This article studies estimation and inference in the autoregressive (AR) models with unspecified and heavy-tailed heteroskedastic noises. A piece-wise locally stationary structure of the noise is constructed to capture various forms of heterogeneity, without imposing any restrictions on the tail index. The new nonstationary AR model allows for not only time-varying conditional features but also unconditional variance and tail index. This makes it appealing in practice, with wide applications in economics and finance. To obtain a feasible inference, we investigate the self-weighted least absolute deviation estimator and derive its asymptotic normality. Since the asymptotic variance relies on an unobserved density, a bootstrap method is proposed to approximate the limiting distribution. Based on the conditional moment condition, a portmanteau test from residuals is further proposed to detect misspecifications in the proposed model. A simulation study and two applications to time series illustrate our inference procedures.