Second Order Inference for Nonstationary Time Series

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The proposed research is on second order inferences for causal nonstationary processes. While a causal stationary process can be viewed as generated by filtering a set of past innovations, one can allow the filter to be time-changing, and henceforth introduce nonstationarity. Two sets of problems are considered. First, for linear models with nonstationary errors, the investigator addresses the estimation of covariance matrices of the least square estimates, as well as general M-estimates. Second, the PI studies the estimation of time-varying covariance functions, time-varying spectrum and covariance matrices of the observed time series. Simultaneous inferences of autocovariance functions and spectra can be used to study their patterns and trends, and are also of interests. The study requires several tools to be developed for nonstationary processes, including empirical processes, Gaussian approximations, strong invariance principles and large deviations for quadratic forms.Stationarity has played an important role in classical time series analysis, which basically says that the overall structure does not change over time. However, in many scientific fields, including economics, engineering, environmental science, finance, and neuroscience etc, it is not realistic to believe the observed time series are stationary. Results from the proposed research will be useful in understanding the nature of the data from various disciplines, making forecasts and conclusions.Furthermore, the second order inferences in the proposal are general and fundamental, and will facilitate further statistical analysis of nonstationary time series.