Hi
Wonder if anyone could point me how I use this method to discover the half life of a mean reverting process. I am looking into pair trading and the time it takes for a cointegrated pair to revert to the norm. --
Stephen Choularton Ph.D., FIoD 9999 2226 0413 545 182 for insurance go to www.netinsure.com.au for markets go to www.organicfoodmarkets.com.au _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
By half-life, do you mean the speed of mean-reversion?
If so, there's a bit of algebraic tomfoolery that's required to discretise the equation and then fit the data to it. I don't have the time right now to go into all the details but it's not hard- you can parameterise the process using simple linear regression. If you need help with that I'll try and get back to you tonight about it. On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: Hi -- _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
In reply to this post by Stephen Choularton-3
Hi Stephen,
You could take a look at http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model for the linear regression method, or take a look at the package "sde" which contains some examples using GMM (not for the Ornstein-Uhlenbeck process, though, only the CIR). The half-life is given as log(2)/mean-reversion speed. Do keep an eye on the partition of the time-axis, e.g. what frequency you are using (daily, yearly) for interpreting the half-life. BR, Bjørn > ------------------------------ > > Message: 2 > Date: Tue, 12 Oct 2010 05:43:32 -0400 > From: Sarbo <[hidden email]> > To: [hidden email] > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > Message-ID: <[hidden email]> > Content-Type: text/plain; charset="utf-8" > > By half-life, do you mean the speed of mean-reversion? > > If so, there's a bit of algebraic tomfoolery that's required to > discretise the equation and then fit the data to it. I don't have the > time right now to go into all the details but it's not hard- you can > parameterise the process using simple linear regression. If you need > help with that I'll try and get back to you tonight about it. > > On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: > > > Hi > > > > Wonder if anyone could point me how I use this method to discover the > > half life of a mean reverting process. > > > > I am looking into pair trading and the time it takes for a > > cointegrated pair to revert to the norm. > > > > -- > > Stephen Choularton Ph.D., FIoD > > > > 9999 2226 > > 0413 545 182 > > > > > > for insurance go to www.netinsure.com.au > > for markets go to www.organicfoodmarkets.com.au > > > > > > _______________________________________________ > > [hidden email] mailing list > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > -- Subscriber-posting only. If you want to post, subscribe first. > > -- Also note that this is not the r-help list where general R questions > should go. > > > -------------- next part -------------- > An HTML attachment was scrubbed... > URL: < > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html > > > -------------- next part -------------- > A non-text attachment was scrubbed... > Name: CoS2010Winner.JPG > Type: image/jpeg > Size: 16091 bytes > Desc: not available > URL: < > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe > > > > ------------------------------ > > _______________________________________________ > R-SIG-Finance mailing list > [hidden email] > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > > End of R-SIG-Finance Digest, Vol 77, Issue 8 > ******************************************** _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
Stephen:
I do mean-reversion trading, and I use a half-life analysis to judge the wisdom of a trade. If the estimated half-life is too long, it doesn't make sense to take the trade. It's a time-vs-risk thing. In the past, I used the log(2)/speed formula mentioned by Bjorn, below. (His link is very useful, BTW.) However, I was very unhappy with the estimates provided by that formula. They did not match my actual trading experience. I did some research on the topic, and got some useful results. I added a momentum term to my model, measuruing the current slope of the spread. The slope answers an important question: is the spread currently reverting (moving towards the mean) or is it averting (moving away from the mean)? The half-life is different, depending upon the current phase (reversion vs. aversion). I found this conditioning term was statistically significant, so I condition my estimate on it. I generate historical data and partition it according to the market's reversion/aversion state. The two partitions shows a different half-life, with the reverting phase having a (much) shorter half-life than the averting phase. Both phases show an exponentally distributed half-life, but with different means. I use those historical estimates now in my trading, instead of the O-U estimate. In retrospect, the problem with the O-U model is that it assumes the mean-reverting process is always reverting. That is not quite correct. In my experience, the spreads can alternate between periods of mean-aversion and mean-reversion. I hope that helps (and I hope it makes sense!). Paul -----Original Message----- From: [hidden email] [mailto:[hidden email]] On Behalf Of Bjorn Skogtro Sent: Tuesday, October 12, 2010 5:34 AM To: [hidden email] Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck Hi Stephen, You could take a look at http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model for the linear regression method, or take a look at the package "sde" which contains some examples using GMM (not for the Ornstein-Uhlenbeck process, though, only the CIR). The half-life is given as log(2)/mean-reversion speed. Do keep an eye on the partition of the time-axis, e.g. what frequency you are using (daily, yearly) for interpreting the half-life. BR, Bjxrn > ------------------------------ > > Message: 2 > Date: Tue, 12 Oct 2010 05:43:32 -0400 > From: Sarbo <[hidden email]> > To: [hidden email] > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > Message-ID: <[hidden email]> > Content-Type: text/plain; charset="utf-8" > > By half-life, do you mean the speed of mean-reversion? > > If so, there's a bit of algebraic tomfoolery that's required to > discretise the equation and then fit the data to it. I don't have the > time right now to go into all the details but it's not hard- you can > parameterise the process using simple linear regression. If you need > help with that I'll try and get back to you tonight about it. > > On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: > > > Hi > > > > Wonder if anyone could point me how I use this method to discover > > the half life of a mean reverting process. > > > > I am looking into pair trading and the time it takes for a > > cointegrated pair to revert to the norm. > > > > -- > > Stephen Choularton Ph.D., FIoD > > > > 9999 2226 > > 0413 545 182 > > > > > > for insurance go to www.netinsure.com.au for markets go to > > www.organicfoodmarkets.com.au > > > > > > _______________________________________________ > > [hidden email] mailing list > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > -- Subscriber-posting only. If you want to post, subscribe first. > > -- Also note that this is not the r-help list where general R > > questions > should go. > > > -------------- next part -------------- An HTML attachment was > scrubbed... > URL: < > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e3 > 2fc7/attachment-0001.html > > > -------------- next part -------------- A non-text attachment was > scrubbed... > Name: CoS2010Winner.JPG > Type: image/jpeg > Size: 16091 bytes > Desc: not available > URL: < > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e3 > 2fc7/attachment-0001.jpe > > > > ------------------------------ > > _______________________________________________ > R-SIG-Finance mailing list > [hidden email] > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > > End of R-SIG-Finance Digest, Vol 77, Issue 8 > ******************************************** [[alternative HTML version deleted]] _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
In reply to this post by Stale-2
just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM.
sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU # ex3.01.R OU.lik <- function(theta1, theta2, theta3){ n <- length(X) dt <- deltat(X) -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) } require(stats4) require(sde) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit summary(fit) # ex3.01.R (cont.) prof <- profile(fit) par(mfrow=c(1,3)) plot(prof) par(mfrow=c(1,1)) vcov(fit) confint(fit) # ex3.01.R (cont.) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 summary(fit2) I hope this helps out stefano On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: > Hi Stephen, > > You could take a look at > > http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model > > for the linear regression method, or take a look at the package "sde" which > contains some examples using GMM (not for the Ornstein-Uhlenbeck process, > though, only the CIR). > > The half-life is given as log(2)/mean-reversion speed. > > Do keep an eye on the partition of the time-axis, e.g. what frequency you > are using (daily, yearly) for interpreting the half-life. > > BR, > Bjørn > > > > > > >> ------------------------------ >> >> Message: 2 >> Date: Tue, 12 Oct 2010 05:43:32 -0400 >> From: Sarbo <[hidden email]> >> To: [hidden email] >> Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck >> Message-ID: <[hidden email]> >> Content-Type: text/plain; charset="utf-8" >> >> By half-life, do you mean the speed of mean-reversion? >> >> If so, there's a bit of algebraic tomfoolery that's required to >> discretise the equation and then fit the data to it. I don't have the >> time right now to go into all the details but it's not hard- you can >> parameterise the process using simple linear regression. If you need >> help with that I'll try and get back to you tonight about it. >> >> On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: >> >>> Hi >>> >>> Wonder if anyone could point me how I use this method to discover the >>> half life of a mean reverting process. >>> >>> I am looking into pair trading and the time it takes for a >>> cointegrated pair to revert to the norm. >>> >>> -- >>> Stephen Choularton Ph.D., FIoD >>> >>> 9999 2226 >>> 0413 545 182 >>> >>> >>> for insurance go to www.netinsure.com.au >>> for markets go to www.organicfoodmarkets.com.au >>> >>> >>> _______________________________________________ >>> [hidden email] mailing list >>> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >>> -- Subscriber-posting only. If you want to post, subscribe first. >>> -- Also note that this is not the r-help list where general R questions >> should go. >> >> >> -------------- next part -------------- >> An HTML attachment was scrubbed... >> URL: < >> https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html >>> >> -------------- next part -------------- >> A non-text attachment was scrubbed... >> Name: CoS2010Winner.JPG >> Type: image/jpeg >> Size: 16091 bytes >> Desc: not available >> URL: < >> https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe >>> >> >> ------------------------------ >> >> _______________________________________________ >> R-SIG-Finance mailing list >> [hidden email] >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >> >> >> End of R-SIG-Finance Digest, Vol 77, Issue 8 >> ******************************************** > > [[alternative HTML version deleted]] > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. ----------------------------------- Stefano M. Iacus Department of Economics, Business and Statistics University of Milan Via Conservatorio, 7 I-20123 Milan - Italy Ph.: +39 02 50321 461 Fax: +39 02 50321 505 http://www.economia.unimi.it/iacus ------------------------------------------------------------------------------------ Please don't send me Word or PowerPoint attachments if not absolutely necessary. See: http://www.gnu.org/philosophy/no-word-attachments.html _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
In reply to this post by Paul Teetor
Just another thought:
If you google around, you may stumble upon some results on first passage times for the OU-process that may be useful to you. These are known in an closed-form solution. I think, like Paul, that polishing the strategy is essential. Finding reverting pairs are the easy part, but optimizing the entry/exit signals are the trick. Bjørn PS: A simple code for calibrating the OU-process. Should've attached this the first time :) #source("ouFit.R") ouFit=function(spread) { n=length(spread) x=spread[1:(n-1)] y=spread[2:n] spread.fit=lm(y~x) coefs=as.numeric(coefficients(spread.fit)) a=coefs[1] b=coefs[2] err=var(residuals(spread.fit)) alpha=-log(a) mu=alpha*b/(1-a) sigma=sqrt(2*alpha*err/(1-a^2)) theta=list(alpha=alpha, mu=mu, sigma=sigma) return(theta) } It won't give you half-life, but you can use the alpha from the above code to find it, in combination with my previous post. 2010/10/12 Paul Teetor <[hidden email]> > Stephen: > > I do mean-reversion trading, and I use a half-life analysis to judge the > wisdom of a trade. If the estimated half-life is too long, it doesn't make > sense to take the trade. It's a time-vs-risk thing. > > In the past, I used the log(2)/speed formula mentioned by Bjorn, below. > (His > link is very useful, BTW.) However, I was very unhappy with the estimates > provided by that formula. They did not match my actual trading experience. > > I did some research on the topic, and got some useful results. I added a > momentum term to my model, measuruing the current slope of the spread. The > slope answers an important question: is the spread currently reverting > (moving towards the mean) or is it averting (moving away from the mean)? > The > half-life is different, depending upon the current phase (reversion vs. > aversion). I found this conditioning term was statistically significant, so > I condition my estimate on it. I generate historical data and partition it > according to the market's reversion/aversion state. The two partitions > shows > a different half-life, with the reverting phase having a (much) shorter > half-life than the averting phase. Both phases show an exponentally > distributed half-life, but with different means. I use those historical > estimates now in my trading, instead of the O-U estimate. > > In retrospect, the problem with the O-U model is that it assumes the > mean-reverting process is always reverting. That is not quite correct. In > my > experience, the spreads can alternate between periods of mean-aversion and > mean-reversion. > > I hope that helps (and I hope it makes sense!). > > Paul > > > -----Original Message----- > From: [hidden email] > [mailto:[hidden email]] On Behalf Of Bjorn > Skogtro > Sent: Tuesday, October 12, 2010 5:34 AM > To: [hidden email] > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > > Hi Stephen, > > You could take a look at > > http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model > > for the linear regression method, or take a look at the package "sde" which > contains some examples using GMM (not for the Ornstein-Uhlenbeck process, > though, only the CIR). > > The half-life is given as log(2)/mean-reversion speed. > > Do keep an eye on the partition of the time-axis, e.g. what frequency you > are using (daily, yearly) for interpreting the half-life. > > BR, > Bjxrn > > > > > > > > ------------------------------ > > > > Message: 2 > > Date: Tue, 12 Oct 2010 05:43:32 -0400 > > From: Sarbo <[hidden email]> > > To: [hidden email] > > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > > Message-ID: <[hidden email]> > > Content-Type: text/plain; charset="utf-8" > > > > By half-life, do you mean the speed of mean-reversion? > > > > If so, there's a bit of algebraic tomfoolery that's required to > > discretise the equation and then fit the data to it. I don't have the > > time right now to go into all the details but it's not hard- you can > > parameterise the process using simple linear regression. If you need > > help with that I'll try and get back to you tonight about it. > > > > On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: > > > > > Hi > > > > > > Wonder if anyone could point me how I use this method to discover > > > the half life of a mean reverting process. > > > > > > I am looking into pair trading and the time it takes for a > > > cointegrated pair to revert to the norm. > > > > > > -- > > > Stephen Choularton Ph.D., FIoD > > > > > > 9999 2226 > > > 0413 545 182 > > > > > > > > > for insurance go to www.netinsure.com.au for markets go to > > > www.organicfoodmarkets.com.au > > > > > > > > > _______________________________________________ > > > [hidden email] mailing list > > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > > -- Subscriber-posting only. If you want to post, subscribe first. > > > -- Also note that this is not the r-help list where general R > > > questions > > should go. > > > > > > -------------- next part -------------- An HTML attachment was > > scrubbed... > > URL: < > > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e3 > > 2fc7/attachment-0001.html > > > > > -------------- next part -------------- A non-text attachment was > > scrubbed... > > Name: CoS2010Winner.JPG > > Type: image/jpeg > > Size: 16091 bytes > > Desc: not available > > URL: < > > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e3 > > 2fc7/attachment-0001.jpe > > > > > > > ------------------------------ > > > > _______________________________________________ > > R-SIG-Finance mailing list > > [hidden email] > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > > > > > End of R-SIG-Finance Digest, Vol 77, Issue 8 > > ******************************************** > > [[alternative HTML version deleted]] > > > -- Up, down, turn around Please dont let me hit the ground Tonight I think Ill walk alone Ill find my soul as I go home. 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In reply to this post by stefano iacus-2
The OU process is Gaussian, but the
market didn't get the memo that *it* has to be Gaussian. On 12/10/2010 16:41, stefano iacus wrote: > just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. > > sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. > > This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU > > > # ex3.01.R > OU.lik<- function(theta1, theta2, theta3){ > n<- length(X) > dt<- deltat(X) > -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) > } > > require(stats4) > require(sde) > set.seed(123) > X<- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit > summary(fit) > > # ex3.01.R (cont.) > prof<- profile(fit) > par(mfrow=c(1,3)) > plot(prof) > par(mfrow=c(1,1)) > vcov(fit) > confint(fit) > > # ex3.01.R (cont.) > set.seed(123) > X<- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 > summary(fit2) > > > I hope this helps out > > stefano > > On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: > >> Hi Stephen, >> >> You could take a look at >> >> http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model >> >> for the linear regression method, or take a look at the package "sde" which >> contains some examples using GMM (not for the Ornstein-Uhlenbeck process, >> though, only the CIR). >> >> The half-life is given as log(2)/mean-reversion speed. >> >> Do keep an eye on the partition of the time-axis, e.g. what frequency you >> are using (daily, yearly) for interpreting the half-life. >> >> BR, >> Bjørn >> >> >> >> >> >> >>> ------------------------------ >>> >>> Message: 2 >>> Date: Tue, 12 Oct 2010 05:43:32 -0400 >>> From: Sarbo<[hidden email]> >>> To: [hidden email] >>> Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck >>> Message-ID:<[hidden email]> >>> Content-Type: text/plain; charset="utf-8" >>> >>> By half-life, do you mean the speed of mean-reversion? >>> >>> If so, there's a bit of algebraic tomfoolery that's required to >>> discretise the equation and then fit the data to it. I don't have the >>> time right now to go into all the details but it's not hard- you can >>> parameterise the process using simple linear regression. If you need >>> help with that I'll try and get back to you tonight about it. >>> >>> On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: >>> >>>> Hi >>>> >>>> Wonder if anyone could point me how I use this method to discover the >>>> half life of a mean reverting process. >>>> >>>> I am looking into pair trading and the time it takes for a >>>> cointegrated pair to revert to the norm. >>>> >>>> -- >>>> Stephen Choularton Ph.D., FIoD >>>> >>>> 9999 2226 >>>> 0413 545 182 >>>> >>>> >>>> for insurance go to www.netinsure.com.au >>>> for markets go to www.organicfoodmarkets.com.au >>>> >>>> >>>> _______________________________________________ >>>> [hidden email] mailing list >>>> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >>>> -- Subscriber-posting only. If you want to post, subscribe first. >>>> -- Also note that this is not the r-help list where general R questions >>> should go. >>> >>> >>> -------------- next part -------------- >>> An HTML attachment was scrubbed... >>> URL:< >>> https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html >>>> >>> -------------- next part -------------- >>> A non-text attachment was scrubbed... >>> Name: CoS2010Winner.JPG >>> Type: image/jpeg >>> Size: 16091 bytes >>> Desc: not available >>> URL:< >>> https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe >>>> >>> >>> ------------------------------ >>> >>> _______________________________________________ >>> R-SIG-Finance mailing list >>> [hidden email] >>> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >>> >>> >>> End of R-SIG-Finance Digest, Vol 77, Issue 8 >>> ******************************************** >> >> [[alternative HTML version deleted]] >> >> _______________________________________________ >> [hidden email] mailing list >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >> -- Subscriber-posting only. If you want to post, subscribe first. >> -- Also note that this is not the r-help list where general R questions should go. > > > ----------------------------------- > Stefano M. Iacus > Department of Economics, > Business and Statistics > University of Milan > Via Conservatorio, 7 > I-20123 Milan - Italy > Ph.: +39 02 50321 461 > Fax: +39 02 50321 505 > http://www.economia.unimi.it/iacus > ------------------------------------------------------------------------------------ > Please don't send me Word or PowerPoint attachments if not > absolutely necessary. See: > http://www.gnu.org/philosophy/no-word-attachments.html > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. > -- Patrick Burns [hidden email] http://www.burns-stat.com http://www.portfolioprobe.com/blog _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
that's another point and I completely agree with you :-)
so the real point is not trying to fit wrong models on the (so to say) "wrong" data stefano On 12 Oct 2010, at 18:00, Patrick Burns wrote: > The OU process is Gaussian, but the > market didn't get the memo that *it* > has to be Gaussian. > > On 12/10/2010 16:41, stefano iacus wrote: >> just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. >> >> sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. >> >> This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU >> >> >> # ex3.01.R >> OU.lik<- function(theta1, theta2, theta3){ >> n<- length(X) >> dt<- deltat(X) >> -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) >> } >> >> require(stats4) >> require(sde) >> set.seed(123) >> X<- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) >> mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), >> method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit >> summary(fit) >> >> # ex3.01.R (cont.) >> prof<- profile(fit) >> par(mfrow=c(1,3)) >> plot(prof) >> par(mfrow=c(1,1)) >> vcov(fit) >> confint(fit) >> >> # ex3.01.R (cont.) >> set.seed(123) >> X<- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) >> mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), >> method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 >> summary(fit2) >> >> >> I hope this helps out >> >> stefano >> >> On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: >> >>> Hi Stephen, >>> >>> You could take a look at >>> >>> http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model >>> >>> for the linear regression method, or take a look at the package "sde" which >>> contains some examples using GMM (not for the Ornstein-Uhlenbeck process, >>> though, only the CIR). >>> >>> The half-life is given as log(2)/mean-reversion speed. >>> >>> Do keep an eye on the partition of the time-axis, e.g. what frequency you >>> are using (daily, yearly) for interpreting the half-life. >>> >>> BR, >>> Bjørn >>> >>> >>> >>> >>> >>> >>>> ------------------------------ >>>> >>>> Message: 2 >>>> Date: Tue, 12 Oct 2010 05:43:32 -0400 >>>> From: Sarbo<[hidden email]> >>>> To: [hidden email] >>>> Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck >>>> Message-ID:<[hidden email]> >>>> Content-Type: text/plain; charset="utf-8" >>>> >>>> By half-life, do you mean the speed of mean-reversion? >>>> >>>> If so, there's a bit of algebraic tomfoolery that's required to >>>> discretise the equation and then fit the data to it. I don't have the >>>> time right now to go into all the details but it's not hard- you can >>>> parameterise the process using simple linear regression. If you need >>>> help with that I'll try and get back to you tonight about it. >>>> >>>> On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: >>>> >>>>> Hi >>>>> >>>>> Wonder if anyone could point me how I use this method to discover the >>>>> half life of a mean reverting process. >>>>> >>>>> I am looking into pair trading and the time it takes for a >>>>> cointegrated pair to revert to the norm. >>>>> >>>>> -- >>>>> Stephen Choularton Ph.D., FIoD >>>>> >>>>> 9999 2226 >>>>> 0413 545 182 >>>>> >>>>> >>>>> for insurance go to www.netinsure.com.au >>>>> for markets go to www.organicfoodmarkets.com.au >>>>> >>>>> >>>>> _______________________________________________ >>>>> [hidden email] mailing list >>>>> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >>>>> -- Subscriber-posting only. If you want to post, subscribe first. >>>>> -- Also note that this is not the r-help list where general R questions >>>> should go. >>>> >>>> >>>> -------------- next part -------------- >>>> An HTML attachment was scrubbed... >>>> URL:< >>>> https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html >>>>> >>>> -------------- next part -------------- >>>> A non-text attachment was scrubbed... >>>> Name: CoS2010Winner.JPG >>>> Type: image/jpeg >>>> Size: 16091 bytes >>>> Desc: not available >>>> URL:< >>>> https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe >>>>> >>>> >>>> ------------------------------ >>>> >>>> _______________________________________________ >>>> R-SIG-Finance mailing list >>>> [hidden email] >>>> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >>>> >>>> >>>> End of R-SIG-Finance Digest, Vol 77, Issue 8 >>>> ******************************************** >>> >>> [[alternative HTML version deleted]] >>> >>> _______________________________________________ >>> [hidden email] mailing list >>> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >>> -- Subscriber-posting only. If you want to post, subscribe first. >>> -- Also note that this is not the r-help list where general R questions should go. >> >> >> ----------------------------------- >> Stefano M. Iacus >> Department of Economics, >> Business and Statistics >> University of Milan >> Via Conservatorio, 7 >> I-20123 Milan - Italy >> Ph.: +39 02 50321 461 >> Fax: +39 02 50321 505 >> http://www.economia.unimi.it/iacus >> ------------------------------------------------------------------------------------ >> Please don't send me Word or PowerPoint attachments if not >> absolutely necessary. See: >> http://www.gnu.org/philosophy/no-word-attachments.html >> >> _______________________________________________ >> [hidden email] mailing list >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >> -- Subscriber-posting only. If you want to post, subscribe first. >> -- Also note that this is not the r-help list where general R questions should go. >> > > -- > Patrick Burns > [hidden email] > http://www.burns-stat.com > http://www.portfolioprobe.com/blog > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. ----------------------------------- Stefano M. Iacus Department of Economics, Business and Statistics University of Milan Via Conservatorio, 7 I-20123 Milan - Italy Ph.: +39 02 50321 461 Fax: +39 02 50321 505 http://www.economia.unimi.it/iacus ------------------------------------------------------------------------------------ Please don't send me Word or PowerPoint attachments if not absolutely necessary. See: http://www.gnu.org/philosophy/no-word-attachments.html _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
In reply to this post by stefano iacus-2
Thanks for this help.
Trying to make sense of it so I have added some notes to the code. I have marked them #?# Delighted if you can tell me if I am write or wrong, add any comments, answers. #?# This appears to be the function that is doing the 'Ornstein-Uhlenbeck #?# process work' particularly via dcOU #?# I have noted in several places that I am after: #?# 'the half-life of the decay equals ln(2)/θ' #?# 'The half-life is given as log(2)/mean-reversion speed.' #?# and I see theta appearing at a number of points in the code. #?# Can you tell me why 3 thetas viz theta1, theta2, theta3 and what they do? #?# eg is one of these the theta I am after? # ex3.01.R OU.lik <- function(theta1, theta2, theta3){ n <- length(X) dt <- deltat(X) -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) } require(stats4) require(sde) #?# random numer generation seed set.seed(123) #?# creation of a data set X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) #?# If I Look at X its like this: #?# Time Series: #?# Start = 0 #?# End = 1000 #?# Frequency = 1 #?# [1] 1.00000000 etc #?# What sort of data object is it and how would I coerce an object with one #?# column from a read.csv into it? mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit summary(fit) #?# This gives: #?# Maximum likelihood estimation #?# Call: #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) #?# Coefficients: #?# Estimate Std. Error #?# theta1 3.355322 0.28159504 #?# theta2 1.106107 0.09010627 #?# theta3 2.052815 0.07624441 #?# -2 log L: 3366.389 #?# What's this telling me? # ex3.01.R (cont.) prof <- profile(fit) par(mfrow=c(1,3)) plot(prof) par(mfrow=c(1,1)) vcov(fit) confint(fit) #?# This provides me with this output using 'fit' from before: #?# > vcov(fit) #?# theta1 theta2 theta3 #?# theta1 0.07929576 0.024620718 0.016634557 #?# theta2 0.02462072 0.008119141 0.005485549 #?# theta3 0.01663456 0.005485549 0.005813209 #?# > confint(fit) #?# Profiling... #?# 2.5 % 97.5 % #?# theta1 2.8448980 3.960982 #?# theta2 0.9433338 1.300629 #?# theta3 1.9147136 2.216113 #?# and 'fit' is: #?# Call: #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) #?# Coefficients: #?# theta1 theta2 theta3 #?# 3.355322 1.106107 2.052815 #?# plus some graphic output #?# Again, what's this telling me. #?# This looks like a further example? # ex3.01.R (cont.) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 summary(fit2) Please excuse the length of this email (and my lack of understanding) Hope you can help and thanks. Stephen Choularton Ph.D., FIoD
On 13/10/2010 2:41 AM, stefano iacus wrote: just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU # ex3.01.R OU.lik <- function(theta1, theta2, theta3){ n <- length(X) dt <- deltat(X) -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) } require(stats4) require(sde) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit summary(fit) # ex3.01.R (cont.) prof <- profile(fit) par(mfrow=c(1,3)) plot(prof) par(mfrow=c(1,1)) vcov(fit) confint(fit) # ex3.01.R (cont.) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 summary(fit2) I hope this helps out stefano On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote:Hi Stephen, You could take a look at http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model for the linear regression method, or take a look at the package "sde" which contains some examples using GMM (not for the Ornstein-Uhlenbeck process, though, only the CIR). The half-life is given as log(2)/mean-reversion speed. Do keep an eye on the partition of the time-axis, e.g. what frequency you are using (daily, yearly) for interpreting the half-life. BR, Bjørn------------------------------ Message: 2 Date: Tue, 12 Oct 2010 05:43:32 -0400 From: Sarbo [hidden email] To: [hidden email] Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck Message-ID: [hidden email] Content-Type: text/plain; charset="utf-8" By half-life, do you mean the speed of mean-reversion? If so, there's a bit of algebraic tomfoolery that's required to discretise the equation and then fit the data to it. I don't have the time right now to go into all the details but it's not hard- you can parameterise the process using simple linear regression. If you need help with that I'll try and get back to you tonight about it. On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote:Hi Wonder if anyone could point me how I use this method to discover the half life of a mean reverting process. I am looking into pair trading and the time it takes for a cointegrated pair to revert to the norm. -- Stephen Choularton Ph.D., FIoD 9999 2226 0413 545 182 for insurance go to www.netinsure.com.au for markets go to www.organicfoodmarkets.com.au _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questionsshould go. -------------- next part -------------- An HTML attachment was scrubbed... URL: < https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html-------------- next part -------------- A non-text attachment was scrubbed... Name: CoS2010Winner.JPG Type: image/jpeg Size: 16091 bytes Desc: not available URL: < https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe------------------------------ _______________________________________________ R-SIG-Finance mailing list [hidden email] https://stat.ethz.ch/mailman/listinfo/r-sig-finance End of R-SIG-Finance Digest, Vol 77, Issue 8 ********************************************[[alternative HTML version deleted]] _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go.----------------------------------- Stefano M. Iacus Department of Economics, Business and Statistics University of Milan Via Conservatorio, 7 I-20123 Milan - Italy Ph.: +39 02 50321 461 Fax: +39 02 50321 505 http://www.economia.unimi.it/iacus ------------------------------------------------------------------------------------ Please don't send me Word or PowerPoint attachments if not absolutely necessary. See: http://www.gnu.org/philosophy/no-word-attachments.html _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go.No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
Hi
I am still trying to sort this one out. Any comments from anyone would be most welcome. Stephen Choularton Ph.D., FIoD
On 14/10/2010 7:29 AM, Stephen Choularton wrote: Thanks for this help. _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
I am doing rolling ADF test on some time series to check mean reversion. When I use short period rolling, I find the residue is not stationary at all. However, when I use horizon longer than 5 years, I find very significant stationary. On the other hand, I find the half life is only around 30 days. Is there anyone who can give me some possible explanation or guide me to some reference? thanks Best, Yihao ________________________________ > Date: Tue, 19 Oct 2010 09:03:55 +1100 > From: [hidden email] > To: [hidden email] > CC: [hidden email] > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > > Hi > > I am still trying to sort this one out. Any comments from anyone would > be most welcome. > > Stephen Choularton Ph.D., FIoD > > > > On 14/10/2010 7:29 AM, Stephen Choularton wrote: > Thanks for this help. > > Trying to make sense of it so I have added some notes to the code. I > have marked them #?# > > Delighted if you can tell me if I am write or wrong, add any comments, > answers. > > > #?# This appears to be the function that is doing the 'Ornstein-Uhlenbeck > #?# process work' particularly via dcOU > #?# I have noted in several places that I am after: > #?# 'the half-life of the decay equals ln(2)/θ' > #?# 'The half-life is given as log(2)/mean-reversion speed.' > #?# and I see theta appearing at a number of points in the code. > #?# Can you tell me why 3 thetas viz theta1, theta2, theta3 and what they do? > #?# eg is one of these the theta I am after? > > # ex3.01.R > OU.lik <- function(theta1, theta2, theta3){ > n <- length(X) > dt <- deltat(X) > -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) > } > > require(stats4) > require(sde) > > #?# random numer generation seed > set.seed(123) > > #?# creation of a data set > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) > #?# If I Look at X its like this: > #?# Time Series: > #?# Start = 0 > #?# End = 1000 > #?# Frequency = 1 > #?# [1] 1.00000000 etc > #?# What sort of data object is it and how would I coerce an object with one > #?# column from a read.csv into it? > > > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit > summary(fit) > > #?# This gives: > > #?# Maximum likelihood estimation > > #?# Call: > #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, > #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) > > #?# Coefficients: > #?# Estimate Std. Error > #?# theta1 3.355322 0.28159504 > #?# theta2 1.106107 0.09010627 > #?# theta3 2.052815 0.07624441 > > #?# -2 log L: 3366.389 > > #?# What's this telling me? > > # ex3.01.R (cont.) > prof <- profile(fit) > par(mfrow=c(1,3)) > plot(prof) > par(mfrow=c(1,1)) > vcov(fit) > confint(fit) > > #?# This provides me with this output using 'fit' from before: > > #?# > vcov(fit) > #?# theta1 theta2 theta3 > #?# theta1 0.07929576 0.024620718 0.016634557 > #?# theta2 0.02462072 0.008119141 0.005485549 > #?# theta3 0.01663456 0.005485549 0.005813209 > #?# > confint(fit) > #?# Profiling... > #?# 2.5 % 97.5 % > #?# theta1 2.8448980 3.960982 > #?# theta2 0.9433338 1.300629 > #?# theta3 1.9147136 2.216113 > > #?# and 'fit' is: > > #?# Call: > #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, > #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) > > #?# Coefficients: > #?# theta1 theta2 theta3 > #?# 3.355322 1.106107 2.052815 > > #?# plus some graphic output > > #?# Again, what's this telling me. > > #?# This looks like a further example? > # ex3.01.R (cont.) > set.seed(123) > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 > summary(fit2) > > > > > Please excuse the length of this email (and my lack of understanding) > > Hope you can help and thanks. > > > > > Stephen Choularton Ph.D., FIoD > > > On 13/10/2010 2:41 AM, stefano iacus wrote: > > just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. > > sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. > > This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU > > > # ex3.01.R > OU.lik <- function(theta1, theta2, theta3){ > n <- length(X) > dt <- deltat(X) > -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) > } > > require(stats4) > require(sde) > set.seed(123) > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit > summary(fit) > > # ex3.01.R (cont.) > prof <- profile(fit) > par(mfrow=c(1,3)) > plot(prof) > par(mfrow=c(1,1)) > vcov(fit) > confint(fit) > > # ex3.01.R (cont.) > set.seed(123) > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 > summary(fit2) > > > I hope this helps out > > stefano > > On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: > > > > Hi Stephen, > > You could take a look at > > http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model > > for the linear regression method, or take a look at the package "sde" which > contains some examples using GMM (not for the Ornstein-Uhlenbeck process, > though, only the CIR). > > The half-life is given as log(2)/mean-reversion speed. > > Do keep an eye on the partition of the time-axis, e.g. what frequency you > are using (daily, yearly) for interpreting the half-life. > > BR, > Bjørn > > > > > > > > > ------------------------------ > > Message: 2 > Date: Tue, 12 Oct 2010 05:43:32 -0400 > From: Sarbo > To: [hidden email] > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > Message-ID: > Content-Type: text/plain; charset="utf-8" > > By half-life, do you mean the speed of mean-reversion? > > If so, there's a bit of algebraic tomfoolery that's required to > discretise the equation and then fit the data to it. I don't have the > time right now to go into all the details but it's not hard- you can > parameterise the process using simple linear regression. If you need > help with that I'll try and get back to you tonight about it. > > On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: > > > > Hi > > Wonder if anyone could point me how I use this method to discover the > half life of a mean reverting process. > > I am looking into pair trading and the time it takes for a > cointegrated pair to revert to the norm. > > -- > Stephen Choularton Ph.D., FIoD > > 9999 2226 > 0413 545 182 > > > for insurance go to www.netinsure.com.au > for markets go to www.organicfoodmarkets.com.au > > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions > > > should go. > > > -------------- next part -------------- > An HTML attachment was scrubbed... > URL: < > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html > > > -------------- next part -------------- > A non-text attachment was scrubbed... > Name: CoS2010Winner.JPG > Type: image/jpeg > Size: 16091 bytes > Desc: not available > URL: < > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe > > > ------------------------------ > > _______________________________________________ > R-SIG-Finance mailing list > [hidden email] > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > > End of R-SIG-Finance Digest, Vol 77, Issue 8 > ******************************************** > > > [[alternative HTML version deleted]] > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. > > > > ----------------------------------- > Stefano M. Iacus > Department of Economics, > Business and Statistics > University of Milan > Via Conservatorio, 7 > I-20123 Milan - Italy > Ph.: +39 02 50321 461 > Fax: +39 02 50321 505 > http://www.economia.unimi.it/iacus > ------------------------------------------------------------------------------------ > Please don't send me Word or PowerPoint attachments if not > absolutely necessary. See: > http://www.gnu.org/philosophy/no-word-attachments.html > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. > > > > > > No virus found in this incoming message. > Checked by AVG - www.avg.com > > > > > > > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. > > > > > > No virus found in this incoming message. > Checked by AVG - www.avg.com > > > > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- > Subscriber-posting only. If you want to post, subscribe first. -- Also > note that this is not the r-help list where general R questions should > go. _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
Wish I could, but as you can see I am having even greater problems
;-)
Stephen Choularton Ph.D., FIoD
On 19/10/2010 12:35 PM, Yihao Lu aeolus_lu wrote: I am doing rolling ADF test on some time series to check mean reversion. When I use short period rolling, I find the residue is not stationary at all. However, when I use horizon longer than 5 years, I find very significant stationary. On the other hand, I find the half life is only around 30 days. Is there anyone who can give me some possible explanation or guide me to some reference? thanks Best, Yihao ________________________________Date: Tue, 19 Oct 2010 09:03:55 +1100 From: [hidden email] To: [hidden email] CC: [hidden email] Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck Hi I am still trying to sort this one out. Any comments from anyone would be most welcome. Stephen Choularton Ph.D., FIoD On 14/10/2010 7:29 AM, Stephen Choularton wrote: Thanks for this help. Trying to make sense of it so I have added some notes to the code. I have marked them #?# Delighted if you can tell me if I am write or wrong, add any comments, answers. #?# This appears to be the function that is doing the 'Ornstein-Uhlenbeck #?# process work' particularly via dcOU #?# I have noted in several places that I am after: #?# 'the half-life of the decay equals ln(2)/θ' #?# 'The half-life is given as log(2)/mean-reversion speed.' #?# and I see theta appearing at a number of points in the code. #?# Can you tell me why 3 thetas viz theta1, theta2, theta3 and what they do? #?# eg is one of these the theta I am after? # ex3.01.R OU.lik <- function(theta1, theta2, theta3){ n <- length(X) dt <- deltat(X) -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) } require(stats4) require(sde) #?# random numer generation seed set.seed(123) #?# creation of a data set X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) #?# If I Look at X its like this: #?# Time Series: #?# Start = 0 #?# End = 1000 #?# Frequency = 1 #?# [1] 1.00000000 etc #?# What sort of data object is it and how would I coerce an object with one #?# column from a read.csv into it? mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit summary(fit) #?# This gives: #?# Maximum likelihood estimation #?# Call: #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) #?# Coefficients: #?# Estimate Std. Error #?# theta1 3.355322 0.28159504 #?# theta2 1.106107 0.09010627 #?# theta3 2.052815 0.07624441 #?# -2 log L: 3366.389 #?# What's this telling me? # ex3.01.R (cont.) prof <- profile(fit) par(mfrow=c(1,3)) plot(prof) par(mfrow=c(1,1)) vcov(fit) confint(fit) #?# This provides me with this output using 'fit' from before: #?# > vcov(fit) #?# theta1 theta2 theta3 #?# theta1 0.07929576 0.024620718 0.016634557 #?# theta2 0.02462072 0.008119141 0.005485549 #?# theta3 0.01663456 0.005485549 0.005813209 #?# > confint(fit) #?# Profiling... #?# 2.5 % 97.5 % #?# theta1 2.8448980 3.960982 #?# theta2 0.9433338 1.300629 #?# theta3 1.9147136 2.216113 #?# and 'fit' is: #?# Call: #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) #?# Coefficients: #?# theta1 theta2 theta3 #?# 3.355322 1.106107 2.052815 #?# plus some graphic output #?# Again, what's this telling me. #?# This looks like a further example? # ex3.01.R (cont.) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 summary(fit2) Please excuse the length of this email (and my lack of understanding) Hope you can help and thanks. Stephen Choularton Ph.D., FIoD On 13/10/2010 2:41 AM, stefano iacus wrote: just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU # ex3.01.R OU.lik <- function(theta1, theta2, theta3){ n <- length(X) dt <- deltat(X) -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) } require(stats4) require(sde) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit summary(fit) # ex3.01.R (cont.) prof <- profile(fit) par(mfrow=c(1,3)) plot(prof) par(mfrow=c(1,1)) vcov(fit) confint(fit) # ex3.01.R (cont.) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 summary(fit2) I hope this helps out stefano On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: Hi Stephen, You could take a look at http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model for the linear regression method, or take a look at the package "sde" which contains some examples using GMM (not for the Ornstein-Uhlenbeck process, though, only the CIR). The half-life is given as log(2)/mean-reversion speed. Do keep an eye on the partition of the time-axis, e.g. what frequency you are using (daily, yearly) for interpreting the half-life. BR, Bjørn ------------------------------ Message: 2 Date: Tue, 12 Oct 2010 05:43:32 -0400 From: Sarbo To: [hidden email] Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck Message-ID: Content-Type: text/plain; charset="utf-8" By half-life, do you mean the speed of mean-reversion? If so, there's a bit of algebraic tomfoolery that's required to discretise the equation and then fit the data to it. I don't have the time right now to go into all the details but it's not hard- you can parameterise the process using simple linear regression. If you need help with that I'll try and get back to you tonight about it. On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: Hi Wonder if anyone could point me how I use this method to discover the half life of a mean reverting process. I am looking into pair trading and the time it takes for a cointegrated pair to revert to the norm. -- Stephen Choularton Ph.D., FIoD 9999 2226 0413 545 182 for insurance go to www.netinsure.com.au for markets go to www.organicfoodmarkets.com.au _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. -------------- next part -------------- An HTML attachment was scrubbed... URL: < https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html -------------- next part -------------- A non-text attachment was scrubbed... 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Iacus Department of Economics, Business and Statistics University of Milan Via Conservatorio, 7 I-20123 Milan - Italy Ph.: +39 02 50321 461 Fax: +39 02 50321 505 http://www.economia.unimi.it/iacus ------------------------------------------------------------------------------------ Please don't send me Word or PowerPoint attachments if not absolutely necessary. See: http://www.gnu.org/philosophy/no-word-attachments.html _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go._______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go.No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
regarding your problem, i think Bjorn's link is very helpful. it even has the matlab code attached. several lines, you can translate it into R and do a comparison on the data, you will have the idea which theta is the rate you should look at. Best, Yihao ________________________________ > Date: Tue, 19 Oct 2010 12:50:46 +1100 > From: [hidden email] > To: [hidden email] > CC: [hidden email] > Subject: Re: [R-SIG-Finance] Mean reversion > > Wish I could, but as you can see I am having even greater problems ;-) > > Stephen Choularton Ph.D., FIoD > > On 19/10/2010 12:35 PM, Yihao Lu aeolus_lu wrote: > > > I am doing rolling ADF test on some time series to check mean reversion. When I use short period rolling, I find the residue is not stationary at all. However, when I use horizon longer than 5 years, I find very significant stationary. On the other hand, I find the half life is only around 30 days. > Is there anyone who can give me some possible explanation or guide me to some reference? thanks > > Best, > Yihao > > > > > > > > ________________________________ > > > Date: Tue, 19 Oct 2010 09:03:55 +1100 > From: [hidden email] > To: [hidden email] > CC: [hidden email] > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > > Hi > > I am still trying to sort this one out. Any comments from anyone would > be most welcome. > > Stephen Choularton Ph.D., FIoD > > > > On 14/10/2010 7:29 AM, Stephen Choularton wrote: > Thanks for this help. > > Trying to make sense of it so I have added some notes to the code. I > have marked them #?# > > Delighted if you can tell me if I am write or wrong, add any comments, > answers. > > > #?# This appears to be the function that is doing the 'Ornstein-Uhlenbeck > #?# process work' particularly via dcOU > #?# I have noted in several places that I am after: > #?# 'the half-life of the decay equals ln(2)/θ' > #?# 'The half-life is given as log(2)/mean-reversion speed.' > #?# and I see theta appearing at a number of points in the code. > #?# Can you tell me why 3 thetas viz theta1, theta2, theta3 and what they do? > #?# eg is one of these the theta I am after? > > # ex3.01.R > OU.lik <- function(theta1, theta2, theta3){ > n <- length(X) > dt <- deltat(X) > -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) > } > > require(stats4) > require(sde) > > #?# random numer generation seed > set.seed(123) > > #?# creation of a data set > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) > #?# If I Look at X its like this: > #?# Time Series: > #?# Start = 0 > #?# End = 1000 > #?# Frequency = 1 > #?# [1] 1.00000000 etc > #?# What sort of data object is it and how would I coerce an object with one > #?# column from a read.csv into it? > > > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit > summary(fit) > > #?# This gives: > > #?# Maximum likelihood estimation > > #?# Call: > #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, > #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) > > #?# Coefficients: > #?# Estimate Std. Error > #?# theta1 3.355322 0.28159504 > #?# theta2 1.106107 0.09010627 > #?# theta3 2.052815 0.07624441 > > #?# -2 log L: 3366.389 > > #?# What's this telling me? > > # ex3.01.R (cont.) > prof <- profile(fit) > par(mfrow=c(1,3)) > plot(prof) > par(mfrow=c(1,1)) > vcov(fit) > confint(fit) > > #?# This provides me with this output using 'fit' from before: > > #?# > vcov(fit) > #?# theta1 theta2 theta3 > #?# theta1 0.07929576 0.024620718 0.016634557 > #?# theta2 0.02462072 0.008119141 0.005485549 > #?# theta3 0.01663456 0.005485549 0.005813209 > #?# > confint(fit) > #?# Profiling... > #?# 2.5 % 97.5 % > #?# theta1 2.8448980 3.960982 > #?# theta2 0.9433338 1.300629 > #?# theta3 1.9147136 2.216113 > > #?# and 'fit' is: > > #?# Call: > #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, > #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) > > #?# Coefficients: > #?# theta1 theta2 theta3 > #?# 3.355322 1.106107 2.052815 > > #?# plus some graphic output > > #?# Again, what's this telling me. > > #?# This looks like a further example? > # ex3.01.R (cont.) > set.seed(123) > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 > summary(fit2) > > > > > Please excuse the length of this email (and my lack of understanding) > > Hope you can help and thanks. > > > > > Stephen Choularton Ph.D., FIoD > > > On 13/10/2010 2:41 AM, stefano iacus wrote: > > just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. > > sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. > > This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU > > > # ex3.01.R > OU.lik <- function(theta1, theta2, theta3){ > n <- length(X) > dt <- deltat(X) > -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) > } > > require(stats4) > require(sde) > set.seed(123) > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit > summary(fit) > > # ex3.01.R (cont.) > prof <- profile(fit) > par(mfrow=c(1,3)) > plot(prof) > par(mfrow=c(1,1)) > vcov(fit) > confint(fit) > > # ex3.01.R (cont.) > set.seed(123) > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 > summary(fit2) > > > I hope this helps out > > stefano > > On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: > > > > Hi Stephen, > > You could take a look at > > http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model > > for the linear regression method, or take a look at the package "sde" which > contains some examples using GMM (not for the Ornstein-Uhlenbeck process, > though, only the CIR). > > The half-life is given as log(2)/mean-reversion speed. > > Do keep an eye on the partition of the time-axis, e.g. what frequency you > are using (daily, yearly) for interpreting the half-life. > > BR, > Bjørn > > > > > > > > > ------------------------------ > > Message: 2 > Date: Tue, 12 Oct 2010 05:43:32 -0400 > From: Sarbo > To: [hidden email] > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > Message-ID: > Content-Type: text/plain; charset="utf-8" > > By half-life, do you mean the speed of mean-reversion? > > If so, there's a bit of algebraic tomfoolery that's required to > discretise the equation and then fit the data to it. I don't have the > time right now to go into all the details but it's not hard- you can > parameterise the process using simple linear regression. If you need > help with that I'll try and get back to you tonight about it. > > On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: > > > > Hi > > Wonder if anyone could point me how I use this method to discover the > half life of a mean reverting process. > > I am looking into pair trading and the time it takes for a > cointegrated pair to revert to the norm. > > -- > Stephen Choularton Ph.D., FIoD > > 9999 2226 > 0413 545 182 > > > for insurance go to www.netinsure.com.au > for markets go to www.organicfoodmarkets.com.au > > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions > > > should go. > > > -------------- next part -------------- > An HTML attachment was scrubbed... > URL: < > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html > > > -------------- next part -------------- > A non-text attachment was scrubbed... > Name: CoS2010Winner.JPG > Type: image/jpeg > Size: 16091 bytes > Desc: not available > URL: < > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe > > > ------------------------------ > > _______________________________________________ > R-SIG-Finance mailing list > [hidden email] > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > > End of R-SIG-Finance Digest, Vol 77, Issue 8 > ******************************************** > > > [[alternative HTML version deleted]] > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. > > > > ----------------------------------- > Stefano M. Iacus > Department of Economics, > Business and Statistics > University of Milan > Via Conservatorio, 7 > I-20123 Milan - Italy > Ph.: +39 02 50321 461 > Fax: +39 02 50321 505 > http://www.economia.unimi.it/iacus > ------------------------------------------------------------------------------------ > Please don't send me Word or PowerPoint attachments if not > absolutely necessary. See: > http://www.gnu.org/philosophy/no-word-attachments.html > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. > > > > > > No virus found in this incoming message. > Checked by AVG - www.avg.com > > > > > > > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. > > > > > > No virus found in this incoming message. > Checked by AVG - www.avg.com > > > > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- > Subscriber-posting only. If you want to post, subscribe first. -- Also > note that this is not the r-help list where general R questions should > go. > > > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. > > > > > > No virus found in this incoming message. > Checked by AVG - www.avg.com > Version: 9.0.862 / Virus Database: 271.1.1/3204 - Release Date: 10/18/10 17:34:00 > > _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
In reply to this post by Yihao Lu aeolus_lu
Hi Folks
I'm using this to find cointegrated stocks on the AX. library(xts) library(quantmod) # quickly re-source this file s <- function() source('meanrev.R') checkPairFromYahoo <- function(sym1, sym2, dateFilter='::') { t.xts <- getCombined(sym1, sym2, dateFilter=dateFilter) cat("Date range is", format(start(t.xts)), "to", format(end(t.xts)), "\n") # Build linear model m <- buildLM(t.xts) # Note beta -- http://en.wikipedia.org/wiki/Beta_(finance) beta <- getBeta(m) cat("Assumed hedge ratio is", beta, "\n") # Build spread sprd <- buildSpread(t.xts, beta) # Test cointegration ht <- testCoint(sprd) cat("PP p-value is", as.double(ht$p.value), "\n") if (as.double(ht$p.value) < 0.05) { cat("###############################################################\n", sym1 ,":", sym2 ," is likely mean-reverting.\n", "###########################################################\n" ) } else { #cat(sym1 ,":", sym2 ," is not mean-reverting.\n") } } getCombined <- function(sym1, sym2, dateFilter='::') { # Grab historical data for both symbols one <- getSymbols(sym1, auto.assign=FALSE) two <- getSymbols(sym2, auto.assign=FALSE) # Give columns more usable names colnames(one) <- c('Open', 'High', 'Low', 'Close', 'Volume', 'Adjusted') colnames(two) <- c('Open', 'High', 'Low', 'Close', 'Volume', 'Adjusted') # Build combined object return(merge(one$Close, two$Close, all=FALSE)[dateFilter]) } buildLM <- function(combined) { return(lm(Close ~ Close.1 + 0, combined)) } getBeta <- function(m) { return(as.double(coef(m)[1])) } buildSpread <- function(combined, beta) { return(combined$Close - beta*combined$Close.1) } testCoint <- function(sprd) { return(PP.test(sprd, lshort = FALSE)) } I run it on batches of stock-pairs and then have a look at those which are cointegrated. Assuming my code is right (and anyone who thinks there is something wrong with it please let me know ;-) Just wondered if anyone simply goes with the results, or if a test of logic is required. I found, for example, that AGL ( a big gas company) was cointegrated with Bunnings Wharehouses (a hardware superstore chain). Can't see the reason for that. AMP (major insurer) cointegrates with AXA (another major insurer). That makes sense and it cointegrates with Westpac (major bank) still some logic but a bit thinner. It also cointegrates with Fortescue Metals (big iron ore operation). Not much logic there. Anyway question is: do you get better results by using informed judgement on these things or just trust the figures? Any comments most welcome. Stephen Choularton Ph.D., FIoD
9999 2226 0413 545 182 for insurance go to www.netinsure.com.au for markets go to www.organicfoodmarkets.com.au On 19/10/2010 12:35 PM, Yihao Lu aeolus_lu wrote: I am doing rolling ADF test on some time series to check mean reversion. When I use short period rolling, I find the residue is not stationary at all. However, when I use horizon longer than 5 years, I find very significant stationary. On the other hand, I find the half life is only around 30 days. Is there anyone who can give me some possible explanation or guide me to some reference? thanks Best, Yihao ________________________________Date: Tue, 19 Oct 2010 09:03:55 +1100 From: [hidden email] To: [hidden email] CC: [hidden email] Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck Hi I am still trying to sort this one out. Any comments from anyone would be most welcome. Stephen Choularton Ph.D., FIoD On 14/10/2010 7:29 AM, Stephen Choularton wrote: Thanks for this help. Trying to make sense of it so I have added some notes to the code. I have marked them #?# Delighted if you can tell me if I am write or wrong, add any comments, answers. #?# This appears to be the function that is doing the 'Ornstein-Uhlenbeck #?# process work' particularly via dcOU #?# I have noted in several places that I am after: #?# 'the half-life of the decay equals ln(2)/θ' #?# 'The half-life is given as log(2)/mean-reversion speed.' #?# and I see theta appearing at a number of points in the code. #?# Can you tell me why 3 thetas viz theta1, theta2, theta3 and what they do? #?# eg is one of these the theta I am after? # ex3.01.R OU.lik <- function(theta1, theta2, theta3){ n <- length(X) dt <- deltat(X) -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) } require(stats4) require(sde) #?# random numer generation seed set.seed(123) #?# creation of a data set X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) #?# If I Look at X its like this: #?# Time Series: #?# Start = 0 #?# End = 1000 #?# Frequency = 1 #?# [1] 1.00000000 etc #?# What sort of data object is it and how would I coerce an object with one #?# column from a read.csv into it? mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit summary(fit) #?# This gives: #?# Maximum likelihood estimation #?# Call: #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) #?# Coefficients: #?# Estimate Std. Error #?# theta1 3.355322 0.28159504 #?# theta2 1.106107 0.09010627 #?# theta3 2.052815 0.07624441 #?# -2 log L: 3366.389 #?# What's this telling me? # ex3.01.R (cont.) prof <- profile(fit) par(mfrow=c(1,3)) plot(prof) par(mfrow=c(1,1)) vcov(fit) confint(fit) #?# This provides me with this output using 'fit' from before: #?# > vcov(fit) #?# theta1 theta2 theta3 #?# theta1 0.07929576 0.024620718 0.016634557 #?# theta2 0.02462072 0.008119141 0.005485549 #?# theta3 0.01663456 0.005485549 0.005813209 #?# > confint(fit) #?# Profiling... #?# 2.5 % 97.5 % #?# theta1 2.8448980 3.960982 #?# theta2 0.9433338 1.300629 #?# theta3 1.9147136 2.216113 #?# and 'fit' is: #?# Call: #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) #?# Coefficients: #?# theta1 theta2 theta3 #?# 3.355322 1.106107 2.052815 #?# plus some graphic output #?# Again, what's this telling me. #?# This looks like a further example? # ex3.01.R (cont.) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 summary(fit2) Please excuse the length of this email (and my lack of understanding) Hope you can help and thanks. Stephen Choularton Ph.D., FIoD On 13/10/2010 2:41 AM, stefano iacus wrote: just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU # ex3.01.R OU.lik <- function(theta1, theta2, theta3){ n <- length(X) dt <- deltat(X) -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) } require(stats4) require(sde) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit summary(fit) # ex3.01.R (cont.) prof <- profile(fit) par(mfrow=c(1,3)) plot(prof) par(mfrow=c(1,1)) vcov(fit) confint(fit) # ex3.01.R (cont.) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 summary(fit2) I hope this helps out stefano On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: Hi Stephen, You could take a look at http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model for the linear regression method, or take a look at the package "sde" which contains some examples using GMM (not for the Ornstein-Uhlenbeck process, though, only the CIR). The half-life is given as log(2)/mean-reversion speed. Do keep an eye on the partition of the time-axis, e.g. what frequency you are using (daily, yearly) for interpreting the half-life. BR, Bjørn ------------------------------ Message: 2 Date: Tue, 12 Oct 2010 05:43:32 -0400 From: Sarbo To: [hidden email] Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck Message-ID: Content-Type: text/plain; charset="utf-8" By half-life, do you mean the speed of mean-reversion? If so, there's a bit of algebraic tomfoolery that's required to discretise the equation and then fit the data to it. I don't have the time right now to go into all the details but it's not hard- you can parameterise the process using simple linear regression. If you need help with that I'll try and get back to you tonight about it. On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: Hi Wonder if anyone could point me how I use this method to discover the half life of a mean reverting process. I am looking into pair trading and the time it takes for a cointegrated pair to revert to the norm. -- Stephen Choularton Ph.D., FIoD 9999 2226 0413 545 182 for insurance go to www.netinsure.com.au for markets go to www.organicfoodmarkets.com.au _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. -------------- next part -------------- An HTML attachment was scrubbed... URL: < https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html -------------- next part -------------- A non-text attachment was scrubbed... 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Iacus Department of Economics, Business and Statistics University of Milan Via Conservatorio, 7 I-20123 Milan - Italy Ph.: +39 02 50321 461 Fax: +39 02 50321 505 http://www.economia.unimi.it/iacus ------------------------------------------------------------------------------------ Please don't send me Word or PowerPoint attachments if not absolutely necessary. See: http://www.gnu.org/philosophy/no-word-attachments.html _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go._______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go.No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
In reply to this post by Yihao Lu aeolus_lu
You have to be careful when interpreting rolling ADF tests because the usual critical values for evaluating the tests are not valid. See the 1992 Journal of Business and Economic Statistics paper by Banerjee, Lumsdaine and Stock.
**************************************************************** * Eric Zivot * * Professor and Gary Waterman Distinguished Scholar * * Department of Economics * * Adjunct Professor of Finance * * Adjunct Professor of Statistics * Box 353330 email: [hidden email] * * University of Washington phone: 206-543-6715 * * Seattle, WA 98195-3330 * * * www: http://faculty.washington.edu/ezivot * **************************************************************** On Mon, 18 Oct 2010, Yihao Lu aeolus_lu wrote: > > I am doing rolling ADF test on some time series to check mean reversion. When I use short period rolling, I find the residue is not stationary at all. However, when I use horizon longer than 5 years, I find very significant stationary. On the other hand, I find the half life is only around 30 days. > Is there anyone who can give me some possible explanation or guide me to some reference? thanks > > Best, > Yihao > > > > > > > > ________________________________ >> Date: Tue, 19 Oct 2010 09:03:55 +1100 >> From: [hidden email] >> To: [hidden email] >> CC: [hidden email] >> Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck >> >> Hi >> >> I am still trying to sort this one out. Any comments from anyone would >> be most welcome. >> >> Stephen Choularton Ph.D., FIoD >> >> >> >> On 14/10/2010 7:29 AM, Stephen Choularton wrote: >> Thanks for this help. >> >> Trying to make sense of it so I have added some notes to the code. I >> have marked them #?# >> >> Delighted if you can tell me if I am write or wrong, add any comments, >> answers. >> >> >> #?# This appears to be the function that is doing the 'Ornstein-Uhlenbeck >> #?# process work' particularly via dcOU >> #?# I have noted in several places that I am after: >> #?# 'the half-life of the decay equals ln(2)/θ' >> #?# 'The half-life is given as log(2)/mean-reversion speed.' >> #?# and I see theta appearing at a number of points in the code. >> #?# Can you tell me why 3 thetas viz theta1, theta2, theta3 and what they do? >> #?# eg is one of these the theta I am after? >> >> # ex3.01.R >> OU.lik <- function(theta1, theta2, theta3){ >> n <- length(X) >> dt <- deltat(X) >> -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) >> } >> >> require(stats4) >> require(sde) >> >> #?# random numer generation seed >> set.seed(123) >> >> #?# creation of a data set >> X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) >> #?# If I Look at X its like this: >> #?# Time Series: >> #?# Start = 0 >> #?# End = 1000 >> #?# Frequency = 1 >> #?# [1] 1.00000000 etc >> #?# What sort of data object is it and how would I coerce an object with one >> #?# column from a read.csv into it? >> >> >> mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), >> method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit >> summary(fit) >> >> #?# This gives: >> >> #?# Maximum likelihood estimation >> >> #?# Call: >> #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, >> #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) >> >> #?# Coefficients: >> #?# Estimate Std. Error >> #?# theta1 3.355322 0.28159504 >> #?# theta2 1.106107 0.09010627 >> #?# theta3 2.052815 0.07624441 >> >> #?# -2 log L: 3366.389 >> >> #?# What's this telling me? >> >> # ex3.01.R (cont.) >> prof <- profile(fit) >> par(mfrow=c(1,3)) >> plot(prof) >> par(mfrow=c(1,1)) >> vcov(fit) >> confint(fit) >> >> #?# This provides me with this output using 'fit' from before: >> >> #?# > vcov(fit) >> #?# theta1 theta2 theta3 >> #?# theta1 0.07929576 0.024620718 0.016634557 >> #?# theta2 0.02462072 0.008119141 0.005485549 >> #?# theta3 0.01663456 0.005485549 0.005813209 >> #?# > confint(fit) >> #?# Profiling... >> #?# 2.5 % 97.5 % >> #?# theta1 2.8448980 3.960982 >> #?# theta2 0.9433338 1.300629 >> #?# theta3 1.9147136 2.216113 >> >> #?# and 'fit' is: >> >> #?# Call: >> #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, >> #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) >> >> #?# Coefficients: >> #?# theta1 theta2 theta3 >> #?# 3.355322 1.106107 2.052815 >> >> #?# plus some graphic output >> >> #?# Again, what's this telling me. >> >> #?# This looks like a further example? >> # ex3.01.R (cont.) >> set.seed(123) >> X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) >> mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), >> method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 >> summary(fit2) >> >> >> >> >> Please excuse the length of this email (and my lack of understanding) >> >> Hope you can help and thanks. >> >> >> >> >> Stephen Choularton Ph.D., FIoD >> >> >> On 13/10/2010 2:41 AM, stefano iacus wrote: >> >> just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. >> >> sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. >> >> This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU >> >> >> # ex3.01.R >> OU.lik <- function(theta1, theta2, theta3){ >> n <- length(X) >> dt <- deltat(X) >> -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) >> } >> >> require(stats4) >> require(sde) >> set.seed(123) >> X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) >> mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), >> method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit >> summary(fit) >> >> # ex3.01.R (cont.) >> prof <- profile(fit) >> par(mfrow=c(1,3)) >> plot(prof) >> par(mfrow=c(1,1)) >> vcov(fit) >> confint(fit) >> >> # ex3.01.R (cont.) >> set.seed(123) >> X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) >> mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), >> method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 >> summary(fit2) >> >> >> I hope this helps out >> >> stefano >> >> On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: >> >> >> >> Hi Stephen, >> >> You could take a look at >> >> http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model >> >> for the linear regression method, or take a look at the package "sde" which >> contains some examples using GMM (not for the Ornstein-Uhlenbeck process, >> though, only the CIR). >> >> The half-life is given as log(2)/mean-reversion speed. >> >> Do keep an eye on the partition of the time-axis, e.g. what frequency you >> are using (daily, yearly) for interpreting the half-life. >> >> BR, >> Bjørn >> >> >> >> >> >> >> >> >> ------------------------------ >> >> Message: 2 >> Date: Tue, 12 Oct 2010 05:43:32 -0400 >> From: Sarbo >> To: [hidden email] >> Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck >> Message-ID: >> Content-Type: text/plain; charset="utf-8" >> >> By half-life, do you mean the speed of mean-reversion? >> >> If so, there's a bit of algebraic tomfoolery that's required to >> discretise the equation and then fit the data to it. I don't have the >> time right now to go into all the details but it's not hard- you can >> parameterise the process using simple linear regression. If you need >> help with that I'll try and get back to you tonight about it. >> >> On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: >> >> >> >> Hi >> >> Wonder if anyone could point me how I use this method to discover the >> half life of a mean reverting process. >> >> I am looking into pair trading and the time it takes for a >> cointegrated pair to revert to the norm. >> >> -- >> Stephen Choularton Ph.D., FIoD >> >> 9999 2226 >> 0413 545 182 >> >> >> for insurance go to www.netinsure.com.au >> for markets go to www.organicfoodmarkets.com.au >> >> >> _______________________________________________ >> [hidden email] mailing list >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >> -- Subscriber-posting only. If you want to post, subscribe first. >> -- Also note that this is not the r-help list where general R questions >> >> >> should go. >> >> >> -------------- next part -------------- >> An HTML attachment was scrubbed... >> URL: < >> https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html >> >> >> -------------- next part -------------- >> A non-text attachment was scrubbed... >> Name: CoS2010Winner.JPG >> Type: image/jpeg >> Size: 16091 bytes >> Desc: not available >> URL: < >> https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe >> >> >> ------------------------------ >> >> _______________________________________________ >> R-SIG-Finance mailing list >> [hidden email] >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >> >> >> End of R-SIG-Finance Digest, Vol 77, Issue 8 >> ******************************************** >> >> >> [[alternative HTML version deleted]] >> >> _______________________________________________ >> [hidden email] mailing list >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >> -- Subscriber-posting only. If you want to post, subscribe first. >> -- Also note that this is not the r-help list where general R questions should go. >> >> >> >> ----------------------------------- >> Stefano M. Iacus >> Department of Economics, >> Business and Statistics >> University of Milan >> Via Conservatorio, 7 >> I-20123 Milan - Italy >> Ph.: +39 02 50321 461 >> Fax: +39 02 50321 505 >> http://www.economia.unimi.it/iacus >> ------------------------------------------------------------------------------------ >> Please don't send me Word or PowerPoint attachments if not >> absolutely necessary. See: >> http://www.gnu.org/philosophy/no-word-attachments.html >> >> _______________________________________________ >> [hidden email] mailing list >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >> -- Subscriber-posting only. If you want to post, subscribe first. >> -- Also note that this is not the r-help list where general R questions should go. >> >> >> >> >> >> No virus found in this incoming message. >> Checked by AVG - www.avg.com >> >> >> >> >> >> >> >> _______________________________________________ >> [hidden email] mailing list >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance >> -- Subscriber-posting only. If you want to post, subscribe first. >> -- Also note that this is not the r-help list where general R questions should go. >> >> >> >> >> >> No virus found in this incoming message. >> Checked by AVG - www.avg.com >> >> >> >> >> _______________________________________________ >> [hidden email] mailing list >> https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- >> Subscriber-posting only. If you want to post, subscribe first. -- Also >> note that this is not the r-help list where general R questions should >> go. > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
In reply to this post by Yihao Lu aeolus_lu
Yihao,
Prof. Zivot is right. The ADF test isn't a great way to test for mean-reversion; it's merely a way to test for stationarity to specify the degrees in an ARIMA model. I dug up some code from my misspent youth as a consultant which you might find useful: StochasticProcessTest <- function(PriceSeries, delta.t, p.value, diagnostics = TRUE, from = NULL, to = NULL, by = NULL, currency = '($)'){ require(stats) require(fSeries) m <- length(PriceSeries) S1 <- PriceSeries[1:(m-1)] S2 <- PriceSeries[2:m] #Fit a GBM process: Y <- diff(log(PriceSeries)); X <- S1 GBMfit <- lm(Y ~ X) #Fit a mean-reverting GBM process: Y <- diff(PriceSeries) / S1; X <- log(S1) MRGBMfit <- lm(Y ~ X) #Fit a Vasicek (O-U) process: Y <- diff(PriceSeries); X <- S1 Vfit <- lm(Y ~ X) #Fit a Cox-Ingersoll-Ross process: Y <- diff(PriceSeries) / sqrt(S1); X <- S1 / sqrt(S1) CIRfit <- lm(Y ~ X) #Gather the parameter estimates: kappa <- as.vector((c(0, -MRGBMfit$coef[2], -Vfit$coef[2], -CIRfit $coef[2])) / delta.t) mu <- as.vector((c(mean(log(S2/S1)), MRGBMfit$coef[1] / kappa[2], Vfit $coef[1] / kappa[3], CIRfit$coef[1] / kappa[4])) / delta.t) sigma <- (c(sd(GBMfit$resid), sd(MRGBMfit$resid), sd(Vfit$resid), sd(CIRfit$resid))) / sqrt(delta.t) tstat <- as.vector(c(summary(GBMfit)$coef[2,3], summary(MRGBMfit)$coef[2,3], summary(Vfit)$coef[2,3], summary(CIRfit)$coef[2,3])) fstat <- list(summary(GBMfit)$fstatistic, summary(MRGBMfit)$fstatistic, summary(Vfit)$fstatistic, summary(CIRfit)$fstatistic) names(fstat) <- c('GBM', 'Mean.Revert.GBM', 'Vasicek', 'CIR') AICs <- c(AIC(GBMfit), AIC(MRGBMfit), AIC(Vfit), AIC(CIRfit)) paramframe <- data.frame(rbind(kappa, mu, sigma, tstat, AICs)) names(paramframe) <- names(fstat) #Now figure out what the actual process is, using AIC: crit <- ifelse(m > 30, qnorm(1 - p.value), qt(1 - p.value, df = n)) tmp <- which.min(AICs) Processes <- names(fstat) Verdict <- Processes[tmp] FinalSummary <- switch(Verdict, GBM = summary(GBMfit), CIR = summary(CIRfit), Mean.Revert.GBM = MRGBMfit, Vasicek = Vfit) fitobj <- switch(Verdict, GBM = GBMfit, CIR = CIRfit, Mean.Revert.GBM = MRGBMfit, Vasicek = Vfit) Output <- list(Parameters = paramframe, Critical.Value = crit, Verdict = Verdict, FinalSummary = FinalSummary, fstat = fstat, fitted.object = fitobj) if (diagnostics){ op <- par(ask = TRUE) on.exit(op) if(!all(c(class(from), class(to)) == 'Date')){ S <- timeSeries(PriceSeries) } else S = timeSeries(PriceSeries, seq(from, to, length.out = m)) plot(S, type = 'l', xlab = 'Date', ylab = paste('Price Series', currency), main = 'Time Series Plot of Data', lwd = 2, col = 'blue') rets <- returns(PriceSeries, 'continuous')[-1] hist(rets, xlab = paste('Log-Returns', currency), col = 'blue', border = 'white', main = 'Histogram of Return Series', freq = FALSE) x <- seq(min(rets), max(rets), length.out = max(m, 1000)) lines(x, dnorm(x, mean(rets), sd(rets)), col = 'magenta', lwd = 2) lines(density(rets), col = 'green', lwd = 2) legend('topright', legend = c('Log-Returns', 'Observed CDF', 'Gaussian Fit'), lwd = rep(2, 3), col = c('blue', 'green', 'magenta')) plot(fitobj, 1:6) } return(Output) } (I don't claim that it's necessarily great code, but it does seem to work.) On Mon, 2010-10-18 at 21:35 -0400, Yihao Lu aeolus_lu wrote: > I am doing rolling ADF test on some time series to check mean reversion. When I use short period rolling, I find the residue is not stationary at all. However, when I use horizon longer than 5 years, I find very significant stationary. On the other hand, I find the half life is only around 30 days. > Is there anyone who can give me some possible explanation or guide me to some reference? thanks > > Best, > Yihao > > > > > > > > ________________________________ > > Date: Tue, 19 Oct 2010 09:03:55 +1100 > > From: [hidden email] > > To: [hidden email] > > CC: [hidden email] > > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > > > > Hi > > > > I am still trying to sort this one out. Any comments from anyone would > > be most welcome. > > > > Stephen Choularton Ph.D., FIoD > > > > > > > > On 14/10/2010 7:29 AM, Stephen Choularton wrote: > > Thanks for this help. > > > > Trying to make sense of it so I have added some notes to the code. I > > have marked them #?# > > > > Delighted if you can tell me if I am write or wrong, add any comments, > > answers. > > > > > > #?# This appears to be the function that is doing the 'Ornstein-Uhlenbeck > > #?# process work' particularly via dcOU > > #?# I have noted in several places that I am after: > > #?# 'the half-life of the decay equals ln(2)/Î¸' > > #?# 'The half-life is given as log(2)/mean-reversion speed.' > > #?# and I see theta appearing at a number of points in the code. > > #?# Can you tell me why 3 thetas viz theta1, theta2, theta3 and what they do? > > #?# eg is one of these the theta I am after? > > > > # ex3.01.R > > OU.lik <- function(theta1, theta2, theta3){ > > n <- length(X) > > dt <- deltat(X) > > -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) > > } > > > > require(stats4) > > require(sde) > > > > #?# random numer generation seed > > set.seed(123) > > > > #?# creation of a data set > > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) > > #?# If I Look at X its like this: > > #?# Time Series: > > #?# Start = 0 > > #?# End = 1000 > > #?# Frequency = 1 > > #?# [1] 1.00000000 etc > > #?# What sort of data object is it and how would I coerce an object with one > > #?# column from a read.csv into it? > > > > > > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit > > summary(fit) > > > > #?# This gives: > > > > #?# Maximum likelihood estimation > > > > #?# Call: > > #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, > > #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) > > > > #?# Coefficients: > > #?# Estimate Std. Error > > #?# theta1 3.355322 0.28159504 > > #?# theta2 1.106107 0.09010627 > > #?# theta3 2.052815 0.07624441 > > > > #?# -2 log L: 3366.389 > > > > #?# What's this telling me? > > > > # ex3.01.R (cont.) > > prof <- profile(fit) > > par(mfrow=c(1,3)) > > plot(prof) > > par(mfrow=c(1,1)) > > vcov(fit) > > confint(fit) > > > > #?# This provides me with this output using 'fit' from before: > > > > #?# > vcov(fit) > > #?# theta1 theta2 theta3 > > #?# theta1 0.07929576 0.024620718 0.016634557 > > #?# theta2 0.02462072 0.008119141 0.005485549 > > #?# theta3 0.01663456 0.005485549 0.005813209 > > #?# > confint(fit) > > #?# Profiling... > > #?# 2.5 % 97.5 % > > #?# theta1 2.8448980 3.960982 > > #?# theta2 0.9433338 1.300629 > > #?# theta3 1.9147136 2.216113 > > > > #?# and 'fit' is: > > > > #?# Call: > > #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, > > #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) > > > > #?# Coefficients: > > #?# theta1 theta2 theta3 > > #?# 3.355322 1.106107 2.052815 > > > > #?# plus some graphic output > > > > #?# Again, what's this telling me. > > > > #?# This looks like a further example? > > # ex3.01.R (cont.) > > set.seed(123) > > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) > > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 > > summary(fit2) > > > > > > > > > > Please excuse the length of this email (and my lack of understanding) > > > > Hope you can help and thanks. > > > > > > > > > > Stephen Choularton Ph.D., FIoD > > > > > > On 13/10/2010 2:41 AM, stefano iacus wrote: > > > > just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. > > > > sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. > > > > This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU > > > > > > # ex3.01.R > > OU.lik <- function(theta1, theta2, theta3){ > > n <- length(X) > > dt <- deltat(X) > > -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) > > } > > > > require(stats4) > > require(sde) > > set.seed(123) > > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) > > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit > > summary(fit) > > > > # ex3.01.R (cont.) > > prof <- profile(fit) > > par(mfrow=c(1,3)) > > plot(prof) > > par(mfrow=c(1,1)) > > vcov(fit) > > confint(fit) > > > > # ex3.01.R (cont.) > > set.seed(123) > > X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) > > mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), > > method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 > > summary(fit2) > > > > > > I hope this helps out > > > > stefano > > > > On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: > > > > > > > > Hi Stephen, > > > > You could take a look at > > > > http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model > > > > for the linear regression method, or take a look at the package "sde" which > > contains some examples using GMM (not for the Ornstein-Uhlenbeck process, > > though, only the CIR). > > > > The half-life is given as log(2)/mean-reversion speed. > > > > Do keep an eye on the partition of the time-axis, e.g. what frequency you > > are using (daily, yearly) for interpreting the half-life. > > > > BR, > > BjÃ¸rn > > > > > > > > > > > > > > > > > > ------------------------------ > > > > Message: 2 > > Date: Tue, 12 Oct 2010 05:43:32 -0400 > > From: Sarbo > > To: [hidden email] > > Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck > > Message-ID: > > Content-Type: text/plain; charset="utf-8" > > > > By half-life, do you mean the speed of mean-reversion? > > > > If so, there's a bit of algebraic tomfoolery that's required to > > discretise the equation and then fit the data to it. I don't have the > > time right now to go into all the details but it's not hard- you can > > parameterise the process using simple linear regression. If you need > > help with that I'll try and get back to you tonight about it. > > > > On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: > > > > > > > > Hi > > > > Wonder if anyone could point me how I use this method to discover the > > half life of a mean reverting process. > > > > I am looking into pair trading and the time it takes for a > > cointegrated pair to revert to the norm. > > > > -- > > Stephen Choularton Ph.D., FIoD > > > > 9999 2226 > > 0413 545 182 > > > > > > for insurance go to www.netinsure.com.au > > for markets go to www.organicfoodmarkets.com.au > > > > > > _______________________________________________ > > [hidden email] mailing list > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > -- Subscriber-posting only. If you want to post, subscribe first. > > -- Also note that this is not the r-help list where general R questions > > > > > > should go. > > > > > > -------------- next part -------------- > > An HTML attachment was scrubbed... > > URL: < > > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html > > > > > > -------------- next part -------------- > > A non-text attachment was scrubbed... > > Name: CoS2010Winner.JPG > > Type: image/jpeg > > Size: 16091 bytes > > Desc: not available > > URL: < > > https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe > > > > > > ------------------------------ > > > > _______________________________________________ > > R-SIG-Finance mailing list > > [hidden email] > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > > > > > End of R-SIG-Finance Digest, Vol 77, Issue 8 > > ******************************************** > > > > > > [[alternative HTML version deleted]] > > > > _______________________________________________ > > [hidden email] mailing list > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > -- Subscriber-posting only. If you want to post, subscribe first. > > -- Also note that this is not the r-help list where general R questions should go. > > > > > > > > ----------------------------------- > > Stefano M. Iacus > > Department of Economics, > > Business and Statistics > > University of Milan > > Via Conservatorio, 7 > > I-20123 Milan - Italy > > Ph.: +39 02 50321 461 > > Fax: +39 02 50321 505 > > http://www.economia.unimi.it/iacus > > ------------------------------------------------------------------------------------ > > Please don't send me Word or PowerPoint attachments if not > > absolutely necessary. See: > > http://www.gnu.org/philosophy/no-word-attachments.html > > > > _______________________________________________ > > [hidden email] mailing list > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > -- Subscriber-posting only. If you want to post, subscribe first. > > -- Also note that this is not the r-help list where general R questions should go. > > > > > > > > > > > > No virus found in this incoming message. > > Checked by AVG - www.avg.com > > > > > > > > > > > > > > > > _______________________________________________ > > [hidden email] mailing list > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > > -- Subscriber-posting only. If you want to post, subscribe first. > > -- Also note that this is not the r-help list where general R questions should go. > > > > > > > > > > > > No virus found in this incoming message. > > Checked by AVG - www.avg.com > > > > > > > > > > _______________________________________________ > > [hidden email] mailing list > > https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- > > Subscriber-posting only. If you want to post, subscribe first. -- Also > > note that this is not the r-help list where general R questions should > > go. > > _______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-sig-finance > -- Subscriber-posting only. If you want to post, subscribe first. > -- Also note that this is not the r-help list where general R questions should go. [[alternative HTML version deleted]] _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
In reply to this post by Stephen Choularton-3
Stephen,
It depends what you mean by "logic".
If you mean statistical logic, I'll defer to Eric Zivot and
Sarbo who are far wiser than I am. I will note, however, that you are testing
for a p-value of 0.05, so I expect 5% of your test results to be misleading. In
other words, for every 20 pairs tested by your batch job, I expect one will be
suspect.
"Spurious cointegration" is a serious problem. I suggest
Googling that topic. You may be suprised what you learn. (The irony, of course,
is that cointegration was supposed to cure "spurious correlation." Oh
well.)
If you mean financial logic, I strongly suggest not blindly
risking money on your statistical test. Some filtering is required. Look for
trades that make sense.
For example, my software reports that the stocks of MSFT and
GOOG form a mean-reverting pair. But I would not trade that spread: too much
idiosyncratic risk. My software also reports that Corn futures and
Soybean Oil futures form a mean-reverting pair. But I would not trade that
spread because the economic connection between corn and bean oil is too
weak.
Hope that helps.
Paul From: [hidden email] [mailto:[hidden email]] On Behalf Of Stephen Choularton Sent: Monday, October 18, 2010 9:46 PM To: [hidden email] Subject: [R-SIG-Finance] cointegration I'm using this to find cointegrated stocks on the AX. library(xts) library(quantmod) # quickly re-source this file s <- function() source('meanrev.R') checkPairFromYahoo <- function(sym1, sym2, dateFilter='::') { t.xts <- getCombined(sym1, sym2, dateFilter=dateFilter) cat("Date range is", format(start(t.xts)), "to", format(end(t.xts)), "\n") # Build linear model m <- buildLM(t.xts) # Note beta -- http://en.wikipedia.org/wiki/Beta_(finance) beta <- getBeta(m) cat("Assumed hedge ratio is", beta, "\n") # Build spread sprd <- buildSpread(t.xts, beta) # Test cointegration ht <- testCoint(sprd) cat("PP p-value is", as.double(ht$p.value), "\n") if (as.double(ht$p.value) < 0.05) { cat("###############################################################\n", sym1 ,":", sym2 ," is likely mean-reverting.\n", "###########################################################\n" ) } else { #cat(sym1 ,":", sym2 ," is not mean-reverting.\n") } } getCombined <- function(sym1, sym2, dateFilter='::') { # Grab historical data for both symbols one <- getSymbols(sym1, auto.assign=FALSE) two <- getSymbols(sym2, auto.assign=FALSE) # Give columns more usable names colnames(one) <- c('Open', 'High', 'Low', 'Close', 'Volume', 'Adjusted') colnames(two) <- c('Open', 'High', 'Low', 'Close', 'Volume', 'Adjusted') # Build combined object return(merge(one$Close, two$Close, all=FALSE)[dateFilter]) } buildLM <- function(combined) { return(lm(Close ~ Close.1 + 0, combined)) } getBeta <- function(m) { return(as.double(coef(m)[1])) } buildSpread <- function(combined, beta) { return(combined$Close - beta*combined$Close.1) } testCoint <- function(sprd) { return(PP.test(sprd, lshort = FALSE)) } I run it on batches of stock-pairs and then have a look at those which are cointegrated. Assuming my code is right (and anyone who thinks there is something wrong with it please let me know ;-) Just wondered if anyone simply goes with the results, or if a test of logic is required. I found, for example, that AGL ( a big gas company) was cointegrated with Bunnings Wharehouses (a hardware superstore chain). Can't see the reason for that. AMP (major insurer) cointegrates with AXA (another major insurer). That makes sense and it cointegrates with Westpac (major bank) still some logic but a bit thinner. It also cointegrates with Fortescue Metals (big iron ore operation). Not much logic there. Anyway question is: do you get better results by using informed judgement on these things or just trust the figures? Any comments most welcome. Stephen Choularton Ph.D., FIoD 9999 2226 0413 545 182 for insurance go to www.netinsure.com.au for markets go to www.organicfoodmarkets.com.au On 19/10/2010 12:35 PM, Yihao Lu aeolus_lu wrote: I am doing rolling ADF test on some time series to check mean reversion. When I use short period rolling, I find the residue is not stationary at all. However, when I use horizon longer than 5 years, I find very significant stationary. On the other hand, I find the half life is only around 30 days. Is there anyone who can give me some possible explanation or guide me to some reference? thanks Best, Yihao ________________________________Date: Tue, 19 Oct 2010 09:03:55 +1100 From: [hidden email] To: [hidden email] CC: [hidden email] Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck Hi I am still trying to sort this one out. Any comments from anyone would be most welcome. Stephen Choularton Ph.D., FIoD On 14/10/2010 7:29 AM, Stephen Choularton wrote: Thanks for this help. Trying to make sense of it so I have added some notes to the code. I have marked them #?# Delighted if you can tell me if I am write or wrong, add any comments, answers. #?# This appears to be the function that is doing the 'Ornstein-Uhlenbeck #?# process work' particularly via dcOU #?# I have noted in several places that I am after: #?# 'the half-life of the decay equals ln(2)/θ' #?# 'The half-life is given as log(2)/mean-reversion speed.' #?# and I see theta appearing at a number of points in the code. #?# Can you tell me why 3 thetas viz theta1, theta2, theta3 and what they do? #?# eg is one of these the theta I am after? # ex3.01.R OU.lik <- function(theta1, theta2, theta3){ n <- length(X) dt <- deltat(X) -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) } require(stats4) require(sde) #?# random numer generation seed set.seed(123) #?# creation of a data set X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) #?# If I Look at X its like this: #?# Time Series: #?# Start = 0 #?# End = 1000 #?# Frequency = 1 #?# [1] 1.00000000 etc #?# What sort of data object is it and how would I coerce an object with one #?# column from a read.csv into it? mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit summary(fit) #?# This gives: #?# Maximum likelihood estimation #?# Call: #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) #?# Coefficients: #?# Estimate Std. Error #?# theta1 3.355322 0.28159504 #?# theta2 1.106107 0.09010627 #?# theta3 2.052815 0.07624441 #?# -2 log L: 3366.389 #?# What's this telling me? # ex3.01.R (cont.) prof <- profile(fit) par(mfrow=c(1,3)) plot(prof) par(mfrow=c(1,1)) vcov(fit) confint(fit) #?# This provides me with this output using 'fit' from before: #?# > vcov(fit) #?# theta1 theta2 theta3 #?# theta1 0.07929576 0.024620718 0.016634557 #?# theta2 0.02462072 0.008119141 0.005485549 #?# theta3 0.01663456 0.005485549 0.005813209 #?# > confint(fit) #?# Profiling... #?# 2.5 % 97.5 % #?# theta1 2.8448980 3.960982 #?# theta2 0.9433338 1.300629 #?# theta3 1.9147136 2.216113 #?# and 'fit' is: #?# Call: #?# mle(minuslogl = OU.lik, start = list(theta1 = 1, theta2 = 0.5, #?# theta3 = 1), method = "L-BFGS-B", lower = c(-Inf, 0, 0)) #?# Coefficients: #?# theta1 theta2 theta3 #?# 3.355322 1.106107 2.052815 #?# plus some graphic output #?# Again, what's this telling me. #?# This looks like a further example? # ex3.01.R (cont.) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 summary(fit2) Please excuse the length of this email (and my lack of understanding) Hope you can help and thanks. Stephen Choularton Ph.D., FIoD On 13/10/2010 2:41 AM, stefano iacus wrote: just for completeness: OU process is gaussian and transitiion density is known in exact form. So maximum likelihood estimation works fine and I suggest to avoid GMM. sde package contains exact transition density for this process (e.g. ?dcOU) which you can use to build the likelihood to pass to mle() function. This example taken from the "inst" directory of the package sde. For the parametrization of the model see ?dcOU # ex3.01.R OU.lik <- function(theta1, theta2, theta3){ n <- length(X) dt <- deltat(X) -sum(dcOU(X[2:n], dt, X[1:(n-1)], c(theta1,theta2,theta3), log=TRUE)) } require(stats4) require(sde) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit summary(fit) # ex3.01.R (cont.) prof <- profile(fit) par(mfrow=c(1,3)) plot(prof) par(mfrow=c(1,1)) vcov(fit) confint(fit) # ex3.01.R (cont.) set.seed(123) X <- sde.sim(model="OU", theta=c(3,1,2), N=1000, delta=1e-3) mle(OU.lik, start=list(theta1=1, theta2=0.5, theta3=1), method="L-BFGS-B", lower=c(-Inf,0,0)) -> fit2 summary(fit2) I hope this helps out stefano On 12 Oct 2010, at 12:33, Bjorn Skogtro wrote: Hi Stephen, You could take a look at http://sitmo.com/doc/Calibrating_the_Ornstein-Uhlenbeck_model for the linear regression method, or take a look at the package "sde" which contains some examples using GMM (not for the Ornstein-Uhlenbeck process, though, only the CIR). The half-life is given as log(2)/mean-reversion speed. Do keep an eye on the partition of the time-axis, e.g. what frequency you are using (daily, yearly) for interpreting the half-life. BR, Bjørn ------------------------------ Message: 2 Date: Tue, 12 Oct 2010 05:43:32 -0400 From: Sarbo To: [hidden email] Subject: Re: [R-SIG-Finance] Ornstein-Uhlenbeck Message-ID: Content-Type: text/plain; charset="utf-8" By half-life, do you mean the speed of mean-reversion? If so, there's a bit of algebraic tomfoolery that's required to discretise the equation and then fit the data to it. I don't have the time right now to go into all the details but it's not hard- you can parameterise the process using simple linear regression. If you need help with that I'll try and get back to you tonight about it. On Tue, 2010-10-12 at 13:47 +1100, Stephen Choularton wrote: Hi Wonder if anyone could point me how I use this method to discover the half life of a mean reverting process. I am looking into pair trading and the time it takes for a cointegrated pair to revert to the norm. -- Stephen Choularton Ph.D., FIoD 9999 2226 0413 545 182 for insurance go to www.netinsure.com.au for markets go to www.organicfoodmarkets.com.au _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. -------------- next part -------------- An HTML attachment was scrubbed... URL: < https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.html -------------- next part -------------- A non-text attachment was scrubbed... Name: CoS2010Winner.JPG Type: image/jpeg Size: 16091 bytes Desc: not available URL: < https://stat.ethz.ch/pipermail/r-sig-finance/attachments/20101012/26e32fc7/attachment-0001.jpe ------------------------------ _______________________________________________ R-SIG-Finance mailing list [hidden email] https://stat.ethz.ch/mailman/listinfo/r-sig-finance End of R-SIG-Finance Digest, Vol 77, Issue 8 ******************************************** [[alternative HTML version deleted]] _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. ----------------------------------- Stefano M. Iacus Department of Economics, Business and Statistics University of Milan Via Conservatorio, 7 I-20123 Milan - Italy Ph.: +39 02 50321 461 Fax: +39 02 50321 505 http://www.economia.unimi.it/iacus ------------------------------------------------------------------------------------ Please don't send me Word or PowerPoint attachments if not absolutely necessary. See: http://www.gnu.org/philosophy/no-word-attachments.html _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go._______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go.No virus found in this incoming message. Checked by AVG - www.avg.com _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. 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In reply to this post by Sarbo
Thank you and thank Prof. Zivot Best, Yihao From: [hidden email] To: [hidden email] Date: Tue, 19 Oct 2010 06:59:37 -0400 Subject: Re: [R-SIG-Finance] Mean reversion Yihao, Prof. Zivot is right. The ADF test isn't a great way to test for mean-reversion; it's merely a way to test for stationarity to specify the degrees in an ARIMA model. I dug up some code from my misspent youth as a consultant which you might find useful: StochasticProcessTest <- function(PriceSeries, delta.t, p.value, diagnostics = TRUE, from = NULL, to = NULL, by = NULL, currency = '($)'){ require(stats) require(fSeries) m <- length(PriceSeries) S1 <- PriceSeries[1:(m-1)] S2 <- PriceSeries[2:m] #Fit a GBM process: Y <- diff(log(PriceSeries)); X <- S1 GBMfit <- lm(Y ~ X) #Fit a mean-reverting GBM process: Y <- diff(PriceSeries) / S1; X <- log(S1) MRGBMfit <- lm(Y ~ X) #Fit a Vasicek (O-U) process: Y <- diff(PriceSeries); X <- S1 Vfit <- lm(Y ~ X) #Fit a Cox-Ingersoll-Ross process: Y <- diff(PriceSeries) / sqrt(S1); X <- S1 / sqrt(S1) CIRfit <- lm(Y ~ X) #Gather the parameter estimates: kappa <- as.vector((c(0, -MRGBMfit$coef[2], -Vfit$coef[2], -CIRfit $coef[2])) / delta.t) mu <- as.vector((c(mean(log(S2/S1)), MRGBMfit$coef[1] / kappa[2], Vfit $coef[1] / kappa[3], CIRfit$coef[1] / kappa[4])) / delta.t) sigma <- (c(sd(GBMfit$resid), sd(MRGBMfit$resid), sd(Vfit$resid), sd(CIRfit$resid))) / sqrt(delta.t) tstat <- as.vector(c(summary(GBMfit)$coef[2,3], summary(MRGBMfit)$coef[2,3], summary(Vfit)$coef[2,3], summary(CIRfit)$coef[2,3])) fstat <- list(summary(GBMfit)$fstatistic, summary(MRGBMfit)$fstatistic, summary(Vfit)$fstatistic, summary(CIRfit)$fstatistic) names(fstat) <- c('GBM', 'Mean.Revert.GBM', 'Vasicek', 'CIR') AICs <- c(AIC(GBMfit), AIC(MRGBMfit), AIC(Vfit), AIC(CIRfit)) paramframe <- data.frame(rbind(kappa, mu, sigma, tstat, AICs)) names(paramframe) <- names(fstat) #Now figure out what the actual process is, using AIC: crit <- ifelse(m > 30, qnorm(1 - p.value), qt(1 - p.value, df = n)) tmp <- which.min(AICs) Processes <- names(fstat) Verdict <- Processes[tmp] FinalSummary <- switch(Verdict, GBM = summary(GBMfit), CIR = summary(CIRfit), Mean.Revert.GBM = MRGBMfit, Vasicek = Vfit) fitobj <- switch(Verdict, GBM = GBMfit, CIR = CIRfit, Mean.Revert.GBM = MRGBMfit, Vasicek = Vfit) Output <- list(Parameters = paramframe, Critical.Value = crit, Verdict = Verdict, FinalSummary = FinalSummary, fstat = fstat, fitted.object = fitobj) if (diagnostics){ op <- par(ask = TRUE) on.exit(op) if(!all(c(class(from), class(to)) == 'Date')){ S <- timeSeries(PriceSeries) } else S = timeSeries(PriceSeries, seq(from, to, length.out = m)) plot(S, type = 'l', xlab = 'Date', ylab = paste('Price Series', currency), main = 'Time Series Plot of Data', lwd = 2, col = 'blue') rets <- returns(PriceSeries, 'continuous')[-1] hist(rets, xlab = paste('Log-Returns', currency), col = 'blue', border = 'white', main = 'Histogram of Return Series', freq = FALSE) x <- seq(min(rets), max(rets), length.out = max(m, 1000)) lines(x, dnorm(x, mean(rets), sd(rets)), col = 'magenta', lwd = 2) lines(density(rets), col = 'green', lwd = 2) legend('topright', legend = c('Log-Returns', 'Observed CDF', 'Gaussian Fit'), lwd = rep(2, 3), col = c('blue', 'green', 'magenta')) plot(fitobj, 1:6) } return(Output) } (I don't claim that it's necessarily great code, but it does seem to work.) On Mon, 2010-10-18 at 21:35 -0400, Yihao Lu aeolus_lu wrote: > I am doing rolling ADF test on some time series to check mean reversion. When I use short period rolling, I find the residue is not stationary at all. However, when I use horizon longer than 5 years, I find very significant stationary. On the other hand, I find the half life is only around 30 days. > Is there anyone who can give me some possible explanation or guide me to some reference? thanks > > Best, > Yihao [[alternative HTML version deleted]] _______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-sig-finance -- Subscriber-posting only. If you want to post, subscribe first. -- Also note that this is not the r-help list where general R questions should go. |
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In reply to this post by Paul Teetor
Hi Paul,
I am very much impressed the way you explain your things. I have on question with regard to Mean Reversion I got various pairs my idea is to identify the indictor which will give me an indication of reversion. Can you highlight any method which would probably help me to identify such reversion. Thanks |
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