Impact of foreign domestic workers on the fertility decision of households: evidence from Hong Kong

Abstract A potential solution to low fertility is the employment of foreign domestic workers (FDWs), who substitute child-rearing and housework duties, thus reducing child-rearing costs. Recent studies argue that the flow of low-skilled foreign workers into the childcare sector influences fertility choice. However, these studies mainly use the availability of FDWs in the local area as the causal inference and focus on Western countries, making it difficult to identify individual direct effects or generalize the findings to other countries. To bridge this research gap and examine the impacts, this study uses household data from the Hong Kong census. Employing ordinary least squares, the inverse probability weighted regression adjustment, and the instrumental variable approach, we find that households that employ live-in FDWs give birth to more children. Moreover, the heterogeneous analysis reveals that women's greater proportional contribution to household income has a positive impact on households' fertility response after employing the FDWs.


Introduction
Total fertility rate in most developed countries, including the United States, Japan, Korea, and China, is lower than 2.10, called the replacement-level fertility [World Bank (2019)]. The causes of this low fertility in developed countries include female participation in the labor market, high costs of child-rearing, and conflicts of time allocation between couples [Balbo et al. (2013)]. In summary, childcare-in terms of both the time allocation and the monetary budget-is costly for rational couples in industrialized countries [Gauthier (2007)].
To promote an increase in fertility, improvements in childcare availability and quality and a decrease in costs are thus required [Rindfuss and Brewster (1996)]. Many governments have attempted to reduce child-rearing costs by providing a children's allowance or expanding access to childcare facilities. However, these interventions have not yet solved the low fertility issue [Gauthier (2007); Balbo et al. (2013)]. Meanwhile, the marketization of child-rearing and housework has received attention from caregivers and experts as a way to reduce child-rearing costs and burden in developed countries. In particular, low-skilled immigrant workers employed at lower salaries-usually at minimum wage-play a key role in the childcare outsourcing industry because households can allocate their budgets and time to other goods and tasks, including labor supply. Thus, outsourcing child-rearing and housework to foreign domestic workers (FDWs) relieves the monetary and psychological burden of potential parents. Recent studies have shown that the flow of FDWs into the childcare outsourcing industry influences fertility behavior in households [Hazan and Zoabi (2015); Furtado (2016); Bar et al. (2018)]. Furthermore, these studies suggest that hiring FDWs serves as an incentive for highly educated or high-earning women to increase their number of births without sacrificing their careers by lowering child-rearing costs based on the "marketization hypothesis" [Freeman and Schettkat (2005)].
Although the relationship between fertility and the inflow of low-skilled workers has been largely explored based on the availability of low-skilled immigrants in the local area, most existing studies have not examined the direct effect of hiring domestic workers on fertility outcomes at the micro-level. Furthermore, research has mainly been conducted in developed Western countries, thus limiting generalization. Conversely, our study aims to detect the direct effects in households to explicate the impact of employing FDWs in Hong Kong. The government gave us access to a 5% random sample of household-level data from the 2016 bi-census, from which we constructed a dataset for the analysis. Hong Kong has a long history of outsourcing household chores to FDWs to relieve the monetary and opportunity costs of child-rearing. In 1973, the government introduced a policy to issue temporary visas for FDWs to increase their supply in the labor market. As a result, the number of FDWs has increased exponentially in the past two decades. Indeed, Hong Kong's long history of employing more live-in FDWs has resulted in their high labor market share in the region [9.3% of the total workforce in 2016; Legislative Council Secretariat (2017a)]; however, studies of the fertility response to the employment of FDWs are lacking. Thus, our analysis explicates the extent to which FDWs in Hong Kong influence fertility behavior using the number of infants (children under five years) in each household as the outcome.
We employ ordinary least squares (OLS) estimation and the inverse probability weighting regression adjustment (IPWRA) to examine the effect of employing FDWs on the fertility choice of households. Furthermore, we presume that the employment of live-in FDWs may be endogenous because of such unobservable traits as a preference for large families, which affect both the employment of FDWs and fertility choice. Therefore, we apply an instrumental variable (IV) approach, which is commonly used in the literature on health behavior, to address potential endogeneity. We use the status of winning the random ballot of the housing support scheme or receiving the housing support provided by public housing organizations as the IVs for FDW employment. Furthermore, given the availability of a large micro-level dataset, we examine the heterogeneity of the impact of outsourcing home duties to low-skilled immigrants using subgroup analyses based on socioeconomic characteristics such as educational attainment and household income (including their female members).
Our specification with related control variables shows that FDW employment triggers an increase in the number of infants in the household in all the analyses by 0.34-0.60 children. Moreover, as our data allow us to track information on the address of the current residence and the residence five years ago for both samples and FDWs, we exploit this information for the causal inference. When we limit the sample to households that have not relocated to a new address as a subsample analysis, we also find that employing identical live-in FDWs for more than five years increases the number of infants in these households by 0.06-2.18 children, which is a robust causal relationship. Moreover, we find that the probability of having two or more children increases in households employing the FDWs. We test the validity of our IVs based on the novel methodology proposed by Conley et al. (2012) and find that even under the condition that our IVs violate the exclusion restriction, our results are still informative. Further, we demonstrate the subgroup analysis using the same estimation methods. We find that outsourcing households of highly educated and high-income-earning women tend to have more children, corroborating the extant literature. Moreover, this tendency is confirmed more robustly in households in which women's incomes contribute more significantly. Therefore, households in Hong Kong, especially those with women who earn more, give birth to more children after they employ live-in FDWs because of the financial relief regarding child-rearing. Our study reveals that outsourcing a share of home and child-rearing duties to FDWs improves fertility in Hong Kong as other studies that support the marketization hypothesis.
This research contributes to the existing literature in two ways. First, it provides insights into the impact of foreign immigrants on the local community in host countries. International labor mobility is an important phenomenon globally and immigrants affect the economy and society in host countries enormously [World Bank (2018)]. In response to this trend, many empirical studies have shown the outcomes of these immigrants. For instance, a large amount of research explicates whether immigrants affect the wages of natives in the local labor market [National Academies of Sciences, Engineering, and Medicine (2017)], influence the crime rate in cities [Ousey and Kubrin (2009)], stimulate the price of immigrant-intensive services in the local market [Cortés (2008)], and affect the academic performance of peers in schools [Ballatore et al. (2018)]. In particular, regarding the flow of FDWs into the outsourcing childcare industry, some studies explicate the impact on female participation in the local labor market. Cortés and Tessada (2011) suggest that the flow of immigrants into the housekeeping sector increases working hours among women at the top of the wage distribution in the United States. Using the Spanish case, Farré et al. (2011) show that the female immigration reduces the price of household services, and it increases the female labor participation among skilled women. Forlani et al. (2015) demonstrate that the share of immigrants working in housekeeping services affects the increase in native female labor supply, especially in countries with low support for family policy. Cortés and Pan (2013) 1 investigate the impacts of the employment of FDWs on the female labor participation in Hong 1 Although Cortés and Pan (2013) use the micro-level census or bi-census data from the Hong Kong government in a similar way to us and employ the number of rooms as their IV for estimating female Kong. They describe that the employment of live-in FDWs increases the female labor participation, especially among the mothers with young children. Also, they conclude that the mothers who have infants and complete the higher education especially benefit the welfare from the employment of the FDWs. Regarding the fertility response, Furtado and Hock (2010), Hazan and Zoabi (2015), and Furtado (2016) for the United States, Raz-Yurovich (2016) and Forlani et al. (2020) for Germany, and Romiti (2018) for the United Kingdom find that there is an incentive to have more children because of the outsourcing of housework. However, these works mainly use the availability of low-skilled immigrants in the local area from macro-level aggregate data to measure the effects of housekeepers on birth choice, except for Raz-Yurovich (2016); thus, it is impossible to identify the direct effect of outsourcing housework on fertility choice in households. Moreover, the focus of the abovementioned studies is on Western countries; thus, we aim to confirm the external validity of the relationship between fertility choice and outsourcing of child-rearing to low-skilled foreign workers.
Second, this research adds to the strand of the literature on the determinants of family planning in developed countries. In Balbo et al.'s (2013) review of fertility choice in developed countries, several aspects are shown to influence family planning at the micro-level, such as household income [Becker and Lewis (1973)] and sex preference [Mills and Begall (2010)], at the mid-micro-level [e.g., social capital; Philipov et al. (2006)], and at the macro-level, such as GDP [Sobotka et al. (2011)] and the availability of childcare assistance [Del Boca (2002)]. However, as research on the impact of outsourcing house duties on fertility choice remains limited, we add to the literature on the effect of the marketization of childcare. Moreover, our model includes the socioeconomic characteristics of households and women to ensure a robust estimation; thus, we prove whether characteristics other than the employment of FDWs affect the fertility outcomes in the household. Through our analysis, we aim to confirm whether accepting immigrants to support child-rearing activities can improve the social structure in countries facing low fertility.
The remainder of this paper is organized as follows. Section 2 describes the research context, including the conceptual framework and situation of FDWs in Hong Kong. Section 3 describes the data and section 4 discusses the estimation strategies employed. Section 5 presents the estimation results, and section 6 concludes.

Research context
As a theoretical consideration of fertility choice, the quality-quantity tradeoff proposed by Becker and Lewis (1973) explains that higher-income families stop having multiple children to guarantee the quality of their child. This concept has predominated the economic analysis on fertility for many decades, but some recent empirical literatures have suggested mixed relationships in the industrialized countries [Black et al. (2005); Angrist et al. (2010); Mogstad and Wiswall (2016); Bagger et al. (2021)]. As Doepke (2015) suggested, this is because public support for high-quality schooling or the restriction of child labor influences the household fertility decision. On the contrary, as some literatures empirically support, the U-shape relationship between fertility and women's education is observed in recent trends in the US. That is, labor force participation, they suggest that the number of rooms may affect having more children. Therefore, we use the status of the randomization-based housing scheme as our instrument in this research.
high-skilled or high-income families increase their fertility, dissenting from the traditional theoretical thoughts. Some works of literature [e.g., Hazan and Zoabi (2015); Bar et al. (2018)] suspect that this trend relies on the "marketization hypothesis" [Freeman and Schettkat (2005)]; that is, higher-income or higher-skilled mothers purchase the housekeeping or babysitting service, in which the lower-skilled immigrants often serve, to reduce their parental burden, and they enable to have more children.
However, the relationship between fertility and the outsourcing of housekeeping and childcare to lower-skilled immigrants is theoretically complex. As mentioned, the employment of FDWs is expected to relieve the monetary and psychological burden on couples rearing children, especially women. Some studies [e.g., Blau and Robins (1989); Balbo et al. (2013); Furtado (2016)] suggest that several mechanisms may be at work when we consider the effect of outsourcing housework on fertility decisions. First, a positive effect on fertility is expected because of relaxing the time budget constraint for couples. By hiring FDWs, couples in effect earn more hours that can be spent on child-rearing activities, and this may lead to an increase in the number of children. Second, a negative effect on fertility is also expected because with an FDW at home, women who used to perform housework full-time now have a higher opportunity cost of staying home and may decide to enter the labor market. That is, at the extensive margin, these women may engage in labor activities and increase labor hours, which is expected to have negative effects on fertility choice. Furthermore, a negative effect on fertility is also expected for women who had been working before the employment of an FDW. Because hiring an FDW allows them to work longer hours, they may have a higher chance of being promoted or more desire to pursue higher positions. This effect at the intensive margin is expected to negatively affect the number of children.
Although other studies have shown that lowering the costs of child-rearing by outsourcing is beneficial for female participation in local labor markets [Cortés and Tessada (2011); Cortés and Pan (2013)], the literature explains that the flow of low-skilled foreign workers into the childcare sector influences the fertility choice in local households, which is our main research interest. Hazan and Zoabi (2015) explain that highly educated women are likely to bear more children by substituting their time spent on housekeeping services and child-rearing. Furtado (2016) uses US census data to show that immigrant inflows are indeed associated with native women's increased likelihood of having a baby, and positive responses come from married women and women with a graduate degree. Bar et al. (2018) suggest that the relationship between income and fertility flattened between 1980 and 2010 in the United States, a time of increasing inequality, as high-income families increased their fertility. According to the authors, this was because of the marketization of parental time costs, which is supplied by low-skilled foreign workers. Using survival analysis and data from the German Socioeconomic Panel, Raz-Yurovich (2016) demonstrates that outsourcing housework is positively associated with an early second birth among German women. Forlani et al. (2020) find that the local availability of household care service, which female immigrants mainly engage in, positively impacts fertility choice, especially in households with skilled women in Germany. Although it cannot evaluate the direct and significant effect on fertility, Romiti's (2018) study suggests that immigrants in the childcare sector affect female labor supply without a fertility tradeoff using panel data from the United Kingdom. However, no studies examine the direct effects of outsourcing housework on the quantum of fertility choice (i.e., number of children) using the employment status of housekeepers based on micro-level household data. Moreover, as most existing research is focused on the United States and Europe, we cannot verify its external validity in Asia.
Like the West, Hong Kong has also faced a low fertility rate for complex reasons. The fertility rate in the city was 5.01 in 1965. However, it has gradually decreased due to economic growth, stagnating at 1.03 in 2000, 1.13 in 2010, and 1.20 in 2015 ( Figure 1). Although many women in Hong Kong intend to bear children in their life-cycle [Wong et al. (2011)], as Basten (2015) claims, they face a potential obstacle for low fertility, namely, a financial disincentive for childbearing and the inflexibility of time allocation to work and child-rearing. First, higher housing and child-rearing costs are crucial issues in the city because properties are so expensive that young families cannot afford to own a house. In addition, childcare costs are said to be higher because parents expect their children to succeed academically. Moreover, to gain a livelihood, couples need to work for longer hours (more than 49 h per week on average). These working environments create tremendous pressure, especially for women, to combine work and child-rearing. We observe these causes and tendencies not only in Hong Kong but worldwide.
Hong Kong has admitted FDWs into local families for approximately half a century. In 1973, the Hong Kong government introduced a temporary visa for live-in FDWs to release local women from housework and allow them to join the labor market. Since then, the number of FDWs has increased exponentially ( Figure 2). For instance, it was 164,000 in 1996 and doubled to 352,000 in 2016. FDWs are typically from the Philippines (54%) and Indonesia (42%), and 98% are women. Of the households that employ live-in FDWs in 2016, 73.9% are married couples with children; thus, FDWs are recognized as crucial components for childcare [Legislative Council Secretariat (2017a)]. One of the reasons is that the female labor participation rate among the working-age population has increased in the past two decades from 47.8% in 1996 to 54.8% in 2016 [HKSAR (2019a); see Figure 2]. Further, in Hong Kong, it is a criminal offense to leave children under 16 years alone at home [HKSAR (2013)]; hence, FDWs are regarded as important guardians on behalf of working parents. The government sets the criteria for employing FDWs for local families, including a minimum household income (HK$15,000) to guarantee the minimum hourly wage of Hong Kong (HK $37.5) and the provision of one room for them to live in the house, although many live-in FDWs share a room with the children of their employers [Chan (2005)]. Moreover, the government permits a two-year contract for each employee and issues visas for FDWs based on each contract. Changing employers in Hong Kong within the two-year contract is prohibited, and the employed FDWs are once required to return to their place of origin at the end of the two-year contract even if they extend their contracts [HKSAR (2018)]. Finally, although some studies have demonstrated that FDWs in Hong Kong affect female labor participation [Cortés and Pan (2013)] and educational performance among children [e.g., Tang and Yung (2016)], the literature on the relationship between FDWs and fertility behavior in Hong Kong is scarce.

Data
We obtain governmental permission to use a 5% random sample of the 2016 bi-census provided by the Census and Statistics Department in Hong Kong. In the sample, 131,434 households and 366,619 individuals are recorded. To control the characteristics of the sample, we limit the population as follows and constructed the dataset accordingly. First, to trace the fertility history of each household precisely, we keep only households with one married woman aged 20-45 years and omit those without such a woman and those with two or more married women (e.g., households living together with a sister and a mother in the age cohort). Moreover, we consider an adequate income level in our survey. Figure 3 depicts the income distribution of households with women in the targeted age cohort. In this dataset, the variance of the income distribution is significant: some households earn more than HK$150,000 monthly, but the median monthly income in 2016 for the targeted age group is just HK$19,000. As some research related to the FDWs in Hong Kong focuses on the low-and middle-income households too [e.g., Tang and Yung (2016)], this study also focuses on relatively low-and middle-income households, and we limit the monthly income of households to HK$15,000 to HK$58,000. There are two reasons for limiting these income thresholds. Firstly, the Hong Kong government limits the minimum threshold to employ one FDW in each household as HK$15,000, as mentioned in section 2. Secondly, the maximum income for multi-person households to apply for the government's housing ownership scheme (HOS), which we use the beneficiary status in our estimation strategy (details in section 4), is HK$58,000. Considering the possibility of benefitting from employing the FDWs and applying the housing scheme, we assume that limiting this income range is reasonable for our estimation.
After refining the sample, we keep 20,391 households in our dataset. Table 1 shows the descriptive statistics. We separate households that had employed FDWs at the time of the survey from those that had not. A t-test shows significant differences between the two groups for several of the variables. Households that employ FDWs are richer, have more family members and 5-10-year-old children, are likely to live in cities, and have more rooms in their houses. Women in households with FDWs have a higher education, earn a high income, contribute more to the household income, and are likely to live without their husbands. Moreover, the first child in households with FDWs is likely to be a daughter. To estimate the pure effect of the employment of FDWs, we control for some socioeconomic characteristics in our estimation.

Estimation strategy
We aim to investigate the impact of outsourcing housekeeping and childcare to FDWs on the household's fertility behavior. Based on the literature review, we hypothesize that the employment of FDWs increases the number of childbirths in households in Hong Note: +P < 0.10, * P < 0.05, ** P < 0.01, *** P < 0.001.
Kong. As we use observational data instead of experimental data, we employ econometric methods for the analysis to identify a causal relationship and ensure internal validity. To test this hypothesis, we examine the following equation in the simplest form as follows: where y i denotes the number of infants (under five years 2 ) in the ith household, and FDWemployment i equals 1 if the ith household employs an FDW and 0 otherwise. X i is the vector of the characteristics including log of household income, the number of children aged over five years and of elderly people, an island residence dummy, the income ratio, 3 the age and educational experience of women, the first child's sex dummy (Female = 1), and the marriage status dummy (Divorce/Separate/Widow = 1). 4 μ i is the error term in the main model. First, we conduct OLS estimations. However, we suspect that fertility outcomes may be influenced by factors that also affect the employment status of FDWs in households, which leads to an endogeneity problem. Our descriptive statistics show systematic differences between households that employ FDWs and those that do not. Thus, we employ two other methods. Firstly, to reduce potential self-selection bias based on the observable characteristics, we use the IPWRA estimator for the causal inference. This estimator is known as a "doubly robust" estimator because it is a consistent estimate if either the treatment model or the outcome model is correctly specified. Under this method, the inverse of the propensity score is used as weights to run regressions of the outcome variable [Wooldridge (2007)]. As we estimate the average treatment effects on the treated (ATT), we employ a weight equal to 1 for treated observations and ps(x)/(1-ps(x) for control observations, where ps refers to the propensity score, and estimate the treatment effects by comparing the averages of the two groups [Hirano et al. (2003)]. The IPWRA estimator can yield causal estimates of the ATT under the assumption that there is only selection on the observables.
Moreover, to reduce endogeneity due to unobserved factors that affect both the employment of FDW and birth choices, we apply the IV approach. A valid excluded IV, z, should be correlated with the endogenous variable [cov (x endo , z) ≠ 0] and not be correlated with the error term in the main model [cov (z,μ) = 0]. We use the 2 Although some literature that examines the role of outsourcing housekeeping work on fertility choice use other indicators of birth choice such as the birth experience in a particular year [e.g., Hazan & Zoabi (2015); Furtado (2016)] or the existence of child with a specific age [e.g., Forlani et al. (2020)], we select the number of children younger than five years (0-4 years) as the outcome of our estimation for two reasons. First, some studies use the number of children in this target cohort to clarify the recent trend of fertility behavior. For example, Furtado (2016) uses the outcome using this cohort as their robustness check. Second, in our later analyses, to eliminate the possible reverse causality between having a child and hiring an FDW, we restrict our sample in such a way that hiring an FDW occurs before the birth of a child. We do this by using a subsample of households that stayed at the same address for at least five years before the data collection year (i.e., our target cohort should be under five years). As a robustness check, we also analyze the impact on the number of children under three years old in Appendix C and those under one year old in Appendix D. 3 We calculate the income ratio as the woman's income divided by household income. If women do not earn any income, the value is zero. On the contrary, if only women earn income or are single, the ratio is 1. 4 The variables "log of household income" and "income ratio" may hold risks of endogeneity because female's labor supply and the birth choice may have interdependency. As a robustness check, we conduct regressions without these variables and also find consistent results (Appendix E). following two IVs: (1) a dummy variable for winning the random ballot for the HOS organized by the Hong Kong Housing Authority and (2) a dummy variable for whether households receive housing subsidies from the government. Below, we explain how these instruments satisfy the necessary conditions for the IVs.
As mentioned above, households in Hong Kong face a chronic scarcity of housing because of high market prices, which have increased since 2004 following the Asian financial crisis and the SARS epidemic. The price index of domestic properties was 150.9 in 2010 and 296.8 in 2015 compared with the 1999 index of 100 [HKSAR (2020)]. Moreover, housing prices in Hong Kong surpass those in other large cities such as Singapore, Shanghai, Los Angeles, and New York [CBRE (2019)]. Thus, families, especially young families, struggle to afford a house. To support local families, the government offers a housing scheme for inhabitants to rent or own their houses at a lower price.
We examine two major schemes: public rental housing (PRH) and the HOS. The Hong Kong Housing Authority provides PRH for low-income families based on eligibility criteria. Demand for PRH exceeds supply; for example, in 2016, there were more than 153,000 applicants for approximately 12,000 flats [Legislative Council Secretariat (2017b)]. Applicants receive an offer of rental housing by random computer batching based on characteristics such as family size and choice of district [Hong Kong Housing Authority (2020)]. On average, applicants have to wait four years for the first offer of houses [Legislative Council Secretariat (2017b)]. Although computer batching is conducted in groups classified by characteristics, the variation in waiting time is similar regardless; thus, some households can move into the offered flat within three years, whereas others wait for six years or more [Hong Kong Housing Authority (2019)]. In 2019, the average waiting period was 5.4 years [Hong Kong Housing Authority (2019)].
Meanwhile, the HOS has been offered since 1978, although there was a moratorium period from 2003 to 2012 [Legislative Council (2011);Li (2016)]. It provides homeownership opportunities to families unable to buy houses in the private sector. If applicants satisfying the conditions win the annual lottery, they can purchase the houses provided by the government at a discounted price (usually 30-40% of the market price). The Hong Kong Housing Authority has many applications every year. For example, 135,000 households applied at the sale in December 2014, the first HOS sale after the moratorium, and this number of applications is 62.5 times larger than the number of sales flats [Forrest and Yip (2015)]. Although the number of applications and flats fluctuates, competition is high [HKSAR (2019b)]. There are other schemes through which the Hong Kong Housing Authority and Hong Kong Housing Society lease or sell subsidized flats, such as the Tenants Purchase Scheme, Private Sector Participation Scheme, and Flats for Sale Scheme, which mainly follow the ballot method (see the eligibility criteria of the HOS and PRH in Appendix A). 5 We use these two variables for homeownership through the HOS and receiving a housing subsidy as our IVs. For our first condition for valid IVs, FDW employment is likely to be strongly correlated with the housing support variables because the number of rooms is an important FDW employment condition. For the second condition, there are two major concerns. The first is whether self-selection bias exists 5 The proportion of households that own their house using the HOS is 10.1% in the full sample and 13.1% in the not-moving sample. The proportion of households receiving the housing subsidy is 43.3% in the full sample and 52.7% in the not-moving sample.
in the application of these support schemes because households that apply for these schemes and those that do not may be systematically different. However, as demand for these schemes is exceptionally high in Hong Kong, this suggests that most sample households will apply. Although the proportion of young and married applicants for subsidized housing schemes is unknown, demand is extremely high. Li (2016) discusses the housing market and policy instruments in Hong Kong and finds that the cohort born after 1980, the target sample in this research, has faced the most difficulties obtaining housing. Furthermore, according to a survey by the Chinese University of Hong Kong [CUHK (2019)], the housing affordability ratio among Hong Kong citizens has fallen as low as −6.13 in 2015, −5.06 in 2016, and −6.62 in 2017 compared with 4.35 in 2002. This result indicates strong demand for housing support in Hong Kong. Moreover, households that receive support are selected by random ballots. Thus, there is no opportunity for the Hong Kong Housing Authority to assess households based on their characteristics or preferences. Considering these suggestions, we do not expect selection bias in applications, while selection into the housing subsidy schemes is also random in our context. 6 The second concern may be the violation of the exclusion restriction assumption, that is, receiving a housing subsidy from these schemes may affect birth choice directly aside from the effect through hiring FDWs (e.g., via increased wealth due to subsidies). 7 While this is difficult to test, we address this concern by conducting the sensitivity test proposed by Conley et al. (2012) to examine whether the IV estimates provide important information even when the exclusion restriction is violated. We report the results of the bound estimation under the assumption that the instruments have non-zero effects on the outcome.
Based on the above discussion, we model the structural equations for the two-stage least squares (2SLS) using cross-sectional data. The first-stage model is as follows: where each IV Z i is the dummy variable for whether the ith household owns a house by winning ballots or is subsidized for housing by the government in addition to the vector X i used in equation (1). Equation (1) is applied to the second stage of 2SLS estimation. Further, to eliminate the potential reverse causality between the employment of FDWs and outcomes, we also conduct estimations based on a subsample that only includes households that have not moved for five years and redefine FDWemployment i as hiring an FDW who has the same address five years before the data collection (5yearsFDWemployment i ). In the bi-census in 2016, respondents were asked where household members, including FDWs, lived five years ago. We take advantage of this question to ensure that FDWs were working in the house before the children were born. This approach can eliminate the possibility of reverse causality in the model using the same instruments. 8 Also, we employ the probit and IV-probit models with a dummy variable of whether the households have more than two infants or not ( = 1 if they have two or more infants in the household) as the outcome variable to check the probability of having multiple children.
In addition to the 2SLS, we employ the IV-Tobit model, particularly when the dependent variable, the number of infants in each household, which is not infinite, is observed only over some interval of its support [Cameron and Trivedi (2010)]. According to our descriptive statistics, the highest number of infants in each household is four. If we use 2SLS under this condition, the results, especially the coefficients, may be upward-biased; hence, we adopt the IV-Tobit model. 9 We use robust standard errors and report 95% confidence intervals because the sample size is large [Wasserstein and Lazar (2016)].
The first-stage estimation for each instrument in the models shows that the excluded variables are significant at the 0.1% level and strongly correlated with the endogenous variable in the benchmark model (FDWemployment i in Appendix B). There is a difference in the signs of the coefficients between two excluded variables: positive for winning the HOS and negative for receiving housing support. This is attributed to the number of rooms provided in the scheme; that is, the average number of rooms is different between the HOS (3.73 rooms) and the housing scheme (3.06 rooms), while the average number of rooms in the full sample is 3.27 rooms. Cortés and Pan (2013) suggest that the number of rooms affects the decision to employ FDWs because of the governmental regulation in Hong Kong. Therefore, as the housing scheme affects the number of rooms, it also affects the decision to employ FDWs. Although the dummy variable for winning for HOS is not significant in the first stage of the limited-sample model (5yearsFDWemployment i in Appendix B), we find that the F-test for the excluded variables is statistically significant. Moreover, the p-values of the Anderson-Rubin Wald F-test of both models are under 0.01. Thus, these instruments are appropriate for our estimation with the endogenous variable (see the first-stage estimation in Appendix B).
Using the large micro-level household dataset from the census, we can also examine the individual impacts of FDWs on fertility choice. Thus, in addition to the estimation using all the samples in the dataset, we conduct subgroup analyses based on the socioeconomic characteristics of the samples to examine the heterogeneous treatment effects. We examine the effects of women's educational level and income, the income ratio, and household income, which are likely to affect FDW outcomes [Hazan and Zoabi (2015); Furtado (2016)]. Tables 2-5 present the summary statistics of each subgroup based on these characteristics. Regarding the educational attainment of women (Table 2), we divide the sample into five levels: primary school, lower secondary school, higher secondary school, associate degree, and bachelor/graduate level. The mean educational attainment in the full sample is 11.851 years, which represents a higher secondary school level. Furthermore, we investigate the heterogeneity in household income in two strata: low and high, and in five groups: 0-20%, 21-40%, 41-60%, 61-80%, and 81-100% of the mean (Table 3). The mean monthly income of the sample is HK$32,341 and the median monthly income is HK $30,500; thus, the income distribution is upward-biased. Similar to the household income subgroup analysis, women's income is divided into two strata and five groups based on income rank (Table 4). About 32.9% of women in our sample have no 9 Some studies exploring the effects on the number of biological children employ the Tobit or IV-Tobit regression model [Fort et al. (2016); Marcén et al. (2018)]. Around 72% of the women in our sample have no children under five years, 23.6% have one, and 4.4% have two or more. income (i.e., they do not perform market activities outside the home). On the contrary, 54.8% earn more than HK$10,000 per month. Finally, we investigate the heterogeneous treatment effects of the income ratio (Table 5). An income ratio of zero suggests that women do not earn outside the home; on the contrary, an income ratio of one implies that only the women, mainly single parents, contribute to household income. In our sample, 7,579 women (37.2%) contribute more than half of household income.  Table 6 presents the main results estimated using the OLS, IPWRA, 2SLS, and IV-Tobit models. The OLS estimation suggests that the employment of FDWs is positively associated with the number of infants (0.360 children in column 1) in the household at the 0.1% significance level, although this cannot be interpreted causally. If we take into account the confounder on the treatment and outcome and apply the IPWRA estimation, the result is consistent (0.341 children in column 2). Moreover, we also observe the same tendency, namely, positive and significant results, if we employ the 2SLS and IV-Tobit models (0.601 children in columns 3 and 4). This finding corresponds with the theoretical consideration proposed by d' Albis et al. (2017) that easier access to market-based childcare services and the expectation of a lower price allow parents to purchase such services, and this encourages childbearing. Columns 5 and 6 show the results of the impacts of employing the FDWs on having the multiple children, and we find that the probability of having more than two children increases in the FDWs-employed households. In our context, feasible access to FDWs may relax the burden of child-rearing. We also replace the outcome variable with the number of children less than one or three years old and find consistent results (Appendix C and D).

Main analysis
Regarding the other covariates, household income negatively affects fertility outcomes in each model (−0.048 to −0.089 children). In terms of the sex of the first child in the household, son preference has a long history in Chinese culture because the husband is dominant in the family [Bo (2018)]. However, since Hong Kong was colonized by the United Kingdom until 1997, integrating it into international culture, such son preference has diminished and is now regarded as a feudal tradition. Our results show that if the first child is a daughter, these families are more likely to bear a second child (0.344-0.359 children). One of the reasons is that the number of migrants from Mainland China has increased and they have a stronger preference for sons than Hong Kong-born residents [Basten and Verropoulou (2013)]. Moreover, as expected, women's age also negatively affects the number of births. In addition, if the household has children aged five years or older, there is a disincentive to bear a child.
However, as a reverse causality may exist in the above analysis, we undertake another regression by restricting our sample to households that had not relocated in the five years before the survey year (12,404 households) using the address information. Of this sample, 367 households have employed the same live-in FDW for more than +P < 0.10, *P < 0.05, **P < 0.01, ***P < 0.001. Robust-standard error in parenthesis. 95% CI in bracket. About (1), the OLS estimation is represented. About (2), we show the average treatment effect on the treated using the IPWRA estimator. This estimator compares the outcome between treatment group (FDW employing households) and control group (Not-employment group) based on each treatment and control regression estimation. The control variables of outcome model in (2) are the same as (1); on the contrary, the treatment model includes the dummy variables of HOS and housing subsidy with the control variables of the outcome model. For brevity, we only represent the ATT results here. For (3), we employ the 2SLS model. For (4), we employ the IV-Tobit model. About (5) and (6), we replace the outcome with the dummy variable whether they have more than two infants in the family. The coefficients represent the marginal effects. +P < 0.10, *P < 0.05, **P < 0.01, ***P < 0.001. Robust-standard error in parenthesis. 95% CI in bracket. About (1), the OLS estimation is represented. About (2), we show the average treatment effect on the treated using the IPWRA estimator. This estimator compares the outcome between treatment group (FDW employing households) and control group (Not-employment group) based on each treatment and control regression estimation. The control variables of outcome model in (2) are the same as (1); on the contrary, the treatment model includes the dummy variables of HOS and housing subsidy with the control variables of the outcome model. For brevity, we only represent the ATT results here. For (3), we employ the 2SLS model. For (4), we employ the IV-Tobit model. About (5) and (6), we replace the outcome with the dummy variable whether they have more than two infants in the family. The coefficients represent the marginal effects.
five years. Table 7 presents the regression results. The OLS and IPWRA estimations confirm the increase in children at more than the 10% significance level (0.047-0.061 children in columns 1 and 2). The 2SLS estimation and IV-Tobit model show that if FDWs have been employed for more than five years, households had more children (2.050-2.183 children in columns 3 and 4) at under the 1% significance level, which is robust as it is free of possible reverse causality. When we demonstrate the probability of having two or more infants in the household using the probit model approach, the results are significantly positive (0.359-4.054 as the marginal effects in columns 5 and 6); thus, the FDWs prompt the fertility choice in the households. Our IV models show relatively larger coefficients of the treatment of employing FDWs compared to OLS models because these estimators are the local average treatment effects (LATE), which is the average treatment effects (ATE) of compliers in the population [Imbens and Angrist (1994)]. Under this assumption, the IV estimators often exceed the estimates by OLS, and the LATEs do not necessarily inform the exact effect sizes of the ATE in the total samples [Angrist and Evans (1998); Oreopoulos (2006)]. 10 Further, this effect of FDWs on fertility choice in our model is a conservative estimate of the true impact, as we code FDW employment equal to one when the household employs the same FDW for more than five years and zero otherwise. This indicates that those households that have employed the same live-in FDW for under five years or different live-in FDWs over time are coded as zero. Regarding the other covariates, the results are similar to those in Table 6. Hence, the results of this analysis show that FDW employment has a positive impact on increasing the number of children in the household.

Heterogeneity analysis
From our large sample, we explore the heterogeneous effects using subgroup analysis. The models in the subgroup analyses employ the 2SLS models and the same control variables presented in OLS and IV models in Tables 6 and 7, which are less likely to be endogenous. Table 8 shows the treatment effect (FDWemployment or 5yearsFDWemployment) on the number of infants born within five years for each subgroup. Regarding educational attainment (columns 1-5), we find that FDW employment has a robust impact, especially in the higher education groups. In particular, we observe that it has a significant impact on the number of births among women who completed above higher secondary school level (e.g., 0.664-3.887 children in bachelor-level group), concurring with the findings of Hazan and Zoabi (2015) and Furtado (2016). For higher-skilled women, as shown by previous studies, FDW employment may relieve the constraints of fertility choice because the budget constraint is altered based on the marketization hypothesis. Next, we focus on women's income in the household. Columns 6-12 suggest that women who earn more and employ FDWs are more likely to have more babies. Bar et al. (2018) also suggest similar implications using US data. We thus confirm that the marketization of childcare is effective for the fertility choice among high-income 10 We also examined whether the households that employed the same FDWs and stayed at the same address for more than five years statistically differ from those who did not hire the FDWs and stayed for the same period. We find that there is no significant difference in the number of infants (Appendix F). Furthermore, the F-test for checking the multicollinearity of covariances also shows insignificant results on the null hypothesis. Those results are available upon request.   Note: +P < 0.10, *P < 0.05, **P < 0.01, ***P < 0.001. Robust-standard error in parenthesis. 95% CI in bracket. The outcome is the number of infants (less than five years old). We employ 2SLS model in the estimation controlling log-monthly household income, number of elderly persons, number of pupils in the household, HHs that live in islands, age of females, education years of females, divorce/widow/separate dummy, First order's gender dummy and ratio of couple's income. About the treatment effects of FDWs working for more than five years, we limit the sample who has not moved for five years.
women. Furthermore, to check the robustness of the results, we investigate the income ratio (columns 13-19). We confirm that higher-income women and those making the top 40% contribution to household income have more children after exploiting the services provided by FDWs. Notably, the result for the top 20% income earners is more robust (0. 941-1.929 children; column 19). This implies that a woman's income and her contribution to household income are crucial determinants of the fertility decision when households employ FDWs. We also find that both low-and high-income households employing FDWs have a positive impact on fertility behavior at the 5% significance level (columns 20-26). In addition, we observe a positive impact in all five income groups. This tendency is the same for the model using FDW employment over five years. In summary, the employment of FDWs is effective at raising the rate of having children in households regardless of household income, especially if we control for the related variables. Hence, all things being equal, FDW-employing households, even lower-income ones, may increase their number of children.
To clarify the mechanism through which income and FDW employment affect the fertility decision, we create a 2 × 2 analysis between low and high household income and low and high income ratio (columns 27-30). Our higher contribution group, in which women earn an income of more than 40% of the household total (the minimum ratio of the higher income ratio group is 40.03%), is responsive to the fertility choice against access to FDWs regardless of the absolute value of household income. We observe the same tendency in the five-year estimation. These results imply that the income contribution by women rather than absolute household income is a crucial component for the fertility response to outsourcing housekeeping and child-rearing to immigrants.

Sensitivity analysis
To confirm the validity of our IV results, we conduct the sensitivity test developed by Conley et al. (2012) known as the plausibly exogenous test. This test assesses whether our IVs continue to provide informative results when relaxing the strong assumption of the exclusion restriction. We rewrite the 2SLS equations in equations (1) and (2), including the additional terms of the IVs in the second stage, as follows: In the previous analysis, we assumed that θ j = 0 in equation (4), and this resulted in point estimates for the coefficients of FDWemployment i and 5yearsFDWemployment i . According to Conley et al. (2012), researchers can remove the assumption that θ j = 0 and select possible values for θ in a range of flexible ways, allowing the instruments to have small direct effects on the outcome, which violates the assumption of the exclusion restriction. Under this weak condition, we estimate the bounds of the effect of the endogenous variable, namely, FDWemployment i and 5yearsFDWemployment i . We employ the union of confidence intervals approach in the plausibly exogenous test and specify the possible restrictions for u j [ [−k, + k]∀ j, where κ > 0 is arbitrarily small. The size of κ is the direct effect of the instruments on the number of infants and the degree of the violation of the exclusion restriction. Table 9 shows the results of the plausibly exogenous test in which we estimate the bounds for the estimator of interest β using 95% confidence intervals with the alternative values of κ = 0.0001, 0.0005, 0.001, 0.005, and 0.01. The results are from the 2SLS estimator controlled by the same vector as in column 3 of Tables 6 and 7. We find that the estimated bounds do not fluctuate with κ as much and that no bounds of our estimation include zero or negative coefficients. Moreover, the 95% confidence interval is [0.423, 0.778] for the effect of employing FDWs (column 3 in Table 3) and [1.338, 2.762] for the effect of employing FDWs for more than five years (column 3 in Table 4) when we assume that the IVs do not violate the exclusion restriction. Given that the union of confidence intervals approach proposed by Conley et al. (2012) is considered to be conservative, this indicates that the results presented earlier are robust even if we relax the strong exclusion restriction necessary for the estimation employing the IVs. Thus, we can conclude that our central inference is informative and valid.

Conclusion
We explore the impact of employing FDWs in Hong Kong on fertility outcomes using the micro data from the census. Using OLS, the IPWRA, and the IV approach with variables related to the housing schemes, we find that households that employ FDWs have more children. To validate the robustness of this finding, we exploit past and current address information and explicate the causal relationship between the long-term employment of FDWs and fertility response. We find that households that had employed the same FDW for five years before the survey year robustly increased the number of children. Furthermore, the subgroup analysis reveals that highly educated women have a positive impact on the fertility response in households. Regarding income characteristics, we find that the employment of FDWs in Notes: The outcome is the number of infants (less than five years old). We employ 2SLS model in the estimation controlling log-monthly household income, number of elderly persons, number of pupils in the household, HHs that live in islands, age of females, education years of females, divorce/widow/separate dummy, first order's gender dummy and ratio of couple's income. We employ robust standard error. About the treatment effects of FDWs working for more than five years, we limit the sample who has not moved for five years. Bounds are estimated using the STATA package "plausexog".
households is prevalent when women have a higher income and contribute more to household income. In summary, FDWs in Hong Kong have a positive impact on households' fertility response, especially in "career women's households". Moreover, we confirm that relaxing the exclusion restriction does not nullify our inference that employing FDWs positively affects birth choice using a sensitivity analysis. As mentioned in the research context, there are two potential fertility-related outcomes when employing FDWs in households: an increase in births due to easing budget constraints or a decrease in births due to female participation in the labor market. Our results show that FDWs in Hong Kong have recently served to reduce the costs of child-rearing in households rather than increasing its opportunity cost. In particular, for households in which women have a higher education or contribute more income to the household, the fertility response is more responsive to FDW employment. This implication supports the concept of the marketization hypothesis [Freeman and Schettkat (2005); Hazan and Zoabi (2015); Bar et al. (2018)], meaning that women purchase and outsource their housework to migrant workers so that they are able to give births without sacrificing their careers. The first-stage estimation of our IV approach suggests that households with higher female income ratios are more likely to employ FDWs. That is, they have a strong preference for outsourcing home duties and have children after hiring FDWs. We consider these outcomes for two main reasons: to mitigate conflicts with career advancement and to exert higher bargaining power in the household. First, high-skilled women face difficulties handling both home duties and labor supply because of time constraints. FDWs relieve their pressure by combining these tasks, and this can motivate women to have a child to maintain work/life balance. Second, if the woman's income share in the household is high, her intra-household bargaining power may rise because her income provides direct bargaining power and because such women tend to acquire the knowledge or skills to handle the work environment, which may raise their intra-household bargaining power further [Hoddinott and Haddad (1995); Doss (2013)]. Some recent studies find that households decide their intra-household resource allocation based on not only the unitary model but also the collective model, which allows for different preferences and bargaining power among household members [e.g., Browning and Chiappori (1998); Attanasio et al. (2012)]. In particular, women prefer to consume health-and education-related resources more than men, and households with female high bargaining power consume more such items [Hoddinott and Haddad (1995); Quisumbing and Maluccio (2003)]. These explanations apply to the preferences and perceived costs of outsourcing housekeeping and child-rearing; that is, households with women who contribute to household income more and have higher decision-making power are likely to outsource household duties. Although fertility decisions are jointly made with partners, women can have a much stronger influence on the fertility choice in the household than men [Rasul (2008); Stein et al. (2014); Doepke and Kindermann (2019)]. The employment of FDWs who relieve the burden of child-rearing thus promotes the willingness of households that have women with higher bargaining power to have more children. Both mechanisms may drive fertility decisions in households with higher-skilled and higher-income women.
In comparison with the above description, we discuss the tradeoff between quality and quantity of children. The conventional economic intuition suggests that the parents will spend less on the quality of children if they invest in the quantity more. Although our data do not allow us to investigate the quality or skills of the infants, the existing literature implies that there is a positive relationship between the employment of FDWs and the quality or skill enhancement among children in Hong Kong. Tse et al. (2009) tested the English reading ability among the fourth-grade students in Hong Kong and find that the children in the FDW-employing households get higher scores. This tendency is robust among the children that the parent's socioeconomic status is relatively high. Tang and Yung (2016) explore the impact of employing the FDWs on the educational performance of children using the household data from the census and the unique survey. They reveal that the probability of late schooling decreases at 3-4% among elementary students whose mothers work and employ Filipino live-in FDWs. Tang (2019) discovers that the probability that children can speak English increases by 12-25% of their mothers works outside and the FDWs speak English in their home. Moreover, Leung (2012) and Dulay et al. (2017) also present that the FDWs are likely to improve the communication skills or the language ability of the children. Through these works, we understand that the FDWs play a crucial role in the skill enhancement of the children, and this structure is robust among the households with working mothers or high-income parents. In other words, our findings and the previous literature suggest that the higher-income households purchase and outsource the housekeeping services, and guarantee both the quality and quantity of children, relying on the marketization hypothesis. On the contrary, it is relatively difficult to ensure both the quantity and the quality of children from the households with lower income because such a concept may not apply to these groups from the findings. As De la Croix and Doepke (2003) imply the linkage of inequality and the differentials of the fertility between rich and poor based on quality-quantity tradeoff, we need to explicate the differential of the quality and the quantity under the marketization hypothesis in the future.
Our analysis has important implications for countries that have faced or will face low fertility. First, as Furtado (2016) suggests, immigrants increase labor supply in host countries, especially in low-skilled industries, and immigration increases the fertility rate, which can maintain the strength of the future workforce. While the literature finds such possibilities in the United States and Europe, this study explicates that they are also applicable in Asia. Asian countries such as Japan, Thailand, Singapore, China, and Korea have faced an extremely low fertility rate for decades. Other emerging countries also face the risk of a declining fertility rate [United Nations (2019)]. Some countries have started to issue temporary visas for low-skilled immigrants to strengthen the labor force in industries including the housekeeping sector, and these immigration policies could also be effective for demographic issues. Moreover, the findings of our subgroup analysis imply that considering both the direct and the indirect costs of child-rearing is important for implementing family policies. As mentioned in the Introduction, the outcomes of existing policies to solve demographic issues such as children's allowance and parental leave are mixed. Although these policies can compensate for some of the costs for mothers, they may be unable to mitigate conflicts between career success and maternity in their daily lives. According to Doepke and Kindermann (2019), subsidizing market-based childcare services is more efficient at increasing the fertility rate than other policies such as parental leave and tax credits. In other words, FDWs help parents not only by performing home duties at lower cost but also by easing the pressure on them, especially high-earning women, thereby allowing them to balance their jobs and household responsibilities. Therefore, alleviating both the monetary (direct) and the opportunity (indirect) costs of child-rearing is important for "having it all." However, this study has its limitations. To understand the mechanism by which FDWs influence the fertility decision of each household, it uses cross-sectional data, which cannot fully control for unobserved characteristics in the estimation. Another limitation is that we do not have a complete history of the FDW employment of households. While we eliminate potential reverse causality by using subsample analyses and IV strategies, future studies should seek to collect specific information on this variable, as this is not available in census data.