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Will You Still Want Me Tomorrow? The Dynamics of Families' Long-Term
Care Arrangements
by Bridget Hiedemann, Michelle Sovinsky, and Steven Stern
Abstract
With data from the Assets and Health Dynamics Among the Oldest Old Survey, we estimate
dynamic models of three dimensions of families' elder-care arrangements: the use of each potential
care arrangement, namely care provided by a spouse, care provided by an adult child, formal home
health care, and/or institutional care; the selection of the primary care arrangement; and hours in
each care arrangement. Our results indicate that both observed heterogeneity and positive true state
dependence contribute to the persistence of care arrangements. Evidence of positive true state
dependence for most or all modes of care in all models suggests that inertia generally dominates
caregiver burnout. Our results indicate that formal care decisions depend on the cost and quality
of care. As a result of inertia, the effectiveness of long-term care policy depends on timing: initial
caregiving decisions are more sensitive than subsequent decisions to economic incentives.
JEL Classification: C51, C61, J14
Keywords: Dynamic Models, Long-Term Care, Home Health Care, Informal Care
Bridget Hiedemann is a professor of Economics at Seattle University. Michelle Sovinsky is a
professor of Economics at University of Mannheim and CEPR. Steven Stern is a professor of
Economics at Stony Brook University. The corresponding author is Steven Stern
(steven.stern@stonybrook.edu). We would like to thank Jan Boone, Liliana Pezzin, and
participants at the 2010 International Conference on Evidence-Based Policy in Long-Term Care,
the 2011 Annual Meetings of the Population Association of America, the University of Leuven
Public Economics Workshop, the University of Pennsylvania Structural Workshop in Honor of
Ken Wolpin, the 2013 Annual Southern Economic Association meetings, the 2013 Annual
Econometric Society meetings, and seminars at Boston College, IRDES (Paris, 2011), the
University of Michigan, Peking University, and Tsinghua University for helpful comments. All
remaining errors are ours. Most of the data used in this article is public, and some is restricted. The
public part can be obtained beginning six months after publication through three years hence from Steven
Stern, Stony Brook University, steven.stern@stonybrook.edu. One can apply for the restricted part with
the owners of the HRS/AHEAD data.
I. Introduction
In light of population aging and high disability rates among the elderly (Butler 1997;
Spillman and Long 2007), many families face decisions concerning long-term care arrangements
for disabled elderly relatives. With the assistance of family members, most notably spouses and
adult children, many disabled elderly individuals remain in the community (Shirey and Summer
2000). Others rely exclusively on formal home health care or a combination of formal home health
care and informal care provided by relatives and friends (Mack and Thompson 2005). Institutional
care represents another major source of care for this population (Burwell and Jackson 1994; Family
Caregiver Alliance 2015).
Long-term care arrangements have profound economic, social, and psychological
implications. Komisar and Thompson (2007) report that national spending on long-term care for
elderly and disabled individuals exceeded $200 billion in 2005. Medicaid and Medicare
respectively covered approximately 49 and 20 percent of these expenses, while private health and
long-term care insurance covered roughly seven percent. Individuals and their families financed
about 18 percent of long-term care services, while the remaining five percent was financed by
other private and public sources (Komisar and Thompson 2007). Most informal care provided by
family members is unpaid, but the opportunity costs in terms of foregone earnings, household
production, human capital accumulation (Skira 2015, hereafter Sk), and leisure are often
substantial. Moreover, the provision of informal care can be psychologically burdensome for
caregivers (Martin 2000; Byrne, et al. 2009, hereafter BGHS), and institutional care often entails
high social and psychological costs for elderly individuals (Macken 1986; Pezzin, Kemper, and
Reschovsky 1996; Coe and Van Houtven 2009).
The aging of the population and the profound implications of care arrangements for elderly
individuals, their families, and society highlight the importance of developing appropriate public
policies concerning long-term care arrangements for the elderly. Although an extensive literature
examines families' long-term care decisions, most studies neglect the intertemporal dimensions of
care. Using data from five waves of the Assets and Health Dynamics Among the Oldest Old Survey
collected between 1995 and 2004, we contribute to the long-term care literature by developing and
estimating three dynamic models of families' elder care arrangements. These models distinguish
among care provided by a spouse, care provided by an adult child or child-in-law, formal home
health care, and institutional care while also allowing for the possibility that the elderly individual
remains independent.
The modeling and estimation of dynamic models of long-term care is still in its infancy. 1 Even
the modeling and estimation of family decision-making in a dynamic environment is relatively
new. This paper explores five issues associated with modeling and estimation of dynamic models
of long-term care with the purpose of helping future researchers make informed decisions about
modeling and policy. First, our models capture several dimensions of families' care arrangements
that appear in the literature, namely the use of each potential care arrangement (for example, Pezzin
and Schone 1999a, hereafter PSa; Aykan 2002; Heitmueller and Michaud 2006, hereafter HM;
Grabowski and Gruber 2007; Sk), the selection of the primary care arrangement (for example,
Stern 1995; Hoerger, Picone, and Sloan 1996; Hiedemann and Stern 1999, hereafter HS; Engers
and Stern 2002, hereafter ES; Rainer and Siedler 2009; Sk), and hours in each potential care
arrangement (for example, Sloan, Picone, and Hoerger 1997, hereafter SPH; Checkovich and Stern
2002, hereafter CS; Van Houtven and Norton 2004; Brown 2006; Stabile, Laporte, and Coyte
2006; Pezzin, Pollak, and Schone 2007, hereafter PPS; BGHS). We estimate these three
dimensions of care arrangements separately because a) most of the literature considers only one
dimension of informal care (for example, the selection of the primary caregiver) and b) we want
to understand each dimension of informal care provision before allowing for strategic interactions
as in BGHS. Second, our dynamic framework links care arrangements over time by allowing for
state dependence (in effect, persistence in care arrangements) while distinguishing between
spurious state dependence due to observed and unobserved heterogeneity and true state
dependence due to inertia or caregiver burnout. To capture the possibility of inertia, our models
allow for positive true state dependence.2 For example, our models distinguish between
persistence in care arrangements attributable to a family's preferences (for example, an aversion to
institutional care) and true state dependence stemming from the high costs of transitioning from
one care arrangement to another (for example, into or out of institutional care). Third, we evaluate
the costs and benefits of using different measures of individual wealth and income in an
environment where wealth is measured with significant measurement error over time and missing
income data reflects selection bias. Fourth, we quantify geographic mobility with particular
emphasis on evidence of endogeneity. And fifth, we compare estimated policy effects across
different models.
Our results indicate that both observed heterogeneity and true state dependence contribute
to the persistence of care arrangements, thus highlighting the importance of a framework that links
care arrangements over time. Our findings suggest that inertia (positive true state dependence)
dominates caregiver burnout and that the use of formal care arrangements depends on the cost and
quality of care. Our results provide important policy implications. The effects of market conditions
and public policies on the use of formal home health care and institutional care are smaller and
less statistically significant in our dynamic model than in an otherwise identical static model. This
pattern suggests that the measured effects in the dynamic model reflect flows while those in the
static model reflect a stock of present and future flows. Thus, the timing of policy aimed at long-
term care is crucial as it has a larger effect when the caregiving decision is first made and that this
decision exhibits persistence in part because of inertia.
The outline of the paper is as follows. In section II, we present a brief review of the long-
term care literature. In section III, we describe the data and present descriptive statistics on the
frequency of care arrangements and intertemporal patterns of care. In section IV, we present our
estimation methodology, a new approach to controlling for initial conditions à la Heckman (1981),
and results of our three dynamic models. We discuss the policy implications of our results in
section V. We present robustness checks and conclude in sections VI and VII.
II. Literature Review
Although predominantly empirical, the long-term care literature offers several formal economic
models. Given the complexities inherent in families' long-term decisions, none captures all
dimensions of decision-making within families. The models vary with respect to the assumptions
concerning family members' preferences, the number of children participating in the decision-
making process, and the scope of care decisions considered.
Allowing for the possibility that preferences vary across family members, several papers
present game-theoretic models (SPH; HS; PSa; CS; ES; Brown 2006; PPS; BGHS). Other models
are based on the assumption of common preferences; for example, Hoerger, Picone, and Sloan
(1996) and Stabile, Laporte, and Coyte (2006) rely on the assumption of a single family utility
function. In the Kotlikoff and Morris (1990) model, the parent and child solve separate
maximization problems if they live separately but maximize a weighted average of their individual
utility functions subject to their pooled budget constraint if they live together. In contrast to our
previous work (for example, HS, ES, BGHS), this paper abstracts from the possibility that family
members have different preferences concerning care arrangements in order to focus on the
dynamic dimension of care.
Several models accommodate all adult children in the decision-making process (HS; CS;
ES; Van Houtven and Norton 2004; Brown 2006; BGHS). Others simplify modeling and/or
estimation by focusing on families that include only one child (Kotlikoff and Morris 1990) or two
adult children (PPS) or by assuming that only one child participates in the family's long-term care
decisions (SPH; PSa; Sk). In this paper, we restrict our sample to families with at most four
children, but we treat each child as a potential caregiver.
The models in this literature also vary with respect to the dimension (for example, primary
care arrangement) and modes (for example, informal care provided by an adult child) of care
decisions examined. Models presented in HS and ES focus on the family's selection of the primary
care arrangement including informal care provided by an adult child, institutional care, or
continued independence. CS and Brown (2006) model the quantity of informal care provided by
each adult child. Similarly, SPH, PSa, Stabile, Laporte, and Coyte (2006), and BGHS model the
provision of informal care and formal home health care. Stabile, Laporte, and Coyte (2006)
distinguish between publicly and privately financed home health care. Van Houtven and Norton
(2004) model children's provision of informal care and parents' use of formal care, defined broadly
as nursing home care, home health care, hospital care, physician visits, and outpatient surgery.
Hoerger, Picone, and Sloan (1996) and PPS focus on living arrangements of sick or disabled
elderly individuals (for example, independent living in the community or residence in an
intergenerational household). Distinguishing among care provided by a spouse, care provided by
an adult child or child-in-law, formal home health care, and institutional care, this paper examines
three dimensions of families' care arrangements: the use of each potential mode of care, the
selection of the primary care arrangement, and hours in each arrangement.
Although the provision of elder care is an inherently dynamic process, most of the
literature abstracts from the intertemporal dimensions of care. Exceptions include Börsch-Supan,
Kotlikoff, and Morris (1991) (hereafter BKM), Garber and MaCurdy (1990) (hereafter GM),
Dostie and Léger (2005) (hereafter DL), HM, Gardner and Gilleskie (2012) (hereafter GG), and
Sk. Using a framework that accounts for unobserved heterogeneity and state dependence, HM
explore the causal links between employment and informal care of sick, disabled, or elderly
individuals over time. In a dynamic model of savings and Medicaid enrollment decisions, GG
jointly estimate long-term care arrangements, savings/gifting behavior, insurance coverage, and
health transitions. Their approach incorporates unobserved permanent and time-varying
heterogeneity. Sk focuses on how care provision by a child affects the human capital accumulation
process of that child. The other three studies focus on living arrangements of elderly individuals.
BKM examine transitions among living independently, living with adult children, and living in an
institution. GM model transitions from living in the community to residing in a nursing home and
vice versa as well as transitions from one of these two living arrangements to death. Accounting
for unobserved heterogeneity as well as state and duration dependence, DL examine transitions
among independent living, cohabitation, nursing home residence, and death.
Following DL, HM, GG, and Sk, our models account for unobserved heterogeneity and
state dependence. Distinguishing among four modes of care, our models encompass a broader
range of care arrangements than those in the existing literature. Examining three care dimensions
of elder care decisions -- the use of each potential mode of care, the selection of the primary care
arrangement, and hours in each arrangement, we also provide a richer description of long-term
care dynamics.
III. Data
To examine families' care arrangements over time, we use data from the 1995, 1998, 2000, 2002,
and 2004 waves of the Assets and Health Dynamics Among the Oldest Old (AHEAD)/ Health and
Retirement (HRS) survey. With an emphasis on the joint dynamics of health and demographic
characteristics, this nationally representative longitudinal survey provides a particularly rich
source of information concerning long-term care arrangements. Selection criteria for the initial
AHEAD/HRS survey, conducted in 1993, include age and living arrangements. In particular, this
initial wave contains 6047 households with non-institutionalized individuals aged 70 years or
older. However, subsequent waves retain all living respondents, thus enabling the study of elderly
individuals in the community as well as nursing home residents. Spouses of respondents are also
respondents even if they would not otherwise qualify on the basis of their own age, thus increasing
the sample size for the initial wave to 8222 respondents. Although AHEAD/HRS oversamples
Florida residents, this oversampling introduces no estimation bias assuming that residential
location is exogenous. AHEAD/HRS also oversamples black and Hispanic households.
After excluding observations with missing values for variables used in our analysis,
individuals who participated in only one wave of the survey, individuals who provided inconsistent
responses, individuals who married or remarried over the course of the survey, families with more
than four children, and mixed-race couples, our sample consists of 3353 individuals including
spouses of original respondents. In addition to 914 married couples (where each individual
represents a respondent), the sample includes 267 unmarried men and 1258 unmarried women.
The preponderance of women (nearly two thirds of the sample) and the higher marriage rates
among men (77.4 percent of men compared to 42.1 percent of women) reflect differences in life
expectancy by gender and age differences between husbands and wives. Fifty-three percent of
elderly households participate in all five waves of the survey.
Our models include characteristics that influence an elderly individual's caregiving needs,
opportunities, and preferences. The need for care may increase with age and activity limitations;
accordingly, our models control for the elderly individual's age, problems with activities of daily
living (ADLs), and problems with instrumental activities of daily living (IADLs). The presence of
a spouse may reduce an elderly individual's need for assistance from adult children or from formal
care providers, particularly if the spouse is relatively young and healthy; thus, our models control
for the elderly individual's marital status, the spouse's age, and the spouse's activity limitations.
Since patterns of care may differ for men and women and across white, black, and Hispanic
families, our models control for gender as well as race/ethnicity. Moreover, to capture potential
differences in care arrangements for mothers/wives relative to fathers/husbands by race and
ethnicity (Martin 2000; Hiedemann 2012), each of our models also includes interactions between
gender and race/ethnicity.
Assets and income are potentially important characteristics that influence an elderly
individual's caregiving needs and opportunities because the ability to purchase care may reduce an
individual's dependence on relatives. Unfortunately, there are several problems with the asset data
reported in AHEAD. The first problem concerns large, spurious changes in assets within families
across time due to changes in the survey structure (for details, see Hurd, Juster, and Smith 2003;
Juster et al. 2007). Since transitions are very important in a dynamic model, the large variation in
asset changes is problematic. Hill (2006) also finds unreasonable variation in changes in assets in
HRS.3 Second, among wealthier individuals, 1993 assets are understated by a factor of two. Third,
income and asset reports in the second wave are inconsistent. Fourth, mean assets double between
the second and third waves. Fifth, financial measures, particularly those related to equity in a
second home, are under-reported (Hurd, Juster, and Smith 2003; Juster et al. 2007). Finally,
income measures are under-reported or mis-reported (Hurd, Juster, and Smith 2003). In the
absence of good asset and income data, our models include the elderly individual's educational
attainment as a proxy for her financial resources. We test whether assets and income, as measured
in AHEAD/HRS, affect family decisions and explore how best to use the data by conducting
Lagrange Multiplier tests.4
Table 1 displays descriptive statistics for the respondents for the first year of data.5 As a
consequence of the exclusion of nursing home residents from the initial wave and the inclusion of
spouses regardless of age, the characteristics of our sample differ from those of a random sample
of individuals aged 72 years and over.6 Respondents range in age from 49 to 103 years with a
mean of 78 years and a standard deviation of six years. On average, the respondents report
difficulty with 0.54 activities of daily living (ADL) such as eating, dressing, or bathing. But the
sample exhibits considerable variation with regard to ADL problems; while some individuals
report no problems with activities of daily living, others report problems with as many as all six
ADLs. Similarly, the respondents report an average of 0.43 problems with instrumental activities
of daily living (IADLs), such as using a telephone, taking medication, handling money, shopping,
or preparing meals; here too the sample displays considerable variation, with respondents reporting
a range of zero to five IADL problems.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 1 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
In addition to 2906 individuals (86.7 percent of the sample) who identify as non-Hispanic
white, the sample includes 324 individuals (9.7 percent of the sample) who identify as non-
Hispanic black and 123 individuals (3.7 percent of the sample) who identify as Hispanic. Although
the original sample includes individuals with other racial/ethnic identities, none of these
individuals remained in the sample after applying the selection criteria. With respect to education,
33.2 percent of respondents have a high school diploma but not a college degree, and 31.0 percent
report having a college or graduate degree.
The elderly households in our sample report a total of 4489 adult children and 3318 children-
in-law. Since each member of this generation is a potential caregiver, our models include
demographic characteristics of the adult children and children-in-law. These characteristics reflect
a potential caregiver's opportunity costs of time, effectiveness in the caregiving role, and/or
caregiving burden. Specifically, our models control for the adult child's or child-in-law's years of
schooling, employment status, marital status, family size (number of children), age, and gender.
Despite the potential endogeneity of employment status (see, for example, Ettner 1996), this
exploratory analysis abstracts from the younger generation's employment status in order to focus
on the intertemporal dimension of care. In other work (BGHS), we model adult children's
caregiving decisions as part of a broader utility maximization framework that includes hours of
work. As discussed extensively in Hiedemann (2012), the role of child gender in elder care
provision may vary by race and ethnicity; thus, our models also interact child gender with race and
ethnicity. Finally, co-residence with or proximity to an elderly parent or parent-in-law may
facilitate care provision. As discussed in Konrad et al. (2002) and Rainer and Seidler (2009),
location may be endogenous. However, Johar and Maruyama (2012) and Stern (2014) show that
the long-term game described in Konrad et al. (2002) may not be supported by the data, and Stern
(1995) shows that, even after controlling for endogeneity, geographical distance explains variation
in informal care arrangements. Accordingly, our models include measures of distance and co-
residence, and we conduct likelihood ratio tests to infer whether location is endogenous.
As shown in the second panel of Table 1, the younger generation displays near gender
balance: 51.1 percent are daughters or daughters-in-law. The average child or child-in-law is
almost 49 years old with nearly 14 years of schooling. These individuals report 29.8 hours of labor
market work per week, but this figure understates mean labor market activity because weekly work
hours are truncated at 40. On average, the adult children and children-in-law of the elderly
respondents have 2.2 children, but it is worth noting that some of these children belong to both a
child and a child-in-law. A small proportion (3.3 percent) of the adult children and children-in-law
reside with the elderly respondents, and 35.5 percent live within 10 miles of the elderly
respondents.
In addition to demographic characteristics and activity limitations, market conditions and
public policies may influence families' care arrangements for elderly individuals. Our models
control for several dimensions of the market for formal care in the elderly individual's or couple's
state of residence: the average weekly cost of full-time home health care (16 hours a day for seven
days or 112 hours per week), nursing home staff hours per nursing home resident per day in
facilities with Medicare or Medicaid beds, nursing home beds per individual above 70 years, and
a measure of the overall level of disability among nursing home residents. As discussed in
Harrington, Carrillo, and LaCava (2006), this disability measure (average ADL score) is a
composite score that reflects nursing home residents' needs for assistance with three ADLs, namely
eating, toileting, and transferring. Each nursing home resident was assigned a score from one to
three for each of these ADLs, increasing in the amount of assistance needed. A summary score
ranging from three to nine was compiled for each facility; facility scores were then summarized
for each state.7
The market for formal home health care and institutional care varies by state. The statistics
presented in the third panel describe the market conditions facing elderly households in our sample
during the first year of data.8 On average, these households reside in states where the weekly cost
of full-time home health care ranges from $699 to $1081 with a mean of $872. These are real
costs, deflated with state-specific price deflators (Bureau of Economic Analysis, 1999). The
elderly households in our sample live in states with 2.4 to 3.6 nursing home staff hours per nursing
home resident per day and 2.6 (=100exp(-3.637)) to 10.3 beds per 100 individuals over 70 years.
On average, these households reside in states where the facility score ranges from 5.2 to 6.7 with
a mean of 5.8 and a standard deviation of 0.31.
Many households rely on public assistance, most notably Medicaid, to cover their long-
term care expenses. Eligibility for Medicaid is linked to actual or potential receipt of cash
assistance under the Supplemental Security Income (SSI) program or the former Aid to Families
with Dependent Children program. Elderly individuals or couples are eligible for SSI payments if
their monthly countable income (income less $20) and countable resources fall below a certain
threshold. Income limits for Medicaid eligibility vary widely by state; given the lack of state-level
data for some years and the high correlation of a state's income limits across time, our models
include only 1993 income limits.9 In most states, individuals or couples whose incomes exceed
the limits for Medicaid eligibility qualify for assistance if their medical expenses are high relative
to their incomes. States with a medically needy program allow households to deduct medical
expenses from income when determining eligibility for Medicaid coverage of nursing home care
or formal home health care. Thus, our models also control for the presence of a medically needy
program.
The bottom panel of Table 1 presents the 1993 average Medicaid income limits facing elderly
individuals in our sample as well as the proportion of sampled households residing in states with
a medically needy program. Individuals face monthly income limits ranging from $238 to $724
with a mean of $446; couples face monthly income limits ranging from $311 to $1110 with a mean
of $673. Over 95 percent of the households in our sample reside in states that had a medically
needy program in 1993.
As discussed in more detail later, we present three dynamic models of families' long-term
care decisions. In particular, we model the family's decision whether to use each potential care
arrangement (section IV.A), the family's selection of the primary care arrangement (section IV.E,
and hours spent in each care arrangement (section IV.G). Our models distinguish among several
modes of care -- institutional care, formal home health care, informal care provided by a spouse,
and informal care provided by a child or child-in-law -- while allowing for the possibility that an
elderly individual does not receive any of these modes of care.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 2 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Eighty-nine percent of the elderly individuals in our sample receive no care during the first
year. Among those relying on at least one mode of care, informal care arrangements are more
common than formal care arrangements. More specifically, as shown in Table 2, 7.2 percent of
respondents receive care from a spouse, and 5.5 percent receive care from an adult child or child-
in-law. While 1.4 percent of respondents rely on formal home health care, only 0.9 percent receive
nursing home care. Similarly, informal care arrangements are more common than formal
arrangements as the primary mode of care.
Spousal caregivers tend to provide substantially more care than do formal home health care
providers or adult children. On average, during the first year of data, spousal caregivers provide
60.7 hours of care per week. In contrast, the average amount of formal home health care is 29.4
hours per week among those who rely on this mode of care. The comparable figure for care
provided by adult children or children-in-law is 14.3 hours per week. The correlation between the
use of any care and hours of care is high (0.961), but it is statistically significantly different from
one. In contrast, the correlations between use of any care or hours of care and the use of a primary
caregiver are considerably lower, at 0.284 and 0.038, respectively. Collectively, the magnitudes of
these correlations suggest that modeling each dimension of care may yield unique insights
concerning families' care arrangements.
Conditioning on using a particular mode of care (for example, spouse), the correlation between
the primary care arrangement and hours of care increases substantially for all modes of care except
care provided by an adult child; for example, the correlation between the choice of the primary
caregiver and hours in care increases from 0.038 overall to 0.806, conditional on the reliance on
any spousal care. Conditional on using a particular mode of care, the relatively high correlations
between the primary care arrangement (in the case of spousal care and the two modes of formal
care) and hours of care imply that the decision to rely on a particular mode of care dominates the
decision concerning the amount of care for spousal care and formal care arrangements.
As discussed earlier, we observe each elderly individual in our sample for at least two and
at most five different time periods. Table 3 shows the number of observed transitions out of a
potential 73,816 transitions into and out of each potential care arrangement. We observe 401 (out
of 3,451+401 possible) transitions into spousal care (a transition rate of over 10 percent) and 254
transitions out of (289+254 possible) spousal care (a transition rate of over 46 percent). Transition
rates into non-spousal care arrangements range from just over one percent (child or child-in-law)
to just under one percent (institutional care). We observe a transition rate of 26 percent out of
institutional care, a rate of almost 43 percent out of care by a particular child or child-in-law care,
and a rate of over 67 percent out of formal home care.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 3 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
IV. Dynamic Models of Long-Term Care Arrangements
In most families that include an elderly individual receiving long-term care, one caregiver
provides all or nearly all of the care. However, shared caregiving is not uncommon, particularly in
large families (CS). Thus, models of families' primary care arrangements as well as models that
allow for multiple care arrangements offer valuable insights. However, discrete-choice models
cannot capture marginal effects within a particular arrangement as discrete-choice models only
involve decisions of whether or not the care arrangement was chosen (not the intensity of the
choice). Thus, modeling time spent in each care arrangement may be more informative than
modeling the discrete dimensions of care. Also, modeling time in each arrangement may reveal
rich substitution patterns across modes of care. Unfortunately, however, our data on hours of care
are bracketed, and these data probably contain significant measurement error. Given the value of
each of these three types of models, we model all three dimensions of families' care arrangements
for an elderly individual in a particular time period: the use of each potential care arrangement, the
selection of the primary care arrangement, and the hours spent in each care arrangement.
Consistent with most of the literature, we estimate these three dimensions of care separately.
Separate models enable us to examine whether time dependence of care arrangements varies across
these three dimensions of care. For example, caregiver burnout may be more relevant for the
primary caregiver than for other caregivers.
Each of our models distinguishes among several modes of care: institutional care, formal
home health care, informal care provided by the spouse, and informal care provided by an adult
child or child-in-law, while also allowing for the possibility that an elderly individual receives no
formal or informal care in a particular period. In each model, the family makes decisions taking
into account characteristics of the potential care arrangements. In contrast to our previous work
(for example, HS, ES, BGHS), we abstract from the possibility that family members have different
preferences and from details about the decision-making process.
Care arrangements may persist as a result of the family's preferences or constraints or as a
result of inertia. For example, a family's aversion to institutional care may lead to persistence in
care arrangements. Differences across family members with respect to their caregiving
effectiveness or their opportunity costs of time may also contribute to persistence in care
arrangements. Accordingly, our models control for observable factors as well as several types of
unobserved heterogeneity that may lead to persistence in care arrangements (in effect, spurious
state dependence). Moreover, the costs of transitioning from one care arrangement to another may
enhance the value of the current arrangement. The lifestyle changes required to enable an adult
child to provide care or an elderly individual's attachment to a formal home health aide may lead
to inertia in care arrangements. Similarly, moving to a nursing home requires substantial lifestyle
changes as well as disinvestments that may be difficult to reverse such as selling a home. To
capture the possibility of inertia, our models allow for positive true state dependence.
Alternatively, care arrangements may evolve over time as conditions change or as a
caregiver experiences burnout. For example, an elderly individual's care arrangements may evolve
as her health or that of her spouse deteriorates, her spouse dies, or formal care becomes more
expensive. Accordingly, our models control for relevant time-varying characteristics that may
affect a family's caregiving decisions. Our models also allow the set of potential care arrangements
to vary over time in response to changes in family structure. In addition, adult children may rotate
the role of primary caregiver (in effect, the caregiver providing the most care) as a way to share
the burden or as the caregiver experiences burnout. To allow for the possibility of caregiver
burnout, our models allow for negative true state dependence.
We develop and estimate three dynamic models of care. Two of these are discrete-choice
models, while the third is a continuous choice model with frequent corner solutions. In the Multiple
Caregiver Model, the family decides whether to use each potential care arrangement (institutional
care, formal home health care, care provided by the spouse, and/or care provided by each particular
child). This model allows for the possibility that the elderly individual relies on more than one
concurrent caregiver or caregiving arrangement. In the Primary Caregiver Model, the family
selects the primary care arrangement from among all available alternatives. Finally, in the Hours
of Care Model, the family determines hours in each potential care arrangement. Like the Multiple
Caregiver Model, this model allows for multiple care arrangements.
In all of our models, we assume that each family has an underlying latent value for each
potential care arrangement. More formally, consider family that consists of one or two elderly
individuals, adult children, and up to children-in-law. Elderly individual (= father or
mother) may require care at time . If she is married, her spouse may provide some or all of her
care. In addition, each adult child or adult child-in-law is a potential caregiver. Depending on the
model, the family decides whether to rely on each potential care arrangement, selects the primary
care arrangement, or determines how much of each arrangement to use. Define the + 4
caregiving alternatives as: no care, care provided by a spouse, formal home health care, care in a
nursing home, and informal care from each of the children or their spouses.
The latent value of care alternative to individual in family at time is denoted by
(1)
=++++.
The vector includes exogenous characteristics of the elderly individual.10 In particular,
includes demographic characteristics and activity limitations that may influence an elderly
individual's caregiving needs, opportunities, and preferences. The vector includes exogenous
characteristics of the potential care arrangements, namely demographics characteristics of the adult
children and children-in-law and market conditions or public policies in the elderly individual's or
household's state of residence.
The observed variable corresponding to the latent variable is given by . As discussed
in the following subsections, the exact definition of the corresponding observed variable varies
with the model specification. The inclusion of allows past choices to influence the current
value of alternative and thus captures the true dynamic component of long-term care decision-
making. To distinguish between true state dependence (as captured by ) and persistence in care
arrangements due to unobserved heterogeneity (in effect, spurious state dependence), we allow for
unobserved correlation across time (as captured by ). The is an idiosyncratic error with
an assumed distribution that varies across models for computational convenience. We refer to
as true state dependence, which is alternative-specific in our models.
We decompose the non-idiosyncratic portion of the random components of families' long-
term care decisions, , into three types of unobserved heterogeneity:
(2) =++
where we assume (0,1), (0,1), and 0, . The and
terms are alternative-specific factor loadings.11 Some elderly individuals may have preferences
for certain care options that are not observed to the econometrician and hence not captured by
or . For example, a family may avoid institutional care due to a particularly strong philosophical
or cultural reason. Such individual/family-alternative-specific correlation across time is captured
by . In addition, there may be individual- or family-specific characteristics that influence all
care alternatives across time but are unobserved by the econometrician. For example, any
unobserved characteristic of the parent that affects the parent's need for care is captured in .
Similarly, high levels of wealth may enhance the value of all care alternatives by enabling families
to purchase higher-quality formal care and/or to alleviate the burden associated with informal care
(for example, by purchasing other time-intensive services such as housekeeping). Finally,
allows for individual/time-specific unobserved heterogeneity, such as temporary health conditions
unrelated to ADL or IADL limitations. For ease of exposition, we suppress the family subscript in
the following subsections.
Two other issues associated with estimating dynamic models are initial conditions and
duration dependence. We discuss initial conditions in section IV.B.12 In this paper, we choose not
to model duration dependence because of the limited number of waves in the data. Dostie and
Léger (2005) model duration dependence and find negative duration dependence in living
arrangements, but they use a much longer panel from the PSID. Other studies, for example, Roth
et al. (2001), Gaugler et al. (2005a, 2005b), and Perren, Schmid, and Wettstein (2006), model
duration of caregiving, but they observe caregivers significantly more frequently than in the
HRS/AHEAD sample. Researchers using HRS/AHEAD data have not modeled duration
dependence because of the relatively small number of waves available (for example, GG; Sk).
A. Multiple Caregiver Model
In our Multiple Caregiver Model, the family decides whether to use each potential care
arrangement by taking into account characteristics of the elderly individual, characteristics of the
care arrangement, and whether the individual relied on that arrangement in the previous period.
Excluding the dynamic component, this approach is similar to that of CS, Brown (2006), and
BGHS. In this model, we assume that the family selects each arrangement with a positive latent
value without considering interactions across care alternatives.
More technically, we estimate a dynamic multivariate probit model where the baseline
latent value of alternative is given in equation (1). We assume (0,1) and define ()
as the joint distribution of . Family uses alternative to provide care for individual at time
if and only if
> 0; in effect,
= 1
> 0.
Let
() = +++
where equals one if alternative was chosen last period and =,,,,, is the
vector of parameters to estimate. Let be a dummy variable indicating whether individual is
living at time .13 Then, the likelihood contribution for an elderly individual is
=
1(
)
().
It is straightforward to simulate the likelihood contribution for each observation.14 As is true for
all the models in the paper, estimating the asymptotic covariance matrix is standard.
B. Initial Conditions Methodology
Initial conditions problems are common to estimation of all dynamic models. In particular,
in the other dynamic long-term care papers, HM find no significant correlation between errors in
the initial and subsequent periods, while Sk find important effects associated with initial
conditions. In this section, we suggest a new methodology for controlling for initial data year
conditions à la Heckman (1981).15
We couch our discussion in terms of the Multiple Caregiver model presented above where
decisions are made at time =,+ 1, . . ,0,1, . . , , but the same methodology can be used in
all three models. We observe {,,}
for each observation where =
(,,,,,,..,,,). This is an example of the classic initial data year conditions problem in
that we must decide how to model the stochastic process for at time 0.
Imagine that we know or can estimate inverse transition matrices,
(3) () = Pr[] and
= Pr[ ].
In some cases, such as age, is degenerate. In others cases such as ADL problems, we
can easily estimate () with AHEAD/HRS. Also we assume that, at , = 0, a
reasonable assumption in our case since almost all respondents live independently at Then, given
{(), ()}
, we can easily simulate {
,
}
(working from = 0
back to =). The simulator may not be continuous, but that does not matter as long as
{(), ()}
does not depend on .
Given a draw of {
,
}
, we can now compute
Pr [,,{
,
}
,,,]
iteratively for =,,...,1, where is a draw of ,
is a draw of , and is a draw
of and = (,,..,). Note that the and draws are used both in the initial period
in the data as well as all subsequent periods thus allowing for the standard initial data year
conditions bias described, for example, in Heckman (1981). Let
() =
+
++
+
.
Define
= Pr = 1
,
,,,
=
and then
(4)
= Pr = 1
,
,,,
=
+
+
+1
for =,,..,0. We can add either
or 1
to the conditional likelihood
function to control for the initial data year conditions. The advantage of this method for this
problem is that it involves adding no parameters to the model. This is important here because we
do not observe much variation in the first period of data to estimate a set of extra parameters with
any precision. The method relies on the existence of a time in the not-too-distant past where it
is reasonable to assume that = 0. This methodology has some significant similarities to Ham
and Lalonde (1996), though, in their work, there is not the same natural starting point assumption.
C. Estimates of Transition Matrices
Using the proposed initial data year conditions methodology requires estimating the inverse
transition matrices defined in equation (3). Table 4 includes a list of all of the inverse transition
models estimated.16 For each model, we include as regressors a constant, dummies for female,
black, and Hispanic, age-80, and (age-80)² and simulate data backwards from the age of the
respondent in the initial wave back to age =60.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 4 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Probit estimates for inverse transition into marriage are reported in Table 5. For example,
the estimate associated with female in the first panel (-1.456) says that, holding constant other
exogenous variables, women are approximately 1.456(1.229) =0.273 less likely than are men
to have a spouse at , conditional on not being in the sample at + 1; 17 in effect, women who die
between and + 1 are less likely to have had a spouse at than men who die between and +
1. The finding that women are less likely than men to be married at the end of life is consistent
with differences in life expectancy by gender and age differences between husbands and wives.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 5 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Based on the probit estimates in the first panel of Table 5, Figure 1 shows how inverse
transition probabilities for marriage at conditional on not being in the sample at + 1 vary with
demographic characteristics. In general, women have significantly lower inverse transition rates
than men, and blacks and Hispanics have lower inverse transition rates. Age effects increase in
magnitude (absolute value) for those with low inverse transition rates because of the nonlinearity
of the probit structure. While this particular example is interesting in its own right, it is not relevant
for controlling for the initial data year conditions problem because someone who is not in the
sample in the initial period (and possibly there the period before) is not included in the data.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Figure 1 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Consider the results in the second panel of Table 5 for the probability of being married at
conditional on being in the sample but not married at + 1; such results are relevant to
controlling for initial data year conditions. Note that the estimate of the effect of female is negative
(-0.426), implying that women are approximately 0.426(1.191) = 0.084 less likely to have a
spouse at t conditional on not having one at + 1 than are men, in effect, it is more likely that an
unmarried man lost his wife within the last period than that an unmarried woman lost her
husband.18 At first glance, this finding may seem counterintuitive in light of men's higher death
rates. To enhance intuition, consider the following implication of this finding: conditional on being
unmarried at + 1, women are more likely than men to have been unmarried at time . This finding
is consistent with gender differences in life expectancy and age differences between men and
women; in effect, among widows and widowers, on average women lost their spouses longer ago.
A summary of the significant estimates for the other inverse transitions is reported in Table
4.19 For example, as shown in Table 4, conditional on not being in the sample at time + 1, black
elderly individuals are less likely than white elders to be married at time , controlling for other
exogenous variables. Conditional on being unmarried at time + 1, black and Hispanic elders are
less likely than their white counterparts to be married at time . Also, not surprisingly, conditional
on being unmarried at time + 1, having a spouse at time depends negatively on age.
Conditional on the number of ADL problems experienced by the elderly individual at time
+ 1, the estimated number of ADL problems at time is higher for women than for men and for
blacks and Hispanics than for whites. Also, as expected, the estimated number of ADL problems
at time depends positively on age, conditional on the number of ADL problems at time + 1.
Similarly, conditional on the number of IADL problems experienced at time + 1, the estimated
number of IADL problems at time is higher for women than for men and for blacks than whites.
The estimated number of IADL problems at time also depends positively on age, conditional on
the number of IADL problems at time + 1.
The spouse's age, years of education, and activity limitations at time depend on the
respondent's demographic characteristics, conditional on the spouse's absence from the sample at
time + 1. For example, conditional on the spouse's absence from the sample at time + 1, on
average, women have older spouses than do men at time . The spouse's activity limitations at time
also depend on demographic characteristics, conditional on the number of activity limitations at
time + 1. For example, conditional on the number of ADL problems experienced by the spouse
at time + 1, husbands (women's spouses) experience fewer ADL problems than do wives.
Finally, several characteristics of the elderly individual's children at time depend on the
elderly individual's demographic characteristics, conditional on the presence of a child at time +
1, the child's employment status at time + 1, or the child's marital status at time + 1. For
example, conditional on the lack of a child at time + 1, the probability of a child at time depends
negatively on the parent's age. The probability that a child worked at time depends negatively on
the parent's age, regardless of whether the child works at time + 1. Conditional on working at
time + 1, the probability that the child worked at time is lower in black and Hispanic families
than in white families.
In the Heckman (1981) approach to initial conditions, one adds a large number of extra
parameters that allow the likelihood contribution of the initial data year condition to be essentially
unrelated to the likelihood contributions of subsequent periods except for potential correlation of
errors. In other words, the joint density of the initial data period and subsequent periods can be
decomposed into a) the density of subsequent periods conditional on the initial period multiplied
by b) the density of the initial period. Given all of the extra parameters used to parameterize (b),
the parameters associated with (a) are identified essentially by the joint density of subsequent
periods conditional on the initial period. In our methodology, the restriction imposed on the initial
data year condition is that the same parameters that affect behavior in subsequent periods must
explain behavior in the initial period as well, though in a different way because transitions between
time periods and 0 are not observed in the data. Thus, the structural parameters are identified
by the product of (a) and (b). The fact that the estimates with and without the initial data year
conditions correction are closely tied suggests that our specification of the process to reach the
initial data year condition is modeled in a reasonable way.
D. Multiple Caregiver Results
Table 6 presents the multivariate probit results. Several demographic characteristics
significantly influence the value of each potential care arrangement.20 Controlling for marital
status, age, activity limitations, educational attainment, and several characteristics of the spouse,
families value each mode of care statistically significantly more highly for men than for women.
Although inconsistent with some of the findings in the literature (for example, McGarry 1998;
Pezzin and Schone 1999b; CS), the implication that families value informal care more highly for
elderly men than for elderly women is consistent with the implications of the game-theoretic
analysis in BGHS; specifically, the findings presented in BGHS suggest that care provided to
mothers is less effective (albeit also less burdensome) than care provided to fathers.
Activity limitations and age significantly influence the value of care arrangements. The
value of each mode of care depends positively and, with one exception, statistically significantly
on the number of ADL and IADL problems experienced by an elderly individual. Controlling for
activity limitations and the age of the spouse, as the individual ages, spousal care becomes less
valuable, while the other modes of care become more valuable.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 6 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Consistent with the literature (for example, Stern 1995; Hoerger, Picone, and Sloan 1996;
Rainer and Siedler 2009), our results imply that spouses are an important source of care for one
another. At first glance, the sign and the statistical significance of the relationships between marital
status and institutional care and between marital status and care provided by an adult child suggests
that families value these care arrangements significantly more highly for unmarried than for
married elders.21 However, some spousal characteristics mitigate the direct effect of marital status.
In fact, for most combinations of spousal characteristics within the range of our sample, families
value institutional care more highly for married than for unmarried individuals. An exception
concerns individuals whose spouses are relatively young (under about 70 years), healthy (no
activity limitations), and highly educated (college degree). Whether families value care provided
by an adult child more or less highly for married individuals than for their otherwise identical
unmarried counterparts depends on the characteristics of the spouse. For example, unless the
spouse is very old (about 88 years or older), the estimates suggest that families value care provided
by a child less highly for a married individual whose college-educated spouse is healthy than for
an unmarried but otherwise identical individual. In the case of elderly individuals whose spouses
are about 71 years or older, less healthy (one ADL and one IADL problem) and less educated (no
high school degree), however, the estimates suggest that families value care provided by an adult
child more highly than if the individual were unmarried but otherwise identical. For all
combinations of spousal characteristics in our sample, families value formal home health care
more highly for married than for unmarried individuals.
Our results also shed light on the role of adult children's characteristics in families' long-
term care arrangements for elderly individuals. Consistent with the literature (for example, SPH;
Wolf, Freedman, and Soldo 1997; CS; ES), child gender is associated with informal care provision
-- at least among white families.22 Our results suggest that white families value daughters more
than sons as caregivers, after controlling for other demographic characteristics. Children who live
near but not with their elderly parents are valued more highly as caregivers than are those living
more than 10 miles away.23
Several market conditions and public policies in the elderly individual's state of residence
also significantly influence care arrangements. After controlling for activity limitations and other
relevant factors, the attractiveness of formal home health care depends negatively on the average
wages of home health care providers and positively on the generosity of a state's income limits
facing couples (but not single individuals) for Medicaid coverage of formal care. Institutional care
is a more attractive option in states with greater nursing home staff hours per nursing home
resident.
Although our models explicitly control for health limitations that relate to ADLs and
IADLs, the person-time-choice factor loadings () may capture the role of temporary health
conditions unrelated to ADL or IADL limitations, while the person-choice factor loadings ()
may capture the role of chronic health conditions unrelated to ADL or IADL limitations. Estimates
of the person-time-choice factor loadings indicate that unobservable person-time-specific
heterogeneity influences the value of each non-spousal care arrangement. Thus, the results suggest
that temporary health conditions unrelated to activity limitations may influence the relative
attractiveness of each non-spousal mode of care and hence may induce changes in care
arrangements over time. Meanwhile estimates of the person-choice factor loadings indicate that
more permanent sources of unobserved heterogeneity such as chronic health conditions also
influence the value of each care alternative. In addition to chronic health conditions, the person-
choice factor loadings may capture the effect of income and wealth. Collectively these results
highlight the importance of controlling for unobserved heterogeneity in modeling families' care
arrangements over time.
Moreover, the results provide evidence of true positive state dependence or inertia across
all modes of care. This finding probably reflects the substantial economic and psychological costs
associated with transitions into and out of care arrangements. To the extent that caregiver burnout
contributes to negative true state dependence, its effect is dominated by inertia. While it would be
very useful to separately estimate the two effects, there is no information in the transition data that
would allow for separate identification.24 However, there are other sources of data with direct
measures of burnout that might allow for a decomposition of the total effect into its two
components (for example, Pezzin, Kemper and Reschovsky 1996; Coe and Van Houtven 2009).
The psychology literature contains much work on direct measurement of burnout (for example,
Lawton et al. 1991; McFall and Miller 1992; Goode et al. 1998; Li, Seltzer, and Greenberg 1999;
Seltzer and Li 2000; Roth et al. 2001; Gaugler et al. 2005a,b; Hirst 2005; Perren, Schmid, and
Wettstein 2006). The problem with applying this literature or the data behind the literature is that
the investigated burnout is over much shorter periods of time than that used in this paper.
We also estimated a static multivariate probit model where care arrangements in the
previous period do not influence current care arrangements (in effect, we restrict = 0). Most of
the parameter estimates associated with the static model are consistent in sign with those of the
dynamic model, but, not surprisingly, their magnitudes tend to be larger and more statistically
significant. For example, the relationship between the generosity of a state's Medicaid policy and
the value of formal home health care is larger and more statistically significant in the static model.
Perhaps some characteristics matter more in the initial choice of the care arrangement than in the
current decision conditional on past decisions. Also, in the model with dynamics, the measured
effects are associated with flows, while, in the static model, the measured effects are associated
with a stock of present and future flows (for example, Berkovec and Stern 1991).
To illustrate more formally how the effects may be associated with flows consider a simple
dynamic programming problem with choices each period (1,2, . . , ) and utility flow
(,) + () where is a vector of exogenous state variables25 and26
().
Then the value of choosing is
() = ,+()+(),
and, given the error distribution assumption,
Pr[=]=(,) + [()]
(,) + [()]
Identification of relies on covariation of with frequency of choice being chosen (for all ),
and27
Pr[=]
()
=
(,) +
[)].
Almost always, [()]/ has the same sign as (,)/ because
() is just a weighted sum of future utility flows, many of which will have = for >.
In other words, if increases (,) at time , then it will do so also at future times when
choice is chosen. Let be the sample analog of [=]/. The estimation procedure
is going to match with the value of [=]/ implied by the model and parameters. In
a dynamic model, where 0 < < 1, some of the theoretical derivative can be captured in
()/, while, in a static model, where = 0, all of the theoretical value
has to be explained by (,)/. In general, if
()=
(,)
with multiplication factor > 0,28 then the estimate of in the static model and the estimate in
the dynamic model must satisfy
(,) = =(+ 1)
(,)
which implies
=(+ 1) > .
Evidence of inertia in care arrangements and the sensitivity of parameter estimates across
our static and dynamic models underscore the importance of developing models that capture
intertemporal patterns of care. As discussed earlier, most of the models in this literature are static
(for example, SPH; PSa; ES; PPS; BGHS). While a few studies present dynamic models (GM;
BKM; DL; HM; GG), our models encompass a broader range of care arrangements.
E. Primary Caregiver Model
Much of the long-term care literature focuses on the selection of the primary care
arrangement for an elderly individual (for example, HS; ES), but all of the existing models of the
primary care arrangement are static in nature. In our Primary Caregiver Model, the family selects
the primary care arrangement for an elderly individual in a particular time period taking into
account the characteristics of the potential care recipient, the characteristics of the potential care
arrangements, and the primary care arrangement selected the previous period. The primary care
arrangement is the arrangement with the highest latent value. If the value of each potential care
arrangement is less than the value of remaining independent, the individual receives no care.29
More technically, we estimate a multinomial mixed logit model (McFadden and Train
2000) where the baseline latent value to alternative is given in equation (1) and .
Assume the family chooses the alternative that provides the highest latent value; in effect,
= 1(
,)
where the set of care alternatives at time is denoted . Let
(5)
() = +++
with parameters given by . The lagged care decision, , is a dummy variable equal to one if
care arrangement was the primary arrangement in the previous period. Then the likelihood
contribution for family is
(6) = exp
exp
()
where () is the joint distribution of the unobservables. There is no closed form solution to
equation (6), so we estimate the model using maximum simulated likelihood estimation. The
simulated likelihood contribution is
=1
{
(
,,
,)}
{
,,
,}
where (
,,
) are errors simulated from their respective densities (BKM; Hajivassiliou,
McFadden, and Ruud 1996). As discussed earlier, we control for initial data year conditions as
described in section IV.B.
F. Primary Caregiver Results
Table 7 presents multinomial mixed logit parameter estimates for the choice of the primary
care arrangement. The roles of demographic characteristics and activity limitations in the decision
to use a particular mode of care are often similar to their roles in the selection of the primary care
arrangement. For example, as in the Multiple Caregiver Model, controlling for marital status, age,
activity limitations, educational attainment, and several characteristics of the spouse, families
value each mode of care more highly for men than for women. Also, as an elderly individual
develops more activity limitations, the value of each mode of care tends to increase as an
arrangement (in Table 6) and as the primary care arrangement (in Table 7).
As in the Multiple Caregiver Model, our results imply that marital status and spousal
characteristics influence families' elder care arrangements. Marital status per se is significantly
associated with the value of informal care provided by an adult child and formal home health care
but not with institutional care. However, the spouse's age and activity limitations (as measured by
the number of ADL problems and/or the number of IADL problems) influence the value of each
mode of care, while the spouse's educational attainment influences the value of care provided by
an adult child. Consistent with the implications of our Multiple Caregiver Model, our estimates
suggest that families tend to place more value on both modes of formal care as the primary
arrangement for married than for unmarried elderly individuals. An exception concerns individuals
whose spouses who are relatively young and healthy; for example, all else equal, the estimates
suggest that a family would value formal home health care more highly for an unmarried individual
than for a married individual whose 60-year old, college-educated spouse has no activity
limitations. Again, whether families value care provided by an adult child more or less highly for
married individuals than for their otherwise identical unmarried counterparts depends on the
characteristics of the spouse.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 7 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
As in the Multiple Caregiver Model, characteristics of the younger generation influence
families' care arrangements. Again, white families value daughters statistically significantly more
than sons as caregivers. Proximity to or co-residence with elderly parents positively influences an
adult child's value as primary caregiver. Although marriage significantly enhances an adult child's
value as a caregiver, marriage significantly reduces her value as the primary caregiver.
Market conditions and public policies significantly influence the value of formal care
arrangements in both discrete-choice models. For example, the value of formal home health care
as an arrangement or as the primary arrangement depends negatively on the cost of home health
aide workers, and the value of institutional care depends positively on the nursing home staff hours
per resident in the individual's state of residence.
As discussed earlier, the person-time-choice factor loadings () may capture the role of
temporary health conditions unrelated to ADL or IADL limitations, while the person-choice factor
loading () may capture the role of chronic health conditions unrelated to activity limitations. As
in the Multiple Caregiver Model, estimates for the person-time-choice factor loadings indicate that
person-time-specific heterogeneity significantly influences the values of non-spousal care
arrangements. Thus, the results of this model suggest that temporary health conditions unrelated
to ADL and IADL limitations may change the relative attractiveness of each non-spousal mode of
care and hence may induce a change in the individual's primary care arrangement. Estimates for
the person-choice factor loadings indicate that unobservable person-specific characteristics
significantly influence the value of care provided by an adult child. This finding may reflect
variation in caregiver burden depending on the presence or severity of chronic health conditions.
As in the Multiple Caregiver Model, the results of the Primary Caregiver Model provide
evidence of true state dependence in elder care arrangements. The signs of the relevant parameter
estimates suggest that inertia contributes to care arrangements in the case of informal care provided
by a spouse or an adult child and in the case of institutional care. Intuitively, the high costs of
transitioning from one care arrangement to another or the accumulation of human capital specific
to caregiving may lead families to maintain their primary care arrangement for an elderly relative.
While three of the four modes of care display positive state dependence, formal home health care
displays negative state dependence.30 However, since a family could replace a burned-out home
health aide, negative state dependence probably cannot be attributed to caregiver burnout in the
case of formal home health care. Instead, negative state dependence may reflect a tendency to rely
on formal home health care 1) for acute rather than chronic health conditions, 2) as a temporary
measure until an informal caregiver becomes available, or 3) until the parent needs institutional
care. Thus, while both discrete-choice models provide evidence of inertia in informal modes of
care and institutional care, the models differ with regard to their implications concerning the use
of formal home health care over time. Collectively the two sets of results suggest that inertia
dominates caregiver burden in families' decisions to rely on a formal home health aide in
conjunction with other modes of care but that the reliance on a formal home health aide as the
primary caregiver tends to be temporary.
G. Hours of Care Model
An important dimension of caregiving decisions concerns how much care each caregiver
provides. Following SPH, Wolf, Freedman, and Soldo (1997), Pezzin and Schone (1999b), CS,
and BGHS, we next consider the continuous choice associated with care arrangements. As
discussed earlier, families may rely on more than one mode of care. For example, an elderly
individual may receive informal care provided by a child together with formal care provided by a
home health aide (Bolin et al. 2008). As this example suggests, various caregiving alternatives --
and the amount provided -- may be substitutable to some extent. Moreover, the quantity of care
received in the past could influence the value associated with the quantity of that care alternative
provided today (US Department of Health and Human Services 1999). Accordingly, our Hours of
Care Model allows for the possibility of multiple care arrangements, while linking arrangements
over time.
We estimate a dynamic multivariate tobit model, where we augment the baseline latent
value of care in equation (1) to allow for substitution across modes of care as well as different
effects of true state dependence. We treat nursing home care differently than other care
arrangements as we do not observe hours of care when a parent is in a nursing home. Let =2
for nursing home, =1 for formal care, = 0 for care by spouse, and > 0 for care by the th
child. Specifically, the latent value associated with the amount of time spent using the th care
arrangement is
(7)
= ++ 1()
+1(2),,
+1(> 0) + ++
In terms of its substitution effect, we distinguish between nursing homes and other alternatives.
For alternatives other than nursing home care, >2, the observed continuous value of
caregiving is given by =(0,
) and (0, ). For nursing home care, =
2, the observed binary measure of caregiving is given by ,,= 1(,,
> 0) because we do
not observe a continuous measure of nursing home care hours. Similar to CS, substitution in total
care provided across alternatives is captured by 1()
and 1(
2),,.31 The terms capture true state dependence in caregiving where the hours in mode
depends both on whether was chosen in the previous period (captured by the threshold value of
true state dependence, ) as well as the quantity of alternative provided in the previous
period (captured by the marginal value of true state dependence, ). The parameters to
estimate are and = (,,,,,,,) .
Let
= ++ 1()
+
1(2),,+1(> 0) + +.
The likelihood contribution for an individual is
(8) =(,)
()
where () denotes the joint distribution of the unobservables,
,= 1
()
,
,
,= ,,
,,,
,
,= 1
.
.
Note that, for >2, the likelihood contribution is a conditional Tobit term, and, for =2, it
is a conditional probit term. We simulate equation (8) by drawing values of from its distribution
(with antithetic acceleration). Again, we control for initial model year conditions by constructing
from equation (4) in the appropriate fashion and adding it to the conditional likelihood
function.
H. Hours of Care Results
Table 8 presents the results of our dynamic multivariate tobit model. The results of this
model reinforce the importance of controlling for unobserved heterogeneity. Estimates for the
person-time-choice factor loadings indicate that person-time-specific heterogeneity significantly
influences the values of both modes of informal care as well as institutional care. Thus, these
results suggest that temporary health conditions unrelated to ADL and IADL limitations may
change the relative attractiveness of these modes of care and hence may induce changes in the
amount of informal care or the use of institutional care over time. Estimates for the person-choice
factor loadings are not statistically significant.
All modes of care exhibit statistically significant true state dependence. The quantity of
each mode of non-institutional care in the current period depends positively on whether that mode
was used in the previous period -- a threshold inertia effect, while the quantity of each mode of
informal care depends positively on the quantity used in the previous period, a marginal inertia
effect. Thus, the results again indicate that, to the extent that caregiver burden influences long-
term care arrangements, its impact tends to be dominated by inertia. Not surprisingly, the reliance
on institutional care depends positively on whether the individual was institutionalized in the
previous period.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 8 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Contrary to our expectations and to the literature (for example, CS; Van Houtven and
Norton 2004), the estimated substitution effects indicate that the quantity of each mode of non-
institutional care and the use of institutional care depend positively on the total amount of care
currently received from other sources. The positive effects associated with informal care provided
by a child may reflect competition among children for a bequest (Bernheim, Shleifer, and Summers
1985). However, a bequest motive of this sort would not explain the positive substitution effects
associated with the other modes of care. Instead, the number of ADL and IADL problems may not
adequately capture long-term care needs, in which case our estimated substitution effects may be
biased. More generally, the positive substitution effects may reflect parent-specific unobserved
heterogeneity not captured by our model specification.
Characteristics that significantly increase (decrease) the latent value of using a particular mode
of care often significantly increase (decrease) hours spent in that mode of care. For example,
activity limitations tend to increase the attractiveness of formal as well as informal care
arrangements. Likewise, hours in each arrangement depend positively on the number of activity
limitations. Consistent with expectations and mostly consistent with the implications of the
discrete-choice models, adult children who live with or near their elderly parents provide more
assistance than do their geographically distant counterparts.
A few market conditions and public policies significantly influence the quantity of formal care
received by the elderly individual. Consistent with our other models, the quantity of formal home
health care depends negatively on the average wages of home health care providers. The quantity
of formal home health care depends positively on the generosity of a state's Medicaid income limits
facing individuals. As expected, the likelihood of selecting institutional care depends positively on
daily nursing home staff hours per resident.
V. Policy Implications
For each of our dynamic models, we present the marginal effects associated with the
characteristics that have the most relevance from a public policy perspective. In particular, Table
9 displays the marginal effects associated with the market conditions and public policies in the
elderly individual's state of residence.
The statistical significance of the estimated marginal effects suggests that the cost of formal
home health care influences all three dimensions of families' long-term care decisions. However,
while the magnitude of the effect on hours is economically significant among families relying on
formal home health care, the magnitudes of the other marginal effects suggest that changes in the
cost of formal home health care would need to be relatively large in order to induce practically
significant changes in the discrete dimensions of care. As indicated in the top panel of Table 9, as
the weekly cost of full-time formal home health care increases by $100, the predicted probability
that the elderly individual receives this mode of care falls by 0.41 percentage points. As shown in
the middle panel, the cost of formal home health care also influences the selection of the primary
care arrangement. In response to a $100 increase in the weekly cost of full-time care, the predicted
probability of relying on formal home health care as the primary arrangement falls by 0.19
percentage points, while the predicted probability of living independently increases by 0.14
percentage points.32 Finally, as shown in the bottom panel, conditional on receiving formal home
health care, an increase of $100 in its weekly cost is associated with a 15 percentage point reduction
(about 16.8 hours) in the predicted quantity of formal home health care.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 9 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
The existing literature provides mixed evidence concerning the effects of state-level
Medicaid policy on nursing home utilization. Cutler and Sheiner (1994) report a positive and
statistically significant relationship between the presence of a medically needy program and the
probability of nursing home use. Similarly, ES suggest that the presence of a medically needy
program enhances the value of nursing home care. In contrast, Aykan (2002) does not find a
statistically significant relationship between the presence of a medically needy program and the
likelihood of institutional care. Likewise, Grabowski and Gruber (2007) (hereafter GrGr) report
that the availability of a medically needy program is not statistically significantly associated with
nursing home use. Consistent with Cutler and Sheiner and ES, the presence of a medically needy
program has a very small but positively and statistically significant effect on the use of institutional
care in our discrete-choice models. Our results indicate that the presence of a medically needy
program is associated with statistically significant increases in predicted hours of formal home
health care.
Care arrangements depend on the generosity of the state's income limits for Medicaid
eligibility. In response to a $1000 increase in the monthly income limit facing unmarried
individuals, the predicted probabilities that a lone elder relies on formal home health care or
institutional care respectively increase by 13.4 and 22.6 percentage points, while the predicted
probabilities that she relies on these arrangements as the primary arrangement increase by 6.5 and
3.3 percentage points.33 The positive and statistically significant relationship between income
limits and nursing home use is consistent with economic theory and GG but inconsistent with
GrGr. It is worth noting that GrGr set income limits to zero in states with spend-down provisions,
while we use the actual income limits for all states. Among lone elders who receive formal home
health care, predicted hours increase by 45.4 (=0.27168) hours per week as the income limit
increases by $1000. The effects of income limits are smaller but are sometimes negative for
married individuals.
The use of institutional care depends on the quality and availability of care in the elderly
individual's state of residence. For example, the predicted probability of relying on institutional
care depends positively on the availability of nursing home beds. Surprisingly, however, the
predicted probabilities of relying on institutional care as an arrangement or as the primary
arrangement depend negatively on nursing home staff hours per resident.
VI. Specification and Robustness Checks
In this section, we address several limitations of our data and models. First, we explore whether
including income and wealth (even though the data are of poor quality) would improve the fit of
our models. Second, we test whether state/mode-specific effects and other state-specific effects
not included in our models influence care arrangements. Third, we examine whether the distance
between an elderly parent and an adult child is endogenous. Finally, we present specification
checks by allowing for a richer covariance structure.
A. Income and Wealth Data
As we discussed in section III, the measures of income and assets/wealth in the
AHEAD/HRS may not be reliable. Therefore, we chose not to let our models include income or
wealth effects. Here, we present several alternative specifications to test whether adding income
or wealth would significantly improve the fit of our models. In light of the findings in GG that an
important effect of wealth operates through its interactions with state Medicaid rules, we also
examine interactions of income and wealth with policy characteristics. To facilitate comparison
with GG and BGHS, we use 1998 and 1993 Medicaid income and asset limits as well as dummy
variables indicating the presence of a medically needy program. For each model, we estimate
several additional specifications. In addition to income or wealth, these specifications include
income or wealth interacted with Medicaid income or asset limits as well as the presence of a
medically needy program. One of the wealth specifications excludes observations with missing
information on wealth. Since the lack of information concerning wealth may not be random, the
remaining wealth specifications include a dummy variable for missing wealth data.34 While GG
treat the Medicaid asset limit as a discrete cutoff, our approach accommodates the possibility that
individuals just above the limit may spend down and thus behave similarly to those who are just
below the limit.35 In particular, we include "smoothed" assets in our third wealth specification.36
To reduce the influence of large fluctuations in reported wealth, the final specification replaces the
individual's current wealth with her average wealth over the observed time frame.
Table 10 presents the results of Lagrange Multiplier tests for each model specification.37
The top panel displays the results concerning income and its interactions, while the bottom panel
displays the results concerning wealth and its interactions. For each specification, we conduct a
joint chi-square test as well as separate chi-square tests for each restriction. Consistent with GG,
the results indicate that the fit of the Multiple Caregiver Model would be improved by including
income and its interactions with the 1998 Medicaid policy characteristics: the score statistics
associated with income interacted with the presence of a medically needy program and income
interacted with Medicaid income limits are individually and jointly statistically significant at the
5 percent level of significance. However, the corresponding score statistics associated with 1993
Medicaid policy characteristics are neither individually nor jointly significant. Moreover, the score
statistics associated with income itself are not statistically significant. For the Primary Caregiver
Model, the score statistics associated with income and its interactions with 1998 Medicaid policy
characteristics are jointly statistically significant. While the score statistic associated with income
and its interaction with the Medicaid income limit is individually statistically significant, the score
statistics associated with income itself and with income and its interaction with the presence of a
medically needy program are not. In the Hours of Care Model, income and its interactions with
Medicaid policy are jointly significant for both Medicaid limit years. However, only the
interactions between income and the presence of a medically needy program and income and
Medicaid income limits in 1993 have a statistically significant score statistic. For all three models,
the score statistics associated with wealth and its interactions are jointly statistically significant,
but most of the individual score statistics associated with wealth or its interactions are not
statistically significant.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 10 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
The mixed evidence concerning the interactions between income and the presence of a medically
needy program or between income and Medicaid income limits may be attributable to the poor
quality of the income data and the nonlinearity of the interactions between income and policy
characteristics. For example, we would expect minimal effects of a medically needy program on
[ ]/ except near the relevant income limit. We did not allow for such
nonlinearity; nor did GG. Furthermore, if there are higher non-response rates for income associated
with higher incomes near the relevant limit, our test statistics may be biased towards zero.
Similarly, the mixed evidence concerning the interaction between wealth and policy characteristics
may partially reflect the poor quality of the wealth data. Collectively the results provide mixed
evidence in favor of including measures of income and wealth even when interacted with Medicaid
policy characteristics.
B. State-Specific Effects
Although our models include several market conditions and public policies in the elderly
individual's or couple's state of residence, the models may not capture all state-specific/mode-
specific or other state-specific effects. For example, other nursing home regulations, the supply of
home health aides, or state-specific variation in cultural attitudes may influence families' care
decisions. Accordingly, our next set of specification tests concerns the possibility of state-
specific/mode-specific fixed effects. Using observations from states where there is more than one
elderly household in our sample, we perform Lagrange Multiplier tests to test the null hypothesis
of no state-specific/mode-specific fixed effects. In particular, we amend the baseline model in
equation (1) to
(9)
=+++
++
where is a dummy variable equal to 1 if and only if parent in family lives in state . Under
the null hypothesis, = 0 for all choices and all states .38 The overall
is 1850.7 which is
statistically significant. However, the vast majority of the individual score statistics are statistically
insignificant, and there is no obvious pattern associated with those that are significant.39
Next, we compare our state-specific/mode-specific score statistics for the two formal
modes of care to 1997 utilization rates based on data presented in LeBlanc, Tonner, and Harrington
(2000). We translate their data into log(nursing home utilization per 1000 individuals aged 70 or
over) and log(number of Medicaid waivers per 1000 individuals aged 70 or over). After excluding
states with missing state-specific dummy score statistics and/or utilization rates along with one
outlier,40 we use data from the remaining 31 states to compute the correlations between the score
statistics and the utilization rates. For the Multiple Caregiver Model, the correlations are roughly
0.38 for nursing home care and 0.09 for formal home health care. The correlations are smaller for
the other two models. These results imply that observed variation across states in nursing home
use is not tied to the state-specific dummies in equation (9), and we find only weak evidence that
state-specific characteristics not included in our models affect nursing home usage.
C. Geographic Distance
Geographic distance between elderly individuals and their adult children may be
endogenous. For example, children may move closer to their parents or vice versa as their parents
age or develop health problems. For families with at least one child, Table 11 presents the overall
location transitions of children relative to parents based on the three possible responses in
AHEAD/HRS: co-residence, living within 10 miles, and living more than 10 miles away. For each
family size, the probabilities on the diagonals reveal strong persistence in geographic distance. For
example, among families where the parent and child are initially more than 10 miles apart, the
proportion later co-residing ranges from 0.008 to 0.029, depending on family size; similarly, the
proportion later living within 10 miles ranges from 0.005 to 0.011.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 11 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Next we examine location transitions in response to increased caregiving needs. Table 12
displays location transitions among families where at least one child initially lives more than 10
miles from her parent(s) and subsequently co-resides with her parent(s). For each of these families,
we follow all initially distant children who provide no care in the first year until one becomes the
primary caregiver (if ever). Column (1) reports the proportion of children who transition to
coresidence with the parent among families where the parent experiences an increase in the number
of ADL problems. Column (2) indicates the proportion of those children from Column 1 who
become the primary caregiver. Columns (3) and (4) indicate the comparable proportions among
families where the parent experiences no change in the number of ADL problems. Under the null
hypothesis that changes in child location are exogenous, we would expect the proportions in
columns (1) and (2) to be small, those in columns (1) and (3) to be similar, and those in columns
(2) and (4) to be similar. For families with one or two children, the proportions in the first two
columns are statistically significantly larger than those that would be predicted by a model with
the transition rates presented in Table 11. Also, for such families, the proportion of children who
become the primary caregiver (Column 2) is statistically significantly larger than what would be
predicted by a model where location transitions are exogenous. We find similar results
corresponding to increases in the number of IADL problems. These results suggest that location
transitions may be endogenous. To further examine the nature of the endogeneity, without the need
to re-estimate all the alternative-specifications, we construct a likelihood-ratio test to decompose
the effect of location into initial (exogenous) location and contemporaneous (possibly endogenous)
location.
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 12 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Recall that our original model captures the effect of location on care provision as
+
where is a dummy variable that equals one if child lives within 10 miles of the parent in
year and is a dummy variable that equals one if child lives with the parent in year . To
further explore the possible endogeneity of location, we construct a likelihood ratio test to
decompose the effect of location into two components: an initial, presumably exogenous location
and a current, possibly endogenous component. Here we estimate a version of the Multiple
Caregiver Model that specifies the effect of location on care provision as
+ ()++ ().
Under the null hypothesis that the model is correctly specified, =, and =.
Alternatively, if location transitions have a different effect on care than does initial location, then
, and . As shown in Table 13, the likelihood-ratio test suggests that we should
reject the null hypothesis that the model is correctly specified, but the Wald tests suggest we should
not reject the null. However, the similarity of the initial and transition parameters suggests that any
endogeneity in location transitions would cause small bias (conditional on the exogeneity of initial
location). Thus, as long as initial location is exogenous, we do not find strong evidence that an
estimate associated with current location is affected by potential endogeneity issues.41 To
summarize, a) there is some evidence of endogeneity of geographic location; b) location transitions
occur infrequently enough to make it impossible to correct for bias associated caused by
endogeneity; and c) there is also evidence that any such bias is very small in magnitude.42
= = = = = = = = = = = = = = = = = = = = = = = = = = =
Insert Table 13 about here
= = = = = = = = = = = = = = = = = = = = = = = = = = =
VII. Conclusions
We contribute to the long-term care literature by developing and estimating three dynamic
models of families' care arrangements for the elderly. Our dynamic framework links care
arrangements over time by allowing for true and spurious state dependence. Controlling for
observable characteristics of the potential care recipients and the potential care providers and
allowing for several types of unobserved heterogeneity, we isolate the impact of true state
dependence. In theory, state dependence could be positive or negative, depending on the relative
importance of caregiver burden and inertia. Our results provide strong evidence of positive state
dependence in care arrangements. Thus, to the extent that caregiver burnout contributes to long-
term care arrangements, its effects are generally dominated by inertia.
In addition, our results provide important policy implications. The effects of market
conditions and public policies on the use of formal home health care and institutional care are
smaller and less statistically significant in our dynamic model than in an otherwise identical static
model. This pattern suggests that the measured effects in the dynamic model reflect flows while
those in the static model reflect a stock of present and future flows. Nevertheless, even after
allowing for state dependence, families' care decisions depend on the cost, quality, and availability
of formal modes of care.
As an early step towards understanding the dynamics of families' long-term care
arrangements, this work abstracts from the possibility that family members have different
preferences. In future work, we plan to examine the strategic aspects of long-term care decisions
in a dynamic setting by developing and estimating dynamic game-theoretic models. The
appropriate game-theoretic model depends on the dimension of care examined. For example, in
the interest of parsimony, a model of families' selection of the primary care arrangement may
abstract from the possibility of other caregivers. On the other hand, a model that allows for multiple
caregivers might focus on each family member's decision whether to provide care. A model of the
continuous dimension of care arrangements might focus on hours within each arrangement while
explaining the pattern of a primary caregiver. The models presented here provide useful insights
concerning the development of appropriate structural dynamic models.
Appendix 1. Endogenous Location
To get a sense of the magnitude of bias, even if the first location is endogenous, consider
a simple linear model,
=+ ()++
where is analogous to initial location and is location in each period = 1,2, . . , . Assume
that
= 0;
= ;
= ||;
=
||;
= ; and
= 1(=).
We allow for serial correlation in both and because they are prevalent in the data and
might affect the bias. On the other hand, we ignore the fact that there are other explanatory
variables in the model and that is really a nonlinear function of explanatory variables and
errors. Then, the OLS estimate of (,) has
=
1
1
,()
1
,()1
()²
,
•
1
,
1
()
,
=
+
1
1
,()
1
,()1
()²
,
1
,+
1
()
,+ ()
=
+
211
1
121
111
1
1
+
.
In our data, = 5 and is very large (see Table 11) which implies that 1
is very
small. Thus
221
1 0
+
5
. 8
=
22[
+
5]. 8
. 8
If 0 and/or / is small, then
.2
1
2
which implies that the bias for should be twice as large as that for . Since the difference
between them in Table 13 is small, under :=, the bias for both must be small, both in an
absolute sense and relative to the estimate in Table 13. Alternatively, if
has the same sign
as and is not that small, then the bias for becomes even smaller relative to that for , and
the same analysis holds.
To approximate the potential bias due to endogenous location, consider a simpler model,
(10) = ++
(0, ²)
where is a vector of exogenous explanatory variables and
(11) = 1 with probability 1 (1)(1)
0 with probability (1 )(1 )
with
(12) = (+).
One should think of as a dummy equal to 1 iff lives near the parent, as the probability that
lives near the parent at an early age, and as the probability that the child moves closer to the
parent. We allow to depend upon the error in equation (10) causing both and to be
endogenous.43 The μ parameter should be set so that matches transition probabilities from
far to near seen in the data. Since these transition probabilities are very small, must be a large,
negative number. If we ignore the endogeneity of and use OLS to estimate (,), then
=
¹
¹
¹
¹
=
¹
¹
¹
¹
¹(++)
¹(++
=
+
¹
¹
¹
¹
and the asymptotic bias is proportional to
=(,)
(13) = {(+) + (+)}
()
We can use a first order Taylor series approximation,
(+) + (+)
() + () + [() + ()]
and then plug it into the formula for ,
= {()+()+[()+()]}
()
= () + ()
and, for equation (13),
(,) {() + () + }
()
= {() + ()}+[() + ()]²
= [()+()]
= ²(1 )[()()].
Since is a large negative number,44 ()() is also very small. At the values of ()
consistent with the data (0.02), ()()0.03. Thus,
( ) 0.03²(1 )
which is very small. If () is small, then the bias term for is (¹)/
(¹) where the denominator is
¹
=²
{()+()+}1
()
= ²[() + ()]² + .
Plugging in our estimate of (), the bias is
0.03²(1 )
²[0.03 + 0.97]² + 3=0.03(1 )
[0.03 + 0.97]² + 3²
which should be small relative to .
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1 See Sovinsky and Stern (2016) for an overview and Mazzocco (2007) for some recent work.
2 We define positive state dependence as a situation where the probability of staying increases
with duration (in effect, inertia). Note that our definition is in contrast to the survival analysis
literature which defines state dependence as the probability of leaving (Lancaster 1990).
3 Hill (2006) performs an experiment with later waves of HRS where respondents are told how
they answered the asset questions in the last wave; this results in a significant reduction in the
variance of asset changes.
4 In addition, Long-Term Care Insurance (LTCI) may be an important determinant of care
arrangement choices. In our sample, only 10 percent of respondents have long-term care
insurance, so we did not include LTCI ownership as a characteristic. However, we performed
Lagrange Multiplier tests to determine if LTCI ownership has an impact on care choices. The
test statistics are not significant in any of the models, indicating that LTCI ownership does not
have a significant impact on long-term care choices.
5 For most respondents, the first year of data used in our analysis is 1995; for some, it is later.
6 The AHEAD data project surveys respondents aged 70 or older in the first wave from 1993.
7 Wages for home health aide workers were obtained from PHI (2007). The nursing home data
were obtained from Grabowski et al. (2004) and Harrington, Carrillo, and LaCava (2006).
8 We are using these data for all years.
9 It was not possible for us to find state Medicaid eligibility criteria for all state-year
combinations in our sample. Our data are available at http://jhr.uwpress.org/.
10 We augment the continuous choice model to allow for substitution across types of care.
11 We estimated a number of different specifications in which we experimented with different
error structures.
12 See, also, for example, Alessie, Hochguertel, and van Soest (2004) and Aguirregabiria and
Mira (2010).
13 We do not distinguish between attrition and death.
14 For all models, we use antithetic acceleration in simulation. Geweke (1988) shows that, for
maximum likelihood estimation, for a large class of models, if antithetic acceleration is
implemented during simulation, then the loss in precision is of order 1/ (where is the
number of observations), which requires no adjustment to the asymptotic covariance matrix.
15 See also Dostie and Léger (2005) for another application using Heckman (1981) specific to
dynamic family caregiving models.
16 We do not estimate parent marriage at conditional on being married at + 1 because we do
not observe enough remarriages in the data. We also do not estimate child distance at
conditional on child distance at + 1 because child migration is very rare in the data. See
section (5.3) or the "child distance tabs" in the online appendix at http://jhr.uwpress.org/ for
estimates of child mobility.
17 Let be the set of coefficients whose estimates are reported in Table 5. Note that ()
=
(). () is evaluated at 1.229, the mean value of based on the reported means in Table
1 and the parameter estimates in the first panel of Table 5. For the purposes of this exercise, we
treat discrete explanatory variables as if they were continuous.
18 The value -1.191 is the mean of based on the sample means from Table 1 and the estimates
in the second panel of Table 5.
19 See http://jhr.uwpress.org/ for point estimates.
20 In this section, our discussion focuses on the relationship between each characteristic and the
latent value of using a particular mode of care relative to the outcome where the individual does
not use that mode of care. In a separate section, we present and discuss the marginal effects
associated with the most policy-relevant variables.
21 One cannot include a variable for marital status in the spousal care value because a spouse can
provide care only if she exists; thus the MLE of such a parameter would be infinity.
22 Again, our discussion focuses on the relationship between each characteristic and the latent
value of using a particular mode of care relative to the outcome where the individual does not
use that mode of care. Later we present and discuss selected marginal effects.
23 Later in the paper, we address the potential endogeneity of geographic distance. In particular,
we test whether the role of the child's initial (exogenous) location relative to the parent differs
from the role of the child's current (potentially endogenous) location relative to the parent.
24 BGHS use information on the well-being of the parent to separately identify burnout and
caregiver quality in a static model.
25 We can generalize to allow for endogenous stochastic state variables and for state variables
that are not specific to a choice.
26 The i.i.d. EV assumption is made to simplify analysis. The basic point does not rely on it.
27 Given the iidEV assumption, the proportionality factor is
[=](1 [=]).
28 If, for example, is an absorbing state, then
=
= ((1 )/(1 )).
If = 0.95, then 20 as , and 5.3 at = 5.
29 In a structural dynamic model, the family would also consider the dynamic effect of choices
today on the value of choices in the future.
30 In the raw data, the transition rate out of formal care is 25 percent higher than for any other
care alternative.
31 CS does not distinguish between the effects of nursing home care and alternatives.
32 The marginal effects associated with the Primary Caregiver Model sum to one. As a result of
rounding error the marginal effects associated with independent living may differ from one
minus the sum of the marginal effects reported in Table 7.
33 Notice that, to obtain the impact of a simulating a $100 increase in monthly income, one can
divide by 10.
34 Wealth includes all assets. We use SSI Medicaid income limits for singles or couples,
depending on the individual's marital status. Although adding dummy variables is a common
way to handle missing values, Abbrevaya and Donald (2011) suggest that this approach may not
always provide the desired result.
35 Norton (1995) reports evidence that welfare aversion among the elderly actually leads to the
opposite pattern as individuals seek to avoid Medicaid eligibility.
36 Let be the wealth of family at time , and let be the state-specific asset limit. The
smoothed asset limit rule we apply is
=(+ 1 ) if
(+ 1 ) if .
37 We conduct a variety of Lagrange Multiplier tests to explore alternative ways to include
income and wealth. The value of these tests is that they allow us to examine the relative fit of a
variety of specifications that include income and wealth without requiring us to re-estimate the
model for each specification.
38 We include only states where there is more than one observation.
39 The methodology and the results can be found at http://jhr.uwpress.org/.
40 We exclude Oregon because the value of log (utilization of nursing homes per capita 70 years
or older) is 6 standard deviations below the mean, while all other states used are within one
standard deviation.
41 We also tested whether location was endogenous in the Primary Caregiver Model. The results
are consistent with those presented here.
42 We include more details in Appendix 1.
43 This is equivalent to adding a different error in equation (11) and allowing it to be correlated
with . There would be another scaling factor in equation (11), but we can ignore this without
loss of generality.
44 Since all that matters is the difference of the density and the distribution and the distribution is
identified in the data, the inclusion of a scaling factor would be irrelevant.
