Estimation and Inference of Special Types of the Coefficients in Geographically and Temporally Weighted Regression Models

Abstract

Geographically and temporally weighted regression (GTWR) models have been widely used to explore spatiotemporal nonstationarity where all the regression coefficients are assumed to be varying over both space and time. In reality, however, constant, only temporally varying, and only spatially varying coefficients might also be possible depending on the underlying effects of the explanatory variables on the response variable. Therefore, the development of inference and estimation methods for such special types of the coefficients is essential to the deep understanding of spatiotemporal characteristics of the regression relationship. In this article, an average-based approach, relying on a modified estimation of the conventional GTWR models, is proposed to calibrate the GTWR models with the special types of the coefficients, on which a statistical test is formulated to simultaneously infer constant, temporally varying, and spatially varying coefficients. The simulation study shows that the test method is of valid Type I error and satisfactory power and the average-based estimation method yields more accurate estimators for the special types of the coefficients. A real-life example based on Beijing house prices is given to demonstrate the applicability of the test and estimation methods as well as the extensibility of the test in model selection.

地理和时间加权回归（GTWR）模型已被广泛用于时空非平稳性的研究。非平稳性假设所有回归参数在空间上和时间上都是变化的。然而, 根据解释变量对响应变量的潜在影响, 回归参数也可能是常量、仅随时间或空间变化。因此, 这些特殊参数的推理和估计方法, 对于深入理解回归关系的时空特征至关重要。基于对传统GTWR模型的改进估计, 本文提出了一种基于平均值的方法, 对含有特殊参数的GTWR模型进行了校正, 开展了能同时推断常量参数、时间变化参数和空间变化参数的统计检验。模拟研究表明, 本文的检验方法具备有效的第一类统计错误和令人满意的统计功效, 基于平均值的估计方法能产生更准确的特殊参数估计。根据北京房价实例, 本文证明了该检验和估计方法的适用性及其在模型选择中的可扩展性。

Los modelos de regresión geográfica y temporalmente ponderada (GTWR) han sido ampliamente usados para explorar la no estacionariedad espaciotemporal, donde todos los coeficientes de regresión se supone varían tanto en el espacio como en el tiempo. No obstante, en la realidad los coeficientes que solo varían temporal y espacialmente también podrían ser posibles dependiendo de los efectos subyacentes de las variables explicativas sobre la variable de respuesta. En consecuencia, el desarrollo de métodos de inferencia y estimación para tales tipos especiales de coeficientes es esencial para la comprensión profunda de las características espaciotemporales de la relación de regresión. En este artículo, con apoyo en una estimación modificada de los modelos convencionales GTWR, se propone un enfoque basado en la media, para calibrar los modelos GTWR con los tipos especiales de los coeficientes, sobre los cuales se formula una prueba estadística para inferir de manera simultánea coeficientes constantes, temporal y espacialmente variables. El estudio de la simulación muestra que el método de prueba registra un error válido de Tipo I y potencia satisfactoria, en tanto que el método de la estimación basado en la media genera estimadores más precisos para los tipos especiales de los coeficientes. Se ilustra esto con un ejemplo de la vida real basado en los precios de la vivienda en Beijing, para mostrar la aplicabilidad de los métodos de prueba y estimación, lo mismo que la extensibilidad de la prueba en la selección de los modelos.

Key Words:

• bootstrap
• geographically and temporally weighted regression
• residual sum of squares
• spatiotemporal nonstationarity

关键词:

• bootstrap
• 地理和时间加权回归
• 残差平方和
• 时空非平稳性。

Palabras clave:

• bootstrap
• no estacionariedad espaciotemporal
• regresión geográfica y temporalmente ponderada
• suma residual de cuadrados

Acknowledgments

The authors would like to thank Homelink Real Estate Agency Company for its authorization to use the raw Beijing secondhand house price data in the real-life example. The authors also thank the two anonymous reviewers for their valuable comments and constructive suggestions which led to significant improvement in the article.

Supplemental Material

Supplemental simulation study for this article can be accessed on the publisher’s site at: http://dx.doi.org/10.1080/24694452.2022.2092443. The supplemental material is the performance assessment of the proposed test and the average-based estimation method under a larger sample size or a larger model error variance.

The R codes for performing the proposed test and estimation methods in the simulation study are available in figshare.com via the private link http://doi.org/10.6084/m9.figshare.16850962. The raw Beijing secondhand house price data used in this study belong to Homelink, a private real estate agency company. The data are not publicly available unless authorized by Homelink.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [11871056] and [71874195].

Notes on contributors

Zhi Zhang

ZHI ZHANG is currently a PhD Candidate in the Department of Statistics, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, China. E-mail: [email protected]. Her research interests include spatiotemporal analysis and statistical modeling.

Chang-Lin Mei

CHANG-LIN MEI is a Professor in the School of Science, Xi’an Polytechnic University, Xi’an, China. E-mail: [email protected]. His research interests include nonparametric regression and spatial data analysis.

Hua-Yi Yu

HUA-YI YU is an Associate Professor in the School of Public Administration and Policy, Renmin University of China, Beijing, China. E-mail: [email protected]. His research interests include urban economics, real estate economics, and applied spatial econometrics.