Will You Still Want Me Tomorrow?

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 Checkovich and Stern (2002), Brown (2006), and Byrne et al. (2009). 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 j is given in Equation 1. We assume ε_nij~iidN(0,1) and define F ^ω(·) as the joint distribution of ω. Family n uses alternative j to provide care for individual i at time t if and only if y ^·ijt > 0; in effect,

Let

where y_{ijt –1} equals one if alternative j was chosen last period, and θ = (β, γ, α, δ, λ, σ η) is the vector of parameters to estimate. Let a_it be a dummy variable indicating whether individual i is living at time t.¹³ Then, the likelihood contribution for an elderly individual i is

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-termcare papers, Heitmueller andMichaud (2006) find no significant correlation between errors in the initial and subsequent periods, while Skira (2015) finds 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 t = T_b , T_b + 1, .., 0, 1, .., T_e , but the same methodology can be used in all three models. We observe for each observation i where Z_it = (Z_i,1,t, Z_i,2,t, .., Z_i,Jn+4,t). This is an example of the classic initial data year conditions problemin that we must decide how to model the stochastic process for y_ijt at time 0.

Imagine that we know or can estimate inverse transition matrices,

and

In some cases, such as age, X_it j X_{it +1} is degenerate. In others cases such as ADL problems, we can easily estimate p_x (X_it j X_{it +1}) with AHEAD/HRS. Alsowe assume that, at T_b , y_ijt = 0, a reasonable assumption in our case since almost all respondents live independently at T_b . Then, given , we can easily simulate (working from t = 0 back to t = T_b ). The simulator may not be continuous, but that does not matter as long as does not depend on θ.

Given a draw of , we can now compute

iteratively for t = T_b , T_{b +1}, .., –1, where is a draw of u_i , is a draw of v_it , and is a draw of η_i and η_i = (η_i1, η_i2, .., η_{iJn + 4}). 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

and then

for t = T_{b +2}, T_{b +3}, ..,0.We can add either 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 T_b in the not-too-distant past where it is reasonable to assume that y_ijTb = 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 T_b = 60.

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 1.456ϕ(1.229) = 0.273 less likely than are men to have a spouse at t, conditional on not being in the sample at t + 1.¹⁷ In effect, women who die between t and t + 1 are less likely to have had a spouse at t than men who die between t and t + 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.

Based on the probit estimates in the first panel of Table 5, Figure 1 shows how inverse transition probabilities for marriage at t conditional on not being in the sample at t + 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.

Consider the results in the second panel of Table 5 for the probability of being married at t conditional on being in the sample but not married at t + 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 0.426ϕ(-1.191) = 0.084 less likely to have a spouse at t conditional on not having one at t + 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 t + 1, women are more likely than men to have been unmarried at time t. This finding is consistent with gender differences in life expectancy and age differences betweenmen 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 t + 1, black elderly individuals are less likely than white elders to be married at time t, controlling for other exogenous variables. Conditional on being unmarried at time t + 1, black and Hispanic elders are less likely than their white counterparts to be married at time t. Also, not surprisingly, conditional on being unmarried at time t + 1, having a spouse at time t depends negatively on age.

Conditional on the number of ADL problems experienced by the elderly individual at time t + 1, the estimated number of ADL problems at time t is higher for women than for men and for blacks and Hispanics than for whites. Also, as expected, the estimated number ofADLproblems at time t depends positively on age, conditional on the number of ADL problems at time t + 1. Similarly, conditional on the number of IADL problems experienced at time t + 1, the estimated number of IADL problems at time t is higher for women than for men and for blacks than whites. The estimated number of IADL problems at time t also depends positively on age, conditional on the number of IADL problems at time t + 1.

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Figure 1

Estimated Probability for Parent Married Conditional on Not in Origin

The spouse’s age, years of education, and activity limitations at time t depend on the respondent’s demographic characteristics, conditional on the spouse’s absence from the sample at time t + 1. For example, conditional on the spouse’s absence from the sample at time t + 1, on average, women have older spouses than do men at time t. The spouse’s activity limitations at time t also depend on demographic characteristics, conditional on the number of activity limitations at time t + 1. For example, conditional on the number of ADL problems experienced by the spouse at time t + 1, husbands (women’s spouses) experience fewer ADL problems than do wives.

Finally, several characteristics of the elderly individual’s children at time t depend on the elderly individual’s demographic characteristics, conditional on the presence of a child at time t + 1, the child’s employment status at time t + 1, or the child’s marital status at time t + 1. For example, conditional on the lack of a child at time t + 1, the probability of a child at time t depends negatively on the parent’s age. The probability that a child worked at time t depends negatively on the parent’s age, regardless of whether the child works at time t + 1. Conditional on working at time t + 1, the probability that the child worked at time t 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 T_b 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; Checkovich and Stern 2002), 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 Byrne et al. (2009); specifically, the findings presented in Byrne et al. (2009) 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.

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 healthcare 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, Sloan, Picone, and Hoerger 1997; Wolf, Freedman, and Soldo 1997; Checkovich and Stern 2002; Engers and Stern 2002), 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 healthcare depends negatively on the average wages of home healthcare 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 (λ_j) may capture the role of temporary health conditions unrelated to ADL or IADL limitations, while the person–choice factor loadings (δ_j) 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 nonspousal care arrangement. Thus, the results suggest that temporary health conditions unrelated to activity limitations may influence the relative attractiveness of each nonspousal 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 estimate the two effects separately, there is no information in the transition data that would allow for separate identification. ²⁴However, there are other sources of data with direct measures of burnout that might allowfor 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, 2005b; 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 α_j = 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 healthcare 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 and utility flow u(d_t , X_dt α) + ε_t(d_t ), where X_dt is a vector of exogenous state variables²⁵ and²⁶

Then the value of choosing d_t is

and, given the error distribution assumption,

Identification of α relies on covariation of X_j with frequency of choice j being chosen (for all j), and²⁷

Almost always, has the same sign as because V(d_{t +1}) is just a weighted sum of future utility flows, many of which will have d_τ = j for τ > t. In other words, if X_j α increases u( j, X_j α) at time t, then it will do so also at future times when choice j is chosen. Let be the sample analog of . The estimation procedure is going to match , with the value of ∂Pr[d_t = j]/∂SX_j implied by the model and parameters. In a dynamic model, where 0 < β < 1, some of the theoretical derivative can be captured in , while, in a static model, where β = 0, all of the theoretical value has to be explained by ∂u( j, X_j α)/∂X_j . In general, if

with multiplication factor m > 0,²⁸ then the estimate of α in the static model and the estimate in the dynamic model must satisfy

which implies

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, Sloan, Picone, and Hoerger 1997; Pezzin and Schone 1999a; Engers and Stern 2002; Pezzin, Pollak, and Schone 2007; Byrne et al. 2009). While a few studies present dynamic models (Garber and MaCurdy 1990; Börsch-Supan, Kotlikoff, and Morris 1991; Dostie and Léger 2005; Heitmueller and Michaud 2006; Gardner and Gilleskie 2012), 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, Hiedemann and Stern 1999; Engers and Stern 2002), 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 j is given in Equation 1 and ε_ijt~ iidEV. Assume the family chooses the alternative that provides the highest latent value; in effect,

where the set of care alternatives at time t is denoted S_it . Let

with parameters given by θ. The lagged care decision, y_{ijt –1} is a dummy variable equal to one if care arrangement j was the primary arrangement in the previous period. Then the likelihood contribution for family n is

where F ^ω(·) 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

where are errors simulated from their respective densities (Börsch-Supan, Kotlikoff, and Morris 1991; 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’ eldercare arrangements. Marital status per se is significantly associated with the value of informal care provided by an adult child and formal home healthcare 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 healthcare 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.

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 coresidence 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 healthcare 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 (λ_j) may capture the role of temporary health conditions unrelated toADLor IADL limitations, while the person– choice factor loading (δ_j) 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 nonspousal 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 nonspousal 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 eldercare 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 healthcare 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 healthcare. Instead, negative state dependence may reflect a tendency to rely on formal home healthcare (i) for acute rather than chronic health conditions, (ii) as a temporary measure until an informal caregiver becomes available, or (iii) 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 healthcare 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 Sloan, Picone, and Hoerger (1997), Wolf, Freedman, and Soldo (1997), Pezzin and Schone (1999b), Checkovich and Stern (2002), and Byrne et al. (2009), 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 (U.S. 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 allowfor 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 j = –2 for nursing home, j = –1 for formal care, j = 0 for care by spouse, and j > 0 for care by the jth child. Specifically, the latent value associated with the amount of time spent using the jth care arrangement is

In terms of its substitution effect, we distinguish between nursing homes and other alternatives. For alternatives other than nursing home care, j>-2, the observed continuous value of caregiving is given by and . For nursing home care, j = –2, the observed binary measure of caregiving is given by because we do not observe a continuous measure of nursing home care hours. Similar to Checkovich and Stern (2002), substitution in total care provided across alternatives is captured by and .³¹ The α_j terms capture true state dependence in caregiving where the hours in mode j depends both on whether j was chosen in the previous period (captured by the threshold value of true state dependence, α_thresh) and on the quantity of alternative j provided in the previous period (captured by the marginal value of true state dependence, α_marg). The parameters to estimate are σ_e, and .

Let

The likelihood contribution for an individual i is

where F_ω(·) denotes the joint distribution of the unobservables,

Note that, for j > -2, the likelihood contribution is a conditional tobit term, and, for j = –2, it is a conditional probit term. We simulate Equation 8 by drawing values of ω 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 bothmodes of informal care, as well as institutional care. Thus, these results suggest that temporary health conditions unrelated to ADL and IADL limitationsmay 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 noninstitutional 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 eachmode 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.

Contrary to our expectations and to the literature (for example, Checkovich and Stern 2002; Van Houtven and Norton 2004), the estimated substitution effects indicate that the quantity of each mode of noninstitutional 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 healthcare depends negatively on the average wages of home healthcare providers. The quantity of formal home healthcare 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.

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 Gardner and Gilleskie (2012) 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 Gardner and Gilleskie (2012) and Byrne et al. (2009), 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 formissing wealth data.³⁴WhileGardner and Gilleskie (2012) 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 Gardner and Gilleskie (2012), 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.

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 vPr[NHcare]/vIncome except near the relevant income limit. We did not allow for such nonlinearity, nor did Gardner and Gilleskie (2012). Furthermore, if there are higher nonresponse 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 statespecific/ 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/modespecific fixed effects. In particular, we amend the baseline model in Equation 1 to

where d_nis is a dummy variable equal to 1 if and only if parent i in family n lives in state s. Under the null hypothesis, τ_js = 0 for all choices and all states s.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 healthcare. 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: coresidence, 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 coresiding 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.

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 coresides 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 reestimate 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.

Recall that our original model captures the effect of location on care provision as

where z_{ikt 6} is a dummy variable that equals one if child k lives within 10 miles of the parent i in year t and z_{ikt 7} is a dummy variable that equals one if child k lives with the parent i in year t. 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, γ₆ = γ_6a, and γ₇ = γ_7a. Alternatively, if location transitions have a different effect on care than does initial location, then γ₆sγ_6a, and γ₇sγ_7a. 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, (i) there is some evidence of endogeneity of geographic location, (ii) location transitions occur infrequently enough to make it impossible to correct for bias associated caused by endogeneity, and (iii) there is also evidence that any such bias is very small in magnitude.⁴²