Unit root test

To avoid spurious regression due to the nonstationarity of the data, a stationarity test of the variables was performed using the ADF unit root test method, and the test results are shown in Table 2. Table 2 shows that all the variables pass the Augmented Dickey–Fuller (ADF) test at the significance level of 1 or 5%, thereby rejecting the null hypothesis of the existence of a unit root and indicating that all the variables are stationary and can be included in the regression analysis.

Benchmark regression

Table 3 reports the impact of educational inequality on families’ multichild behavior. Columns (1) and (3) report the estimated results regarding the random effects (RE), and columns (2) and (4) report the estimated results concerning the fixed effects (FE). Column (1) and column (2) reveal that regardless of which estimation model is used, the estimation coefficient of Pilotrt remains significantly negative. The applicability of the RE and the FE are further analyzed via the Hausman test. The results reveal that the P value of the statistics associated with the model is much lower than 0.01, thus indicating that the FE are correlated with other explanatory variables in the model; thus, the hypothesis regarding the RE is rejected. Therefore, the results regarding the FE presented in column (4) of Table 3 are primarily used as an example to analyze the results. As shown in column (4), the estimated coefficient of Pilotrt remains significantly negative after the individual-, family- and city-level control variables are added, thus indicating that the “super middle school” phenomenon in cities has a significant negative effect on families’ multichild behavior; accordingly, H1 is confirmed.

Endogeneity test

This paper further used the instrumental variable method to reduce the degree of endogeneity bias as a test of the robustness of the empirical results. In line with the research conducted by Xia and Lu (2019), the total number of Jinshi (successful candidates on the highest-level imperial examinations) during the Ming and Qing Dynasties for each city was used to construct the instrumental variables. Reasonable instrumental variables should satisfy both correlation and externality requirements. In terms of relevance, the formation of a city’s culture requires long-term accumulation, and developed ancient imperial areas are considered to be the core areas for Chinese culture and education (Shen 2004). Therefore, previous studies have often used the number of Jinshi in the Ming and Qing Dynasties to reflect the cultural heritage of and educational changes in various cities; furthermore, this factor is closely related to the “super high school” phenomenon in cities. With respect to exogeneity, historical data regarding the total number of Jinshi people in the Ming and Qing Dynasties can meet the relevant conditions to a large extent. Given that cross-sectional data concerning the total number of urban Jinshi residents during the Ming and Qing Dynasties are available^{Footnote 6}, this paper introduces the interaction term (Jinshi) between the original variable and the year dummy variable into the model as an instrumental variable and uses a two-stage regression method to estimate the model. The regression results regarding the instrumental variables are shown in Table 4.

Column (1) reports the regression results for the first stage. Jinshi is significantly positively correlated with Pilotrt at the 1% level, thus indicating that the higher the total number of Jinshi in the Ming and Qing Dynasties in a city is, the higher the probability of the existence of a “super high school” phenomenon in the city in question; thus, the instrumental variables meet the correlation conditions. In addition, the F statistic is 387.553, which is greater than the critical threshold of 16.38 at the 10% level with regard to the Stock–Yogo weak recognition test; thus, the possibility of a “weak instrumental variable” is excluded. The value of the LM statistic (rho) is 0.000 is less than 0.01, thus rejecting the null hypothesis, which is not identifiable. All the test results indicate that the selected instrumental variables are suitable.

Column (2) reports the results of the two-stage regression. The estimated coefficient of Child_num is significantly negative and larger than that of the benchmark regression, thus indicating that the influence of Pilotrt on Child_num is underestimated in the benchmark regression due to the influence of endogeneity. Overall, the results of the estimation of the instrumental variables reveal that the existence of a “super high school” phenomenon in a city significantly inhibits the number of family births in that city, whereas ignoring the endogeneity problem can cause this effect to be underestimated.

Robustness test

To mitigate the randomness of the research results to the greatest extent possible, this paper tests the robustness of the estimated results of the benchmark regression by reducing the sample range and replacing both the measurement model and instrumental variables.

Reducing the sample range

In light of the problem of survival analysis, that is, the fact that the number of births produced by young women at the observation point does not represent their eventual number of births, it is possible to overestimate the effect of the “super high school” phenomenon in cities on the number of births in families in this case. Therefore, this paper relies on the research conducted by Zhu and Zhao (2022) by positing that the survival analysis problem can be addressed by limiting the age of married women; thus, the samples are recalculated to include married women between the ages of 31 and 40 years and subjected to a robustness test. The regression results are presented in Table 5a.

The regression results presented in column (2) of Table 5a reveal that the estimated coefficient of Pilotrt is significantly negative at the 1% level, thus indicating that after addressing the survival analysis problem, the “super high school” phenomenon in the city continues to have a significant negative effect on the number of family births, and the conclusions obtained in this context are consistent with the benchmark regression results.

Replacing the measurement model

Given that the dependent variable is a counting variable, this paper further uses the Poisson model for the robustness test. Table 5a (3) and (4) report the regression results regarding the effect of the “super high school” phenomenon on the number of family births according to the Poisson model. The estimated coefficients of Pilotrt are all significantly negative, thus indicating that the urban “super high school” phenomenon has a significant negative effect on the number of family births; this finding confirms the robustness of the estimated results.

Replacing the instrumental variables

In addition to the data concerning the number of Jinshi in the Ming and Qing Dynasties, some scholars have used the number of Confucian academies to reflect the cultural characteristics of a city. Therefore, this paper introduces the interaction term (Shuyuan) between the number of Confucian academies and the annual dummy variable in the model as an instrumental variable for the robustness test; the results are presented in Table 5b.

Column (1) presents the first-stage regression results. Shuyuan has a significantly positive effect on the existence of “super high schools” in cities, thus indicating a strong correlation between the number of Confucian academies and the existence of “super high schools” in cities. In addition, the F statistic is greater than the critical threshold at the 10% level with regard to the Stock–Yogo weak recognition test, and the LM statistic corresponds to a value of less than 0.01. Column (2) presents the second-stage regression results, according to which the estimated coefficient of Pilotrt is significantly negative at the 5% level, thus indicating that the presence of the “super high school” phenomenon in a city inhibits the number of family births in that city, thereby providing further support for the robustness of the benchmark regression results.

Performing other robustness tests

To ensure the reliability of the results, a robustness test is performed to take the following aspects into account. First, the winsorization process is applied to the continuous variables at the 5% and 95% levels. Second, this paper further introduces the interactive region and time fixed effects with the aim of controlling for the influence of regional characteristic factors, which may change dynamically over time, on the results. Finally, in light of the influence of regional differences on the estimation results, this paper divides the total sample into three subsamples: East, Central and West.

The results of winsorization are presented in Table 5c (1), and the estimated coefficient of Pilotrt is significantly negative at the 1% level. The results regarding the fixed effects are shown in Table 5c (2), and the estimated coefficient of Pilotrt does not change substantially; that is, the previous estimated result is reliable. The results pertaining to the regional dimension are shown in columns (3)–(5), respectively, of Table 5c. The estimated coefficients of Pilotrt are significantly negative, thus indicating that the conclusions obtained are consistent with the benchmark regression results after taking into account the influence of regional factors on the number of family births.

Mechanism analysis

According to the benchmark regression results, the “super high school” phenomenon in cities can reduce the number of family births. To verify the mechanism underlying the effect of this phenomenon on the number of family births in further detail, on the basis of the preceding theoretical analysis, this section tests whether this phenomenon in cities has a negative effect on the number of family births through increased family education costs, which are divided into two dimensions, i.e., the explicit and implicit costs of education for families. The regression results are presented in Table 6.

Column (1) reports the estimated effect of the “super high school” phenomenon on the explicit costs of education faced by families. The estimated coefficient of Pilotrt is significantly positive at the 5% level, thus indicating that this phenomenon increases the total annual education expenditure of families, that is, the explicit costs of education faced by families. Therefore, the “super high school” phenomenon in cities can reduce the number of family births by increasing the explicit costs of education faced by families.

Columns (2) and (3) report the estimated results of the effect of the “super high school” phenomenon in cities on the implicit education costs faced by families. The estimated coefficient of Pilotrt presented in column (2) is significantly negative at the 10% level, thus indicating that this phenomenon significantly reduces the labor availability of women and causes them to face income loss (i.e., an opportunity cost). The estimated coefficient of Pilotrt presented in column (3) is significantly positive at the 5% level, thus indicating that this phenomenon significantly improves women’s participation in children’s education; that is, it increases their time and energy investments in their children’s education. Therefore, the “super high school” phenomenon in cities can reduce the number of family births by increasing the costs of implicit education for families, thus supporting H2.

Heterogeneity analysis

In light of the heterogeneity of the influence of the “super high school” phenomenon on the number of family births, this paper conducts a subsample test on the basis of female education level, urban and rural classification and urban grade. The estimated results are presented in Tables 7a and 7b.

Education level

Previous studies have suggested that the influence of female education level on the number of family births is influenced by the sum of the substitution and income effects (Amin and Behrman 2014). The substitution effect refers to the fact that the higher the education level women have obtained, the greater the opportunity costs they face with respect to having children. The income effect refers to the fact that the higher the education level women have obtained, the more likely they are to earn a high income and the less economic cost pressure they face due to bearing children. Therefore, this study divides the total sample into three groups, i.e., “primary education level”, “middle education level” and “higher education level”, with the goal of investigating the influence of the “super high school” phenomenon on the number of family births among women with different education levels.

Table 7a reports the differences in education level according to the effect of the “super high school” phenomenon on the number of family births in cities. With the exception of column (3), the estimated coefficients of all Pilotrt variables are significantly negative, thus indicating that for women with a primary education level or a middle education level, this phenomenon has an inhibitory effect on the number of family births. According to the regression results presented in column (3), the estimated coefficient of Pilotrt is negative but not significant; that is, for women with a higher education level, the influence of this phenomenon on the number of family births is statistically equivalent to zero.

Urban and rural classification

In this paper, the total sample is divided into rural and urban households on the basis of household registration location. The estimated results are reported in Table 7b. The estimated coefficient of Pilotrt presented in column (1) is negative but not significant, thus indicating that the “super high school” phenomenon in urban areas has no obvious inhibitory effect on the number of births among rural families. The estimated coefficient of Pilotrt presented in column (2) is significantly negative at the 1% level, thus indicating that for urban families, this phenomenon significantly inhibits the number of family births.

Urban grade

Given that cities at different levels exhibit tremendous differences in terms of their levels of economic development, educational resources and other aspects, these differences may cause the influence of the “super high school” phenomenon on the number of family births to be heterogeneous. Therefore, in line with the research conducted by Li and Yang (2019), this study divides the total sample into two groups, i.e., “general cities” and “key cities”, according to whether the cities in question are municipalities that are directly under the central government, provincial capitals or subprovincial cities; the regression results are shown in Columns (3) and (4) of Table 7b, respectively. The estimated coefficient of Pilotrt is significantly negative at the 1% level, thus indicating that the “super high school” phenomenon has an inhibitory effect on the number of family births in cities of different grades and that the impact on “key cities” is significantly stronger than the corresponding impact on “general cities”.