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. 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. 2.4 A general sharp bound on L 2 convergence rates ‖ ̂ h − ˜ h ‖ L $^{2}$( X ) = O$_{p}$ (√ K/n ) and ‖ ˜ h − h$_{0}$ ‖ L $^{2}$( X ) = O$_{p}$ ( ‖ h$_{0}$ − h$_{0}$$_{,K}$ ‖ L $^{2}$( X ) ) . $^{7}$Our result on sup-norm stability of wavelet basis is used by Belloni et al. (2014) for the optimal sup-norm rate of the wavelet LS estimator. $^{8}$Our proof strategy for the i.i.d. data case is similar to that in a draft note by Chen and Huang (2003), who applied Huang (2003b)’s sup-norm stability of spline basis to obtain the optimal sup-norm rate for spline series LS estimator under the condition E [ | ϵ$_{i}$ | 2+ $^{δ}$] < ∞ for some δ > d/p . The authors carelessly set δ = 2, but did not like the strong condition E [ | ϵ$_{i}$ | $^{4}$] < ∞ and hence did not finish the paper. The note was circulated among some colleagues and former students of the authors, and is available upon request. 16