The Impact of Rural Labor Migration on Elderly Health from the Perspective of Gender Structure: A Case Study in Western China

Aging and rural labor migration have become two major demographic features in China. Using data of 400 rural households in Sichuan, China in 2015, this study constructs an ordered probit model containing instrumental variables to analyze the effect of rural labor migration on the health of the elderly in the family, from the perspective of the gender structure of migrant labor. The results indicate that the overall impact of labor migration on the elderly’s health is positive, and labor migration in the family has different effects on the health of the elderly by gender. Specifically, the results indicate the following: (1) the joint migration of both male and female labor or the migration of only male labor in a household can have a positive impact; (2) if only female labor migrates, the impact is negative; and (3) although the effect is negative, the migration of only female labor has a more prominent impact on the elderly’s self-rated health, whereas the migration of only male labor has a more significant effect on the elderly’s activities of daily living. Our findings suggest that the differential influence of labor migration by gender on rural elderly health should be considered to ensure the welfare of the elderly.


Introduction
Aging and migration of agricultural labor, especially of young and middle-aged people, have become two of the most prominent demographic characteristics in developing countries [1][2][3][4][5]. Moreover, there is growing attention given to the impact of labor migration on the health of elderly. As a result of the lack of a pension system [6][7][8], family support remains the main pension model in developing countries such as China, especially in rural areas [9,10], by means of the intergenerational support of family members for the security of the elderly. The research suggests that labor migration not only exacerbates rural population aging, but also changes the traditional family intergenerational support structure [11,12], and thus affects the health of left-behind elderly.
Intergenerational support mainly includes three parts, economic, instrumental, and emotional support [13,14]. Current studies generally address the effects of change caused by labor migration on the elderly health via two paths. The first is through economic support. On the one hand, some scholars believe that labor migration can improve the overall ability of the family to provide economic support [15], and thus improve the living standard of family members, including the medical service level of the elderly [16], which has a positive impact on the health of the elderly [17,18]. On the other hand, some studies argue that labor migration in rural families weakens economic support for the elderly and increases their workload, thus, harming the elderly's health [19], because migrant labor may pay more attention to their own development, and may not necessarily provide economic support to the elderly left behind [19,20]. Moreover, even if the migrant labor remits to the elderly left behind, little is known about the extent to which remittances are directed to the elderly [21]; especially when the migrants have children left at home, the elderly often have to raise grandchildren in exchange for economic support from their children [12,22], which further aggravates the labor burden of the elderly [23]. The second path is through instrumental and emotional support. Most scholars who focus on this path believe that labor migration reduces the accessibility of instrumental and emotional support that the elderly receive and has a negative impact on the elderly's health [21,24]. Furthermore, in China, most studies show that elderly with only one child [25] and living alone [26] are most affected [8,15,27,28]. In summary, existing studies have shown that labor migration creates disruptions in family care systems [11] and affects the health of left-behind elderly, however, the existing conclusions are not consistent, and there are still some important limitations that may affect the comprehensive understanding of the correlation between labor migration and elderly health.
First, not much research has been undertaken to analyze the impact of labor migration on the health of the elderly from the perspective of gender structure [28]. In the traditional Chinese rural family division of labor, men tend to seek work through migration as the main breadwinners [29], whereas women are more likely to act as caregivers [30], and intergenerational support, especially economic support, should be provided mainly by sons [31]. Nevertheless, currently, rural female labor is increasingly inclined to seek work through migration for higher income [22] and the gender differences in economic support for the elderly by sons and daughters have been narrowing [30,32]. Moreover, although son preference is still highly prevalent in rural China, daughters tend to offer a greater benefit to parents who have reached old age, being associated with greater filial piety, better relationships, and satisfactory care [33]. Furthermore, while married sons usually assume social responsibility for supporting the family's elderly, their spouses (daughters-in-law) are, in fact, the main contributors to daily care [34]. In other words, daughters and daughters-in-law often play a more important role than sons do in daily instrumental and emotional support for the elderly [35]. Therefore, it can be assumed that the migration of male labor, which is a traditional source of economic support, can improve the economic support for the elderly, but might not have a significant negative impact on the elderly with respect to instrumental support and emotional support. Although the migration of female labor can also increase economic support for the elderly, it may remove a source of caregiving and companionship for the elderly who are left behind [36,37], which will weaken the family's daily instrumental and emotional support for the elderly. Thus, differences due to migrant gender of the impact of labor migration on the health of the elderly need further quantitative analysis.
Secondly, only a few earlier studies have considered the differences in the effects of labor migration on different aspects of the elderly's health, however, health is a multidimensional concept, and the effects of labor migration on the physical and mental health of the elderly differ [21,38]. For example, Adhikari et al. [16] found that the migration of adult children has a significant negative effect on the mental health of the elderly but no significant effect on physical health. Furthermore, most research to date has focused mainly on the elderly's physical health, neglecting the interactions between physical and mental health [15,25,39], however, if the elderly feel isolated when left behind, their physical health might also deteriorate [40]. Therefore, the comprehensive health statue of the elderly other than physical health should also be taken into account.
In addition, endogeneity problems of the models in many studies have not been adequately addressed [18,27,40]. According to the new economics of labor migration (NELM) [41], a decision about labor migration is not an independent individual decision, but a common one for the entire family. Furthermore, farmers are rational, and make their decisions about labor distribution under the principle of maximizing family interests and minimizing risks. Therefore, the health status of the elderly affects the migration decision of family labor [18,42]. Giles and Mu [42] showed that poor parental health reduces the likelihood of rural-urban migration for male children in China, because the elderly in ill health tend to require more instrumental support than do those who are well, however, it is also possible that some elderly in ill health expect their family members to seek work through migration in order to earn higher income to pay for their medical expenses and improve their health. Consequently, the migration of family labor force is a self-selection decision rather than a random event, which is a problem of reverse causation with the health of the elderly, and thus an endogeneity problem is evident.
Adopting survey data of peasant households in Sichuan Province, this study uses the activities of daily living (ADL) to measure the physical health of the elderly, and self-rated health (SRH) to measure the comprehensive physical and mental health of the elderly. Then, this study constructs an econometric model to explore the differential influence of labor migration by gender on the health of the elderly and uses instrumental variables to solve the endogeneity problem of the model. The present research covers the entire labor force in the family. Moreover, this study does not specifically use a quantitative model to identify the path of labor migration that affects the health of the elderly, but instead explores the overall effect of labor migration on the health of the elderly. The key scientific questions to be answered are as follows: (1) Overall, how does labor migration in the family affect the health of the elderly? and (2) What is the difference between the effects of labor migration by gender on the health of the elderly? In addition, the following two hypotheses are proposed: Hypothesis 1 (H1). The effect of female labor migration on the health of the elderly is negative.
Hypothesis 2 (H2). The effect of male labor migration on the health of the elderly is positive.

Study Area and Data Source
Sichuan Province, an important exporter of labor to other parts of China [43], is located in the southwest of China. In 2017, the population of Sichuan aged 65 years and above was 11.57 million, accounting for 13.94% of the resident population, and the province had the second highest number of elderly people in the country [44]. Meanwhile, due to relatively lower overall economic development [45], the local government's ability to pay for old-age welfare is weaker than that in wealthy regions [46], exacerbating the rural elderly's reliance on the intergenerational support of family members in Sichuan.
The data used in this study mainly come from a questionnaire survey conducted among farmers of Sichuan Province in April 2016 (data for 2015). The survey sample was determined using a combination of stratified sampling and equal probability random sampling, and the household was taken as the final sample level. First, all 183 counties in Sichuan Province were presented in descending order according to per capita gross value of industrial output (GVIO) and divided into five equivalent groups. The GVIO was used based on Rozelle [47], who showed that it is one of the best predictors of both living standard and development potential and is often more reliable than net rural per capita income [48][49][50][51]. One county was randomly selected from each group, leaving a total of five counties, Guangyuan, Shehong, Guangan, Jiangyou, and Zigong. Secondly, the townships in all the sample counties were divided into two groups in accordance with the GVIO (taking the county median as the criterion of division, one group has per capita GVIO above the median and the other group below it), and one township was randomly selected from each group, leaving 10 townships in total. Finally, the same method was adopted to select the final 20 sample villages and then 20 households were randomly extracted according to the roster with reference to every village. Through the abovementioned procedure, a total of 400 households were selected. The distribution of the sample area is shown in Figure 1. health status data of the elderly, including ADL and SRH, as well as personal demographic characteristics, household data, and village data that may affect the health status of the elderly.
In the sample selection, we focused only on family members who are registered together. We further restricted the sample to parents aged above 50 years and deleted observations with key missing variables. The main sample included 272 households and 508 elderly.

Dependent Variables
The research object of this study is the elderly aged 50 years and over in the family and the key to the study is the measurement of the health status of the elderly. Although the definition of an elderly person as aged 50 years and over does not meet the demographic criteria, in rural China, people aged 50 years and over usually have adult children and are no longer able to engage in heavy manual labor. In this way, the elderly population affected by the labor migration in the family can be investigated to the maximum extent, which is more in line with the actual situation in China's rural areas [25,52].
In this study, health status was measured from two aspects, physical health and comprehensive physical and mental health. The ADL was selected as the indicator of physical health. The ADL can reflect the self-care ability of the elderly, which is an important indicator of health status [53]. In the survey, we collected eight items, namely, walking, standing, carrying heavy objects, squatting, bathing, cooking, dressing, and going to the toilet. All items can be accomplished independently by the definition of normal function; one item cannot be completed at all or two items need completion with the help of others for impaired function; and two or more items cannot be completed at all or three or more need completion with the help of others for dysfunction. SRH was used to measurement comprehensive physical and mental health. In the survey, SRH was obtained through the question "Personal Health Status in 2015" and was divided into five grades, the higher the score, the better the health status. Because SRH is a subjective evaluation, it reflects the mental health of the elderly to a large extent based on the evaluation of their objective physical health level. Many studies have shown that SRH can effectively predict the mortality rate of respondents [27,54], which is a simple and easy available comprehensive indicator. The survey was conducted in the form of face-to-face interviews using a predesigned questionnaire. To ensure the survey data was true and reliable, all investigators received rigorous training and carried out research before the survey was formally conducted. The survey collected the health status data of the elderly, including ADL and SRH, as well as personal demographic characteristics, household data, and village data that may affect the health status of the elderly.

Independent Variables and Controlled Variables
In the sample selection, we focused only on family members who are registered together. We further restricted the sample to parents aged above 50 years and deleted observations with key missing variables. The main sample included 272 households and 508 elderly.

Dependent Variables
The research object of this study is the elderly aged 50 years and over in the family and the key to the study is the measurement of the health status of the elderly. Although the definition of an elderly person as aged 50 years and over does not meet the demographic criteria, in rural China, people aged 50 years and over usually have adult children and are no longer able to engage in heavy manual labor. In this way, the elderly population affected by the labor migration in the family can be investigated to the maximum extent, which is more in line with the actual situation in China's rural areas [25,52].
In this study, health status was measured from two aspects, physical health and comprehensive physical and mental health. The ADL was selected as the indicator of physical health. The ADL can reflect the self-care ability of the elderly, which is an important indicator of health status [53]. In the survey, we collected eight items, namely, walking, standing, carrying heavy objects, squatting, bathing, cooking, dressing, and going to the toilet. All items can be accomplished independently by the definition of normal function; one item cannot be completed at all or two items need completion with the help of others for impaired function; and two or more items cannot be completed at all or three or more need completion with the help of others for dysfunction. SRH was used to measurement comprehensive physical and mental health. In the survey, SRH was obtained through the question "Personal Health Status in 2015" and was divided into five grades, the higher the score, the better the health status. Because SRH is a subjective evaluation, it reflects the mental health of the elderly to a large extent based on the evaluation of their objective physical health level. Many studies have shown that SRH can effectively predict the mortality rate of respondents [27,54], which is a simple and easy available comprehensive indicator.

Independent Variables and Controlled Variables
The main influencing factor of focus in this study is labor migration in the family. Among the variables, labor force was defined as the population in the family aged between 16 years and 50 years and excluded the population who were still at school or unable to work owing to illness, pregnancy, or disability. Migration was defined as working outside the county, where household registration was located for more than 3 months [55][56][57][58][59].
In addition to labor migration, the health of the elderly may be affected by many other factors. In order to avoid deviation of the model caused by missing variables, based on existing research [16,21,24,25,[60][61][62], this study added control variables at three levels, individual, household, and village. At the individual level, age, gender, marital status, education level, health insurance, and work conditions of the elderly were controlled. The control variables at the household level included the living arrangements of the elderly, whether the elderly need to take care of their grandchildren, the number of children of the elderly, and the fixed assets of the family. Finally, the type of water source was used to reflect the sanitation environment of the village, and the existence of a senior activity center for the elderly in the village reflected the social convenience of the elderly.

Basic Ordered Probit Model
The two dependent variables in this study, ADL and SRH, were all ordered data, and therefore this study used the ordered probit model, which is widely used in research estimation. The model is set up as follows: where, y i is the dependent variable, that is, ADL or SRH of the elderly; y i * is an unobservable continuous variable behind y i , named the latent variable, and the observed y i is determined by y i * ; X i is a series of independent and control variables that affect the health of the elderly; and µ 1 < µ 2 < · · · < µ J−1 are parameters to be estimated, known as tangent points.

Selection of Instrumental Variables and IV Ordered Probit Model
In order to avoid deviation of the estimation result caused by endogeneity problem, the instrumental variables were introduced into the basic ordered probit model in this study.
Two instrumental variables were selected. The first one is the proportion of migrant labor in the village where the household is located, and the selection of this variable was based on the migration network theory. The theory holds that the migration network of a village can reduce the cost of the migration of other members and plays a very important role in driving the labor force out of the village. Therefore, this variable is directly related to the labor migration of individual families, but not to the health status of the elderly in individual families; moreover, many studies show that this is an effective instrumental variable [3,23,25]. The second instrumental variable is the structure change of the employed population, that is, the change of the proportion of the agricultural population to the employed population. The selection of this variable was based on Lian, Li, and Huang [27] who mainly considered the promotion of China's dual economic structure to the migration of agricultural labor in the past two decades; the more labor is released from the agricultural sector in a region, the more labor migrates and engages in non-agricultural industries, and the higher is the possibility of labor  1 Due to the larger variation of the original data of "household fixed assets" in the variables, the logarithmic processing is carried out to make the model more stable. 1 US dollar ≈ 6.2284 yuan in 2015.
All models and estimates were performed using the extended regression model framework in software Stata 15, and the limited information maximum likelihood (LIML) method was adopted. Table 1 shows the descriptive statistics of the variables involved in the models. The average ADL score of the sample elderly is 2.14, which is between normal and impaired, and the overall condition is not bad, however, the average score of SRH is only 2.94, which is between bad and general. As compared with the average score of ADL, the overall health self-evaluation of the elderly is lower than the actual physical health status. In addition, the households of 69% of the sampled elderly have at least one migrant laborer; among them, the proportion of households with both male and female migrant labor is 31% and that with only male migrant labor is also 31%, but only 6% of the sampled elderly's households have only female migrant labor. It is conjectured that male labor is still the main force behind labor migration and provision of economic support to Sichuan rural families.

Descriptive Statistics Analysis
For both ADL and SRH, the elderly from households with migrant labor have higher average scores than do those without migrant labor in the family (2.44−2.35 = 0.09 and 2.97−2.92 = 0.05). The average score for the elderly with both male and female migrant labor in the family is higher than that for the elderly with only male or female migrant labor in the family. The elderly with only male migrant labor in the family has a higher average score than do those with only female migrant labor in the family. Moreover, the migration of female labor leads to a greater difference in the SRH scores of the elderly, while the ADL scores of the elderly differ more under the influence of male labor migration (Figures 2 and 3). From the descriptive statistical analysis, it is preliminarily speculated that labor migration in the family is beneficial overall to the health of the elderly. The positive effect of the joint migration of both male and female labor is greater than that of male labor migration alone and is greater than the effect of the migration of female labor alone. The influence of female labor migration on the SRH of the elderly is greater, but this effect may be negative, while male labor migration has a more significant effect on the ADL of the elderly. Therefore, the econometric models were used for further verification. Table 1 shows the descriptive statistics of the variables involved in the models. The average ADL score of the sample elderly is 2.14, which is between normal and impaired, and the overall condition is not bad, however, the average score of SRH is only 2.94, which is between bad and general. As compared with the average score of ADL, the overall health self-evaluation of the elderly is lower than the actual physical health status. In addition, the households of 69% of the sampled elderly have at least one migrant laborer; among them, the proportion of households with both male and female migrant labor is 31% and that with only male migrant labor is also 31%, but only 6% of the sampled elderly's households have only female migrant labor. It is conjectured that male labor is still the main force behind labor migration and provision of economic support to Sichuan rural families.
For both ADL and SRH, the elderly from households with migrant labor have higher average scores than do those without migrant labor in the family (2.44−2.35 = 0.09 and 2.97−2.92 = 0.05). The average score for the elderly with both male and female migrant labor in the family is higher than that for the elderly with only male or female migrant labor in the family. The elderly with only male migrant labor in the family has a higher average score than do those with only female migrant labor in the family. Moreover, the migration of female labor leads to a greater difference in the SRH scores of the elderly, while the ADL scores of the elderly differ more under the influence of male labor migration (Figures 2 and 3). From the descriptive statistical analysis, it is preliminarily speculated that labor migration in the family is beneficial overall to the health of the elderly. The positive effect of the joint migration of both male and female labor is greater than that of male labor migration alone and is greater than the effect of the migration of female labor alone. The influence of female labor migration on the SRH of the elderly is greater, but this effect may be negative, while male labor migration has a more significant effect on the ADL of the elderly. Therefore, the econometric models were used for further verification.   In addition, the average age of the sample elderly is 63.63 years, of whom 85% had spouses; the average number of years of education of the sample elderly is only 4.25; the average number of children of the sample elderly is 2.16, but less than half, or 47%, of the elderly live with their children; 79% of the sample elderly still need to work other than housework; 48% of the sample elderly need  In addition, the average age of the sample elderly is 63.63 years, of whom 85% had spouses; the average number of years of education of the sample elderly is only 4.25; the average number of children of the sample elderly is 2.16, but less than half, or 47%, of the elderly live with their children; 79% of the sample elderly still need to work other than housework; 48% of the sample elderly need to take care of their grandchildren; and 95% of the sample elderly have medical insurance, showing that the New Rural Cooperative Medical Insurance system has been comprehensively promoted up to now. Among the sample villages, only 25% has senior activity centers, and nearly half, or 46%, has tap water. Table 2 presents the econometric model results of the relationship between rural labor migration and the health level of the elderly in the family. Among them, models one to four show the econometric model results of the relationship between rural labor migration and the ADL level of the elderly in the family, while models fjive to eight show the econometric model results of the relationship between labor migration and the SRH level of the elderly in the family. According to the test statistics of models (Wald chi 2 ), models one to eight are all significant at the 0.01 level, indicating that at least one of the independent variables has a significant relationship to the dependent variable in every model.

Econometric Model Results
From the regression results of the endogenous variables (Migra, Both, Fonly, and Monly) to the instrumental variables (Pro, Trans) in each model, it is observed that the regression coefficients are all significant at the level of 0.05 and 0.01, indicating that the selected instrumental variables have a strong correlations with the endogenous variables, and can explicate the endogenous variables well. Among them, the coefficients of Pro are all significantly positive, indicating that migration network can indeed promote labor migration. If there is more migrant labor in the village, there is a greater possibility that the family has migrant labor. The coefficients of Trans are all significantly negative, which is consistent with Lian et al. [27]. The faster the proportion of agricultural labor to total labor declines, the more labor is released from the agricultural sector in the region, and the more agricultural labor needs to transfer employment, and thus agricultural labor in the region is more likely to migrate.
In Table 2, corr(e.x, e.y) is the residual value of the two regression equations in the same model. One is the regression equation of the endogenous variables to the instrumental variables, and the other is the regression equation of the dependent variables to all the independent variables. The residual values are all non-zero at a significance level of 0.05 or 0.01, which indicates that the labor migration situation in the models is indeed an endogenous variable, and our treatment is reasonable.
The results of models one to eight show that the coefficients of the concerned variable Migra are all significantly positive, indicating that labor migration in the family is overall positive for the health of the elderly, whether for ADL or SRH. Moreover, the coefficients of Both and Monly are both significantly positive, while those of Fonly are significantly negative. This finding shows that when there is both male and female migrant labor or only male migrant labor in the family, the impact on the health of the elderly is positive, but if the family has only female migrant labor, there is a significant negative impact on the health of the elderly.
In terms of controlled variables, Edu, Work, and Asset are significantly positively for the health of the elderly; Age and the health level of the elderly are significantly negatively correlated, that is, the older the elderly are, the worse is their general health status; Gender, Childnum, and Centre have positive effects on the health of the elderly, but the impact is not significant; Grandchil is negatively related to the health of the elderly, which indicates that taking care of the grandchildren damages the health of the elderly, especially measured by ADL; and Live has a positive effect on the health of the elderly, but the influence is significant only in models five to eight, that is, living with children can significantly improve the SRH level of the elderly, but the positive effect on ADL is not obvious. It is worth noting that although the significance of the influence of Minsu is unstable, the effect of ADL and SRH on the elderly is negative, which indicates that the crowding-out effect of medical insurance on children's intergenerational support is obvious in rural areas of Sichuan.

Robustness Check
In the previous subsection, ADL and SRH are used to characterize the physical and comprehensive physical and mental health of the elderly, respectively. The effects of labor migration, especially the migration by gender in the family on the health of the elderly, are investigated using an IV ordered probit model. To test the robustness of the models further, the following two strategies are used to perform the robustness test.
First, in the Chinese culture, if the elderly have multiple children, adult children with the same household registration usually bear the responsibility for supporting the elderly, but other adult children also provide intergenerational support, especially economic support. In addition, when there are multiple laborers in the family, even if there is migrant labor, the laborers engaged in agriculture or part-time work at home can still provide the necessary instrumental and emotional support for the elderly. Therefore, the new controlled variable Esupp (economic support obtained from children outside the household registration in 2015, logarithm) and Labor (whether there are other laborers left behind after migration, 1 = yes and 0 = no) are added to the models to control the intergenerational support of the elderly received other than migrant labor, and the models are still estimated using the IV ordered probit model. Second, the elderly with lower levels of physical and mental health in general are more likely to get sick. Therefore, Sick (whether got sick in 2015, 1 = yes and 0 = no) is used as a health indicator to test the robustness of the role of labor migration. Since Sick is a binary variable, it is estimated using the IV probit model. Table 3 shows the robustness test results. The regression coefficient of Fonly on the SRH of the elderly is not significant, but the regression coefficients of the other concerned variables are significant and the direction of influence on the health of the elderly is consistent with the previous results. Thus, it can be concluded that the results of this study are robust and reliable.

Analysis of the Marginal Effect of Labor Migration
Because the correlation coefficient of the ordered probit model has no direct meaning, only limited information can be obtained from the significance and direction of variables, and therefore this study further analyzed the effects of labor migration by gender on the health of the elderly in the family by calculating the marginal effects. In this study, the discrete marginal effects of the endogenous explanatory variables were calculated, that is, the other variables were kept at the mean value, and the probability of the explained variable was calculated for each value when Both = 1, Fonly = 1, Monly = 1 and Both = 0, Fonly = 0, Monly = 0; the difference between these two probabilities is the discrete marginal effect of the variable. This marginal effect means that when other exogenous explanatory variables are at an average level, if only the gender structure of the labor migration changes, the probability of the ADL and SRH scores for the elderly in the family changes. The results of marginal effect calculation are shown in Table 4. Robust standard errors are in parentheses. *, **, and *** represent significant difference at: p = 0.10, p = 0.05, and p = 0.01. From the results of Table 4, first, it can be seen that the marginal effects of Both and Monly are basically the same, which can significantly reduce the probability of ADL being dysfunctional or impaired (∆Both = −0.052/−0.024; ∆Monly = −0.012/−0.013), and improve the probability of ADL having normal function (∆Both = 0.075; ∆Monly = 0.025). At the same time, these two variables can reduce the probability of the elderly's SRH being very bad and bad (∆Both = −0.020/−0.042; ∆Monly = −0.007/−0.036), and improve the probability of SRH being good and very good (∆Both=0.040/0.007; ∆Monly = 0.031/0.015). Second, Fonly significantly reduces the probability of ADL having normal function (∆Fonly = −0.018) and SRH being very good (∆Fonly = −0.044), however, it can also reduce the probability of ADL being dysfunctional (∆Fonly = −0.028) and SRH being very bad (∆Fonly = −0.013). Third, it can be observed from differences of the marginal effects that the marginal contribution of Both to the elderly's ADL is more prominent than that of Monly (∆Both − ∆Monly = 0.051 when ADL = 3), but Monly makes a greater marginal contribution to the elderly's SRH (∆Both − ∆Monly = −0.008 when SRH = 5). In addition, although the marginal effects of Fonly are all negative (when ADL=3 and SRH=5), in terms of absolute values, Fonly has a greater effect on the SRH of the elderly than Monly (|−0.044|−|0.015|) does, and the effect on the ADL of the elderly is not as great as that of Monly (|−0.018|−|0.025|).
In summary, joint migration of both male and female labor or migration of male labor only in the family is more likely to improve the probability of good health of the elderly. Moreover, the ADL of the elderly with both male and female migrant labor in the family has a higher probability of having normal function, while the SRH of the elderly with only male migrant labor in the family has a higher probability of being very good. When there is only female migrant labor in the family, the probability that the health of the elderly is evaluated as extreme (either very good or very poor) is reduced, that is, the health condition of the elderly is not very good, but it is not likely to become very bad. Moreover, the migration of female labor is more likely to affect the elderly's SRH, while the migration of male labor tends to play a more important role in the ADL of the elderly.

Discussion
This study focused on the effect of rural labor migration on the health of the elderly in the family from the perspective of the gender structure of migrant labor and discussed other factors that may affect the health of the elderly.
The results showed that, without considering the specific impact path, the migration of family labor was overall beneficial to the health of the elderly, which was consistent with earlier research [3,17,18,39]. At the same time, as hypothesized in H1 and H2 in the present study and as demonstrated by Chen [13] and Liu [37], the migration of female labor had a negative impact on the health of the elderly, but the effect of male labor migration on the health of the elderly was positive. This may be because male labor is usually the main force of rural households' labor migration [29], and most of the family's daily care responsibilities are mainly borne by female labor, so that the migration of male labor does not have a significant impact on the instrumental support provided to the elderly. On the contrary, male migrant labor usually can earn higher income than females can, and the growth in household income is conducive to improving the living conditions of the elderly, however, when female labor migrate alone, the positive marginal effect of economic growth brought by their migration might not be enough to counteract the negative marginal effect of the lack of instrumental support and emotional support on the health of the elderly. Interestingly, however, this study found that if male and female labor in the family migrated together, it could significantly improve the health of the elderly. This may be because this situation can further increase the overall economic income of the family and reduce the family's dependency ratio, thereby improving the material living standards of family members and reducing the labor burden of the elderly. Thus, even in the absence of instrumental support provided by female labor, as the positive marginal effect of increased economic support is more prominent, the overall effect on the health of the elderly will be positive.
In addition, the analysis of the marginal effects showed that the positive effect of the joint migration of both male and female labor on the ADL of the elderly was the most obvious, whereas for SRH, the positive effect of male labor migration was the most significant. Furthermore, although the direction of influence was different, the migration of female labor was more likely to affect the SRH of the elderly (negative), and the migration of male labor was more likely to affect the elderly's ADL (positive). This result may be because ADL measures the objective physiological health level of the elderly, and the main role of the increase in economic support brought about by labor migration is improvement in the material living standards and medical conditions of the elderly, so that the effect on physical health is more significant. SRH, as a subjective evaluation of the elderly's own health level, is affected by the elderly's mental health and state. Therefore, although the overall impact of family labor migration on the health of the elderly is positive as compared with the migration of male labor, the migration of female labor has a greater impact on the psychological status of the elderly. The elderly may still prefer the care and companionship of female family members in daily life. However, the regression coefficient of Fonly on the SRH of the elderly is not significant in robustness check 1, this may be because other left-behind laborers in the household can to some extent make up for the drain of instrumental and emotional support caused by the migration of female labor. Furthermore, the results show that when only female labor in the family migrated, the overall impact on the health of the elderly was negative, but the health assessment of the elderly did not tend to be "very bad", the reason for this may be that with the popularity of the out-migration employment of female labor in rural China, especially in a relatively backward area like Sichuan, the elderly's acceptance of the phenomenon was increasing, and that as household incomes further increased, the negative impact of female labor migration on the health of the elderly would decline.
By comparison, it is preliminarily speculated that the impact of male labor migration on the health of the elderly is usually realized through the economic support path, whereas the role of female labor migration is realized more through the instrumental support path and emotional support path. Therefore, the migration of male labor is more likely to have a positive impact on the physical health of the elderly, but the migration of female labor is more likely to have a negative impact on the mental health of the elderly. In general, however, when the positive marginal effect of economic support brought by labor migration is large enough, it can counteract the negative effect of lack of instrumental support. This reflects that, under the current low levels of economic development in rural areas of Sichuan Province, raising the level of economic security should be an important way to improve the welfare system for the elderly at the present stage.
The novel contributions of this study in the literature are as follows. (1) Taking Sichuan as the sample region, this study specifically discussed the relationship between labor migration and the health of the elderly in rural areas of western China; (2) from the perspective of the gender structure of migrant labor, the different effects of female and male labor migration on the health of the elderly were investigated, and the possible paths and directions of the impact of labor migration by gender on the health of the elderly were tentatively discussed; and (3) by measuring the physical health of the elderly and their comprehensive physical and mental health at the same time, the endogeneity problem of the model was dealt with and a corresponding robustness test was undertaken.
This study has certain limitations that can be addressed in future research. First, the role of family labor migration on the health of the elderly may have a "hysteresis effect", and therefore the results may be more accurate if the data of health status of the elderly are provided one year after labor migration. Secondly, the health status of the elderly and labor migration status in the family are both dynamic, but, owing to data availability, this study used only one-stage cross-sectional data. Future research could use panel data to undertake a more complete analysis of the impact of labor force migration by gender on the health of the elderly. Third, as the focus of the study is the gender structure of family's migrant labor, several variables such as the relationship of migrant labor to the elderly, duration of migration, remittances of the migrant labor to the household, and income of migrant labor were not included in the original analysis and need to be examined in any future research.

Conclusions and Recommendations
From the perspective of the gender structure of migrant labor in the family, this study applied data of rural households in Sichuan Province in 2015 collected through a survey and used a IV ordered probit model to explore the impact of labor migration on the health of rural elderly. The results are summarized as follows: 1.
In general, labor migration has a positive effect on the health of the elderly in rural households. The average ADL score is 0.09 (2.44−2.35) higher and the average SRH score is 0.05 (2.97−2.92) higher for the elderly with migrant labor in the family than for those without.

2.
Labor migration in the family has different effects on the health of the elderly by gender. In terms of the direction of influence, the joint migration of both male and female labor, or the migration of male labor only, has a positive effect on the health of the elderly, but the migration of female labor only has a negative impact. With regard to the difference of the influence on physical and mental health, the migration of female labor only has a more prominent effect on the SRH of the elderly, whereas the migration of male labor only has a more significant effect on the ADL of the elderly.
Some policy suggestions are offered based on the research results. First, compared with the lack of instrumental support and emotional support in the daily life of the elderly caused by labor migration in the family, the improvement of income level and economic support has a more prominent effect on the health status of the rural elderly. Therefore, with the irreversible trend of industrialization and urbanization in China, there should be support for the employment transfer of the surplus labor supply of rural youth, but at the same time, the government must further improve the welfare system of the elderly in rural areas. Secondly, the companionship of family members is of great importance for the mental health of the elderly, especially the support and companionship of female family members in daily life, and thus while encouraging rural labor force (especially female labor) to transfer employment and increase household income, policies should be adopted to achieve local or near-transfer rather than migration. In addition, the elderly welfare system can be improved to encourage the elderly to migrate with the other family members.