In this work, we deal with the problem of estimation of the time-varying coefficients of a linear system, where the input and output are locally stationary processes. In our approach, we propose two types of estimators, kernel and wavelet estimators. They are time-domain estimators in the sense that they involve Yule–Walker type equations and ordinary least squares method. We provide some simulation results and briefly discuss the (asymptotic) statistical properties of the estimators.

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 α-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 autocorrelation cannot be defined. However, since the α-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 behaviour 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.