Some new asymptotic theory for least squares series: Pointwise and uniform results

By Alexandre Belloni, Victor Chernozhukov, D. Chetverikov, Kengo Kato. 2012

255 citations recorded by the source corpus.

In this work we consider series estimators for the conditional mean in light of three new ingredients: (i) sharp LLNs for matrices derived from the non-commutative Khinchin inequalities, (ii) bounds on the Lebesgue factor that controls the ratio between the L8 and L2-norms, and (iii) maximal inequalities for processes whose entropy integrals diverge at some rate. These technical tools allow us to contribute to the series literature, specifically the seminal work of Newey (1997), as follows. First, we weaken considerably the condition on the number k of approximating functions used in series estimation from the typical k2/n ? 0 to k/n ? 0, up to log factors, which was available only for spline and local polynomial partition series before. Second, under the same weak conditions we derive L2 rates and pointwise central limit theorems results when the approximation error vanishes. Under an incorrectly specified model, i.e. when the approximation error does not vanish, analogous results are also shown. Third, under stronger conditions we derive uniform rates and functional central limit theorems that hold if the approximation error vanishes or not. That is, we derive the strong approximation for the entire estimate of the nonparametric function. Finally, we derive uniform rates and inference results for linear functionals of interest of the conditional expectation function such as its partial derivative or conditional average partial derivative.

Subfields: Applied Mathematics, Control and Systems Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/4629c0b55bb92313d01c1be4541d37e6fee309cb

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Citations in the corpus

Minimal dispersion approximately balancing weights: asymptotic properties and practical considerations (2017, methodology): “This condition is satisfied by many classes of basis functions including the regression spline, trigonometric polynomial, and wavelet bases (Newey, 1997; Horowitz et al., 2004; Chen, 2007; Belloni et al., 2015; Fan et al., 2016).”
Two-Step Estimation and Inference with Possibly Many Included Covariates (2018, background): “See Belloni et al. (2015) and Cattaneo, Farrell and Feng (2018) for more discussion and examples.”
On cross-validated Lasso in high dimensions (2016, background): “5) [0, 5] [6, 10] [11, 15] [16, 20] [21, 25] [26, p] [31, p] [36, p] (n, p)=(100, 40) 0.”
Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions (2014, methodology): “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 supxE[|ǫi|2+δ |Xi = x] <∞ for some δ > d/p.7 By contrast, we require a weaker unconditional moment condition…”
Estimation and Inference about Heterogeneous Treatment Effects in High-Dimensional Dynamic Panels. (2017, background): “…2‖ )1/2 .iv √ logNm Nm ( ETr( 1 Nm N∑ i=1 m∑ p=1 1 q2 (B∗i2p) 2) )1/2 .v √ logNm Nm 1 q sup ip ( TrE(B∗i2p)2 )1/2 . where i holds by Symmetrization Lemma (Lemma 2.3.6 in Van der Vaart and Wellner (1996)), ii by Lemma 6.1 from Belloni et al. (2015), iii by Jensen inequality for the function x→ √ x.”
A unifying approach for doubly-robust $\ell_1$ regularized estimation of causal contrasts (2019, background): “Interestingly, as pointed out by Belloni et al. (2014), the class Sn (φ, t, α, l) , l = 1 or 2, is so large that one can find for each n a function cn (Z) ∈ Sn (φ, α, k) such that E [ {cn (Z)− ĉn (Z)}(2) ] converges to non-zero constant. Specifically, let p (n) = 2n and consider the function cn (Z) ”
Inference in Linear Regression Models with Many Covariates and Heteroscedasticity (2015, methodology): “For further technical details on series-based methods see, for example, Newey (1997), Chen (2007), Cattaneo and Farrell (2013), and Belloni et al. (2015), and references therein.”
Matching for Balance (2019, background): “They are satisfied by many classes of basis functions including regression spline, trigonometric polynomial, and wavelet bases (Newey 1997, Horowitz et al. 2004, Chen 2007, Belloni et al. 2015, Fan et al. 2016).”
Towards optimal doubly robust estimation of heterogeneous causal effects (2020, methodology): “In standard regimes the tuning parameter k would have to be chosen of smaller order than n (e.g., k log k/n → 0 as in Belloni et al. [2015] and elsewhere) in order for Condition 1 to hold, making the variance contribution from nuisance estimation asymptotically negligible.”
Simultaneous inference for Best Linear Predictor of the Conditional Average Treatment Effect and other structural functions (2017, methodology): “The second building block of our paper is the literature on least squares series estimation (Newey (2007), Belloni et al. (2015)), which establishes the pointwise and the uniform limit theory for least squares series estimation.”