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I have a data set of daily EUR/USD rate for time period 2000-2018.

EUR/USD daily rate 2000-2018

My goal is to model future behaviour of this financial time series using Ornstein-Uhlenbeck model: $$d X_t = \alpha (\theta - X_t) d t + \sigma d B_t$$ In order to calibrate the model I used MLE (Maximum Likelihood estimation) to obtain estimators for model parameters $\alpha, \theta$ and $\sigma$. Bellow are resulting estimators:

$$\hat{\alpha} = 0.00103 \\ \hat{\theta} = 1.27334 \\ \hat{\sigma} = 0.00647 $$

Whereas estimated value for $\theta$ seems to be pretty reasonable (long-term mean to which the process tends to revert), value for $\alpha$ being close to $0$ does not make much sense to me. As a result the first mean-reverting part of the equation basically drops out and what is left is "weak" (due to small value of $\sigma$) Brownian motion $d X_t = \hat{\sigma} d B_t$. Consequently my simulated process is not modelling the real process at all as can be seen in the picture below:Simulated O_U process vs. real EUR/USD daily rate process

I am aware of the fact that this problem might be related to lack of stationarity in my data (p-value for ADF test was around 0.6, meaning we did not reject null hypothesis of non-stationarity).

Is this the only reason for poor modelling power of my calibrated model? If so, what are the best ways to tackle the problem of non-stationarity? Are there any other possible reasons for $\alpha$ being close to 0? Also, is O-U model even suitable for modelling exchange rate processes? If not, what models would be more approriate?

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regarding lack of stationarity: yes, OU process is only viable for stationary processes, and it makes sense that EURUSD data are non-stationary, few rare events can lead to sharp moves: Greece crises, Trump election, etc..the distribution of returns has fat tails and the volatility is heteroskedastic

Most simple financial model is the random walk , AR(1) process, you can also try ARIMA ,GARCH models,Merton's jump diffusion model, the last two take care of the heteroskedasticity of the data

FYI instead of using MLE to estimate the OU parameters you can also use linear regression, the results should be the same:

How can I estimate the Ornstein-Uhlenbeck paramters of some mean reverting data that I have on R?

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