We consider the $AR(1)$ time series model $X_t-\beta X_{t-1}=\xi_t$, $\beta^{-p}n\mathbb{N}mallsetminus\{1\}$, when $X_t$ has Beta distribution $\mathrm{B}(p,q)$, $p\in(0,1]$, $q>1$. Special attention is given to the case $p=1$ when the marginal distribution is approximated by the power law distribution closely connected with the Kumaraswamy distribution $\operatorname{Kum}(p,q)$, $p\in(0,1]$, $q>1$. Using the Laplace transform technique, we prove that for $p=1$ the distribution of the innovation process is uniform discrete. For $p\in(0,1)$, the innovation process has a continuous distribution. We also consider estimation issues of the model.