Statistical inference on $AR(p)$ models with non-i.i.d. innovations

The autoregressive process is one of the fundamental and most important models that analyze a time series. Theoretical results and practical tools for fitting an autoregressive process with i.i.d. innovations are well-established. However, when the innovations are white noise but not i.i.d., those tools fail to generate a consistent confidence interval for the autoregressive coefficients. Focus on an autoregressive process with \textit{dependent} and \textit{non-stationary} innovations, this paper provides a consistent result and a Gaussian approximation theorem for the Yule-Walker estimator. Moreover, it introduces the second order wild bootstrap that constructs a consistent confidence interval for the estimator. Numerical experiments confirm the validity of the proposed algorithm with different kinds of white noise innovations. Meanwhile, the classical method(e.g., AR(Sieve) bootstrap) fails to generate a correct confidence interval when the innovations are dependent. According to Kreiss et al. \cite{10.1214/11-AOS900} and the Wold decomposition, assuming a real-life time series satisfies an autoregressive process is reasonable. However, innovations in that process are more likely to be white noises instead of i.i.d.. Therefore, our method should provide a practical tool that handles real-life problems.