Hi all, We followed some books and sample codes and did some EMM estimation, only to find it won't be able to generate forecast. This is because in the stochastic volatility models we are estimating, the volatilities are latent variables, and we want to forecast 1-step ahead or h-step ahead volatilities. So it is nice to have the system estimated, but we couldn't get it to forecast at all. There is a "Reprojection" Method described in the original EMM paper, but let's say we reproject to a GARCH(1,1) model, then only the GARCH(1, 1) parameters are significant, which basically means we degrade the SV model into a GARCH model. There is no way to do the forecast... Could anybody give some pointers? Thanks!
EMM: how to make forecast using EMM methods?
11 messages · elton wang, Mark Leeds, Patrick Burns +4 more
I've heard opinions that GARCH/SV volatility models are not better on forecasting than simple exponential moving average volatilities or even rolling window historical vol. Any practitioners mind comment?
--- Michael <comtech.usa at gmail.com> wrote:
Hi all, We followed some books and sample codes and did some EMM estimation, only to find it won't be able to generate forecast. This is because in the stochastic volatility models we are estimating, the volatilities are latent variables, and we want to forecast 1-step ahead or h-step ahead volatilities. So it is nice to have the system estimated, but we couldn't get it to forecast at all. There is a "Reprojection" Method described in the original EMM paper, but let's say we reproject to a GARCH(1,1) model, then only the GARCH(1, 1) parameters are significant, which basically means we degrade the SV model into a GARCH model. There is no way to do the forecast... Could anybody give some pointers? Thanks!
_______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
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I can't say much about Garch/SV being better or worse but I know that's there an approximate functional equivalence between exponential smoothing and a regular moving average ( i.e: rolling window ). It's something like lambda = 1/(2n +1) or something like that but I don't remember. It's in any decent technical analysis book and it's true empirically because I've played around with it in the past. -----Original Message----- From: r-sig-finance-bounces at stat.math.ethz.ch [mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of elton wang Sent: Thursday, February 28, 2008 4:36 PM To: r-sig-finance at stat.math.ethz.ch; r-help Subject: Re: [R-SIG-Finance] EMM: how to make forecast using EMM methods? I've heard opinions that GARCH/SV volatility models are not better on forecasting than simple exponential moving average volatilities or even rolling window historical vol. Any practitioners mind comment?
--- Michael <comtech.usa at gmail.com> wrote:
Hi all, We followed some books and sample codes and did some EMM estimation, only to find it won't be able to generate forecast. This is because in the stochastic volatility models we are estimating, the volatilities are latent variables, and we want to forecast 1-step ahead or h-step ahead volatilities. So it is nice to have the system estimated, but we couldn't get it to forecast at all. There is a "Reprojection" Method described in the original EMM paper, but let's say we reproject to a GARCH(1,1) model, then only the GARCH(1, 1) parameters are significant, which basically means we degrade the SV model into a GARCH model. There is no way to do the forecast... Could anybody give some pointers? Thanks!
_______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
____________________________________________________________________________ ________ Looking for last minute shopping deals? _______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
Michael, If I understand correctly, you've used some EMM algorithms to estimate the parameters of a stochastic volatility model. If this is the case you should now be able to use Monte Carlo methods to generate forecasts from your model. That is, you will generate random variables (according to the specifications of your model), feed them into your model and hence simulate your stochastic volatility process. Note sure what references you have been using but perhaps these would be helpful: Gallant, Hsieh and Tauchen (1997). "Estimation of stochastic volatility models with diagnostics", Journal of Econometrics, 81, 159-192. Andersen, T.G. H.-J. Chung, and B.E. Sorensen (1999). "Efficient Method of Moments Estimation of a Stochastic Volatility Model: A Monte Carlo Study," Journal of Econometrics, 91, 61-87. Best, -- G -----Original Message----- From: r-sig-finance-bounces at stat.math.ethz.ch [mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of Michael Sent: Thursday, February 28, 2008 12:56 PM To: r-sig-finance at stat.math.ethz.ch; r-help Subject: [R-SIG-Finance] EMM: how to make forecast using EMM methods? Hi all, We followed some books and sample codes and did some EMM estimation, only to find it won't be able to generate forecast. This is because in the stochastic volatility models we are estimating, the volatilities are latent variables, and we want to forecast 1-step ahead or h-step ahead volatilities. So it is nice to have the system estimated, but we couldn't get it to forecast at all. There is a "Reprojection" Method described in the original EMM paper, but let's say we reproject to a GARCH(1,1) model, then only the GARCH(1, 1) parameters are significant, which basically means we degrade the SV model into a GARCH model. There is no way to do the forecast... Could anybody give some pointers? Thanks! _______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
Hi Guy, Thanks for your help! Yes, we have the coefficient estimated using EMM. And we followed those papers. Just want to check my understanding about your suggestion: Do you mean that after we obtain the estimated coefficients, we run one simulation to obtain the whole sequence of latent variable (the volatility time series, from time 0 to time t+1), where time t is today, and t+1 is tomorrow(one step forecast); And that's one simulation. And we run such simulation for N times, let's say N=10000, and obtain 10000 such volatility time series, each ending at time t+1, and then we take average of the 10000 data points at t+1, the average will be the mean-forecast of the volatility tomorrow(i.e. that's the one step forecast that we want)... Am I right in doing these procedures? Thanks On Thu, Feb 28, 2008 at 4:30 PM, Guy Yollin
<guy.yollin at rotellacapital.com> wrote:
Michael, If I understand correctly, you've used some EMM algorithms to estimate the parameters of a stochastic volatility model. If this is the case you should now be able to use Monte Carlo methods to generate forecasts from your model. That is, you will generate random variables (according to the specifications of your model), feed them into your model and hence simulate your stochastic volatility process. Note sure what references you have been using but perhaps these would be helpful: Gallant, Hsieh and Tauchen (1997). "Estimation of stochastic volatility models with diagnostics", Journal of Econometrics, 81, 159-192. Andersen, T.G. H.-J. Chung, and B.E. Sorensen (1999). "Efficient Method of Moments Estimation of a Stochastic Volatility Model: A Monte Carlo Study," Journal of Econometrics, 91, 61-87. Best, -- G -----Original Message----- From: r-sig-finance-bounces at stat.math.ethz.ch [mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of Michael Sent: Thursday, February 28, 2008 12:56 PM To: r-sig-finance at stat.math.ethz.ch; r-help Subject: [R-SIG-Finance] EMM: how to make forecast using EMM methods? Hi all, We followed some books and sample codes and did some EMM estimation, only to find it won't be able to generate forecast. This is because in the stochastic volatility models we are estimating, the volatilities are latent variables, and we want to forecast 1-step ahead or h-step ahead volatilities. So it is nice to have the system estimated, but we couldn't get it to forecast at all. There is a "Reprojection" Method described in the original EMM paper, but let's say we reproject to a GARCH(1,1) model, then only the GARCH(1, 1) parameters are significant, which basically means we degrade the SV model into a GARCH model. There is no way to do the forecast... Could anybody give some pointers? Thanks!
_______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
(This is not being sent to R-help. It is considered impolite to cross-post messages, especially on topics that are purely financial.) As always, the answer is, "That depends." The key question is the time frame of the prediction. If the prediction is for a month or more, then there's unlikely to be much advantage in a fancy model. If the time frame is a few days, then something like a garch model will vastly outperform a rolling window. How much a garch model would outperform an exponential smooth depends on the smoothing parameter (an exponential smooth is a degenerate form of a garch model). As far as I know, there is not a clear winner between garch models and stochastic volatility models, but with some evidence that garch might be better. Corrections to this impression are certainly welcome. Patrick Burns patrick at burns-stat.com +44 (0)20 8525 0696 http://www.burns-stat.com (home of S Poetry and "A Guide for the Unwilling S User")
elton wang wrote:
I've heard opinions that GARCH/SV volatility models are not better on forecasting than simple exponential moving average volatilities or even rolling window historical vol. Any practitioners mind comment? --- Michael <comtech.usa at gmail.com> wrote:
Hi all, We followed some books and sample codes and did some EMM estimation, only to find it won't be able to generate forecast. This is because in the stochastic volatility models we are estimating, the volatilities are latent variables, and we want to forecast 1-step ahead or h-step ahead volatilities. So it is nice to have the system estimated, but we couldn't get it to forecast at all. There is a "Reprojection" Method described in the original EMM paper, but let's say we reproject to a GARCH(1,1) model, then only the GARCH(1, 1) parameters are significant, which basically means we degrade the SV model into a GARCH model. There is no way to do the forecast... Could anybody give some pointers? Thanks!
_______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
____________________________________________________________________________________ Looking for last minute shopping deals? _______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
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Hi Michael, Yes, this is what I'm suggesting. Bear in mind, your model estimation process should have also resulted in volatility estimates for t-1, t-2, etc. Your simulation will require one or more of these terms as input (in addition to the random innovations) since your stochastic volatility model will have lagged volatility terms. Good luck. -- G -----Original Message----- From: Michael [mailto:comtech.usa at gmail.com] Sent: Thursday, February 28, 2008 5:46 PM To: Guy Yollin; r-help; r-sig-finance at stat.math.ethz.ch Subject: Re: [R-SIG-Finance] EMM: how to make forecast using EMM methods? Hi Guy, Thanks for your help! Yes, we have the coefficient estimated using EMM. And we followed those papers. Just want to check my understanding about your suggestion: Do you mean that after we obtain the estimated coefficients, we run one simulation to obtain the whole sequence of latent variable (the volatility time series, from time 0 to time t+1), where time t is today, and t+1 is tomorrow(one step forecast); And that's one simulation. And we run such simulation for N times, let's say N=10000, and obtain 10000 such volatility time series, each ending at time t+1, and then we take average of the 10000 data points at t+1, the average will be the mean-forecast of the volatility tomorrow(i.e. that's the one step forecast that we want)... Am I right in doing these procedures? Thanks On Thu, Feb 28, 2008 at 4:30 PM, Guy Yollin
<guy.yollin at rotellacapital.com> wrote:
Michael, If I understand correctly, you've used some EMM algorithms to
estimate
the parameters of a stochastic volatility model. If this is the case you should now be able to use Monte Carlo methods
to
generate forecasts from your model. That is, you will generate random variables (according to the specifications of your model), feed them into your model and hence simulate your stochastic volatility process. Note sure what references you have been using but perhaps these would
be
helpful: Gallant, Hsieh and Tauchen (1997). "Estimation of stochastic
volatility
models with diagnostics", Journal of Econometrics, 81, 159-192. Andersen, T.G. H.-J. Chung, and B.E. Sorensen (1999). "Efficient
Method
of Moments Estimation of a Stochastic Volatility Model: A Monte Carlo Study," Journal of Econometrics, 91, 61-87. Best, -- G -----Original Message----- From: r-sig-finance-bounces at stat.math.ethz.ch [mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of Michael Sent: Thursday, February 28, 2008 12:56 PM To: r-sig-finance at stat.math.ethz.ch; r-help Subject: [R-SIG-Finance] EMM: how to make forecast using EMM methods? Hi all, We followed some books and sample codes and did some EMM estimation, only to find it won't be able to generate forecast. This is because in the stochastic volatility models we are
estimating,
the volatilities are latent variables, and we want to forecast 1-step ahead or h-step ahead volatilities. So it is nice to have the system estimated, but we couldn't get it to forecast at all. There is a "Reprojection" Method described in the original EMM paper, but let's say we reproject to a GARCH(1,1) model, then only the GARCH(1, 1) parameters are significant, which basically means we degrade the SV model into a GARCH model. There is no way to do the forecast... Could anybody give some pointers? Thanks!
_______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
But I doubt this is not a one-step forecast. For one-step cast, you only need start from today's value and simulate one step ahead. you need to use the orignal innovations as of today instead of simulating from day 1.
--- Guy Yollin <guy.yollin at rotellacapital.com> wrote:
Hi Michael, Yes, this is what I'm suggesting. Bear in mind, your model estimation process should have also resulted in volatility estimates for t-1, t-2, etc. Your simulation will require one or more of these terms as input (in addition to the random innovations) since your stochastic volatility model will have lagged volatility terms. Good luck. -- G -----Original Message----- From: Michael [mailto:comtech.usa at gmail.com] Sent: Thursday, February 28, 2008 5:46 PM To: Guy Yollin; r-help; r-sig-finance at stat.math.ethz.ch Subject: Re: [R-SIG-Finance] EMM: how to make forecast using EMM methods? Hi Guy, Thanks for your help! Yes, we have the coefficient estimated using EMM. And we followed those papers. Just want to check my understanding about your suggestion: Do you mean that after we obtain the estimated coefficients, we run one simulation to obtain the whole sequence of latent variable (the volatility time series, from time 0 to time t+1), where time t is today, and t+1 is tomorrow(one step forecast); And that's one simulation. And we run such simulation for N times, let's say N=10000, and obtain 10000 such volatility time series, each ending at time t+1, and then we take average of the 10000 data points at t+1, the average will be the mean-forecast of the volatility tomorrow(i.e. that's the one step forecast that we want)... Am I right in doing these procedures? Thanks On Thu, Feb 28, 2008 at 4:30 PM, Guy Yollin <guy.yollin at rotellacapital.com> wrote:
Michael, If I understand correctly, you've used some EMM
algorithms to estimate
the parameters of a stochastic volatility model. If this is the case you should now be able to use
Monte Carlo methods to
generate forecasts from your model. That is, you will generate random variables
(according to the
specifications of your model), feed them into
your model and hence
simulate your stochastic volatility process. Note sure what references you have been using but
perhaps these would be
helpful: Gallant, Hsieh and Tauchen (1997). "Estimation of
stochastic volatility
models with diagnostics", Journal of
Econometrics, 81, 159-192.
Andersen, T.G. H.-J. Chung, and B.E. Sorensen
(1999). "Efficient Method
of Moments Estimation of a Stochastic Volatility
Model: A Monte Carlo
Study," Journal of Econometrics, 91, 61-87. Best, -- G -----Original Message----- From: r-sig-finance-bounces at stat.math.ethz.ch [mailto:r-sig-finance-bounces at stat.math.ethz.ch]
On Behalf Of Michael
Sent: Thursday, February 28, 2008 12:56 PM To: r-sig-finance at stat.math.ethz.ch; r-help Subject: [R-SIG-Finance] EMM: how to make forecast
using EMM methods?
Hi all, We followed some books and sample codes and did
some EMM estimation,
only to find it won't be able to generate
forecast.
This is because in the stochastic volatility
models we are estimating,
the volatilities are latent variables, and we
want to forecast 1-step
ahead or h-step ahead volatilities. So it is nice to have the system estimated, but
we couldn't get it to
forecast at all. There is a "Reprojection" Method described in the
original EMM paper,
but let's say we reproject to a GARCH(1,1) model,
then only the
GARCH(1, 1) parameters are significant, which
basically means we
degrade the SV model into a GARCH model. There is
no way to do the
forecast... Could anybody give some pointers? Thanks!
_______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list
https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first. _______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
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Hi Guy and Elton, Thanks for the replies. However this is exactly the weird thing about EMM. It simulates the latent variables when doing the estimating itself. So there is no clear estimates of the latent variables itself. That's to say, I don't have a "today's value" to work out the 1-step ahead forecast... That's kind of strange... Any thoughts?
On Fri, Feb 29, 2008 at 7:37 AM, elton wang <ahala2000 at yahoo.com> wrote:
But I doubt this is not a one-step forecast. For one-step cast, you only need start from today's value and simulate one step ahead. you need to use the orignal innovations as of today instead of simulating from day 1. --- Guy Yollin <guy.yollin at rotellacapital.com> wrote:
> Hi Michael, > > Yes, this is what I'm suggesting. Bear in mind, > your model estimation > process should have also resulted in volatility > estimates for t-1, t-2, > etc. > > Your simulation will require one or more of these > terms as input (in > addition to the random innovations) since your > stochastic volatility > model will have lagged volatility terms. > > Good luck. > > -- G > > > -----Original Message----- > From: Michael [mailto:comtech.usa at gmail.com] > Sent: Thursday, February 28, 2008 5:46 PM > To: Guy Yollin; r-help; > r-sig-finance at stat.math.ethz.ch > Subject: Re: [R-SIG-Finance] EMM: how to make > forecast using EMM > methods? > > Hi Guy, > > Thanks for your help! Yes, we have the coefficient > estimated using > EMM. And we followed those papers. > > Just want to check my understanding about your > suggestion: > > Do you mean that after we obtain the estimated > coefficients, > > we run one simulation to obtain the whole sequence > of latent variable > (the volatility time series, from time 0 to time > t+1), > > where time t is today, and t+1 is tomorrow(one step > forecast); > > And that's one simulation. > > And we run such simulation for N times, let's say > N=10000, > > and obtain 10000 such volatility time series, each > ending at time t+1, > > and then we take average of the 10000 data points at > t+1, > > the average will be the mean-forecast of the > volatility tomorrow(i.e. > that's the one step forecast that we want)... > > Am I right in doing these procedures? > > Thanks > > > > On Thu, Feb 28, 2008 at 4:30 PM, Guy Yollin > <guy.yollin at rotellacapital.com> wrote:
> > Michael, > > > > If I understand correctly, you've used some EMM
> algorithms to > estimate
> > the parameters of a stochastic volatility model. > > > > If this is the case you should now be able to use
> Monte Carlo methods > to
> > generate forecasts from your model. > > > > That is, you will generate random variables
> (according to the
> > specifications of your model), feed them into
> your model and hence
> > simulate your stochastic volatility process. > > > > Note sure what references you have been using but
> perhaps these would > be
> > helpful: > > > > Gallant, Hsieh and Tauchen (1997). "Estimation of
> stochastic > volatility
> > models with diagnostics", Journal of
> Econometrics, 81, 159-192.
> > > > Andersen, T.G. H.-J. Chung, and B.E. Sorensen
> (1999). "Efficient > Method
> > of Moments Estimation of a Stochastic Volatility
> Model: A Monte Carlo
> > Study," Journal of Econometrics, 91, 61-87. > > > > Best, > > > > -- G > > > > > > > > > > -----Original Message----- > > From: r-sig-finance-bounces at stat.math.ethz.ch > > > > [mailto:r-sig-finance-bounces at stat.math.ethz.ch]
> On Behalf Of Michael
> > Sent: Thursday, February 28, 2008 12:56 PM > > To: r-sig-finance at stat.math.ethz.ch; r-help > > > > > > Subject: [R-SIG-Finance] EMM: how to make forecast
> using EMM methods?
> > > > Hi all, > > > > We followed some books and sample codes and did
> some EMM estimation,
> > only to find it won't be able to generate
> forecast.
> > > > This is because in the stochastic volatility
> models we are > estimating,
> > the volatilities are latent variables, and we
> want to forecast 1-step
> > ahead or h-step ahead volatilities. > > > > So it is nice to have the system estimated, but
> we couldn't get it to
> > forecast at all. > > > > There is a "Reprojection" Method described in the
> original EMM paper,
> > but let's say we reproject to a GARCH(1,1) model,
> then only the
> > GARCH(1, 1) parameters are significant, which
> basically means we
> > degrade the SV model into a GARCH model. There is
> no way to do the
> > forecast... > > > > Could anybody give some pointers? > > > > Thanks! > > > > > > > > _______________________________________________ > > R-SIG-Finance at stat.math.ethz.ch mailing list > >
> > -- Subscriber-posting only. > > -- If you want to post, subscribe first. > >
> > _______________________________________________ > R-SIG-Finance at stat.math.ethz.ch mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. > -- If you want to post, subscribe first. >
____________________________________________________________________________________ Be a better friend, newshound, and _______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.
For simple SV models (e.g. log normal ar(1)), the model can be written in state space form and the the Kalman filter may be used to forecast the latent volatility. See Harvey, Ruiz and Shephard's paper in ReStud for details. However, the Kalman filter is only the best linear forecast. In general, the SV models are non-linear and non-gaussian state space models and the optimal forecasting algorithms are given by the particle filter. I have a short paper that describes how to do this on my webpage http://faculty.washington.edu/ezivot/research/Creal_Gu_Zivot_2007.pdf -----Original Message----- From: r-sig-finance-bounces at stat.math.ethz.ch [mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of Guy Yollin Sent: Thursday, February 28, 2008 4:30 PM To: Michael; r-sig-finance at stat.math.ethz.ch Subject: Re: [R-SIG-Finance] EMM: how to make forecast using EMM methods? Michael, If I understand correctly, you've used some EMM algorithms to estimate the parameters of a stochastic volatility model. If this is the case you should now be able to use Monte Carlo methods to generate forecasts from your model. That is, you will generate random variables (according to the specifications of your model), feed them into your model and hence simulate your stochastic volatility process. Note sure what references you have been using but perhaps these would be helpful: Gallant, Hsieh and Tauchen (1997). "Estimation of stochastic volatility models with diagnostics", Journal of Econometrics, 81, 159-192. Andersen, T.G. H.-J. Chung, and B.E. Sorensen (1999). "Efficient Method of Moments Estimation of a Stochastic Volatility Model: A Monte Carlo Study," Journal of Econometrics, 91, 61-87. Best, -- G -----Original Message----- From: r-sig-finance-bounces at stat.math.ethz.ch [mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of Michael Sent: Thursday, February 28, 2008 12:56 PM To: r-sig-finance at stat.math.ethz.ch; r-help Subject: [R-SIG-Finance] EMM: how to make forecast using EMM methods? Hi all, We followed some books and sample codes and did some EMM estimation, only to find it won't be able to generate forecast. This is because in the stochastic volatility models we are estimating, the volatilities are latent variables, and we want to forecast 1-step ahead or h-step ahead volatilities. So it is nice to have the system estimated, but we couldn't get it to forecast at all. There is a "Reprojection" Method described in the original EMM paper, but let's say we reproject to a GARCH(1,1) model, then only the GARCH(1, 1) parameters are significant, which basically means we degrade the SV model into a GARCH model. There is no way to do the forecast... Could anybody give some pointers? Thanks! _______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first. _______________________________________________ R-SIG-Finance at stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. -- If you want to post, subscribe first.