Regression discontinuity designs with unknown state-dependent discontinuity points: estimation and testing
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Lixiong Yang
Abstract
This paper extends Regression discontinuity designs with unknown discontinuity points developed by (Porter, J., and P. Yu. 2015. “Regression Discontinuity Designs with Unknown Discontinuity Points: Testing and Estimation.” Journal of Econometrics 189: 132–147.) to allow for state-dependent discontinuity points. We discuss the estimation of the model, and propose test statistics for treatment effect and state dependency in the discontinuity points. We conduct Monte Carlo simulations to compare the proposed estimator with these based on the constant discontinuity RDD and the classic fuzzy RDD, and find that overlooking the state dependency can lead to biased estimates of treatment effects, while the proposed estimator works well and is robust when applied to constant discontinuity RDDs. Monte Carlo experiments also point out that the sizes and powers of the proposed test statistics are generally satisfactory. The model is illustrated with an empirical application.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 71803072
Funding statement: National Natural Science Foundation of China, Funder Id: 10.13039/501100001809, Grant Number: 71803072.
Acknowledgement
The author thanks two anonymous referees and the editor for very valuable comments and suggestions on previous versions of the paper. Remaining errors are my own.
Appendix
Robustness of the simulation results
This Appendix discusses the robustness of the simulation results to different choices of kernel and bandwidth. We compare the simulation results based on the uniform and triangular kernels, and examine the sensitivity of the simulation results to the choice of bandwidth by choosing the bandwidth h in an ad-hoc way, setting h = 0.1,0.2 and 0.5.
Table 6–Table 8 report the simulation results based on different choices of kernel and bandwidth. Table 6 presents the summary statistics (i.e. mean and standard deviation) for the estimates by applying the proposed estimation procedure with state-dependent discontinuity points. Table 7 reports the summary statistics (i.e. mean and standard deviation) for the parameter estimates based on the Porter and Yu’s (2015) procedure assuming a constant discontinuity point. In Table 8, we compare the proposed estimator with the classical fuzzy RDD estimator. These results support that the simulation results are not sensitive to the choices of kernel and bandwidth. And thus, these simulation results confirm the robustness of the main findings in the paper.
Estimates of the parameters using the estimation procedure proposed in Section 2.
| Treatment effect discontinuity point | Coefficients | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| bandwidth | Mean | Std.dev | Mean | Std.dev | Mean | Std.dev | Mean | Std.dev | Mean | Std.dev | |
| Panel A: Uniform kernel | |||||||||||
| C = 1 | a0 | α = a0 | γ0 = 1 | γ1 = 0 | β0 = 1 | β1 = 1 | |||||
| h = 0.1 | 0.5 | 0.504 | 0.083 | 1.003 | 0.095 | 0.000 | 0.013 | 1.017 | 0.322 | 0.981 | 0.354 |
| h = 0.2 | 0.5 | 0.498 | 0.055 | 1.000 | 0.000 | 0.000 | 0.000 | 0.996 | 0.099 | 1.005 | 0.118 |
| h = 0.5 | 0.5 | 0.499 | 0.034 | 1.000 | 0.000 | 0.000 | 0.000 | 1.001 | 0.021 | 1.000 | 0.029 |
| Entire sample | 0.5 | 0.500 | 0.022 | 1.000 | 0.000 | 0.000 | 0.000 | 1.000 | 0.009 | 1.000 | 0.009 |
| C | a0 | α = a0 | γ0 = 1 | γ1 = 0.5 | β0 = 1 | β1 = 1 | |||||
| h = 0.1 | 0.5 | 0.498 | 0.042 | 1.000 | 0.111 | 0.500 | 0.021 | 0.999 | 0.048 | 1.002 | 0.038 |
| h = 0.2 | 0.5 | 0.499 | 0.030 | 1.000 | 0.000 | 0.500 | 0.000 | 1.002 | 0.031 | 1.000 | 0.027 |
| h = 0.5 | 0.5 | 0.500 | 0.023 | 1.000 | 0.000 | 0.500 | 0.000 | 1.001 | 0.016 | 0.999 | 0.015 |
| Entire sample | 0.5 | 0.500 | 0.021 | 1.000 | 0.000 | 0.500 | 0.000 | 1.000 | 0.009 | 1.000 | 0.008 |
| Panel B: Triangular kernel | |||||||||||
| C = 1 | a0 | α = a0 | γ0 = 1 | γ1 = 0 | β0 = 1 | β1 = 1 | |||||
| h = 0.1 | 0.5 | 0.498 | 0.087 | 1.000 | 0.045 | 0.000 | 0.022 | 0.991 | 0.407 | 1.011 | 0.441 |
| h = 0.2 | 0.5 | 0.497 | 0.062 | 1.000 | 0.000 | 0.000 | 0.000 | 0.996 | 0.127 | 1.005 | 0.152 |
| h = 0.5 | 0.5 | 0.502 | 0.037 | 1.000 | 0.000 | 0.000 | 0.000 | 0.998 | 0.038 | 1.000 | 0.028 |
| Entire sample | 0.5 | 0.500 | 0.023 | 1.000 | 0.000 | 0.000 | 0.000 | 0.999 | 0.009 | 1.000 | 0.010 |
| C = 1 + 0.5S | a0 | α = a0 | γ0 = 1 | γ1 = 0.5 | β0 = 1 | β1 = 1 | |||||
| h = 0.1 | 0.5 | 0.501 | 0.048 | 1.000 | 0.000 | 0.500 | 0.000 | 0.997 | 0.055 | 1.001 | 0.047 |
| h = 0.2 | 0.5 | 0.500 | 0.035 | 1.000 | 0.000 | 0.500 | 0.000 | 1.002 | 0.036 | 0.999 | 0.031 |
| h = 0.5 | 0.5 | 0.501 | 0.025 | 1.000 | 0.000 | 0.500 | 0.000 | 1.001 | 0.019 | 0.999 | 0.018 |
| Entire sample | 0.5 | 0.500 | 0.021 | 1.000 | 0.000 | 0.500 | 0.000 | 1.000 | 0.009 | 1.000 | 0.008 |
The simulations were written in the GAUSS programming language. The number of replications is 5000.
Estimates of the parameters using the estimation procedure assuming a constant discontinuity point.
| Treatment effect discontinuity point | Coefficients | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| bandwidth | Mean | Std.dev | Mean | Std.dev | Mean | Std.dev | Mean | Std.dev | Mean | Std.dev | |
| Panel A: Uniform kernel | |||||||||||
| C = 1 | a0 | α = a0 | γ0 = 1 | γ1 = 0 | β0 = 1 | β1 = 1 | |||||
| h = 0.1 | 0.5 | 0.501 | 0.082 | 1.000 | 0.000 | – | – | 1.001 | 0.328 | 0.994 | 0.357 |
| h = 0.2 | 0.5 | 0.499 | 0.058 | 1.000 | 0.000 | – | – | 0.998 | 0.104 | 1.002 | 0.125 |
| h = 0.5 | 0.5 | 0.500 | 0.035 | 1.000 | 0.000 | – | – | 1.000 | 0.022 | 1.000 | 0.031 |
| Entire sample | 0.5 | 0.500 | 0.023 | 1.000 | 0.000 | – | – | 1.000 | 0.008 | 1.000 | 0.006 |
| C = 1 + 0.5S | a0 | α = a0 | γ0 = 1 | γ1 = 0.5 | β0 = 1 | β1 = 1 | |||||
| h = 0.1 | 0.5 | 0.043 | 0.116 | 1.028 | 0.256 | – | – | 0.892 | 0.492 | 1.353 | 0.536 |
| h = 0.2 | 0.5 | 0.023 | 0.109 | 1.041 | 0.272 | – | – | 0.949 | 0.267 | 1.319 | 0.241 |
| h = 0.5 | 0.5 | 0.022 | 0.121 | 1.093 | 0.284 | – | – | 0.993 | 0.146 | 1.234 | 0.106 |
| Entire sample | 0.5 | 0.189 | 0.173 | 1.033 | 0.279 | – | – | 1.002 | 0.020 | 1.018 | 0.048 |
| Panel B: Triangular kernel | |||||||||||
| C = 1 | a0 | α = a0 | γ0 = 1 | γ1 = 0 | β0 = 1 | β1 = 1 | |||||
| h = 0.1 | 0.5 | 0.502 | 0.090 | 1.000 | 0.000 | – | – | 1.003 | 0.411 | 0.976 | 0.449 |
| h = 0.2 | 0.5 | 0.499 | 0.065 | 1.000 | 0.000 | – | – | 0.999 | 0.136 | 1.002 | 0.162 |
| h = 0.5 | 0.5 | 0.500 | 0.039 | 1.000 | 0.000 | – | – | 1.000 | 0.028 | 1.000 | 0.039 |
| Entire sample | 0.5 | 0.500 | 0.024 | 1.000 | 0.000 | – | – | 1.000 | 0.011 | 1.000 | 0.010 |
| C = 1 + 0.5S | a0 | α = a0 | γ0 = 1 | γ1 = 0.5 | β0 = 1 | β1 = 1 | |||||
| h = 0.1 | 0.5 | 0.035 | 0.119 | 1.023 | 0.263 | – | – | 0.911 | 0.515 | 1.259 | 0.523 |
| h = 0.2 | 0.5 | 0.025 | 0.119 | 1.055 | 0.288 | – | – | 0.956 | 0.276 | 1.267 | 0.222 |
| h = 0.5 | 0.5 | 0.033 | 0.111 | 1.071 | 0.277 | – | – | 0.990 | 0.161 | 1.241 | 0.115 |
| Entire sample | 0.5 | 0.083 | 0.183 | 1.115 | 0.283 | – | – | 1.015 | 0.070 | 1.151 | 0.073 |
The simulations were written in the GAUSS programming language. The number of replications is 5000.
Estimation of treatment effect based on state-dependent RDDs and classical fuzzy RDDs.
| State-dependent RDD | Fuzzy RDD | State-dependent RDD | Fuzzy RDD | ||||||
|---|---|---|---|---|---|---|---|---|---|
| bandwidth | a0 | Mean | Std.dev | Mean | Std.dev | Mean | Std.dev | Mean | Std.dev |
| Panel A: Uniform kernel | |||||||||
| C = 1 | C = 1 + 0.1S | ||||||||
| h = 0.1 | 0.5 | 0.499 | 0.081 | 0.500 | 0.129 | 0.502 | 0.046 | 0.053 | 38.158 |
| h = 0.2 | 0.5 | 0.500 | 0.056 | 0.501 | 0.088 | 0.500 | 0.037 | 0.189 | 24.075 |
| h = 0.5 | 0.5 | 0.499 | 0.035 | 0.501 | 0.053 | 0.500 | 0.031 | 0.497 | 0.135 |
| Entire sample | 0.5 | 0.499 | 0.026 | 0.501 | 0.038 | 0.500 | 0.027 | 0.500 | 0.055 |
| C = 1 + 0.5S | C = 1 + S | ||||||||
| h = 0.1 | 0.5 | 0.501 | 0.041 | 0.061 | 34.355 | 0.500 | 0.041 | 0.235 | 53.988 |
| h = 0.2 | 0.5 | 0.500 | 0.031 | 0.295 | 22.581 | 0.502 | 0.031 | 0.907 | 50.517 |
| h = 0.5 | 0.5 | 0.500 | 0.023 | 0.356 | 20.323 | 0.500 | 0.021 | 0.347 | 46.394 |
| Entire sample | 0.5 | 0.503 | 0.032 | 0.381 | 9.438 | 0.501 | 0.015 | 0.411 | 46.207 |
| Panel B: Triangular kernel | |||||||||
| C = 1 | C = 1 + 0.1S | ||||||||
| h = 0.1 | 0.5 | 0.498 | 0.091 | 0.499 | 0.136 | 0.501 | 0.059 | 0.033 | 43.112 |
| h = 0.2 | 0.5 | 0.501 | 0.063 | 0.500 | 0.093 | 0.498 | 0.049 | 0.606 | 22.145 |
| h = 0.5 | 0.5 | 0.499 | 0.056 | 0.499 | 0.058 | 0.500 | 0.036 | 0.488 | 0.858 |
| Entire sample | 0.5 | 0.500 | 0.023 | 0.501 | 0.041 | 0.500 | 0.023 | 0.499 | 0.069 |
| C = 1 + 0.5S | C = 1 + S | ||||||||
| h = 0.1 | 0.5 | 0.500 | 0.048 | 0.312 | 17.657 | 0.499 | 0.048 | 0.133 | 55.676 |
| h = 0.2 | 0.5 | 0.500 | 0.034 | 0.678 | 22.479 | 0.500 | 0.034 | 1.601 | 52.811 |
| h = 0.5 | 0.5 | 0.500 | 0.025 | 0.326 | 15.918 | 0.500 | 0.023 | 1.347 | 52.333 |
| Entire sample | 0.5 | 0.500 | 0.021 | 0.411 | 13.277 | 0.500 | 0.018 | 0.232 | 55.222 |
The simulations were written in the GAUSS programming language. The number of replications is 5000.
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Supplementary Material
The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/snde-2017-0059).
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