Estimation and Inference in Functional Time Series Analysis

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This proposal aims to extend the theory and applications of functional time series analysis (FTSA). Functional data analysis (FDA) came into prominence in the 1990's, and the early developments focused on simple random samples of observations that can be framed as curves or functions. Functional data are, however, often obtained sequentially by breaking nearly continuous time records into smaller segments. For example, high frequency records of pollution levels may be segmented to form a series of daily pollution curves. Other examples include sequentially observed functions that describe physical phenomena, as in functional magnetic resonance imaging, where functions describing blood flow in the brain are computed over time. The assumption of a simple random sample is often too strong in these cases, and a central issue then becomes how to account for and utilize temporal dependence in such complex data. FTSA provides theory and methodology for addressing this issue. The research outlined in this proposal expands the knowledge of FTSA in two primary directions:******(1) Estimation of the long run covariance operator:******A covariance object used in the study of functional time series is the long run covariance operator, which describes the second order behavior of the sample mean function and incorporates information about the dependence within the series. To date, the theoretical and empirical properties of estimators of the long run covariance operator have been only lightly investigated. A data driven bandwidth selection procedure for nonparametric estimators of the long run covariance is proposed below that bridges a significant methodological gap in FTSA. This addresses a difficulty in the analysis of sequentially observed summary functions that describe how biological agents interact with each other, as frequently arise in the study of agent based models, and applications along these lines are proposed.******(2) Differentiating between structural breaks and integration with functional time series:******Many methods used to forecast time series data rely on the assumption of stationarity. In case of traditional time series, testing this assumption has been thoroughly studied in the statistics and econometrics literature, with the most widely used tests belonging to the Dickey-Fuller and KPSS families. When trend stationarity is rejected, it is often because the trend changes within the sample (structural break), or the error process is itself non-stationary (integration), and a wealth of literature exists on differentiating between the two possible sources of non-stationarity. Recently, tests for stationarity with functional time series have been developed, however methods for identifying specific sources of non-stationarity remain unstudied. The proposed research culminates in methodology for differentiating between structural breaks and "unit roots" with functional time series data.**