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We provide a simple distribution regression estimator for treatment effects in the difference-in-differences (DiD) design. Our procedure is particularly useful when the treatment effect differs across the distribution of the outcome variable. Our proposed estimator easily incorporates covariates and, importantly, can be extended to settings where the treatment potentially affects the joint distribution of multiple outcomes. Our key identifying restriction is that the counterfactual distribution of the treated in the untreated state has no interaction effect between treatment and time. This assumption results in a parallel trend assumption on a transformation of the distribution. We highlight the relationship between our procedure and assumptions with the changes-in-changes approach of Athey and Imbens (2006). We also reexamine the Card and Krueger (1994) study of the impact of minimum wages on employment to illustrate the utility of our approach., Comment: 32 pages, 4 figures, 4 tables; minor edits and new empirical application with respect to previous version

Rank-rank regression is commonly employed in economic research as a way of capturing the relationship between two economic variables. It frequently features in studies of intergenerational mobility as the resulting coefficient, capturing the rank correlation between the variables, is easy to interpret and measures overall persistence. However, in many applications it is common practice to include other covariates to account for differences in persistence levels between groups defined by the values of these covariates. In these instances the resulting coefficients can be difficult to interpret. We propose the conditional rank-rank regression, which uses conditional ranks instead of unconditional ranks, to measure average within-group income persistence. The difference between conditional and unconditional rank-rank regression coefficients can then be interpreted as a measure of between-group persistence. We develop a flexible estimation approach using distribution regression and establish a theoretical framework for large sample inference. An empirical study on intergenerational income mobility in Switzerland demonstrates the advantages of this approach. The study reveals stronger intergenerational persistence between fathers and sons compared to fathers and daughters, with the within-group persistence explaining 62% of the overall income persistence for sons and 52% for daughters. Smaller families and those with highly educated fathers exhibit greater persistence in economic status., Comment: 41 pages, 3 figures, 9 tables; minor edits on main text and updated simulation results

We propose an instrumental variable framework for identifying and estimating causal effects of discrete and continuous treatments with binary instruments. The basis of our approach is a local copula representation of the joint distribution of the potential outcomes and unobservables determining treatment assignment. This representation allows us to introduce an identifying assumption, so-called copula invariance, that restricts the local dependence of the copula with respect to the treatment propensity. We show that copula invariance identifies treatment effects for the entire population and other subpopulations such as the treated. The identification results are constructive and lead to practical estimation and inference procedures based on distribution regression. An application to estimating the effect of sleep on well-being uncovers interesting patterns of heterogeneity.

The Arellano-Bond estimator is a fundamental method for dynamic panel data models, widely used in practice. However, the estimator is severely biased when the data's time series dimension $T$ is long due to the large degree of overidentification. We show that weak dependence along the panel's time series dimension naturally implies approximate sparsity of the most informative moment conditions, motivating the following approach to remove the bias: First, apply LASSO to the cross-section data at each time period to construct most informative (and cross-fitted) instruments, using lagged values of suitable covariates. This step relies on approximate sparsity to select the most informative instruments. Second, apply a linear instrumental variable estimator after first differencing the dynamic structural equation using the constructed instruments. Under weak time series dependence, we show the new estimator is consistent and asymptotically normal under much weaker conditions on $T$'s growth than the Arellano-Bond estimator. Our theory covers models with high dimensional covariates, including multiple lags of the dependent variable, common in modern applications. We illustrate our approach by applying it to weekly county-level panel data from the United States to study opening K-12 schools and other mitigation policies' short and long-term effects on COVID-19's spread.

Using CPS data for 1976 to 2022 we explore how wage inequality has evolved for married couples with both spouses working full time full year, and its impact on household income inequality. We also investigate how marriage sorting patterns have changed over this period. To determine the factors driving income inequality we estimate a model explaining the joint distribution of wages which accounts for the spouses' employment decisions. We find that income inequality has increased for these households and increased assortative matching of wages has exacerbated the inequality resulting from individual wage growth. We find that positive sorting partially reflects the correlation across unobservables influencing both members' of the marriage wages. We decompose the changes in sorting patterns over the 47 years comprising our sample into structural, composition and selection effects and find that the increase in positive sorting primarily reflects the increased skill premia for both observed and unobserved characteristics., Comment: 63 pages; 13 figures; 36 tables

We consider the estimation of a dynamic distribution regression panel data model with heterogeneous coefficients across units. The objects of primary interest are specific functionals of these coefficients. These include predicted actual and stationary distributions of the outcome variable and quantile treatment effects. Coefficients and their functionals are estimated via fixed effect methods. We investigate how these functionals vary in response to changes in initial conditions or covariate values. We also identify a uniformity issue related to the robustness of inference to the unknown degree of heterogeneity, and propose a cross-sectional bootstrap method for uniformly valid inference on function-valued objects. Employing PSID annual labor income data we illustrate some important empirical issues we can address. We first quantify the impact of a negative labor income shock on the distribution of future labor income. We also examine the impact on the distribution of labor income from increasing the education level of a chosen group of workers. Finally, we demonstrate the existence of heterogeneity in income mobility, and how this leads to substantial variation in individuals' incidences to be trapped in poverty. We also provide simulation evidence confirming that our procedures work well.

We provide estimation methods for nonseparable panel models based on low-rank factor structure approximations. The factor structures are estimated by matrix-completion methods to deal with the computational challenges of principal component analysis in the presence of missing data. We show that the resulting estimators are consistent in large panels, but suffer from approximation and shrinkage biases. We correct these biases using matching and difference-in-differences approaches. Numerical examples and an empirical application to the effect of election day registration on voter turnout in the U.S. illustrate the properties and usefulness of our methods.

In ordinary quantile regression, quantiles of different order are estimated one at a time. An alternative approach, which is referred to as quantile regression coefficients modeling (QRCM), is to model quantile regression coefficients as parametric functions of the order of the quantile. In this paper, we describe how the QRCM paradigm can be applied to longitudinal data. We introduce a two-level quantile function, in which two different quantile regression models are used to describe the (conditional) distribution of the within-subject response and that of the individual effects. We propose a novel type of penalized fixed-effects estimator, and discuss its advantages over standard methods based on $\ell_1$ and $\ell_2$ penalization. We provide model identifiability conditions, derive asymptotic properties, describe goodness-of-fit measures and model selection criteria, present simulation results, and discuss an application. The proposed method has been implemented in the R package qrcm., Comment: 71 pages, 2 figures, includes appendix, R companion package available at https://cran.r-project.org/web/packages/qrcm/index.html

We examine the impact of annual hours worked on annual earnings by decomposing changes in the real annual earnings distribution into composition, structural and hours effects. We do so via a nonseparable simultaneous model of hours, wages and earnings. Using the Current Population Survey for the survey years 1976--2019, we find that changes in the female distribution of annual hours of work are important in explaining movements in inequality in female annual earnings. This captures the substantial changes in their employment behavior over this period. Movements in the male hours distribution only affect the lower part of their earnings distribution and reflect the sensitivity of these workers' annual hours of work to cyclical factors., Comment: 44 pages, 20 figures, includes web appendix

The widespread use of quantile regression methods depends crucially on the existence of fast algorithms. Despite numerous algorithmic improvements, the computation time is still non-negligible because researchers often estimate many quantile regressions and use the bootstrap for inference. We suggest two new fast algorithms for the estimation of a sequence of quantile regressions at many quantile indexes. The first algorithm applies the preprocessing idea of Portnoy and Koenker (1997) but exploits a previously estimated quantile regression to guess the sign of the residuals. This step allows for a reduction of the effective sample size. The second algorithm starts from a previously estimated quantile regression at a similar quantile index and updates it using a single Newton-Raphson iteration. The first algorithm is exact, while the second is only asymptotically equivalent to the traditional quantile regression estimator. We also apply the preprocessing idea to the bootstrap by using the sample estimates to guess the sign of the residuals in the bootstrap sample. Simulations show that our new algorithms provide very large improvements in computation time without significant (if any) cost in the quality of the estimates. For instance, we divide by 100 the time required to estimate 99 quantile regressions with 20 regressors and 50,000 observations., Comment: 29 pages, 3 figures, 4 tables; for associated Stata package, see https://sites.google.com/site/blaisemelly/home/computer-programs/fast

Chernozhukov et al. (2018) proposed the sorted effect method for nonlinear regression models. This method consists of reporting percentiles of the partial effects in addition to the average commonly used to summarize the heterogeneity in the partial effects. They also proposed to use the sorted effects to carry out classification analysis where the observational units are classified as most and least affected if their causal effects are above or below some tail sorted effects. The R package SortedEffects implements the estimation and inference methods therein and provides tools to visualize the results. This vignette serves as an introduction to the package and displays basic functionality of the functions within., Comment: 15 pages, 6 figures, 8 tables

The relationship between democracy and economic growth is of long-standing interest. We revisit the panel data analysis of this relationship by Acemoglu, Naidu, Restrepo and Robinson (forthcoming) using state of the art econometric methods. We argue that this and lots of other panel data settings in economics are in fact high-dimensional, resulting in principal estimators -- the fixed effects (FE) and Arellano-Bond (AB) estimators -- to be biased to the degree that invalidates statistical inference. We can however remove these biases by using simple analytical and sample-splitting methods, and thereby restore valid statistical inference. We find that the debiased FE and AB estimators produce substantially higher estimates of the long-run effect of democracy on growth, providing even stronger support for the key hypothesis in Acemoglu, Naidu, Restrepo and Robinson (forthcoming). Given the ubiquitous nature of panel data, we conclude that the use of debiased panel data estimators should substantially improve the quality of empirical inference in economics., Comment: 8 pages, 2 tables, includes supplementary appendix

We analyze the role of selection bias in generating the changes in the observed distribution of female hourly wages in the United States using CPS data for the years 1975 to 2020. We account for the selection bias from the employment decision by modeling the distribution of the number of working hours and estimating a nonseparable model of wages. We decompose changes in the wage distribution into composition, structural and selection effects. Composition effects have increased wages at all quantiles while the impact of the structural effects varies by time period and quantile. Changes in the role of selection only appear at the lower quantiles of the wage distribution. The evidence suggests that there is positive selection in the 1970s which diminishes until the later 1990s. This reduces wages at lower quantiles and increases wage inequality. Post 2000 there appears to be an increase in positive sorting which reduces the selection effects on wage inequality., Comment: 52 pages, 19 figures, change of title with respect to version v4

We develop a distribution regression model under endogenous sample selection. This model is a semi-parametric generalization of the Heckman selection model. It accommodates much richer effects of the covariates on outcome distribution and patterns of heterogeneity in the selection process, and allows for drastic departures from the Gaussian error structure, while maintaining the same level tractability as the classical model. The model applies to continuous, discrete and mixed outcomes. We provide identification, estimation, and inference methods, and apply them to obtain wage decomposition for the UK. Here we decompose the difference between the male and female wage distributions into composition, wage structure, selection structure, and selection sorting effects. After controlling for endogenous employment selection, we still find substantial gender wage gap -- ranging from 21% to 40% throughout the (latent) offered wage distribution that is not explained by composition. We also uncover positive sorting for single men and negative sorting for married women that accounts for a substantive fraction of the gender wage gap at the top of the distribution., Comment: 86 pages, 4 tables, 40 figures, includes supplement

A common problem in econometrics, statistics, and machine learning is to estimate and make inference on functions that satisfy shape restrictions. For example, distribution functions are nondecreasing and range between zero and one, height growth charts are nondecreasing in age, and production functions are nondecreasing and quasi-concave in input quantities. We propose a method to enforce these restrictions ex post on point and interval estimates of the target function by applying functional operators. If an operator satisfies certain properties that we make precise, the shape-enforced point estimates are closer to the target function than the original point estimates and the shape-enforced interval estimates have greater coverage and shorter length than the original interval estimates. We show that these properties hold for six different operators that cover commonly used shape restrictions in practice: range, convexity, monotonicity, monotone convexity, quasi-convexity, and monotone quasi-convexity. We illustrate the results with two empirical applications to the estimation of a height growth chart for infants in India and a production function for chemical firms in China., Comment: 42 pages, 5 figures, 3 tables, v5 includes changes in the main text

This paper provides a method to construct simultaneous confidence bands for quantile functions and quantile effects in nonlinear network and panel models with unobserved two-way effects, strictly exogenous covariates, and possibly discrete outcome variables. The method is based upon projection of simultaneous confidence bands for distribution functions constructed from fixed effects distribution regression estimators. These fixed effects estimators are debiased to deal with the incidental parameter problem. Under asymptotic sequences where both dimensions of the data set grow at the same rate, the confidence bands for the quantile functions and effects have correct joint coverage in large samples. An empirical application to gravity models of trade illustrates the applicability of the methods to network data., Comment: 71 pages, 8 figures, 3 tables, includes supplementary appendix

We consider identification and estimation of nonseparable sample selection models with censored selection rules. We employ a control function approach and discuss different objects of interest based on (1) local effects conditional on the control function, and (2) global effects obtained from integration over ranges of values of the control function. We derive the conditions for the identification of these different objects and suggest strategies for estimation. Moreover, we provide the associated asymptotic theory. These strategies are illustrated in an empirical investigation of the determinants of female wages in the United Kingdom., Comment: 53 pages, 4 tables, 5 figures

Many applications involve a censored dependent variable and an endogenous independent variable. Chernozhukov et al. (2015) introduced a censored quantile instrumental variable estimator (CQIV) for use in those applications, which has been applied by Kowalski (2016), among others. In this article, we introduce a Stata command, cqiv, that simplifes application of the CQIV estimator in Stata. We summarize the CQIV estimator and algorithm, we describe the use of the cqiv command, and we provide empirical examples., Comment: 12 pages, 1 table, associated software can be found at https://ideas.repec.org/c/boc/bocode/s457478.html. arXiv admin note: text overlap with arXiv:1104.4580. We have updated the command to report standard errors and bootstrap percentile-t confidence intervals

We propose strategies to estimate and make inference on key features of heterogeneous effects in randomized experiments. These key features include best linear predictors of the effects using machine learning proxies, average effects sorted by impact groups, and average characteristics of most and least impacted units. The approach is valid in high dimensional settings, where the effects are proxied (but not necessarily consistently estimated) by predictive and causal machine learning methods. We post-process these proxies into estimates of the key features. Our approach is generic, it can be used in conjunction with penalized methods, neural networks, random forests, boosted trees, and ensemble methods, both predictive and causal. Estimation and inference are based on repeated data splitting to avoid overfitting and achieve validity. We use quantile aggregation of the results across many potential splits, in particular taking medians of p-values and medians and other quantiles of confidence intervals. We show that quantile aggregation lowers estimation risks over a single split procedure, and establish its principal inferential properties. Finally, our analysis reveals ways to build provably better machine learning proxies through causal learning: we can use the objective functions that we develop to construct the best linear predictors of the effects, to obtain better machine learning proxies in the initial step. We illustrate the use of both inferential tools and causal learners with a randomized field experiment that evaluates a combination of nudges to stimulate demand for immunization in India., Comment: 81 pages, 8 figures, 17 tables, includes Online Appendix, minor revision with respect to previous version

Triangular systems with nonadditively separable unobserved heterogeneity provide a theoretically appealing framework for the modelling of complex structural relationships. However, they are not commonly used in practice due to the need for exogenous variables with large support for identification, the curse of dimensionality in estimation, and the lack of inferential tools. This paper introduces two classes of semiparametric nonseparable triangular models that address these limitations. They are based on distribution and quantile regression modelling of the reduced form conditional distributions of the endogenous variables. We show that average, distribution and quantile structural functions are identified in these systems through a control function approach that does not require a large support condition. We propose a computationally attractive three-stage procedure to estimate the structural functions where the first two stages consist of quantile or distribution regressions. We provide asymptotic theory and uniform inference methods for each stage. In particular, we derive functional central limit theorems and bootstrap functional central limit theorems for the distribution regression estimators of the structural functions. These results establish the validity of the bootstrap for three-stage estimators of structural functions, and lead to simple inference algorithms. We illustrate the implementation and applicability of all our methods with numerical simulations and an empirical application to demand analysis., Comment: 45 pages, 4 figures, 1 table, we have added grant funding acknowledgement to v3

This article reviews recent advances in fixed effect estimation of panel data models for long panels, where the number of time periods is relatively large. We focus on semiparametric models with unobserved individual and time effects, where the distribution of the outcome variable conditional on covariates and unobserved effects is specified parametrically, while the distribution of the unobserved effects is left unrestricted. Compared to existing reviews on long panels (Arellano and Hahn 2007; a section in Arellano and Bonhomme 2011) we discuss models with both individual and time effects, split-panel Jackknife bias corrections, unbalanced panels, distribution and quantile effects, and other extensions. Understanding and correcting the incidental parameter bias caused by the estimation of many fixed effects is our main focus, and the unifying theme is that the order of this bias is given by the simple formula p/n for all models discussed, with p the number of estimated parameters and n the total sample size., Comment: 40 pages, 1 table

Multinomial choice models are fundamental for empirical modeling of economic choices among discrete alternatives. We analyze identification of binary and multinomial choice models when the choice utilities are nonseparable in observed attributes and multidimensional unobserved heterogeneity with cross-section and panel data. We show that derivatives of choice probabilities with respect to continuous attributes are weighted averages of utility derivatives in cross-section models with exogenous heterogeneity. In the special case of random coefficient models with an independent additive effect, we further characterize that the probability derivative at zero is proportional to the population mean of the coefficients. We extend the identification results to models with endogenous heterogeneity using either a control function or panel data. In time stationary panel models with two periods, we find that differences over time of derivatives of choice probabilities identify utility derivatives "on the diagonal," i.e. when the observed attributes take the same values in the two periods. We also show that time stationarity does not identify structural derivatives "off the diagonal" both in continuous and multinomial choice panel models., Comment: 23 pages

Extremal quantile regression, i.e. quantile regression applied to the tails of the conditional distribution, counts with an increasing number of economic and financial applications such as value-at-risk, production frontiers, determinants of low infant birth weights, and auction models. This chapter provides an overview of recent developments in the theory and empirics of extremal quantile regression. The advances in the theory have relied on the use of extreme value approximations to the law of the Koenker and Bassett (1978) quantile regression estimator. Extreme value laws not only have been shown to provide more accurate approximations than Gaussian laws at the tails, but also have served as the basis to develop bias corrected estimators and inference methods using simulation and suitable variations of bootstrap and subsampling. The applicability of these methods is illustrated with two empirical examples on conditional value-at-risk and financial contagion., Comment: 32 pages, 4 tables, 7 figures, forthcoming in the Handbook of Quantile Regression

The R package quantreg.nonpar implements nonparametric quantile regression methods to estimate and make inference on partially linear quantile models. quantreg.nonpar obtains point estimates of the conditional quantile function and its derivatives based on series approximations to the nonparametric part of the model. It also provides pointwise and uniform confidence intervals over a region of covariate values and/or quantile indices for the same functions using analytical and resampling methods. This paper serves as an introduction to the package and displays basic functionality of the functions contained within., Comment: 12 pages, 5 figures

The Counterfactual package implements the estimation and inference methods of Chernozhukov, Fern\'andez-Val and Melly (2013) for counterfactual analysis. The counterfactual distributions considered are the result of changing either the marginal distribution of covariates related to the outcome variable of interest, or the conditional distribution of the outcome given the covariates. They can be applied to estimate quantile treatment effects and wage decompositions. This paper serves as an introduction to the package and displays basic functionality of the commands contained within., Comment: 15 pages, 4 figures

We present the Stata commands probitfe and logitfe, which estimate probit and logit panel data models with individual and/or time unobserved effects. Fixed effect panel data methods that estimate the unobserved effects can be severely biased because of the incidental parameter problem (Neyman and Scott, 1948). We tackle this problem by using the analytical and jackknife bias corrections derived in Fernandez-Val and Weidner (2016) for panels where the two dimensions ($N$ and $T$) are moderately large. We illustrate the commands with an empirical application to international trade and a Monte Carlo simulation calibrated to this application., Comment: 29 pages, 3 tables, ado and help files for the commands available at http://econpapers.repec.org/software/bocbocode/s458279.htm and http://econpapers.repec.org/software/bocbocode/s458278.htm

Quantile and quantile effect functions are important tools for descriptive and causal analyses due to their natural and intuitive interpretation. Existing inference methods for these functions do not apply to discrete random variables. This paper offers a simple, practical construction of simultaneous confidence bands for quantile and quantile effect functions of possibly discrete random variables. It is based on a natural transformation of simultaneous confidence bands for distribution functions, which are readily available for many problems. The construction is generic and does not depend on the nature of the underlying problem. It works in conjunction with parametric, semiparametric, and nonparametric modeling methods for observed and counterfactual distributions, and does not depend on the sampling scheme. We apply our method to characterize the distributional impact of insurance coverage on health care utilization and obtain the distributional decomposition of the racial test score gap. We find that universal insurance coverage increases the number of doctor visits across the entire distribution, and that the racial test score gap is small at early ages but grows with age due to socio economic factors affecting child development especially at the top of the distribution. These are new, interesting empirical findings that complement previous analyses that focused on mean effects only. In both applications, the outcomes of interest are discrete rendering existing inference methods invalid for obtaining uniform confidence bands for observed and counterfactual quantile functions and for their difference -- the quantile effects functions., Comment: 38 pages, 6 figures, 4 tables

The partial (ceteris paribus) effects of interest in nonlinear and interactive linear models are heterogeneous as they can vary dramatically with the underlying observed or unobserved covariates. Despite the apparent importance of heterogeneity, a common practice in modern empirical work is to largely ignore it by reporting average partial effects (or, at best, average effects for some groups). While average effects provide very convenient scalar summaries of typical effects, by definition they fail to reflect the entire variety of the heterogeneous effects. In order to discover these effects much more fully, we propose to estimate and report sorted effects -- a collection of estimated partial effects sorted in increasing order and indexed by percentiles. By construction the sorted effect curves completely represent and help visualize the range of the heterogeneous effects in one plot. They are as convenient and easy to report in practice as the conventional average partial effects. They also serve as a basis for classification analysis, where we divide the observational units into most or least affected groups and summarize their characteristics. We provide a quantification of uncertainty (standard errors and confidence bands) for the estimated sorted effects and related classification analysis, and provide confidence sets for the most and least affected groups. The derived statistical results rely on establishing key, new mathematical results on Hadamard differentiability of a multivariate sorting operator and a related classification operator, which are of independent interest. We apply the sorted effects method and classification analysis to demonstrate several striking patterns in the gender wage gap., Comment: 62 pages, 9 figures, 8 tables, includes appendix with supplementary materials