A NONPARAMETRIC TEST OF SIGNIFICANT VARIABLES IN GRADIENTS

We propose a nonparametric test of significant variables in the partial derivative of a regression mean function. The derivative is estimated by local polynomial estimation and the test statistic is constructed through a variation-based measure of the derivative in the direction of variables of interest. We establish the asymptotic null distribution of the test statistic and demonstrate that it is consistent. Motivated by the null distribution, we propose a wild bootstrap test, and show that it exhibits the same null distribution, whether the null is valid or not. We perform a Monte Carlo study to demonstrate its encouraging finite sample performance. An empirical application is conducted showing how the test can be applied to infer certain aspects of regression structures in a hedonic price model.

Optimal bandwidth choice for robust bias-corrected inference in regression discontinuity designs

Summary Modern empirical work in regression discontinuity (RD) designs often employs local polynomial estimation and inference with a mean square error (MSE) optimal bandwidth choice. This bandwidth yields an MSE-optimal RD treatment effect estimator, but is by construction invalid for inference. Robust bias-corrected (RBC) inference methods are valid when using the MSE-optimal bandwidth, but we show that they yield suboptimal confidence intervals in terms of coverage error. We establish valid coverage error expansions for RBC confidence interval estimators and use these results to propose new inference-optimal bandwidth choices for forming these intervals. We find that the standard MSE-optimal bandwidth for the RD point estimator is too large when the goal is to construct RBC confidence intervals with the smaller coverage error rate. We further optimize the constant terms behind the coverage error to derive new optimal choices for the auxiliary bandwidth required for RBC inference. Our expansions also establish that RBC inference yields higher-order refinements (relative to traditional undersmoothing) in the context of RD designs. Our main results cover sharp and sharp kink RD designs under conditional heteroskedasticity, and we discuss extensions to fuzzy and other RD designs, clustered sampling, and pre-intervention covariates adjustments. The theoretical findings are illustrated with a Monte Carlo experiment and an empirical application, and the main methodological results are available in R and Stata packages.

Power calculations for regression-discontinuity designs

In this article, we introduce two commands, rdpow and rdsampsi, that conduct power calculations and survey sample selection when using local polynomial estimation and inference methods in regression-discontinuity designs. rdpow conducts power calculations using modern robust bias-corrected local polynomial inference procedures and allows for new hypothetical sample sizes and bandwidth selections, among other features. rdsampsi uses power calculations to compute the minimum sample size required to achieve a desired level of power, given estimated or user-supplied bandwidths, biases, and variances. Together, these commands are useful when devising new experiments or surveys in regression-discontinuity designs, which will later be analyzed using modern local polynomial techniques for estimation, inference, and falsification. Because our commands use the communitycontributed (and R) package rdrobust for the underlying bandwidths, biases, and variances estimation, all the options currently available in rdrobust can also be used for power calculations and sample-size selection, including preintervention covariate adjustment, clustered sampling, and many bandwidth selectors. Finally, we also provide companion R functions with the same syntax and capabilities.

Local polynomial estimation of time-dependent diffusion parameter for discretely observed SDE models

Extending the results of Yu, Yu, Wang and Lin [10], we study the local polynomial estimation of the time-dependent diffusion parameter for time-inhomogeneous diffusion models. Considering the diffusion parameter being positive, we obtain the local polynomial estimation of the diffusion parameter by taking the diffusion parameter to be local log-polynomial fitting. The asymptotic bias, asymptotic variance and asymptotic normal distribution of the volatility function are discussed. A real data analysis is conducted to show the performance of the estimations proposed.

Two-Stage Local Polynomial Regression Method for Image Heteroscedastic Noise Removal

In this paper, we introduce the extension of local polynomial fitting to the linear heteroscedastic regression model and its applications in digital image heteroscedastic noise removal. For better image noise removal with heteroscedastic energy, firstly, the local polynomial regression is applied to estimate heteroscedastic function, then the coefficients of regression model are obtained by using generalized least squares method. Due to non-parametric technique of local polynomial estimation, we do not need to know the heteroscedastic noise function. Therefore, we improve the estimation precision, when the heteroscedastic noise function is unknown. Numerical simulations results show that the proposed method can improve the image quality of heteroscedastic noise energy.