Expectations Among the Elderly About Nursing Home Entry.
Data Sources. Waves 1 and 2 of the Assets and Health Dynamics Among the Oldest Old (AHEAD) survey.
Study Design. We model expectations about nursing home entry as a function of expectations about leaving a bequest, living at least ten years, health condition, and other observed characteristics. We use an instrumental variables and generalized least squares (IV-GLS) method based on Hausman and Taylor (1981) to obtain more efficient estimates than fixed effects, without the restrictive assumptions of random effects.
Principal Findings. Expectations about nursing home entry are reasonably close to the actual probability of nursing home entry. Most of the variables that affect actual entry also have significant effects on expectations about entry. Medicaid subsidies for nursing home care may have little effect on expectations about nursing home entry; individuals in the lowest asset quartile, who are most likely to receive these subsidies, report probabilities not significantly different from those in other quartiles. Application of the IV-GLS approach is supported by a series of specification tests.
Conclusions. We find that expectations about future nursing home entry are consistent with the characteristics of actual entrants. Underestimation of risk of nursing home entry as a reason for low levels of long-term care insurance is not supported by this analysis.
Key Words. Nursing home entry, expectations, instrumental variables, panel data
In this study we examine the expectations of the elderly about future nursing home entry. Although the characteristics of nursing home residents are well known (Norton 2000), how the elderly form expectations about future entry is not very well understood. If the elderly understand the risk factors behind nursing home entry, the presence (or absence) of chronic and time-invariant risk factors should affect expectations in the same way they affect actual entry. If the elderly do not fully understand the risk factors behind entry, the covariates that explain expectations may not be consistent with actual entry.
Understanding the expectations of the elderly about nursing home entry is important for several reasons. First, although nursing home stays are both common and expensive, relatively few elderly purchase private long-term care insurance (Norton and Newhouse 1994; Sloan and Norton 1997; Norton 2000). Expectations may explain this puzzle. In addition to the usual reasons for lack of a developed insurance market, adverse selection and moral hazard, one of the reasons long-term care insurance is rare may be that people underestimate their probability of nursing home entry. If this is true, the significance and quantitative importance of risk factors known to be associated with nursing home stays would be less important in predicting expected entry compared to actual entry.
Second, changes in expectations about nursing home care may lead to changes in financial behavior. Individuals who expect to enter a nursing home may set up trusts, or may try to transfer assets to their children to qualify for Medicaid reimbursement by avoiding asset spend-down (Taylor, Sloan, and Norton 1999). For this sort of behavior to occur, individuals must have accurate expectations about entering a nursing home.
Third, accurate expectations about nursing home entry may also indirectly improve quality of care (Nyman 1989). A competitive nursing home market depends on consumers being informed about both cost and quality of care. Individuals who expect to need nursing home care in the future are more likely to search for high quality or low cost nursing homes, or both, than individuals who suddenly need care without having foreseen the need.
The expectation measure we use is the probability the respondent places on entering a nursing home in the future. Eliciting probabilities of a discrete event that will occur some time in the future has the desirable feature that the answer should contain all information the respondent has about the future. However, in practice, there is some concern that the concept of probability may vary by individual (Dominitz and Manski 1997). Some respondents may be optimistic about the future and thus consistently underestimate the probability of adverse events and consistently overestimate the probability of favorable events. Furthermore, Walley (1991) reviews cases where individuals consistently respond in the lower and upper ends of the probability tails when faced with questions about probabilities, determining that numerical probabilities elicited in surveys may be consistently biased toward extremes. Failure to control for the unobserved heterogeneity related to expectation formulation can lead to a loss of effici ency. In addition, the unobserved heterogeneity can lead to bias when one expectation is the dependent variable and a second expectation is included as an explanatory variable. In turn, this bias could hamper our understanding of expectations and lead to erroneous policy choices.
In our study, we model expectations about entering a nursing home as a function of expectations about living beyond ten years, expectations about leaving a bequest, health shocks, and other characteristics. The explanatory variables for expectations about living and leaving a bequest may be correlated with the unobserved heterogeneity that determines expectations about nursing home use, and thus the random effect estimates may be inconsistent. Furthermore, unobserved heterogeneity may also include unmeasured differences in health status and other factors, which clearly affect expectations and may be correlated with individual characteristics such as age and gender. Although we can difference out this form of unobserved heterogeneity using fixed effects, we would lose too much information because there are only two observations per person in our data set. Monte Carlo evidence shows that even if the individual effects are uncorrelated with the explanatory variables, the number of observations per individual mu st be at least eight to justify the use of fixed effects estimation with its loss of degrees of freedom (Johnson and Skinner 1988). Hotz and Miller (1988) also caution against using fixed effects when the number of observations per individual is small. In addition, fixed effects estimates would reveal nothing about the effects of marital status, number of children, and gender--all of which are known to affect nursing home entry.
In this study, we apply an econometric method developed by Hausman and Taylor (1981) that--unlike random effects--does not require independence of unobserved and observed characteristics, allows researchers to study variables that are fixed over time, is consistent, and is more efficient than fixed effects. The Hausman-Taylor method uses deviations of time-varying variables from their means as instruments for explanatory variables that may be correlated with individual effects. The method does not require additional instruments that often are not available.
The Hausman-Taylor approach is related to other panel IV approaches that have appeared in the health services literature. The fundamental difference between the Hausman-Taylor approach and methods based solely on exclusion restrictions is that Hausman and Taylor use moment conditions to identify the parameter estimates. Specifically, if the explanatory variables are correlated with the individual effect but not the time-varying random error, they can be transformed into deviations from the mean and used as an instrument. The moment condition refers to any variable that, when measured in deviations from the mean, is uncorrelated with the individual effect. We test the validity of the moment condition using the tests developed by Hausman and Taylor. The implicit structural assumption is that deviations from the mean of the explanatory variables can be validly excluded from the main equation. Thus, the moment condition can be reinterpreted as an exclusion restriction. As should be done with usual IV methods base d on exclusion restrictions, we test whether this exclusion restriction is valid using the standard overidentification tests studied in Staiger and Stock (1997).
The Hausman-Taylor approach is more efficient than fixed effects because it uses more variation to identify the parameters. Fixed effects estimation uses within-group variation in the variables to identify all parameter estimates. In contrast, the Hausman-Taylor approach uses within-group variation to identify only the parameters associated with the variables that are correlated with the error component. The parameters of the strictly exogenous variables are identified using both within- and between-group variation in the variables. Additional efficiency is gained by using the within-group variation of all exogenous variables to identify the parameters associated with the variables correlated with the error component. The greater efficiency of the Hausman-Taylor estimate does come with a trade-off: the approach imposes more structural assumptions on the model by requiring that at least some of the exogenous variables are uncorrelated with the fixed error component. As described below, this assumption can be t ested, although in some applications the tests are of low power.
In another article that uses IV estimates in a panel framework, McClellan and Newhouse (1997) used fixed effects to difference out unobserved fixed hospital, demographic, and time effects to identify the cost-effectiveness of medical technology. Their main equation uses instruments that are interactions between hospital and time for adoption of related technologies. One fundamental difference between the approach in our article and the approach applied by McClellan and Newhouse is that they have multiple types of heterogeneity and difference them out in an initial step. The other difference is that they are not concerned with efficiency because they have an extremely large data set. We must be concerned about efficiency because we have a short panel with relatively few individuals. Our method instruments out the correlation between the included expectation variables and the individual-specific random effects.
This technique is appropriate in a variety of applications in health services research. For example, consider a model of inpatient length of stay, with repeated observations at each hospital, as a function of hospital, demographic, insurance, and market characteristics. Although fixed effects estimation would control for all time-invariant hospital characteristics, a researcher might want to study the effect of teaching status or ownership on length-of-stay decisions. Random effects estimation is likely to be inconsistent because the hospital-specific part of the error component, say unobserved hospital quality, will be correlated with hospital-specific characteristics. The technique discussed in this study enables the researcher to obtain consistent and efficient estimates of the effect of both the time-invariant variables and the time-varying variables correlated with the individual effects. Furthermore, the estimation procedure can be easily implemented in many statistical software packages with limited a dditional programming. A sample Stata program is available from the corresponding author upon request.
Consider the following equation to predict the expectation of entering a nursing home in the next five years:
Nursing [Home.sub.i] = [[beta].sub.0] + [[beta].sub.1] [Characteristics.sub.it] + [[beta].sub.2][Expectations.sub.it] +
[delta]Fixed [Characteristics.sub.i] + Fixed [Error.sub.i] + Random [Error.sub.it], (1)
where Nursing Home is the continuous expectation of entering a nursing home, Characteristics is a vector of time-varying independent variables uncorrelated with both the fixed and random error components, Expectations is a vector that includes time-varying expectations about living and leaving a bequest that may be correlated with the fixed component but not the random error, Fixed Characteristics is a time-invariant independent variable assumed to be uncorrelated with both the fixed and random error, [[beta].sub.0], [[beta].sub.1], [[beta].sub.2], and [delta] are parameters to be estimated, and i and t represent individuals and time. In contrast to Hausman and Taylor (1981), there is no correlation between the time-invariant characteristics and the fixed error components.
With only two observations per individual, a natural question is whether panel techniques are any more appropriate than OLS. We test for the presence of the fixed error component using the Breusch-Pagan (1980) Lagrange multiplier test. If the error component exists, random effects estimation will yield consistent and efficient estimates as long as the fixed error components are uncorrelated with Expectations. We can test this assumption using a specification test derived by Hausman and Taylor (1981). If the fixed error components are correlated with Expectations, fixed effects estimation yields consistent estimates of [[beta].sub.0], [[beta].sub.1], and [[beta].sub.2]. However, fixed effects models do not estimate [delta] because the time-invariant variables in Fixed Characteristics are differenced out of the model. In addition, fixed effects result in a loss of efficiency, with a loss of one degree of freedom per individual. Thus, fixed effects estimation is not appropriate if a researcher is interested in how time-invariant variables affect health outcomes or has few observations per individual.
An example of the effect of differencing follows. In our application, Characteristics includes several health conditions. For example, the variable for adverse events may represent three types of people: (1) those who have had a heart attack before the first observation in our data set; (2) those who have a heart attack between the first and second observations; and (3) those who have never had a heart attack. Fixed effects estimation treats the people in categories 1 and 3 identically. Variation in the health condition variable represents only changes in health condition from one period to the next when using fixed effects. Although the fixed effects estimate represents the effect of having a heart attack between the first and second observations, it ignores the fact that a significant number of people had a heart attack before the first observation. Such events are likely to affect expectations about nursing home entry.
Hausman and Taylor (1981) developed an alternative to both fixed and random effects estimation, which we will call the instrumental variable generalized least squares (IV-GLS) method. The IV-GLS method is more efficient than fixed effects, estimates parameters for time-invariant variables, and does not make the restrictive assumptions of random effects. The first step is to perform a GLS transformation. As with random effects estimation, the IV-GLS method requires that all of the variables be transformed prior to estimation using an estimate of [theta], which is a number between 0 and 1 representing the degree of between-group variation. Like the random effects transformation, if all the variation is within group, [theta] = 0 and the transformation is equivalent to fixed effects. If all the variation is between group, [theta] = 1 and the transformation leaves the data unchanged and is equivalent to OLS. The transformation for variables that vary over time is [[X.sup.0].sub.it] = [X.sub.it] - (1 - [theta])[X.sub.i.], where [X.sub.i.] is the mean value of Characteristics or Expectations (denoted [Z.sub.it]) for each individual, and the transformation of Fixed Characteristics (denoted [Z.sub.i]) is [[Z.sup.[theta]].sub.i] = [theta][Z.sub.i]. The only difference between the random effects transformation and the IV-GLS transformation is the way [theta] is calculated (see Hausman and Taylor 1981 for the calculation of [theta]).
After transforming the variables by [theta], the IV-GLS approach is implemented by estimating Equation 1 using two-stage least squares regression. In the first stage, the [theta]-transformed independent variable [[Expectations.sup.[theta]].sub.it] is estimated as a function of [theta]-transformed exogenous and instrumental variables. The instruments are the time-varying variables measured in deviations from the mean. Hausman and Taylor show that the time-varying variables, Characteristics and Expectations, measured in deviations from the individual's mean, are valid instruments. While Expectations is correlated with the fixed error component, the variable's deviations from the mean are uncorrelated. Recall that the fixed effect is interpreted as the way people form expectations. If this interpretation is correct, we expect that the fixed effect will shift Expectations each year by the same magnitude, and thus Expectations measured in deviations from the mean will not be correlated with the fixed effect. This is the assumption we use to identify the estimates, and it is the same assumption necessary for the consistency of fixed effects. Unlike 2SLS, which requires one or more variables to be excluded from the main equation, the IV-GLS estimates rely on this moment condition for identification.
As with traditional IV approaches, variables not included in Equation 1 can also be used as instruments, subject to the usual requirement that the instruments be correlated with Expectations but uncorrelated with the error terms. Amemiya and MaCurdy (1986) and Breusch, Mizon, and Schmidt (1989) have extended the Hausman-Taylor estimator to include more instruments based on additional moment conditions and therefore larger efficiency gains. However, the assumptions for the validity of the additional instruments are stronger than the Hausman-Taylor assumptions and cannot be used in this application because we have only two time periods. See Baltagi and KhantiAkom (1990) for a comparison of these estimators.
The validity of the instruments implied by the moment conditions and traditional instruments based solely on exclusion criteria needs to be tested. First, we test the consistency of the IV-GLS estimator using the Hausman and Taylor (1981) test statistic, which compares the IV-GLS estimates to the fixed effects estimates. Fixed effects estimates are consistent under both the null and alternative hypotheses in this test. However, the IV-GLS estimates are only consistent under the null hypothesis. If either type of instrument implied by the moment conditions is not validly excluded, the test rejects the null hypothesis because the estimates would deviate significantly from the fixed effects estimates. In addition, we use the two types of tests for overidentification described in Staiger and Stock (1997). The first test is an LM test, calculated by regressing the residuals from the second-stage IV-GLS estimate on the instruments. The (unadjusted) [R.sup.2] from this regression is then multiplied by the number of observations. This test statistic is compared to the critical value of the [[chi].sup.2] distribution with degrees of freedom equal to the number of excluded instruments less the number of endogenous variables (see Davidson and MacKinnon 1993 for more details). Staiger and Stock recommend using a second test, proposed by Basmann (1961), when the instruments are weak.
The validity of the IV-GLS estimator also requires that the instruments describe the variation in the instrumented variable, [[X.sup.[theta]].sub.2it] (Bound, Jaeger, and Baker 1995; Staiger and Stock 1997). One measure of the degree of correlation is the joint significance of the instruments in the first stage. A large F-statistic implies that the instruments are good at explaining variation in the dependent variable. A small F-statistic, say 3 or less, implies that the instruments do a poor job of explaining variation of the endogenous variable and caution should be used in interpreting the results of the estimation and the other specification tests.
Thus far we have described a method that can be used to obtain efficient and consistent estimates of [[beta].sub.2] when Expectations is correlated with the fixed error component, but not with the random error. However, in our application, Expectations may also be correlated with the random error due to simultaneous determination of expectations about leaving a bequest and nursing home entry. We test whether this type of endogeneity exists between expectations about bequests and nursing home entry. To do so, we perform the Durbin-Wu-Hausman test described in Staiger and Stock (1997). The IV-GLS estimates described above are always consistent and efficient under the null hypothesis. If the bequest variable is correlated with random error in addition to the fixed component, the IV-GLS results become inconsistent. However, consistent estimates can be obtained by estimating Equation 1 without using [Expectations.sub.it] - [Expectations.sub.i]. as instruments (see Coruwell, Schmidt, and Wyhowski 1992). This estim ator is less efficient under the null hypothesis than IV-GLS due to the loss of two instruments, but remains consistent under the alternative.
In summary, the estimation approach is as follows:
1. Apply the Breusch-Pagan test for error components to evaluate whether panel techniques are more appropriate than OLS.
2. Calculate the Hausman-Taylor test statistic to compare random effects to fixed effects estimation.
3. If the null hypothesis (random effects estimates are consistent) is rejected, estimate [theta] and transform the data using [theta].
4. Estimate Equation 1 using two-stage least squares on the [theta]-transformed data with Characteristics and Expectations measured in deviations from the mean as instruments in Equation 1.
5. Check the F-tests from the first stage and calculate both overidentification tests to assess whether the instruments explain the instrumented variable and are validly excluded.
6. Repeat the Hausman-Taylor test, replacing random effect with IV-GLS estimates. If the null hypothesis is not rejected, use IV-GLS; if it is rejected, use fixed effects.
7. Estimate Equation 1 without using [Expectations.sub.it] - [Expeclations.sub.i]. as an instrument, then perform a Durbin-Wu-Hausman test comparing the IV-GLS estimates to this estimate.
Application of these steps allows one to choose the most efficient and consistent estimator. In some applications, the explanatory variables may be uncorrelated with the fixed error component and random effects may be the preferred estimator based on the results of the test in step 2. In other applications, all of the explanatory variables maybe correlated with the fixed error component, and the Hausman-Taylor method would be rejected in favor of fixed effects based on the test in step 6. In our application, we find that some, but not all, of the explanatory variables are uncorrelated with the fixed error component so that the Hausman-Taylor method produces a consistent and efficient estimator.
We use data from the Study of Assets and Health Dynamics Among the Oldest Old (AHEAD) sponsored by the National Institute on Aging. Currently, there are two waves of data available: wave 1, completed in February 1994, and wave 2, completed in May 1996. We supplemented the AHEAD data with information on the price of an annual stay in a nursing home in each state.
The most important variables in our analysis are respondents' expectations about the future. We limited the sample to those who answered the expectation, health status, and asset questions in both years, leaving us with a balanced panel of 6,752 observations. We imputed missing values for number of children, annual nursing home cost, age mother died, and age father died using out-of-sample predictions from regressions of these variables on all other exogenous variables. We imputed about 27 percent of the values of nursing home cost, 14 percent of the values of number of children, 2.2 percent of the values for the age the mother died, and 1.7 percent of the values for the age the father died. We used wave 1 locations when merging in nursing home price because we were unable to observe where people lived in wave 2. In addition, we estimated individuals' assets in wave 2 by replacing the range elicited in the survey with the mean. The number of valid observations represent about half of the AHEAD sample due to a low response rate of the expectations, health status, and asset questions. Of the persons who answered the questions in wave 1, 80.6 percent answered the questions in wave 2; 16.1 percent remained in AHEAD but failed to answer at least one of the questions in wave 2; and the remaining 3.4 percent did not participate in the AHEAD survey because they died, because they entered a nursing home, or for other unknown reasons.
The dependent variable in the analysis is the answer to the question, "Of course nobody wants to go to a nursing home, but sometimes it becomes necessary. What do you think are the chances that you will move to a nursing home in the next five years?" The answer to this question is scaled between 0 and 100. The mean answer is 16.1 (see Table 1, which divides the reported expectation values by 100). The two other expectation questions used in the analysis are, "What do you think are the chances that you (or your husband/wife/partner) will leave a financial inheritance?" and "What do you think are the chances you will live to be at least 100, 95, 90, 85, 80?" In the latter question, the years applied to persons aged 86-90, 81-85, 76-80, 71-75, and 70 or less, respectively. The mean answer to the bequest question is 56.2 and the mean answer to the age question is 47.3. These two variables are expected to be correlated with the fixed component of the error and are thus represented by Expectations in Equation 1.
We include expectations about leaving a bequest as an explanatory variable because, not only does it represent a bequest motive, it also proxies for the relationship with children who might be potential caregivers. We expect that this expectation will be negatively correlated with expectation about nursing home entry. As we note below, we test whether this expectation is endogenous (correlated with the random error).
Besides expectation variables, the explanatory variables in our analysis are based on previous studies of actual nursing home entry. If individuals form realistic expectations about entering a nursing home, we would expect that the covariates that are significant predictors of actual entry will also describe expectations. Numerous studies of the probability of nursing home entry have been done, and demographic characteristics were found to be significant predictors of nursing home entry. Marital status is an important predictor of nursing home entry due to the ability of married partners to substitute home care for nursing home care. However, even controlling for marital status, men are less likely to enter nursing homes than women. Age also has been shown as a significant predictor of entry due possibly to a longer exposure period (Murtaugh, Kemper, and Spillman 1990). Race has been significant in many studies (see Norton 2000). Blacks seem less likely to enter a nursing home due to cultural differences, lo cational preferences, or differences in access or opportunity cost (Headen 1992). The number of children has also been shown as a significant predictor of nursing home entry (Reschovsky 1996). This appears to be due to home care provided by children as a substitute for nursing home care.
The primary determinant of nursing home entry is health status. The number of activities of daily living (ADLs) and instrumental activities of daily living (IADLs) have been consistent predictors of nursing home demand (Garber and MaCurdy 1993; Liu, McBride, and Coughlin 1994; Reschovsky 1996). Those with more troubles with these activities are more likely to enter a nursing home. Chronic risk factors of nursing home entry are incontinence, arthritis, and psychiatric problems. In addition, health conditions that have sudden onset, such as respiratory problems, strokes, hip injuries, and injuries due to falls are all correlated with actual nursing home entry. Other health conditions, such as angina, heart attacks, and cancer, may also have sudden onset but are significant predictors of other types of long-term care, such as hospice or home health.
The variables that vary over time but are uncorrelated with unobserved heterogeneity are four types of health conditions and indicator variables for asset quartiles. Nursing home--related health events include strokes, lung problems (e.g., emphysema), falls, and hip fractures. If an individual has experienced any one of these events at any time in the past the variable equals 1, and 0 otherwise. About 44 percent of individuals experienced at least one of these events. Other health events equals 1 if the individual has known cancer or heart problems (e.g., angina and heart attacks) and 0 otherwise. Approximately 43 percent of individuals had experienced at least one of these events. Chronic risk factors is a count variable that represents whether the person has experienced incontinence, arthritis, or other pain. The number of ADLs and IADLs is also a count variable. The mean number of chronic risk factors is 0.95 and the mean number of ADLs and IADLs is 0.70.
We created indicator variables for asset quartiles because we expect the effect of assets on nursing home entry to vary by quartile. This is explained by Medicaid coverage of the nursing home expenses for those with assets below a ceiling. The variables that do not vary over time are gender, age, race, marital status, education, number of children, and annual nursing home cost. Gender equals 1 if the respondent is male and 0 if female. Race equals 1 if the individual is not Caucasian and 0 if Caucasian. Marital status equals 1 if the individual is married and 0 otherwise. The sample is 41 percent male, 10 percent nonwhite, and 54 percent married. Age and education are reported in years, and annual nursing home cost is reported in thousands of dollars. The average person is 76 years old and has about 12 years of education. Age is treated as fixed for each person because everybody ages the same amount between the waves. We do include a wave 2 dummy variable in all specifications to control for this aging and o ther time-specific characteristics. The average person has 2.82 children. The average annual nursing home cost is just over $29,000 per year.
The instruments used in the analysis are all time-varying variables measured in deviations from the mean. We do not use the full set of instruments suggested by Hausman and Taylor because we only have two waves of data. However, in addition to the Hausman-Taylor instruments, we use age mother died and age father died as instruments because they are likely to be correlated with expectations about living but uncorrelated with nursing home entry. The average mother died at 74 years old and the average father died at 71 years old.
An example of how the moment conditions used to identify the estimates work in practice may be useful. Table 2 shows the correlation matrix of untransformed expectations and expectations that are measured in deviations from the mean. In the untransformed expectations, those who report a lower probability of entering a nursing home report a higher probability of leaving a bequest and a higher probability of living. We would expect this result if the formation of expectations included an individual-specific component (e.g., optimism). When we transform the expectations we difference out the individual-specific components, and the probability of living is now positively correlated with probability of entering a nursing home. In addition, when the expectations about living and leaving a bequest are transformed into deviations from the mean, they are highly correlated with the untransformed expectations about living and leaving a bequest but uncorrelated with expectations about entering a nursing home. This result , which will be tested more rigorously below, supports the use of transformed expectations as instruments because they are correlated with the instrumented variable and, by construction, uncorrelated with the individual effect.
Table 3 displays summary statistics by the level of expectations about living and by levels of one of the instruments, deviation from the mean in expectations about living. Note that the mean expectation about entering a nursing home is highest for those with a low expected probability of living and lowest for those with a high expected probability of living. As discussed above, we would expect this trend if formation of expectations included an individual-specific component (e.g., optimism). The IV estimates rely on changes in the level of expectations and other exogenous variables for identification. It is these changes that do not incorporate the effect of the fixed error component but are likely to result from meaningful differences in the environment under which expectations are formed. In addition, a clear trend exists in the variables measuring health characteristics. Those with more risk factors have a lower expected probability of living than those with a high probability of living. However, these g roups look more similar when examining across levels of expectations measured in deviations from the means.
The specification tests support the choice of IV-GLS as the preferred model for our study (see Table 4). The Breusch-Pagan test for the existence of error components rejects the null hypothesis of no error component, with a p-value of less than 0.001. This test implies that some kind of error component model (fixed effects, random effects, or IV-GLS) should be used, rather than OLS. Next we tested whether the random effects estimates were consistent using a Hausman-Taylor test. In this case, the null hypothesis is that random effects estimation is consistent. We reject the null hypothesis with a p-value of less than 0.001, implying that there is a significant difference between random and fixed effects. Because the test implies that random effects estimation is inconsistent, and fixed effects would lose too many degrees of freedom and exclude important variables from the model, the use of the IV-GLS method as an alternative to fixed or random effects is warranted.
Next we test the validity of our instruments. Recall that the instruments in our IV-GLS specification include all of the time-varying right-hand-side variables measured in deviations from the mean plus age mother died and age father died. These instruments are significant in the first-stage reduced-form regression with expectations about living as the dependent variable (p [less than] .001). The instruments are also significant in the first-stage reduced-form regression for expectations about bequests (p [less than] .001). The F-statistics from the first-stage regressions used to predict expectations about living and expectations about bequests are both greater than 400, implying extremely strong instruments.
Based on the last set of tests, the instruments are clearly correlated with the endogenous variables; however, to be valid instruments they must also be validly excluded from the main equation. The results of the tests for valid exclusions show that we are unable to reject the null hypothesis of exclusion with a p-value of 0.17 for both the residual-based and Basmann tests. The other form of overidentification test suggested by Hausman and Taylor reveals that the null hypothesis of consistency of the IV-GLS cannot be rejected (p = 0.17). These tests point to IV-GLS using the Hausman-Taylor instruments as the preferred specification.
Finally, the Durbin-Wu-Hausman test compares the IV-GLS estimates with and without expectations measured in deviations from the mean as instruments. We are unable to reject the null hypothesis of no difference between the estimates with a p-value of 0.78. This result suggests that we cannot reject the null hypothesis of exogeneity, that is, of the expectation of leaving a bequest. In addition, we estimated the model without expectations about bequests as an explanatory variable and found that the results were qualitatively identical and quantitatively similar.
Before focusing on the preferred specification (IV-GLS), we note how the parameter estimates change across specifications; these changes reinforce the rationale for preferring IV-GLS (see Table 5). First, consider the coefficient on the expectation of living ten years or more. In the OLS and random effects estimates, this coefficient is significant and very large in magnitude. In contrast, the coefficient is insignificant and small in magnitude in the fixed effects estimate. As shown in Table 4, the specification test comparing the random and fixed effects suggests that the expectation of living ten years or more is correlated with the fixed component in the equation. Once we instrument for the expectation of living in the IV-GLS equation, the correlation is removed and no longer has a significant effect on the expected probability of nursing home entry. Thus, application of the Hausman-Taylor method has an important effect on the interpretation of the expectation about living. Moreover, our confidence in th is conclusion is bolstered by the specification tests, which indicate that we have strong, valid instruments for expectations about living and leaving a bequest.
Second, the IV-GLS estimator should be more efficient than the fixed effects estimator. This prediction is confirmed by the fact that, in virtually every case, the standard errors are smaller for IV-GLS than for the fixed effects estimator. However, despite the gain in efficiency, expectations about living and making a bequest are still insignificant in the IV-GLS estimates.
Third, the results clearly illustrate the shortcomings of the fixed effects estimates for this study. Coefficients cannot be estimated for the fixed demographic variables and the annual nursing home cost. Moreover, none of the health conditions have significant effects in the fixed effects estimates.
Fourth, comparing the preferred IV-GLS estimates to the OLS and random effects estimates, most of the coefficients are similar, with the exception of the expectation variables. Apparently, these variables are not correlated with either the expectation variables or the fixed components. Thus, even if the equations are improperly specified by omitting the fixed component (OLS) or by not instrumenting the included expectation variables (OLS and random effects), the coefficients on the other variables are not affected.
The signs on all of the demographic variables are consistent with actual nursing home entry. Marital status, age, education, and the number of children are all statistically significant in the preferred IV-GLS specification. The magnitude of the parameter estimates across each regression is very similar. This suggests that these variables are uncorrelated with the fixed error component. Similarly the signs of nursing home-related adverse events, chronic risk factors, and the number of ADLs and IADLs are consistent with actual nursing home entry. Each of these parameter estimates are significant in the IV-GLS specification. In contrast to actual nursing home entry, gender and race/ethnicity do not have significant effects on expectations about entry, although the parameter coefficients have the same sign as for actual entry. Of the financial information, only annual nursing home cost is a significant predictor of expectations. The negative sign suggests that expectations are cost sensitive. It appears that ass ets do not affect expectations about nursing home entry.
Our model shows that elderly individuals do not underestimate the probability of entering a nursing home in a five-year period. Murtaugh, Kemper, and Spillman (1995) show that for a typical 75 year old in good health, the chance of entering a nursing home in the next five years is only 6 percent. For those in worse health, the probability ranges from 17 percent (major illness) to 27 percent (ADL limitation) to 44 percent (cognitive impairment). The overall probability of being admitted to a nursing home is about 12 percent for persons aged 75. We predict that a typical 75-year-old unmarried white female with 12 years of education and two children has an expectation of 12.9 percent of entering the nursing home. If that person has a chronic risk factor, the expectation increases to 14.8 percent, and if she has limitations in three ADLs the expectation increases to 16.5 percent. So we find that the healthiest may overestimate the risk, while the sickest may underestimate the risk.
Contrary to what some have proposed, underestimation of risk of nursing home entry does not seem to be a problem, at least for the elderly individuals in our sample. This does not completely rule out underestimation of risk as a cause for the lack of purchase of long-term insurance, because such insurance usually has to be bought while a person is relatively healthy and not too aged. It is possible that expectations at this stage are less accurate than for the older individuals in our sample. Still, we believe that our evidence that elderly individuals have accurate expectations about nursing home entry increases the plausibility of the hypothesis that younger individuals also have accurate expectations. The results suggest that inaccurate expectations are unlikely to be a major explanation of the lack of demand for private longterm care insurance.
Overall, we found that expectations about nursing home entry are driven by many of the same factors that affect actual nursing home entry. This result casts some doubt on the theory that systematic underprediction of the probability of entering a nursing home explains why relatively few individuals purchase long-term care insurance to cover nursing home care. Accurate formation of expectations about nursing home entry is a prerequisite for Medicaid spend-down, and the results suggest that this necessary--but not sufficient--condition holds. However, some of the other results' implications for spend-down are less clear. Individuals who expect to leave bequests are neither more nor less likely to expect to enter a nursing home than individuals who do not expect to leave bequests. And individuals in the lowest asset quartile, who are most likely to be affected by Medicaid subsidies and the possibility for spend-down, do not have a significantly different expected probability of entering a nursing home.
In this study, the IV-GLS method is preferred to other commonly used methods. The expectations about living and leaving a bequest were found not to matter in our preferred specification, but we would have concluded otherwise based on the results from the other methods. In addition, the results of this paper would be inconclusive if we relied on the relatively inefficient fixed effects estimates. The efficiency gain from the IV-GLS technique gave us the ability to test the hypotheses properly. As in any empirical exercise, the tests of the underlying assumptions are extremely important for model choice and validating assumptions. The instruments used here perform extremely well when subjected to the standard tests for overidentification and correlation with the endogenous variables.
This research was funded by the Agency for Health Care Policy and Research and the National Institute on Aging under grant no. R01-HS09515. The authors thank Mark McClellan and anonymous referees for numerous comments that served to improve the focus and clarity of the article.
Address correspondence to Richard C. Lindrooth, Ph.D., Research Assistant Professor, Institute for Health Services Research and Policy Studies, Northwestern University, 629 Noyes Street, Evanston, IL 60208-4170. Thomas J. Hoerger, Ph.D. is Senior Economist, Research Triangle Institute, Research Triangle Park, NC. Edward C. Norton, Ph.D. is Associate Professor, University of North Carolina at Chapel Hill, Chapel Hill, NC. This article, submitted to Health Services Research on November 29, 1999, was revised and accepted for publication on August 8, 2000.
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|Author:||Lindrooth, Richard C.; Hoerger, Thomas J.; Norton, Edward C.|
|Publication:||Health Services Research|
|Date:||Dec 1, 2000|
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