In this article, I propose a method to estimate the counterfactual distribution of an outcome variable when the treatment is endogenous, continuous, and its effect is heterogeneous. The types of counterfactuals considered are those in which the change in treatment intensity can be correlated with the individual effects or when some of the structural functions are changed by some other group’s counterparts. I characterize the outcome and the treatment with a triangular system of equations in which the unobservables are related by a copula that captures the endogeneity of the treatment, which is nonparametrically identified by inverting the quantile processes that determine the outcome and the treatment. Both processes are estimated using existing quantile regression methods, and I propose a parametric and a nonparametric estimator of the copula. To illustrate these methods, I estimate several counterfactual distributions of the birth weight of children, had their mothers smoked differently during pregnancy. 

In this paper, we assess how different need-based grant assignment rules target various students and how they impact their performance in the first year of university, including dropout rates. To do so, we first predict individual outcomes with a fixed-amount grant and without it. Then, we conduct counterfactual analyses using different rules while maintaining a constant number of grants. These rules fall into two categories: those based on efficiency, that target students with significant performance improvements due to the grant, and those based on merit, that focus on high performing students, \textit{i.e}. those with the smallest probability of dropping out if they are awarded a grant. Using administrative data from all public Italian universities, we find that the first type of rules achieves the maximum reduction in dropout rates of the entire population. In contrast, the second type of rules minimizes dropout rates amongst grant recipients, at the cost of increasing the overall number of dropouts.

We study the effect of the COVID-19 pandemic during the first semester of 2020 on the labor market outcomes of elderly workers, using data from the Survey of Health, Ageing and Retirement in Europe (SHARE). We measure the gender gap in the conditional mean of the probability of experiencing a job interruption, of changing the number of hours worked, and of working from home. We control for a rich set of observable characteristics, including several measures of cognitive and non-cognitive ability. We apply decomposition methods to distinguish, on the one hand, the part of the gap that is due to gender differences in the endowments of the determinants of the outcome in question and, on the other, to gender differences in the effects of these determinants. We find that there is no gender gap in the probability of experiencing a job interruption nor in the probability of working fewer hours than before the pandemic. In contrast, there were significant differences in the probability of increasing the amount of worked hours or working remotely, which were larger for females in both cases. For the latter variable, the difference is largely attributable to different endowments between men and women. However, the gap in the probability of working longer hours is mostly attributable to the coefficients component.

I study the identification and estimation of a nonseparable triangular model with an endogenous binary treatment. I impose neither rank invariance nor rank similarity on the unobservable term of the outcome equation. Identification is achieved by using continuous variation of the instrument and a shape restriction on the distribution of the unobservables, which is modeled with a copula. The latter captures the endogeneity of the model and is one of the components of the marginal treatment effect, making it informative about the effects of extending the treatment to untreated individuals. The estimation is a multi-step procedure based on rotated quantile regression. Finally, I use the estimator to revisit the effects of Work First Job Placements on future earnings.

In a binary choice panel data framework, probabilities of the outcomes of several individuals depend on the correlation of the unobserved heterogeneity. I propose a random effects estimator that models the correlation of the unobserved heterogeneity among individuals in the same cluster using a copula. I discuss the asymptotic efficiency of the estimator relative to standard random effects estimators, and to choose the copula I propose a specification test. The implementation of the estimator requires the numerical approximation of high-dimensional integrals, for which I propose an algorithm that works for Archimedean copulas that does not suffer from the curse of dimensionality. This method is illustrated with an application of labor supply in married couples, finding that about one half of the difference in probability of a woman being employed when her husband is also employed, relative to those whose husband is unemployed, is explained by correlation in the unobservables.

Existing estimates of the public/private wage gap allow for possible sorting of individuals into one sector, but they rely on parametric assumptions that may introduce substantial bias in the parameter of interest. Solutions are semi and nonparametric approaches. For Italy, the latter methods yield a gap of approximately 20-21%, whereas the bias from parametric assumptions is as large as 10%.

In this paper I study the manipulation of test scores in the Italian education system. Using an experiment consisting in the random assignment of external monitors to classrooms, I apply a new methodology to study the nature and extent of manipulation of test scores at different levels in primary and secondary education, and I propose a correction method. The results show frequent manipulation, which is not associated with an increase in the correlation of the answers after I control for mean test scores. The manipulation is concentrated in the South and Islands region, and it tends to favor female and immigrant students. Finally, the negative correlation between the amount of manipulation and the number of missing answers in open ended questions relative to multiple choice questions suggests that teachers are more responsible for the manipulation than students.

I present a method to jointly estimate social spillovers in the classroom and the distributions of teacher and student effects. This method builds on Graham (2008) and is based on the covariance and higher order moments restrictions of the test scores and requires the random assignment of teachers and students to classrooms. Using the Tennessee Project STAR dataset, I find sizable spillovers in kindergarten classrooms and departures from normality of the teacher and student ability distributions. I also find that reducing class size has a positive effect on mean performance but it increases the inequality. Based on these estimates, I perform several input-neutral policy counterfactuals involving teachers and students assignment rules, and changing the distribution of class sizes. For the latter, I derive an optimal class size distribution rule, which increases mean test scores and reduces the overall variance.

This paper addresses computational challenges in estimating Quantile Regression with Selection (QRS). The estimation of the parameters that model self-selection requires the estimation of the entire quantile process several times. Moreover, closed-form expressions of the asymptotic variance are too cumbersome, making the bootstrap more convenient to perform inference. Taking advantage from recent advancements in quantile regression methods, along with some specific characteristics of the QRS estimator, I propose streamlined algorithms for the QRS estimator. These algorithms significantly reduce computation time through preprocessing techniques and quantile grid reduction for the estimation of the copula parameter. The proposed methods demonstrate improved precision without sacrificing computational efficiency, offering practical solutions for estimators with a non-differentiable and non-convex criterion functions such as those based on copulas. Finally, I show the optimization enhancements with several simulations.

The selection of employees in the Italian central bank is conducted through a competitive exam. In this paper I analyze its effectiveness in selecting the most able candidates and whether there is gender discrimination. To accomplish this, a multi-equation model is employed, which combines the scores of different exam questions, the choices made by candidates regarding which questions to answer, and individual unobserved heterogeneity. The results indicate that the exam performs well in filtering out less able candidates, as those who progress to subsequent stages tend to exhibit higher levels of ability compared to the initial pool of applicants. Moreover, a measure of the unobserved ability of hired candidates tends to be positively correlated to work performance. Furthermore, there is no evidence to suggest that the decline in the proportion of women who pass the exam, relative to the number of female applicants, can be attributed to discrimination. Finally, I run some simulations showing how certain modifications to the exam structure could potentially enhance the selection process by increasing the average ability of the selected candidates.

I address the decomposition of the differences between the distribution of outcomes of two groups when individuals self-select themselves into participation. I differentiate between the decomposition for participants and the entire population, highlighting how the primitive components of the model affect each of the distributions of outcomes. Additionally, I introduce two ancillary decompositions that help uncover the sources of differences in the distribution of unobservables and participation between the two groups. The estimation is done using existing quantile regression methods, for which I show how to perform uniformly valid inference. I illustrate these methods by revisiting the gender wage gap, finding that changes in female participation and self-selection have been the main drivers for reducing the gap.

Researchers interested in the estimation of peer and network effects, even if these are algebraically identified, still need to address the problem of correlated effects. In this paper we characterize the identification conditions for consistently estimating all the parameters of a spatially autoregressive or linear-in-means model when the structure of social or peer effects is exogenous, but the observed and unobserved characteristics of agents are cross-correlated over some given metric space. We show that identification is possible if the network of social interactions is non-overlapping up to enough degrees of separation, and the spatial matrix that characterizes the co-dependence of individual unobservables and peers’ characteristics is known up to a multiplicative constant. We propose a GMM approach for the estimation of the model’s parameters, and we evaluate its performance through Monte Carlo simulations. Finally, we show that in a classical empirical application about classmates our approach might estimate statistically nonsignificant peer effects when conventional approaches register them as significant.

In many contexts, individual interact with each other, creating interdependence in their final choices. Standard random effects models assume independence among individuals, which leads to inconsistent estimates of the probability of joint and conditional events. I propose a random effects estimator in which there is dependence among the unobserved heterogeneity of individuals connected in a general network. The dependence is modeled with a parametric copula, and allows to consistently estimate the probability of joint and conditional events with a statistically coherent model. The estimator is computed by maximizing a pairwise composite likelihood function, which requires bidimensional numerical integration, which works for a well-defined class of copulas.

This paper analyzes the distributional effect of class size on academic achievement using the data and the empirical design in Angrist and lavy (1999), thereby extending their results. To do so, instrumental variable quantile regression is used. The instrument takes advantage of  discontinuities in the rule that determines class size in Israel. This way one can see the effect that this variable has at different quantiles of the distribution, hence taking into account heterogeneity in the effects. Then a counterfactual distributions estimation method under endogeneity is proposed. To do so, one needs to rearrange estimated quantile curves in order to have the monotonicity property of these curves, which allows to obtain the adjusted quantile of each observation. The results found show that class size has in general, but not always, a negative effect on  grades, and this affects almost the entire distribution. Moreover, the counterfactual analysis shows that decreasing the marginal class size would lead to an increase in class grades for almost the entire distribution in most cases.

Il lavoro descrive gli assetti di contrattazione collettiva più adottati dalle piccole imprese dell’industria, valutandone le ricadute in termini di livelli e dispersione dei salari, sia nel complesso sia nelle diverse aree del Paese. Sebbene le retribuzioni mediane nei contratti dell’artigianato siano più contenute di quelle dei contratti Confindustria, la differenza nei livelli retributivi tra i due tipi di contratto è minima per le retribuzioni basse, potenziale punto di ingresso per una persona in cerca di impiego. Nel 2019, il primo decile della distribuzione dei salari dei contratti artigianato era dell’11,5% inferiore rispetto a quello dei contratti Confindustria; il divario cresce monotonicamente lungo la distribuzione e raggiunge il massimo al nono decile (17%). Tale tendenza è ancor più marcata nel Mezzogiorno, dove le retribuzioni basse mostrano uno scostamento particolarmente contenuto e inferiore rispetto a quello registrato nelle altre aree del Paese. Nel complesso, i risultati sottolineano la presenza di una soglia minima sotto la quale le retribuzioni non vengono fissate, pur in assenza di un salario minimo legale in Italia.