2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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? $\endgroup$ – Jan Sila Jul 5 '16 at 12:55
  • $\begingroup$ 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 $\endgroup$ – vkrouglov Jul 5 '16 at 13:00
  • $\begingroup$ 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. $\endgroup$ – vkrouglov Jul 5 '16 at 13:02
  • $\begingroup$ @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.. $\endgroup$ – Jan Sila Jul 5 '16 at 13:13
  • 1
    $\begingroup$ @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. $\endgroup$ – vkrouglov Jul 7 '16 at 12:38
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ +1 for the nice answer. I think this is indeed the way to go $\endgroup$ – Quantuple Jul 6 '16 at 7:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.