Estimation and Inference for Multi-Kink Quantile Regression

Abstract

This article proposes a new Multi-Kink Quantile Regression (MKQR) model which assumes different linear quantile regression forms in different regions of the domain of the threshold covariate but are still continuous at kink points. First, we investigate parameter estimation, kink points detection and statistical inference in MKQR models. We propose an iterative segmented quantile regression algorithm for estimating both the regression coefficients and the locations of kink points. The proposed algorithm is much more computationally efficient than the grid search algorithm and not sensitive to the selection of initial values. Second, asymptotic properties, such as selection consistency of the number of kink points and asymptotic normality of the estimators of both regression coefficients and kink effects, are established to justify the proposed method theoretically. Third, a score test based on partial subgradients is developed to verify whether the kink effects exist or not. Test-inversion confidence intervals for kink location parameters are also constructed. Monte Carlo simulations and two real data applications on the secondary industrial structure of China and the triceps skinfold thickness of Gambian females illustrate the excellent finite sample performances of the proposed MKQR model. A new R package MultiKink is developed to easily implement the proposed methods.

Keywords:

• Change-point detection
• Hypothesis testing
• Kink regression
• Model selection
• Quantile regression

Supplementary Material

The convergence of the BRISQ Algorithm, the Wild bootstrap algorithm for P-Values, the proof of Proposition 3.1 for the null asymptotic distribution of the smoothed rank score (SRS) test statistic and additional simulation results are included in a separate online supplemental file.

Acknowledgments

We thank the Editor, the Associate Editor and two referees for their encouragements and insightful comments which have substantially improved the article. We also thank Weiping Bao and Xiang Li for technical support on cluster computing.

Notes

1 In fact, $R_n(\delta )$ is the partial subgradient of the quantile objective function with respect to β₁ evaluated at ${\beta }_1=0$ and $\mathit{\alpha }=\widehat{\mathit{\alpha }}$ up to a constant for the model $Q_Y(\tau ;\mathit{\theta }|{\mathbf{W}}_t)={\alpha }_0+{\alpha }_1{X}_t+{\beta }_1(X_t-\delta )I(X_t\le \delta )+{\mathit{\gamma }}^{\mathrm{T}}{\mathbf{Z}}_t,$ which is essentially same as the model (2.1) with K = 1 after simple reparameterization.

2 We also separately test the existence of kink effects between Y_t and $Z_{t1}$, Y_t and $Z_{t2}$ at different quantiles. The resulting p-values are all greater than 0.1 across all quantiles indicating no kink effect on FE and FAI.

3 Professor Hollis B. Chenery at Harvard University believed that modern economic growth can be understood as a comprehensive transformation of the economic structure. He divided the structural transformation process of GDP per capita into three stages: Initial, Intermediate and Post-industrial stages, corresponding to the GDP per capita less than 1495 dollars, 1495–11214 dollars and greater than 11214 dollars.

Additional information

Funding

Zhong’s research was supported by the National Natural Science Foundation of China (11922117, 11771361), Fujian Provincial Science Fund for Distinguished Young Scholars (2019J06004), Basic Scientific Center Project of NNSFC (71988101) and the 111 Project (B13028). Zhang’s research was supported by National Natural Science Foundation of China (11931014).