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The asymptotic bias and variance of a general class of local polynomial estimators of M-regression functions are studied over the whole compact support of the multivariate covariate under a minimal assumption on the support. The support assumption ensures that the vicinity of the boundary of the support will be visited by the multivariate covariate. The results show that like in the univariate case, multivariate local polynomial estimators have good bias and variance properties near the boundary. For the local polynomial regression estimator, we establish its asymptotic normality near the boundary and the usual optimal uniform convergence rate over the whole support. For local polynomial quantile regression, we establish a uniform linearization result which allows us to obtain similar results to the local polynomial regression. We demonstrate both theoretically and numerically that with our uniform results, the common practice of trimming local polynomial regression or quantile estimators to avoid “the boundary effect” is not needed.

Local polynomial regression is extremely popular in applied settings. Recent developments in shape-constrained nonparametric regression allow practitioners to impose constraints on local polynomial estimators thereby ensuring that the resulting estimates are consistent with underlying theory. However, it turns out that local polynomial derivative estimates may fail to coincide with the analytic derivative of the local polynomial regression estimate which can be problematic, particularly in the context of shape-constrained estimation. In such cases, practitioners might prefer to instead use analytic derivatives along the lines of those proposed in the local constant setting by Rilstone and Ullah (1989). Demonstrations and applications are considered.

We provide straightforward new nonparametric methods for testing conditional independence using local polynomial quantile regression, allowing weakly dependent data. Inspired by Hausman's (1978) specification testing ideas, our methods essentially compare two collections of estimators that converge to the same limits under correct specification (conditional independence) and that diverge under the alternative. To establish the properties of our estimators, we generalize the existing nonparametric quantile literature not only by allowing for dependent heterogeneous data but also by establishing a weak consistency rate for the local Bahadur representation that is uniform in both the conditioning variables and the quantile index. We also show that, despite our nonparametric approach, our tests can detect local alternatives to conditional independence that decay to zero at the parametric rate. Our approach gives the first nonparametric tests for time-series conditional independence that can detect local alternatives at the parametric rate. Monte Carlo simulations suggest that our tests perform well in finite samples. We apply our test to test for a key identifying assumption in the literature on nonparametric, nonseparable models by studying the returns to schooling.

This chapter develops a novel bootstrap procedure to obtain robust bias-corrected confidence intervals in regression discontinuity (RD) designs. The procedure uses a wild bootstrap from a second-order local polynomial to estimate the bias of the local linear RD estimator; the bias is then subtracted from the original estimator. The bias-corrected estimator is then bootstrapped itself to generate valid confidence intervals (CIs). The CIs generated by this procedure are valid under conditions similar to Calonico, Cattaneo, and Titiunik’s (2014) analytical correction – that is, when the bias of the naive RD estimator would otherwise prevent valid inference. This chapter also provides simulation evidence that our method is as accurate as the analytical corrections and we demonstrate its use through a reanalysis of Ludwig and Miller’s (2007) Head Start dataset.

A regression kink design (RKD or RK design) can be used to identify casual effects in settings where the regressor of interest is a kinked function of an assignment variable. In this chapter, we apply an RKD approach to study the effect of unemployment benefits on the duration of joblessness in Austria, and discuss implementation issues that may arise in similar settings, including the use of bandwidth selection algorithms and bias-correction procedures. Although recent developments in nonparametric estimation (Calonico, Cattaneo, & Farrell, 2014; Imbens & Kalyanaraman, 2012) are sometimes interpreted by practitioners as pointing to a default estimation procedure, we show that in any given application different procedures may perform better or worse. In particular, Monte Carlo simulations based on data-generating processes that closely resemble the data from our application show that some asymptotically dominant procedures may actually perform worse than “sub-optimal” alternatives in a given empirical application.

Regression discontinuity designs have become popular in empirical studies due to their attractive properties for estimating causal effects under transparent assumptions. Nonetheless, most popular procedures assume i.i.d. data, which is unreasonable in many common applications. To fill this gap, we derive the properties of traditional local polynomial estimators in a fixed- G setting that allows for cluster dependence in the error term. Simulation results demonstrate that accounting for clustering in the data while selecting bandwidths may lead to lower MSE while maintaining proper coverage. We then apply our cluster-robust procedure to an application examining the impact of Low-Income Housing Tax Credits on neighborhood characteristics and low-income housing supply.

In this paper, we study a partially linear dynamic panel data model with fixed effects, where either exogenous or endogenous variables or both enter the linear part, and the lagged-dependent variable together with some other exogenous variables enter the nonparametric part. Two types of estimation methods are proposed for the first-differenced model. One is composed of a semiparametric GMM estimator for the finite-dimensional parameter θ and a local polynomial estimator for the infinite-dimensional parameter m based on the empirical solutions to Fredholm integral equations of the second kind, and the other is a sieve IV estimate of the parametric and nonparametric components jointly. We study the asymptotic properties for these two types of estimates when the number of individuals N tends to ∞ and the time period T is fixed. We also propose a specification test for the linearity of the nonparametric component based on a weighted square distance between the parametric estimate under the linear restriction and the semiparametric estimate under the alternative. Monte Carlo simulations suggest that the proposed estimators and tests perform well in finite samples. We apply the model to study the relationship between intellectual property right (IPR) protection and economic growth, and find that IPR has a non-linear positive effect on the economic growth rate.

This chapter reviews recent developments in the density discontinuity approach. It is well known that agents having perfect control of the forcing variable will invalidate the popular regression discontinuity designs (RDDs). To detect the manipulation of the forcing variable, McCrary (2008) developed a test based on the discontinuity in the density around the threshold. Recent papers have noted that the sorting patterns around the threshold are often either the researcher’s object of interest or may relate to structural parameters such as tax elasticities through known functions. This, in turn, implies that the behavior of the distribution around the threshold is not only informative of the validity of a standard RDD; it can also be used to recover policy-relevant parameters and perform counterfactual exercises.

The purpose of this paper is to use the local correlation technique to measure flight to quality, which is defined as a pronounced and generally rapid increase in risk aversion. Flight to quality between American, British, German, Japanese, and Hong Kong spot equity indices and index futures is examined.

The technique of non‐linear local correlation is employed to detect flight to quality in both spot and futures markets. The use of this methodology allows us to properly process both normally or non‐normally distributed time series. In addition, the estimation of local correlation minimizes the theoretical restrictions resulting from the selection of conditional events and the use of linear regression.

As market risk grows, an increase in flight to quality is documented. For example, a crash in the US stock market results in the flight of capital to the Treasury bond market. Evidence of flight to quality from domestic and foreign spot equity markets to US Treasury bonds is provided. Furthermore, flights to quality from domestic and foreign index futures to US bond futures are revealed. The strength of the reaction from one market to the other is measured and reported. Surprisingly, the authors observe that when market risk becomes extremely high, flight to quality diminishes.

To the best of the authors' knowledge, this is the first study that examines flight to quality in the futures markets by applying local correlation analysis. This study broadens the application of local polynomial regression and local correlation analysis.