The core explanatory variable in this study is older adults’ disability risk. We chose the older adults’ ADL as the agent, specifically measured as follows. We chose six basic daily-life items in the questionnaire to represent the agent, and the specific questions were as follows:

The corresponding options for answers 1 to 4 are “No, I do not have any difficulty”; “I have difficulty but can still do it”; “yes, I have difficulty and need help”; and “I cannot do it.” Finally, the ADL value was obtained based on the scores for the six items, and specific classification criteria were defined according to the scores. Further details are provided in Table 1. The statistical results show that the proportion of severe disabilities is only 0.53% and, with a reduction in disability level, its proportion in the total sample increases; for example, the proportion of the population with mild disabilities has reached 18.61%.

According to the calculated ADL index and characteristics that conformed to the ranking variable, we established a panel-ordered logit benchmark test model. The model settings were as follows:

The ADL in Equation (1) is recorded as the ability of the older adult to take care of themselves according to the agency index of disability risk. is a city-size variable. City size was obtained from the population size of the city where the interviewees were located at the time of the survey. During processing, we used logarithmic transformation. The statistical results show that the mean value of the logarithm of city size was 6.1987. In addition, to ensure comparability of the estimated results, we synchronized the statistics for the city categories, specifically dividing them into prefecture-level, provincial capital, and sub-provincial cities and municipalities directly under the central government. Because the sub-provincial cities intersect with the provincial capital and prefecture-level cities, the statistics will be based on the highest level and will not be repeated.

The statistical results show that the proportion of prefecture-level cities in the sample is 73.82%, that of provincial capital cities is 5.79%, and that of sub-provincial cities and municipalities directly under the Central Government is 20.39%. X refers to the control variables. This study mainly controls for individual and family variables such as gender, age, education level, marriage, registered residence, medical insurance, per-capita household income, and macro city-characteristic variables such as city category, regional air quality, per-capita GDP, green coverage, regional financial expenditure, population density, and the urban-sprawl index. Time represents the time-fixed effect, Id is the individual fixed effect, ε is the random interference term, and the symbol and significance of the coefficient, α 1 , are the key points of the benchmark model. The urban-sprawl index in the benchmark model is calculated as follows:

Accordingly, i refers to the city and S I refers to the urban-sprawl index. The right side of the Equation refers to 1 minus the change value, which is the ratio of the change rate of the regional population size to the change rate of the regional built-up area. In this study, the year 2000 is selected as the base period, where P P i 0 refers to the regional population at the base period and Δ P P i 0 refers to the change value of the regional population compared to that of the base period at the time of investigation; AB D i 0 refers to the area of the built-up area in the base period, and Δ A B D i 0 refers to the change value of the built-up area in the survey compared to that of the base period. Table 2 shows that the average urban-sprawl index (SI) is 0.8673. Statistical results for the other variables are presented in Table 2. Most variable names are abbreviated, and their specific meanings are provided in the definitions.

In addition, considering the existence of a potential transmission mechanism for the impact of city size on older adults’ disability risk, it is necessary to investigate the potential transmission mechanism of the benchmark test model. According to the stepwise analysis method, the potential transmission mechanism model in this study was set as follows:

Equation (3) examines whether city size significantly affects the mediation variable, M I . Equation (4) examines the significance and coefficient changes after introducing intermediate variables into the benchmark model; the significance of the coefficients β 1 , χ 1 , and χ 2 under Models (3) and (4) is the focus of the mediation effect. If the three are significant, this indicates that a mediation effect must exist, and there is a full mediation. If β 1 is insignificant, this implies that a mediation effect does not exist. If β 1 is significant, and one of χ 1 and χ 2 is significant, then there is partial mediation. The mechanism diagram for city size and older adults’ disability risk is shown in Figure 1.

The article data were selected from the China Health and Retirement Longitudinal Study (CHARLS) database survey data for (9, 26). The CHARLS survey database covers 122 out of 28 provinces in China (the survey sample does not include Hainan Province, Tibet Autonomous Region, and Ningxia Autonomous Region), and the main survey data are from people aged 45 years and above, which can better reflect the basic characteristics of China’s older adult population. In terms of survey time, the survey data published for each issue were scattered each year. For example, the survey samples in 2018 covered those in 2016, 2017, and 2018. Therefore, the city size information of the current period was determined according to the actual survey year and month of the sample to achieve accurate matching between macro and micro survey data. In this study, the main population was aged 60 years and above. Unbalanced panel data were constructed through data screening and selection, and 17,981 effective samples covering the follow-up survey were obtained.

The descriptive statistics of the core variables of this study, including city size, individual family characteristics, and city characteristics, are shown in Table 2.

First, based on the benchmark model, combined with macro survey data, we empirically examined the impact of city size on the ADL of older adults using the stepwise regression analysis method. Table 3 Models (1)–(3) are the results of controlling different variables, and the results in Model (1) show that city size has a significant positive correlation with the ADL level of older adults; that is, city size can significantly improve the ADL level of older adults. Gender, age, education level, marital status, registered residence, prefecture-level cities, and provincial capital cities in the control variables have significant effects, among which prefecture-level cities (PLC) and provincial capital city (PC) categories are more significant than sub-provincial cities and municipalities (SPCM). Model (2) is the test result of introducing urban macro characteristic variables based on Model (1). The results show that after controlling for the macro characteristic variables of the city, city size still has a significant positive impact on the ADL of older adults.

However, AQI, GDP per capita (GPC), and population density (PD) in urban macro characteristics have significant effects on the ADL of older adults. Model (3) further introduces the urban sprawl index (SI) based on Model (2), and the results show that the city size still has a significant positive impact on the ADL of older adults. The SI also has a significant negative impact on older adults’ ADL; that is, the higher the SI, the smaller, the older adults’ ADL, and the higher their disability rate. The SI reflects the ratio of the dynamic change in population to the change in urban built-up areas. The larger the value, the more diffuse the city; therefore, the results in Model (3) show that the larger the city expansion, the worse it is for older adults to improve their ADL. Because the benchmark model uses the panel-ordered logit model, its coefficients do not directly reflect the effect. Therefore, based on Model (3), we synchronously reported the OR value of each variable and its 95% confidence interval (CI), as shown in the results in Model (4) in Table 3.

The results in Model (4) show that the OR value of city size is 1.3286; that is, compared with PC and PLC, the ADL of older adults living in SPCM will be significantly higher, the probability of their ADL is 1.3286 times higher than the former, and the 95% CI is [1.082243, 1.631089]. In addition, the OR value of the SI was 0.7965, and the 95% CI was [0.675246, 0.939536], which means that compared with the low SI group, the ADL of the older adults under the high SI group was only 76.65% of the former, which was significantly worse. This result further shows that SI is not conducive to improving ADL in older adults and even increases the overall disability risk of older adults in the region.

As the benchmark-model test may include some factors (e.g., missing key variables), some missing key variables will affect both city size and older adults’ ADL, resulting in biased estimation results. Meanwhile, there is also an “environmental migration effect” (34); that is, residents will selectively migrate to live in big cities because of the better living environment and public-service facilities there, and people will choose to migrate according to their own health level. The probability of people with poor health quality, especially those with high disability risk, migrating to big cities will be reduced, leading to a two-way impact of city size on older adults’ ADL. The Hausman test results reported in Model (2) of Table 4 show that the p value is less than 0.5, which rejects the basic assumption of no difference between the instrumental-variable and benchmark models, indicating that the former method should be used. Therefore, given the estimation error caused by the above endogenous problems, we attempt to deal with the endogenous problems using the instrumental-variable method. From the perspective of existing studies, several instruments deal with variables related to city size. Scholars have chosen both geographical dimensions and historical city sizes (1, 6).

In this study, city size for 122 cities in the 1982 survey sample was used as the instrumental variable for processing. The reason is that city size in 1982 is significantly related to that during the survey period, and city size in 1982 does not directly affect older adults’ ADL at the time of the survey. Thus, theoretically, it is consistent with the selection criteria for instrumental variables. Model (1) of Table 4 is the result of testing the exogenous instrumental variables.

The results show that when the 1982 city size was introduced into the benchmark model, it did not significantly impact older adults’ ADL. This result is consistent with the exogenous assumptions, and it also shows that the instrumental variable does not affect older adults’ ADL in other ways than only indirectly through city size; that is, the instrumental variable is effective. Models (2) and (3) of Table 4 are the results obtained after processing the instrumental variables. The results of the first stage show that city size in 1982 had a significant and positive impact on city size at the time of the survey. The results of the second stage show that after the treatment of the instrumental variables, city size still had a significant and positive impact on older adults’ ADL; that is, the benchmark test results are robust. In addition, to ensure the robustness of the instrumental-variable processing results, based on the exogenous test of the instrumental variables, we also selected the weak instrumental-variable method (Twostepweakiv) for testing, which showed confidence sets (if calculated) based on the 100 points in [−0.199,505, 0.381,228] and a partition cutoff of 5% based on the grid.

Meanwhile, as shown in Model (4) of Table 4, the robust K_ 2sls, LC_2sls, and confidence sets (CS) of AR (Arellano-Bond test) are [0.0351, 0.1466], indicating that the instrumental variable is valid.

The effect of city size on the ADL (disability risk) of the older adult shows heterogeneity due to group differences. Based on benchmark research, we attempted to examine the group heterogeneity of city size affecting the disability risk of older adults from four aspects: age (AGE), registered residence (RR), income (INCOME), and education level (EDU).

In terms of AGE, with the rapid development of China’s aging population, the incidence of disability population in China is increasing, and its size is also expanding rapidly. Therefore, it is of great practical significance to study the group heterogeneity in the impact of urban size on the disability risk of older adults from the perspective of age. According to the age division of the welfare policy for the older adult in China, we take 80 years old as the group boundary of the older adults, the younger ones are under 80 years old, and the older ones are over 80 years old. Figure 2 shows the effect of city size and its 95% CI in the test results of population heterogeneity. The results show that, in terms of AGE heterogeneity, the disability risk of older adults aged 80 years and above is more affected by city size than that of young adults. It also shows that with an increase in the age of older adults, their physical functions decline, and when the risk of disability gradually increases, they need more long-term care. In the context of the gradual weakening of family pension security, older adults are affected more by the more perfect long-term care service security system in big cities. They can obtain higher service satisfaction from big cities, thus showing group heterogeneity.

In terms of RR, we mainly divided RR into urban and rural areas according to the characteristics of registered residences of urban and rural populations in China. The empirical test results are presented in Figure 2. It can be seen from Figure 2 that, compared with urban older adults, the disability risk of rural older adults is more affected by city size. With the rapid development of urbanization and population mobility in China, more and more rural older adults are gradually migrating to urban areas under the trend of urbanization and enjoying the benefits of urban public service resources. Especially, in large cities, the radiation of urbanization enables rural residents to obtain more long-term care security service resources, thereby improving their ADL and reducing their disability risk. Under the same circumstances, the disability risk of urban older adults is less affected by the expansion of the city size than that of rural older adults.

In terms of INCOME, we divide household income into five levels: 20% of the highest income, 20% of the upper-middle income, 20% of the middle income, 20% of the lower-middle income, and 20% of the lowest income. In this study, the middle-income and lower-income groups are defined as the low-income group, while the upper-middle-income and highest-income groups are defined as the high-income group. The test results in Figure 2 show that compared with the high-income group, the older adults in the low-income group had a higher impact effect from the expansion of city size. Compared with the high-income group, the low-income group not only has a relative disadvantage in income but also has lower long-term care security. For example, high-income older adults can meet their disability risk security needs by self-purchasing commercial long-term care insurance or hiring domestic workers to provide basic care services. Therefore, the effect of improving self-care ability due to the expansion of city size was lower.

For EDU, we define people with education levels below junior high school as the low education group and those above junior high school as the high education group. This is because the current education level of older adults in China is still low. The test results in Figure 2 show that, compared with the low education group, the older adults in the high education group had a higher ADL effect from city size expansion. Due to the lower information barriers, such as access to basic city service facilities and understanding of long-term care service security policies, the highly educated population can better utilize existing service resources in large cities, thus showing strong group heterogeneity.

To ensure the robustness of the benchmark test results, we further conducted a robustness test by selecting and replacing the explained variables, control samples, and individual clustering and eliminating follow-up samples. The results are presented in Models (1)–(6) in Table 5. Models (1) and (2) show the test results of using the method of replacing the explained variable. We selected self-assessment health and single physical-disability variables to replace ADL variables; the values of 1–5 indicate that the health status is worsening; in the single physical-disability variable, 1 indicates disability and 0 indicates none. The test results in Models (1) and (2) show that after replacing the explained variables, city size still significantly affects older adults’ self-assessment health and single physical-disability rate; that is, the test results of the basic model are robust, and Model (3) is the result obtained after excluding the abnormal influence of municipal cities. This is because the levels of municipal cities are much higher than those of ordinary prefecture-level cities and their administrative levels are equivalent to those of provinces, thus affecting the estimation results.

However, the results remain stable after excluding the observations from municipal cities, and Model (4) is the test result after controlling for the PLC sample. Without considering other types of cities, PLC city size still significantly impacts older adults’ ADL. Model (5) is the test result after clustering individuals, while Model (6) is the result after excluding follow-up samples. Both models show that the results of the significant impact of city size on older adults’ ADL are robust.

The benchmark test examined only the direct impact of city size on older adults’ ADL, while the potential mechanism of the impact requires further discussion. At a micro level, older adults’ ADL results from dynamic changes in their health; that is, in most cases, chronic or major diseases cause the risk of disability. Therefore, we explore the potential mediating mechanism of the effect of city size from a health perspective, that is, to investigate older adults’ disease-relief and emotional effects. At a macro level, a change in city size will bring about macro impact effects, such as the number of doctors per thousand in the city and the number of employees in the secondary industry in the region, especially the NEMI. Therefore, we examine the macro mediation effect of city size in terms of two aspects: the health-improvement and industrial-structure optimization effects. Among them, the health-improvement effect selects the DPP in a region as the proxy variable, TWPR is selected as the agent for the disease-remission effect, DOD is chosen as the proxy for the emotional effect, and the industrial-structure optimization effect selects the logarithm of the NEMI as the agent.

First, based on the selection of the above four mediation variables, we examined whether mediation effects existed. The results are presented in Table 6. The results in Table 6 Model (1) show that in the first stage, city size has a significant and negative impact on the DPP, TWPR, and NEMI in a city and a significant positive impact on older adults’ DOD. This result indicates that city size affects the above mediation variables. In Table 6, the results of the second stage show that after controlling for the relevant mediation variables in the model, city size and each mediation variable play significant roles in simultaneously influencing older adults’ ADL, indicating that a mediation effect exists. Meanwhile, when the health-improvement effect is considered, the impact effect of city size increases from 0.2841 to 0.2883.

However, when the disease-mitigation, emotional, and employment-structure optimization effects are considered, the impact effect of city size decreases to 0.1960, 0.2498, and 0.2684, respectively, which further confirms the existence of disease-mitigation, emotional, and employment-structure optimization effects.

Second, to further investigate whether a mediation effect exists and its contribution, we used the KHB method. The results are presented in Table 7, which shows that the four mediation variables are significant; that is, a mediation effect exists. Meanwhile, according to the estimated results of the model, the total contribution rate of the four mediation variables to the city-size effect was 35.17%; that is, the total mediation effect was 35.17%. Among them, the emotional and employment-structure optimization effects were the two most critical, with contribution rates of 11.78% and 17.77%, respectively, while that of the health-improvement effect was only 0.77%. Overall, 35.17% of the impact of city size on older adults’ ADL was transmitted through the health-improvement, disease-mitigation, emotional, and employment-structure optimization effects.

Finally, due to the existence of the urban-sprawl effect, to effectively analyze the role of optimal city size, we further investigated the moderation effect of the urban-sprawl index and city air-pollution intensity. The results are presented in Table 8. Models (1) and (2) in Table 8 control for city size × SI and city size × AQI_100 days, respectively, while Model (3) controls for both. The results show that city size × SI and city size × AQI_ 100 days all show significant and negative effects; that is, when the city size reaches a certain level, the higher the urban sprawl and the more severe the city-pollution intensity, the more significant the positive impact of city size on older adults’ ADL, thus inhibiting the health and welfare effects of city size. These findings further show that the larger a city, the better it is; however, it requires a reasonable layout and healthy development. Meanwhile, the city must be alert to the risk of air-quality deterioration during its expansion.