$AR(1)$ TIME SERIES WITH APPROXIMATED BETA MARGINAL

Bozidar V. Popovic

Abstract: We consider the $AR(1)$ time series model $X_t-\beta X_{t-1}=\xi_t$, $\beta^{-p}\in\mathbb{N}\smallsetminus\{1\}$, 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.

Keywords: Beta distribution, Kumaraswamy distribution, approximated Beta distribution, Kummer function of the first kind, first order autoregressive model

Classification (MSC2000): 62M10; 33C15, 66F10, 60E10

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