Convergence of Probability Measures. By Patrick Billingsley. Pp. xii, 253. 117s. 1968. (John Wiley & Sons, Inc.)

This paper provides an introduction and “user guide” to Regression Discontinuity (RD) designs for empirical researchers. It presents the basic theory behind the research design, details when RD is likely to be valid or invalid given economic incentives, explains why it is considered a “quasi-experimental” design, and summarizes different ways (with their advantages and disadvantages) of estimating RD designs and the limitations of interpreting these estimates. Concepts are discussed using examples drawn from the growing body of empirical research using RD. ( JEL C21, C31)

/pdf/regression-discontinuity-designs-in-economics-5foiogfcmd.pdf

Regression discontinuity designs in economics

We investigate conditions sufficient for identification of average treatment effects using instrumental variables. First we show that the existence of valid instruments is not sufficient to identify any meaningful average treatment effect. We then establish that the combination of an instrument and a condition on the relation between the instrument and the participation status is sufficient for identification of a local average treatment effect for those who can be induced to change their participation status by changing the value of the instrument. Finally we derive the probability limit of the standard IV estimator under these conditions. It is seen to be a weighted average of local average treatment effects.

Evolution and Rationality Some Recent Game-Theoretic Results. Identification and Estimation of Local Average Treatment Effects

https://sticerd.lse.ac.uk/dps/em/em577.pdf

Dynamic panels with threshold effect and endogeneity

This article explores estimation and inference in a regression kink model with an unknown threshold. A regression kink model (or continuous threshold model) is a threshold regression constrained to be everywhere continuous with a kink at an unknown threshold. We present methods for estimation, to test for the presence of the threshold, for inference on the regression parameters, and for inference on the regression function. A novel finding is that inference on the regression function is nonstandard since the regression function is a nondifferentiable function of the parameters. We apply recently developed methods for inference on nondifferentiable functions. The theory is illustrated by an application to the growth and debt problem introduced by Reinhart and Rogoff, using their long-span time-series for the United States.

/pdf/regression-kink-with-an-unknown-threshold-jpc7pqbfiy.pdf

Regression Kink With an Unknown Threshold

Regression discontinuity designs with unknown discontinuity points: Testing and estimation

Threshold Regression with Endogeneity

This article studies computation, estimation, inference, and testing for linearity in threshold regression with a threshold boundary. We first put forward a new algorithm to ease the computation of...

Threshold Regression With a Threshold Boundary

This paper develops a general procedure to check the bootstrap validity in M-estimation. We apply the procedure in discontinuous threshold regression to show the inconsistency of the nonparametric bootstrap for inference on the threshold point. Especially, the conditional weak limit of the nonparametric bootstrap is shown not to exist. By comparing with two other boundaries in the literature, we show the fact that the threshold point is a boundary of the covariate that makes its bootstrap inference so different. The remedies to the bootstrap failure in the literature are summarized, and the nonparametric posterior interval is suggested by some simulation studies.

/pdf/the-bootstrap-in-threshold-regression-4nnn9h7133.pdf

The Bootstrap in Threshold Regression

In nonparametric regression, the derivative estimation has attracted much attention in recent years due to its wide applications. In this paper, we propose a new method for the derivative estimation using the locally weighted least absolute deviation regression. Different from the local polynomial regression, the proposed method does not require a finite variance for the error term and so is robust to the presence of heavy-tailed errors. Meanwhile, it does not require a zero median or a positive density at zero for the error term in comparison with the local median regression. We further show that the proposed estimator with random difference is asymptotically equivalent to the (infinitely) composite quantile regression estimator. The proposed method is also extended to estimate the derivatives at the boundaries and to estimate the higher-order derivatives. For the equidistant design, we derive theoretical results for the proposed estimators, including the asymptotic bias and variance, consistency, and asymptotic normality. Finally, we conduct simulation studies to demonstrate that the proposed method has better performance than the existing methods in the presence of outliers and heavy-tailed errors.

/pdf/robust-estimation-of-derivatives-using-locally-weighted-4r28ye2zk3.pdf

Robust Estimation of Derivatives Using Locally Weighted Least Absolute Deviation Regression

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