Locally stationary processes with stable and tempered stable innovations - Chen, Shu Wei Chou (2020)
In the literature, the class of locally stationary processes assumes that there is a time-varying spectral representation, i.e. the existence of finite second moment. In this work, we first propose the stable locally stationary process by modifying the innovations into stable distributions, which has heavy tail, 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 stable family of distributions, as a generalization of the Gaussian distribution, is closed under linear combination, which includes the possibility of handling asymmetry and thicker tails, the proposed model presents the same tail behavior throughout the time. We carry out simulations to study the performance of the indirect inference and compare it to the existing methodology, blocked Whittle estimation. When the process has stable innovations, the indirect inference presents more promising results than the existing methodology because of infinite variance. Next, we consider the locally stationary process with tempered stable innovations, whose center is similar to that of a stable distribution, but its tails are lighter (semi-heavy tail) and all moments are finite. We present some theoretical results of this model and propose a two-step estimation to estimate the parametric form of the model. Simulations suggest that the time-varying structure can be estimated well, but the parameters related to the innovation are biased for small time series length. However, the bias disappears when time series length increases. Finally, an empirical application is illustrated.