Bounded AR(1) Process

Question

I am trying to estimate an AR(1) model for a stochastic process Xt that is bounded in some interval (a,b]. Is there a simple transformation I can do on Xt that ensures my AR(1) model does not violate those bounds when I simulate it? Maybe something like a sigmoid function?

Answer 1

Yes, you can use a sigmoid function $\sigma:\mathbb{R} \to [0,1]$ (and rescale it) so that you can use errors $(e_n)_{n \in \mathbb{N}}$ which are $e_n:\Omega \to \mathbb{R}$ without restricting their distribution, but obtaining a bounded AR(p) 'type' process (in the sense that it depends on the previous $p$ terms). For example the one lag version could be $$X_n=\sigma(\phi_0+\phi_1X_{n-1}+e_n)$$ However, mind that the distribution of $(X_n)_{n \in \mathbb{N}}$ might become quite atypical, which is not always desirable, and its autocorrelation function esoteric. This is the empirical density by using $\sigma(x)=(1+e^{-x})^{-1}$ (which bounds the process in $(0,1)$) and choices $\phi_0=-0.2,\phi_1=-0.9,\sigma_e=0.8$ with Gaussian errors: