3.1 Estimating the effects of the probability to look after a parent on fertility

This section discusses modeling a couple's decision relating to their family size. It is assumed that couples decide on their family size at the time of their marriage.Footnote ⁹ To be specific, the dependent variable to be explained is the number of children born to a woman who is aged 45 years or older as we assume that women of this age have completed their childbearing activities. The focus of the analysis is whether the estimated probability of the couple being the primary care provider for at least one parent or parent in law, P _i, has any effects on the wife's fertility.

As the existing literature [e.g., Melkersson and Rooth (Reference Melkersson and Rooth2000) and Santos Silva and Covas (Reference Santos Silva and Covas2000)] has argued, there may be a significant difference between childless couples and couples with at least one child. Both Melkersson and Rooth (Reference Melkersson and Rooth2000) and Santos Silva and Covas (Reference Santos Silva and Covas2000) argue that the outcome of zero children may result from two distinct sources: a couple choosing to have no children or a couple being physically unable to have children at all, whereas the outcome of one or more children is just a result of a couple's choice. Given this difference between childless couples and couples with children, one way of modeling this is to use a Poisson logit hurdle model. The first-stage model is a logit model that examines the decision of whether to have a child or not, and in the second stage for couples who decide to have children a standard Poisson regression model is used to explain the number of children they have. Another reason for preferring the Poisson hurdle model is that the Poisson model which assumes that the mean and the variance of the number of children are the same cannot explain the observed under-dispersion in the number of children.

If the number of children that couple i has is denoted by y_i, then assuming a Poisson hurdle model means the probability of having 0, 1, …, N children is given by [see Winkelmann (Reference Winkelmann2003) and Cameron and Trivedi (Reference Cameron and Trivedi2013)]:

(5a)$${\rm Prob}( {y_i = 0{\rm \vert }x_i} ) = w_i$$

(5b)$${\rm Prob}( {y_i = j{\rm \vert }x_i} ) = ( {1-w_i} ) \lambda _i^j \exp ( {-\lambda_i} ) /[ ( 1 \! - {\rm exp}( \!\! - \! \lambda _i) ) j!] , \;\quad j = 1, \;2, \;3,... \;$$

where w_i is given by a logistic function, namely,$\;w_i = 1/( {1 + \exp ( {x_i^{\prime} \gamma } ) } )$ and $\lambda _i = \exp ( {x_i^{\prime} \beta } )$.Footnote ¹⁰ In this model, $E( {y_i{\rm \vert }y_i > 0} ) = 1 + {\rm exp}( {x_i^{\prime} \beta } )$ and E(y _i > 0) = 1 − w _i, so that for variables that are continuous $\partial E( {y_i{\rm \vert }y_i > 0} ) /\delta x_i = \exp ( {x_i^{\prime} \beta } ) \beta$, ∂E(y _i > 0)/δx _i =  w _i(1 − w _i)γ, and $\partial E( {y_i} ) /\partial x_i = \exp ( {x_i^{\prime} \beta } ) \beta \;( {1-w_i} ) \lambda _i^j$. It should be noted that the standard Poisson model and this Poisson hurdle model are non-nested models, so that one way to choose between them is to use an information criterion like the Akaike information criterion (AIC) or the Bayesian information criterion (BIC).

If we think of the bivariate choice problem of having no children or some children, the model in equations (5a) and (5b) will imply this choice is determined by a logit model with Prob(y _i = 0|x _i) = w _i and obviously Prob(y _i > 0|x _i) = 1 − w _i. In fact, as McDowell (Reference McDowell2003) indicates an equivalent way to estimating the model in equations (5a) and (5b) by maximum likelihood is to first estimate a logit model with Prob(y _i = 0|x _i) = w _i and Prob(y _i > 0|x _i) = 1 − w _i, and then for y _i > 0 to estimate a Poisson model.

The vector of control variables, x_i, includes the probability of the couple looking after their parents and/or parents-in-law, P _i, that is discussed in detail in section 3.2, and other control variables such as the husband's age at the time of his marriage, the wife's age at the time of her marriage, the husband's education level, the wife's education level, an urban dummy, and seven regional dummies. In addition, we include a 0–1 dummy variable which takes the value 1 if husband is an only child and 0 otherwise to control for some of the effects of being the eldest son.Footnote ¹¹ The a priori expectation is that a higher probability of the couple looking after their parents and/or parents-in-law will tend to reduce the number of children the couple has.

3.2 Estimating the probability of having to look after a parent

As indicated in Table 2, in Japan the primary care providers for frail aging parents are typically other family members rather than private care providers. If a parent or parent-in-law has died before the survey, the 1998 National Family Research of Japan (NFRJ98) survey contains information on whether the respondent reports that he/she was the primary care provider for that person before they died. For each deceased father, mother, father-in-law, or mother-in-law, the NFRJ98 survey asks respondents whether the relevant parent or parent-in-law required care for a period before they died, and if the answer is yes, respondents are then asked “(t)o what extent were you involved in caregiving and nursing?” The respondent picks an answer from one of the following five choices: (1) I was the primary person involved in care providing and nursing; (2) I was involved, though not as the primary care provider; (3) I was somewhat involved; (4) I was not much involved; and (5) I had very little opportunity to be involved because of the person's sudden death. Based on cases where the relevant parent or parent-in-law had already died by the time of the survey, Table 3 provides information on whether the parent or parent-in-law required care, and when care was required who provided what level of care to whom.

Table 3. Who provides how much care to whom?

Notes: (1) The columns labeled “Requiring care” report the percentage of care givers who reported that the relevant care receiver required care before his/her death and the sample size on which this calculation is based. The columns labeled “Care level conditional on requiring care” report the percentage of respondents who said they provided a particular level of care for a specified parent or parent-in-law conditional on having reported that the specified parent or parent-in-law required care for a period of time before his/her death. In both cases, the information is only available if the relevant parent or parent-in-law has died by the time of the survey, and in the case of the “Care level,” the care giver reported that the relevant parent or parent-in-law required care for a period before his or her death. This is one of the important reasons for differences in the sample sizes for identical care givers.

(2) This table contains estimates of whether care was required for the two types of caregivers and four types of care receivers based on respondents' answers to questions 32, 33, 34, and 35 for his/her father, mother, father-in-law, and mother-in-law, respectively, on the 1998 NFRJ survey. This table contains estimates of the care level provided by the two types of caregivers and four types of care receivers based on respondents' answers to supplementary questions 1 for questions 32, 33, 34, and 35 for his/her father, mother, father-in-law, and mother-in-law, respectively, on the 1998 NFRJ survey. It should be noted that these questions are only asked on the 1998 NFRJ survey.

Source: Computed from the 1998 NFRJ survey.

Perhaps the most surprising finding in Table 3 compared to the conventional wisdom is that for their own parents both the husband and wife report that they are just as likely to be the primary care provider. For example, 16.7% of husbands report they were the primary carers for their fathers. On the contrary, 14.7% of wives were the primary carers for their fathers. Wives are more likely to be the primary carers for their own mothers than their husbands (24%), but still 18.5% of husbands were the primary carers for their own mothers. In the Japanese context, it is not surprising to find that husbands do not look after their in-laws as the primary care provider (less than 4% for in-laws), but their wives do and they are more likely to do so than for the case of the wife looking after her own parents! In total, 35.2% of wives were the primary carers for their father-in-law and 47.3% of them were the primary carers for their mother-in-law. The proportions of wives who became the prime carers for their parent-in-laws are much higher than that of their own parents. In the case of wives, there is a strong cultural expectation that they will be strongly involved in caring for their husbands' parents.

In order to measure the future burden of caring for aged parents, we construct a probability of having to look after a parent or parent-in-law in the future. For the ith couple, let p _ijk be the probability of individual j in couple i looking after parent k where j denotes either the husband (h) or wife (w) and k refers to the husband's (h) or wife's (w) parent(s). For example, for the husband of the ith couple, the probability of looking after his own parent(s) (husband's parents) is denoted by p _ihh and the probability of looking after his parent(s)-in-law (wife's parent) is denoted by p _ihw.

In order to estimate the probability of looking after a parent in the future, the information in NFRJ98 is more appropriate than the information on whether the respondent is currently involved in caregiving and nursing a parent as many of the respondents do not yet have aging frail parents. Another advantage of this information is that when computing the probability of looking after a parent we can take account of those cases where a parent's death was sudden and the respondent did not have to provide any care to that parent.

Denoting the husband and wife by h and w, respectively, we construct four 0–1 dummy variables care _ijk (j = h, w, k = h, w) which take the value one if the ith respondent was the husband (j = h) or wife (j = w) and “the primary person involved in care providing and nursing” for k's parents where k denotes the husband (h) or the wife (w), and 0 otherwise, and denote the associated probability of care _ijk = 1 by p _ijk, then the following probit model is used to model p _ijk and explain the two outcomes for care _ijk, for each value of j and k, namely,Footnote ^12, Footnote ¹³

(6)$$p_{ijk} = \Pr ( {care_{ijk} = 1} ) = \Pr ( {CARE_{ijk} + \eta_{ijk} > 0} ) , \;\quad j = h, \;w; \;\,k = h, \;w; \;\,i = 1, \;\ldots , \;N_{\,jk}$$

where η _ijk is a normally, identically and independently distributed random variable with expected value zero and variance 1, and N_jk is the available sample size for the relevant combination of j and k.

For CARE _ijk, the following four models are assumed:

(7)$$CARE_{ihh} = \beta _{10} + \beta _{11}siblings\_ h_i + \beta _{12}one\_parent\_ h_i + \beta _{13}birth\_year\_ h_i + \beta _{14}educ\_ h_i\;$$

(8)$$CARE_{ihw} = \beta _{20} + \beta _{21}siblings\_ w_i + \beta _{22}one\_parent\_ w_i + \beta _{23}birth\_year\_ h_i + \beta _{24}educ\_ h_i$$

(9)$$CARE_{iww} = \beta _{30} + \beta _{31}siblings\_ w_i + \beta _{32}one\_parent\_ w_i + \beta _{33}birth\_year\_ w_i + \beta _{34}educ\_ w_i$$

(10)$$CARE_{iwh} = \beta _{40} + \beta _{41}siblings\_ h_i + \beta _{42}one\_parent\_ h_i + \beta _{43}birth\_year\_ w_i + \beta _{44}educ\_ w_i$$

where $siblings\_ j_i$ is the number of js siblings [husband (j = h) or wife (j = w)],Footnote ¹⁴ $one\_parent\_ j_i$ is a 0–1 dummy variable which takes the value one if one of js parents is already deceased at the time of his/her marriage, and 0 otherwise [husband (j = h) or wife (j = w)], $birth\_year\_ j_i$ is the year of the birth of individual j [husband (j = h) or wife (j = w)], and $educ\_ j_i$ is the years of education for individual j [husband (j = h) or wife (j = w)]. As Table 2 indicates, in around 25% of all cases a cohabiting spouse is the care provider, so the variable $one\_parent\_ j_i$ highlights the fact that if one of the parents is already deceased at the time of a couple's marriage, the probability of having to look after a parent should increase. $birth\_year\_ j_i$ is included to control for cohort effects, and $educ\_ j_i$ is used as a proxy for income. Although parental income at the time of marriage may be an important predictor of the probability of their children having to provide future care, such information is not available in our data set.Footnote ¹⁵ When estimating equations (7)–(10), the values of the variables at the time of the couple's marriage are used. Similarly, when the probability of looking after a parent is calculated the explanatory variables used in equations (7)–(10) are measured at the time of the couple's marriage.

In equations (7)–(10), one key issue is the sign of β _i1, i = 1, 2, 3, 4, that is, the impact of the number of siblings. There are a number of possibilities depending on what assumptions are made. If we assume that the couple in question is completely selfish, then they will not care about the provision of care for their parents or for their parents-in-law by themselves or by their siblings, and so the number of siblings that a couple has should be irrelevant, that is, β _i1 = 0, i = 1, 2, 3, 4 [the other coefficients in equations (7)–(10) would also be expected to be zero]. If care provision is driven by a “warm glow” motivation, then the couple may provide care to their parents (or parents-in-law) depending on the strength of the warm glow motivation. In this case, the provision or lack of provision of care by siblings is again irrelevant. One simple story for why β _i1 > 0 is that if there is only one primary care provider, then on a probabilistic basis with more siblings, the likelihood of being the primary care provider can be expected to be smaller. A more sophisticated story would assume that the couple and the couple's siblings are all altruistic with respect to their parents (and parents-in-law), so that the provision of care for their parents (parents-in-law) by them and their siblings can be viewed as being a version of a public goods game. Assuming that both parents and children only care about the total contribution made by the children, increases in the number of siblings will reduce the individual contributions made to the first-generation parents by each second-generation sibling.Footnote ¹⁶

Once estimates of the parameters of equations (7)–(10) have been obtained, using these estimated coefficients and a different sample of all the available married men and women, respectively, including those who yet to experience the death of their parents or parents-in-law in the 1998 and 2008 NFRJ surveys estimates of p _ihh and p _ihw, and p _iww and p _iwh can be obtained by inserting the values of explanatory variables and estimated parameters into equations (6)–(10).

Caregiving and nursing of a frail parent should be seen as the joint product of couple. For example, a husband may not be the primary care provider if he is the breadwinner of the household, but his wife can be the primary care provider. Thus, the probability of the couple having to look after at least one of their parents or parents-in-law as a couple, P_i, can be written asFootnote ¹⁷:

(11)$$P_i = 1-( {1-p_{ihh}} ) \times ( {1-p_{ihw}} ) \times ( {1-p_{iww}} ) \times ( {1-p_{iwh}} ) .$$

Here, (1 − p _ihh) is the husband's probability of not becoming the primary care provider for his parents. Therefore, (1 − p _ihh) × (1 − p _ihw) × (1 − p _iww) × (1 − p _iwh) is the probability that neither the husband nor the wife are the primary care provider for any of their parents or parents-in-law.

4.1 Sample selection

We estimate our models explaining the number of children for households that satisfy the following eight criteria. First, couples where the wife's age is 45 years or older in order to focus on those women who have completed child bearing. Second, we focus on married respondents where the husband's age at the time of the marriage is 18 years old or older and the wife's age at the time of the marriage is 16 years old or older. This is because the Japanese legal age for marriage is 18 for men and 16 for women. Third, we only use respondents who are currently married and who have never been divorced or widowed. Divorcees or widows may have children from their previous marriage, but the NFRJ surveys do not contain information on their previous marriages. Fourth, we also exclude an observation if the wife's age at her marriage was 45 or older as she is unlikely to have any child. Fifth, we drop observations which report a deceased child as this can have an impact on later fertility decisions. Sixth, in order to control for outliers, observations where there are 10 or more siblings (99% quantile) are excluded. Seventh, we exclude all marriages from 1997 onward to eliminate the actual or anticipated effects of the Long-term Care Insurance Act (Kaigo Hoken Ho) which passed into law in December 1997 and came into effect on April 1, 2000. Eighth, we exclude all observations which do not contain all the information on all the variables required for estimating the relevant model. After imposing these selection criteria, 4,045 observations remain. Descriptive statistics for this sample are summarized in Table 4. Using the mean (2.2) and the standard deviation of the number of children (0.88) reported in Table 4, we can see that the variance of this variable (0.77) is just one-third of the size of the mean, a strong indicator of under-dispersion.

Table 4. Descriptive statistics for the samples used to estimate the logit, Poisson, and Poisson-logit hurdle models