Edgeworth Expansions in Curved Cross Section Autoregression

Edgeworth expansions are developed for the finite sample distribution of the least squares estimator in a time series parametric first order autoregression with Hilbert space curves of cross section data. The main result extends to this functional data environment the Edgeworth expansion in the corresponding scalar time series AR(1). In doing so, the results show how function-valued cross section data, and hence general forms of cross section dependence, affect the finite sample distribution of the serial correlation coefficient. Autoregressions with functional fixed effect intercepts are included and the results therefore relate to dynamic panel autoregression with individual effects. The primary impact of the use of high-dimensional curved cross section data is to reduce the variation in scalar regression estimation and provide some improvement in the accuracy of the usual asymptotic approximation to the finite sample distribution. Limit results for the expansions under full cross section dependence matching the scalar time series case and independence matching the dynamic panel case are given as special cases. The findings are supported by numerical computations of the exact distributions and the approximations.