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The density weighted average derivative (DWAD) of a regression function is a canonical parameter of interest in economics. Classical first-order large sample distribution theory for kernel-based DWAD estimators relies on tuning parameter restrictions and model assumptions that imply an asymptotic linear representation of the point estimator. These conditions can be restrictive, and the resulting distributional approximation may not be representative of the actual sampling distribution of the statistic of interest. In particular, the approximation is not robust to bandwidth choice. Small bandwidth asymptotics offers an alternative, more general distributional approximation for kernel-based DWAD estimators that allows for, but does not require, asymptotic linearity. The resulting inference procedures based on small bandwidth asymptotics were found to exhibit superior finite sample performance in simulations, but no formal theory justifying that empirical success is available in the literature. Employing Edgeworth expansions, this paper shows that small bandwidth asymptotic approximations lead to inference procedures with higher-order distributional properties that are demonstrably superior to those of procedures based on asymptotic linear approximations.

Yurinskii's coupling is a popular theoretical tool for non-asymptotic distributional analysis in mathematical statistics and applied probability, offering a Gaussian strong approximation with an explicit error bound under easily verifiable conditions. Originally stated in $\ell^2$-norm for sums of independent random vectors, it has recently been extended both to the $\ell^p$-norm, for $1 \leq p \leq \infty$, and to vector-valued martingales in $\ell^2$-norm, under some strong conditions. We present as our main result a Yurinskii coupling for approximate martingales in $\ell^p$-norm, under substantially weaker conditions than those previously imposed. Our formulation further allows for the coupling variable to follow a more general Gaussian mixture distribution, and we provide a novel third-order coupling method which gives tighter approximations in certain settings. We specialize our main result to mixingales, martingales, and independent data, and derive uniform Gaussian mixture strong approximations for martingale empirical processes. Applications to nonparametric partitioning-based and local polynomial regression procedures are provided, alongside central limit theorems for high-dimensional martingale vectors., Comment: 57 pages, 1 figure

In the context of treatment effect estimation, this paper proposes a new methodology to recover the counterfactual distribution when there is a single (or a few) treated unit and possibly a high-dimensional number of potential controls observed in a panel structure. The methodology accommodates, albeit does not require, the number of units to be larger than the number of time periods (high-dimensional setup). As opposed to modeling only the conditional mean, we propose to model the entire conditional quantile function (CQF) without intervention and estimate it using the pre-intervention period by a l1-penalized regression. We derive non-asymptotic bounds for the estimated CQF valid uniformly over the quantiles. The bounds are explicit in terms of the number of time periods, the number of control units, the weak dependence coefficient (beta-mixing), and the tail decay of the random variables. The results allow practitioners to re-construct the entire counterfactual distribution. Moreover, we bound the probability coverage of this estimated CQF, which can be used to construct valid confidence intervals for the (possibly random) treatment effect for every post-intervention period. We also propose a new hypothesis test for the sharp null of no-effect based on the Lp norm of deviation of the estimated CQF to the population one. Interestingly, the null distribution is quasi-pivotal in the sense that it only depends on the estimated CQF, Lp norm, and the number of post-intervention periods, but not on the size of the post-intervention period. For that reason, critical values can then be easily simulated. We illustrate the methodology by revisiting the empirical study in Acemoglu, Johnson, Kermani, Kwak and Mitton (2016).

Factor and sparse models are two widely used methods to impose a low-dimensional structure in high-dimensions. However, they are seemingly mutually exclusive. We propose a lifting method that combines the merits of these two models in a supervised learning methodology that allows for efficiently exploring all the information in high-dimensional datasets. The method is based on a flexible model for high-dimensional panel data, called factor-augmented regression model with observable and/or latent common factors, as well as idiosyncratic components. This model not only includes both principal component regression and sparse regression as specific models but also significantly weakens the cross-sectional dependence and facilitates model selection and interpretability. The method consists of several steps and a novel test for (partial) covariance structure in high dimensions to infer the remaining cross-section dependence at each step. We develop the theory for the model and demonstrate the validity of the multiplier bootstrap for testing a high-dimensional (partial) covariance structure. The theory is supported by a simulation study and applications.

Optimal pricing, i.e., determining the price level that maximizes profit or revenue of a given product, is a vital task for the retail industry. To select such a quantity, one needs first to estimate the price elasticity from the product demand. Regression methods usually fail to recover such elasticities due to confounding effects and price endogeneity. Therefore, randomized experiments are typically required. However, elasticities can be highly heterogeneous depending on the location of stores, for example. As the randomization frequently occurs at the municipal level, standard difference-in-differences methods may also fail. Possible solutions are based on methodologies to measure the effects of treatments on a single (or just a few) treated unit(s) based on counterfactuals constructed from artificial controls. For example, for each city in the treatment group, a counterfactual may be constructed from the untreated locations. In this paper, we apply a novel high-dimensional statistical method to measure the effects of price changes on daily sales from a major retailer in Brazil. The proposed methodology combines principal components (factors) and sparse regressions, resulting in a method called Factor-Adjusted Regularized Method for Treatment evaluation (\texttt{FarmTreat}). The data consist of daily sales and prices of five different products over more than 400 municipalities. The products considered belong to the \emph{sweet and candies} category and experiments have been conducted over the years of 2016 and 2017. Our results confirm the hypothesis of a high degree of heterogeneity yielding very different pricing strategies over distinct municipalities.