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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 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.

The bootstrap provides a simple and powerful means of assessing the quality of estimators. However, in settings involving large data sets—which are increasingly prevalent—the calculation of bootstrap-based quantities can be prohibitively demanding computationally. Although variants such as subsampling and the m out of n bootstrap can be used in principle to reduce the cost of bootstrap computations, these methods are generally not robust to specification of tuning parameters (such as the number of subsampled data points), and they often require knowledge of the estimator's convergence rate, in contrast with the bootstrap. As an alternative, we introduce the 'bag of little bootstraps' (BLB), which is a new procedure which incorporates features of both the bootstrap and subsampling to yield a robust, computationally efficient means of assessing the quality of estimators. The BLB is well suited to modern parallel and distributed computing architectures and furthermore retains the generic applicability and statistical efficiency of the bootstrap. We demonstrate the BLB's favourable statistical performance via a theoretical analysis elucidating the procedure's properties, as well as a simulation study comparing the BLB with the bootstrap, the m out of n bootstrap and subsampling. In addition, we present results from a large-scale distributed implementation of the BLB demonstrating its computational superiority on massive data, a method for adaptively selecting the BLB's tuning parameters, an empirical study applying the BLB to several real data sets and an extension of the BLB to time series data.

A content analysis of 2 years of Psychological Science articles reveals inconsistencies in how researchers make inferences about indirect effects when conducting a statistical mediation analysis. In this study, we examined the frequency with which popularly used tests disagree, whether the method an investigator uses makes a difference in the conclusion he or she will reach, and whether there is a most trustworthy test that can be recommended to balance practical and performance considerations. We found that tests agree much more frequently than they disagree, but disagreements are more common when an indirect effect exists than when it does not. We recommend the bias-corrected bootstrap confidence interval as the most trustworthy test if power is of utmost concern, although it can be slightly liberal in some circumstances. Investigators concerned about Type I errors should choose the Monte Carlo confidence interval or the distribution-of-the-product approach, which rarely disagree. The percentile bootstrap confidence interval is a good compromise test.

We propose a novel technique to boost the power of testing a high-dimensional vector H : θ = 0 against sparse alternatives where the null hypothesis is violated by only a few components. Existing tests based on quadratic forms such as the Wald statistic often suffer from low powers due to the accumulation of errors in estimating high-dimensional parameters. More powerful tests for sparse alternatives such as thresholding and extreme value tests, on the other hand, require either stringent conditions or bootstrap to derive the null distribution and often suffer from size distortions due to the slow convergence. Based on a screening technique, we introduce a \"power enhancement component,\" which is zero under the null hypothesis with high probability, but diverges quickly under sparse alternatives. The proposed test statistic combines the power enhancement component with an asymptotically pivotal statistic, and strengthens the power under sparse alternatives. The null distribution does not require stringent regularity conditions, and is completely determined by that of the pivotal statistic. The proposed methods are then applied to testing the factor pricing models and validating the cross-sectional independence in panel data models.

In this article, we consider bootstrapping the Lasso estimator of the regression parameter in a multiple linear regression model. It is known that the standard bootstrap method fails to be consistent. Here, we propose a modified bootstrap method, and show that it provides valid approximation to the distribution of the Lasso estimator, for all possible values of the unknown regression parameter vector, including the case where some of the components are zero. Further, we establish consistency of the modified bootstrap method for estimating the asymptotic bias and variance of the Lasso estimator. We also show that the residual bootstrap can be used to consistently estimate the distribution and variance of the adaptive Lasso estimator. Using the former result, we formulate a novel data-based method for choosing the optimal penalizing parameter for the Lasso using the modified bootstrap. A numerical study is performed to investigate the finite sample performance of the modified bootstrap. The methodology proposed in the article is illustrated with a real data example.

This paper considers inference on functional of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. These models are often ill-posed and hence it is difficult to verify whether a (possibly nonlinear) functional is root-n estimable or not. We provide computationally simple, unified inference procedures that are asymptotically valid regardless of whether a functional is root-estimable or not. We establish the following new useful results: (1) the asymptotic normality of a plug-in penalized sieve minimum distance (PSMD) estimator of a (possibly nonlinear) functional; (2) the consistency of simple sieve variance estimators for the plug-in PSMD estimator, and hence the asymptotic chi-square distribution of the sieve Wald statistic; (3) the asymptotic chi-square distribution of an optimally weighted sieve quasi likelihood ratio (QLR) test under the null hypothesis; (4) the asymptotic tight distribution of a non-optimally weighted sieve QLR statistic under the null; (5) the consistency of generalized residual bootstrap sieve Wald and QLR tests; (6) local power properties of sieve Wald and QLR tests and of their bootstrap versions; (7) asymptotic properties of sieve Wald and SQLR for functionals of increasing dimension. Simulation studies and an empirical illustration of a nonparametric quantile IV regression are presented.

Matching estimators are widely used in empirical economics for the evaluation of programs or treatments. Researchers using matching methods often apply the bootstrap to calculate the standard errors. However, no formal justification has been provided for the use of the bootstrap in this setting. In this article, we show that the standard bootstrap is, in general, not valid for matching estimators, even in the simple case with a single continuous covariate where the estimator is root-N consistent and asymptotically normally distributed with zero asymptotic bias. Valid inferential methods in this setting are the analytic asymptotic variance estimator of Abadie and Imbens (2006a) as well as certain modifications of the standard bootstrap, like the subsampling methods in Politis and Romanoǐ (1994).

The Fleiss’ kappa statistic is a well-known index for assessing the reliability of agreement between raters. It is used both in the psychological and in the psychiatric field. Unfortunately, the kappa statistic may behave inconsistently in case of strong agreement between raters, since this index assumes lower values than it would have been expected. The aim of this paper is to propose a new method to avoid this paradox through permutation techniques. Furthermore, we study the problem of kappa confidence intervals and, in particular, we suggest to use Bootstrap confidence intervals free of paradoxes.

This article examines the problem of estimation in a quantile regression model when observations are missing at random under independent and nonidentically distributed errors. We consider three approaches of handling this problem based on nonparametric inverse probability weighting, estimating equations projection, and a combination of both. An important distinguishing feature of our methods is their ability to handle missing response and/or partially missing covariates, whereas existing techniques can handle only one or the other, but not both. We prove that our methods yield asymptotically equivalent estimators that achieve the desirable asymptotic properties of unbiasedness, normality, and -consistency. Because we do not assume that the errors are identically distributed, our theoretical results are valid under heteroscedasticity, a particularly strong feature of our methods. Under the special case of identical error distributions, all of our proposed estimators achieve the semiparametric efficiency bound. To facilitate the practical implementation of these methods, we develop an iterative method based on the majorize/minimize algorithm for computing the quantile regression estimates, and a bootstrap method for computing their variances. Our simulation findings suggest that all three methods have good finite sample properties. We further illustrate these methods by a real data example. Supplementary materials for this article are available online.

In the absence of relevant prior experience, popular Bayesian estimation techniques usually begin with some form of ‘uninformative’ prior distribution intended to have minimal inferential influence. The Bayes rule will still produce nice looking estimates and credible intervals, but these lack the logical force that is attached to experience‐based priors and require further justification. The paper concerns the frequentist assessment of Bayes estimates. A simple formula is shown to give the frequentist standard deviation of a Bayesian point estimate. The same simulations as required for the point estimate also produce the standard deviation. Exponential family models make the calculations particularly simple and bring in a connection to the parametric bootstrap.