# How to estimate parameters for 2 correlated Ornstein-Uhlenbeck processes with maximum likelihood?

I would like to use maximum likelihood to estimate the parameters of two correlated Ornstein-Uhlenbeck processes from empirical data.

Do you have any good references for this? If you have any hints as to how to code it in Matlab, that would also be great.

I suppose I can just use the log-likelihood function for multivariate processes and then I need mean and variance of an Ornstein-Uhlenbeck process, e.g. as described in this answer. Right?

However, even once I have coded it in Matlab, I would still have to reference where I got the formulas from, and there it would be helpful to have some (academic) papers.

I am familiar with the paper by Schwartz and Smith 2000 (Short-term variations and long-term dynamics in commodity prices, Management Science), but this one is on 1 O-U and 1 Brownian Motion.

• Could you please give us the multivariate SDE that you want to estimate. Otherwise it is not clear if you want two 1D O-U with correlated Brownians or a proper 2D O-U process. – Kiwiakos May 7 '16 at 17:51
• The Brownians are correlated with $dW_1 dW_2 = \rho dt$. The two O-U processes are $dX_1 = k_1 (\mu_1-X_1)dt + \sigma_1 dW_1$ and $dX_2 = k_2 (\mu_2-X_2)dt + \sigma_2 dW_2$ . I managed to solve it with a multivariate log-likelihood and the help of getting the covariance of the two processes (from here). – LenaH May 13 '16 at 11:12
• The only problem that remains is that $k_1$ and $k_2$ are consistently over-estimated, which is a common problem to which I didn't find any solution in the literature. I am wondering whether I should use least squares instead of likelihood... – LenaH May 13 '16 at 11:12
• Since the model is Gaussian MLE and OLS are equivalent. If the true $k$ is close to zero then you have a have a 'unit root' problem and the sampling distribution of the parameter is Dickey-Fuller rather than Student-t. This is skewed, therefore it is known and expected that you are more likely to overestimate. – Kiwiakos May 13 '16 at 11:45
• What about estimating the parameters of a VAR(1) model for a bivariate time series? Since Vasicek model is equivalent to AR(1) in discrete time, I think this should work. Please, correct if I'm wrong. – Egodym Jun 6 '16 at 21:15

## 1 Answer

I don't know what exactly you want but have a look at the article Calibrating the Ornstein-Uhlenbeck (Vasicek) model.

You calibrate the first one in stand alone, then the second one in stand alone, finally you can compute correlation on the residuals of the increments knowing your parameters.

• This would assume that the proccesses are independent, though. Yet they are correlated so I for sure have to use the multivariate log-likelihood approach. I already found the expected value and variance of the Ornstein-Uhlenbeck process. Now I am still trying to figure out how to get the covariance of two O-U processes. – LenaH Apr 8 '16 at 13:30
• I think you do not understand my answer. Do the mean reversion and volatility parameter of one process depends on the other ? – MJ73550 Dec 18 '16 at 7:02