How do I obtain the standard errors for Ornstein Uhlenbeck parameter estimates?

Question

I have used least squares estimation to obtain estimates for parameters to be used in Ornstein Uhlenbeck process. Now, I would like to compute the standard errors of estimates.

$𝑑𝑆𝑡=𝜆(𝜇−𝑆𝑡)𝑑𝑡+𝜎𝑑𝑊𝑡$ (Ornstein Uhlenbeck process and parameters to be estimated)

The regression is as follows:

$S(t+1)=a*S(t)+b+𝜖$

The estimates for parameters are obtained from the regression coefficients as follows:

$𝜆=−(ln𝑎)/Δ𝑡, 𝜇=𝑎/(1−𝑏)$

and

$𝜎=sd(𝜖𝑡) * \sqrt{2𝜆/(1−𝑒^{−2𝜆Δ𝑡})}$

I can obtain the standard errors for a and b, but how would I be able to obtain standard errors for the parameters?

Any help is greatly appreciated, I apologize the messy layout of formulas.

Answer 1

If you are reporting to physicists or chemicists, I would recommend the Gaussian error propagation law, because that is what they usually apply in such cases: $$(\Delta f)^2 = \left(\frac{\partial f}{\partial a} \Delta a\right)^2 + \left(\frac{\partial f}{\partial b} \Delta b\right)^2 + \ldots$$ If you are reporting to statisticians, a resampling method like the bootstrap or the jacknife might raise fewer eyebrows ;-) The jackknife is easier and yields reproducibale results, whilst the bootstrap yields a random (albeit not much different) result.