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rate by spline, local polynomial partition and wavelet series LS estimators for i.i.d. data. However, de Jong (2002)’s result (20) is proved under the strong bounded conditional fourth moment condition of sup$_{x}$ E [ | ϵ$_{i}$ | 4 | X$_{i}$ = x ] < ∞ and the side condition ζ 2 $_{K,n}$K/n = o (1). 

Recently, Cattaneo and Farrell (2013) proved that a local polynomial partitioning regression esti- mator can attain the optimal sup-norm rate under i.i.d. data and the conditional moment condition sup$_{x}$ E [ | ϵ$_{i}$ | 2+ δ | X$_{i}$ = x ] < ∞ for some δ ≥ max(1 , d/p ). Belloni et al. (2014) show that spline, local polynomial partition and wavelet LS estimators attain the optimal sup-norm rate under i.i.d. data and the conditional moment condition sup$_{x}$ E [ | ϵ$_{i}$ | 2+ δ | X$_{i}$ = x ] < ∞ for some δ > d/p . 7 By contrast, we require a weaker unconditional moment condition E [ | ϵ$_{i}$ | 2+( d/p $^{)}$] < ∞ for spline and wavelet LS estima- tors to attain the optimal uniform convergence rate, allowing for both i.i.d. data and weakly dependent regressors. 8 It remains an open question whether one could obtain the optimal sup-norm convergence rate without imposing a finite higher-than-second unconditional moment of the error term, however. 

## 2.4 A general sharp bound on L 2 convergence rates 

In this subsection, we present a simple but sharp upper bound on the L 2 (or root mean square) convergence rates of series LS estimators with an arbitrary basis and weakly dependent regressors. 

Recall that B$_{K,w}$ = clsp { b$_{K}$$_{1}$w$_{n}$, . . . , b$_{KK}$w$_{n}$ } is a general weighted linear sieve space and ˜ h = P$_{K,w,n}$h$_{0}$ is defined in (12). Let h$_{0}$$_{,K}$ be the L $^{2}$( X ) orthogonal projection of the conditional mean function h$_{0}$ onto B$_{K,w}$ . 

Lemma 2.5 Let Assumptions 1(i), 2(i)(ii), 4(iii) and 5 hold. Then: 

Lemmas 2.5, 2.1 and 2.2 immediately imply the following result. 

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