On a Generalized Mixed AR(1) Time Series Model

S. Nadarajah, B.V. Popovic, M.M. Ristic

2011, v.17, №4, 637-650

ABSTRACT

We consider the mixed $AR(1)$ time series model \begin{align*} X_t = \begin{cases} a X_{t-1} +\xi_t, & \mbox{w.p.} \quad p_0, b X_{t-1} + \xi_{t}, & \mbox{w.p.} \quad p_1, \xi_t, & \mbox{w.p.} \quad p_2, c X_{t-1}, & \mbox{w.p.} \quad p_3 , \end{cases} \end{align*} for $a, b, c \in (0,1)$, where $X_t$ has the beta distribution ${\rm B}(p,q)$, $p, q>0$. Using the Laplace transform technique, we prove that the distribution of the innovation is a continuous one. We also consider estimation issues of the model.

Keywords: beta distribution,first order autoregressive model,Kummer function of the first kind