Informal Care and Formal Home Care Use
in Europe and the United States
Alberto Hollya , Thomas M. Lufkina ,
Edward C. Nortonb , Courtney Harold Van Houtvenc
Version date : July 1, 2010
This is a preliminary version. Comments are most welcome.
Please do not quote without the authors’explicit consent.
a
b
c
Institute of Health Economics and Management and Faculty of Business and
Economics, University of Lausanne
Department of Health Management and Policy and Department of Economics,
University of Michigan
Center for Health Services Research in Primary Care, Durham Veterans A¤airs
Medical Center and Department of Medicine, Division of General Internal
Medicine, Duke University Medical Center
Correspondence: Alberto Holly
[email protected]
University of Lausanne, IEMS, Vidy, CH-1015 Lausanne, Switzerland
Tel: +41 21 692 3482, Fax: +41 21 692 3665
Abstract
The provision of informal care by adult children is an important form of longterm care for older individuals and can reduce the use of medical services if they
are substitutes. We examine how informal care by all children and formal care
interact, which is critically important given demographic trends and the many
policies proposed to promote informal care. The purpose of this study is to compare the United States and European countries, by merging data from the U.S.
Health and Retirement Study (HRS) with its European counterpart, the Survey
of Health, Ageing and Retirement in Europe (SHARE). We argue that the institutional setting is di¤erent across the Atlantic, as European home care schemes
are predominantly publicly run, whereas the market plays a bigger role in the
United States. We use a fexible simultaneous equations approach that allows for
a di¤erent relationship between informal and formal home care in the two regions,
using copulas. We …nd that in Europe it is predominantly the supply of formal
home care that in‡uences children’s decisions to provide informal care, while in the
United States parents’decisions to use formal home care are based on the amount
of informal care received and the amount of informal care provided by children is
dependent on the amount of formal care.
1
Introduction
The provision of informal care by adult children is an important form of long-term
care for older individuals. Informal care can reduce the use of formal health care
if they are substitutes and enhance the use if they are complements. Rapid aging
of populations in developed countries make this question of critical importance to
health policy. Demand for informal care is growing with the ranks of the elderly,
while the supply is shrinking given smaller average family size and increased labor
force participation by women. Therefore, it is critically important to understand
the relationship between informal and formal care to predict the likely e¤ects on
future health care expenditures.
We address how informal care by all children and formal home care interact,
building on recent research that has highlighted the importance of informal care
(Van Houtven and Norton, 2004, 2008; Bolin et al., 2008). This published work
demonstrated that informal care is a substitute for formal long-term care in the
United States and in Europe. The results, however, are mixed for other types
of formal care. Van Houtven and Norton (2008) …nd that informal care reduces
1
inpatient expenditures in the U.S. while Bolin et al. (2008) …nd a complement
relationship for Europe.
Comparing the U.S. and Europe is challenging because of the large di¤erences
in relevant institutions. The three published papers all modelled informal care and
formal care simultaneously, and used instrumental variables to obtain consistent
estimates in the face of endogeneity. Furthermore, the papers focused on how
informal care a¤ects formal care, not the other way around. A close examination
of the institutions in both Europe and the United States leads us to question this
assumption. Therefore we propose a model that ‡exibly allows for either case and
let the data decide how best to model the relationship.
In Europe formal home care is mainly provided by public or not-for-pro…t institutions. In general, provision of care is only dependent on need and not on income
or wealth. Therefore individuals in poorer health will use the services provided to
them at low or no cost by the state or services covered by their insurance. They
may turn to their children and relatives for support in the form of informal care
only in cases where this formal home care is not su¢ cient. However because most
European countries face budget constraints, the provision of services is limited and
only a part of the needs will be met. In some cases formal care decisions are made
by the authorities after explicitly accounting for the availability of informal care.
Individuals who can bene…t from informal care provided by their children might
not receive as much formal care as those who do not have children who can take
care of them. These two e¤ects go in opposite directions. It is therefore not clear,
in Europe, whether it is the amount of informal care that in‡uences formal care
provision or the opposite
In the United States however, publicly provided home care is much more limited, generally being available to individuals who are permanently disabled or
homebound (Medicare) or meet strict asset limitations (Medicaid). These public
programs do not typically account for the availability of informal care available to
a parent. For this reason it may be that adult children in the U.S. wait to see if
formal care will be provided publicly, and then alter their own supply decisions
accordingly.
In addition, given that publicly-provided home care in the U.S. is quite limited
compared to Europe, the private formal home care market is much more developed
there. Formal home care can be purchased privately, or, to a much smaller extent,
can be …nanced through private long-term care insurance plans (10% of persons
age 65 and over have long-term care insurance, Finkelstein and McGarry, 2006).
Thus, the existence of a private market means that potential caregivers have more
‡exibility. In the case of parents covered under long-term care insurance, children
can decide on the amount of informal care to provide, taking into account the
formal home care bene…t available to the parent. Nevertheless, if there is no
2
long-term care insurance and no Medicare or Medicaid coverage, the purchase
of long-term care can be very costly. Hence, it is very likely that parents will
purchase formal care services only if they do not receive help from their children.
Thus, informal caregivers in the U.S. will consider the potentially high cost of
formal care when making their own informal care decisions, because formal care
most likely will be …nanced by the parent or the parent’s family, which in turn
may a¤ect a child’s bequest.
In both the U.S. and European settings, the simultaneity of the care decisions
between parents and children implies endogeneity in the model. We expect there
to be di¤erences in the way informal care and formal care decisions are made
in the two settings, but we are agnostic about the direction of causality in our
estimation procedures. Using the Survey of Health, Ageing and Retirement in
Europe (SHARE) and the U.S. Health and Retirement Study (HRS) we test the
hypothesis that Europeans and Americans behave similarly by explicitly allowing
the direction of causality to di¤er between Europe and the United States with a
simultaneous equations model based on copulas. This allows to model the variables
of interest, informal and formal home care, as two-part margins with generalized
gamma distributions in the positive part, thus accounting for skewness and a large
mass at zero, while at the same time permitting a ‡exible dependence structure
between them.
The results for the United States imply that informal care by the recipient’s
children reduces the probability of home health use, while in Europe it is rather
the formal care use of the parent that is being taken into consideration by children
when deciding to provide informal care. Among users of care, we …nd that the
amounts of the two types of care are much more in‡uenced by each other in the
U.S. than in Europe, and that the amount of formal care more strongly in‡uences
the amount of informal care than the reverse, on both sides of the Atlantic.
2
2.1
Conceptual framework
The relationship between informal and formal care
Prior studies on the relationship between formal and informal care in the United
States do not seem to de…nitively establish whether the two types of care are
substitutes or complements. For example Langa et al. (2001) …nd that people with
greater social support have a higher probability of entering a nursing home and that
paid home care tends to be used predominantly by elderly living with their children,
3
while Kemper (1992) …nds that the availability of family caregivers decreased the
reliance on formal care and increased the use of informal care. Most of these earlier
studies do not take into account the endogeneity of formal and informal care. The
few studies that control for endogeneity (Lo Sasso and Johnson, 2002; Greene,
1983) suggest that informal care and formal care are substitutes. More recently,
Van Houtven and Norton (2004) analyze the e¤ect of informal care by children on
formal care use. Their model is based on the parent’s optimization process given
her children’s choice of informal care supply: the children …rst determine their
optimal level of informal care provision, and then parents choose optimally their
formal care consumption, based on the amount of informal care that they receive.
They …nd that informal care reduces home health care use, using data from the
HRS and the Asset and Health Dynamics Among the Oldest-Old Panel Survey
(AHEAD). Furthermore they detect endogeneity and use instrumental variables
to instrument for informal care.
There is a growing literature on the relationship between informal care and
formal care in Europe. Using SHARE, Bolin et al. (2008) apply the same model
as Van Houtven and Norton (2004) to the European context. More formally, they
use a probit model to analyze the probability of home care use, one measure of
formal care. The quantities are then estimated using OLS, conditional on positive
use. They …nd that endogeneity is detected in the model of home health care.
Furthermore they show that, without correcting for endogeneity, informal care is
a complement to formal home care, whereas after taking endogeneity into account
they become substitutes.
All of these studies have assumed the same direction of causality from informal
to formal care. However, it is not clear a priori that it is the provision of informal
care by children that has an impact on their parents’formal care use, since di¤erent
institutional settings and markets for formal home care should imply di¤erent
incentives both for informal caregivers and for care recipients.
2.2
Theoretical framework
The literature takes three directions to model informal care by children. The …rst
de…nes a family utility function, as in Pezzin et al. (1996). This is rather restrictive,
as it imposes that parents and children have the same objective function, which is
rather unlikely, especially if they do not live in the same household. The second
considers the strategic interactions between the children and treats the parent’s
utility as a public good to be maximized by all children (for example Checkovich
and Stern, 2002), while the third models the relationship between parents and
children as two agents, either by letting only one child provide care (for example
4
Hiedemann and Stern, 1999) or by explicitly imposing only one parent and one
child (for example Pezzin and Schone, 1999; Sloan et al., 1997). We will pursue in
this last direction for the theoretical framework, by building on a model developed
by Chiappori (1992). For simplicity we will also restrict ourselves to the case of
only one parent and one child for the theoretical model, but this framework could
be extended to allow for more children.
The child maximizes her utility function subject to time and budget constraints.
Her utility is a function of her own consumption C c , her leisure L and the amount
of informal care IC provided to the parent, given her own characteristics K c ,
such as demographic characteristics, employment status and distance from the
parent and the parent’s health status H and dependence level D. There are two
justi…cations for the inclusion of informal care in the child’s utility function. An
altruistic child is concerned about her parent’s well being, which is in‡uenced by
the amount of informal care provided. In this case informal care has an indirect
e¤ect on the child’s utility level. However informal care can also have a direct
e¤ect on the child’s utility if she feels satisfaction from providing care. This e¤ect
is often referred to as "process utility", as in Brouwer et al. (2005). The child must
allocate her time between work W , leisure L and the provision of informal care; and
her resources, that is labor income and a transfer from the parent (either positive
if the parent compensates the child for her informal care provision or negative if
the child substitutes informal care provision by a payment aimed at buying other
types of care on the market1 ) between consumption and the costs associated with
the provision of care. Her optimization problem is thus to maximize her utility
U c = U c (C c ; L; IC; K c ; H; D), subject to a time constraint W + L + IC T and a
budget constraint C c + pIC IC !W + S, where T is the total time endowment of
the child, pIC is the opportunity cost of providing informal care (including direct
costs), ! is the wage rate and S is a (non-monetary) transfer from the parent to
the child.
The parent maximizes her own utility function subject to a budget constraint.
Her utility is a function of her own consumption C p , including insurance, and
utilization of both informal and formal care F C, given K p , her demographic characteristics and health and dependence levels, H and D. In most applications
the insurance status is endogenous. This e¤ect is however weakened here since
the individuals considered are over 70 and are likely to have enrolled in their
insurance plan before 65. The same arguments as before apply regarding the direct and indirect e¤ects of care, which could have an indirect e¤ect through the
improvement of the health status or a direct e¤ect as the parent bene…ts from
receiving care and attention from children. Because we are concerned with retired
individuals, we can assume that the parent’s income Y is …xed. She maximizes
1
See Bonsang (2007) for a discussion on intergenerational time and …nancial transfers.
5
her utility U p = U p (C p ; IC; F C; K p ; H; D) subject to her own budget constraint
C p + pF C F C Y S. The solution to this problem has the form:
IC = fIC (F C; Y; C p ; C c ; T; L; K p ; K c ; H; D)
F C = fF C (IC; Y; C p ; C c ; T; L; K p ; K c ; H; D)
The two types of care are therefore jointly determined and will thus have to be
estimated simultaneously.
2.3
Institutional setting
Institutional factors are a major determinant of informal and formal care use,
but since welfare systems are more developed in Europe than in the United States
(Börsch-Supan, 2007), this leads to heterogeneity on the supply side of the informal
and formal care markets. In the United States, publicly provided home care is
limited. The Medicare program provides treatment to the elderly in need of acute
and care and short-term rehabilitation, but has no home health and nursing home
care for those who are not severely impaired or homebound. For those who meet
strict …nancial criteria, the Medicaid program covers the costs of care after copayments and deductibles. Therefore, there is no universal right to public services
in the United States. However these public programs do not typically account for
the availability of informal care available to a parent. For this reason it may be
that adult children in the U.S. wait to see if formal care will be provided publicly,
and then take their own supply decisions accordingly.
In addition, given that publicly provided home care is quite limited, the private
formal home care market is much more developed in the U.S. Formal home care
can be purchased privately, or, to a much smaller extent, can be …nanced through
private long-term care insurance plans (10% of persons aged 65 and over have
long-term care insurance, as mentioned by Finkelstein et al., 2005). This implies
that access to professional services is easier, albeit only for those who can a¤ord
it. The existence of a private market means also that potential caregivers have
more ‡exibility. In the case of parents covered under long-term care insurance,
children can decide on the amount of informal care to provide, taking into account
the formal home care bene…t available to the parent. Nevertheless, if there is no
long-term care insurance and no Medicare or Medicaid coverage, the purchase of
formal home care can be very costly. Hence, adult children who know their parent’s
…nancial situation, insurance and disability status will know that the formal care
options are limited, and will factor these limitations into their own informal care
supply decisions. Clearly, a few European countries have an emerging private
6
formal home care market (such as Italy which has a large ‘grey’market for formal
home care using immigrant labor, as described by Lamura et al., 2006), but in
all cases these markets are not as developed as in the U.S. This also implies that
in the U.S. it is more likely the use of formal care that in‡uences the amount of
informal care supplied by the children.
Most European countries o¤er public provision of health care, either through a
national health service, a social insurance scheme or a compulsory private insurance
system. All of these systems include some long-term care as part of the basic
coverage, and formal home care is mainly provided by public or not-for-pro…t
institutions.
For example, Germany has a social insurance system covering about 90% of the
population, which includes long-term care provision. Bene…ts are granted after a
medical assessment of needs and are distributed in cash, in kind, or a combination
of the two. Austria also has a system of long-term care allowances based on need
of care and independent of income or wealth. In Spain, some autonomous regions
provide home care free of charge, and so there are large disparities across regions.
The Netherlands have a national insurance scheme that covers a (large) share of
the costs related to formal care (Portrait et al., 2000), and in Sweden, most care
for the elderly is …nanced through taxes, and provided by public organizations.
For a more detailed description of the di¤erent health care systems, see OECD
(2005).
In general, provision of care is only dependent on need and not on income
or wealth, which implies that those who need care should not worry about the
…nancial consequences of receiving formal home care. This could have an in‡uence
on the informal care decisions of children, as they could decide to provide care to
their parents only after observing the amount of formal care provided to them,
and thus this would imply that formal care that in‡uences the amount of informal
care provided by children. However in most European countries, social insurance
schemes are under a lot of pressure to balance their budget, and thus the provision
of services is limited and only the needs of the more dependent elderly will be met.
In some cases formal care provision decisions are made by the authorities after
explicitly accounting for the availability of informal care, as in the Netherlands,
where the needs of the elderly are assessed and the individual is generally put on
a waiting list (Portrait et al., 2000). Supplementary or complementary insurance
schemes provide additional coverage and allow for more use of services. This
implies that individuals who can bene…t from informal care provided by their
children will not receive as much formal care as those who do not have children who
can take care of them. Using the European Community Household Panel (ECHP),
Viitanen (2007) shows that increased subsidies to formal home care lead to a
decrease in the provision of informal care outside of the carer’s household, which is
7
consistent with both hypotheses, either that in Europe children reduce their care
provision if their parents receive more formal care, or that supply constraints play
a role in the relationship between formal and informal care.
Another key di¤erence between Europe and the U.S. is the amount of public
support for informal caregivers. In some European countries informal caregivers
are paid or receive pension contributions if they provide informal care in lieu of
active labor force participation. Payments can be substantial, depending on the
severity of the care recipient. By contrast, in the U.S. informal caregivers at most
receive a modest tax credit, and very few actually do. Hence, based on di¤erences
in compensation for informal caregivers, the motivations to provide informal care
are also potentially very di¤erent across settings. Rather than from the state, in
the U.S. any …nancial motivation to provide informal care will be due to the potential …nancial gains provided by the parent (consistent with a strategic bequest or
exchange motive). Although it is illegal to divide bequests unevenly across children
in most European countries, a strategic bequest or exchange motive may similarly
exist (Angelini, 2007). There is however also a much higher likelihood that informal caregivers will be compensated directly. Compounding this di¤erence, is that
long-term care costs in the U.S. are much more likely to be both …nanced by the
parent and more likely to be substantial. Thus in the U.S. informal caregivers will
consider the potentially high cost of formal care when making their own informal
care decisions, since formal care most likely will be …nanced by the parent or the
parent’s family, and thus will reduce the amount of eventual bequests.
In conclusion, the direction of causality is not clear a priori, and we will therefore specify the most possible ‡exible model, in order to account for the di¤erent
possible relations between informal and formal home care.
3
Econometric speci…cation
The variables of interest are hours of informal care and formal care. Both take
only non-negative values and have skewed distributions with a large mass at zero.
In order to take this into account, we model our variables of interest using use
hurdle or two-part models (Duan et al., 1983). This is one of the most common
approaches to deal with corner solution outcomes (Wooldridge, 2002, p. 536) in
the health economics litterature. This is also the approach chosen by Van Houtven
and Norton (2004) and Bolin et al. (2008), who studied the e¤ect of the number of
hours of informal care on di¤erent types of formal care use. As the informal care
variable was endogenous, they used used instrumental variables (IV), which did
8
not allow exploration of the reverse e¤ect of formal care on informal care. Moreover, they both only used a binary indicator of formal home care use, and were
thus limited to an IV-probit estimator for this type of formal care. We extend
the analysis by using a continuous variable of formal home care and simultaneous
equations. This approach has two main advantages. First, it allows to study the
direct e¤ects of both informal care on formal care, and vice versa, thus ensuring
that the simultaneity of the parents and children’s decisions is taken into account.
Second, it also accounts for potential common unobservables such as health characteristics (Charles and Sevak, 2005) or preferences for care (Bonsang, 2009) that
could simultaneously a¤ect the use of formal and informal home care. This is
achieved by imposing an unobserved dependence between the two types of care in
order to avoid possible endogeneity biases.
As we are in a simultaneous equations context, we extend the univariate hurdle
model to the bivariate case. Our approach is close to the work of Yen et al. (2009),
who extend the censored model (Tobit type I) model to the bivariate case, using
generalized log-Burr distributions, and Deb et al. (2009), who develop another
version of the bivariate hurdle model, using standard gamma distributions for the
second part. The main di¤erence with our model is that theirs does not allow for
a simultaneous equations speci…cation.
As in the univariate case, the …rst part is a probit predicting the probability
of having a positive number of hours of care. It is speci…ed as:
Pi (yi > 0) = (yj
0
i
+ x0
0
i)
(1)
where yi and yj are the number of hours of informal and formal care (i; j 2
fIC; F Cg ; i 6= j), ( ) is the standard normal cumulative distribution function, x
is the matrix of covariates, including an intercept, and i0 and the i0 ’s are coe¢ cients to be estimated.
In the second part, we use the generalized gamma distribution in order to account for the skewness of the informal care and home care variables, in stead of
the more widely used regression with a log transformed variable. Manning et al.
(2005) have advocated the use of the generalized gamma distribution in health
economics, because it includes the standard gamma, lognormal, exponential and
Weibull distributions as special cases, and it is more robust to misspeci…cation
than them. The parametrization they use is the same as Stata’s (StataCorp LP,
2009b, p. 361). It is preferred over the standard parametrization due to its numerical stability in a maximum likelihood estimation (Pentsak, 2007, p. 237). The
probability density function is speci…ed as a function of the parameters i+ , i+ , i
and i that are to be estimated:
i
gi yi ;
+
i ;
+
i ;
i;
i jyi
> 0; x; yj =
i yi
9
p
i
i
( i)
exp [zi
p
i
ui ]
if
i
6= 0 (2)
where i = j i j 2 , zi = sign( i ) ln(yi ) yj i+ x0 i+
i , and ui = i exp (j i j zi ),
and with i; j 2 fIC; F Cg ; i 6= j. It simpli…es to the standard gamma distribution
when = , to the Weibull when = 1, to the exponential when = = 1 and
the lognormal when tends to zero, and in this case we have:
gi yi ;
+
i ;
+
i ; 0;
i jyi
> 0; x; yj =
1
p
i yi
exp zi2 2
2
(3)
The cumulative density function Gi associated with gi is:
Gi yi ;
+
i ;
+
i ;
i;
i jyi > 0; x; yj =
Zyi
8
<
I ( i ; ui )
if
(zi )
if
:
1 I ( i ; ui ) if
gi (tjt > 0; x)dt =
0
i
i
i
>0
=0
<0
(4)
where I(R ; ) is called the incomplete gamma function. It is de…ned as I(a; x) =
x
1 / (a) 0 e t ta 1 dt.
The two-part model can then be written as:
P (yi = 0jx; yj )
if yi = 0
P (yi > 0jx; yj ) gi (yi jyi > 0; x; yj ) if yi > 0
fi (yi jx; yj ) =
(5)
For maximum likelihood analysis it is however more convenient to write this density
as:
fi (yi jx; yj ) = [P (yi = 0jx; yj )]1(yi =0) [P (yi > 0jx; yj ) gi (yi jyi > 0; x; yj )]1(yi >0) (6)
In our case, with
fi (yi ;
0
i;
6= 0, we obtain:
0 +
i; i ;
or equivalently when
fi (yi ;
0
i;
0 +
i; i ;
+
i ;
i;
i jx; yj )
"
=
yj
0
i
x0
i
yj
0
i
+ x0
0
i
i
yj
0
i
x0
1(yi =0)
0
i
#1(yi >0)
p
exp zi i ui
(7)
p
i yi
i ( i)
= 0:
+
i ; 0;
i jx; yj )
=
yj i0
10
+
0
i
1(yi =0)
x0 i0
exp [zi2 / 2]
p
i yi 2
1(yi >0)
(8)
The corresponding cumulative distribution function is given by:
Fi (yi ;
0
i;
0 +
i; i ;
+
i ;
i;
i jx; yj )
=
= Pi (yi = 0jx) + Pi (yi > 0jx)
=
8
<
:
(
(
(
yj i0
yj i0
yj i0
x0 i0 )
x0 i0 )
x0 i0 )
+
+
+
Zyi
1
Zyi
fi (tjx)dt
gi (tjt > 0; x)dt
0
(yj i0
(yj i0
(yj i0
+ x0
+ x0
+ x0
0
i ) I ( i ; ui )
0
(zi )
i)
0
I ( i ; ui )]
i ) [1
if
if
if
i
i
i
>0
=0
<0
(9)
for y 0.
The simultaneous equations are implemented by de…ning a bivariate distribution function, starting from the two-part distributions that we speci…ed previously
for informal and formal care. This is done with the help of a copula, which is a
function that links a multivariate distribution to its one-dimensional margins, as
Sklar’s theorem (Sklar, 1973) shows. More formally, it implies that for two variables yi and yj with marginal distributions Fi and Fj there exists a copula C such
that:
(10)
C(Fi (yi ); Fj (yj ); ) = F (yi ; yj )
where is a dependence parameter and F is the joint distribution function of
(yi ; yj ). Furthermore the copula C is unique if the margins Fi (yi ) and Fj (yj ) are
continuous. We can thus express the bivariate distribution of the two types of care
in terms of its marginal distributions and a copula function.This approach naturally extends to higher dimensional distributions. For a more detailed discussion
on the topic, see for example Nelsen (1999), Joe (1997) and Trivedi and Zimmer
(2005).
If we de…ne
@C(u; v; )
@u
@C(u; v; )
Cj (u; v; ) =
@v
Ci (u; v; ) =
and
Cij (u; v; ) =
@C(u; v; )
@u@v
then the density f of the distribution F is given by:
f (yi ; yj ) = Cij (Fi (yi ); Fj (yj ); )
fi (yi )
11
fj (yj )
(11)
where fi (yi ) and fj (yj ) are the densities of Fi (yi ) and Fj (yj ), as speci…ed in equations (7) and (9) respectively. Putting everything together, we obtain the following
likelihood contribution for an observation (yin ; yjn ), n = 1; :::; N :
Ln = [Cij (Fi (yin ); Fj (yjn ); )
fi (yin )
fj (yjn )]1(yin >0)1(yjn >0)
[Ci (Fi (y1n ); Fj (0); )
fi (yi )]1(yin >0)1(yjn =0)
[Cj (Fi (0); Fj (yjn ); )
fj (yjn )]1(yin =0)1(yjn >0)
[C(Fi (0); Fj (0); )]1(yin =0)1(yjn =0)
(12)
and log likelihood:
`n = 1(yin > 0)1(yjn > 0) [ln (Cij (Fi (yin ); Fj (yjn ); )) + ln (fi (yin )) + ln (fj (yjn ))]
+ 1(yin > 0)1(yjn = 0) [ln (Ci (Fi (y1n ); Fj (0); )) + ln (fi (yi ))]
+ 1(yin = 0)1(yjn > 0) [ln (Cj (Fi (0); Fj (yjn ); )) + ln (fj (yjn ))]
+ 1(yin = 0)1(yjn = 0) ln (C(Fi (0); Fj (0); ))
(13)
There are many di¤erent copula functions, each with di¤erent properties. We
experimented with the Clayton, Farlie-Gumbel-Morgenstern (FGM), Frank and
Gumbel copulas, which all allow di¤erent dependence structures. As discussed by
Trivedi and Zimmer (2005), the Clayton copula only supports positive dependence
and displays strong left tail and weak right tail dependence, the FGM copula
is suitable only for limited positive or negative dependence, the Frank copula
allows both negative and positive dependence and the Gumbel copula only allows
positive dependence and displays strong right tail and weak left tail dependence.
The distribution functions and densities of all these copulas are reported in the
Appendix.
The parameters i0 ; i0 ; i+ ; i+ ; i ; i ; j0 ; j0 ; j+ ; j+ ; j ; j and are all estimated
together by direct maximum likelihood, using Stata 11’s modi…ed Newton-Raphson
algorithm with numerical derivatives (StataCorp LP, 2009a, p. 1069).
4
Data and variables
We combine data from both SHARE and HRS. SHARE was designed to be comparable to HRS (and to the English Longitudinal Study of Ageing (ELSA)). The data
from SHARE were collected in 11 European countries and Israel in 2004/2005 and
cover a sample of non-institutionalized persons over 50 as well as their partners.
For a more detailed description of SHARE, we refer the reader to Börsch-Supan
12
and Jürges (2005) and to Juster and Suzman (1995) for an overview of HRS. The
data from Greece and Switzerland were discarded because a problem during the
collection of the data made the variables on formal home care unusable.
Because it is di¢ cult to assess the e¤ect of help provided by the spouse or
partner for individuals living as couples and since some questions are only asked
at the household level (in particular those regarding help provided by individuals
living outside the household, but also …nancial questions for example), the analysis
is restricted to individuals who live alone (single, divorced, widow or separated), are
not institutionalized and have at least one child (o¤spring, stepchild or adopted).
We further restrict the analysis to individuals aged 70 or over since people under
that age are not frequent users of home care. Moreover this restriction makes the
sample comparable to the one used in Van Houtven and Norton (2004).
Most variables are directly comparable between the two surveys, however it is
not the case of our main variables of interest. In the HRS, there is a question about
the use of formal home care in the "health care utilization" section (section N). The
question only asks whether the respondent has had any medically trained person
at home to help them since the last interview or the last two months if it is a …rst
interview. There is unfortunately no follow-up question on the amount of formal
home care received. The HRS has another section on helpers (section G), and it
is from there that we recover the relevant information, by considering that helpers
who were paid are professionals providing formal home care. The recall period is
one month in this case. The informal care variables is constructed in the same
way, with one month recall, this time by using helpers who are children (natural
children, stepchildren or adopted children) or grandchildren of the respondent who
do not live in the same household.
SHARE contains similar information on informal care. The recall period however is 12 months, so we divide the number of hours by 52 in order to have hours
of care per week, which is easier to understand than hours per month or even by
year. The variables from HRS are adjusted accordingly. The selection criteria are
the same in both cases. The formal home care variable is the total number of hours
of "professional or paid nursing or personal care" and "professional or paid home
help, for domestic tasks that [the respondent] could not perform [herself] due to
health problems" in the last 12 months (Survey of Health, Ageing and Retirement
in Europe, 2005, question HC032). Here again, we divide the number of hours in
order to account for the di¤erent recall periods. The formal care variable is thus
not de…ned exactly in the same way in both samples.
The explanatory variables are divided into …ve categories. The …rst is composed of demographic and socioeconomic characteristics of the respondent, such
as age, the number of years of education, income and wealth categories These last
to variables represent quintiles computed by country, in order to avoid cost of liv-
13
ing di¤erences. Insurance status (Medicare or Medicaid bene…ciary and long-term
insurance status for the Americans; and complementary and/ or supplementary
insurance for the Europeans) also belongs to this group. The insurance variables
serve as exclusion restriction, since they are only included in the formal care equations. Racial variables are widely used in studies on the United States, but they
are not collected in Europe, and thus were not included.
The second is composed of the health variables such as the number of limitations in (instrumental) activities of daily living (I)ADLs. IADLs are activities
related to independent living and include cooking, managing money, shopping for
groceries or personal items, performing housework, and using a telephone. This
variable ranges from 0 to 7. ADLs are activities related to personal care and include bathing or showering, dressing, getting in or out of bed, using the toilet, and
eating. This variable is coded from 0 to 6. The health variables also include the
number of chronic conditions and self-reported health. The number of chronic conditions is an index which takes values from 0 to 6. It includes physical conditions
such as a heart attack or any other heart problem, a stroke or cerebral vascular disease, high blood pressure or hypertension, diabetes, cancer, a chronic lung
disease and arthritis. Self-reported health is a subjective measure of well-being.
The respondent is asked to assess her own health on a scale between 1 (excellent)
and 5 (poor). We also include a binary variable taking the value 1 if there was
a proxy respondent during the interview. This variable is a measure of mental
health (see for example Van Houtven and Norton, 2004), because it controls for
cognitive limitations.
The third group is composed of behavioral variables indicating whether the
respondent is a current smoker, and the frequency with which she drinks (0, never
to 6 almost every day).
The fourth group is made of variables concerning the children as informal
caregivers, such as the number of children, the number of children living close
(under 10 miles in the U.S. and under 25 kilometers away from the respondent’s
house in Europe), and the number of children working. These variables also serve
as exclusion restrictions, as they do not appear in the formal care equations.
Finally the last group includes geographical dummies, taking the value 1 if the
interview was conducted in the United States, in Northern (DK, SE), Central (AT,
BE, DE, FR, NL), or Southern (ES, IT) Europe respectively and 0 otherwise. The
omitted category in the analysis is the U.S.
See Table 1 for a complete list of all the variables, along with descriptive statistics, Table 2 for more details of the care variables and Survey of Health, Ageing
and Retirement in Europe (2005) and Survey of Health, Ageing and Retirement
in Europe (2009) for a more detailed description of the variables.
–Insert Tables 1 and 2 about here –
14
From Table 2 it appears that Europeans use more care on the extensive margin,
while Americans have a bigger consumption of care on the intensive margin. We
see that about 36% of Europeans use informal care and 28% use formal care,
as opposed to 17% and 7% respectively for Americans. On the other hand, the
average number of hours of care for users are smaller in Europe than in the U.S.
with about 9 hours of informal care and 6 hours of formal home care per week for
Europeans and 15 and 31 hours respectively for Americans.
5
Results and discussion
Since all models we have estimated have the same number of parameters, it is easy
to choose the one with the best …t without the help of penalized log likelihood
criteria such as the Akaike or Bayes-Schwarz information criteria (AIC and BIC,
respectively). We simply choose the model with the biggest log likelihood. The
preferred speci…cation is the Gumbel copula, with a log likelihood of 6698:97, as
can be seen in Table 3.
–Insert Table 3 about here –
The bivariate Gumbel copula takes the form
C(u; v; ) = exp
h
( ln (u)) + ( ln (v))
i1=
(14)
The dependence parameter is restricted to the region [1; 1), which implies that
the Gumbel copula only allows positive dependence. Moreover, in this case we
have
C(u; v; 1) = = uv
C(u; v; 1) = M = min(u; v)
where
is the product copula, which corresponds to stochastic independence,
and M is the Fréchet-Hoe¤ding upper bound (Nelsen, 1999, p. 9). The Gumbel
copula does not however attain the Fréchet lower bound W . The three copulas
, M and W are important in understanding the dependence of two variables
in empirical work, as it can be shown that for every copula C(u; v; ), we have
W C(u; v; ) M . Therefore the ability of a given copula to represent di¤erent
15
degrees of association can be examined in terms of the extent to which it covers
the interval between the lower and upper Fréchet bounds.
As a consequence the Gumbel copula can only model positive dependence. This
…nding was con…rmed when using copulas that allow both negative and positive
dependence, like the FGM or Frank copulas. We conclude that informal and
formal care have a positive association, with an estimated value of of 1:263 for
the Gumbel copula. We found that this value was statistically di¤erent from 1 at
the 5% level, which indicates that our copula does not reduce to a product copula,
and therefore that the two variables of interest are not independent. This con…rms
the presence of endogeneity, and justi…es our simultaneous estimation of the two
care equations.
It is not easy to compare di¤erent values of for di¤erent copulas, so other
measures of association like Kendall’s tau are used in stead. It is de…ned as the
probability of concordance minus the probability of discordance between two independent draws from a bivariate distribution (Nelsen, 1999, p. 126), and for the
Gumbel copula we can compute it from with = 1 1= . As = 1:080 which
gives = 0:074 in our case. This result could be expected, given that both types of
care would be similarly in‡uenced by unobserved components of the health state
and dependence level of the recipient of care. Moreover, as was mentioned earlier,
the Gumbel copula displays strong right tail and weak left tail dependence, and
its density is totally positive of order 2 (TP2) (Joe, 1997, p. 142). This implies
that large (small) values of informal care hours are associated with large (small)
values of formal home care hours. This result is consistent with the idea that the
elderly with small needs use little or no formal or informal care, while those with
the largest needs do not rely on only one source of care, and thus use a lot of both
informal and formal care. This however does not imply that the two types of care
are not substitutes, as will be seen below.
Estimates for the parameters and give = 0:628 and = 1:436 for informal
care and = 0:474 and = 1:412 for formal care. Tests of = 1, = 1 and =
are all rejected at the 1% level, which indicates that for both informal and formal
care the generalized gamma does not reduce to the standard gamma, lognormal,
exponential or Weibull distributions in the continuous part.
Parameter estimates ( i0 ; i0 ; i+ ; i+ ; j0 ; j0 ; j+ ; j+ ) are reported in Tables 6, 7,
8 and 9, and Tables 4 and 5 are simpli…ed versions, showing only the coe¢ cients
on the care variables.
–Insert Tables 4 and 5 about here –
As far as the care variables are concerned, it can be seen from the hurdle part
( i0 ; j0 ) of Tables 4 and 5 that formal care and informal care are substitutes. Moreover formal care has a much larger e¤ect (approximately twice the size, depending
16
on the copula speci…cation) on the probability of informal care provision in Europe
than in the U.S., with coe¢ cients ranging from 0:016 to 0:007 with the Gumbel
copula. On the other hand, if we turn to the e¤ect of informal care on the probability that the parent receives formal care, we see a reverse pattern, where the
e¤ect is larger in the United States than in Europe, with coe¢ cients of 0:013 and
0:021 for Europe and the U.S., respectively, in the Gumbel model. This seems
to con…rm our initial intuition that European children take the amount of formal
care provided to the parent into account before making their caregiving choices,
while American parents seek formal care when the amount of informal care that
they receive does not cover their needs. The second part of the hurdle speci…cation ( i+ ; j+ ), the continuous part, gives less striking results. First it appears that
the substitution e¤ects are larger in the U.S. than in Europe, with coe¢ cients of
0:007 for both types of care in Europe, and 0:022 and 0:014 for formal and
informal care respectively in the United States. The second observation is that the
Clayton, FGM and Frank speci…cations give a positive, non statistically signi…cant
coe¢ cient for the e¤ect of amount of formal care on the hours of informal care
in Europe, while the Gumbel model gives a statistically signi…cant result (at the
10% level). The same holds for the e¤ect of formal care on informal care, still in
Europe, the only di¤erence being that the Gumbel no longer gives a signi…cant
result. This last result con…rms the …ndings of (Bonsang, 2009), who …nds a little
or no e¤ect of the intensity of informal care on the use of formal home care by
elderly Europeans, depending on the measure of formal care used. Moreover, a
similar e¤ect appears in the United States, where the e¤ect of informal care on
formal care disapears in the Clayton, FGM and Frank speci…cations. Trivedi and
Zimmer (2009) show that the choice of a copula is important in order to capture
the dependence structure, and in this is veri…ed in our case. As discussed earlier,
the fact that the Gumbel copula is the preferred model shows that the informal
care variables displays a strong upper tail dependence. This dependence is not
properly accounted for by the other three copulas, since the Clayton copula generates dependence in the lower tail but not in the upper tail (Jondeau et al., 2007, p.
251) and the FGM and Frank copula displays no particular dependence patterns.
–Insert Tables 6, 7,8 and 9 about here –
Results for the other explanatory variables ( i0 ; i+ ; j0 ; j+ ) can be found in
Tables 6, 7, 8 and 9. We will only comment the results from the model with the
Gumbel copula, since it is the preferred speci…cation. The demographic characteristics of the parent seem to play a bigger role in the hurdle part, as the age
variable is positive and signi…cant in the equation of both types of care, while gender only in‡uences the probability that children provide care. The parent’s level of
17
education has a positive impact on the amount of formal care used, which would
suggest that more educated care recipients are better at "navigating the system".
Income has a negative impact on the probability to receive informal care and a
positive impact on formal care provision. However, amon those who receive informal care, richer parents receive more care from their children. Wealth however is
not signi…cant in any equation.
The health variables, having a proxy respondent, self-rated health (a larger
values implies a worse health status), the number of ADL and IADL limitations
and the number of conditions enter all equations signi…cantly, with a positive sign ,
except for our indicator of mental health, having a proxy respondent, which is not
signi…cant in the …rst part of the formal care equation. Moreover Self-perceived
health does not in‡uence the amount of formal care received, and the number
of ADL limitations has no impact on the amount of informal care provided by
children. This probably comes from the fact that personal care is mostly provided
by more skilled professionals, and also from our aggregated measure of informal
care which encompasses di¤erent types of interactions with the parent, ranging
from personal care like bathing or getting out of bed, to help with household tasks
like cleaning or preparing meals and even help with paperwork or managing bills.
The number of conditions only in‡uences the probability to receive any type of
care, bet not the quantities. It can be noted that our indicators of health behavior,
being a smoker, and the frequency of alcohol consumption, are not signi…cant in
any equation.
The variables describing the children indicate that proximity is a good predictor
of care provision, while for a parent, having more children has a positive impact on
the amount of informal care received and on the other hand, more children working
imply less hours of care. The insurance variables show that being enrolled in the
Medicaid program and having a complementary or supplementary insurance has a
positive impact on the probability to receive formal home care, while only Medicare
has an impact on the number of hours. It is interesting to note that long-term
care insurance does not statistically in‡uence the provision of formal home care.
This might be due to the small number of individuals who have such an insurance
coverage. Finally we see that Europeans tend to have a bigger probability than
Americans to receive both informal and formal home care, but that among the
recipients, the Americans tend to bene…t from more hours of care, except for
individuals living in the South of Europe, who seem to have patterns of informal
care provision comparable to those in the U.S.
18
6
Conclusion
In this paper we analyze the relationship between informal care and formal home
care in Europe the United States, by merging two large surveys of the 50+ population, the HRS and SHARE. We show that the two types of care are substitutes,
and that endogeneity is detected. Moreover we explain that institutional di¤erences must be taken into account, otherwise the direction of causality between
informal care provision and formal home care use could be misspeci…ed. The main
di¤erence is that European countries have predominantly public long-term-care
schemes, which o¤er their services at little or no cost, and thus are used by more
individuals. They are however a¤ected by supply shortages in some areas, and
thus Europeans who receive care use on average less formal care than the American elderly in the same conditions. In the United States the sector of home care
is more market oriented and thus allows for more ‡exibility. As Americans have
to pay more for formal care, it appears that there are fewer recipients of formal
care there, and that elderly Americans base their decision to use formal home
care services much more on the amount of informal care that they receive than
Europeans in the same situation. Moreover, public schemes have di¤erent welfare
considerations than privately run services, and thus we see that there are more
users of formal and informal care in Europe, while among users, it is in the U.S.
that the consumption is bigger. On a more technical note, our results con…rm
that using copulas with more ‡exible dependence patterns is useful, and that the
speci…c choice of copula should be determined by the dependence structure of the
data.
Acknowledgements
The authors are listed in alphabetical order.
The authors would like to thank Owen O’Donnell and participants of the
Health and Labour Economics seminar of the University of Lausanne for
helpful suggestions, as well as Jacques Huguenin, Florian Pelgrin, Norma B.
Coe, Lucy White, Brigitte Dormont and Michel Mougeot for comments and
suggestions on previous versions of this paper.
This paper uses data from SHARE release 2.3.0, as of November 13th 2009.
SHARE data collection in 2004-2007 was primarily funded by the European
19
Commission through its 5th and 6th framework programmes (project numbers QLK6-CT-2001- 00360; RII-CT- 2006-062193; CIT5-CT-2005-028857).
Additional funding by the US National Institute on Aging (grant numbers
U01 AG09740-13S2; P01 AG005842; P01 AG08291; P30 AG12815; Y1-AG4553-01; OGHA 04-064; R21 AG025169) as well as by various national
sources is gratefully acknowledged (see http://www.share-project.org for a
full list of funding institutions).
The HRS (Health and Retirement Study) is sponsored by the National Institute on Aging (grant number NIA U01AG009740) and is conducted by the
University of Michigan.
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Appendix A
Copula functions
Clayton copula
C(u; v;
@C(u; v;
@u
@C(u; v;
@v
2
@ C(u; v;
@u@v
1=
)= u v
1
)
=u 1
u v
)
)
=v
1
1
u v
= (1 + ) (uv)
1
1
23
1 1=
1 1=
u v
1
2 1=
Farlie-Gumbel-Morgenstern (FGM) copula
C(u; v;
@C(u; v;
@u
@C(u; v;
@v
@ 2 C(u; v;
@u@v
) = uv [1 + (1 u) (1
)
= v [1 + (1 2u) (1
)
)
= u [1 + (1
= 1 + (1
v)]
v)]
u) (1
2u) (1
2v)]
2v)
Frank copula
C(u; v; ) =
1
"
ln 1 +
e
u
1
e
v
e
1
e u e v 1
1) + (e u 1) (e v
e v e u 1
1) + (e u 1) (e v
e (u+v) e
1
@C(u; v; )
=
@u
(e
@C(u; v; )
=
@v
(e
2
@ C(u; v; )
=
@u@v
[(e
1) + (e
u
u
e + ve
1=
1
1) (e
v
#
1)
1)
1)]2
Gumbel copula
C(u; v; ) = exp
h
i
with u
e=
@C(u; v; )
u
e 1+
1+1=
= C(u; v; )
u
e + ve
@u
u
ve 1+
@C(u; v; )
1+1=
= C(u; v; )
u
e + ve
@v
v
2
@ C(u; v; )
(e
uve) 1+
= C(u; v; )
1+ + u
e + ve
@u@v
uv
Appendix B
Kendall’s tau
Clayton copula
(C(u; v; )) =
+2
24
ln (u) and ve =
1=
u
e + ve
ln (v)
2+1=
Farlie-Gumbel-Morgenstern (FGM) copula
(C(u; v; )) =
2
9
Frank copula
(C(u; v; )) = 1
4
1
D1 ( )
where
Dk (x) =
8
>
>
<
>
>
:
k
xk
Rx
0
kjxj k
1+k jxjk
tk
dt
et 1
R0
x
if x
tk
dt
et 1
0
if x < 0
is the Debeye function.
Kendall’s tau can be approximated by a Maclaurin series expansion (Nelsen,
1999, p. 150) for moderate values of :
(C(u; v; )) '
1
900
1
9
3
+
1
52920
Gumbel copula
(C(u; v; )) = 1
1
25
5
:::
Table 1: Descriptive statistics
Mean
Std. dev. Min. Max.
Informal care
3.106705 12.68846
0
168
Formal care
1.968429 12.24475
0
168
Age (years)
79.21234 6.286817
70
106
Female
.7712025 .420125
0
1
Education (years)
9.632753 4.190882
0
20
Income cat. (0 –4)
.8018987 1.066333
0
4
Wealth cat. (0 –4)
1.362658 1.329576
0
4
Proxy respondent
.078481 .2689696
0
1
Self-rated health (1=excel. –5=poor) 3.127532 1.061515
1
5
ADL limitations (n)
.4139241 .9755782
0
6
IADL limitations (n)
.6901899 1.280988
0
7
Conditions (n)
1.702848 1.261349
0
6
Current smoker
.0924051 .2896427
0
1
Days/week drinks (n)
1.244462 2.346753
0
7
Children (n)
2.807595 1.843601
1
18
Children living close (n)
1.224051 1.286127
0
14
Children working (n)
1.909494 1.373806
0
11
LTC insurance
.2126582 .4092526
0
1
Medicaid
.0575949 .2330128
0
1
Compl./ suppl. insurance
.2056962 .4042734
0
1
Region:
–U.S.
.4439873 .4969313
0
1
–Northern Europe
.1335443 .3402159
0
1
–Central Europe
.3490506 .4767454
0
1
–Southern Europe
.0734177 .2608622
0
1
Observations
3160
26
Table 2: Descriptive statistics, care variables
Mean
Europe:
–Informal care
3.405773
–1(Informal care > 0)
.3625498
–Informal care | Informal care > 0 9.393944
–Formal care
1.737741
–1(Formal care > 0)
.2868526
–Formal care | Formal care > 0
6.057959
U.S.:
–Informal care
2.732177
–1(Informal care > 0)
.1717748
–Informal care | Informal care > 0 15.90558
–Formal care
2.257323
–1(Formal care > 0)
.0719886
–Formal care | Formal care > 0
31.35668
Table 3:
Std. dev.
Min.
Max.
n
12.24369
.4808732
18.90914
9.410107
.4524209
16.81991
0
0
.0192308
0
0
.0192308
168
1
168
168
1
168
1757
1757
637
1757
1757
504
13.21924
.3773191
28.46784
15.059
.2585612
47.51609
0
0
.2307692
0
0
.4615385
168
1
168
168
1
168
1403
1403
241
1403
1403
101
parameters and log likelihoods
Copula
Clayton
FGM
parameter
Frank
Gumbel
0.148**
0.098
0.209
1.080***
(2.680)
(0.757)
(0.787)
(33.919)
Log likelihood –6699.06 –6703.39 –6703.37 –6698.97
t statistics in parentheses, * p<0.10, ** p<0.05, *** p<0.01
27
Table 4: Informal care
Copula
Clayton
FGM
Frank
Gumbel
Hurdle part
Formal care (Europe) –0.01137*** –0.00955** –0.00961** –0.01611***
(–3.188)
(–2.531)
(–2.537)
(–3.539)
Formal care (U.S.)
–0.00636** –0.00542** –0.00546** –0.00733***
(–2.485)
(–2.033)
(–2.042)
(–2.610)
Continuous part
Formal care (Europe)
0.00113
0.00177
0.00170
–0.00749*
(0.230)
(0.352)
(0.338)
(–1.670)
Formal care (U.S.)
–0.01991*** –0.01951*** –0.01955*** –0.02171***
(–5.184)
(–5.003)
(–5.009)
(–5.994)
t statistics in parentheses, * p<0.10, ** p<0.05, *** p<0.01
Table 5: Formal care
Copula
Clayton
Hurdle part
Informal care (Europe)
Informal care (U.S.)
FGM
Frank
Gumbel
–0.00615*
–0.00431
–0.00439
–0.01309***
(–1.944)
(–1.296)
(–1.309)
(–2.784)
–0.01702*** –0.01551*** –0.01558*** –0.02095***
(–3.562)
(–3.234)
(–3.239)
(–4.418)
Continuous part
Informal care (Europe)
0.00262
0.00375
0.00367
(0.538)
(0.744)
(0.726)
Informal care (U.S.)
–0.01138
–0.01120
–0.01123
(–1.493)
(–1.466)
(–1.470)
t statistics in parentheses, * p<0.10, ** p<0.05, *** p<0.01
28
–0.00659
(–1.305)
–0.01419*
(–1.955)
29
t statistics in parentheses
p < 0:1,
p < 0:05,
p < 0:01
Table 6: Clayton copula
Hurdle part
Informal care
Formal care
Formal care (Europe)
–0.011
(–3.19)
Formal care (U.S.)
–0.006
(–2.48)
Informal care (Europe)
–0.006
(–1.94)
Informal care (U.S.)
–0.017
(–3.56)
Age (years)
0.025
(5.07)
0.052
(9.28)
Female
0.252
(3.65)
0.106
(1.33)
Education (years)
–0.010
(–1.34) 0.003
(0.29)
Income cat. (0 –4)
–0.083
(–2.91) 0.056
(1.75)
Wealth cat. (0 –4)
0.021
(0.92)
0.010
(0.35)
Proxy respondent
0.480
(4.55) –0.032
(–0.27)
Self-rated health (1=excel. –5=poor) 0.131
(4.20)
0.149
(4.01)
ADL limitations (n)
0.076
(2.29)
0.255
(7.09)
IADL limitations (n)
0.307
(11.04) 0.287
(9.53)
Conditions (n)
0.079
(3.09)
0.115
(3.80)
Current smoker
0.042
(0.44)
0.091
(0.83)
Days/week drinks (n)
–0.016
(–1.32) 0.011
(0.82)
Children (n)
–0.018
(–0.75)
Children living close (n)
0.113
(4.36)
Children working (n)
0.023
(0.79)
LTC insurance
–0.057
(–0.69)
Medicaid
0.467
(2.92)
Compl./ suppl. insurance
0.439
(5.14)
Region:
–U.S.
ref.
ref.
–Northern Europe
0.678
(7.25)
1.406
(11.67)
–Central Europe
0.738
(9.07)
1.265
(10.22)
–Southern Europe
0.148
(1.14)
0.719
(4.40)
Constant
–4.025
(–8.96) –7.232
(–13.68)
ref.
–1.524
–0.831
–0.988
–0.402
(–6.11)
(–3.20)
(–2.82)
(–0.38)
(–5.47)
(–2.94)
(0.76)
(1.88)
ref.
–1.034
–0.499
0.198
1.684
(0.54)
(–1.49)
(1.20)
(–0.98)
(2.33)
(0.52)
(1.41)
(2.79)
(1.29)
(3.13)
(5.19)
(0.73)
(–0.19)
(–0.15)
(–0.94)
(1.77)
(–0.50)
care
Continuous part
Informal care
Formal
0.001
(0.23)
–0.020
(–5.18)
0.003
–0.011
–0.003
(–0.32) 0.013
–0.114
(–0.83) –0.144
0.002
(0.12)
0.040
0.121
(2.36)
0.030
–0.059
(–1.38) 0.077
0.496
(3.45)
0.506
0.132
(2.26)
0.089
–0.007
(–0.15) 0.162
0.209
(5.13)
0.241
–0.026
(–0.58) 0.038
–0.233
(–1.27) –0.041
0.020
(0.83) –0.004
0.082
(1.87)
0.029
(0.62)
–0.224
(–4.17)
–0.145
0.579
–0.079
30
t statistics in parentheses
p < 0:1,
p < 0:05,
p < 0:01
Table 7: FGM copula
Hurdle part
Informal care
Formal care
Formal care (Europe)
–0.010
(–2.53)
Formal care (U.S.)
–0.005
(–2.03)
Informal care (Europe)
–0.004
(–1.30)
Informal care (U.S.)
–0.016
(–3.23)
Age (years)
0.024
(5.00)
0.051
(9.21)
Female
0.252
(3.65)
0.106
(1.33)
Education (years)
–0.010
(–1.32) 0.002
(0.28)
Income cat. (0 –4)
–0.085
(–3.00) 0.053
(1.64)
Wealth cat. (0 –4)
0.020
(0.88)
0.011
(0.41)
Proxy respondent
0.461
(4.37) –0.056
(–0.47)
Self-rated health (1=excel. –5=poor) 0.134
(4.27)
0.151
(4.08)
ADL limitations (n)
0.070
(2.10)
0.252
(7.00)
IADL limitations (n)
0.301
(10.79) 0.276
(9.20)
Conditions (n)
0.080
(3.11)
0.115
(3.80)
Current smoker
0.035
(0.37)
0.084
(0.76)
Days/week drinks (n)
–0.015
(–1.27) 0.012
(0.88)
Children (n)
–0.015
(–0.65)
Children living close (n)
0.113
(4.37)
Children working (n)
0.020
(0.67)
LTC insurance
–0.060
(–0.72)
Medicaid
0.469
(2.93)
Compl./ suppl. insurance
0.443
(5.18)
Region:
–U.S.
ref.
ref.
–Northern Europe
0.677
(7.24)
1.381
(11.52)
–Central Europe
0.733
(9.02)
1.241
(10.07)
–Southern Europe
0.139
(1.07)
0.694
(4.25)
Constant
–4.000
(–8.90) –7.176
(–13.61)
ref.
–1.544
–0.865
–1.022
–0.300
(–6.18)
(–3.33)
(–2.92)
(–0.28)
(–5.52)
(–2.99)
(0.71)
(1.92)
ref.
–1.043
–0.508
0.185
1.725
(0.74)
(–1.47)
(1.13)
(–1.00)
(2.28)
(0.56)
(1.45)
(2.84)
(1.26)
(3.11)
(5.03)
(0.75)
(–0.16)
(–0.15)
(–0.90)
(1.75)
(–0.47)
care
Continuous part
Informal care
Formal
0.002
(0.35)
–0.020
(–5.00)
0.004
–0.011
–0.003
(–0.36) 0.012
–0.113
(–0.83) –0.148
0.001
(0.11)
0.039
0.123
(2.39)
0.032
–0.059
(–1.39) 0.080
0.491
(3.42)
0.514
0.131
(2.24)
0.087
–0.012
(–0.24) 0.160
0.205
(5.06)
0.233
–0.026
(–0.57) 0.039
–0.230
(–1.25) –0.034
0.020
(0.85) –0.003
0.084
(1.91)
0.030
(0.63)
–0.226
(–4.19)
–0.139
0.574
–0.074
31
t statistics in parentheses
p < 0:1,
p < 0:05,
p < 0:01
Table 8: Frank copula
Hurdle part
Informal care
Formal care
Formal care (Europe)
–0.010
(–2.54)
Formal care (U.S.)
–0.005
(–2.04)
Informal care (Europe)
–0.004
(–1.31)
Informal care (U.S.)
–0.016
(–3.24)
Age (years)
0.024
(5.00)
0.051
(9.21)
Female
0.252
(3.65)
0.106
(1.33)
Education (years)
–0.010
(–1.32) 0.002
(0.28)
Income cat. (0 –4)
–0.085
(–2.99) 0.053
(1.64)
Wealth cat. (0 –4)
0.020
(0.88)
0.011
(0.40)
Proxy respondent
0.461
(4.37) –0.055
(–0.47)
Self-rated health (1=excel. –5=poor) 0.134
(4.27)
0.151
(4.08)
ADL limitations (n)
0.070
(2.10)
0.252
(7.00)
IADL limitations (n)
0.301
(10.79) 0.276
(9.20)
Conditions (n)
0.080
(3.11)
0.115
(3.80)
Current smoker
0.035
(0.37)
0.084
(0.77)
Days/week drinks (n)
–0.015
(–1.27) 0.012
(0.88)
Children (n)
–0.015
(–0.65)
Children living close (n)
0.113
(4.37)
Children working (n)
0.020
(0.67)
LTC insurance
–0.060
(–0.72)
Medicaid
0.468
(2.93)
Compl./ suppl. insurance
0.443
(5.18)
Region:
–U.S.
ref.
ref.
–Northern Europe
0.677
(7.24)
1.381
(11.52)
–Central Europe
0.733
(9.02)
1.241
(10.07)
–Southern Europe
0.139
(1.08)
0.694
(4.25)
Constant
–4.000
(–8.90) –7.178
(–13.61)
ref.
–1.543
–0.864
–1.021
–0.305
(–6.18)
(–3.33)
(–2.92)
(–0.29)
(–5.52)
(–2.99)
(0.71)
(1.92)
ref.
–1.042
–0.508
0.186
1.723
(0.73)
(–1.47)
(1.14)
(–1.00)
(2.28)
(0.56)
(1.45)
(2.84)
(1.27)
(3.11)
(5.03)
(0.75)
(–0.16)
(–0.14)
(–0.90)
(1.75)
(–0.47)
care
Continuous part
Informal care
Formal
0.002
(0.34)
–0.020
(–5.01)
0.004
–0.011
–0.003
(–0.36) 0.012
–0.113
(–0.83) –0.148
0.001
(0.11)
0.039
0.123
(2.39)
0.032
–0.059
(–1.39) 0.080
0.492
(3.42)
0.514
0.131
(2.24)
0.087
–0.011
(–0.23) 0.160
0.206
(5.06)
0.233
–0.026
(–0.57) 0.039
–0.229
(–1.25) –0.034
0.020
(0.85) –0.003
0.084
(1.91)
0.030
(0.63)
–0.226
(–4.19)
–0.139
0.575
–0.074
32
t statistics in parentheses
p < 0:1,
p < 0:05,
p < 0:01
Table 9: Gumbel copula
Hurdle part
Informal care
Formal care
Formal care (Europe)
–0.016
(–3.54)
Formal care (U.S.)
–0.007
(–2.61)
Informal care (Europe)
–0.013
(–2.78)
Informal care (U.S.)
–0.021
(–4.42)
Age (years)
0.025
(5.08)
0.052
(9.45)
Female
0.255
(3.71)
0.112
(1.42)
Education (years)
–0.010
(–1.33) 0.002
(0.18)
Income cat. (0 –4)
–0.077
(–2.71) 0.062
(1.95)
Wealth cat. (0 –4)
0.020
(0.87)
0.006
(0.23)
Proxy respondent
0.470
(4.42) –0.016
(–0.13)
Self-rated health (1=excel. –5=poor) 0.132
(4.24)
0.158
(4.28)
ADL limitations (n)
0.085
(2.52)
0.246
(6.89)
IADL limitations (n)
0.303
(10.90) 0.285
(9.51)
Conditions (n)
0.079
(3.10)
0.113
(3.77)
Current smoker
0.046
(0.49)
0.090
(0.83)
Days/week drinks (n)
–0.014
(–1.18) 0.012
(0.91)
Children (n)
–0.016
(–0.69)
Children living close (n)
0.113
(4.39)
Children working (n)
0.021
(0.74)
LTC insurance
–0.058
(–0.71)
Medicaid
0.451
(2.85)
Compl./ suppl. insurance
0.442
(5.21)
Region:
–U.S.
ref.
ref.
–Northern Europe
0.684
(7.36)
1.363
(11.47)
–Central Europe
0.731
(9.03)
1.228
(10.07)
–Southern Europe
0.147
(1.15)
0.701
(4.37)
Constant
–4.030
(–9.01) –7.232
(–13.84)
ref.
–1.554
–0.849
–0.986
–0.161
(–6.24)
(–3.31)
(–2.84)
(–0.15)
(–5.50)
(–2.77)
(0.99)
(1.67)
ref.
–1.019
–0.461
0.255
1.480
(–1.30)
(–1.95)
(1.06)
(–1.03)
(2.12)
(0.31)
(1.24)
(2.73)
(1.29)
(2.80)
(5.59)
(0.65)
(–0.07)
(–0.02)
(–0.88)
(1.68)
(–0.72)
care
Continuous part
Informal care
Formal
–0.007
(–1.67)
–0.022
(–5.99)
–0.007
–0.014
–0.001
(–0.09) 0.011
–0.126
(–0.94) –0.152
0.004
(0.34)
0.036
0.138
(2.72)
0.018
–0.056
(–1.34) 0.068
0.493
(3.48)
0.492
0.134
(2.35)
0.089
0.016
(0.32)
0.145
0.199
(4.97)
0.259
–0.030
(–0.67) 0.034
–0.165
(–0.92) –0.016
0.018
(0.79) –0.001
0.085
(1.96)
0.021
(0.46)
–0.214
(–4.04)
–0.135
0.551
–0.113