/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

On robust estimation of the location parameter

We consider statistical inference for regression when data are grouped into clus- ters, with regression model errors independent across clusters but correlated within clusters. Examples include data on individuals with clustering on village or region or other category such as industry, and state-year dierences-in-dierences studies with clustering on state. In such settings default standard errors can greatly overstate es- timator precision. Instead, if the number of clusters is large, statistical inference after OLS should be based on cluster-robust standard errors. We outline the basic method as well as many complications that can arise in practice. These include cluster-specic �xed eects, few clusters, multi-way clustering, and estimators other than OLS.

/pdf/a-practitioner-s-guide-to-cluster-robust-inference-1xbcejtio0.pdf

A Practitioner’s Guide to Cluster-Robust Inference

Many empirical questions in economics and other social sciences depend on causal effects of programs or policies. In the last two decades, much research has been done on the econometric and statistical analysis of such causal effects. This recent theoreti- cal literature has built on, and combined features of, earlier work in both the statistics and econometrics literatures. It has by now reached a level of maturity that makes it an important tool in many areas of empirical research in economics, including labor economics, public finance, development economics, industrial organization, and other areas of empirical microeconomics. In this review, we discuss some of the recent developments. We focus primarily on practical issues for empirical research- ers, as well as provide a historical overview of the area and give references to more technical research.

/pdf/recent-developments-in-the-econometrics-of-program-42vjewfvjw.pdf

Recent developments in the econometrics of program evaluation

Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in the probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration inequality for maxima of Gaussian random vectors, which derives a useful upper bound on the Levy concentration function for the maximum of (not necessarily independent) Gaussian random variables. The bound is universal and applies to vectors with arbitrary covariance matrices. This anti-concentration inequality plays a crucial role in establishing bounds on the Kolmogorov distance between maxima of Gaussian random vectors. These results have immediate applications in mathematical statistics. As an example of application, we establish a conditional multiplier central limit theorem for maxima of sums of independent random vectors where the dimension of the vectors is possibly much larger than the sample size.

Comparison and anti-concentration bounds for maxima of Gaussian random vectors

Modern construction of uniform condence e and Nickl (2010). This condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process (equivalently, for the supremum of a Gaussian process with the same covariance function as that of the studentized empirical process). The principal contribution of this paper is to remove the need for this classical condition. We show that a considerably weaker sucient condi- tion is derived from an anti-concentration property of the supremum of the approximating Gaussian process, and we derive an inequality lead- ing to such a property for separable Gaussian processes. We refer to the new condition as a generalized SBR condition. Our new result shows that the supremum does not concentrate too fast around any value. We then apply this result to derive a Gaussian multiplier bootstrap procedure for constructing honest condence bands for nonparametric density estimators (this result can be applied in other nonparametric problems as well). An essential advantage of our approach is that it ap- plies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist (or is un- known). This is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data-driven fash- ion, which is needed for adaptive constructions of the condence bands. Furthermore, our approach is asymptotically honest at a polynomial rate { namely, the error in coverage level converges to zero at a fast, polynomial speed (with respect to the sample size). In sharp contrast, the approach based on extreme value theory is asymptotically honest only at a logarithmic rate { the error converges to zero at a slow, loga- rithmic speed. Finally, of independent interest is our introduction of a new, practical version of Lepski's method, which computes the optimal, non-conservative resolution levels via a Gaussian multiplier bootstrap method.

/pdf/anti-concentration-and-honest-adaptive-confidence-bands-laefuld7b4.pdf

Anti-concentration and honest, adaptive confidence bands

Under the assumption that individuals know the conditional distributions of signals given the payoff-relevant parameters, existing results conclude that as individuals observe infinitely many signals, their beliefs about the parameters will eventually merge. We first show that these results are fragile when individuals are uncertain about the signal distributions: given any such model, a vanishingly small individual uncertainty about the signal distributions can lead to a substantial (non-vanishing) amount of differences between the asymptotic beliefs. We then characterize the conditions under which a small amount of uncertainty leads only to a small amount of asymptotic disagreement. According to our characterization, this is the case if the uncertainty about the signal distributions is generated by a family with "rapidly-varying tails" (such as the normal or the exponential distributions). However, when this family has "regularly-varying tails" (such as the Pareto, the log-normal, and the t-distributions), a small amount of uncertainty leads to a substantial amount of asymptotic disagreement.

/pdf/fragility-of-asymptotic-agreement-under-bayesian-learning-2bzbvdhx2f.pdf

Fragility of Asymptotic Agreement Under Bayesian Learning

Most modern supervised statistical/machine learning (ML) methods are explicitly designed to solve prediction problems very well. Achieving this goal does not imply that these methods automatically deliver good estimators of causal parameters. Examples of such parameters include individual regression coefficients, average treatment effects, average lifts, and demand or supply elasticities. In fact, estimates of such causal parameters obtained via naively plugging ML estimators into estimating equations for such parameters can behave very poorly due to the regularization bias. Fortunately, this regularization bias can be removed by solving auxiliary prediction problems via ML tools. Specifically, we can form an orthogonal score for the target low-dimensional parameter by combining auxiliary and main ML predictions. The score is then used to build a de-biased estimator of the target parameter which typically will converge at the fastest possible 1/root(n) rate and be approximately unbiased and normal, and from which valid confidence intervals for these parameters of interest may be constructed. The resulting method thus could be called a "double ML" method because it relies on estimating primary and auxiliary predictive models. In order to avoid overfitting, our construction also makes use of the K-fold sample splitting, which we call cross-fitting. This allows us to use a very broad set of ML predictive methods in solving the auxiliary and main prediction problems, such as random forest, lasso, ridge, deep neural nets, boosted trees, as well as various hybrids and aggregators of these methods.

Double/Debiased Machine Learning for Treatment and Causal Parameters

Counterfactual distributions are important ingredients for policy analysis and decomposition analysis in empirical economics. In this article we develop modeling and inference tools for counterfactual distributions based on regression methods. The counterfactual scenarios that we consider consist of ceteris paribus changes in either the distribution of covariates related to the outcome of interest or the conditional distribution of the outcome given covariates. For either of these scenarios we derive joint functional central limit theorems and bootstrap validity results for regression-based estimators of the status quo and counterfactual outcome distributions. These results allow us to construct simultaneous confidence sets for function-valued effects of the counterfactual changes, including the effects on the entire distribution and quantile functions of the outcome as well as on related functionals. These confidence sets can be used to test functional hypotheses such as no-effect, positive effect, or stochastic dominance. Our theory applies to general counterfactual changes and covers the main regression methods including classical, quantile, duration, and distribution regressions. We illustrate the results with an empirical application to wage decompositions using data for the United States. 
As a part of developing the main results, we introduce distribution regression as a comprehensive and flexible tool for modeling and estimating the \textit{entire} conditional distribution. We show that distribution regression encompasses the Cox duration regression and represents a useful alternative to quantile regression. We establish functional central limit theorems and bootstrap validity results for the empirical distribution regression process and various related functionals.

Victor Chernozhukov

Papers

Comparison and anti-concentration bounds for maxima of Gaussian random vectors

Anti-concentration and honest, adaptive confidence bands

Fragility of Asymptotic Agreement Under Bayesian Learning

Double/Debiased Machine Learning for Treatment and Causal Parameters

Inference on Counterfactual Distributions