/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

We extend conformal inference to general settings that allow for time series data. Our proposal is developed as a randomization method and accounts for potential serial dependence by including block structures in the permutation scheme. As a result, the proposed method retains the exact, model-free validity when the data are i.i.d. or more generally exchangeable, similar to usual conformal inference methods. When exchangeability fails, as is the case for common time series data, the proposed approach is approximately valid under weak assumptions on the conformity score.

Exact and Robust Conformal Inference Methods for Predictive Machine Learning With Dependent Data

We propose a notion of conditional vector quantile function and a vector quantile regression. A conditional vector quantile function (CVQF) of a random vector $Y$, taking values in $\mathbb{R}^{d}$ given covariates $Z=z$, taking values in $\mathbb{R}^{k}$, is a map $u\longmapsto Q_{Y|Z}(u,z)$, which is monotone, in the sense of being a gradient of a convex function, and such that given that vector $U$ follows a reference non-atomic distribution $F_{U}$, for instance uniform distribution on a unit cube in $\mathbb{R}^{d}$, the random vector $Q_{Y|Z}(U,z)$ has the distribution of $Y$ conditional on $Z=z$. Moreover, we have a strong representation, $Y=Q_{Y|Z}(U,Z)$ almost surely, for some version of $U$. The vector quantile regression (VQR) is a linear model for CVQF of $Y$ given $Z$. Under correct specification, the notion produces strong representation, $Y=\beta (U)^{\top}f(Z)$, for $f(Z)$ denoting a known set of transformations of $Z$, where $u\longmapsto\beta(u)^{\top}f(Z)$ is a monotone map, the gradient of a convex function and the quantile regression coefficients $u\longmapsto\beta(u)$ have the interpretations analogous to that of the standard scalar quantile regression. As $f(Z)$ becomes a richer class of transformations of $Z$, the model becomes nonparametric, as in series modelling. A key property of VQR is the embedding of the classical Monge–Kantorovich’s optimal transportation problem at its core as a special case. In the classical case, where $Y$ is scalar, VQR reduces to a version of the classical QR, and CVQF reduces to the scalar conditional quantile function. An application to multiple Engel curve estimation is considered.

/pdf/vector-quantile-regression-an-optimal-transport-approach-djxh44afye.pdf

Vector quantile regression: An optimal transport approach

In parametric models a sufficient condition for local identification is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix. We show that there are corresponding sufficient conditions for nonparametric models. A nonparametric rank condition and differentiability of the moment conditions with respect to a certain norm imply local identification. It turns out these conditions are slightly stronger than needed and are hard to check, so we provide weaker and more primitive conditions. We extend the results to semiparametric models. We illustrate the sufficient conditions with endogenous quantile and single index examples. We also consider a semiparametric habit-based, consumption capital asset pricing model. There we find the rank condition is implied by an integral equation of the second kind having a one-dimensional null space.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1822264_code331494.pdf?abstractid=1822264&mirid=2

Local Identification of Nonparametric and Semiparametric Models

In this supplementary appendix we provide additional results, omitted proofs and extensive simulations that complement the analysis of the main text (arXiv:1201.0224). 1. Intuition for the Importance of Double Selection To build intuition, we discuss the case where there is only one control; that is, p = 1. This scenario provides the simplest possible setting where variable selection might be interesting. In this case, Lasso-type methods act like conservative t-tests which allows the properties of selection methods to be explained easily. With p = 1, the model is

Supplementary appendix for \inference on treatment effects after selection amongst high-dimensional controls"

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation σ or nor does it need to pre-estimate σ. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate σ{(s/n) log p}1/2 in the prediction norm, and thus matching the performance of the lasso with known σ. These performance results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions. We formulate the square-root lasso as a solution to a convex conic programming problem, which allows us to implement the estimator using efficient algorithmic methods, such as interior-point and first-order methods.

Victor Chernozhukov

Papers

Exact and Robust Conformal Inference Methods for Predictive Machine Learning With Dependent Data

Vector quantile regression: An optimal transport approach

Local Identification of Nonparametric and Semiparametric Models

Supplementary appendix for \inference on treatment effects after selection amongst high-dimensional controls"

Square-Root Lasso: Pivotal Recovery of Sparse Signals via Conic Programming