

Summary

This document frames a statistical modeling question about fitting an ARMA mean process to data generated by an ARMA-GARCH process. It asks whether the ARMA parameter estimates remain asymptotically optimal when conditional variance changes over time, and whether estimating the ARMA model first and fitting GARCH to its residuals afterward is valid. It also raises a practical concern: sequential estimation may affect the variance and uncertainty of the overall procedure.

The discussion is motivated by workflows that use a first model as a filter before fitting a second, sometimes because joint estimation is unavailable for a chosen model combination. The document provides no answer, derivation, simulation, or empirical evidence, so it does not establish the consistency or efficiency of either procedure. Its value is in identifying questions a researcher should resolve, including how first-stage estimation uncertainty carries into residual-based second-stage inference.

Key ideas

• The question concerns ARMA parameter estimation when errors have GARCH conditional variance.
• Sequentially fitting a mean model and then a variance model raises uncertainty propagation concerns.
• A first-stage filter may be used for computational convenience when joint estimation is unavailable.
• The document poses these issues but supplies no conclusion or supporting evidence.

Tags

# What kind of errors arise when I fit ARMA(1,1) to data generated from ARMA(1,1)-GARCH(1,1) process?


# What kind of errors arise when I fit ARMA(1,1) to data generated from ARMA(1,1)-GARCH(1,1) process?












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.