An Exponential Endogenous Switching Regression with Correlated Random Coefficients
Abstract
:1. Introduction
2. Exponential Regime Switching CRC Model
- (i)
- (ii)
3. Estimating the Average Treatment Effects
4. Specification Tests
4.1. Tests for Endogeneity of Treatment
4.2. Model Selection Test
5. Monte Carlo Simulations
5.1. Data-Generating Processes
5.2. Simulation Results
6. An Application: Oregon’s Health Insurance Experiment
7. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
1 | One can use the copula method to generate a multivariate distribution in which marginal effects are scaled distribution. However, the copula method was not used here, since we only want to have pairs of errors with some predetermined correlation without further specifying the dependence structure. Dependence can also be created by using some common standard normal random variables from which two different chi-squared random variables are constructed. We obtained the desired correlation by changing the “parameters” through trial and error. Codes will be provided upon request. |
2 | The intercept values are sufficiently large to prevent any negative dependent variable. In 10,000 replications, there are at most one or two negative dependent variables. They were dropped and the effect is negligible. |
3 | Terza (2009) assumes the joint normality of (). On the contrary, my model does not assume the jointicity. By linear projection, and is orthogonal or independent with v. The only requirement is the normality of and . However, the CLT makes the approximation more accurate as the number of covariates increases. My model presupposes the existence of covariates with CRC. Therefore, the more appropriate the model is, the more accurate the approximation is. Finally, according to the simulation results for errors in Table 1 and Table 2, under large samples, the performances of CRC model is better than the linear model. So, again, when the model is appropriate, it outperforms the preexisting linear method. |
4 |
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L-CRC | E-nCRC | E-CRC | L-CRC | E-nCRC | E-CRC | ||||
---|---|---|---|---|---|---|---|---|---|
n = 5000 | bias | 0.133 | 0.024 | 0.003 | 0.189 | 0.056 | 0.039 | ||
mc. st. dev | 0.454 | 0.485 | 0.658 | 0.454 | 0.460 | 0.590 | |||
RMSE | 0.473 | 0.486 | 0.658 | 0.492 | 0.463 | 0.591 | |||
median | 0.863 | 0.953 | 0.965 | 0.811 | 0.942 | 0.963 | |||
MAD | 0.320 | 0.317 | 0.371 | 0.302 | 0.303 | 0.309 | |||
IR | 1.138 | 1.197 | 1.471 | 1.146 | 1.117 | 1.368 | |||
n = 10,000 | bias | 0.137 | 0.024 | 0.006 | 0.190 | 0.044 | 0.013 | ||
mc. st. dev | 0.318 | 0.325 | 0.394 | 0.328 | 0.319 | 0.370 | |||
RMSE | 0.347 | 0.326 | 0.394 | 0.379 | 0.322 | 0.370 | |||
median | 0.879 | 0.980 | 0.986 | 0.815 | 0.959 | 0.963 | |||
MAD | 0.215 | 0.219 | 0.252 | 0.221 | 0.208 | 0.231 | |||
IR | 0.797 | 0.819 | 0.932 | 0.829 | 0.775 | 0.916 |
L-CRC | E-nCRC | E-CRC | L-CRC | E-nCRC | E-CRC | ||||
---|---|---|---|---|---|---|---|---|---|
n = 5000 | bias | 0.257 | 0.017 | 0.188 | 0.303 | 0.020 | 0.262 | ||
mc. st. dev | 0.607 | 0.577 | 0.964 | 0.638 | 0.574 | 0.932 | |||
RMSE | 0.659 | 0.577 | 0.982 | 0.706 | 0.574 | 0.968 | |||
median | 0.763 | 0.928 | 0.998 | 0.763 | 0.998 | 1.125 | |||
MAD | 0.385 | 0.346 | 0.462 | 0.442 | 0.383 | 0.472 | |||
IR | 1.528 | 1.499 | 2.060 | 1.661 | 1.508 | 1.810 | |||
n = 10,000 | bias | 0.240 | 0.046 | 0.085 | 0.318 | 0.019 | 0.128 | ||
mc. st. dev | 0.386 | 0.373 | 0.543 | 0.441 | 0.384 | 0.500 | |||
RMSE | 0.454 | 0.375 | 0.550 | 0.544 | 0.384 | 0.517 | |||
median | 0.753 | 0.923 | 1.009 | 0.694 | 0.967 | 1.062 | |||
MAD | 0.281 | 0.256 | 0.309 | 0.279 | 0.235 | 0.302 | |||
IR | 1.005 | 0.933 | 1.300 | 1.081 | 0.951 | 1.154 |
L-CRC | E-nCRC | E-CRC | L-CRC | E-nCRC | E-CRC | ||||
---|---|---|---|---|---|---|---|---|---|
n = 5000 | bias | 0.012 | 0.016 | 0.028 | 0.003 | 0.021 | 0.021 | ||
mc. st. dev | 0.283 | 0.290 | 0.311 | 0.275 | 0.289 | 0.298 | |||
RMSE | 0.283 | 0.290 | 0.312 | 0.275 | 0.290 | 0.299 | |||
median | 1.026 | 1.011 | 1.030 | 1.028 | 1.007 | 1.035 | |||
MAD | 0.200 | 0.186 | 0.193 | 0.175 | 0.185 | 0.200 | |||
IR | 0.733 | 0.716 | 0.773 | 0.706 | 0.701 | 0.726 | |||
n = 10,000 | bias | 0.020 | 0.022 | 0.001 | 0.021 | 0.033 | 0.002 | ||
mc. st. dev | 0.219 | 0.195 | 0.243 | 0.214 | 0.197 | 0.228 | |||
RMSE | 0.220 | 0.196 | 0.243 | 0.215 | 0.200 | 0.228 | |||
median | 0.987 | 1.026 | 1.013 | 0.988 | 1.032 | 1.010 | |||
MAD | 0.145 | 0.129 | 0.157 | 0.143 | 0.135 | 0.153 | |||
IR | 0.552 | 0.498 | 0.593 | 0.529 | 0.518 | 0.575 |
Variable | Mean | Std. Dev. | Min | Max |
---|---|---|---|---|
vafter | 0.997 | 2.410 | 0 | 22 |
vbefore | 0.774 | 1.863 | 0 | 17 |
famsize | 1.210 | 0.409 | 1 | 3 |
OHP | 0.241 | 0.427 | 0 | 1 |
lottary | 0.391 | 0.488 | 0 | 1 |
Panel A. | Panel B. | Panel C. | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
OLS | 2SLS | E-nCRC | E-CRC | OLS | 2SLS | E-nCRC | E-CRC | OLS | 2SLS | E-nCRC | E-CRC | |||
0.445 *** | 0.342 *** | 0.329 | 0.524 | 0.428 *** | 0.330 *** | 0.450 | 0.175 | 0.427 *** | 0.300 *** | 0.168 | −0.079 | |||
(0.027) | (0.097) | (1.129) | (3.085) | (0.026) | (0.093) | (0.519) | (1.734) | (0.025) | (0.089) | (0.389) | (2.118) | |||
R1: | 0.728 *** | 0.731 *** | 0.272 *** | 0.313 * | 0.721 *** | 0.724 *** | 0.311 *** | 0.380 * | 0.688 *** | 0.693 *** | 0.390 *** | 0.066 | ||
(0.009) | (0.009) | (0.010) | (0.167) | (0.010) | (0.010) | (0.012) | (0.228) | (0.011) | (0.012) | (0.017) | (0.285) | |||
−0.252 *** | −0.249 *** | −0.329 *** | −0.417 | −0.251 *** | −0.248 *** | −0.384 *** | −0.372 | −0.250 *** | −0.246 *** | −0.406 *** | −2.172 | |||
(0.028) | (0.028) | (0.084) | (1.825) | (0.027) | (0.027) | (0.076) | (1.728) | (0.025) | (0.025) | (0.077) | (1.837) | |||
−0.240 *** | −0.367 *** | −0.788 * | ||||||||||||
(0.064) | (0.089) | (0.449) | ||||||||||||
−1.111 | −1.004 | −1.657 * | ||||||||||||
(0.897) | (0.794) | (0.875) | ||||||||||||
R0: | 0.333 *** | 0.728 *** | 0.408 *** | 0.586 ** | 0.507 *** | 0.863 *** | ||||||||
(0.010) | (0.140) | (0.011) | (0.287) | (0.015) | (0.302) | |||||||||
−0.435 *** | −0.089 | −0.374 *** | −0.507 | −0.418 *** | −2.915 * | |||||||||
(0.067) | (2.567) | (0.072) | (2.442) | (0.060) | (1.700) | |||||||||
0.022 | −0.058 | −0.035 | ||||||||||||
(0.095) | (0.139) | (0.143) | ||||||||||||
−0.863 | −1.361 | −1.523 | ||||||||||||
(1.065) | (1.360) | (1.190) |
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Keay, M.-J. An Exponential Endogenous Switching Regression with Correlated Random Coefficients. Econometrics 2022, 10, 1. https://doi.org/10.3390/econometrics10010001
Keay M-J. An Exponential Endogenous Switching Regression with Correlated Random Coefficients. Econometrics. 2022; 10(1):1. https://doi.org/10.3390/econometrics10010001
Chicago/Turabian StyleKeay, Myoung-Jin. 2022. "An Exponential Endogenous Switching Regression with Correlated Random Coefficients" Econometrics 10, no. 1: 1. https://doi.org/10.3390/econometrics10010001