# Nonparametric Estimation of a Conditional Quantile Function in a Fixed Effects Panel Data Model

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

**STEP**

**1.**

**STEP**

**2.**

- (1)
- Estimate ${\varphi}_{u}\left(t\right)$ by$${\widehat{\varphi}}_{u}\left(t\right)=\frac{1}{NT}\sum _{i=1}^{N}\sum _{s=1}^{T}{e}^{\iota t{\widehat{u}}_{is}}.$$
- (2)
- (3)
- By the deconvolution method, we estimate ${f}_{\u03f5}(\xb7)$ by$${\widehat{f}}_{\u03f5}\left(z\right)=\frac{1}{2\pi}{\int}_{-\infty}^{\infty}{e}^{-\iota tz}{\mathsf{\Phi}}_{k}\left(\frac{t}{{T}_{n}}\right){\widehat{\varphi}}_{\u03f5}\left(t\right)dt,\phantom{\rule{7.22743pt}{0ex}}z\in \mathcal{R},$$$${\mathsf{\Phi}}_{k}\left(t\right)=\left\{\begin{array}{cc}0,\hfill & \left|t\right|\ge \frac{1}{2},\hfill \\ \frac{1}{2},\hfill & \left|t\right|=\frac{1}{2},\hfill \\ 1,\hfill & \left|t\right|\le \frac{1}{2}.\hfill \end{array}\right.$$

**STEP**

**3.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

## 3. Monte Carlo Simulation

## 4. Extension: Conditional Heteroskedastistic Error Case

**Remark**

**5.**

**Remark**

**6.**

**STEP**

**1.**

**STEP**

**2.**

**STEP**

**3.**

**Remark**

**7.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1. | For ease of exposition, we assume ${X}_{it}$ is univariate, the extension to multivariate case can be carried over straightforwardly. |

2. | This can be achieved by using de-mean data for the dependent variable, i.e., replacing ${Y}_{it}$ by ${Y}_{it}-{\left(NT\right)}^{-1}{\sum}_{j=1}^{N}{\sum}_{s=1}^{T}{Y}_{js}$ in Equation (1). For notational simplicity, we still use ${Y}_{it}$ to denote the dependent variable although it is actually the de-mean version of it. |

3. | Recently, Fang et al. (2018) proposes a new nonparametric method for estimating a conditional quantile function with cross-sectional data. We refer readers to Fang et al. (2018) for a detailed discussion. |

4. | For ease of exposition, we drop the subscript $it$ in ${Q}_{{\u03f5}_{it}}\left(\tau \right)$ and use ${Q}_{\u03f5}\left(\tau \right)$ to denote the $\tau $-th quantile of ${\u03f5}_{it}$ in general, since ${\u03f5}_{it}$ is an $i.i.d.$ sequence. |

5. | The consistency can be straightforwardly shown using similar arguments as in Fang et al. (2018) and Horowitz and Markatou (1996). |

6. | There is no rule-of-thumb to choose the optimal bandwidth in the deconvolution method. In practice, researchers can try different bandwidths as a robust check to see how results vary across the different bandwidths. |

7. | Fang et al. (2018) also considers the same form of heteroskedastic error as described here. |

8. | This implies the conditional density of ${\u03f5}_{it}$ given ${X}_{it}=x$ is symmetric, since, given that ${\u03f5}_{it}{|}_{{X}_{it}=x}=\sigma \left(x\right){\eta}_{it}$, the symmetry of ${\eta}_{it}$ is equivalent to the symmetry of ${\u03f5}_{it}$. |

**Figure 1.**Recovered densities across different bandwidths and homoskedastic symmetric normal errors.

Sample Size $(\mathit{N},\mathit{T})$ | Estimators | ||||||
---|---|---|---|---|---|---|---|

$\mathit{MSE}(\widehat{\mathit{m}})$ | $\mathit{MSE}({\widehat{\mathit{Q}}}_{\mathbf{\u03f5}}(\mathbf{\tau}))$ | $\mathit{MSE}({\widehat{\mathit{q}}}_{\mathbf{\tau}})$ | |||||

$\mathbf{\tau}=\mathbf{0.2}$ | $\mathbf{\tau}=\mathbf{0.3}$ | $\mathbf{\tau}=\mathbf{0.4}$ | $\mathbf{\tau}=\mathbf{0.2}$ | $\mathbf{\tau}=\mathbf{0.3}$ | $\mathbf{\tau}=\mathbf{0.4}$ | ||

$(100,10)$ | 0.0149 | 0.0037 | 0.0022 | 0.0006 | 0.0204 | 0.0185 | 0.0163 |

$(200,10)$ | 0.0092 | 0.0012 | 0.0007 | 0.0002 | 0.0107 | 0.0102 | 0.0096 |

$(400,10)$ | 0.0048 | 0.00051 | 0.00028 | 0.000082 | 0.0052 | 0.0050 | 0.0048 |

Sample Size $(\mathit{N},\mathit{T})$ | Estimators | ||||||
---|---|---|---|---|---|---|---|

$\mathit{MSE}(\widehat{\mathit{m}})$ | $\mathit{MSE}({\widehat{\mathit{Q}}}_{\mathbf{\u03f5}}(\mathbf{\tau}))$ | $\mathit{MSE}({\widehat{\mathit{q}}}_{\mathbf{\tau}})$ | |||||

$\mathbf{\tau}=\mathbf{0.2}$ | $\mathbf{\tau}=\mathbf{0.3}$ | $\mathbf{\tau}=\mathbf{0.4}$ | $\mathbf{\tau}=\mathbf{0.2}$ | $\mathbf{\tau}=\mathbf{0.3}$ | $\mathbf{\tau}=\mathbf{0.4}$ | ||

$(100,10)$ | 0.0139 | 0.0423 | 0.0235 | 0.0065 | 0.0642 | 0.0433 | 0.0235 |

$(200,10)$ | 0.0094 | 0.0210 | 0.0128 | 0.0036 | 0.0304 | 0.0222 | 0.0130 |

$(400,10)$ | 0.0048 | 0.0091 | 0.0063 | 0.0019 | 0.0104 | 0.0112 | 0.0067 |

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## Share and Cite

**MDPI and ACS Style**

Yan, K.X.; Li, Q.
Nonparametric Estimation of a Conditional Quantile Function in a Fixed Effects Panel Data Model. *J. Risk Financial Manag.* **2018**, *11*, 44.
https://doi.org/10.3390/jrfm11030044

**AMA Style**

Yan KX, Li Q.
Nonparametric Estimation of a Conditional Quantile Function in a Fixed Effects Panel Data Model. *Journal of Risk and Financial Management*. 2018; 11(3):44.
https://doi.org/10.3390/jrfm11030044

**Chicago/Turabian Style**

Yan, Karen X., and Qi Li.
2018. "Nonparametric Estimation of a Conditional Quantile Function in a Fixed Effects Panel Data Model" *Journal of Risk and Financial Management* 11, no. 3: 44.
https://doi.org/10.3390/jrfm11030044