Under the pressure of serious environmental pollution and energy shortage, China needs to improve its energy efficiency to alleviate these problems. Environmental regulation is an important constraint on economic development, which has an impact on energy efficiency. Meanwhile, energy efficiency is a reference factor for adjusting environmental policies, which has an impact on environmental regulation. Therefore, the relationship between environmental regulation and energy efficiency needs to be further studied under a unified framework. Based on Chinese provincial panel data, we first use a stochastic frontier model to estimate the energy efficiency of China’s 30 provinces from 2004 to 2019, and then employ a spatial simultaneous equation model to study the spatial spillover effects of environmental regulation and energy efficiency and their interactions. The results show that: (1) Both energy efficiency and environmental regulation have significantly positive spatial spillover effects. Specifically, an overall increase of 1% in energy efficiency in the surrounding areas can promote an improvement in the local energy efficiency by about 1.0404%, and an overall increase of 1% in environmental regulation in the surrounding areas can lead to an increase of about 0.6075% in the local environmental regulation. (2) The impact of environmental regulation on energy efficiency is significantly positive; i.e., under the current situation in China, an increase of 1% in environmental regulation can promote local energy efficiency by about 0.2777%. (3) The impact of energy efficiency on environmental regulation is significantly positive; i.e., a 1% increase in energy efficiency may stimulate local governments to strengthen their environmental regulation by 1.5981%. Accordingly, some targeted policy suggestions are given.

The urban-rural gap of elementary education seriously affects social fairness, so the study on the urban-rural gap of elementary education can help promote social fairness in China. China’s urbanization is an important variable affecting the social process, and its impact on the urban-rural gap of elementary education is worthy of further study. Based on China’s provincial panel data from 2006 to 2017, this paper uses the Theil index to measure the urban-rural gap in different aspects of elementary education and uses principal component analysis (PCA) to construct a comprehensive index to objectively measure the urban-rural gap of overall elementary education in China. Our results show that the gap between urban and rural areas in China’s elementary education showed an upward trend from 2006 to 2010, and the gap decreased year by year from 2010 to 2017. Then we used the panel smooth transition regression (PSTR) model to study the impact of urbanization on the urban-rural gap of elementary education in China and find evidence that urbanization has a nonlinear effect on the urban-rural gap of elementary education. That is, in economically underdeveloped areas, urbanization exacerbates the gap between urban and rural areas in elementary education, while in economically developed areas, urbanization narrows the gap between urban and rural areas in elementary education. Therefore, developing economy and continuing to promote urbanization are effective measures to narrow the gap between urban and rural areas in elementary education.

This paper deals with spatial data that can be modelled by partially linear varying coefficient spatial autoregressive models with Bayesian P-splines quantile regression. We evaluate the linear and nonlinear effects of covariates on the response and use quantile regression to present comprehensive information at different quantiles. We not only propose an empirical Bayesian approach of quantile regression using the asymmetric Laplace error distribution and employ P-splines to approximate nonparametric components but also develop an efficient Markov chain Monte Carlo technique to explore the joint posterior distributions of unknown parameters. Monte Carlo simulations show that our estimators not only have robustness for different spatial weight matrices but also perform better compared with quantile regression and instrumental variable quantile regression estimators in finite samples at different quantiles. Finally, a set of Sydney real estate data applications is analysed to illustrate the performance of the proposed method.

Understanding the coupled and coordinated development of China’s urban and rural basic education systems is crucial for fostering their interaction and synergistic growth. Using China’s provincial panel data from 2011 to 2023, this study measures the coupled and coordinated development level of urban–rural basic education (CCD-URBE) via the entropy weight method, G1-method and coupling coordination degree model. On this basis, the Dagum Gini coefficient decomposition method, traditional and spatial Markov chain models, as well as convergence test models are employed for empirical research. The results show that: (1) During the study period, the CCD-URBE across the nation and the four major regions improves significantly. Both intra-regional and inter-regional disparities show a consistent downward trend. Inter-regional disparities are the main source of the overall disparities, and the contribution rate of transvariation density to the overall disparities exhibits the most significant increase. (2) The CCD-URBE demonstrates strong stability, as most regions tend to maintain their original CCD-URBE grades. Meanwhile, neighborhood grades moderate the local transition probability significantly. Neighborhoods with high CCD-URBE promote the upward improvement of the local CCD-URBE, while those with low CCD-URBE inhibit it. (3) The CCD-URBE across the nation and the four major regions shows obvious trends of σ-convergence, absolute β-convergence, and conditional β-convergence. The central region, which has lower CCD-URBE, exhibits higher convergence speed. Based on these findings, targeted policy implications are derived.