Julien Bergeot
email: julien.bergeot@u-cergy.fr.

Roméo Fontaine

The Heterogeneous Effect of Retirement on Informal Care Behavior

It is often argued that the increased labor market participation of seniors threatens family support provided to dependent elderly people. The purpose of this paper is to assess the causal effect of retirement on the frequency of care provided by individuals aged 55-69 to their elderly parent. Using data from the Survey of Health, Ageing and Retirement in Europe (SHARE), we estimate an endogenous switching model that allows the retirement effect to be heterogeneous with respect to observed and unobserved characteristics. To tackle the possible endogeneity of selection into retirement, we use the heterogeneity of retirement rules between and within European countries. On average, being retired does not significantly impact the probability of providing care but significantly increases the frequency of care conditional on being caregiver. The same pattern is observed regardless of the individual observed characteristics, even if the provision of informal care appears to be less sensitive to the retirement status when the child cannot rely on the other parent to provide care or when both parents are in poor health. These results suggest that pension system reforms should not affect the number of caregivers. Some adverse effects on the intensity of involvement among caregivers are nevertheless expected.

Introduction

The family is an important resource in the production of long-term care for elderly people who can no longer carry out certain daily life activities on their own. In OECD countries, we estimated that 13% of individuals aged 50 or more were providing care at least on a weekly basis in 2015 (OECD, 2017), mostly to a family member [START_REF] Colombo | Help wanted? Providing and paying for long-term care[END_REF].

Demand for informal care is expected to increase in the coming decades because of population aging. However, in a context where most European countries have implemented pension system reforms to increase the effective retirement age, it is often argued that the increased labor market participation of seniors threatens the potential availability of informal caregivers [START_REF] Colombo | Help wanted? Providing and paying for long-term care[END_REF], thus making informal care provision for disabled elderly people living at home more difficult [START_REF] Kingston | Is late-life dependency increasing or not? A comparison of the Cognitive Function and Ageing Studies (CFAS)[END_REF]. This argument is based on the implicit assumption that workers, who have less free time, are less likely to provide informal care to their parents. However, there is no empirical evidence to support this hypothesis.

Previous articles exploring the relationship between retirement and informal care have considered caregiving to an elderly person as a determinant of labor supply [START_REF] Jacobs | Caregiving intensity and retirement status in Canada[END_REF][START_REF] Jacobs | The impact of informal caregiving intensity on women's retirement in the United States[END_REF][START_REF] Meng | Informal caregiving and the retirement decision[END_REF][START_REF] Van Houtven | The effect of informal care on work and wages[END_REF][START_REF] Niimi | Does providing informal elderly care hasten retirement? Evidence from Japan[END_REF]. These studies analyze the indirect costs of informal caregiving, alongside others which examine the effect of caregiving on the health status of caregivers [START_REF] Coe | Caring for mom and neglecting yourself? The health effects of caring for an elderly parent[END_REF][START_REF] Bom | Health effects of caring for and about parents and spouses[END_REF].

Our approach is different, and considers the reverse pathway of causation: to what extent is caregiving determined by retirement? This paper is thus more in line with the literature that examines how labor market status conditions informal care [START_REF] Stern | Estimating Family Long-Term Care Decisions in the Presence of Endogenous Child Characteristics[END_REF][START_REF] He | Does formal employment reduce informal caregiving?[END_REF], even if retirement has never been examined specifically. In this paper, we aim to address the following questions: What is the causal effect of retirement on caregiving behavior? How heterogeneous is this effect?

We use a pooled sample of individuals aged 55-69 from the 1st, 2nd and 6th waves of the Survey of Health, Ageing and Retirement in Europe (SHARE). Our measure of informal care behavior is the frequency of care provision to parents living outside the household. To estimate the causal effect of retirement on informal care behavior, we estimate an endogenous switching model that allows the retirement effect to be heterogeneous with respect to observed and unobserved characteristics. Finally, we use the heterogeneity in early and official retirement ages across the European countries to instrument the retirement status.

To our knowledge, this paper is the first to estimate the causal effect of retirement on informal care behavior. Its main contributions can be summarized as follows. First, our results suggest that, on average, care provision (regardless of frequency) does not significantly depend on whether the individual is retired or not. Second, we show nevertheless that being retired significantly decreases the probability of providing care monthly or less frequently and increases the probability of providing weekly and daily care. Hence, retirement tends to increase the level of provision only among people who would have been caregivers even if they were not retired. Consequently, from a public policy perspective, our results suggest that pension system reforms increasing retirement ages should have no significant impact on the number of caregivers, but some adverse effects on the level of involvement among caregivers are expected. Third, the retirement effect is relatively homogenous with respect to the observed individual characteristics, even though the provision of informal care appears to be less sensitive to the retirement status when the child cannot rely on the other parent to provide care or when both parents are in poor health.

Literature Review

Some papers have examined the effect of informal care provision on the likelihood of retiring. 1 Overall, this literature indicates that the impact is small: few individuals are forced to leave the labor market because of their role as caregivers.

To the best of our knowledge, no papers investigate the reversal causality, i.e. the specific effect of retirement on caregiving behavior, although some study the effect of labor supply on the supply of informal care to an elderly person. In this literature, the evidence is still unclear since some papers have identified a negative effect and others an effect that is positive or nonsignificant. [START_REF] Spitze | Employment and filial relations: Is there a conflict?[END_REF] investigate the impact of working hours on different care outcomes.

They estimate an OLS model and do not find any significant effect of employment on caregiving activity. [START_REF] Stern | Estimating Family Long-Term Care Decisions in the Presence of Endogenous Child Characteristics[END_REF] adopts an instrumental variable (IV) approach with panel data using two waves (1982 and 1984) of the US National Long-Term Care Survey. Their results suggest that work status does not significantly affect care provision, and that work status may be treated as exogenous. [START_REF] He | Does formal employment reduce informal caregiving?[END_REF] They do not find any significant effect of labor market participation on the probability of being a caregiver, but a negative effect when considering working time. Börsch-Supan et al. (1992), with data from Massachusetts, estimate a Tobit model and find a significant positive effect of employment (treated as exogenous) on time spent with parents.

1 These papers contribute to the vast literature examining the effect of care provision to an elderly person on labor market participation and working time. [START_REF] Lilly | Labor market work and home care's unpaid caregivers: a systematic review of labor force participation rates, predictors of labor market withdrawal, and hours of work[END_REF] and Bauer and Souza-Poza (2015) are worth reading for a broad literature review.

Some papers also study this pathway of causation within a dynamic framework. [START_REF] Carmichael | Who will care? Employment participation and willingness to supply informal care[END_REF] use data from 15 waves (1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005) of the British Household Panel Survey (BHPS). They find that while being employed one year (t) has no significant effect on the probability of caring less than 20 hours a week the following year (in t+1), it reduces the probability of caring more than 20 hours a week in t+1. Again with the BHPS (2000)(2001)(2002)(2003)(2004)(2005), [START_REF] Michaud | A dynamic analysis of informal care and employment in England[END_REF] find that the likelihood of providing extra-residential and co-residential care is increased by 2.3 percentage points and 0.9 points, respectively, in the year following a transition from employment to non-employment. [START_REF] Boaz | Paid Work and Unpaid Help by Caregivers of the Disabled and Frail Elders[END_REF], [START_REF] Pavalko | Women's caregiving and paid work: Causal relationships in late midlife[END_REF], [START_REF] Spiess | Midlife Caregving and Employment. An Analysis of Adjustements, in Work Hours and Informal Care for Female Employees in Europe[END_REF] and [START_REF] Berecki-Gisolf | Transitions into informal caregiving and out of paid employment of women in their 50s[END_REF], who study both pathways of causation, confirm the main message of the literature: an exogenous increase in care provision negatively and, in most cases, significantly affects labor supply, whereas an exogenous variation of labor supply has an unclear but generally non-significant effect on care provision, except above a certain threshold of working time.

As previously mentioned, these studies do not identify to what extent retirement affects the provision of informal care. Identifying the specific effect of retirement is nevertheless of particular interest because the retirement decision concerns individuals who are also more at risk of providing care for an elderly parent. Moreover, retirement is usually a unique and permanent transition that may entail an enduring change in lifestyle.2 Last but not least, understanding the effect of retirement on informal care is a critical issue to evaluate the potential effects of the ongoing reforms that tend to increase the retirement age.

Empirical approach

Econometric framework

To assess the impact of retirement on caregiving behavior, we use the Neyman-Fisher-Cox-Rubin model of potential outcomes described by [START_REF] Aakvik | Estimating treatment effects for discrete outcomes when responses to treatment vary: an application to Norwegian vocational rehabilitation programs[END_REF] to estimate a treatment effect for discrete outcomes when responses to treatment vary.

We apply this general framework to study the impact of retirement on the frequency of caregiving. For each individual i, we consider two potential caregiving outcomes, 𝐹𝑟𝑒𝑞 0𝑖 and 𝐹𝑟𝑒𝑞 1𝑖 , corresponding, respectively, to the potential frequency of care in the non-retired and retired states. It is assumed that 𝐹𝑟𝑒𝑞 0𝑖 and 𝐹𝑟𝑒𝑞 1𝑖 are defined for everyone. We nevertheless only observe 𝐹𝑟𝑒𝑞 𝑖 = (1 -𝑅 𝑖 )𝐹𝑟𝑒𝑞 0𝑖 + 𝑅 𝑖 𝐹𝑟𝑒𝑞 1𝑖 , where 𝑅 𝑖 determines which regime the agent experiences, with 𝑅 𝑖 being equal to 1 if 𝑖 is retired and 0 otherwise.

As measured in the SHARE survey, the frequency of care provision corresponds to an ordinal polytomous variable that can take 5 values corresponding to different care frequencies: 0) no care, 1) less than monthly, 2) almost monthly, 3) almost weekly and 4) almost daily.

The potential frequencies of care in the non-retired and retired states are assumed to be generated by a latent index as follow (𝑓𝑜𝑟 𝑓 = 0,1,2,3,4):

𝐹𝑟𝑒𝑞 0𝑖 = 𝑓 𝑖𝑓𝜏 𝑓-1 < 𝐹𝑟𝑒𝑞 0𝑖 * ≤ 𝜏 𝑓 𝑤𝑖𝑡ℎ 𝐹𝑟𝑒𝑞 0𝑖 * = 𝑋 𝑖 𝛽 0 + 𝜀 0𝑖 (1)
𝐹𝑟𝑒𝑞 1𝑖 = 𝑓 𝑖𝑓𝜏 𝑓-1 < 𝐹𝑟𝑒𝑞 1𝑖 * ≤ 𝜏 𝑓 𝑤𝑖𝑡ℎ 𝐹𝑟𝑒𝑞 1𝑖 * = δ 𝑓 + 𝑋 𝑖 𝛽 1 + 𝜀 1𝑖 (2)
where 𝑋 𝑖 is a vector of observed random variables, (𝜀 0𝑖 , 𝜀 1𝑖 ) are unobserved random terms and (𝛽 0 , 𝛽 1 ) are vectors of parameters (to be estimated). 𝜏's are the cut-off points to be estimated3 that determine the category to which a frequency belongs when they are crossed by the latent.

𝛿's (for 𝑓 = 0,1,2,3,4) are also parameters to be estimated. They measure the common effect of retirement on the latent frequency of care. We partially relax the well-known parallel regression assumption by allowing 𝛿 𝑓 to differ according to 𝑓, thus allowing the effect of retirement to vary freely across cut-off points. 4 For all other covariates, the parallel regression assumption is maintained.

We assume that the assignment or decision rule for the indicator 𝑅 𝑖 is generated by a latent variable 𝑅 𝑖 * such that:

𝑅 𝑖 * = 𝑍 𝑖 𝛽 𝑅 + 𝑢 𝑅𝑖 (3) 𝑅 𝑖 = 1 𝑖𝑓 𝑅 𝑖 * ≥ 0, 𝑅 𝑖 = 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
where 𝑍 𝑖 is a vector of observed random variables, 𝑢 𝑅𝑖 is an unobserved random term, and 𝛽 𝑅 is a vector of parameters to be estimated.

We account for a possible endogenous selection in the retired and non-retired states by assuming that (𝜀 0𝑖 , 𝜀 1𝑖 , 𝑢 𝑅𝑖 ) are distributed as a trivariate normal with mean vector zero and covariance matrix 𝛴 where:

𝛴 = ( 1 𝜌 0,1 𝜌 0,𝑅 𝜌 0,1 1 𝜌 1,𝑅 𝜌 0,𝑅 𝜌 1,𝑅 1 )
with the variance of 𝜀 0𝑖 , 𝜀 1𝑖 , 𝑢 𝑅𝑖 normalized to 1.

The endogeneity of retirement status may arise from several sources. Reverse causality may be a first concern if individuals decide to withdraw from the labor market due to their caregiving role. Omitted variables, such as family values or labor market attachment, may also have an impact on both retirement status and caregiving behavior. Therefore, systematic differences in the caregiving outcome variable between retired and non-retired may reflect not only the effect of retirement, but also differences generated by the selection process [START_REF] Abadie | Semiparametric instrumental variable estimation of treatment response models[END_REF].

As specified, the model defined by equations 1-3 belongs to the general class of endogenous switching models [START_REF] Maddala | Switching regression models with exogenous and endogenous switching[END_REF]. It is estimated by maximum likelihood. Note that the potential care frequencies in the retired and non-retired states are not observed simultaneously. As a consequence, the covariance 𝜌 0,1 cannot be estimated. 5 This also implies that the likelihood function involves only 2-dimensional integrals despite the trivariate dimension of the parametric distribution of the residuals. Finally, even if the model is identified by construction through nonlinearities, our identification also relies on exclusion restrictions with variables that are included in 𝑍 but not in 𝑋 (see Section 4.4.).

Treatment parameters

Following [START_REF] Aakvik | Estimating treatment effects for discrete outcomes when responses to treatment vary: an application to Norwegian vocational rehabilitation programs[END_REF], we use the estimated parameters of the model to assess the extent to which the effect of retirement on caregiving frequency varies among groups of individuals. The average treatment effect of retirement (ATE) on the probability of providing care with frequency 𝑓 among individuals who are observationally identical, denoted

△ 𝑓 𝐴𝑇𝐸 (𝑥), is equal to: △ 𝑓 𝐴𝑇𝐸 (𝑥) = 𝐸[𝑃(𝐹𝑟𝑒𝑞 1 = 𝑓|𝑋 = 𝑥)] -E[𝑃(𝐹𝑟𝑒𝑞 0 = 𝑓|𝑋 = 𝑥] = 𝐸[Φ(𝜏 𝑓 -δ 𝑓 -𝑥𝛽 1 ) -Φ(𝜏 𝑓-1 -δ 𝑓 -𝑥𝛽 1 )] -E[Φ(𝜏 𝑓 -𝑥𝛽 0 ) -Φ(𝜏 𝑓-1 -𝑥𝛽 0 )] for 𝑓 = 0,1,2,3,4. 6
It measures the effect of retirement on a person selected randomly from the population of individuals with a given value of 𝑋. Note that the identification of the average treatment effect relies on the identification of the bivariate distributions (𝐹𝑟𝑒𝑞 0 * , 𝑅 * ) and (𝐹𝑟𝑒𝑞 1 * , 𝑅 * ), and does not require knowledge of the full trivariate distribution.

The extent to which the ATEs vary among individuals may be investigated by analysing the effects of change in X, say in a covariate 𝑥 𝑘 , on the ATE integrated over the distribution of

X, 𝐸 𝑋 ( 𝜕△ 𝑓 𝐴𝑇𝐸 (𝑥) 𝜕𝑥 𝑘
), with:

𝜕 △ 𝑓 𝐴𝑇𝐸 (𝑥) 𝜕𝑥 𝑘 = 𝐸 [-𝛽 1𝑘 (𝜑(𝜏 𝑓 -δ 𝑓 -𝑥𝛽 1 ) -𝜑(𝜏 𝑓-1 -δ 𝑓 -𝑥𝛽 1 ))] -𝐸 [-𝛽 0𝑘 (𝜑(𝜏 𝑓 -𝑥𝛽 0 ) -𝜑(𝜏 𝑓-1 -𝑥𝛽 0 ))] for 𝑓 = 0,1,2,3,4

Data

We use pooled data from the 1 st , 2 nd and 6 th waves of the Survey of Health, Ageing and Retirement in Europe (SHARE), collected respectively in 2004-2005, 2006-2007 and 2015 from individuals aged 50 or over in 17 countries. 7 SHARE is a multidisciplinary database of micro-data on health, socio-economic status and intergenerational transfers. Note that we do not exploit the panel dimension of SHARE as we conduct a cross-sectional analysis.

We restrict our sample to individuals aged between 55 and 69 years and with at least one parent alive. We focus on extra-residential care because there is no information on the frequency of care provided to household members in the survey 8 ; those with missing values 7 The questions about informal care provision are different in the 4th and the 5th waves. After comparing the percentages of caregivers across waves we considered that they were too different to include waves 4 and 5 in the analysis. 8 In Section 8, we show that retirement does not have a significant effect on the probability of co-residing with the parent(s). This suggests that this sample restriction does not generate a sample selection bias in our baseline estimates.

on the outcome or any control variable and those living with their parents are dropped. We thus obtain a pooled sample of 14,082 observations. Finally, we make one additional restriction: we exclude individuals who do not report the same geographical distance to each parent. As we are not able to know whether the parents are living together, we have assumed that parents were living together when the distance reported for both parents is the same. It is more difficult to deal with parents located at a different distance, especially as it concerns few observations (179). Our results do not change substantively when we include this sample in our analyses.

Retirement

Retirement is defined with the self-reported adult child's occupational status. This definition of retirement is widely used in studies using SHARE survey data to study the effect of retirement on health [START_REF] Rohwedder | Mental retirement[END_REF][START_REF] Godard | Gaining weight through retirement? Results from the SHARE survey[END_REF]Mazzonna andPeracchi 2012, 2017).

Outcome variable

To capture caregiving behavior, we consider the frequency of care provision: an ordered polytomous variable, 𝐹𝑟𝑒𝑞, with different values for care provided almost daily (𝐹𝑟𝑒𝑞 = 4), almost weekly (𝐹𝑟𝑒𝑞 = 3), almost monthly (𝐹𝑟𝑒𝑞 = 2), less than monthly (𝐹𝑟𝑒𝑞 = 1)

and no care provided (𝐹𝑟𝑒𝑞 = 0). If informal care is provided to two parents, we consider the maximum frequency reported.

Control variables

We control for some characteristics of the parents, including age 9 and health reported by the child as a proxy for care needs. Health is measured in the analysis through a polytomous variable that takes different values for very good or excellent (Health1), fair or good (Health2), and poor (Health3). When only one parent is alive, we also control for the gender of the living parent. When both parents are alive we also control for the second parent's health status and age. These variables are interacted with a dummy that takes the value of 1 if the two parents are alive, and 0 otherwise. The effect of parents' health status and age is assumed to be equal regardless of the gender and number of living parents.

Concerning the adult child's characteristics, we control for the distance from the parent which may be an important determinant of the child's availability to provide care. This variable takes the following values: less than 1 km (Dist1), 1-25 km (Dist2) and more than 25 km (Dist3). The educational levels are lower secondary or less (Educ1), upper secondary (Educ2)

and post-secondary (Educ3). We also control for age, conjugal status (more precisely whether the adult child is single, i.e. widow, divorced or never married), self-reported health status and number of children. Finally, we control for the numbers of sisters and brothers to account for the existence of other potential caregivers who might replace the adult child, but also for the fact that care is more often provided by daughters.

Two control variables may be endogenous. The first is the distance from the parent, given that an adult child may relocate to live closer to a parent who needs long-term care. This fact is examined by [START_REF] Stern | Estimating Family Long-Term Care Decisions in the Presence of Endogenous Child Characteristics[END_REF] and the endogeneity bias appears to be very limited. More recently, [START_REF] Hiedemann | Will you still want me tomorrow? The dynamics of families' long-term care arrangements[END_REF] used panel data (AHEAD/HRS) from 1995 to 2004 and found that there are very few transitions in geographical distance through time. But they also found that parents' needs for informal care have a predictive power on transitions and the children who move closer are more likely to provide informal care, suggesting that the distance is actually endogenous. The second is the health of the adult child, which is also potentially endogenous because caregiving can be detrimental to health. There is a large literature on this subject (see, for example, Bauer and Souza-Poza 2015). We estimated the models after removing these variables from the list of controls and the results remained unchanged.

Finally, we include wave dummies to account for the pooled structure of the data. We also add country dummies to account for country-specific effects such as differences in long-term care financing. An interaction of the wave and country dummies is also introduced. The sample distribution is shown in the Online Appendix (Table A.1).

Excluded variables

Our identification strategy exploits the fact that for individuals reaching legal retirement age the individual financial incentive to retire increases strongly, creating a likely exogenous variation in the probability of being retired. Indeed, in most European countries, eligibility for public pensions is conditioned on age. There are generally two age thresholds: the Early Retirement Age (ERA), the age at which it is possible to claim pension benefits with possible deductions, and the Official Retirement Age (ORA), the age at which it is possible to claim pension benefits without deduction. 10 They differ across countries, especially the ERA, and by gender, as men become eligible for public pensions at an older age than women in many countries. Due to retirement reforms that, in most cases, have raised the ERA, this age also differs within countries across time.

Hence, due to heterogeneity in ERA and ORA, some individuals may be either below or above the ERA and the ORA for a given age. Figure 1 outlines the observed proportions of retirees by age among three subsamples: those under the ERA, those above the ERA but under the ORA and those above the ORA. As expected, and after controlling for age, those reaching the ERA or ORA are more likely to be retired. Estimates from our main model (see Online Appendix) confirm this result after controlling for others covariates.

10 Theses ages are reported in the Online Appendix (Table A2)

We thus use as excluded instruments two dummy variables. The first is equal to one if the age of the adult child 𝑖 in the country 𝑐 in wave 𝑡 is between the ERA and the ORA in force in the country 𝑐 in wave 𝑡. The other is equal to one if the age is higher or equal to the ORA:

1 𝐸𝑅𝐴<𝑎𝑔𝑒<𝑂𝑅𝐴 = 1 𝑖𝑓 𝐸𝑅𝐴 𝑐𝑡 ≤ 𝑎𝑔𝑒 𝑖𝑐𝑡 < 𝑂𝑅𝐴 𝑐𝑡 1 𝑎𝑔𝑒>𝑂𝑅𝐴 = 1 if age ict ≥ ORA ct
Theses discontinuities in eligibility rules are assumed to create plausibly exogenous variations in the retirement decision. This exclusion restriction holds if we assume there are no discontinuities in the caregiving outcome variables at the ERAs and ORAs, except for the indirect effect of retirement at these ages.

1 𝐸𝑅𝐴<𝑎𝑔𝑒<𝑂𝑅𝐴 and 1 𝑎𝑔𝑒>𝑂𝑅𝐴 are also used in the selection equation to create a piecewise age trend. The effect of age on the propensity to be retired is therefore allowed to differ depending on whether the individual's age is lower than ERA, between ERA and ORA or higher than ORA. 11 Columns (2) to (5) report the mean ATE(x) after controlling for covariates but imposing some restrictive assumptions. Results reported in Column (2) are based on the two following restrictive assumptions: (i) the retirement status is exogenous; (ii) the effect of retirement on caregiving does not depend on observed (and unobserved) characteristics. We thus impose prior to the estimation: 𝜌 0,𝑅 = 𝜌 1,𝑅 = 0 and 𝛽 0 = 𝛽 1 . The estimated model is thus an ordered probit model. Results reported in Column (3) are based on a specification in which retirement is still assumed to be exogenous (𝜌 0,𝑅 = 𝜌 1,𝑅 = 0), but has a different effect according to the observed characteristics (𝛽 0 ≠ 𝛽 1 ). In this case, the estimated model belongs to the class of exogenous switching ordred probit models. Results reported in Column (4) relax the assumption that the retirement status is exogenous (𝜌 0,𝑅 and 𝜌 1,𝑅 𝜖 [-1; 1]). The common treatment effect assumption is nevertheless imposed (𝛽 0 = 𝛽 1 and ρ 0,R = 𝜌 1,𝑅 ). In this case, the estimated model belongs to the class of IV ordered probit models. Results reported in Column ( 5) are based on a specification in which retirement is assumed to be endogenous but has no unobserved heterogeneity (𝜌 0,𝑅 = 𝜌 1,𝑅 𝜖 [-1; 1]), and has a different effect according to the observed characteristics (𝛽 0 ≠ 𝛽 1 ).

Results

Mean average treatment effects

We can observe that the point estimates of the mean ATE(x)'s are not sensitive to the specification, even if we observe a loss of efficiency when retirement is treated as endogenous. The correlations of the error terms are found to be not significantly different from zero in specification (4), ( 5) and ( 6), so we cannot reject the exogeneity of the retirement status, which is in line with previous literature. We thus focus on the effect of retirement under this assumption for the rest of the paper.

We find that on average, retirement has no effect on the probability of not providing care.

Hence, care provision does not seem to depend on whether the individual is retired or not.

Being retired significantly increases the frequency of care conditional on being caregiver.

This suggests that being retired tends to increase the frequency of care among people who would have been caregivers even if they were not retired. Indeed, being retired significantly decreases the probability of providing care monthly or less frequently, while it increases weekly and daily care provision by approximately 4 percentage points on average. This effect is quite large in relative terms as the baseline percentage of individuals providing care almost weekly or daily is 18.47.

These results are in line with literature, which generally finds that labor market participation does not have a contemporaneous causal effect on care involvement [START_REF] He | Does formal employment reduce informal caregiving?[END_REF][START_REF] Stern | Estimating Family Long-Term Care Decisions in the Presence of Endogenous Child Characteristics[END_REF]. Our result on the causal effect of retirement on the conditional frequency of care provision is also in line with studies that consider the reverse pathway of causation [START_REF] Jacobs | Caregiving intensity and retirement status in Canada[END_REF](Jacobs et al., , 2016)).

We recall that in all our specifications, we investigate the effect of retirement status on the frequency of care without imposing the parallel regression assumption. When the 𝛿 𝑓 's (for 𝑓 = 0,1,2,3,4) are constrained to be equal, we do not observe any significant effect of retirement status on the frequency of care. Hence, imposing the parallel regression assumption would have led us to wrongly conclude the absence of an effect.

Heterogeneity in average treatment effects

As we do not reject the exogeneity assumption of retirement status, we investigate here the observed heterogeneity in the effect of retirement using estimates obtained from the exogenous switching model (Specification 3).12 Table 2 reports the estimates of the marginal effect on the ATEs for each control variable. 13 We observe that the majority of the covariates do not have any significant effect on the ATEs.

This means that the retirement effect is quite homogenous regardless of observed individual characteristics. Broadly speaking, the main conclusion drawn from the estimated mean ATE is therefore still relevant: for a majority of individuals, while being retired does not significantly influence the probability of providing care, it significantly decreases the probability of providing care weekly or less frequently and increases the probability of providing weekly or daily care.

The strong homogeneity of the effect of retirement on the provision of informal care is particularly true with regard to the observed characteristics of the child. Most of these characteristics do not impact significantly the mean ATEs. When they do (such as age or education level), the marginal effect is of low magnitude. There is however one exception.

Our results suggest that single men differ significantly from the average (and from single women in particular). Figures 2a and2b illustrate their specificity. Figure 2a reports the average predicted probabilities of providing care (regardless of frequency), according to the child's gender and marital status. Being retired does not significantly affect informal care provision, except among single men, for whom it unexpectedly decreases the probability of providing care by 9 percentage points (from 28% to 19%). This result is also observed in descriptive statistics 14 and is robust to the omission or the inclusion of other covariates. We have also tested the effect of retirement on social participation. The results suggest that retirement also reduces other types of social activities for this particular group. This specificity with regard to the effect of retirement on informal care involvement is thus likely effect because the coefficients measure the association of each covariate with the latent variable outcomes (𝐹𝑟𝑒𝑞 0𝑖 * and 𝐹𝑟𝑒𝑞 1𝑖 * ) whose relation with the observed outcomes (𝐹𝑟𝑒𝑞 0𝑖 and 𝐹𝑟𝑒𝑞 1𝑖 ) depends on the 𝛿 𝑓 's. 13 To save space, and because our main conclusions are not affected by this simplification, Table 2 reports results after aggregation of the probabilities of providing care monthly or less often on one side and weekly or daily on the other. 14 Among single men (n=897, 6% of the sample), 20.71% of those who are retired and 30.96% of those who are not retired provide care.

to reveal a more general difference in the way retirement reshapes social participation for single men. One potential channel is related to the drop in mental health of single men after retirement [START_REF] Picchio | The Mental Health Effects of Retirement[END_REF]. Another potential mechanism is the activitycomplementarity hypothesis put forward by the literature that examines the relationship between the different forms and types of productive activities at older ages. 15 These mechanisms are nevertheless unlikely to fully explain this large decrease. Further research is therefore required to clarify this result.

Figure 2b reports the average predicted probabilities of providing weekly or daily care conditional on being a caregiver, according to the child's gender and marital status.

Conditional on being a caregiver, being retired is associated with a significant increase in the probability of providing weekly or daily care regardless of gender and marital status, even if the effect is smaller among single men.

The effect of retirement on the provision of informal care appears to be more heterogeneous with regard to the parents' characteristics.

It differs in particular according to the number of living parents. Filial care is less sensitive to the labor market position when the child cannot rely on the other parent to provide care.

Indeed, when only one parent is alive, being retired does not significantly affect the probability of being a caregiver, whereas retirement does increase participation in care provision when both parents are alive (by 5 pp on average). Conditional on being a caregiver, we observe an increase in the average probability of providing care weekly or daily regardless 15 The activity-complementarity hypothesis states that multiple activities performed in parallel may complement each other, thus leading to an overall greater productive engagement [START_REF] Hank | Volunteer work, informal help and care among the 50+ in Europe: Further evidence for "linked" productive activities at older ages[END_REF][START_REF] Lancee | Social connectedness and the transition from work to retirement[END_REF]. Le [START_REF] Bihan-Youinou | Travailler et prendre soin d'un parent âgé dépendant[END_REF] explain for instance that working allows compensating the emotional burden and stress experienced by caregivers. To the best of our knowledge, no paper investigates the association between various types of activities in later life by distinguishing gender and marital status.

the number of living parents, but the magnitude of the effect is also lower when only one parent is alive

The health status of the living parents also appears as an important predictor of caregiving behavior. For those having only one living parent, both the probability of providing care and the conditional probability of providing weekly or daily care increases with a deterioration of his/her health status, regardless of the child's retirement status (Figures 3a and3b). The conditional frequency of care is nevertheless slightly less sensitive to the parent's care need for retired children. In particular, provision of informal care is more homogenous between retired and non-retired children when the parent is in poor health, suggesting that adult children need to provide care for their parents in poor health regardless of their labor market position.

For individuals with two parents, the same pattern is observed: caregiving involvement of workers is more driven by the care needs of the parents and/or the ability of one parent to provide care to the other than for retirees. In particular, when both parents are in poor health, those participating in the labor market are as involved as retirees (Figure 4a). On the contrary, when both parents are in very good or excellent health, retirees are characterised by a significantly higher probability of being a caregiver. The same result is observed, at a smaller magnitude, when one parent needs care (poor health) while the other is a potential caregiver (in good health).

With regard to geographical distance, as expected, we observe a negative effect of distance on the propensity to be caregiver (Figure 5a) and on the conditional propensity to provide weekly or daily care (Figure 5b), regardless of retirement status. Moreover, we do not observe any significant variation in the effect of retirement on the probability of providing care. Our results show nevertheless that the ATEs on the probability of providing weekly or daily care differ significantly by distance. The greater the distance, the greater the effect of retirement on the probability of providing weekly or daily care.

Sensitivity analysis and additional results

In this section, we investigate the sensitivity of the estimated effects to alternative sample restrictions and sources of heterogeneity in the ATEs. We summarize here the main results.

More details are provided in the online appendix.

One of our sample restrictions excludes individuals co-residing with their parent(s), as we do not have any information on the frequency of the care they provide. This sample restriction could potentially bias our results if the decision to co-reside is endogenous to retirement status. To address this concern, we check whether retirement is associated with co-residence through a simple Probit model (assuming that retirement is exogenous) and a Bivariate Probit model, allowing the retirement status to be endogenous. In both cases, estimations results do not provide any evidence of a significant effect of retirement status on the probability to coreside with a parent, suggesting that our sample restriction is unlikely to affect our main results.

Another possible source of concern is the inclusion in our sample of individuals who are selfemployed. One might argue that official retirement ages are less relevant for self-employed workers. We have therefore conducted a robustness test by excluding the self-employed from the sample. Results remain unchanged.

Covariates included in the estimation model do not capture all potential sources of heterogeneity. Some characteristics are missing either because we do not have any information on the survey or because they are likely to be endogenous.

One missing dimension is related to the economic situation of the child. SHARE offers information on household income and wealth. We chose not to include these variables as covariates in our main specification for two main reasons. First, there is a high proportion of non-responses. This would require reducing significantly our sample size, with a risk of selection bias. Second, the exogeneity of the child's economic situation is a source of concern because income and wealth are likely to be affected by the trade-off between informal care involvement and labor market participation [START_REF] Van Houtven | The effect of informal care on work and wages[END_REF][START_REF] Schmitz | Informal care and long-term labor market outcomes[END_REF][START_REF] Heitmueller | The earnings of informal carers: Wage differentials and opportunity costs[END_REF]. To investigate how economic background affects the average treatment effect, we conduct an additional analysis reducing our sample to individuals who participated in retrospective SHARE surveys (waves 3 or 7). We use two available variables associated with salaries and retirement pensions [START_REF] Mincer | Schooling, Experience, and Earnings[END_REF]: the number of worked years before 50 and the main job sector. They are also unlikely to be endogenous with regard to the current informal care provision. In addition, we control for homeownership. The inclusion of these control variables does not modify our results. Moreover, these covariates do not significantly affect the ATE of retirement and confirm that the effect of retirement is rather not sensitive to the characteristic of the children (see Online Appendix for detailed results).

The effect of retirement transition might also be different whether one has a full-time or parttime job. We do not account for this in the main analyse, and it is arguably a limitation. To explore this issue, we added to the list of covariate a dummy equal to 1 if the child is working full-time, 0 otherwise. We are unfortunately unable to instrument this variable and estimation results rely on the assumption that working time is exogenous. Hence, we must be cautious when interpreting these results. We first estimated the model assuming the parallel regression assumption (for the full-time employment dummy) and found no difference in the effect of retirement for those who work part-time or full-time. We then relax the parallel regression assumption for the full-time employment dummy. We find that retirement has still no effect on the probability to provide informal care regardless of working time. However, retirement is associated with an increase in the frequency of care for those who work full-time, but we do not observe a significant effect for those who work part-time. In others words, the increase in the frequency of care associated with retirement seems to be driven by full-time workers.

Further research is nevertheless needed to test the robustness of this result after accounting for the endogeneity of working time.

Finally, one might consider that people provide informal care not only to parent(s) but also to parents-in-law. Such care is rare as only 3% of individuals in a couple declare providing care to a parent-in-law in the survey. We nevertheless take this possibility into account by reestimating our model with an outcome variable indicating the frequency of care provided to a parent or a parent-in-law and an additional covariate measuring the number of living parentsin-law. For children providing care for both parents and parents-in-law, we retain the higher frequency of care as an outcome. The mean ATEs of retirement remains unchanged (see Online Appendix for detailed results).

Policy implications

The implications of our results can be better interpreted with a policy simulation aimed at assessing to what extend a 2-year increase in the ERA and ORA would reduce filial care.

Based on our estimation results, this simulated increase in the ERA and ORA is expected to result in a 2.64 pp decline in the average probability of being retired. Table 3 reports the indirect effect on the frequency of care. 16 As expected, we observe an increase in the probability of providing care monthly or less often and a decrease in the probability of providing care weekly or daily. However, given the small proportion of individuals directly 16 See Online Appendix for more detail on the calculation.

affected by the simulated reform, and the relatively low sensitivity of filial care to retirement status, the magnitude of the effect on care behavior is marginal. Converted to number of hours, the expected decrease is about 50 minutes per year on average in our sample, but 35 hours per year among those who switch from retired to non-retired due to the reform. Extrapolated to the European population aged 50 to 69 with at least one living parent, the decrease in informal care would represent approximately 29.1 million informal care hours per year. As a consequence of this decline in filial care, one would expect an increase in the use of home care services. 17 The existing literature suggests however a relatively small substitution effect between informal care and formal care [START_REF] Bolin | Informal and formal care among single-living elderly in Europe[END_REF][START_REF] Van Houtven | Informal care and health care use of older adults[END_REF]. From these estimations, an increase of 2.4 million formal care hours per year would be expected, representing less than 1 500 full-time equivalents. Although this calculation is only approximate, it shows that the impact of an increase in the legal retirement age on long-term care provision is unlikely to be a major source of concern at the macroeconomic level.

Conclusion

In the context of increasing care needs, family support is often seen as a cost-effective way of preventing institutionalization and enabling dependent elderly people to remain at home [START_REF] Zigante | Informal care in Europe -Exploring formalization, availability and quality[END_REF].

Many argue that informal care provision for dependent older adults will decline, however, because the labor market participation of seniors is increasing in response to pension system reforms that have raised the pensionable ages and the minimum contribution years required for a full pension (European Commission, 2018). However, the extent to which, at the individual level, retirement affects the level of informal care provided by seniors to their elderly parents has never been directly addressed.

The purpose of this paper is to fill the knowledge gap by estimating the causal effect of retirement on informal care behavior. Our empirical approach allows us to deal with the possible endogeneity of selection in retirement and to investigate to what extent the retirement effect is heterogeneous. Our approach presents some limitations, however. First, some potentially important variables are missing from the data, such as the use of formal care, the parent's disability level or the parent's place of residence. In particular, some parents may live in a nursing home. Second, individuals co-residing with their parent are excluded from the analysis because of a lack of information on the frequency of their involvement.

In spite of these limitations, our results help to clarify the nature of the alleged decline in informal care provision due to the increase in the effective retirement age. First, being retired or not has no impact, on average, on the propensity to provide care. However, a significant decrease in the probability of providing weekly or daily care is observed. Our results suggest that this pattern is globally homogenous, even if the provision of care is less sensitive to the child's retirement status in some situations (single parent in poor health, both parents in poor health, geographical proximity).

Accordingly, the decrease in the proportion of retired adult children should not reduce the number of caregivers, but rather the level of involvement among caregivers. At the macroeconomic level, our analysis nevertheless suggests that the adverse effects of a policy aimed at extending working lives, either through an increase either in unmet long-term care needs or in demand for home-care services or nursing home places [START_REF] Van Houtven | Informal care and health care use of older adults[END_REF][START_REF] Mommaerts | Are coresidence and nursing homes substitutes? Evidence from Medicaid spend-down provisions[END_REF] are limited.

  use a sample of women from the 1996, 2001 and 2004 waves of the Survey of Income and Program Participation. They estimate an IV probit model with local unemployment rates to instrument the labor supply.

  Table1reports the mean ATE(x), i.e. the average treatment effects integrated over the distribution of X, obtained from different specifications. Column (1) reproduces descriptive statistics through a model without covariates and assuming that retirement is exogenous.

	Column (6) reports results from our endogenous switching model.

11

We test various age trend specifications and retain a quadratic age trend for each piecewise.

Table 2 .

 2 Marginal effect on average treatment effects For individuals with only one parent, the marginal effect is computed assuming that the second parent is the same age as the first, with health=health1 (2); (3) Marginal effects are constrained to be equal for both parents * p<0.1, ** p<0.05, *** p<0.01. Clustered standard errors at the individual level in parentheses. This table reports the marginal effect of each observed characteristics on the mean average treatment effect (ATE(X)) for no care provided (1), care provided less than monthly or monthly (2) and care provided weekly or daily (3). Results are based on specification 3 with retirement assumed to be exogenous.

	No Care	Annual or monthly	Weekly of daily
	(1)	(2)	(3)

Source: SHARE, own calculations

Figure 1.

Observed proportion of retirees by age

  Source: SHARE, waves 1, 2 and 6, own calculations.Note: The lines show a quadratic fit with 95% CI. ERA and ORA stand for early and official retirement ages respectively.

		100
		90
		80
	%) (in retirees of	60 70 50	subsample under ERA subsample above ERA
	Proportion	30 40	but under ORA subsample above ORA
		20
		10
		0
		55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
		Age

Figures 2a and 2b. Retirement effect by gender and marital status of the child 2a

  . Average probability of providing care 2b. Average probability of providing weekly or daily care, conditional on providing care Source : SHARE, waves 1, 2 and 6, own calculations Note : Figures2a and 2breport average predicted probabilities from specification (3) (exogenous switching model), with 95% confidence interval.

	Figures

3a and 3b Retirement effect by parent's health status, among those having one living parent 3a

  . Average probability of providing care 3b Average probability of providing weekly or daily care, conditional on providing care Source: SHARE, waves 1, 2 and 6, own calculations Note: Figures3a and 3breport average predicted probabilities from specification (3) (exogenous switching model), with 95% confidence interval.

	55%			100%
	50%			90%
	45%			80%
	15% 20% 25% 30% 35% 40%			30% 40% 50% 60% 70%
	10%			20%
	5%			10%
	0%			0%
	single	single man	man in	woman in	single	single man	man in	woman in
	woman		couple	couple	woman		couple	couple
	retirement 0	retirement 1	retirement 0	retirement 1
	55%			100%
	50%			90%
	45%			80%
	15% 20% 25% 30% 35% 40%			30% 40% 50% 60% 70%
	10%			20%
	5%			10%
	0%			0%
	1 (very	2 (Fair/good)	3 (Poor)	1 (very
	good/excellent)			good/excellent)
	retirement 0	retirement 1

Figures 4a and 4b. Retirement effect by parent's health status, among those having two living parents 4a

  . Average probability of providing care 4b Average probability of providing weekly or daily care, conditional on providing care Legend : Health 1 : very good or excellent ; health2 : fair or good ; health 3 : poor Source: SHARE, waves 1, 2 and 6, own calculations Note: Figures4a and 4breport average predicted probabilities from specification 3 (exogenous switching model), with 95% confidence interval.

	Figures

5a and 5b. Retirement effect by distance from the parent 5a

  . Average probability of providing care 5b. Average probability of providing weekly or daily care, conditional on providing care Source: SHARE, waves 1, 2 and 6, own calculations Note : Figures5a and 5breport average predicted probabilities from specification (3) (exogenous switching model), with 95% confidence interval.

	55%					100%
	50%					90%
	45%					80%
	15% 20% 25% 30% 35% 40%					30% 40% 50% 60% 70%
	10%					20%
	5%					10%
	0%					0%
	Health1	Health1	Health1	Health2	Health2	Health3	Health1	Health1	Health1	Health2	Health2	Health3
	&	&	&	&	&	&	&	&	&	&	&	&
	Health1	Health2	Health3	Health2	Health3	Health3	Health1	Health2	Health3	Health2	Health3	Health3
		retirement 0	retirement 1			retirement 0	retirement 1
	55%					100%
	50%					90%
	45%					80%
	15% 20% 25% 30% 35% 40%					30% 40% 50% 60% 70%
	10%					20%
	5%					10%
	0%					0%
	1 (less than 1km) 2 (between 1	3 (more than 25	1 (less than	2 (between 1
			and 25 km)	km)	1km)	and 25 km)
		retirement 0	retirement 1

Literature shows that some individuals unretired and return to work. See for instance[START_REF] Maestas | Back to work expectations and realizations of work after retirement[END_REF].

Except for 𝜏 -1 and 𝜏

that are assumed to be respectively equal to -∞ and +∞.

The equality of the 𝛿 𝑓 's can be tested through a likelihood ratio test. In all our specifications, the parallel regression assumption, i.e. the hypothesis that the 𝛿 𝑓 are equal for all values of f, is always rejected at the 1% level.

This does not mean that it is equal to zero[START_REF] Maddala | Limited-Dependent and Qualitative Variables in Econometrics[END_REF].

𝛷 is the cumulative normal distribution function and later 𝜑 will be the normal density distribution function.

We tested to introduce age squared, but its coefficient was never significant so we use linear age in the model.

Coefficient estimates from Specification 3 (exogenous switching model) and Specification 6 (endogenous switching model) are reported in TableA3of the Online Appendix. Note that the comparison of the estimated coefficients in retired and non-retired states only gives an approximate idea of the heterogeneity of the treatment

This back-of-the-envelope calculation focuses on a potential increase in the use of home-care services. Others effects are possible, such as an increase in nursing home entry or an increase in informal care from others providers.

(Fair/good)

(Poor) retirement 0 retirement 1

(more than 25 km) retirement 0 retirement 1

Acknowledgements

The authors are grateful to Rosalind Bell-Aldeghi, Carole Bonnet, Eve Caroli, Olivier Donni, Claudio Lucifora, Dominique Meurs, Lidia Panico, Patrick Puhani, Arthur van Soest and Marianne Tenand for

The SHARE data collection has been primarily funded by the 00360), FP6 (SHARE-I3: RII-CT-2006-062193, COMPARE: CIT5-CT-2005-028857, SHARELIFE: CIT4-CT-2006-028812) and FP7 (SHARE-PREP: N°211909, SHARE-LEAP: N°227822, SHARE M4: N°261982). Additional funding from the German Ministry of Education and Research, the Max Planck Society for the Advancement of Science, the U.S. National Institute on Aging (U01_AG09740-13S2, P01_AG005842, P01_AG08291, P30_AG12815, R21_AG025169, Y1-AG-4553-01, IAG_BSR06-11, OGHA_04-064, HHSN271201300071C) and from various national funding sources is gratefully acknowledged (see www.share-project.org).

 -23269.903 -19277.646 -19228.436 -19277.641 -19228.398 -19228Note : 𝐸(△ 𝑓𝑟𝑒𝑞=𝑓 𝐴𝑇𝐸 ) is the mean average treatment effect of retirement on the frequency 𝑓, with 𝑓 = 0 for no care, 𝑓 = 1 for care less than monthly, 𝑓 = 2 for monthly care, 𝑓 = 3 for weekly care and 𝑓 = 4 for daily care.

Standard errors are clustered at the individual level. This table reports the average treatment effect of retirement on the probabilities of providing no care, less than monthly care, monthly care, weekly care and daily care. They are obtained controlling for the parents' age, gender and health status (reported by the child), but also for the child's age (squared), gender, self-assessed health status, distance from the parent, number of siblings, number of children, educational level and conjugal status. In addition, we add wave and country dummies (interacted).

Source: SHARE, own calculation.