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Suppose we estimate the regression model $$\triangle y_{t}=\alpha + \beta y_{t-1}+\varepsilon_{t}$$ This is actually quite similar to the Dickey-Fuller test. If $\beta=0$, then the process has a unit root. Let's proceed assuming that $\beta<0$, i.e. that the process is stationary. The first equation is also similar to the continuous time ...


What you are saying might be correct for discrete time processes. In continuous time the process $$ dX_t = X_t^2 dW_t,\quad X_0 > 0 $$ is stationary but not mean reverting.


Similar to Juan Gil's answer but a bit differently I would say the following based on this: The OU process $$dX_t = \kappa(\theta-X_t)dt + \sigma dW_t$$ can be (Euler-Maryuama discretization) discretized at times $n \Delta t,n=1,\ldots,\infty $ which gives with $t = k \Delta t$ $$ X_{k+1} - X_k = \kappa \theta \Delta t -\kappa X_k \Delta t + \sigma (W_{k+1} ...


For a Ornstein-Uhlenbeck process, the maximum likelihood parameters are the ones from least squares regression. If your process is: $$ dX=\kappa (\theta-X)dt+\sigma dW $$ you can do a linear regression in the form $$ \frac{dX}{dt}=a+bX+\epsilon $$ So your parameters will be: $$ \kappa=-b $$ $$ \theta=-\frac{a}{b} $$ $$ \sigma=std(\epsilon dt) $$

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