[R-SIG-Finance] Vasicek model estimation via linear regression

[Schaeffer, Derek BGI SF]

Hi Marco,

The discrete time Vasicek model at partition size dt (e.g. 1/252) is
given by:

X(t) - X(t-dt) = [a*b] *  dt - [b * X(t-dt)] * dt + s_e * e,   e ~
N(0,1)

Run the following regression:

X(t) = A + B X(t-1) + v,  v ~ N(0,s_v^2)

The continuous-time parameters can be recovered from the regression
estimates using:

b   = -A/B
a   = -ln(1 + B)/dt
s_e = s_v * sqrt(2 * ln(1 + B) / dt) / sqrt( (1+B)^2 - 1)
 
Best,
Derek M. Schaeffer, Ph.D. 
Principal 
Senior Research Officer

TEL   415 894 6427
CELL  415 516 9558
FAX   415 618 1824
derek.schaeffer at barclaysglobal.com

Barclays Global Investors 400 Howard Street San Francisco, CA 94105



-----Original Message-----
From: r-sig-finance-bounces at stat.math.ethz.ch
[mailto:r-sig-finance-bounces at stat.math.ethz.ch] On Behalf Of Zanella
Marco
Sent: Wednesday, June 17, 2009 9:41 AM
To: r-sig-finance at stat.math.ethz.ch
Subject: [R-SIG-Finance] Vasicek model estimation via linear regression

Hi,
I have to check mean reversion with a Vasicek model for a time series.
As you certainly know Vasicek process is discribed by following formula:

dXt = a(b-Xt)dt + sdWt   (1) 

http://en.wikipedia.org/wiki/Vasicek_model

To estimate the parameters in my data I can use this expression:

Xt - Xt-1 = a(b-Xt-1)Dt + se   (2) 

where:
Xt: time series at time t
Xt-1: time series at time t-1
a: unknown parameter
b: unknown parameter
Dt: in my case I can assimilate it = 1
S: standard deviation
e: error ~N(0,1)

Basically, I want to estimate unknown parameters a and b using a linear
regression. Usually I work on linear regression with lm() function but I
don't undertand how formulate my (2) model in to lm command. Can anyone
give me some suggestions?

Thanks in advance.

Regards,

Marco

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