# Smoothed Maximum Score Estimation of Discrete Duration Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modelling

## 3. The Estimator

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

**Assumption**

**5.**

**Assumption**

**6.**

**Assumption**

**7.**

**Assumption**

**8.**

**Assumption**

**9.**

**Assumption**

**10.**

**Assumption**

**11.**

**Lemma**

**1.**

**Proposition**

**1.**

## 4. Simulation Exercise

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of Lemma 1 (a).**

**Lemma**

**A1.**

**Proof**

**of**

**Lemma**

**A1.**

**Proof**

**of**

**Proposition**

**1.**

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1 | |

2 | Some normalization of the parameter space is necessary. We find it most convenient to impose a unit coefficient on ${X}_{is}^{*}$ immediately. |

3 | The model is easily reformulated to incorporate functions of the ${Y}_{j}$’s, $j\le s$ as conditioning variables. |

4 | For example, Cameron and Heckman (1998) defined S as the upper limit to years of education. In practice, for programming purposes, it suffices to set S equal to the longest duration in the dataset being used. In the simulations reported in Section 4, the maximum duration was 37. |

5 | This has two aspects: one is that it implies that estimates of the other $\beta $’s are all to scale and that we know the sign of the first coefficient. |

6 | In this case, the random sampling assumption should be interpreted as referring to the stochastic elements of ${x}_{s}$. |

7 | |

8 | This corresponds to Manski for $S=1$. |

No. of Observations | Spec (1) | Spec (2) | ||
---|---|---|---|---|

Normal Error | Normal, Heteroscedastic Error | |||

500 | 1000 | 500 | 1000 | |

Using second order kernel | ||||

True value | 1.000 | 1.000 | 1.000 | 1.000 |

Estimates | ||||

Mean | 1.013 | 0.982 | 1.034 | 1.001 |

Standard dev. | 0.114 | 0.081 | 0.094 | 0.063 |

RMSE | 0.115 | 0.083 | 0.100 | 0.063 |

Skewness | 0.452 | 0.481 | 0.491 | 0.308 |

Kurtosis | 3.167 | 3.305 | 4.226 | 3.652 |

Using normal cdf as continution probability | ||||

True value | 1.000 | 1.000 | 1.000 | 1.000 |

Estimates | ||||

Mean | 1.017 | 1.003 | 0.937 | 0.939 |

Standard dev. | 0.093 | 0.032 | 0.063 | 0.045 |

RMSE | 0.094 | 0.032 | 0.090 | 0.076 |

Skewness | 0.260 | 0.114 | 0.163 | −0.082 |

Kurtosis | 2.712 | 2.924 | 2.900 | 2.970 |

No. of Observations | Spec (1) | Spec (2) | ||
---|---|---|---|---|

Normal Error | Normal, Heteroscedastic Error | |||

500 | 1000 | 500 | 1000 | |

Using second order kernel | ||||

True value | 2.000 | 2.000 | 2.000 | 2.000 |

Estimates | ||||

Mean | 2.359 | 2.112 | 1.737 | 1.790 |

Standard dev. | 3.426 | 2.356 | 1.854 | 1.340 |

RMSE | 3.441 | 2.356 | 1.871 | 1.355 |

Skewness | 0.126 | 0.181 | −0.280 | −0.065 |

Kurtosis | 4.233 | 3.986 | 8.149 | 4.617 |

Using normal cdf as continution probability | ||||

True value | 2.000 | 2.000 | 2.000 | 2.000 |

Estimates | ||||

Mean | 2.577 | 2.343 | 1.813 | 1.633 |

Standard dev. | 1.544 | 1.010 | 0.830 | 0.623 |

RMSE | 1.647 | 1.066 | 0.850 | 0.722 |

Skewness | 0.126 | −0.042 | 0.304 | 0.433 |

Kurtosis | 3.150 | 3.092 | 2.894 | 3.344 |

No. of Observations | Spec (1) | Spec (2) | ||
---|---|---|---|---|

Normal Error | Normal, Heteroscedastic Error | |||

500 | 1000 | 500 | 1000 | |

Using second order kernel | ||||

True value | −1.000 | −1.000 | −1.000 | −1.000 |

Estimates | ||||

Mean | −2.147 | −1.554 | 0.685 | 0.042 |

Standard dev. | 14.302 | 9.805 | 6.804 | 3.911 |

RMSE | 14.334 | 9.810 | 7.003 | 4.043 |

Skewness | 0.182 | −3.846 | 2.400 | 1.283 |

Kurtosis | 7.844 | 43.237 | 21.338 | 8.266 |

Using normal cdf as continution probability | ||||

True value | −1.000 | −1.000 | −1.000 | −1.000 |

Estimates | ||||

Mean | −4.032 | −2.678 | −1.435 | −0.922 |

Standard dev. | 6.113 | 3.609 | 1.798 | 1.273 |

RMSE | 6.819 | 3.977 | 1.848 | 1.274 |

Skewness | −0.929 | −0.655 | −1.135 | −1.576 |

Kurtosis | 4.467 | 4.051 | 5.005 | 9.012 |

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## Share and Cite

**MDPI and ACS Style**

Reza, S.; Rilstone, P.
Smoothed Maximum Score Estimation of Discrete Duration Models. *J. Risk Financial Manag.* **2019**, *12*, 64.
https://doi.org/10.3390/jrfm12020064

**AMA Style**

Reza S, Rilstone P.
Smoothed Maximum Score Estimation of Discrete Duration Models. *Journal of Risk and Financial Management*. 2019; 12(2):64.
https://doi.org/10.3390/jrfm12020064

**Chicago/Turabian Style**

Reza, Sadat, and Paul Rilstone.
2019. "Smoothed Maximum Score Estimation of Discrete Duration Models" *Journal of Risk and Financial Management* 12, no. 2: 64.
https://doi.org/10.3390/jrfm12020064