Alexandre Belloni, Victor Chernozhukov, Denis Chetverikov, Kengo Kato
arXiv 3 Dec 2012 · Statistics — Methodology · publishedJournal of Econometrics (2015) · 227 citations (OpenAlex)
arXiv:1212.0442 · PDF · DOI · OpenAlex · Extracted main text
In applications it is common that the exact form of a conditional expectation is unknown and having flexible functional forms can lead to improvements. Series method offers that by approximating the unknown function based on $k$ basis functions, where $k$ is allowed to grow with the sample size $n$. We consider series estimators for the conditional mean in light of: (i) sharp LLNs for matrices derived from the noncommutative Khinchin inequalities, (ii) bounds on the Lebesgue factor that controls the ratio between the $L^\infty$ and $L_2$-norms of approximation errors, (iii) maximal inequalities for processes whose entropy integrals diverge, and (iv) strong approximations to series-type processes. These technical tools allow us to contribute to the series literature, specifically the seminal work of Newey (1997), as follows. First, we weaken the condition on the number $k$ of approximating functions used in series estimation from the typical $k^2/n \to 0$ to $k/n \to 0$, up to log factors, which was available only for spline series before. Second, we derive $L_2$ 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. We derive uniform rates, Gaussian approximations, and uniform confidence bands for a wide collection of linear functionals of the conditional expectation function.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Newey, W.K (1997) Convergence rates and asymptotic normality for series estimators | 1.000 | 15 | 5 | 100% |
| 2 | Cattaneo, M. and M. Farrell (2013) Optimal convergence rates, Bahadur representation, and asymptotic normality of partitioning estimators | 1.000 | 7 | 3 | 100% |
| 3 | Huang, J.Z (2003) Local asymptotics for polynomial spline regression | 1.000 | 5 | 3 | 100% |
| 4 | Chen, X (2007) Large sample sieve estimation of semi-nonparametric models | 0.928 | 4 | 3 | 100% |
| 5 | Huang, J.Z (2003) Asymptotics for polynomial spline regression under weak conditions | 0.811 | 4 | 2 | 100% |
| 6 | Chernozhukov, V., S. Lee, and A. Rosen (2013) Intersection bounds: estimation and inference self | 0.737 | 3 | 3 | 67% |
| 7 | Stone, C.J (1994) The use of polynomial splines and their tensor products in multivariate function estimation | 0.737 | 3 | 2 | 100% |
| 8 | Chernozhukov, V., D. Chetverikov and K. Kato (2012) Gaussian approximation of suprema of empirical processes self | 0.644 | 4 | 2 | 50% |
| 9 | Rudelson, M (1999) Random vectors in the isotropic position | 0.644 | 4 | 2 | 50% |
| 10 | Chen, X. and T. Christensen (2015) Optimal uniform convergence rates for sieve nonparametric instrumental variables regression, forthcoming in Journal of Econometr… | 0.644 | 2 | 2 | 100% |
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