  In this article, we introduce and study a one sided tempered stable first
order autoregressive model called TAR(1). Under the assumption of stationarity
of the model, the marginal probability density function of the error term is
found. It is shown that the distribution of the error term is infinitely
divisible. Parameter estimation of the introduced TAR(1) process is done by
adopting the conditional least square and method of moments based approach and
the performance of the proposed methods are evaluated on simulated data. Also
we study an autoregressive model of order one with tempered stable innovations.
Using appropriate test statistic it is shown that the model fit very well on
real and simulated data. Our models generalize the inverse Gaussian and
one-sided stable autoregressive models existing in the literature.
