This paper develops and applies new asymptotic theory for estimation and inference in parametric autoregression with function valued cross section curve time series. The study provides a new approach to dynamic panel regression with high dimensional dependent cross section data. Here we deal with the stationary case and provide a full set of results extending those of standard Euclidean space autoregression, showing how function space curve cross section data raises efficiency and reduces bias in estimation and shortens confidence intervals in inference. Methods are developed for high-dimensional covariance kernel estimation that are useful for inference. The findings reveal that function space models with wide-domain and narrow-domain cross section dependence provide insights on the effects of various forms of cross section dependence in discrete dynamic panel models with fixed and interactive fixed effects. The methodology is applicable to panels of high dimensional wide datasets that are now available in many longitudinal studies. An empirical illustration is provided that sheds light on household Engel curves among ageing seniors in Singapore using the Singapore life panel longitudinal dataset.
