  The class of locally stationary processes assumes that there is a
time-varying spectral representation, that is, the existence of finite second
moment. We propose the $\alpha$-stable locally stationary process by modifying
the innovations into stable distributions and the indirect inference to
estimate this type of model. Due to the infinite variance, some of interesting
properties such as time-varying auto-correlation cannot be defined. However,
since the $\alpha$-stable family of distributions is closed under linear
combination which includes the possibility of handling asymmetry and thicker
tails, the proposed model has the same tail behavior throughout the time. In
this paper, we propose this new model, present theoretical properties of the
process and carry out simulations related to the indirect inference in order to
estimate the parametric form of the model. Finally, an empirical application is
illustrated.
