Their encouragement, guidance and ability to spark new ideas have been instrumental to my progress. From sleepovers to hiking adventures and board games, their company added immense joy to my journey.
Introduction
The available results all rely on the random censoring assumption, and are not directly applicable to the copula-based setting. Organization of the article: Section 1.2 introduces the framework for endogenous censoring and the treatment effect parameters.
Setup and Parameter of Interest
Model Framework
In Section 1.6, we illustrate the final sample performance of our proposed estimators and the bootstrap confidence sets via Monte Carlo simulations. Since the observed durationY and the censoring indicator are sparse deterministic functions of T and C, the above assumption immediately implies that (Y1,Y0,R1,R0)⊥⊥D|X.
Parameters of Interest
We also note that since the experimental setting can be considered a special case of Assumption 1.1, all our theories presented below will automatically transfer to the randomized controlled trial setup. The answer depends on two factors: (i) the quantification of the level of dependency censorship and (ii) a link from the censorship mechanism to the policy parameters.
Identification
Single-Index Model
This conditional moment restriction will serve as the basis for the identification of the index parameters. This shows that the index parameters can be recovered as the unique minimizer of the minimum distance type criterion, (1.3.2).
Partial Identification through Copula
As a direct implication of the statement, when the true copula is known, or equivalently when. Each type of treatment effects introduced in Section 1.2 consists of treatment responses that respect the FOSD relations of STd.
Estimation and Large Sample Theory
- Single-Index Parameters
- BGF Estimators
- Uniform Linear Expansion
- Weak Convergence
Similar results for the unconditional case would further depend on the existence of the linear representation. It is the linear representation of the first-order Hoeffding projection of the dominant process U.
Multiplier Bootstrap
Bootstrap Confidence Bands
We provide an algorithm for constructing uniform confidence bands of the overall TEBF estimators in what follows. An analogous procedure that produces uniform confidence bands of the conditional TEBF estimators is given in Section 1.9.1.3.2.
Monte Carlo Study
Notes: The left panel depicts the unconditional BGFs for the control group and the right panel illustrates the overall DTEs. Given the one-to-one mapping between θ and Kendall's τ, the two θθθ choices correspond to Kendall's τ lying between [0,1/3] and [0,1/2], respectively.
Empirical Illustration
In the first step of the analysis, we assess the validity of the index sufficiency assumption. The dashed (dot-dash) lines and light gray area depict the upper (lower) bound of the DTE and the corresponding uniform confidence bands, with a Gumbel copula and θθθ= (1,1.5).
Conclusion
Supplementary Appendix
Proofs of Main Results
- Proof of Results from Section 1.3
- Proof of Results from Section 1.4
- Proofs for Results from Section 1.5
We have shown that the triangular array{fni} satisfies conditions (i) - (v) in Lemma 1.6, which implies that ˆGx†n. Proof of Theorem 1.6. The proof is a direct consequence of Theorem 1.4, Corollary 1.3 and the continuous mapping.
Single-Index Estimator
- Single-Index Kernel Estimator
- Test of Single-Index Assumption
However, their proof is carried out, in this weaker condition, if the maximum inequality from Lemma 1.5 is used in the proof, instead of the main conclusion of Sherman (1994). It is also implicit in that of Theorem 1 in Chiang and Huang (2012), and therefore, is omitted.
Auxiliary Results
- Definitions and Additional Results
- Auxiliary Lemmas
- Covariance Functions
- First-Stage Estimator for the QTE
Combining the bounds and applying H¨older's inequality, we conclude by Lemma 2.13 of Pakes and Pollard (1989) that Mθ is a VC class. From the continuous differentiability of K(1)(·), we derive by similar arguments to Lemma 8.4 in Maistre and Patilea (2019) that.
Introduction
This article belongs to the growing literature on the (marginal) unconditional policy effect. 2009b) introduced the method of unconditional quantile regressions (UQR), the study of unconditional policy effects has received much attention. It is more closely related to the verify-out-of-sample model of Chen et al.
Setup
The first part of Assumption 2.1(d) indicates that ns is of the same order of magnitude as na growth. The second part of Assumption 2.1(d) ensures that the pseudo-true pooled population is not a degenerate one conditional on all possible values of Z.
Identification
- Parameter of Interest
- First Step Identification
- Identification with Continuously Distributed Covariates
- Identification with Discrete Covariate
On the other hand, when X is discretely valued, we show via example 2.2 that assumption 2.3 can be fulfilled with a discrete instrument. Since the rank variables are no longer uniquely fixed by strictly increasing quantile functions, we strengthen Assumption 2.4(a) so that conditional independence holds for all the rank variables in the corresponding class.
Estimation and Inference
Estimation Procedure
Our identification relies on a compact support condition, and it is well known that the Prazen-Rosenblatt density estimator is not valid near the support limit. Now, plugging in the nuisance estimators, U QEj(τ, G) can thus be estimated from,. 2.4.7) We summarize the evaluation procedure in the following algorithm.
Large Sample Results
Finally, due to the estimate of qτ, we impose a finite fourth moment condition in Assumption 2.7(f), which is stronger than the usual quadratic integrability condition. From the linear expansions in Theorem 2.3, we conclude that U QE converges at a rate slower than root-n.
Empirical Illustration
Visual inspection of the actual-experience-specific age-income profiles can serve as a preliminary test of the exclusion restriction. Next, we consider the bias caused by using potential experience instead of actual experience.
Concluding Remarks
We note that one-sample UQE estimates based on IPUMS tend to be smaller in magnitude at lower income quantiles than that based on the combined data.
Appendix
Proofs of Lemmas and Theorems in Section 2.3
The result of Theorem 2.1 then follows from the fact that the Hadamard derivative operator of the quantile functional is linear. Ja,t(qτ), the third comes from the definition of Ues, and a change of variable fraxtou, the fourth equality follows from the construction of Φ∗and assumptions 2.4(a) and (b), the fifth line is again by assumption 2.4(a), the eighth follows from the definition of Us and the standard change-of-variable argument, the tenth line is from assumptions 2.1(a)-(c) and Bayes' law.
Asymptotic Linear Representation of UQE Estimators
Supplementary Appendix
Proofs of Lemmas and Theorems from Main Text
The first term represents the estimation error of fYs|R=1, the second term corresponds to the estimation effect ofθ, and the last term accounts for the contribution of bgp. The second inequality of (2.8.24) follows from a second-order Taylor expansion with respect to π(·), which is valid under Assumption 2.7(e) and Assumption 2.9(a).
Auxiliary Lemmas and Proofs
Since G−1(·) is continuously differentiable with a bounded first-order derivative under assumption 2.8(d), according to Lemma 3.9.25 in Van Der Vaart and Wellner (1996), it is Hadamard differentiable. Given the Hadamard differentiability, we then apply the delta method (as in Theorem 3.9.4 in Van Der Vaart and.
Asymptotic Variance Estimators
Finally, we provide estimators for influence functions related to the estimation of nonparametric first steps. This chapter is taken from the working paper “Difference-in-Differences with Compositional Changes” and reproduced with permission from my co-author Pedro H.
Introduction
To answer this question, we compare our derived semiparametric efficiency frontier, which does not impose the assumption of no composition changes, with the semiparametric efficiency frontier derived by Sant'Anna and Zhao (2020), which makes full use of it. The test compares our non-parametric DiD estimator ATT, which is robust against composition changes, with a non-parametric extension of the DR DiD estimator of Sant'Anna and Zhao (2020), which assumes no composition changes.
Difference-in-Differences
Framework
In Section 3.3, we present our nonparametric DR DiD estimators, discuss their large sample properties, and how to choose tuning parameters. Monte Carlo simulations are provided in Section 3.5 and an empirical illustration is considered in Section 3.6.
Identification and Semiparametric Efficiency Bound
That is, any influence function has zero mean, we can take the expected value of ηeff(W) and isolateτ to get the following estimate for ATT. Note that we can rewrite τdras τorestimand augmented by IPW terms that weight the errors of the regression of Y on X between the subgroups defined by (d,t)∈S−, that is,.
Bias-Variance Trade-Off With Respect To Stationarity
These calculations show that the Sant'Anna and Zhao (2020) DR DiD estimator for ATT can be biased when Assumption 3.3 is violated. Proposition 3.1 also highlights that not all violations of Assumption 3.3 result in biases in ATT using Sant'Anna and Zhao's (2020) estimate.
Estimation and Inference
- Rate Doubly Robust
- Local Polynomial Estimation of Nuisance Functions
- Asymptotic Normality
- Bandwidth Selection
We allow varying local polynomial orders for the PS and OR estimators and, in the case of the latter, for different treatment groups. This algorithm exploits analytical expressions for MISE and bypasses the computational burden of the cross-validation method.
Testing for Compositional Changes
We recommend using this procedure when it is small and the size of the data set is significant. Vb is an estimator of the variance of the difference between the two DiD estimators of ATT.
Monte Carlo Simulation Study
Simulation 1: Non-Stationary Covariate Distribution
Bandwidth for the PS function is selected using the log-likelihood criterion, “ML”, and the least squares criterion, “LS”, respectively. Pow.(α)” stands for the mean test statistic, and empirical power of the test with a nominal size α, respectively.
Simulation 2: Stationary Covariate Distribution
Var.”, “Dec.” and “CIL”, stand for the specification, cross-validation criterion, mean simulated bias, median simulated bias, simulated root-mean-square errors, average of the plug-in estimator for the asymptotic variance, 95% coverage probability and 95% confidence interval length, respectively. Size(α)” stands for the mean test statistic, and empirical size of the test with a nominal sizeα, respectively.
Empirical Illustration: the Effect of Tariff Reduction on Corruption
Following Sequeira (2016), we consider four different outcome measures: a binary variable indicating whether a bribe is paid, the logarithmic form, log(x+1), of the amount of the bribe paid, the logarithmic form of the amount of the bribe paid . as part of the value of the shipment, and as part of the weight of the shipment, respectively. When considering the ratio of bribe payment to tonnage, both nonparametric DR DiD estimators report large but insignificant (at the 95% level) ATT estimates.
Concluding Remarks
Our DR DiD estimates suggest that the magnitudes of the effects are about the same as the original paper, indicating that excluding treatment effect heterogeneity and compositional changes are not of primary concern in this particular application. Allowing for compositional change in that setup seems promising, especially since multiple time periods suggest that an assumption of no compositional change may be even more restrictive than in the simple two-period case.
Supplementary Appendix
Proofs for Results from Main Text
Proof of Proposition 3.1: The proof follows directly from LIE as shown in the main text. The first inequality holds under Assumption 3.2(iii), and the second is a consequence of the Cauchy-Schwarz inequality.
Results on Asymptotic Linear Expansion of Local Polynomial Estimators
- Rates of Convergence: U-Statistics
- Asymptotic Linear Expansion of Local Polynomial Estimators
The three quantities represent the bias, the first-order stochastic part and the remaining terms derived from the decomposition of the PS estimator, respectively. Analogous to the PS case, we use B(or)n,d,t,Sn,d,t(or) and R(or)n,d,t to represent the bias, the first-order stochasticity and the .
Auxiliary Lemmas and Results
- Auxiliary Lemmas
- Mean Integrated Squared Error
- Plug-In Estimators
- Cluster-Robust Inference: Bootstrap Procedures
Because of Lemma 3.6(i), the first term on the right-hand side of the previous equation has a mean value of zero. On the role of propensity score for efficient semiparametric estimation of mean treatment effects.
Monte Carlo results for the conditional and overall TEBFs
Summary statistics
Estimation results for conditional and overall TEBFs
Summary statistics
Estimation results for unconditional quantile effects
Monte Carlo results under compositional changes. Sample size: n = 1,000
Difference-in-differences estimation results for Sequeira (2016)
BGFs and DTEs with multiple levels of θ
Estimates of the potential EFS curves
Distributional treatment effect estimates
Marginal distributional shift and marginal quantile shift
Unconditional quantile effect of actual experience on log(Earnings)
