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.

• Sorry, cant put this into comment: What language you working in? The concept you want is called 'moving window'. What is it you are trying to achieve? Looking at the coefficients? – Jan Sila Jul 5 '16 at 12:55
• This is an optimisation question. 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 – vkrouglov Jul 5 '16 at 13:00
• I am also 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. – vkrouglov Jul 5 '16 at 13:02
• @BehrouzMaleki I know, but dont have 50 reps yet. vkrouglov: What do you mean by true values? Never heard of a model having 'true' coefficient values, unless you simulate it, but yours seem to be empirical ...the only convergence I know of GARCH is that their forecasts converges to unconditional volatility. Also I don't think (99% sure) you can update a model coefficients, that is a result of a fit to a dataset, with a new observation without reculculating the model (the fit). That doesnt make sense to me.. – Jan Sila Jul 5 '16 at 13:13
• @Quantuple, that is indeed the way to go. It is almost trivial to prove that if $\theta_i$ is a sequence of updates in a (quasi)Newton scheme then it converges to the true value of parameters. – vkrouglov Jul 7 '16 at 12:38

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.