Cross Section Curve Autoregression: The Unit Root Case

This paper is part of a joint study of parametric autoregression with cross section curve time series, focussing on unit root (UR) nonstationary curve data autoregression. The Hilbert space setting extends scalar UR and local UR models to accommodate high dimensional cross section dependent data under very general conditions. New limit theory is introduced that involves two parameter Gaussian processes that generalize the standard UR and local UR asymptotics. Bias expansions provide extensions of the well-known results in scalar autoregression and fixed effect dynamic panels to functional dynamic regressions. Semiparametric and ADF-type UR tests are developed with corresponding limit theory that enables time series inference with high dimensional curve cross section data, allowing also for functional fixed effects and deterministic trends. The asymptotics reveal the effects of general forms of cross section dependence in wide nonstationary panel data modeling and show dynamic panel regression limit theory as a special limiting case of curve time series asymptotics. Simulations provide evidence of the impact of curve cross section data on estimation and test performance and the adequacy of the asymptotics. An empirical illustration of the methodology is provided to assess the presence of time series nonstationarity in household Engel curves among ageing seniors in Singapore using the Singapore life panel dataset.