Why Refitting Simulated Ornstein-Uhlenbeck Paths Does Not Prove Model Fit

Summary

The document examines a proposed test of whether empirical price data follow an Ornstein-Uhlenbeck (OU) process. The described procedure estimates OU parameters by maximum likelihood, simulates a new path using those estimates, then fits the OU model again to the simulated path. Similar estimates from the empirical and simulated samples are presented in the cited paper as evidence of a good fit, and the question asks what mathematical basis supports that conclusion.

The procedure is a simulation-based comparison, but the document does not provide an answer or establish that closeness of the refitted parameters is a valid goodness-of-fit criterion. A simulated sample generated under the fitted model is expected to reflect that model’s structure; this comparison alone does not show how likely the observed data would be under the model or distinguish it from alternatives. The note raises a methodological question rather than supplying a test, calibration method, or empirical results, so further analysis would need to specify a suitable statistic and reference distribution.

Key ideas

• The proposed procedure estimates OU parameters from empirical data and simulates a path using those estimates.
• It refits the model to the simulated path and compares the resulting parameter estimates with the empirical estimates.
• Similarity between these estimates is presented as evidence of fit, but the note questions the justification.
• A model-generated simulation comparison alone does not establish that the empirical process follows OU dynamics.

Tags

# Testing the fit of an Ornstein-Uhlenbeck process


# Testing the fit of an Ornstein-Uhlenbeck process












I would like to check if a time-series follows an Ornstein-Uhlenbeck process defined by an SDE:

$$dX_t - \lambda (\mu - X_t) dt = \sigma dW_t$$

where

- $\lambda > 0$ is the mean-reversion coefficient

- $\mu$ is the long-term mean

- $\sigma>0$ is the variance

- $W_t$ is the Wiener process

This paper does the following:

> For each pair, we first estimate the parameters for the OU model from empirical price data. Then, we use the estimated parameters to simulate price paths according to the corresponding OU process. Based on these simulated OU paths, we perform another MLE and obtain another set of OU parameters as well as the maximum average log-likelihood $l$. As we can see, the two sets of estimation outputs (the rows names “empirical” and “simulated”) are very close, suggesting the empirical price process fits well to the OU model.

They find the parameters via Maximum Likelihood Estimate (MLE), then simulate a fresh OU process with those parameters and find again the parameters of the simulated process. Then they compare the two sets of parameters, and if they are close, they claim the original process fits well to the OU model.

What is the mathematical justification for this?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.