  We consider a class of nonparametric time series regression models in which
the regressor takes values in a sequence space. Technical challenges that
hampered theoretical advances in these models include the lack of associated
Lebesgue density and difficulties with regard to the choice of dependence
structure in the autoregressive framework. We propose an infinite-dimensional
Nadaraya-Watson type estimator, and investigate its asymptotic properties in
detail under both static regressive and autoregressive contexts, aiming to
answer the open questions left by Linton and Sancetta (2009). First we show
pointwise consistency of the estimator under a set of mild regularity
conditions. Furthermore, the asymptotic normality of the estimator is
established, and then its uniform strong consistency is shown over a compact
set of logarithmically increasing dimension with respect to $\alpha$-mixing and
near epoch dependent (NED) samples. We specify the explicit rates of
convergence in terms of the Lambert W function, and show that the optimal rate
is of logarithmic order, confirming the existence of the curse of infinite
dimensionality.
