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as n,K → ∞ provided that either (a), (b) or (c) is satisfied: 

(a) { ( X$_{i}$,Y$_{i}$ ) } n i =1 is i.i.d.: √$_{nV}$ − 1 / 2 K K − p/d = o (1) and K (2+ δ ) $^{/δ}$(log n ) /n ≲ 1 with δ < 2 ; 

(b) { ( X$_{i}$,Y$_{i}$ ) } n i =1 is exponentially β -mixing: √$_{nV}$ − 1 / 2 K K − p/d = o (1) and K (2+ δ ) $^{/δ}$(log n ) $^{2}$/n = o (1) with δ ≤ 2 ; 

(c) { ( X$_{i}$,Y$_{i}$ ) } n i =1 is algebraically β -mixing at rate γ : √$_{nV}$ − 1 / 2 K K − p/d = o (1) and K (2+ δ ) $^{/δ}$(log K ) /n γ/ (1+ γ ) = o (1) with δ ≤ 2 . 

Conditions for weakly dependent data in Corollary 3.2 parts (b) and (c) are natural extensions of those in part (a) for i.i.d. data, which in turn are much weaker than the well-known conditions in Newey (1997) for the asymptotic normality of nonlinear functionals of spline LS estimators, namely √$_{nK}$ − p/d = o (1), K $^{4}$/n = o (1) and sup$_{x}$ E [ | ϵ$_{i}$ | 4 | X$_{i}$ = x ] < ∞ . 

## 3.2 Asymptotic normality of sieve t statistics for general functionals 

We now turn to the consistent estimation of V$_{K}$ = ‖ v ∗ K ‖ 2 sd and feasible asymptotic inference for f ( h$_{0}$ ). Assumption 10 ‖ v ∗ K ‖ − 1 L $^{2}$( X ) ∥ ∂f ( h ) ∂h [ ˜ b K $_{w}$] − ∂f ( h$_{0}$ ) ∂h [ ˜ b K $_{w}$] ∥ = o (1) uniformly over h ∈ N$_{K,n}$ or B$_{ϵ,}$$_{∞}$ ( h$_{0}$ ) . Note that Assumption 10 is automatically satisfied when f is a linear functional. It is only required 

to establish consistency of ̂ ‖ v ∗ K ‖ sd for a nonlinear functional, and corresponds to Assumption 3.1(iii) of Chen and Liao (2014). 

The first part of the following Lemma establishes the consistency of the sieve variance estimator under both the i.i.d. and general weakly dependent data. 

Lemma 3.2 Let Assumptions 1(i)(ii), 2(i)(ii)(iv), 4(iii), 5, 8 and 10 hold and ‖ ̂ h − h$_{0}$ ‖ ∞ ,w = o$_{p}$ (1) . Then: ∗ 

(1) ∣ ∣$^{̂}$ ‖ v $_{K}$‖$_{sd}$ ‖ v ∗ $_{K}$‖$_{sd}$ − 1 ∣ = o$_{p}$ (1) as n,K → ∞ . (2) 

Further, if Assumptions 2(v) and 9 hold, then: 

̂ V 1 / 2 K d as n,K → ∞ provided that either (a) { ( X$_{i}$,Y$_{i}$ ) } n i =1 is i.i.d., or (b) { X$_{i}$ } n i =1 is weakly dependent with ‖ ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ‖ = o$_{p}$ ( K − 1 / $^{2}$) is satisfied. 23 

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