# 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? Commented Jul 5, 2016 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 Commented Jul 5, 2016 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. Commented Jul 5, 2016 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.. Commented Jul 5, 2016 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. Commented Jul 7, 2016 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.