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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 

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<img src='content_image/313733.jpg'>

<img src='content_image/313734.jpg'>

<img src='content_image/313735.jpg'>

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 

<img src='content_image/313740.jpg'>

<img src='content_image/313741.jpg'>

