Economists are often interested in the mechanisms by which a treatment affects an outcome. We develop tests for the ‘sharp null of full mediation’ that a treatment $D$ affects an outcome $Y$ only through a particular mechanism (or set of mechanisms) $M$. Our approach exploits connections between mediation analysis and the econometric literature on testing instrument validity. We also provide tools for quantifying the magnitude of alternative mechanisms when the sharp null is rejected: we derive sharp lower bounds on the fraction of individuals whose outcome is affected by the treatment despite having the same value of $M$ under both treatments (‘always-takers’), as well as sharp bounds on the average effect of the treatment for such always-takers. An advantage of our approach relative to existing tools for mediation analysis is that it does not require stringent assumptions about how $M$ is assigned. We illustrate our methodology in two empirical applications.

In the value-added literature, it is often claimed that regressing on empirical Bayes shrinkage estimates corrects for the measurement error problem in linear regression. We clarify the conditions needed; we argue that these conditions are stronger than the those needed for classical measurement error correction, which we advocate for instead. Moreover, we show that the classical estimator cannot be improved without stronger assumptions. We extend these results to regressions on nonlinear transformations of the latent attribute and find generically slow minimax estimation rates.

Researchers often use specifications that correctly estimate the average treatment effect under the assumption of constant effects. When treatment effects are heterogeneous, however, such specifications generally fail to recover this average effect. Augmenting these specifications with interaction terms between demeaned covariates and treatment eliminates this bias, but often leads to imprecise estimates and becomes infeasible under limited overlap. We propose a generalized ridge regression estimator, 𝚛𝚎𝚐𝚞𝚕𝚊𝚃𝙴, that penalizes the coefficients on the interaction terms to achieve an optimal trade-off between worst-case bias and variance in estimating the average effect under limited treatment effect heterogeneity. Building on this estimator, we construct confidence intervals that remain valid under limited overlap and can also be used to assess sensitivity to violations of the constant effects assumption. We illustrate the method in empirical applications under unconfoundedness and staggered adoption, providing a practical approach to inference under limited overlap.

We present a novel approach for extrapolating causal effects away from the margin between treatment and non-treatment in sharp regression discontinuity designs with multiple covariates. Our methods apply both to settings in which treatment is a function of multiple observables and settings in which treatment is determined based on a single running variable. Our key identifying assumption is that conditional average treated and untreated potential outcomes are comonotonic: covariate values associated with higher average untreated potential outcomes are also associated with higher average treated potential outcomes. We provide an estimation method based on local linear regression. Our estimands are weighted average causal effects, even if comonotonicity fails. We apply our methods to evaluate counterfactual mandatory summer school policies.

We consider inference on a regression coefficient under a constraint on the magnitude of the control coefficients. We show that a class of estimators based on an auxiliary regularized regression of the regressor of interest on control variables exactly solves a tradeoff between worst-case bias and variance. We derive “bias-aware” confidence intervals (CIs) based on these estimators, which take into account possible bias when forming the critical value. We show that these estimators and CIs are near-optimal in finite samples for mean squared error and CI length. Our finite-sample results are based on an idealized setting with normal regression errors with known homoskedastic variance, and we provide conditions for asymptotic validity with unknown and possibly heteroskedastic error distribution. Focusing on the case where the constraint on the magnitude of control coefficients is based on an $\scriptsize\ell_p$ norm (p ≥ 1), we derive rates of convergence for optimal estimators and CIs under high-dimensional asymptotics that allow the number of regressors to increase more quickly than the number of observations.

We provide an inference procedure for the sharp regression discontinuity design (RDD) under monotonicity, with possibly multiple running variables. Specifically, we consider the case where the true regression function is monotone with respect to (all or some of) the running variables and assumed to lie in a Lipschitz smoothness class. Such a monotonicity condition is natural in many empirical contexts, and the Lipschitz constant has an intuitive interpretation. We propose a minimax two-sided confidence interval (CI) and an adaptive one-sided CI. For the two-sided CI, the researcher is required to choose a Lipschitz constant where she believes the true regression function to lie in. This is the only tuning parameter, and the resulting CI has uniform coverage and obtains the minimax optimal length. The one-sided CI can be constructed to maintain coverage over all monotone functions, providing maximum credibility in terms of the choice of the Lipschitz constant. Moreover, the monotonicity makes it possible for the (excess) length of the CI to adapt to the true Lipschitz constant of the unknown regression function. Overall, the proposed procedures make it easy to see under what conditions on the underlying regression function the given estimates are significant, which can add more transparency to research using RDD methods.

We consider the problem of adaptive inference on a regression function at a point under a multivariate nonparametric regression setting. The regression function belongs to a Hölder class and is assumed to be monotone with respect to some or all of the arguments. We derive the minimax rate of convergence for confidence intervals (CIs) that adapt to the underlying smoothness, and provide an adaptive inference procedure that obtains this minimax rate. The procedure differs from that of Cai and Low (2004), intended to yield shorter CIs under practically relevant specifications. The proposed method applies to general linear functionals of the regression function, and is shown to have favorable performance compared to existing inference procedures.

Shrinkage methods are frequently used to estimate fixed effects. However, the risk properties of existing estimators are fragile to violations of the underlying distributional assumptions. I develop an estimator for the fixed effects that obtains the best possible mean squared error within a class of shrinkage estimators. This class includes conventional estimators, and the optimality does not require distributional assumptions. Importantly, the fixed effects are allowed to vary with time and to be serially correlated, and the shrinkage optimally incorporates the underlying correlation structure in this case. In such a context, I also provide a method to forecast fixed effects one period ahead.

We consider Bayes and Empirical Bayes (EB) approaches for dealing with violations of parallel trends. In the Bayes approach, the researcher specifies a prior over both the pre-treatment violations of parallel trends $\delta_{pre}$ and the post-treatment violations $\delta_{post}$. The researcher then updates their posterior about the post-treatment bias $\delta_{post}$ given an estimate of the pre-trends $\delta_{pre}$. This allows them to form posterior means and credible sets for the treatment effect of interest, $\tau_{post}$. In the EB approach, the prior on the violations of parallel trends is learned from the pre-treatment observations. We illustrate these approaches in two empirical applications.

This paper is concerned with possible model misspecification in moment inequality models. Two issues are addressed. First, standard tests and confidence sets for the true parameter in the moment inequality literature are not robust to model misspecification in the sense that they exhibit spurious precision when the identified set is empty. This paper introduces tests and confidence sets that provide correct asymptotic inference for a pseudo-true parameter in such scenarios, and hence, do not suffer from spurious precision. Second, specification tests have relatively low power against a range of misspecified models. Thus, failure to reject the null of correct specification does not necessarily provide evidence of correct specification. That is, model specification tests are subject to the problem that absence of evidence is not evidence of absence. This paper develops new diagnostics for model misspecification in moment inequality models that do not suffer from this problem.

In this paper, we investigate what can be learned about average counterfactual outcomes as well as average treatment effects when it is assumed that treatment response functions are smooth. We obtain a set of new partial identification results for both the average treatment response and the average treatment effect. In particular, we find that the monotone treatment response and monotone treatment selection bound of Manski and Pepper (2000) can be further tightened if we impose the smoothness conditions on the treatment response. Since it is unknown in practice whether the imposed smoothness restriction is met, it is desirable to conduct a sensitivity analysis with respect to the smoothness assumption. We demonstrate how one can carry out a sensitivity analysis for the average treatment effect by varying the degrees of smoothness assumption. We illustrate our findings by reanalyzing the return to schooling example of Manski and Pepper (2000) and also by measuring the effect of the length of job training on the labor market outcomes.

We analyse cross‐country trends in several aspects of technological progress during 1980–2011 by examining the US patent citations data. Our estimation results on patent quality and citation lags relative to the US reveal the following. The emerging Asian economies of Korea, Taiwan and China have achieved substantial catch‐up. In the case of Korea and Taiwan, progress has been made in both patent quality and citation lags. China has achieved improvement in patent quality but not in citation lag. In contrast, advanced economies of Europe and Japan have displayed steady decline in patent quality, while the US has strengthened its position.