## 1. Introduction

## 2. Heteroskedasticity and Likelihood Function

#### 2.1. Different Sources of Heteroskedasticity

- Heteroskedasticity a la Mazodier and Trognon (1978): The heteroskedasticity is due to the individual effects. Thus, ${\mu}_{i}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}iid(0,{\sigma}_{{\mu}_{i}}^{2})$ and ${\nu}_{it}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}iid(0,{\sigma}_{\nu}^{2})$.
- Heteroskedasticity a la Baltagi (1988) and Wansbeek (1989): the heteroskedasticity is due to the idiosyncratic errors. Thus, ${\mu}_{i}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}iid(0,{\sigma}_{\mu}^{2})$ and ${\nu}_{it}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}iid(0,{\sigma}_{{\nu}_{it}}^{2})$.
- Heteroskedasticity a la Randolph (1988): The heteroskedasticity is due to both the individual effects and the idiosyncratic errors. Thus, ${\mu}_{i}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}iid(0,{\sigma}_{{\mu}_{i}}^{2})$ and ${\nu}_{it}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}iid(0,{\sigma}_{{\nu}_{it}}^{2})$. An alternative specification by Verbon (1980) is to consider that ${\mu}_{i}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}iid(0,{\sigma}_{{\mu}_{i}}^{2})$ and ${\nu}_{it}\phantom{\rule{0.166667em}{0ex}}\sim \phantom{\rule{0.166667em}{0ex}}iid(0,{\sigma}_{{\nu}_{i}}^{2})$.

#### 2.2. Likelihood Function

## 3. Estimation and Tests

#### 3.1. Estimation Requirements

#### 3.2. Test Procedure

## 4. Monte Carlo Experiments

#### 4.1. Power and Empirical Size of the Test

#### 4.2. Bias and Mean Square Error of the Estimates

#### 4.3. Robustness of Validity

## 5. Additional Robustness Checks

#### 5.1. Application Examples and Comparisons

#### 5.2. Quadrature Points Check

#### 5.3. Misspecified Heteroskedasticity: Effects of Applying the Wrong Approach

## 6. Case Study

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. STATA Code for Computing the Marginal Effects

## Appendix B. STATA Code for Generating the Dataset

## Appendix C. STATA Code for Monte Carlo Experiments

## Appendix D. Power of the Test for Different Degrees of Heteroskedasticity

#### Appendix D.1. Testing for the Joint Hypothesis

**Table A1.**Power and size of the LR test based on 5000 replications: case of ${H}_{0}:\phantom{\rule{0.166667em}{0ex}}{\theta}_{\mu}={\theta}_{\nu}=0$.

Setting | ${\mathit{\sigma}}_{\mathit{\mu}}^{2}=2$ | ${\mathit{\sigma}}_{\mathit{\mu}}^{2}=6$ | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{N}=\mathbf{50}$ | $\mathit{N}=\mathbf{500}$ | $\mathit{N}=\mathbf{50}$ | $\mathit{N}=\mathbf{500}$ | ||||||

${\mathit{\theta}}_{\mathit{\mu}}$ | ${\mathit{\theta}}_{\mathit{\nu}}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ |

0 | 0 | 4.48 | 4.88 | 4.44 | 5.42 | 5.58 | 5.52 | 5.6 | 5.6 |

0 | 1 | 12.68 | 91.24 | 99.48 | 100 | 6.6 | 92.24 | 98.42 | 100 |

0 | 2 | 35.88 | 99.92 | 100 | 100 | 31.84 | 100 | 100 | 100 |

0 | 3 | 50.22 | 99.94 | 100 | 100 | 38.42 | 100 | 100 | 100 |

1 | 0 | 22.08 | 39.08 | 99.48 | 100 | 13.36 | 29.3 | 98.46 | 100 |

1 | 1 | 26.56 | 97.56 | 100 | 100 | 14.46 | 90.04 | 99.6 | 100 |

1 | 2 | 46.36 | 100 | 100 | 100 | 49.26 | 100 | 100 | 100 |

1 | 3 | 45.78 | 100 | 100 | 100 | 62.56 | 100 | 100 | 100 |

2 | 0 | 30.79 | 72.34 | 100 | 100 | 21.08 | 43.12 | 100 | 100 |

2 | 1 | 54.14 | 99.26 | 100 | 100 | 23.08 | 75.34 | 100 | 100 |

2 | 2 | 78.28 | 100 | 100 | 100 | 65.62 | 100 | 100 | 100 |

2 | 3 | 79.28 | 100 | 100 | 100 | 87.06 | 100 | 100 | 100 |

3 | 0 | 50.92 | 73.8 | 100 | 100 | 30.08 | 58.46 | 100 | 100 |

3 | 1 | 59.52 | 96.88 | 100 | 100 | 37.72 | 56.46 | 100 | 100 |

3 | 2 | 90.3 | 100 | 100 | 100 | 57.88 | 99.92 | 100 | 100 |

3 | 3 | 95.46 | 100 | 100 | 100 | 93.8 | 100 | 100 | 100 |

#### Appendix D.2. Testing for the Marginal Hypothesis of No Heteroskedasticity in Individual Effects Given Homoskedastic Idiosyncratic Errors

**Table A2.**Power and size of the LR test based on 5000 replications: case of ${H}_{0}:\phantom{\rule{0.166667em}{0ex}}{\theta}_{\mu}=0\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\theta}_{\nu}=0$.

${\mathit{\sigma}}_{\mathit{\mu}}^{2}=2$ | ${\mathit{\sigma}}_{\mathit{\mu}}^{2}=6$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

Setting | $\mathit{N}=\mathbf{50}$ | $\mathit{N}=\mathbf{500}$ | $\mathit{N}=\mathbf{50}$ | $\mathit{N}=\mathbf{500}$ | |||||

$\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | ||

${\theta}_{\mu}$ | |||||||||

0 | 5.26 | 5.46 | 5.06 | 5.02 | 4.42 | 5.08 | 4.76 | 4.72 | |

1 | 24.86 | 47.12 | 99.54 | 100 | 5.88 | 15.76 | 77.84 | 98.68 | |

2 | 35.16 | 71.7 | 100 | 100 | 6.22 | 17.68 | 90.52 | 100 | |

3 | 29.14 | 64.98 | 100 | 100 | 4.94 | 15.28 | 99.32 | 100 | |

${\theta}_{\nu}$ | |||||||||

0 | 5.26 | 5.46 | 5.06 | 5.02 | 4.42 | 5.08 | 4.76 | 4.72 | |

1 | 5.78 | 5.46 | 5.56 | 4.74 | 5.06 | 4.82 | 4.72 | 4.4 | |

2 | 5.26 | 5.06 | 5.58 | 4.46 | 5.56 | 4.62 | 5.02 | 4.42 | |

3 | 5.12 | 4.94 | 4.96 | 4.44 | 5.44 | 4.74 | 4.92 | 4.42 | |

${\theta}_{\mu}$ | ${\theta}_{\nu}$ | ||||||||

0 | 0 | 5.26 | 5.46 | 5.06 | 5.02 | 4.42 | 5.08 | 4.76 | 4.72 |

0 | 1 | 5.78 | 5.46 | 5.56 | 4.74 | 5.06 | 4.82 | 4.72 | 4.4 |

0 | 2 | 5.26 | 5.06 | 5.58 | 4.46 | 5.56 | 4.62 | 5.02 | 4.42 |

0 | 3 | 5.12 | 4.94 | 4.96 | 4.44 | 5.44 | 4.74 | 4.92 | 4.42 |

1 | 0 | 24.86 | 47.12 | 99.54 | 100 | 5.88 | 15.76 | 77.84 | 98.68 |

1 | 1 | 27.2 | 53.2 | 99.6 | 100 | 15.66 | 34.26 | 96.3 | 100 |

1 | 2 | 23.88 | 45.78 | 97.2 | 100 | 22.56 | 42.84 | 98.64 | 100 |

1 | 3 | 14 | 33.56 | 79.58 | 100 | 19.08 | 31.72 | 92.44 | 100 |

2 | 0 | 35.16 | 71.7 | 100 | 100 | 6.22 | 17.68 | 90.52 | 100 |

2 | 1 | 59.46 | 91.98 | 100 | 100 | 24.76 | 59.52 | 100 | 100 |

2 | 2 | 67 | 94.98 | 100 | 100 | 55.48 | 89.68 | 100 | 100 |

2 | 3 | 52.72 | 89.16 | 100 | 100 | 55.9 | 88.36 | 100 | 100 |

3 | 0 | 29.14 | 64.98 | 100 | 100 | 4.94 | 15.28 | 99.32 | 100 |

3 | 1 | 66.46 | 95.96 | 100 | 100 | 21.56 | 59.06 | 100 | 100 |

3 | 2 | 88.42 | 99.8 | 100 | 100 | 65.1 | 97.06 | 100 | 100 |

3 | 3 | 86.9 | 99.66 | 100 | 100 | 83.36 | 99.58 | 100 | 100 |

#### Appendix D.3. Testing for the Marginal Hypothesis of no Heteroskedasticity in Idiosyncratic Errors Given Homoskedastic Individual Effects

**Table A3.**Power and size of the LR test based on 5000 replications: case of ${H}_{0}:\phantom{\rule{0.166667em}{0ex}}{\theta}_{\nu}=0\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\theta}_{\mu}=0$.

${\mathit{\sigma}}_{\mathit{\mu}}^{2}=2$ | ${\mathit{\sigma}}_{\mathit{\mu}}^{2}=6$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

Setting | $\mathit{N}=\mathbf{50}$ | $\mathit{N}=\mathbf{500}$ | $\mathit{N}=\mathbf{50}$ | $\mathit{N}=\mathbf{500}$ | |||||

$\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | $\mathit{T}=\mathbf{5}$ | $\mathit{T}=\mathbf{20}$ | ||

${\theta}_{\nu}$ | |||||||||

0 | 4.64 | 4.48 | 4.44 | 5.1 | 5.56 | 5.6 | 5.6 | 5.6 | |

1 | 20.5 | 95.6 | 99.8 | 100 | 11.12 | 96.62 | 99.5 | 100 | |

2 | 54.28 | 99.98 | 100 | 100 | 52.8 | 100 | 100 | 100 | |

3 | 66.18 | 99.96 | 100 | 100 | 62.46 | 100 | 100 | 100 | |

${\theta}_{\mu}$ | |||||||||

0 | 4.64 | 4.48 | 4.44 | 5.1 | 5.56 | 5.6 | 5.6 | 5.6 | |

1 | 6.6 | 8.2 | 39.06 | 43.18 | 23.96 | 23.98 | 87.02 | 96.4 | |

2 | 16.08 | 23.78 | 77.06 | 96.1 | 15.54 | 34.34 | 99.96 | 99.42 | |

3 | 25.28 | 29.19 | 97.98 | 98.72 | 24.58 | 49.7 | 99.96 | 100 | |

${\theta}_{\mu}$ | ${\theta}_{\nu}$ | ||||||||

0 | 0 | 4.64 | 4.48 | 4.44 | 5.1 | 5.56 | 5.6 | 5.6 | 5.6 |

0 | 1 | 20.5 | 95.6 | 99.8 | 100 | 11.12 | 96.62 | 99.5 | 100 |

0 | 2 | 54.28 | 99.98 | 100 | 100 | 52.8 | 100 | 100 | 100 |

0 | 3 | 66.18 | 99.96 | 100 | 100 | 62.46 | 100 | 100 | 100 |

1 | 0 | 6.6 | 8.2 | 39.06 | 43.18 | 23.96 | 23.98 | 87.02 | 96.4 |

1 | 1 | 11.52 | 96.46 | 99.54 | 100 | 34.2 | 87.7 | 84.38 | 100 |

1 | 2 | 50.92 | 100 | 100 | 100 | 53.16 | 100 | 100 | 100 |

1 | 3 | 60.96 | 100 | 100 | 100 | 76.9 | 100 | 100 | 100 |

2 | 0 | 16.08 | 23.78 | 77.06 | 96.1 | 15.54 | 34.34 | 99.96 | 99.42 |

2 | 1 | 24.22 | 86.38 | 91.18 | 100 | 25.24 | 50.28 | 100 | 100 |

2 | 2 | 45.94 | 100 | 100 | 100 | 34.06 | 99.98 | 100 | 100 |

2 | 3 | 70.08 | 100 | 100 | 100 | 78.88 | 100 | 100 | 100 |

3 | 0 | 25.28 | 29.19 | 97.98 | 98.72 | 24.58 | 49.7 | 99.96 | 100 |

3 | 1 | 33.71 | 56.64 | 98.04 | 99.92 | 32.36 | 59.6 | 100 | 100 |

3 | 2 | 48.88 | 99.98 | 100 | 100 | 51.34 | 99.32 | 100 | 100 |

3 | 3 | 67.52 | 100 | 100 | 100 | 65.98 | 100 | 100 | 100 |

## Appendix E. Application and Comparisons

#### Appendix E.1. Application and Comparison for Data Generated with Individual Effects Heteroskedastic

Variables | $\mathit{DGP}$ | Homoskedastic | Heteroskedastic | Heteroskedastic |
---|---|---|---|---|

Panel Probit | Pooled Probit | Panel Probit | ||

With $(N,T)=(500,5)$ | ||||

$LogL$ | $-1241.7666$ | $-1283.699$ | $-1238.6702$ | |

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | $9.52$ *** | 6.2328 ** | ||

The estimated index function parameters. | ||||

${X}_{1}$ | $0.8$ | $\underset{[0.6008;1.0561]}{0.8285}$ *** | $\underset{[0.6053;1.1348]}{0.87}$ *** | $\underset{[0.6021;1.0557]}{0.8289}$ *** |

${X}_{2}$ | $-2$ | $\underset{[-2.2014;-1.7102]}{-1.9558}$ *** | $\underset{[-2.5151;-1.7146]}{-2.1149}$ *** | $\underset{[-2.2077;-1.7179]}{-1.9628}$ *** |

$intercept$ | $1.5$ | $\underset{[1.2052;1.5922]}{1.3993}$ *** | $\underset{[1.206;1.7702]}{1.4881}$ *** | $\underset{[1.2077;1.5907]}{1.3992}$ *** |

The variance parameters. | ||||

${Z}_{{\mu}_{i}}$ | $0.7$ | $\underset{[0.1485;0.6686]}{0.4086}$ *** | $\underset{[0.1192;1.3103]}{0.7147}$ ** | |

${\lambda}_{0}$ | $-0.8$ | $\underset{[-1.2052;-0.5373]}{-0.8713}$ *** | $\underset{[-1.257;-0.4327]}{-0.8449}$ *** | |

${\sigma}_{\nu}\phantom{\rule{0.166667em}{0ex}}\left(assumed\right)$ | 1 | 1 | 1 | 1 |

With $(N,T)=(500,20)$ | ||||

$LogL$ | $-4500.4005$ | $-4928.991$ | $-4492.8442$ | |

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | $8.81$ *** | $20.5892$ *** | ||

The estimated index function parameters. | ||||

${X}_{1}$ | $0.8$ | $\underset{[0.6114;0.8356]}{0.7235}$ *** | $\underset{[0.5392;0.7671]}{0.6531}$ *** | $\underset{[0.6073;0.8307]}{0.719}$ *** |

${X}_{2}$ | $-2$ | $\underset{[-2.1368;-1.8949]}{-2.0158}$ *** | $\underset{[-1.9884;-1.6852]}{-1.8368}$ *** | $\underset{[-2.1238;-1.8838]}{-2.0038}$ *** |

$intercept$ | $1.5$ | $\underset{[1.4813;1.7012]}{1.5913}$ *** | $\underset{[1.3315;1.5689]}{1.4502}$ *** | $\underset{[1.4651;1.6725]}{1.5688}$ *** |

The variance parameters. | ||||

${Z}_{{\mu}_{i}}$ | $0.7$ | $\underset{[0.0613;0.3028]}{0.182}$ *** | $\underset{[0.39;1.0016]}{0.6953}$ *** | |

${\lambda}_{0}$ | $-0.8$ | $\underset{[-1.0042;-0.6462]}{-0.8252}$ *** | $\underset{[-0.9646;-0.6059]}{-0.7853}$ *** | |

${\sigma}_{\nu}\phantom{\rule{0.166667em}{0ex}}\left(assumed\right)$ | 1 | 1 | 1 | 1 |

#### Appendix E.2. Application and Comparison for Data Generated with Idiosyncratic Errors Heteroskedastic

Variables | $\mathit{DGP}$ | Homoskedastic | Heteroskedastic | Heteroskedastic |
---|---|---|---|---|

Panel Probit | Pooled Probit | Panel Probit | ||

With $(N,T)=(500,5)$ | ||||

$LogL$ | $-1356.3042$ | $-1363.723$ | $-1352.2662$ | |

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | 5.99 ** | $8.0787$ *** | ||

The estimated index function parameters. | ||||

${X}_{1}$ | $0.8$ | $\underset{[0.515;0.9284]}{0.7217}$ *** | $\underset{[0.511;0.9931]}{0.752}$ *** | $\underset{[0.6085;1.167]}{0.8878}$ *** |

${X}_{2}$ | $-2$ | $\underset{[-1.7353;-1.3111]}{-1.5232}$ *** | $\underset{[-2.0197;-1.3763]}{-1.698}$ *** | $\underset{[-2.2846;-1.5283]}{-1.9064}$ *** |

$intercept$ | $1.5$ | $\underset{[0.9026;1.2262]}{1.0644}$ *** | $\underset{[0.9675;1.4364]}{1.2019}$ *** | $\underset{[1.0544;1.599]}{1.3267}$ *** |

The variance parameters. | ||||

${Z}_{{\nu}_{it}}$ | $0.6$ | $\underset{[0.0649;0.6067]}{0.3358}$ *** | $\underset{[0.1352;0.7143]}{0.4247}$ *** | |

${\sigma}_{\mu}$ | $0.45$ | $\underset{[0.2935;0.507]}{0.3857}$ *** | $\underset{[0.3619;0.638]}{0.4805}$ *** | |

With $(N,T)=(500,20)$ | ||||

$LogL$ | $-5230.5929$ | $-5280.282$ | $-5189.8208$ | |

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | $89.12$ *** | $83.3667$ *** | ||

The estimated index function parameters. | ||||

${X}_{1}$ | $0.8$ | $\underset{[0.4553;0.6568]}{0.5561}$ *** | $\underset{[0.5877;0.8665]}{0.7271}$ *** | $\underset{[0.6191;0.9046]}{0.7618}$ *** |

${X}_{2}$ | $-2$ | $\underset{[-1.6786;-1.4698]}{-1.5742}$ *** | $\underset{[-2.2619;-1.8871]}{-2.0745}$ *** | $\underset{[-2.3319;-1.951]}{-2.1414}$ *** |

$intercept$ | $1.5$ | $\underset{[1.1361;1.3086]}{1.2223}$ *** | $\underset{[1.4578;1.7419]}{1.5999}$ *** | $\underset{[1.4917;1.7842]}{1.638}$ *** |

The variance parameters. | ||||

${Z}_{{\nu}_{it}}$ | $0.6$ | $\underset{[0.5062;0.7772]}{0.6417}$ *** | $\underset{[0.4731;0.7329]}{0.603}$ *** | |

${\sigma}_{\mu}$ | $0.45$ | $\underset{[0.3144;0.4007]}{0.3549}$ *** | $\underset{[0.3883;0.4987]}{0.44}$ *** |

#### Appendix E.3. Application and Comparison for Data Generated with Both Individual Effects and Idiosyncratic Heteroskedastic

Variables | $\mathit{DGP}$ | Homoskedastic | Heteroskedastic | Heteroskedastic |
---|---|---|---|---|

Panel Probit | Pooled Probit | Panel Probit | ||

With $(N,T)=(500,5)$ | ||||

$LogL$ | $-1378.8704$ | $-1407.272$ | $-1371.3114$ | |

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | $11.00$ *** | $15.1319$ *** | ||

The estimated index function parameters. | ||||

${X}_{1}$ | $0.8$ | $\underset{[0.4946;0.9175]}{0.7061}$ *** | $\underset{[0.5124;1.1015]}{0.8069}$ *** | $\underset{[0.6023;1.1749]}{0.8886}$ *** |

${X}_{2}$ | $-2$ | $\underset{[-1.6449;-1.2077]}{-1.4263}$ *** | $\underset{[-2.0588;-1.2365]}{-1.6477}$ *** | $\underset{[-2.1511;-1.4324]}{-1.7918}$ *** |

$intercept$ | $1.5$ | $\underset{[0.8499;1.1915]}{1.0207}$ *** | $\underset{[0.8931;1.4836]}{1.1883}$ *** | $\underset{[1.002;1.5365]}{1.2692}$ *** |

The variance parameters. | ||||

${Z}_{{\mu}_{i}}$ | $0.7$ | $\underset{[-0.1847;0.3971]}{0.1062}$ | $\underset{[0.0271;1.2499]}{0.6385}$ ** | |

${\lambda}_{0}$ | $-0.8$ | $\underset{[-1.5411;-0.8204]}{-1.1807}$ *** | $\underset{[-1.2244;-0.3096]}{-0.767}$ *** | |

${Z}_{{\nu}_{it}}$ | $0.6$ | $\underset{[0.1734;0.7628]}{0.4681}$ *** | $\underset{[0.155;0.7411]}{0.448}$ *** | |

With $(N,T)=(500,20)$ | ||||

$LogL$ | $-5245.4829$ | $-5447.011$ | $-5200.2317$ | |

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | $63.51$ *** | $92.7282$ *** | ||

The estimated index function parameters. | ||||

${X}_{1}$ | $0.8$ | $\underset{[0.4553;0.659]}{0.5571}$ *** | $\underset{[0.5743;0.8692]}{0.7218}$ *** | $\underset{[0.6011;0.8804]}{0.7407}$ *** |

${X}_{2}$ | $-2$ | $\underset{[-1.62;-1.4096]}{-1.5148}$ *** | $\underset{[-2.117;-1.6912]}{-1.9041}$ *** | $\underset{[-2.1449;-1.7943]}{-1.9696}$ *** |

$intercept$ | $1.5$ | $\underset{[1.0879;1.271]}{1.1794}$ *** | $\underset{[1.3057;1.6302]}{1.468}$ *** | $\underset{[1.3669;1.6383]}{1.5026}$ *** |

The variance parameters. | ||||

${Z}_{{\mu}_{i}}$ | $0.7$ | $\underset{[-0.0189;0.2485]}{0.1148}$ * | $\underset{[0.4228;1.0902]}{0.7565}$ *** | |

${\lambda}_{0}$ | $-0.8$ | $\underset{[-1.6468;-1.2556]}{-1.4512}$ *** | $\underset{[-1.1258;-0.7013]}{-0.9136}$ *** | |

${Z}_{{\nu}_{it}}$ | $0.6$ | $\underset{[0.4099;0.6896]}{0.5498}$ *** | $\underset{[0.4108;0.6603]}{0.5255}$ *** |

## Appendix F. Estimates for Different Numbers of Quadrature Points

**Table A7.**Changes in Parameters and in log-likelihood with respect to the number of quadrature point Q.

Variables | $\mathit{DGP}$ | $\mathit{Q}=6$ | $\mathit{Q}=8$ | $\mathit{Q}=10$ | $\mathit{Q}=12$ | $\mathit{Q}=14$ | $\mathit{Q}=16$ | $\mathit{Q}=18$ | $\mathit{Q}=20$ |
---|---|---|---|---|---|---|---|---|---|

$LogL$ | $-5233.2859$ | $-5210.0882$ | $-5200.2317$ | $-5195.4211$ | $-5192.3082$ | $-5190.1575$ | $-5189.2627$ | $-5188.4647$ | |

$Wald\phantom{\rule{0.166667em}{0ex}}stat$ | $66.3381$ | $81.5596$ | $94.3733$ | $103.304$ | $109.889$ | $115.235$ | $116.835$ | $117.604$ | |

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | $67.0431$ | $80.8018$ | $92.7282$ | $100.563$ | $106.433$ | $110.666$ | $112.443$ | $114.037$ | |

${X}_{1}$ | $0.8$ | $\underset{[0.5508;0.811]}{0.6809}$ *** | $\underset{[0.579;0.8495]}{0.7143}$ *** | $\underset{[0.6011;0.8804]}{0.7407}$ *** | $\underset{[0.6167;0.9028]}{0.7598}$ *** | $\underset{[0.6292;0.9211]}{0.7751}$ *** | $\underset{[0.6403;0.9375]}{0.7889}$ *** | $\underset{[0.646;0.9464]}{0.7962}$ *** | $\underset{[0.6517;0.9553]}{0.8035}$ *** |

${X}_{2}$ | $-2$ | $\underset{[-1.9899;-1.6612]}{-1.8256}$ *** | $\underset{[-2.0755;-1.7361]}{-1.9058}$ *** | $\underset{[-2.1449;-1.7943]}{-1.9696}$ *** | $\underset{[-2.1978;-1.8365]}{-2.0171}$ *** | $\underset{[-2.2428;-1.8709]}{-2.0568}$ *** | $\underset{[-2.2814;-1.8999]}{-2.0906}$ *** | $\underset{[-2.3029;-1.9145]}{-2.1087}$ *** | $\underset{[-2.3244;-1.9287]}{-2.1265}$ *** |

$Intercept$ | $1.5$ | $\underset{[1.2299;1.4803]}{1.3551}$ *** | $\underset{[1.3098;1.5703]}{1.4401}$ *** | $\underset{[1.3669;1.6383]}{1.5026}$ *** | $\underset{[1.4067;1.6892]}{1.5479}$ *** | $\underset{[1.4384;1.7319]}{1.5852}$ *** | $\underset{[1.4642;1.7681]}{1.6162}$ *** | $\underset{[1.4774;1.7889]}{1.6332}$ *** | $\underset{[1.4901;1.8094]}{1.6497}$ *** |

${Z}_{{\mu}_{i}}$ | $0.7$ | $\underset{[0.4615;1.1903]}{0.8259}$ *** | $\underset{[0.4341;1.1137]}{0.7739}$ *** | $\underset{[0.4228;1.0902]}{0.7565}$ *** | $\underset{[0.4036;1.0696]}{0.7366}$ *** | $\underset{[0.4094;1.0778]}{0.7436}$ *** | $\underset{[0.4241;1.0923]}{0.7582}$ *** | $\underset{[0.425;1.0947]}{0.7598}$ *** | $\underset{[0.4347;1.1046]}{0.7697}$ *** |

${\lambda}_{0}$ | $-0.8$ | $\underset{[-1.3352;-0.8482]}{-1.0917}$ *** | $\underset{[-1.2023;-0.7656]}{-0.984}$ *** | $\underset{[-1.1258;-0.7013]}{-0.9136}$ *** | $\underset{[-1.0733;-0.6474]}{-0.8603}$ *** | $\underset{[-1.0463;-0.6157]}{-0.831}$ *** | $\underset{[-1.0266;-0.5941]}{-0.8104}$ *** | $\underset{[-1.015;-0.5784]}{-0.7967}$ *** | $\underset{[-1.0077;-0.5697]}{-0.7887}$ *** |

${Z}_{{\nu}_{it}}$ | $0.6$ | $\underset{[0.2782;0.5244]}{0.4013}$ *** | $\underset{[0.3529;0.5993]}{0.4761}$ *** | $\underset{[0.4108;0.6603]}{0.5355}$ *** | $\underset{[0.4512;0.7053]}{0.5783}$ *** | $\underset{[0.4842;0.7442]}{0.6142}$ *** | $\underset{[0.5109;0.7761]}{0.6435}$ *** | $\underset{[0.5243;0.7943]}{0.6593}$ *** | $\underset{[0.5372;0.8122]}{0.6747}$ *** |

$\Delta \phantom{\rule{0.166667em}{0ex}}in\phantom{\rule{0.166667em}{0ex}}LogL$ | $0.0044$ | $0.0019$ | $0.0009$ | $0.0006$ | $0.0004$ | $0.0002$ | $0.0002$ | ||

$\Delta \phantom{\rule{0.166667em}{0ex}}in\phantom{\rule{0.166667em}{0ex}}Wald\phantom{\rule{0.166667em}{0ex}}stat$ | $0.2259$ | $0.155$ | $0.0936$ | $0.0632$ | $0.0482$ | $0.0138$ | $0.0065$ | ||

$\Delta \phantom{\rule{0.166667em}{0ex}}in\phantom{\rule{0.166667em}{0ex}}LR\phantom{\rule{0.166667em}{0ex}}stat$ | $0.2022$ | $0.1458$ | $0.0836$ | $0.0578$ | $0.0394$ | $0.0159$ | $0.0141$ | ||

$\Delta \phantom{\rule{0.166667em}{0ex}}in\phantom{\rule{0.166667em}{0ex}}param.$ | $0.162$ | $0.1022$ | $0.0631$ | $0.0335$ | $0.0341$ | $0.0465$ | $0.0533$ | $0.0599$ | |

$Time\phantom{\rule{0.166667em}{0ex}}\left(sec\right)$ | 41 | 53 | 62 | 79 | 96 | 106 | 130 | 133 |

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1. | A user-written Stata’s ado file is provided to deal with these purposes. This ado file is an extension of the existing Stata’s $hetprobit$ and $xtprobit,\phantom{\rule{0.166667em}{0ex}}re$ commands that accounts for each of the types of heteroskedasticity observed in panel one-way error component models in the literature. A Stata code for computing the marginal effects after the proposed estimation procedure is given in the Appendix A. |

2. | The estimation procedure described above has been implemented as a Stata user-written ado file using the Stata’s $d0$ procedure for maximum likelihood estimation (see Gould et al. 2010; Moussa and Delattre 2018). |

3. | For all others applications presented herein, $Q=10$ is used as the number of quadrature points. |

4. | An example of the Stata code for the experiment of the power of the test in presence of heteroskedasticity due to both ${\mu}_{i}$ and ${\nu}_{it}$ with $N=100$ and $T=5$ is provided in the Appendix C. The Appendix B reports the Stata code used to generate the data. |

5. | The empirical size estimated on 5000 replications is significantly different from the nominal size of 5% if it does not range between 4.4% and 5.6%. These thresholds are calculated as $0.05\pm 1.96\sqrt{\frac{0.05\ast 0.95}{5000}}$. |

Settings | ${\mathit{H}}_{0}:\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\mu}}=0\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\nu}}=0$ | ${\mathit{H}}_{0}:\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\nu}}=0\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\mu}}=0$ | ${\mathit{H}}_{0}:\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\mu}}={\mathit{\theta}}_{\mathit{\nu}}=0$ | |
---|---|---|---|---|

Dimensions | Obs. | % | % | % |

Low degree of heteroskedasticity: ${\sigma}_{\mu}^{2}=0.2$, ${\theta}_{\mu}=0.7$ and ${\theta}_{\nu}=0.6$ | ||||

$(N,T)=(50,5)$ | 250 | 12.04 | 23.86 | 14.62 |

$(N,T)=(100,5)$ | 500 | 19.88 | 43.32 | 31.68 |

$(N,T)=(500,5)$ | 2500 | 69.54 | 97.22 | 96.74 |

$(N,T)=(50,10)$ | 500 | 19.9 | 39.74 | 35.16 |

$(N,T)=(100,10)$ | 1000 | 34.04 | 65.62 | 64.72 |

$(N,T)=(500,10)$ | 5000 | 94.14 | 99.98 | 100 |

$(N,T)=(50,20)$ | 1000 | 27.4 | 67.14 | 63.74 |

$(N,T)=(100,20)$ | 2000 | 50.58 | 93.2 | 92.94 |

$(N,T)=(500,20)$ | $10,000$ | 99.26 | 100 | 100 |

High degree of heteroskedasticity: ${\sigma}_{\mu}^{2}=0.2$, ${\theta}_{\mu}=2.1$ and ${\theta}_{\nu}=1.8$ | ||||

$(N,T)=(50,5)$ | 250 | 81.72 | 47.78 | 65.16 |

$(N,T)=(100,5)$ | 500 | 98.44 | 85 | 96.2 |

$(N,T)=(500,5)$ | 2500 | 100 | 100 | 100 |

$(N,T)=(50,10)$ | 500 | 94.72 | 93.7 | 98.12 |

$(N,T)=(100,10)$ | 1000 | 98.88 | 99.9 | 99.96 |

$(N,T)=(500,10)$ | 5000 | 100 | 100 | 100 |

$(N,T)=(50,20)$ | 1000 | 98.36 | 99.98 | 100 |

$(N,T)=(100,20)$ | 2000 | 100 | 100 | 100 |

$(N,T)=(500,20)$ | $10,000$ | 100 | 100 | 100 |

Settings | ${\mathit{H}}_{0}:\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\mu}}=0\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\nu}}=0$ | ${\mathit{H}}_{0}:\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\nu}}=0\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\mu}}=0$ | ${\mathit{H}}_{0}:\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\mu}}={\mathit{\theta}}_{\mathit{\nu}}=0$ | |
---|---|---|---|---|

Dimensions | Obs. | % | % | % |

$(N,T)=(50,5)$ | 250 | 4.82 | 4.98 | 4.54 |

$(N,T)=(100,5)$ | 500 | 4.7 | 5.02 | 4.52 |

$(N,T)=(500,5)$ | 2500 | 4.64 | 5.34 | 4.68 |

$(N,T)=(50,10)$ | 500 | 5.36 | 5.46 | 4.72 |

$(N,T)=(100,10)$ | 1000 | 4.54 | 4.74 | 5.14 |

$(N,T)=(500,10)$ | 5000 | 4.62 | 4.48 | 4.68 |

$(N,T)=(50,20)$ | 1000 | 5.08 | 5.00 | 4.94 |

$(N,T)=(100,20)$ | 2000 | 4.92 | 4.48 | 5.04 |

$(N,T)=(500,20)$ | $10,000$ | 4.58 | 5.54 | 5.04 |

Settings | $(\mathit{N},\mathit{T})=(50,5)$ | $(\mathit{N},\mathit{T})=(50,20)$ | $(\mathit{N},\mathit{T})=(500,5)$ | $(\mathit{N},\mathit{T})=(500,20)$ | |||||
---|---|---|---|---|---|---|---|---|---|

Parameter | DGP | Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE |

Parameters of the index function | |||||||||

${\alpha}_{0}$ | $1.5$ | 0.0009 | 0.2435 | 0.0814 | 0.0554 | 0.0402 | 0.0225 | 0.0606 | 0.0126 |

${\alpha}_{1}$ | $0.8$ | 0.0040 | 0.2474 | 0.0331 | 0.0474 | 0.0237 | 0.0221 | 0.0400 | 0.0062 |

${\alpha}_{2}$ | $-2$ | 0.0072 | 0.4323 | 0.0834 | 0.0814 | 0.0530 | 0.0388 | 0.0909 | 0.0172 |

Parameters of the variances of ${\mu}_{i}$ and ${\nu}_{it}$ | |||||||||

${\lambda}_{0}$ | $-0.8$ | 0.1683 | 1.5406 | 0.0609 | 0.2058 | 0.0517 | 0.0846 | 0.0407 | 0.0177 |

${\theta}_{\mu}$ | $0.7$ | 0.1660 | 1.1061 | 0.0232 | 0.4044 | 0.0412 | 0.1204 | 0.0353 | 0.0316 |

${\theta}_{\nu}$ | $0.6$ | 0.0721 | 0.2369 | 0.0618 | 0.0447 | 0.0456 | 0.0225 | 0.0301 | 0.0119 |

DGP | ${\mathit{H}}_{0}:\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\mu}}=0\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\nu}}=0$ | ${\mathit{H}}_{0}:\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\nu}}=0\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\mu}}=0$ | ${\mathit{H}}_{0}:\phantom{\rule{0.166667em}{0ex}}{\mathit{\theta}}_{\mathit{\mu}}={\mathit{\theta}}_{\mathit{\nu}}=0$ |
---|---|---|---|

Normal | 4.82 | 4.98 | 5.54 |

Student (3) | 6.38 | 7.86 | 8.32 |

Exponential | 17.56 | 7.18 | 21.36 |

Uniform | 5.74 | 7.68 | 8.7 |

Chi-square | 3.24 | 5.8 | 4.5 |

Case | ${\mathit{\mu}}_{\mathit{i}}$ Heteroskedastic | ${\mathit{\nu}}_{\mathbf{it}}$ Heteroskedastic | ${\mathit{\mu}}_{\mathit{i}}$ and ${\mathit{\nu}}_{\mathit{it}}$ Heteroskedastic | |||
---|---|---|---|---|---|---|

Model | (1) | (2) | (3) | (4) | (5) | (6) |

$LogL$ | $-4499.6355$ | $-4492.5752$ | $-5231.2132$ | $-5189.5929$ | $-5235.0043$ | $-5210.8403$ |

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | $7.0066$ *** | $21.1272$ *** | $0.5819$ | $83.8224$ *** | $23.1831$ *** | $71.5111$ *** |

The variance parameters. | ||||||

${Z}_{{\mu}_{i}}$ | $\underset{[0.3888;1.0016]}{0.6952\phantom{\rule{4pt}{0ex}}}$ *** | $\underset{[-0.2497;0.5673]}{0.1588}$ | $\underset{[-0.2652;0.5436]}{0.1392}$ | $\underset{[0.4655;1.1504]}{0.8079}$ *** | ||

${\lambda}_{0}$ | $\underset{[-0.9916;-0.6178]}{-0.8047\phantom{\rule{4pt}{0ex}}}$ *** | $\underset{[-1.3659;-0.8957]}{-1.1308}$ *** | $\underset{[-1.1252;-0.6519]}{-0.8886}$ *** | $\underset{[-1.3771;-0.9513]}{-1.1642}$ *** | ||

${\sigma}_{\mu}$ | $\underset{[0.5369;0.6672]}{0.5985}$ *** | $\underset{[0.5419;0.6669]}{0.6012}$ *** | ||||

${Z}_{{\nu}_{it}}$ | $\underset{[-0.3455;-0.0493]}{-0.1974}$ *** | $\underset{[-0.1627;0.0743]}{-0.0442}$ | $\underset{[0.4726;0.7324]}{0.6025}$ *** | $\underset{[0.4223;0.6734]}{0.5479}$ *** |

Model | (1) | (2) | (3) |
---|---|---|---|

$LogL$ | $-4536.406$ | $-4535.2483$ | $-4535.1225$ |

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | $0.2644$ | $2.5797$ | $2.8313$ |

The variance parameters. | |||

${Z}_{{\mu}_{i}}$ | $\underset{[-0.2406;0.4118]}{0.0856}$ | $\underset{[-0.2428;0.4098]}{0.0835}$ | |

${\lambda}_{0}$ | $\underset{[-0.886;-0.5203]}{-0.7031}$ *** | $\underset{[-0.9357;-0.5548]}{-0.7452}$ *** | |

${\sigma}_{\mu}$ | $\underset{[0.4421;0.5514]}{0.4938}$ *** | ||

${Z}_{{\nu}_{it}}$ | $\underset{[-0.2197;0.0222]}{-0.0987}$ | $\underset{[-0.2194;0.0224]}{-0.0985}$ |

Variables | Homoskedastic Model | Heteroskedastic Model | ||||
---|---|---|---|---|---|---|

$\mathit{Coef}.$ | $\mathit{M}.\mathit{E}{.}^{+}$ | $\mathit{M}.\mathit{E}{.}^{++}$ | $\mathit{Coef}.$ | $\mathit{M}.\mathit{E}{.}^{+}$ | $\mathit{M}.\mathit{E}{.}^{++}$ | |

$age$ | $\underset{[0.0175;0.0228]}{0.0201}$ *** | $\underset{[0.0048;0.0062]}{0.0055}$ *** | $\underset{[0.0061;0.0078]}{0.0069}$ *** | $\underset{[0.0023;0.0032]}{0.0027}$ *** | $\underset{[0.0055;0.0066]}{0.0061}$ *** | $\underset{[0.007;0.0082]}{0.0076}$ *** |

$income$ | $\underset{[-0.1341;0.1278]}{-0.0032}$ | $\underset{[-0.0366;0.0349]}{-0.0009}$ | $\underset{[-0.0463;0.0441]}{-0.0011}$ | $\underset{[-0.0256;0.0237]}{-0.001}$ | $\underset{[-0.0581;0.0157]}{-0.0212}$ | $\underset{[-0.0747;0.0115]}{-0.0316}$ |

$kids$ | $\underset{[-0.2079;-0.0996]}{-0.1538}$ *** | $\underset{[-0.0567;-0.0272]}{-0.0420}$ *** | $\underset{[-0.0717;-0.0344]}{-0.053}$ *** | $\underset{[-0.0433;-0.0238]}{-0.0336}$ *** | $\underset{[-0.064;-0.0353]}{-0.0497}$ *** | $\underset{[-0.0707;-0.039]}{-0.0549}$ *** |

$education$ | $\underset{[-0.0462;-0.0212]}{-0.0337}$ *** | $\underset{[-0.0126;-0.0058]}{-0.0092}$ *** | $\underset{[-0.0159;-0.0073]}{-0.0116}$ *** | $\underset{[-0.0083;-0.0046]}{-0.0065}$ *** | $\underset{[-0.0066;-0.001]}{-0.0038}$ *** | $\underset{[-0.0051;0.0014]}{-0.0018}$ |

$married$ | $\underset{[-0.0477;0.0803]}{0.0163}$ | $\underset{[-0.013;0.0219]}{0.0045}$ | $\underset{[-0.0164;0.0277]}{0.0056}$ | $\underset{[-0.0101;0.011]}{0.0005}$ | $\underset{[-0.0149;0.0163]}{0.0007}$ | $\underset{[-0.0164;0.018]}{0.0008}$ |

$intercept$ | $\underset{[-0.1591;0.2273]}{0.0341}$ | $\underset{[0.0251;0.0864]}{0.0558}$ *** | ||||

The variance parameters: variance of ${\mu}_{i}$ | ||||||

$female$ | $\underset{[-0.1101;-0.0431]}{-0.0766}$ *** | |||||

${\lambda}_{0}$ | $\underset{[-2.1311;-2.0837]}{-2.1074}$ *** | |||||

${\sigma}_{\mu}$ | $\underset{[0.8649;0.9379]}{0.9007}$ *** | |||||

The variance parameters: variance of ${\nu}_{it}$ | ||||||

$age$ | $\underset{[-0.0232;-0.0198]}{-0.0215}$ *** | |||||

$income$ | $\underset{[0.028;0.3916]}{0.2098}$ ** | |||||

$education$ | $\underset{[-0.0691;-0.0529]}{-0.061}$ *** | |||||

$LogL$ | $-16,273.964$ | $-14,019.325$ | ||||

$LR\phantom{\rule{0.166667em}{0ex}}stat$ | $4509.45$ *** |

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