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Why GARCH(1,1) Maximum Likelihood Estimates Need Batch Refitting

Article Quant Q&A · Author: vkrouglov
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Summary

The document asks whether GARCH(1,1) parameters can be updated as each new price arrives, without rerunning the full estimation. Its answer is that maximum likelihood estimates depend on the full likelihood across observations. Because the GARCH recursion makes a new observation affect likelihood values for earlier points, the existing optimum cannot generally be adjusted with a simple score update.

For repeated estimation, the suggested practical shortcuts are to initialize each new fit with the previous parameter estimates or to hold parameters fixed between periodic refits. These approaches reduce effort but do not provide an exact incremental maximum likelihood solution. The discussion also cautions that the data generating process is unobservable, so estimated coefficients are approximations rather than recoverable true parameters. It offers no empirical comparison of the shortcuts or convergence guarantees for them.

Key ideas

  • A new observation can change earlier likelihood contributions in a recursively specified GARCH(1,1) model.
  • Maximum likelihood parameter estimates therefore generally require refitting on the expanded dataset.
  • Using prior estimates as starting values can make repeated full fits easier.
  • Periodic refitting trades update frequency for reduced computation.
  • Estimated parameters approximate an unobserved data generating process.

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VolatilityStatisticsBacktesting
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# Streaming update of the GARCH(1,1) model


# Streaming update of the GARCH(1,1) model












Given the estimate of GARCH(1, 1) model parameters I observe the new price. How to update the estimate with this new information.

Let's assume I know the coefficients that maximize the likelihood given the data up to the time $T$. At time $T+1$ the new price is observed and I wish to update the coefficients without recomputing the full model

I am looking for the asymptotic convergence of the coefficients - at each time step $T$ I am OK to update the coefficients in suboptimal way but I want them to converge to the true values at infinity.

## Answer by Malick (score 3, accepted)

https://quant.stackexchange.com/a/27940

If you estimate your model via Maximum Likelihood method, you are forced to re-estimate the full model. This is due to the fact that estimates are values which maximize the full likelihood, the latter being based on a recursive algorithm which use all observations (including the new one) and implies that a new observation may also impact likelihood values of previous points. There is no way to find a kind of score vector to update your estimates.

> However, if you need to update several times your model, you can facilitate the estimation by fixing the starting values to the previous estimates at each step you re-estimate the model. Another rough method is to assume constant your parameters for a period of $x$ observations, and to re-estimate the model every $x$ points.

When you are talking about the 'true' estimate, don't miss the point that we never observe the true Data Generating Process (DGP) and so that we can't find these 'true' estimates. Your estimates at time t are only an approximation of the DGP at time t.

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