Functional-Coefficient Quantile Regression for Panel Data with Latent Group Structure

Abstract

This article considers estimating functional-coefficient models in panel quantile regression with individual effects, allowing the cross-sectional and temporal dependence for large panel observations. A latent group structure is imposed on the heterogeneous quantile regression models so that the number of nonparametric functional coefficients to be estimated can be reduced considerably. With the preliminary local linear quantile estimates of the subject-specific functional coefficients, a classic agglomerative clustering algorithm is used to estimate the unknown group structure and an easy-to-implement ratio criterion is proposed to determine the group number. The estimated group number and structure are shown to be consistent. Furthermore, a post-grouping local linear smoothing method is introduced to estimate the group-specific functional coefficients, and the relevant asymptotic normal distribution theory is derived with a normalization rate comparable to that in the literature. The developed methodologies and theory are verified through a simulation study and showcased with an application to house price data from U.K. local authority districts, which reveals different homogeneity structures at different quantile levels.

Keywords:

• Cluster analysis
• Functional-coefficient models
• Incidental parameter
• Latent groups
• Local linear estimation
• Panel data
• Quantile regression

Supplementary Materials

The supplemental document contains proofs of the main asymptotic theorems, technical lemmas with proofs, some extensions of the developed method and theory, and additional simulation and empirical results.

Acknowledgments

The authors would like to thank an Associate Editor and two reviewers for the constructive comments, which helped to substantially improve the article. The authors also thank Dr Chaowen Zheng for collecting the empirical data and preparing Figures 4–6. The usual disclaimer applies.

Disclosure Statement

The authors report there are no completing interests to declare.

Notes

1 In the main model, we assume that Z_t is a continuous random variable with a density function satisfying Assumption 2(ii). This ensures that the maximum distance between two consecutive observations of Z_t is of order $O_P(\hspace{0.17em}\log \hspace{0.17em}T/T)$, indicating that there is a large number of observations in any neighborhood of $z\in [0,1]$ when T is sufficiently large. This is crucial for the kernel-based nonparametric estimation methodology and theory to be developed in the subsequent sections.

2 When the index variable is Z_it which varies over i and t, we estimate the functional coefficients at a set of equidistant grid points $u_1,u_2,\dots ,u_m$, where m is a sufficiently large positive integer. Then, we define $\widehat{\Delta }(j,k)=\frac{1}{m}{\sum }_{t=1}^m||{\widehat{\mathit{\beta }}}_j(u_t)-{\widehat{\mathit{\beta }}}_k(u_t)||$.

3 Due to data unavailability, we consider LADs only in England and Wales.

4 Four LADs from England and Wales - Aylesbury Vale, Gloucester, Norwich, and Powys, are excluded due to their outlying values. This gives a total of 335 LADs for the subsequent analysis.

5 More data details can be found in Chen, Shin, and Zheng (2022).

6 The membership of the two groups at $\tau =0.50$ is similar to that of the two groups at $\tau =0.25$ with a large number of overlapping member LADs (e.g., for Group 1, there are 45 overlapping LADs and for Group 2, there are 247 overlapping LADs).

Additional information

Funding

Yang’s research is partially supported by Natural Science Foundation of Zhejiang Province (No. LY22A010006), National Social Science Foundation of China (No. 23BTJ036), National Natural Science Foundation of China and the Fundamental Research Funds for Provincial Universities of Zhejiang. Chen’s research is partially supported by the Economic and Social Research Council in the United Kingdom (No. ES/T01573X/1). D. Li’s research is partially supported by the National Natural Science Foundation of China (No. 72033002). R. Li’s research is partially supported by NIH grants R01AI170249 and R01AI136664.