Thanks Eric for below. There's also a very nice book by Stefano Iacus called "Simulation and Inference for Stochastic<br />Differential Equations" with R examples. It doesn't seem to get much press but I bought it recently and REALLY like it. Just the right level of theory and practicality, atleast for me. It's not a Karatzas type of book or even Oskendal. Much better I think.<br /><br /> Mark<br /><br /><br /><p>On Jun 17, 2009, <strong>Eric Zivot</strong> <ezivot@u.washington.edu> wrote: </p><div class="replyBody"><blockquote style="border-left: 2px solid #267fdb; margin: 0pt 0pt 0pt 1.8ex; padding-left: 1ex">The Euler discretization of the Vasicek model is misspecified (i.e., it is not the exact discretization based on the true transition density) and so the least squares estimates will be biased (see Andrew Lo's Econometric Theory paper on estimating continuous time models from their discrete time counterparts, and Broze, Scaillet and Zakoian's Journal of Empirical Finance paper on estimating continuous time models from their discretized counterparts). This bias is typically not too large. You can eliminate this bias using an indirect inference estimation technique (see the book by Gourieroux and Monfort for an example applied to the Vasicek model). Alternatively, you can estimate the Vasicek model by exact mle using the code in the nice sde package (see also the accompanying book for more explanation). <br />ez<br /><br />****************************************************************<br />* Eric Zivot *<br />* Professor and Gary Waterman Distinguished Scholar *<br />* Department of Economics *<br />* Adjunct Professor of Finance *<br />* Adjunct Professor of Statistics<br />* Box 353330 email: <a href="mailto:ezivot@u.washington.edu" target="_blank" class="parsedEmail">ezivot@u.washington.edu</a> *<br />* University of Washington phone: 206-543-6715 *<br />* Seattle, WA 98195-3330 * *<br />* www: <a href="http://faculty.washington.edu/ezivot" target="_blank" class="parsedLink">http://faculty.washington.edu/ezivot</a> *<br />****************************************************************<br /><br />On Wed, 17 Jun 2009, Schaeffer, Derek BGI SF wrote:<br /><br />><br />> Hi Marco,<br />><br />> The discrete time Vasicek model at partition size dt (e.g. 1/252) is<br />> given by:<br />><br />> X(t) - X(t-dt) = [a*b] * dt - [b * X(t-dt)] * dt + s_e * e, e ~<br />> N(0,1)<br />><br />> Run the following regression:<br />><br />> X(t) = A + B X(t-1) + v, v ~ N(0,s_v^2)<br />><br />> The continuous-time parameters can be recovered from the regression<br />> estimates using:<br />><br />> b = -A/B<br />> a = -ln(1 + B)/dt<br />> s_e = s_v * sqrt(2 * ln(1 + B) / dt) / sqrt( (1+B)^2 - 1)<br />><br />> Best,<br />> Derek M. Schaeffer, Ph.D.<br />> Principal<br />> Senior Research Officer<br />><br />> TEL 415 894 6427<br />> CELL 415 516 9558<br />> FAX 415 618 1824<br />> <a href="mailto:derek.schaeffer@barclaysglobal.com" target="_blank" class="parsedEmail">derek.schaeffer@barclaysglobal.com</a><br />><br />> Barclays Global Investors 400 Howard Street San Francisco, CA 94105<br />><br />><br />><br />> -----Original Message-----<br />> From: <a href="mailto:r-sig-finance-bounces@stat.math.ethz.ch" target="_blank" class="parsedEmail">r-sig-finance-bounces@stat.math.ethz.ch</a><br />> [mailto:<a href="mailto:r-sig-finance-bounces@stat.math.ethz.ch" target="_blank" class="parsedEmail">r-sig-finance-bounces@stat.math.ethz.ch</a>] On Behalf Of Zanella<br />> Marco<br />> Sent: Wednesday, June 17, 2009 9:41 AM<br />> To: <a href="mailto:r-sig-finance@stat.math.ethz.ch" target="_blank" class="parsedEmail">r-sig-finance@stat.math.ethz.ch</a><br />> Subject: [R-SIG-Finance] Vasicek model estimation via linear regression<br />><br />> Hi,<br />> I have to check mean reversion with a Vasicek model for a time series.<br />> As you certainly know Vasicek process is discribed by following formula:<br />><br />> dXt = a(b-Xt)dt + sdWt (1)<br />><br />> <a href="http://en.wikipedia.org/wiki/Vasicek_model" target="_blank" class="parsedLink">http://en.wikipedia.org/wiki/Vasicek_model</a><br />><br />> To estimate the parameters in my data I can use this expression:<br />><br />> Xt - Xt-1 = a(b-Xt-1)Dt + se (2)<br />><br />> where:<br />> Xt: time series at time t<br />> Xt-1: time series at time t-1<br />> a: unknown parameter<br />> b: unknown parameter<br />> Dt: in my case I can assimilate it = 1<br />> S: standard deviation<br />> e: error ~N(0,1)<br />><br />> Basically, I want to estimate unknown parameters a and b using a linear<br />> regression. Usually I work on linear regression with lm() function but I<br />> don't undertand how formulate my (2) model in to lm command. Can anyone<br />> give me some suggestions?<br />><br />> Thanks in advance.<br />><br />> Regards,<br />><br />> Marco<br />><br />> ____________________________________________________________<br />><br />><br />> _______________________________________________<br />> <a href="mailto:R-SIG-Finance@stat.math.ethz.ch" target="_blank" class="parsedEmail">R-SIG-Finance@stat.math.ethz.ch</a> mailing list<br />> <a href="https://stat.ethz.ch/mailman/listinfo/r-sig-finance" target="_blank" class="parsedLink">https://stat.ethz.ch/mailman/listinfo/r-sig-finance</a><br />> -- Subscriber-posting only.<br />> -- If you want to post, subscribe first.<br />><br />><br />> --<br />><br />> This message and any attachments are confidential, pro...{{dropped:10}}<br /><br />_______________________________________________<br /><a href="mailto:R-SIG-Finance@stat.math.ethz.ch" target="_blank" class="parsedEmail">R-SIG-Finance@stat.math.ethz.ch</a> mailing list<br /><a href="https://stat.ethz.ch/mailman/listinfo/r-sig-finance" target="_blank" class="parsedLink">https://stat.ethz.ch/mailman/listinfo/r-sig-finance</a><br />-- Subscriber-posting only.<br />-- If you want to post, subscribe first.<br /></blockquote></div>