We show that spline and wavelet series regression estimators for weakly dependent regressors attain the optimal uniform (i.e. sup-norm) convergence rate (n/ log n) −p/(2p+d) of Stone (1982), where d is the number of regressors and p is the smoothness of the regression function. The optimal rate is achieved even for heavy-tailed martingale difference errors with finite (2 + (d/p))th absolute moment for d/p < 2. We also establish the asymptotic normality of t statistics for possibly nonlinear, irregular functionals of the conditional mean function under weak conditions. The results are proved by deriving a new exponential inequality for sums of weakly dependent random matrices, which is of independent interest. JEL Classification: C12, C14, C32

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This paper makes several important contributions to the literature about nonparametric instrumental variables (NPIV) estimation and inference on a structural function h 0 and its functionals. First, we derive sup-norm convergence rates for computationally simple sieve NPIV (series 2SLS) estimators of h 0 and its derivatives. Second, we derive a lower bound that describes the best possible (minimax) sup-norm rates of estimating h 0 and its derivatives, and show that the sieve NPIV estimator can attain the minimax rates when h 0 is approximated via a spline or wavelet sieve. Our optimal sup-norm rates surprisingly coincide with the optimal root-mean-squared rates for severely ill-posed problems, and are only a logarithmic factor slower than the optimal root-mean-squared rates for mildly ill-posed problems. Third, we use our sup-norm rates to establish the uniform Gaussian process strong approximations and the score bootstrap uniform confidence bands (UCBs) for collections of nonlinear functionals of h 0 under primitive conditions, allowing for mildly and severely ill-posed problems. Fourth, as applications, we obtain the first asymptotic pointwise and uniform inference results for plug-in sieve t-statistics of exact consumer surplus (CS) and deadweight loss (DL) welfare functionals under low-level conditions when demand is estimated via sieve NPIV. Empiricists could read our real data application of UCBs for exact CS and DL functionals of gasoline demand that reveals interesting patterns and is applicable to other markets.

We assume that the state process X = {X t : t ∈ T } is either beta-mixing or rho-mixing. The beta-mixing coefficient between two σ-algebras A and B iswith the supremum taken over all A-measurable finite partitions {A i } i∈I and Bmeasurable finite partitions {B j } j∈J . The beta-mixing coefficients of X are defined asWe say that X is exponentially beta-mixing if β q ≤ Ce −cq for some C c > 0. The rho-mixing coefficients of X are defined asWe say that X is exponentially rho-mixing if ρ q ≤ e −cq for some c > 0. We use the sequence ξ k = sup x G −1/2 b k (x) to bound convergence rates. When X has bounded rectangular support and Q has a density that is bounded away from 0 and ∞, ξ k is known to be O( √ k) for (tensor-product) spline, cosine, and certain wavelet bases and O(k) for (tensor-product) polynomial series (see, e.g., Newey (1997), Chen andChristensen (2015)). It is also possible to derive alternative sufficient conditions in terms of higher moments of) by extending arguments in Hansen (2015) to accommodate weakly dependent data and asymmetric matrices. Timothy M. Christensen: