Applying a Population Flow–Based Spatial Weight Matrix in Spatial Econometric Models: Conceptual Framework and Application to COVID-19 Transmission Analysis

Abstract

This article proposes a novel method for constructing an asymmetric spatial weight matrix and applies it to improve spatial econometric modeling. As opposed to traditional spatial weight matrices that simply consider geographic or economic proximity, the spatial weight matrix proposed in this study is based on large-volume daily population flow data. It can more accurately reflect the socioeconomic interactions between cities over any given period. To empirically test the validity and accuracy of this proposed spatial weight matrix, we apply it to a spatial econometric model that analyzes COVID-19 transmission in Mainland China. Specifically, this matrix is used to address spatial dependence in outcome and explanatory variables and to calculate the direct and indirect effects of all predictors. We also propose a practical framework that combines instrumental variable regressions and a Hausman test to validate the exogeneity of this matrix. The test result confirms its exogeneity; hence, it can produce consistent estimates in our spatial econometric models. Moreover, we find that spatial econometric models using our proposed population flow–based spatial weight matrix significantly outperform those using the traditional inverse distance weight matrix in terms of goodness of fit and model interpretation, thus providing more reliable results. Our methodology not only has implications for national epidemic control and prevention policies but can also be applied to a wide range of research to better address spatial autocorrelation issues. Key Words: COVID-19 transmission, endogeneity, population flow, spatial dependence (autocorrelation), spatial weight matrix.

本文提出了构建非对称空间权重矩阵的新方法，并用于改进空间计量经济建模。不同于传统的仅考虑地理或经济邻近性的空间权重矩阵，本文提出的空间权重矩阵以海量的日人口流量数据为基础，能更准确地反映任何时期城市之间的社会经济互动。为了验证本文的空间权重矩阵的有效性和准确性，我们将其应用于中国大陆COVID-19疾病传播分析的空间计量模型。具体的，该矩阵考虑了结果变量和解释变量的空间相关性，计算了各预测因子的直接和间接影响。我们还提出了一个应用框架，可以结合工具变量回归和Hausman检验，验证该矩阵的外生性。验证结果证实了矩阵的外生性，并能够在空间计量经济模型中提供稳定的估计。此外，我们发现，在拟合优度和模型解释方面，采用基于人口流动的空间权重矩阵的空间计量模型，明显优于基于传统的反距离权重矩阵的空间计量模型，能提供更可靠的结果。我们的方法不仅有益于国家流行病控制和预防政策，也可用于其它研究，能更好地解决空间自相关性的问题。

En este artículo se propone un método novedoso para construir una matriz asimétrica de pesos espaciales, la cual se aplica para mejorar la modelización econométrica espacial. A diferencia de las tradicionales matrices de ponderación espacial que apenas consideran la proximidad geográfica o económica, la matriz de ponderación espacial que se propone en el estudio se basa en datos de flujo de población diario de gran volumen. Esta puede reflejar con mayor precisión las interacciones socioeconómicas entre ciudades, durante un determinado período. Para poner a prueba empíricamente la validez y precisión de la matriz de pesos espaciales propuesta, la aplicamos a un modelo econométrico espacial que analiza la propagación del COVID-19 en la China continental. Específicamente, esta matriz se usa para abordar la dependencia espacial de las variables de resultado y las explicativas, y para calcular los efectos directos e indirectos de todos los predictores. Proponemos, además, una enmarcación práctica que combina regresiones de variables instrumentales y una prueba de Hausman para validar la exogeneidad de la matriz. El resultado de la prueba confirma su exogeneidad; entonces, puede producir estimaciones consistentes en nuestros modelos econométricos espaciales. Además, descubrimos que los modelos econométricos espaciales que usen nuestra matriz propuesta de pesos espaciales basada en flujo de población superan de modo significativo a los que usan la matriz convencional de ponderación de distancia inversa, en términos de buen acoplamiento e interpretación del modelo, generando así resultados de mayor confiabilidad. Nuestra metodología no solo tiene implicaciones para las políticas nacionales de control y prevención de epidemias, sino que también tiene aplicabilidad en una amplia gama de investigaciones con las que se aborden mejor los problemas de autocorrelación espacial.

关键词：:

• COVID-19疾病传播
• 内生性
• 人口流动
• 空间相关性（自相关）
• 空间权重矩阵。

Palabras clave::

• dependencia espacial (autocorrelación)
• endogeneidad
• flujo de población
• matriz de pesos espaciales
• transmisión de COVID-19.

Acknowledgment

This work was supported by the Natural Science Foundation of Guangdong Province [Grant Number: 2021A1515011250].

Notes

1 The moving-out indexes and moving-in indexes are based on the travel intensity between specific city pairs. For example, the moving-out index of Beijing to Tianjin refers to the volume of population flow traveling from Beijing to Tianjin. According to Baidu’s metadata, population flow from city i to city j is considered to be the move-out index for city i and the move-in index for city j.

2 The inverse distance weighted interpolation follows the equation: $${\mathit{Corresponding}\mathrm{ }\mathit{Percentage}}_{ij}=\frac{\frac{1}{{\mathit{Distance}}_{ij}^2}\mathrm{*}(100-\sum _1^A\mathit{Accurate}\mathrm{ }\mathit{Daily}\mathrm{ }{\mathrm{ }\mathit{Moving}\_\mathit{Out}\mathrm{ }\mathit{Index}}_{ia})}{{\sum }_1^V\frac{1}{{\mathit{Distance}}_{iv}^2}},$$ where Distance_ij is the Euclidean distance between city i and city j, A is the number of destination cities with accurate move-out indexes for origin city i, and V is the number of destination cities without accurate move-out indexes for origin city i.

3 According to Vega and Elhorst (2013), there are three other types of spatial econometric models: (1) the SLX (spatial lag of X) model that includes spatial interactions of explanatory variables, (2) the SAC model that includes a spatially lagged dependent and a spatially correlated error term, and (3) the general nesting spatial model that includes all three types of spatial interaction effects.

4 During Period I, population flow between cities in China was not disrupted because it was only announced on 20 January that COVID-19 could be transmitted human to human and no further warnings on travel risks were announced until 23 January. Moreover, Baidu did not publicly release daily population flow data until 11 January. Therefore, in the models for Period I, the spatial weight matrix is built based on the population flow volume during Period I. In the models for Period II, we still use the population flow volume during Period I to account for the fourteen-day incubation period of the virus.

5 Note that the exogeneity of our population flow–based W depends on what outcome variable is used in the model. In our models, daily COVID-19 case number is the outcome variable. In other models using socioeconomic variables as outcome variables, the exogeneity of our population flow–based W might be affected.

6 According to DXY.cn, the COVID-19 infection data they published were reported by thirty-two provincial-level Health Commissions in China.

7 To convert the two indexes into the actual volume of person movements in and out of each city, we use the daily number of people traveling into and out of Hong Kong provided by the Hong Kong Immigration Department to calibrate and calculate the number of people to which each moving-in index and moving-out index unit corresponds. Using these data, we estimate that one index unit in the move-in index and move-out index corresponds to 71,121 person movements. This estimated converting factor is constant across all cities and is used to calculate the actual daily volume of population inflow and outflow of each city.

8 Kriging is a type of statistical technique for optimal spatial prediction that has been used widely in meteorological applications, agriculture, geosciences, and many other disciplines due to its minimized prediction error. Compared with classical kriging methods, the empirical Bayesian kriging is more robust because it accounts for the errors introduced by the estimation of the Semivariogram model (Krivoruchko and Gribov 2019).

9 After January 24, different levels of prevention and control measures (i.e., the shutdown of public transport and public places, the lockdown of residential buildings and neighborhoods, and the setup of checkpoints to control the population entering the city) were implemented by most of the cities, especially for cities in Hubei province and some cities with relatively more confirmed cases like Wenzhou, Hangzhou, and Harbin (Fang, Wang, and Yang 2020).

10 Based on our original population flow–based SWM (before normalization), we calculate the value of each element in symmetric population flow–based SWM as $$w_{ij}=\frac{1}{2}\left({\mathit{Moving}\_\mathit{Out}\mathrm{ }\mathit{Index}}_{ij}+\mathit{Moving}\_\mathit{Out}\mathrm{ }{\mathit{Index}}_{ji}\right)=w_{ji},$$ where the $w_{ij}$ is the spatial weight of unit i toward unit j, $w_{ji}$ is the spatial weight of unit j toward unit i, ${\mathit{Moving}\_\mathit{Out} \mathit{Index}}_{ij}$ is the index that reflects the volume of population traveling from city i to city j, and ${\mathit{Moving}\_\mathit{Out} \mathit{Index}}_{ji}$ is the index that reflects the volume of population traveling from city j to city i.

11 The gravity model used in this article can be expressed as follows: $$w_{ij}=\mathrm{ }\frac{{\mathit{GDP}}_i^{\alpha }{\mathit{GDP}}_j^{\beta }{\mathit{Pop}}_i^{\phi }{\mathit{Pop}}_j^{\omega }}{{{\mathit{Distance}}_{ij}}^{\tau }},$$ where the $w_{ij}$ is the spatial weight of unit i toward unit j, ${\mathit{GDP}}_i$ (${\mathit{GDP}}_j$) is the GDP of city i (city j), ${\mathit{Pop}}_i$ (${\mathit{Pop}}_j$) is the total population of city i (city j), and ${\mathit{Distance}}_{ij}\mathit{\text{ is }}\mathit{the}$ geographic distance between city i and city j. We first select city pairs with accurate move-out indexes in our data set to estimate $\alpha , \beta , \phi , \omega , \mathrm{and}\mathrm{ }\tau$ based on the preceding equation. We then use the estimation results to impute move-out indexes for city pairs without accurate move-out indexes.

Additional information

Notes on contributors

Pengyu Zhu

PENGYU ZHU is an Associate Professor in the Division of Public Policy at the Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR. E-mail: [email protected]. His research interests include big data and urban planning, sustainable transportation, economic development policy, housing and land use policy, and migration and employment.

Jiarong Li

JIARONG LI is a PhD student in the Urban Governance and Design Thrust at Hong Kong University of Science and Technology (Guangzhou), Nansha, Guangzhou, 511400, Guangdong, China. She is also a research assistant at the Center for Applied Social and Economic Research at Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. E-mail: [email protected]. Her research interests include spatial data analysis, urban transportation, and spatial econometrics.

Yuting Hou

YUTING HOU is an assistant professor in the Department of Building and Real Estate at the Hong Kong Polytechnic University, Kowloon, Hong Kong SAR. E-mail: [email protected]. Her research interests include the interactions between land use and transportation, urban and regional economics, and applied spatial analysis.