series LS estimator of the conditional mean function h$_{0}$ is where b$_{K}$$_{1}$, . . . , b$_{KK}$ are a collection of K sieve basis functions and ̂ h ( x ) = b $^{K}$( x ) $^{′}$( B $^{′}$B ) $^{−}$B $^{′}$Y (2) b $^{K}$( x ) = ( b$_{K}$$_{1}$ ( x ) , . . . , b$_{KK}$ ( x )) ′ (3) B = ( b $^{K}$( X$_{1}$ ) , . . . , b $^{K}$( X$_{n}$ )) ′ (4) Y = ( Y$_{1}$, . . . , Y$_{n}$ ) $^{′}$. (5) Choosing a particular class of sieve basis function and the dimension K are analogous to choosing the type of kernel and bandwidth, respectively, in kernel regression techniques. The basis functions are chosen such that their closed linear span B$_{K}$ = clsp { b$_{K}$$_{1}$, . . . , b$_{KK}$ } can well approximate the space of functions in which h$_{0}$ is assumed to belong. When the data { ( X$_{i}$, Y$_{i}$ ) } n i =1 are a random sample it is often reasonable to assume that X is supported on a compact set X ⊂ R $^{d}$. However, in a time-series setting it may be necessary to allow the support X of X to be infinite, as in the example of nonparametric autoregressive regression with a student t distributed error term. See, e.g., Fan and Yao (2003) and Chen (2013) for additional examples and references. To allow for possibly unbounded support X of X$_{i}$ we modify the usual series LS estimator and notion of convergence. First, we weight the basis functions by a sequence of non-negative weighting functions w$_{n}$ : X → { 0 , 1 } given by where D n ⊆ X is compact, convex, and has nonempty interior, and D n ⊆ D n +1 for all n . The resulting series LS estimator is then w$_{n}$ ( x ) =     1 if x ∈ D n 0 otherwise (6) ̂ h ( x ) = b K $_{w}$( x ) $^{′}$( B $^{′}$$_{w}$B$_{w}$ ) $^{−}$B $^{′}$$_{w}$Y (7) 7