Remark 2.6 Let Assumptions 1(i), 2(i)(ii) and λ$_{K,n}$ ≲ 1 hold. Then: (1) ‖ ̂ h − h$_{0}$ ‖ L $^{2}$( X ) = O$_{p}$ ( √ K/n + ‖ h$_{0}$ − h$_{0}$$_{,K}$ ‖ L $^{2}$( X ) ) , (22) provided that either (1.a), (1.b) or (1.c) is satisfied: provided that either (2.a) or (2.b) is satisfied: With i.i.d. data under the condition λ$_{K,n}$ ≲ 1, Newey (1997) derived the same sharp L 2 rate in (22) for series LS estimators under the restriction ζ 2 $_{K,n}$K/n = o (1). Huang (2003a) showed that spline LS es- timator has the same L 2 rate under the much weaker condition K (log K ) /n = o (1). Both Belloni et al. (2014) and our Remark 2.6 part (1.a) extend Huang (2003a)’s weakened condition to other bases sat- isfying ζ$_{K,n}$ ≲ √$_{K}$ (such as trigonometric polynomial and wavelet) for series LS regression with i.i.d. data. In addition, Remark 2.6 part (1.b) shows that the mild condition K (log K ) $^{2}$/n = o (1) suffices for trigonometric polynomial, wavelet, spline and other bases satisfying ζ$_{K,n}$ ≲ √$_{K}$ for exponentially β -mixing regressors. With weakly-dependent data, Chen and Shen (1998) derived L 2 rates for LS regression using vari- ous linear or nonlinear sieves with beta-mixing sequence under higher-than-second moment restriction (see Proposition 5.1 in Chen and Shen (1998)). Huang and Yang (2004) and others derived the op- timal L 2 rate for spline LS regression with strongly mixing sequence assuming a uniformly bounded 17 (2.a) { X$_{i}$ } n i =1 is i.i.d. or exponentially β -mixing: with p > 0 for trigonometric polynomial, spline or wavelet series, and p > d/ 2 for power series; (2.b) { X$_{i}$ } n i =1 is algebraically β -mixing at rate γ : with p > d/ (2 γ ) for trigonometric polynomial, spline or wavelet series; and p > d (2 + γ ) / (2 γ ) for power series. (1.a) { X$_{i}$ } n i =1 is i.i.d., and ζ$_{K,n}$ √ (log K ) /n = o (1) ; (1.b) { X$_{i}$ } n i =1 is exponentially β -mixing, and ζ$_{K,n}$ √ (log n ) $^{2}$/n = o (1) ; (1.c) { X$_{i}$ } n i =1 is algebraically β -mixing at rate γ , and ζ$_{K,n}$ √ (log K ) /n γ/ (1+ γ ) = o (1) . (2) Further, if Assumptions 1 and 6 hold and K ≍ ( n/ log n ) d/ (2 p + d $^{)}$, then ‖ ̂ h − h$_{0}$ ‖ L $^{2}$( X ) = O$_{p}$ ( n − p/ (2 p + d $^{)}$)