On mixed AR(1)AR(1) time series model with approximated beta marginal

We consider the mixed AR(1)AR(1) time series model Xt={αXt−1w.p. αpβXt−1+ξtw.p. 1−αp,α,β∈(0,1), when XtXt has the two parameter beta distribution B2(p,q), p∈(0,1],q>1p∈(0,1],q>1. Special attention is given to the case p=1p=1 when the marginal distribution is approximated by the power law distribution closely connected with the two parameter Kumaraswamy distribution Kum2(p,q),p∈(0,1],q>1. Using the Laplace transform technique, we prove that for p=1p=1 the distribution of the innovation process is uniform discrete. For p∈(0,1)p∈(0,1), the innovation process has a continuous distribution. We also consider estimation issues of the model.