Switching regression estimates of the intergenerational persistence of consumption.
Guo, Sheng
I. INTRODUCTION
Since the seminal work of Friedman (1957) and Modigliani and
Brumberg (1954), the fact that life-cycle consumption is much smoother
than income has been established as one of the cornerstones in
macroeconomics. Economic agents are able to optimize on their life-cycle
consumption via the means of saving and borrowing. Similarly, family
dynasties may be able to optimize on their lifetime consumption across
generations through the channel of intergenerational transfers.
However, despite strong evidence of the massive intergenerational
asset transfers (Kotlikoff and Summers 1981), few studies-except perhaps
Mulligan (1997, 1999) and Waldkirch, Ng, and Cox (2004) (1)--have
explored the intergenerational dynamics of consumption. By contrast,
there is a large body of estimates on intergenerational relationships in
income or earnings, for the United States and for other countries around
the world. (2)
Investigating the intergenerational consumption relationship would
complement our knowledge of the relationship of intergenerational income
or earnings. Consumption is a more direct measure of economic well-being
than income. Furthermore, understanding the intergenerational effects of
parental financial transfers on consumption would be helpful for
sensible public policy design. If parents are transferring resources in
various forms (including financial transfers) in an optimal manner to
promote their offspring's overall well-being, then government
transfer programs that target disadvantageous individuals at specific
life stages through specific channels (e.g., education financial aids,
or food stamp programs) may replace or crowd out parental inputs without
achieving the same optimal effects on the overall well-being of these
individuals.
Broadly, our study is linked to the question of how to interpret
parental financial transfers to their adult children. (3) Financial
transfers from parents can occur in the form of inter vivos (i.e.,
between living persons) gifts, (4) or bequests. (5,6) The literature has
debated on whether post-education financial transfers are driven by
parental altruism, or by an exchange arrangement in return for services
delivered or expected to be delivered from adult children. Previous
studies have found evidence that inter vivos transfers are consistent
with both motives (Cox 1987, 1990), (7) yet bequest transfers consistent
with neither (Tomes 1981; Wilhelm 1996). McGarry (1999) showed that an
altruistic parent makes inter vivos transfers to ease his child's
liquidity constraints (therefore strongly related to her current
income), and arranges bequest transfers in response to the child's
permanent income (therefore only partially related to her current
income).
Our study is more closely linked to the view that treats bequest
receipt as a signal of access to credit markets for human capital
investments. Becker and Tomes (1986) argued that altruistic parents
leave financial bequests to children only after they have made efficient
human capital investments in their children. Under imperfect credit
markets, there are credit constrained parents who cannot self-finance
these investments without forgoing own consumption that has an
opportunity cost higher than the market interest rate. This results in a
lower consumption transmission from parents to children in constrained
families.
Besides signaling access to credit markets, financial transfers
including bequests enable a parent to guard his/her offspring against
any relative downward trending of consumption that arises from relative
downward trending of lifetime income, thus contribute to a higher degree
of persistence in consumption. From this perspective, the total welfare
cost of credit constraints goes beyond what is revealed by education
achievement or lifetime income, and the benefit of being born in a
richer family is not limited to being able to afford elite education.
This study tests the connection between credit constraint and
intergenerational consumption persistence, using bequest receipt as the
signal of constraint status for the parental households. This test has
been featured in Mulligan (1997, 1999). (8) However, compared with
Mulligan's work, we consider the possibility that the variable of
bequest receipts is error-ridden when used as the signal, which may lead
to misclassification of observations in estimation.
We employ switching regressions (SRs) under imperfect sample
separation to correct for the misclassification error. In terms of
methodology, there has been only a couple of studies related to credit
constraint in other contexts using SR with imperfect sample separation
(Garcia, Lusardi, and Ng 1997; Jappelli, Pischke, and Souleles 1998). To
our best knowledge, this is the first study to employ SR in estimating
parameters of intergenerational mobility. (9)
One traditional limitation of SR models is that the error term
under each switching regime has to be in specific classes of parametric
distributions, in particular, the normal distribution. We show that this
does not have to be the case: the SR model of two regimes, under the
Monotonicity Condition (defined in Section IV), is identified when
regime error terms exhibit any arbitrary distributions (see Appendix B).
(10)
Our SR estimates indicate that children raised in credit
constrained parental households are more likely to have consumption
levels similar to those of their parents than children from
unconstrained parental households. Constrained families on average
consume less than unconstrained families, which implies that their lower
consumption (thus lower utility) will perpetuate into future
generations. The SR model fits the data better when compared with the
simple sample splitting procedure. The SR estimates are robust to
whether expected, or actual inheritance, or other various related
variables are used for classifying constrained versus unconstrained
families. The estimates are in contrast with the prediction for
consumption from the theory, indicating the need of more work to deepen
our understanding of the determinants of the intergenerational economic
relationships.
The rest of this study is organized as follows. Section II sketches
the theory based on Becker and Tomes (1986). Section III describes the
data (especially, the two bequest variables), and presents conventional
sample splitting estimates. Section IV sets up the SR model and presents
SR results along with robustness checks. With estimates contradictory to
the theory, Section V discusses possible alternative explanations.
Section VI concludes.
II. THE ECONOMIC MODEL OF INTERGENERATIONAL MOBILITY
The estimation of intergenerational persistence of any kind of
economic status, including consumption, is through the following
regression:
(1) log [X.sub.c] = constant + [beta] log [X.sub.p] + U,
where [X.sub.c] and [X.sub.p] are measurements of some economic
variable of interest, such as consumption or earnings, for parents and
children respectively. In literature, [beta] is often labeled as the
intergenerational persistence, or the degree of intergenerational
regression toward the mean, meaning how much of the economic difference
among parents is bestowed onto their children; correspondingly, 1 -
[beta] is referred to as the intergenerational mobility. Using logarithm
of variables in Equation (1) measures the difference on the relative
rather than the absolute basis.
To interpret the size of [beta] in Equation (1), we present here a
simplified version of Becker-Tomes model assuming a perfect-foresight
economy. Suppose individuals live through two consecutive time periods:
childhood and adulthood. Each parent has exactly one offspring and the
child's childhood overlaps with the parent's adulthood. The
child has no role in human capital investment decision-making. By the
time the child grows up and starts working, the parent is assumed to
pass away.
The parent decides how to allocate his/her resources between: (1)
his/her own consumption; (2) his/her investment in his/her child's
human capital; (3) the amount of financial transfer he/she is willing to
pass onto his/her child. For the sake of simplicity, grandchildren have
no explicit role in the model. The budget constraint for the parent is:
(2a) [C.sup.p] + h + T = I;
(2b) T [greater than or equal to] 0,
where Cp is the level of parental consumption, h is the human
capital investment in his/her child, and T is the financial transfer
from parent to child. Equation (2b) excludes the possibility for the
parent to borrow against the child's future earnings, capturing the
essence of credit constraints in a simple, tractable way. In reality,
credit markets for human capital are imperfect because private education
loan repayment entails limited enforcement for creditors, or because the
private nature of information possessed by adult children in costly job
searching or choosing their work efforts makes contracting on their
future earnings difficult, (11) or because of the possibility of
"moral hazard" from parents in raising their own consumption
by borrowing and leaving substantial debts to their children.
The budget constraint for the adult child is:
(3) [C.sub.c] = (1 +R)T + [Bh.sup.v],
where R is the intergenerational rate of return on financial
assets, and B is the child's innate ability. As we normalize the
labor supply of everyone in the economy to one, the human capital
production function [Bh.sup.v] converts the investment amount and innate
ability into the outcome of the child's earnings, where 0 < v
< 1 captures the characteristic of the diminishing rate of return
from such an investment.
The parent cares about his/her own consumption as well as his/her
child's (12):
(4) [delta]/[delta] - 1 [C.sup.[delta]-1/[delta].sub.p] + [alpha]
[delta]/[delta] - 1 [C.sup.[delta]- 1/[delta].sub.c],
where [alpha](>0) captures the degree of altruism of parent to
child. [delta](>0) is the elasticity of intergenerational consumption
substitution. The parent's optimization problem is to maximize
Equation (4) subject to Equations (2a), (2b), and (3).
Let [DELTA] = 1 if the borrowing constraint Equation (2b) is not
binding (hence the parent transfers some assets to the child), and let
[DELTA] = 0 if otherwise (hence the parent makes no transfer of assets
to the child). If [DELTA] = 1, the efficient human capital investment
amount is solved by equalizing the rate of return between human capital
and nonhuman capital investment,
[vBh.sup.v-1] = 1 + R,
therefore
[h.sup.*] = [(vB/1 + R).sup.1/1-v]
It follows that the threshold income for a family to be
unconstrained, [I.sub.0], can be computed as:
(5) [I.sub.0] = [h.sup.*] [1 + [([alpha]v).sup.-[delta]]
[B.sup.1-[delta]]/[h.sup.*](1-[delta])(1-v)].
Therefore, the function for the indicator ([DELTA]) of being
unconstrained is
(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Moreover, the amount of asset transfer from parent to child when
the family is unconstrained can be solved out and expressed as:
T = I - [h.sup.*] - (1 + R)] ([h.sup.*]/v) [([alpha](1 +
R)).sup.-[delta]]/ 1+(1 + R)[[[alpha](1 + R)].sup.-[delta]].
We solve for the consumption persistence equations for both
constrained and unconstrained families:
(8a)
log [C.sub.c] = log [C.sub.p] + [delta] (log a + log (1 + R)) if
[DELTA] = 1;
(8b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
if [DELTA] = 0,
which suggests a system of regression equations for the consumption
of these two types of families:
(9a) log [C.sub.c] = [[beta].sub.1] log [C.sub.p] + [U.sub.1], if
[DELTA] = 1;
(9b) log [C.sub.c] = [[beta].sub.0] log [C.sub.p] + [U.sub.0] if
[DELTA] = 0.
As 0 < v/v + (1 - v)[delta] < 1, the model predicts
[[beta].sub.1] > [[beta].sub.0] in Equations (9a) and (9b). It is
helpful to understand Equations (9a) and (9b) with the patterns of
intergenerational earnings mobility in mind. For unconstrained (richer)
families, there is more often a downward regression toward the mean in
the earnings of their children; for constrained (poorer) families, there
is more often an upward regression. In the case of a downward regression
in children's earnings, unconstrained parents could bequeath assets
to offset the otherwise implied downward regression in their
children's consumption, and constrained parents could not afford to
do so. In the case of an upward regression in children's earnings,
which would lead to an upward regression in their consumption without
the need of asset transfers, the fact that constrained parents are
unable to borrow against their children's earnings implies that the
upward regression of consumption of their children goes unfettered. To
summarize, the absence of borrowing constraint slows down the degree of
regression toward the mean for intergenerational consumption, whereas
the existence of borrowing constraint prevents such a slowdown to occur.
Thus, [[beta].sub.1] > [[beta].sub.0].
This empirical prediction on consumption ([[beta].sub.1] >
[[beta].sup.0]) is preserved when human capital investments are risky
and this risk cannot be hedged away in financial markets, or there are
heterogeneities in [alpha], B, or v that are not systematically
correlated with family income (Mulligan 1997, 1999). The prediction
would also be preserved in an otherwise identical, two-period model
(with a working period and a retirement period), as long as the
consumption in question is measured for the working period. This is
because the within-period marginal utility of both the parent and the
child is equalized intergenerationally, when a positive asset transfer
occurs. (13) However, it is unclear whether the prediction would be
affected, if the number of children is endogenously determined, or if
assortative mating existing in the marriage market is taken into
consideration.
Although it is not a focus of this study, one may be concerned that
the error terms in Equations (9a) and (9b), which are correlated to I
and hence [C.sub.p] as implied by Equation (6), produce selection bias.
Han and Mulligan (2001) quantitatively investigated this issue for a
variety of numerical values of [delta], and found that this selection
bias does not affect the relative magnitudes of [[beta].sub.1] and
[[beta].sub.0], except when the [delta] is close to 0, then
[[beta].sub.1] and [[beta].sub.0] become difficult to distinguish from
each other. The results of this study show that [[beta].sub.1] and
[[beta].sub.0] are indeed quantitatively and statistically different
from each other.
III. DATA AND SIMPLE SAMPLE SPLITTING ESTIMATES
To estimate Equations (9a) and (9b), we need parents' and
children's consumption at comparable ages and an indicator of
bequest transfer from parents to children. In addition, information on
relevant sociodemographic characteristics is needed in order to hold
these sociodemographic factors constant in the regressions. Mulligan
(1997, 1999) tested the implications from Becker-Tomes model on a sample
of 1781 parent-child pairs from the panel study of income dynamics
(PSID), a longitudinal survey of U.S. individuals and their families.
Starting in 1968, households in PSID were interviewed annually through
1997, and since then were interviewed biannually. When children grew up
and left home to form their own households, these "child
split-off" households were also tracked in subsequent interviews.
We use exactly the same sample as is in Mulligan (1997, 1999) for
comparison of results. In this intergenerational sample, parents were
surveyed in 1968-1972 and adult children were surveyed in 1984-1989 at
comparable ages. Adult children already participated in the job market
by the time of survey. Consumption is constructed as the weighted
average of a household's expenditures on food at home, food away
from home, rent, and the value of the family's house. (14) We refer
readers to Mulligan (1997, 1999) for detailed description of the sample
selection and other aspects of the sample data.
For our purposes, we describe in detail below two types of bequest
receipt variables--expected versus actual inheritances--available in the
PSID data, each plagued with its own source of measurement error.
Gaviria (2002) reported the disparity in estimates of earnings
persistence when he used these two variables to split a PSID sample into
unconstrained and constrained groups. (15) Similarly, we find such a
disparity in estimates of consumption persistence when using these two
inheritance variables, which motivates the adoption of SR framework.
A. Expected Inheritances
In 1984, PSID respondents were asked whether they had received any
inheritances up to 1984,
(k150) Now we are interested in where does people's assets
come from. Have you (or anyone in your family living there) ever
inherited any money or property?
as well as how much they expected to receive in the future (16):
(k157) What about future inheritances--are you fairly sure that you
(or someone in your family living there) will inherit some money or
property in the next ten years (emphasis added)?
As only 9% of adult children in the sample did actually receive any
inheritances at some point prior to 1984, for the sake of convenience we
shall label the constructed variable from these two questions as the
expected inheritance. This variable was used in Mulligan (1997) to
classify the original parental households of 1968-1972 into constrained
versus unconstrained groups. The justification for splitting up the
sample by expected inheritances is that children who expected sizable
inheritances from parents were unlikely to have had difficulty obtaining
financial support for schooling, quality health care, and other forms of
human capital investment. (17) Specifically, Mulligan used a fixed
cut-off value of $25,000 for expected inheritances to split the sample:
those who expected to receive more than $25,000 are from unconstrained
families, and those who did not are from constrained families. (18,19)
Based upon the expected inheritance survey questions mentioned earlier,
if an interviewee was fairly sure that his/her wealthy parents would
leave him/her a sizable bequest, but not sure that they would pass away
in the next 10 years, he/she would choose to answer "no"
instead of "yes." In addition, the expected inheritance survey
questions are not clear on whether gifts are supposed to be included.
This sort of response error originates from the ambiguity in how
respondents had interpreted the survey question. (20) With these caveats
in mind, we examine another piece of inheritance information-actual
inheritances/gifts-from the same database.
B. Actual Inheritances/Gifts
In 1984-1999 (once in every 5 years), and in 2001 and 2003,
retrospective, follow-up questions regarding actual inheritances and
gifts (21) received are introduced in the survey. In the PSID 1989
survey, the question of actual inheritances and gifts posed to the
respondent is: (22)
(G228) Some people's assets come from gifts and inheritances.
During the last five years, have you (or anyone in your family
living there) received any large gifts or inheritances of money or
property worth $ 10,000 or more?
We use the sum of inflation-adjusted actual financial transfers
received over the years up to
2003 to divide the observations into the unconstrained versus
constrained group, adopting the same threshold value $25,000. (23) About
79.1% of these adult children have received zero or have missing values
up to 2003. Figure 1 plots the distribution density of financial
transfers received by those grown children who have received positive
inheritances/gifts, from which we observe that $25,000 is near the mode
and mean of the distribution. Table 1 shows that a majority of adult
children in the sample have neither anticipated nor actually received
inheritances/gifts over the period of 1984-2003, and the proportion of
those with actual inheritances/gifts more than $25,000 is below 10%.
[FIGURE 1 OMITTED]
This actual inheritance variable has its own caveats. One is the
attrition. Although each year the attrition rate of the PSID sample is
fairly small (<5%) over the years, many cases of missing values have
accumulated for actual inheritance/gift variable. Attrition affects the
classification of an observation, for we code these attrition cases as
if their actually received inheritances/gifts are less than $25,000,
which is not necessarily true. The misclassification resulting from this
is analogous to the response error associated with expected inheritance.
We examined whether attrition causes systematic discrepancy of some of
the relevant variables for observations that have attrited in later
years, as opposed to the ones that have not, by conducting
Wilcoxon-Mann-Whitney tests. We found that observations from families
with low consumption, with sons, and with single parents are more likely
to disappear over the years, which favors our treatment of observations
with missing values of inheritances to be more likely in the constrained
group. (24)
The other caveat is that actual inheritances/gifts may contain
financial surprises. Some parents happened to experience financial
windfalls at later ages (25); in such cases, the actual inheritance/gift
would diverge from what parents earlier intended to bequeath to
children. This has similar impacts on estimation as those from
respondents misreporting their expected inheritance. In any case, we
certainly cannot rule out the sorts of aforementioned measurement error
embedded in the variable.
Table 2 presents summary statistics for groups split by both
inheritance variables. Parents' ages are statistically, but not
economically different between the subsample of sizable
inheritances/gifts and the other. In families where adult children
expected to or have received sizable inheritances/gifts, they enjoyed
higher income, higher consumption, and more schooling years, and their
parents also enjoyed higher levels of consumption and income.
C. Simple Sample Splitting Estimates
Based upon the binary variable constructed from expected
inheritances ([D.sub.e] = 1 if a child expected a total inheritance
amount of more than $25,000; [D.sub.e] = 0 if otherwise). Mulligan
(1997) estimated Equations (9a) and (9b) for the subsample of [D.sub.e]
= 1 versus [D.sub.e] = 0 directly, the procedure we name as the
"simple sample splitting" to differentiate from SR that will
be considered later. Children's household consumption is the
dependent variable. Parental household consumption is the primary
independent variable of interest, with covariates controlling for
life-cycle effects. (26) These covariates include the child's
gender, the parental household head's and the adult children's
age quadratics, their marriage status for the period when parents and
adult children are respectively observed. For the sake of comparison, we
follow his choice of covariates in our SR estimation. (27)
The main finding from Mulligan (1997) is that the unconstrained
families do not seem to exhibit a higher degree of consumption
persistence. In fact, if anything, the unconstrained families have a
lower degree of persistence in consumption than the constrained ones
([[??].sub.1] = 0.45 vs. [[??].sub.0] = 0.55, see the two rightmost
columns in Table 3), contrary to the prediction of our theoretical
model. However, we obtain the opposite results when turning to the
variable of actual inheritances/gifts to split the sample. Table 4
presents linear regression estimates from splitting the sample according
to this variable ([D.sub.a] = 1 if a child has actually received more
than $25,000 inheritances/gifts; [D.sub.a] = 0 if otherwise). Now we
obtain something in line with the theory: we find [[??].sub.1] = 0.63
for those likely to be unconstrained ([D.sub.a] = 1) as opposed to
[[??].sub.0] = 0.52 for those likely to be constrained ([D.sub.a] = 0).
(28)
IV. SR ESTIMATES AND ROBUSTNESS CHECKS
Therefore, the estimates of intergenerational mobility of
consumption are sensitive to the variable, the expected or actual
inheritance, which is used for classification. Guided by the theory,
expected inheritances seem to be a better measure than actual
inheritances to be used in the empirical test, for actual inheritances
may be affected by parental later-life market luck that is less
associated with earlier credit availability for investing in children.
This justification, however, can be overshadowed, if the mis-measurement
caused by the response error in expected inheritances is severe enough.
A sound empirical approach is called upon to explicitly address the
embedded measurement error.
Statistically, the insignificance between estimated [[??].sub.1]
and [[??].sub.0] can possibly be attributed to the attenuation bias
caused by the misclassification error. Interested readers can refer to
Appendix A that proves how the attenuation bias is generated by
classical errors-in-variables and by setting arbitrary cutoffs to divide
the sample.
A. SR Estimates
Adopt the notation
(10) Pr([[DELTA].sub.i] = 1|[D.sub.i] = 1) = [p.sub.1],
Pr([[DELTA].sub.i] = 0|[D.sub.i] = 0) = [p.sub.0],
where [[DELTA].sub.i] is the true underlying indicator and
[D.sub.i] is the observed indicator with misclassification error. The
simple sample splitting estimates are only consistent when [p.sub.1] =
[p.sub.0] = 1.
If Equation (10) is parameterized into
(11) Pr([[DELTA].sub.i] = 1|[D.sub.i]) = F ([[gamma].sub.0] +
[[gamma].sub.1][D.sub.i]),
where g = ([[gamma].sub.0], [[gamma].sub.1]) is the vector of
parameters, then Equation (11) is called the switching equation in the
context of the SR framework and can be viewed as the probability
equation of predicting [[DELTA].sub.i] from the knowledge of [D.sub.i].
(29)
With misclassification, 0 < [p.sub.k] < 1 (k = 0,1), the
likelihood function derived from Equation (9) will be
(12a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
(12b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The identification of parameters in the likelihood function
Equations (12a) and (12b) requires: (1) ([U.sub.1], [U.sub.0]) (called
"regime error terms") in Equations (9a) and (9b) belong to a
specific family of distributions whose finite-mixture can be identified
up to subscripts, notably normal distributions (Yakowitz and Spragins
1968); (2) [p.sub.1] + [p.sub.0] > 1 (named as the Monotonicity
Condition following Hausman, Abrevaya, and Scott-Morton 1998), (30)
namely, relying on the imperfect proxy D is better than without it to
predict [DELTA], a condition already implicitly present in the cited
literature in Section
I. As shown in Appendix B, the Monotonicity Condition helps anchor
the interpretation of subscripts, and thus completes the identification
of parameters in the model. In this study, the Monotonicity Condition
stipulates that those with a larger size of inheritance/gift are more
likely to be in the unconstrained group of dynasties, an assumption
inherited from the Becker-Tomes model.
Formally, following the literature (Kiefer 1978, 1979; Lee and
Porter 1984; Quandt 1972; Quandt and Ramsey 1978), we define the
switching regression (SR) model as follows:
DEFINITION 1. The system of two-regime Equations (2a) and (2b),
along with the misclassification errors defined in Equation (10), is a
switching regression model, if:
1. The possibility of misclassification is non-trivial-0 <
[p.sub.k] < 1 (k = 0, 1);
2. The Monotonicity Condition holds--[p.sub.1] + [P.sub.0] > 1;
3. The regime error terms (U,, U0) follow one of the finite-mixture
identifiable distributions.
We proved, in Appendix B, that the last assumption in the
definition above can be relaxed, in that [U.sub.1] and [U.sub.0] can
follow any arbitrary distributions and the model is still identified.
This proof relies on the finding in Ferguson (1983) that any arbitrary
distribution on the real line can be indefinitely approximated by a
mixture of a countable number of normal distributions. This extension of
identifiability of finite-mixture models is of particular practical
interest, for the consistency of the maximum likelihood estimator (MLE)
hinges critically on the correct specification of the distributions of
error terms. Our identification result ensures that under our specific
assumptions, if the distributions of error terms are misspecified, it is
very likely that the MLE algorithm will not converge or yield sensible
estimates. Therefore, one can adjust the number of mixtures upwards or
downwards for [U.sub.1] or [U.sub.0] until obtaining the best fit of the
data. (31) The practical procedure for implementation of SR estimation
is relegated to Appendix C. (32)
Using the same intergenerational sample from PSID, the SR estimates
shown in Table 3 differ remarkably from those if the constructed
indicator D is used directly. According to our theoretical model,
children anticipating sizable inheritance receipts are more likely to be
in unconstrained families. However, our estimates indicate that
constrained families have a higher consumption persistence rate of 1.05
as opposed to 0.44 for unconstrained families, larger than the previous
conventional estimates. Moreover, the difference is statistically
significant. The coefficient for the unconstrained families is almost
identical compared with that in the sample splitting ordinary least
squares (OLS), for the majority of the population is unconstrained based
upon our estimation.
Meanwhile, the interpretation of Pr([DELTA] = 1|D = 1) - Pr([DELTA]
= 1|D = 0) reveals that the families whose children expect more
inheritance are 7.4% more likely to be unconstrained than the others.
The evidence taken as a whole suggests that those unconstrained families
comprise over 80% of the population, which, surprisingly, is fairly
close to Jappelli's (1990) findings that 19% of families are
rationed in the credit market from directly observed data. (33) We
caution that this interpretation holds only if we still regard
intergenerational transfer as the indicator of credit constraints.
Now, we turn to the SR estimation employing the actual
inheritance/gift splitting indicator. Table 4 presents results both from
linear regressions of simple sample splitting and from SR ([D.sub.a] = 1
if a child received more than $25,000 inheritances/gifts; [D.sub.a] = 0
if otherwise). In contrast to simple sample splitting estimates, the SR
estimates are almost identical to the ones using expected inheritance
splitting indicator: 0.44 for unconstrained and 1.02 for constrained.
Without receiving sizable inheritances/gifts, the family will be
unconstrained with probability .84; for families receiving sizable
inheritances/gifts, this probability increases to .93.
It may be useful to plot against each other the raw data of
children's consumption, the simulated data of children's
consumption based upon SR parameters, and the predicted values of
children's consumption based upon simple splitting estimates. These
are presented in Figures 2 and 3. For each parent-child observation, we
take the values of covariates, including the value of one of the
inheritance indicators, as are given in the data, and generate
consumption value according to our estimated coefficients and estimated
distributions of the random errors. The resulting two figures show that
the simulated data from SR fit raw data better than those from sample
splitting OLS, especially in capturing the tails of the distribution. In
general, we believe that our SR estimates, in its precision and the fit
of data, represent an improvement over those in Mulligan (1997, 1999).
B. Robustness Checks
We performed a number of robustness checks. First, as SR does not
treat each observation as definitely in one underlying group or the
other, less sensitivity should be observed by arbitrarily choosing a
threshold value (such as $25,000) in SR estimates than in simple sample
splitting estimates. We checked this aspect of robustness by looking
into actual inheritances/gifts, for more non-missing, continuous values
are available in actual inheritances/gifts than in expected
inheritances. Table 5 shows the results. The cutoffs for actual
inheritances/gifts are varied from $0 to $50,000 to see how the
estimates would be affected. The most contrasting simple sample
splitting estimates among all thresholds are the ones at the threshold
of $40,000: 0.69 versus 0.52. However, SR shows less sensitivity in
estimates from varying threshold values: for unconstrained, it is always
around 0.44; and for constrained ones, it is always around 1.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
Second, we put the logarithm of actual inheritances/gifts into the
switching equation of the regression. In Tables 3 and 4, the coefficient
of the bequest receipt indicator is not statistically significant. This
may be attributed to less variation in the dummy variable resulting from
construction. In addition, a level of transfers from a particular
household that signals a non-binding borrowing constraint does not
necessarily indicate a non-binding borrowing constraint for another
household. A uniform threshold across households with different levels
of bequest receipts is not reflective of this difference. Direct
deployment of the bequest receipt amount in a regression thus helps
address this concern. We chose the actual inheritances/gifts variable
for this exercise, for it has fewer bracketed or zero/missing values
than the expected inheritance variable. Table 6 presents the results.
The persistence rate of consumption for the unconstrained, (0.44) and
for the constrained (0.98) is not much different from the estimates
obtained before. The coefficient for the continuous actual inheritance
variable is negative and now statistically significant at 5% level. If
both the dummy variable of expected inheritances and the logarithm of
actual inheritances/gifts are put into the switching equation (not
shown), the coefficient for the latter is still negative and significant
at the 10% level.
Third, one may be concerned about the effects of the attrition of
adult children from PSID surveys on our results. A total of 221 adult
children of our sample were not observed in the 1994 survey; 753 were
not observed in 2003. We estimated the same SR model on the sample
exclusive of all these 753 observations. The persistence rate of
consumption for the unconstrained is 0.45 and for the constrained is
1.17, the latter slightly greater than the benchmark. The coefficient
for the actual inheritance variable is again negative and statistically
significant at the 5% level.
Fourth, we conducted the SR analysis for various subsamples of our
data set. We estimated it on the subset of families wherein fathers are
present in 1967-1971, the subset of sons only, the subset excluding
Survey of Economic Opportunity (SEO) observations, (34) and the subset
of families without parents cohabitation change in 1967-1971. For these
subsamples, the estimates for the unconstrained families range from 0.45
to 0.49 and for the constrained families from 0.98 to 1.33.
Qualitatively, these results do not change the conclusion derived from
the entire sample.
Last, we repeated the SR analysis by invoking a richer set of
indicators in the switching equation. These indicators are determinants
rather than direct measures of intergenerational transfers. Given that
parental altruism and children's ability are not observed and are
not controlled for by using these determinants, SR results using these
indicators as switching variables may suffer from omitted variable
biases. (35) Table 7 presents the sets of included variables and their
associated results. We still obtain a higher consumption persistence
rate for constrained families than for unconstrained families, although
the magnitude for constrained families has dropped somewhat (as low as
0.76 in one case). Nonetheless, all switching variables predict the
probability of being in one group versus the other as what we would
expect: parents with adequate savings, owning one or more cars, or the
mother having a college degree are more likely to be unconstrained;
parental households with a nonwhite head, a head aged 50 and above, or
more children in schools are more likely to be in the constrained group.
V. DISCUSSION
Recall that our total consumption is a predicted measure out of a
few individual components. The variance of predicted consumption is less
than the true variance, which may generate upward bias when predicted
consumption is used as one of the regressors. (36) The [R.sup.2] for
this consumption prediction regression is .724 (Skinner 1987). A
back-of-envelope calculation, starting from our estimates of 0.44 and
1.02, yields 0.32 versus 0.72 after taking the [R.sup.2] for this
consumption prediction regression into consideration. The 0.32 means
that only 3.3% ([approximately equal to] 0.32 (3)) of difference in
consumption between two great-grandparent households is predictably
transmitted to their descendants of current generation (any other
difference in current generation will be attributed to unpredicted
"shocks" that have occurred during this time period); for the
coefficient .72, this percentage is 37.3% ([approximately equal to].72
(3)).
Why are our estimates contradictory to the theoretical predictions,
that is, 0.32 is related to unconstrained families instead, whereas 0.72
is related to constrained families? We examine several alternative
interpretations of this finding under the Becker-Tomes framework.
First, could it be caused by any kind of unobservable heterogeneity
in data, especially, the parents' preference? Mulligan (1997)
argued that if the gap of intergenerational mobility between the
unconstrained and constrained groups is to be eliminated, the parental
altruism has to be somehow negatively correlated with parental
resources. If parental altruism is merely randomly heterogeneous, Han
and Mulligan (2001, Figure 5) provided simulation evidence showing that
the persistence rate for constrained families may overtake that for
unconstrained ones with a tiny margin, and the degree of
intergenerational substitution elasticity for consumption, at the same
time, has to be sufficiently small. Their quantitative evidence is not
remotely adequate to account for the difference as large as 0.32 versus
0.72.
Second, could it be caused by the fact that unconstrained parents
spend a smaller fraction of consumption on foods? It is well known that
the share of consumption on food will decline when income is increased.
Our consumption measure is constructed not only by food expenditure;
however, food expenditure is an essential component. For simplicity,
suppose consumption is predicted from food expenditure alone. Let
[f.sub.i,t] be the food expenditure for the family i in generation t,
[tau] the average food expenditure share of total consumption (which is
presumably less for unconstrained families), and [[xi].sub.i,t] the
idiosyncratic part of food expenditure share, then
[f.sub.i,t] = [tau][[xi].sub.i,t]
the logarithmic version of which is
log [f.sub.i,t] = log [tau] + log [[xi].sub.i,t] + log [C.sub.i,t].
Now, if the prediction based on log [f.sub.i,t] rather than log
[C.sub.i,t] itself, is directly used in intergenerational consumption
persistence regressions, what matters is the variance and co-variance of
log %l t within each of unconstrained and constrained groups, not the
relative magnitude of x between these two groups. In other words, it has
more to do with the variation in food shares within each group, rather
than the level of food shares. More evidence is needed to consider this
possibility. (37)
Third, could it be caused by failure of the Monotonicity Condition?
Is it possible that those who received sizable inheritances/gifts are
actually children of parents who were once borrowing constrained? These
parents could not have spent more on their children early on, and chose
sizable bequests/gifts later to compensate for their disadvantaged
children. If so, actual inheritances should be positively correlated
with measures of adult children's economic or financial needs,
conditional on inheritances they had already expected to receive. For
suggestive evidence, we refer to Table 8. This table presents the
correlations of actual inheritances/gifts with measures of adult
children's economic well-being, (38) conditional on parental income
and the expected inheritance dummy variable. (39)
As shown in Table 8, the size of actual inheritances/gifts is
strongly positively correlated with children's education and
wealth, conditional on expected inheritances and parental income. Owning
a house is often associated with various financial and liquidity
advantages (Cooper 2013; Engelhardt 1996; Robst, Deitz, and McGoldrick
1999; Sheiner 1995); however, homeowners among adult children received
more inheritances than non-homeowners. Similarly, having more kids to
raise demands more financial resources, but adult children with more
kids received less transfers from their own parents. All these evidences
suggest that compensation motive, if there is any, is of secondary
effect and cannot reverse the signaling power of bequest receipts as an
indicator of parents not being borrowing constrained, that is, our
original interpretation of the Monotonicity Condition.
Last, could it be because earnings are more persistent in
constrained families, and that consumption simply tracks each
generation's earnings without intergenerational linkage? To address
this concern, we conducted similar analysis for earnings and wages. The
SR model failed to detect the existence of two groups from our data of
earnings or wages. (40) The algorithm either never converged, or, even
if it converged, the estimated coefficients for the two groups bore
little difference. This is so even when we included the same sets of
variables in the switching equation as those we have used for the
consumption persistence estimation, or when we experimented with
non-normal distributions for the error terms. Thus, our results on
consumption cannot be attributed to consumption simply tracking
generational earnings or wages.
In this regard, Waldkirch, Ng, and Cox (2004) estimated a
structural model that assumes that the consumption of each generation is
a function of its own permanent income, but there will be correlations
of consumption between parents and children even after parsing out the
effects of own income on consumption. Interestingly, they found that,
for families whose adult children did not receive financial transfers
from parents or did not receive financial help towards the down payment
of a house--would be "constrained" families according to our
definition--the estimated transmission degree of this residual
consumption is higher than that for the entire sample (0.55-0.61 vs.
0.45) (Waldkirch, Ng, and Cox 2004, Table 5, p. 373). (41) This goes in
the same direction as what has been shown for the consumption
persistence from our sample, and one suspects that the difference would
be even larger if SR is called upon to correct for possible noise in the
transfers data they use.
VI. CONCLUSION
This study applies SR to estimate the intergenerational consumption
persistence for credit constrained and unconstrained families, in order
to test a related implication of the Becker-Tomes model. Our focus is
the issue that, if a family's access to credit markets for
children's human capital investments--as signaled by bequest
receipt--is imperfectly measured and contains misclassification error,
simple sample splitting regression estimates will suffer from
attenuation bias. By employing SR to account for this misclassification
error, our estimates reveal that the intergenerational consumption
persistence is higher for credit constrained families than that for
unconstrained families. This result indicates that adult children of
credit constrained families are more likely to have consumption levels
similar to their parents' than are children of unconstrained
families. Estimates from SR fit data better and are robust over a number
of various specifications.
Under the settings of our model, we relax the parametric
assumptions for identification, which are often imposed by the
traditional SR literature, although for our data, the normal
distribution turns out to fit well for each underlying group. Our
approach does require, however, that the misclassification error in
[D.sub.i] is uncorrelated with error terms in each of the two regime
equations. This assumption appears innocuous given our focus on data
issues and we have not been alarmed to evidence suggesting the
otherwise. This assumption can be relaxed if additional instrumental
variables for the true status (Lewbel 2007; Mahajan 2006) or
independently repeated measurements of the true status (Hu 2010) are
available. (42)
Hence, why are our estimates contradictory to the theoretical
predictions? We have discussed several possible explanations that may
address this discrepancy between evidence and theory; however, none of
them appears particularly attractive. More future research is needed to
address this discrepancy.
ABBREVIATIONS
CDF: Cumulative Distribution Function
CEX: Consumer Expenditure Survey
MLE: Maximum Likelihood Estimator
PSID: Panel Study of Income Dynamics
SEO: Survey of Economic Opportunity
SR: Switching Regression
doi: 10.1111/ecin.12094
Online Early publication May 1, 2014
APPENDIX
APPENDIX A: THE MISCLASSIFICATION RESULTING FROM MEASUREMENT ERROR
This section demonstrates the misclassification caused by
measurement error (as a result of response error, for example) in
inheritance (expected or actual), T. In our definition of unconstrained
families, positive inheritance is a one-to-one mapping to the
unconstrained status in intergenerational investment for a particular
observation indexed i
(A1) Pr ([[DELTA].sub.i] = 1|[T.sub.i]) > 0) = 1.
Suppose instead of observing T, we observe an error-ridden variable
[T.sup.*] = T - [epsilon], where [epsilon] is variation free of T. As
[T.sup.*] cannot be negative in our setting, [epsilon] [less than or
equal to] T. Therefore for a particular value of T, the distribution of
e is a truncated one, the probability density function of which is
denoted by [f.sub. [epsilon]|[epsilon] [less than or equal to] T](*).
Using [T.sup.*] instead of T to classify gives us
(A2) Pr (A, = I \[T.sup.*] > 0) .
Using the dummy indicator D to represent the constraint status by
employing [T.sup.*], for any particular [[epsilon].sub.i], we have
(A3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [F.sub.T](T) is the CDF of T. As [[epsilon].sub.i] is
unobservable, we integrate over its support for those
with [T.sub.i]
(A4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
whose value is between 0 and 1 under regular assumptions about the
distributions of [F.sub.T](.) and [F.sub.[epsilon]](.).
We integrate (A4) over [T.sub.i] for the subsample [D.sub.i] = 1,
as [T.sub.i] is not directly observed:
(A5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
which is still between 0 and 1. This means the subsample of
[D.sub.i] = 1 will be a mixed group including both [D.sub.i] = 1 and
[D.sub.i] = 0 observations. Lee and Porter (1984) have proved that such
a misclassification will lead to attenuation bias in estimated
[[beta].sub.1].
Studies on liquidity constraints (Mulligan 1997; Runkle 1991;
Zeldes 1989) arbitrarily specify a positive cut-off value instead of 0.
Therefore instead of Equation (A2), we have
(A6) Pr([[DELTA].sub.i] = 1|[T.sup.*.sub.i] > [bar.T])
where [bar.T] is some positive number. Correspondingly, Equation
(A4) now becomes
(A7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
and
(A8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
It is easy to prove that
(dp ([bar.T]; [T.sub.i]))/d[bar.T]) [greater than or equal to] 0.
Thus, when the threshold is lifted, we should expect the subsample
[D.sub.i] as defined to enclose more and more genuinely [[DELTA].sub.i]
= 1 observations, and the attenuation bias for p, would be alleviated.
However, also associated with lifting thresholds, the sample size of D,
= 1 is shrinking, which may lead to imprecise and less robust estimates.
APPENDIX B: PROOF OF IDENTIFICATION OF TWO-REGIME SWITCHING
REGRESSIONS WITH ARBITRARY REGIME ERROR TERM DISTRIBUTIONS
Yakowitz and Spragins (1968) establish the result that finite
mixtures of normals can be identified up to "label switching."
Ferguson (1983) proves that any arbitrary distribution on the real line
can be indefinitely approximated by a mixture of a countable number of
normal distributions subject to label switching, i.e., for any density
function f(x),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
These two results, coupled with the Monotonicity Condition,
underlie our sketch of proof of identification.
To fix ideas, suppose each of the error terms in Equations (9a) and
(9b), [U.sub.1] and [U.sub.0], can be adequately described by a
two-component normal mixtures:
(A9a) f([u.sub.1]) = [c.sub.1][phi] ([u.sub.1]|[[mu].sub.1],
[[sigma].sup.2.sub.1]) + (1 - [c.sub.1]) [phi] ([u.sub.1]|[[mu].sub.2],
[[sigma].sup.2.sub.2]);
(A9b) f([u.sub.0]) = [c.sub.0]([phi]) ([u.sub.0]|[[mu].sub.3],
[[sigma].sup.2.sub.3]) + (1 - [c.sub.0]) [phi] ([u.sub.0]|[[mu].sub.4],
[[sigma].sup.2.sub.4]).
"Label switching" means that, for instance, in Equation
(A9a), we do not really care which weight is labeled as c, and which as
1 - [c.sub.1], or which is labeled as (Pl, a,) and which as
([[mu].sub.0], [[sigma].sub.0]), because there is no meaningful
interpretation attached to each label. A simple rule, such as [c.sub.1]
[less than or equal to] 0.5, or [[mu].sub.1] [less than or equal to]
[[mu].sub.2], would help anchor the labels if so desired. Another
notable fact from either Equation (A9a) or (A9b) is that even without
label identification, the matching between weights and normals is never
confused. We also exploit this fact in what follows.
This irrelevance of labels is subject to change when we refer the
label 1 to the unconstrained group as in our intergenerational mobility
model. As a result of the imperfect classification by D, the subgroup of
observations D = 1 includes cases drawn from both [U.sub.1] and
[U.sub.0]. Consequently, the distribution of the error term for D = 1,
denoted by [[??].sub.1], is in turn a mixture of [U.sub.1], and
[U.sub.0] (recall that [p.sub.1] = Pr([DELTA] = 1|D = 1)):
(A10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Although the label switching between [c.sub.i] and 1 - [c.sub.i] (i
= 0,1) is innocuous, we have to ascertain which is [p.sub.1] as opposed
to 1 - [p.sub.1] because of the meaning of the label 1 in [p.sub.1].
Equation (A10) is a mixture of four normals that can still be
identified up to labels, thus four weights are obtained alongside with
four different sets of ([[mu].sub.i], [[sigma].sub.1]), denoted by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. However, nothing is
known about which of them corresponds to which of ([p.sub.1][c.sub.1],
[p.sub.1](1 - [c.sub.1]), (1 - [p.sub.1])[c.sub.0], (1 - [p.sub.1])(1 -
[c.sub.0])).
Similarly, the distribution of the error term for D = 0 has a
similar form ([p.sub.0] = Pr([DELTA] = 0|D = 0)):
(A11)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
from which we can obtain [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE
IN ASCII]. Notice that the same subscript i in [[??].sub.i] and
[[??].sub.i] indicates that they are associated with the same normal
component of the mixture.
An examination of the ratios of [[??].sub.i] and [[??].sub.i] (i =
1, 2, 3, 4) reveals that they can only take either value of: [p.sub.1]/1
- [p.sub.0] and 1 - [p.sub.1]/[p.sub.0], from which we can solve out two
unknowns. The solution of these two unknowns (denoted by [[??].sub.i]
and [[??].sub.0]) still suffers from the unidentification of labels,
because we are yet to distinguish between the following two
possibilities:
(A12a) [[??].sub.i] = [p.sub.1], [[??].sub.0] = [p.sub.0];
(A12b) [[??].sub.1] = 1 - [p.sub.1], [[??].sub.0] = 1 - [p.sub.0].
Here is when the Monotonicity Condition shows its power: only one
of Equations (A12a) and (A12b) will satisfy the condition, which helps
anchor the labels of [p.sub.1] and [p.sub.0]. After this step, given the
information of [p.sub.1] and [p.sub.0], [c.sub.i] (i = 1,0) can be
subsequently recovered from the fact that [c.sub.i] and 1 - [c.sub.i]
are associated with the same normal components that [p.sub.j] or 1 -
[p.sub.j] (j = 1, 0) is associated with, from either Equation (A10) or
(A11). Therefore the underlying distributions of [U.sub.1] and [U.sub.0]
are completely recovered. The same line of reasoning in this proof can
be extended to the case of more than two normal mixture components in
[U.sub.1] or [U.sub.0].
APPENDIX C: IMPLEMENTATION OF SWITCHING REGRESSION ESTIMATION
In order to illustrate, we assume that [U.sub.j] (j = 0, 1) are
normals:
[U.sub.j] ~ N (0, [[sigma].sup.2.sub.j]) (j = 0,1).
Non-normal distributions of [U.sub.j] only involve decomposing it
further into mixture of normals. Hence, the likelihood function is
(A13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [[phi].sub.j](*) is the PDF of [U.sub.j].
To make the model parsimonious, we write [p.sub.1] and [p.sub.0] as
a binary function of [D.sub.i], F([D.sub.i]), such that F(1) = [p.sub.1]
and F(0) = 1 - [p.sub.0]. We choose F([D.sub.i]) = 1/1
+exp([[gamma].sub.0] + [[gamma].sub.1][D.sub.i]). Given the functional
form of F(*), if [gamma].sub.1] < 0, then F(1) > F(0), which means
receiving a sizable inheritance/gift will be more likely to be
classified into the group whose estimates are indexed by 1. Therefore,
by the Monotonicity Condition, this group should be labeled as the
unconstrained group. F(*) can include more than one switching variable,
as long as the Monotonicity Condition is applicable to at least one of
these variables to ensure identifiability. The likelihood function now
becomes
(A14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
A well-known challenge of maximizing the likelihood function such
as Equation (A 14) is that, if a goes to zero, the value of likelihood
function explodes, which does not constitute a valid estimate. There are
at least two ways of getting around this issue. For the first, Kiefer
(1978) proves the likelihood that Equation (A 14) has a consistent and
asymptotically efficient root, and suggests using the method of moment
generating functions as laid out in Quandt and Ramsey (1978) to find out
the initial consistent estimate. Schmidt (1982) improves Quandt and
Ramsey's (1978) estimator by demonstrating that the generalized
method of moments applied to the aforementioned moment generating
function performs better. For the second, Hathaway (1985) shows that
simple constraints on [[sigma].sub.i]. such as the relative ratios of
either one to the other cannot be too small, can help rule out spurious
local maximizers. That is, if the constraints
(A15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
are imposed, where c is a sufficiently small number and j, k are
indexing any two of regime error terms, the maximum likelihood problem
is well defined in optimization, and the global solution is strongly
consistent.
We follow Hathaway's (1985) method for its simplicity in
implementation. We pick the constraint parameter c = 0.0067 which is
adequately small for the iterations to converge. After obtaining the
initial consistent estimates through this step, we feed them as initial
values into a subsequent, more refined MLE step.
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(1.) Waldkirch, Ng, and Cox (2004) examined the intergenerational
correlation in consumption as a result of the intergenerational linkages
in income and tastes in a structural econometric framework, which in
substance significantly differs from our model of interpreting and
estimating the intergenerational consumption correlation.
(2.) See the references in Mulligan (1997), and more recently,
Chadwick and Solon (2002), Gaviria (2002), Ermisch, Francesconi, and
Siedler (2006), Mazumder (2005), Blanden, Gregg, and Macmillan (2007),
Bratsberg et al. (2007), and Lee and Solon (2009).
(3.) Altonji, Hayashi, and Kotlikoff (1997) reported that, for
instance, in PSID 1988 sample, the mean age of adult children who
received positive transfer from parents is 29, and the mean age of their
parents is 58; only 2.9% of these children were still in school at the
time of transfer.
(4.) According to Altonji, Hayashi, and Kotlikoff (1997), the mean
was $1507.8 for the subset of PSID 1988 sample with positive amount of
inter vivos transfer money. According to McGarry (1999), the mean of
positive amount of inter vivos transfer money to each child (age 18 and
over) was $3,013 from the Health and Retirement Study (HRS) 1992 survey,
and was $4,215 from the 1993 Assets and Health Dynamics of the Oldest
Old (AHEAD) survey.
(5.) "Inheritance" is more often used from the viewpoint
of the recipient of a bequest. We use "bequests" or
"inheritances" interchangeably throughout the study.
(6.) Hurd and Smith (2002) provided the size and distribution of
actual bequests received by the children of the elderly surveyed in the
1993 AHEAD survey who passed away prior to the 1995 wave: more than 40%
of children received nothing when their last surviving parent died; the
mean size of inheritance is $18,600, and only one in ten children
collected $54,000 or more.
(7.) However, Altonji, Hayashi, and Kotlikoff (1997) found that the
magnitude of inter vivos transfer is only 13% of what the parental
altruism model implies.
(8.) Separately, Grawe (2010) tested the connection between credit
constraint and family size effects, also using bequest receipt as the
signal variable. He found contradictory evidence to the theoretical
predictions.
(9.) Nevertheless, the same framework has been used in a number of
studies in other fields of economics, and Maddala (1986) provided an
excellent survey by then. For instance, Lee and Porter (1984) used the
SR model to test the price behavior under firm collusion in the
industry, whereby the binary variable of whether firms are in collusion
or not is at best imperfectly observed. Recent work by Kopczuk and
Lupton (2007) employed the SR framework to signify the existence of
significant bequest motives for the elderly that is difficult to detect
from data otherwise.
(10.) For the data we specifically examine, however, a normal
distribution for each regime turns out to be adequate.
(11.) Lochner and Monge-Naranjo (2011) surveyed the literature that
has incorporated these elements into quantitative models.
(12.) This assumption, an alternative to assuming parents care
about children's earnings/income, was invoked in some of the
studies previously reviewed (Altonji, Hayashi, and Kotlikoff 1997; Cox
1987, 1990; McGarry 1999; Tomes 1981).
(13.) For a sketchy illustration, assume that there are two periods
(1 and 2) for both the parent and the child, and the second period of
the parent overlaps with the first period of the child. For simplicity,
assume the gross rate of return is one and the discount rate between
periods is zero, a is still the parental altruism. The parent is
maximizing U([C.sub.p,1]) + U([I.sub.p]-T-[C.sub.p,1]) +
[alpha]U([C.sub.c+1]) + [alpha]U/ ([I.sub.c] + T- [C.sub.c,1]), while
the child is maximizing the sum of the last two terms. The
within-lifetime Euler equation for the parent is U'([C.sub.p,1]) =
U'([C.sub.p,2]), and for the child, U'([C.sub.c,1]) =
U'([C.sub.c,2]). When T>0, U'([C.sub.p,2]) =
[alpha]U'([C.sub.c,2]). Therefore, U'([C.sub.p,1]) =
[alpha]U'([C.sub.c,1]), which leads to an equation identical to
(9).
(14.) The weights are taken from Skinner's (1987) study which
estimates the weights of these aforementioned individual consumption
components by regressing total consumption on these individual
components from Consumer Expenditure Survey (CEX) data. This measure of
"weighted consumption" is also employed in Waldkirch, Ng, and
Cox (2004).
(15.) Actually, his construction criterion for splitting the sample
based on actual inheritances is a mixed one: whether children reported
receiving more than $ 10,000 inheritances/gifts in 1984-1989 or whether
their parents had more than $100,000 in wealth in 1988. Families
satisfying either of these two conditions will be regarded as
unconstrained. Using this indicator, Gaviria showed that the earnings or
wage mobility is indeed higher in unconstrained than in constrained
families in linear regressions, just as the Becker-Tomes model predicts.
Our conventional linear regression results by using the indicator of
actual inheritances also agree with the prediction of the Becker-Tomes
model. Notwithstanding, Gaviria did acknowledge the limitation in
relying on the wealth information: wealthy parents may fail to invest
optimally in their children if they are not altruistic enough.
(16.) More "unfolding brackets" questions about the
amounts of inheritances would follow, if the respondent answered
"Yes" to either of these two questions.
(17.) The parent's expectation about how much he/she is to
bequeath to his/her child is more relevant based upon the model.
Therefore, the implicit assumption here is that children's
expectation coincides with parents' expectation.
(18.) Note that tying the theoretical construct of borrowing
constraint with a certain low range of observable financial or economic
variables has a long history in the literature (Chetty 2008; Gaviria
2002; Mulligan 1997, 1999; Runkle 1991; Zeldes 1989). Implicitly, what
this assumes is that the sample units falling within the defined range
are "more likely" to be borrowing constrained, a presumption
to be explicitly formalized in this study.
(19.) Answering "no" or having all missing values in
anticipated inheritance will be treated as zero. The key here is not
about the distinction between zero and missing values, but about the
group with large size of inheritance versus all else, that is, all we
need is that the group with sizable inheritances is more likely to be
unconstrained than otherwise, including those with missing values. The
possibility that missing-value observations are otherwise more likely to
be unconstrained does not sit well with available evidence. Same applies
to the actual inheritances/gifts measure to be introduced next. There
are only a dozen missing observations for expected inheritances, in
contrast to over 700 missing observations for actual inheritances/gifts.
Mulligan (1997, Table 8.6, Columns 3 and 4) obtained the results almost
identical to OLS ones by a Tobit model with regard to the expected
inheritance mea sure. We experimented with a Tobit model of the
consumption persistence regressions for the actual inheritance mea sure
and also obtain results almost identical to those from OLS. Furthermore,
in the robustness check (refer to p. 22), we estimated the SR model
restricted to non-missing observations for the actual inheritance
variable, and obtained similar results.
(20.) To investigate this issue, we examined the variable of
parents' vital status (Deceased, Alive, or N/A) as of 1984 and as
of 1994 of the current sample. We found that the distribution of
parental vital status for children who answered "Yes" to the
expected inheritance question is roughly the same as that for those who
answered "No." The majority of respondents anticipating that
they would receive inheritances in years 1984-1994 had both of their
parents alive in 1984 as well as in 1994, the same as the pattern for
respondents who indicated that they were not anticipating any
inheritances for the same period. Among the few respondents who had
neither parent alive at the time of survey in 1984, some still expressed
their anticipation of inheritances from somewhere. These suggest the
data on expected inheritance are probably error-ridden due to response
error. Notice that we do not claim that these expressed expectations
from data are irrational, as there might be true surprises when it comes
to the discrepancy between expected and actual inheritances. Our data do
not allow us to set apart whimsical expectations from true surprises to
rational expectations.
(21.) Or "actual financial transfers," which we will use
interchangeably.
(22.) Once again, more "unfolding brackets" questions
will follow regarding the size and the receiving year, if a respondent
answers "Yes" to the survey question below.
(23.) This is the same variable used in Grawe (2010) to study the
connection between credit constraints and family size effects.
(24.) Results are available upon request. Furthermore, SR results
changed little when those attrition observations are excluded (see
Subsection IV.B).
(25.) We will explore this issue in the discussion section of our
results. Nonetheless, missing observations for parents in their
retirement years do not allow us to offer a complete answer.
(26.) Grawe (2006) discussed the estimation bias in
intergenerational earnings persistence resulting from the deviation of
observed earnings from lifetime earnings that varies with age.
(27.) Notably, Mulligan (1997) did not include family size as one
of the covariates. Implicitly, the Becker-Tomes model treats the number
of children as a choice variable that is likely to be endogenous.
(28.) These two sets of estimates are both statistically
significant at 0.01 level in and by themselves. However, the
difference between 0.45 and 0.55, or the difference between 0.63
and 0.52, is not statistically significant at 0.10 level.
(29.) In practice, the predictor [D.sub.i] can be generalized to a
vector of variables, as long as the Monotonicity Condition is applicable
to at least one of the variables in [D.sub.i].
(30.) Hausman, Abrevaya, and Scott-Morton (1998) used this term to
describe the restriction on misclassification error in the dependent
variable of discrete choice models.
(31.) This is by no means to substitute for a formal statistical
test of the number of mixtures or parameter values of the mixture
components. However, as Garel (2007) has noted, theoretical results of
testing against more than two-component mixtures are difficult to
obtain.
(32.) Even though ([U.sub.1], [U.sub.0]) in (2) can be of any
arbitrary distributions, we found normal distributions are adequate for
our data in estimation. Expanding [U.sub.1] or [U.sub.0] further into
mixture of two normals would lead to the estimate of one of the weights
over 0.99.
(33.) The data in Jappelli's study did not include details
about categories of the loans applied by these families, for example,
children's college education loans, as opposed to mortgage loans,
therefore it is not clear whether and to what extent these loans are
related to children's human capital investments.
(34.) SEO oversamples low income households.
(35.) For this point, one only needs to check the terms involved in
[I.sub.0] of the choice Equation (7), and the terms defined in [U.sub.1]
and [U.sub.0] of the outcome Equation (2).
(36.) See Guo (2010) for a discussion about econometric issues
involved in using the predicted consumption measure.
(37.) We can investigate household consumption surveys (such as
CEX, or recent years of PSID) for this issue, but again, the challenge
is whether financial transfer variables (preferable to parental income
or wealth) are available to divide the observations into the constrained
versus unconstrained, even with more detailed consumption data.
(38.) Note that these variables are mostly in negative
correspondence with the degree of financial needs, for example, higher
income means less need of financial help, other things equal.
(39.) Since information is not available for most of these parents
at the time of bequeathing, this regression should be interpreted with
caution. A positive correlation between, say, parental wealth close to
the time of bequeathing and children's wellbeing, would bias the
estimates towards being more significant than the otherwise. Yet,
insofar as late-age parental wealth is positively correlated with
parental income and children's expected inheritances, this bias
will be partially mitigated.
(40.) Han and Mulligan (2001) used simulations to show that, should
there be much heterogeneity in earnings ability in the population, it
would not be easy to detect earnings persistence between constrained and
unconstrained families in regressions, even in the absence of the
misclassification issue.
(41.) Waldkirch, Ng, and Cox (2004, Table 5) also presented the
results related to their defined "liquidity constrained"
cases. But it is defined from the life-cycle viewpoint of adult children
instead of intergenerational viewpoint of parents. And they simply
define those with low income as liquidity constrained.
(42.) The variables of expected bequests and actual bequests, to
the extent that they are repeated signal measures of binding credit
constraint, are statistically highly correlated in our data. Therefore,
Hu's (2010) approach cannot be directly implemented here.
TABLE 1
Distribution of Expected and Actual Inheritance/Gift by Size
[$25,000,
Expected Inheritance +[infinity]) (0, $25,000) 0/(missing)
as of 1984 Total 166 (9.3%) 171 (9.6%) 1444(81.1%)
Actual inheritance/
gift received
Prior to 1984
[$25,000, [infinity]) 10 (0.6%) 5 (0.3%) 39 (2.2%)
(0, $25,000) 12(0.7%) 14 (0.8%) 81 (4.5%)
0/(missing) 144(8.1%) 152 (8.5%) 1324 (74.3%)
1984-1994
[$25,000, [infinity]) 26(1.5%) 13 (0.7%) 70 (3.9%)
(0, $25,000) 25 (1.4%) 26(1.5%) 123 (6.9%)
0/(missing) 115(6.5%) 132 (7.4%) 1251 (70.2%)
1994-2003
[$25,000, [infinity]) 16 (0.9%) 8 (0.4%) 39 (2.2%)
(0, $25,000) 8 (0.4%) 9 (0.5%) 52 (2.9%)
0/(missing) 142 (8.0%) 154 (8.6%) 1353 (76.0%)
In total
[$25,000, [infinity]) 39 (2.2%) 20(1.1%) 106(6.0%)
(0, $25,000) 24(1.3%) 31 (1.7%) 152 (8.5%)
0/(missing) 103 (5.8%) 120(6.7%) 1186 (66.6%)
Notes: Figures in each cell include number of observations
accompanied by the corresponding fraction relative to the whole
sample size.
TABLE 2
Summary Statistics of Relevant Variables by Expected and Actual
Inheritance/Gift Size in PSID Intergenerational Sample
Expected Inheritance [greater
than or equal to] $25,000
Actual
Inheritance Actual
[greater than or Inheritance
Variable All equal to] $25,000 < $25,000
Parent's age (a) 40.3 43.0 42.0
(7.4) (7.9) (7.3)
Parent's income (b) 28661.59 41407.35 29366.85
(19880.37) (27667.92) (17224.92)
Parent's 17224.65 21386.75 17644.44
consumption (b)
(7577.35) (8167.39) (7187.16)
Parent's wage (c) 10.18 13.44 10.07
(7.41) (7.85) (6.10)
Parent's education 10.44 12.39 10.73
achievement (d)
(3.67) (4.08) (3.59)
Child's age (e) 31.3 31.9 31.8
(2.6) (2.6) (2.6)
Child's income (b) 27283.13 40043.55 32050.94
(19729.82) (16881.34) (25822.68)
Child's 13334.61 19162.79 14779.99
consumption (b)
(7274.90) (8560.21) (8894.92)
Child's wage (c) 8.24 9.34 9.21
(5.36) (5.22) (7.59)
Child's education 13.21 14.29 13.36
achievement (d)
(2.17) (2.10) (2.23)
Expected Inheritance < $25,000
Actual
Inheritance Actual
[greater than or Inheritance
Variable equal to] $25,000 <$25,000
Parent's age (a) 41.6 39.9
(7.8) (7.3)
Parent's income (b) 36942.06 27452.50
(17970.01) (19523.75)
Parent's 21635.12 16694.53
consumption (b)
(9220.81) (7312.35)
Parent's wage (c) 13.33 9.82
(6.77) (7.49)
Parent's education 12.04 10.21
achievement (d)
(3.41) (3.62)
Child's age (e) 30.6 31.3
(2.9) (2.6)
Child's income (b) 33447.17 25791.74
(20496.73) (18688.22)
Child's 16274.73 12721.57
consumption (b)
(8270.92) (6756.71)
Child's wage (c) 9.48 8.00
(5.48) (5.04)
Child's education 14.18 13.08
achievement (d)
(2.15) (2.14)
Notes: This table presents the mean and standard deviation (in
parenthesis) of the variables for the entire sample as well as
for each of the four subgroups defined by the sizes of expected
inheritance and actual inheritance.
(a) The parental household head's age as of 1967.
(b) In thousand dollars.
(c) In dollars.
(d) Years of schooling.
(e) Child's age as of 1987.
TABLE 3
Switching and Simple Sample Splitting Regressions of
Intergenerational Consumption Persistence: Classification
Based on Expected Inheritances
Consumption Persistence Regression: Classification
According to Expected Inheritances
Switching Regression
[DELTA] = [DELTA] =
Estimation Methods 1 (b) 0 (b)
Parental consumption (d) 0.4394 *** 1.0527 ***
(0.0237) (0.1425)
Daughter dummy -0.0400 * 0.3524 ***
(0.0207) (0.1146)
Parental marital status -0.0210 *** 0.0205
(0.0066) (0.0362)
Child's marital status 0.4465 *** 1.3008 ***
(0.0246) (0.1438)
Parent's age (x [10.sup.-1]) (e) -0.1130 -0.1810
(0.1157) (0.7600)
Parent's age squared (x [l0.sup.-3]) (e) 0.1287 0.2534
(0.1359) (0.8964)
Child's age (x [10.sup.-1]) (e) -1.0510 5.2938
(0.8404) (4.1824)
Child's age squared (x [l0.sup.-3]) (e) 2.0703 -8.2160
(1.3425) (6.6167)
(Intercept) 6.4907 *** -10.540
(1.3313) (6.7839)
[[sigma].sub.U] 0.3547 *** 0.6616 ***
(0.0093) (0.0392)
[[gamma].sub.0] -1.0040 ***
(0.1304)
[[gamma].sub.1] -0.3290
(0.4255)
Maximized log-likelihood -1109.17
Pr[([DELTA]|[D.sub.e] = 1).sup.f] 0.9167 0.0933
Pr[([DELTA]|[D.sub.e] = 0).sup.f] 0.8423 0.1577
Simple Sample Splitting (a)
[D.sub.e] = [D.sub.e] =
Estimation Methods 1 (c) 0 (c)
Parental consumption (d) 0.4491 *** 0.5487 ***
(0.0770) (0.0358)
Daughter dummy 0.0282 -0.0232
(0.0655) (0.0280)
Parental marital status -0.0101 -0.0076
(0.0225) (0.0117)
Child's marital status 0.6222 *** 0.6058 ***
(0.0990) (0.0345)
Parent's age (x [10.sup.-1]) (e) 0.7487 * -0.1703
(0.4141) (0.1604)
Parent's age squared (x [l0.sup.-3]) (e) -0.8822 ** 0.1922
(0.4516) (0.1856)
Child's age (x [10.sup.-1]) (e) 0.4896 -0.2741
(3.2374) (1.2167)
Child's age squared (x [l0.sup.-3]) (e) -0.1292 0.6997
(5.1189) (1.9581)
(Intercept) 1.7653 4.2185 **
(5.1462) (1.9292)
[[sigma].sub.U]
[[gamma].sub.0]
[[gamma].sub.1]
Maximized log-likelihood
Pr[([DELTA]|[D.sub.e] = 1).sup.f]
Pr[([DELTA]|[D.sub.e] = 0).sup.f]
Notes: Sample size 1,781; dependent variable: adult child's
logarithm of consumption; standard error in parentheses; expected
inheritance indicator is used for identifying which set of
parameters in the switching regression corresponds to the regime
of borrowing constrained ([DELTA] = 0). See text for details.
(a) Linear regressions for each subsample with standard error
clustered by parental family identifiers.
(b) 1, unconstrained; 0, constrained.
(c) 1, expected inheritance greater than $25,000 (219 cases); 0,
expected inheritance less than $25,000 (1,562 cases).
(d) Consumption is the logarithm of multi-year average of Skinner
(1987) consumption measure.
(e) Parent's age is the household head's age as of 1967; child's
age is the child's age as of 1987.
(f) Probability of being "unconstrained" or "constrained"
conditional on the value of the indicator [D.sub.e]; calculated
from [PHI]([gamma]0 + [gamma]l[D.sub.e]), where [PHI](.) is the
cumulative distribution function (CDF) of standard normal
distribution.
*** Statistical significance of a coefficient at .01; ** statistical
significance of a coefficient at .05; * statistical significance of a
coefficient at .10.
TABLE 4
Switching and Simple Sample Splitting Regressions of
Intergenerational Consumption Persistence: Classification
Based on Actual Inheritances/Gifts
Consumption Persistence Regression: Classification
According to Actual Inheritances/Gifts
Switching Regression
[DELTA] = [DELTA] =
Variable 1 (b) 0 (b)
Parental consumption (d) 0.4398 *** 1.0176 ***
(0.0236) (0.1376)
Daughter dummy -0.0400 * 0.3263 ***
(0.0206) (0.1151)
Parental marital status -0.0220 *** 0.0240
(0.0066) (0.0360)
Child's marital status 0.4474 *** 1.2824 ***
(0.0245) (0.1437)
Parent's age (x [l0.sup.-1]) (e) -0.1130 -0.2170
(0.1148) (0.7710)
Parent's age squared (x [l0.sup.-3]) (e) 0.1285 0.2854
(0.1346) (0.9063)
Child's age (x [lO.sup.-1]) (e) -1.1060 4.4107
(0.8388) (4.1255)
Child's age squared (x [l0.sup.-3]) (e) 2.1587 -6.7650
(1.3412) (6.4890)
(Intercept) 6.5716 *** -8.762
(1.3291) (6.5981)
[[sigma].sub.U] 0.3534 *** 0.6656 ***
(0.0092) (0.0372)
[[gamma].sub.0] -0.9940 ***
(0.1267)
[[gamma].sub.1] -0.4700
(0.5883)
Maximized log-likelihood -1108.76
Pr([DELTA]|[D.sub.a] = l) (f) 0.9284 0.0716
Pr([DELTA]|[D.sub.a] = 0) (f) 0.8399 0.1601
Simple Sample Splitting (a)
[D.sub.a] = [D.sub.a] =
Variable 1 (c) 0 (c)
Parental consumption (d) 0.6254 *** 0.5229 ***
(0.0948) (0.0363)
Daughter dummy -0.0086 -0.0289
(0.0713) (0.0272)
Parental marital status -0.0506 * -0.0065
(0.0306) (0.0112)
Child's marital status 0.5212 *** 0.6110 ***
(0.1056) (0.0339)
Parent's age (x [l0.sup.-1]) (e) -0.2391 -0.0691
(0.3791) (0.1609)
Parent's age squared (x [l0.sup.-3]) (e) 0.2538 0.0675
(0.4260) (0.1858)
Child's age (x [lO.sup.-1]) (e) 1.0696 -0.4862
(2.8681) (1.2236)
Child's age squared (x [l0.sup.-3]) (e) -1.0324 1.0615
(4.5347) (1.9639)
(Intercept) 1.4808 4.5718 **
(4.3258) (1.9509)
[[sigma].sub.U]
[[gamma].sub.0]
[[gamma].sub.1]
Maximized log-likelihood
Pr([DELTA]|[D.sub.a] = l) (f)
Pr([DELTA]|[D.sub.a] = 0) (f)
Notes: Sample size 1,781; dependent variable: adult child's
logarithm of consumption; standard error in parentheses; actual
inheritance indicator is used for identifying which set of
parameters in the switching regression corresponds to the regime
of borrowing constrained ([DELTA] = 0). See text for details.
(a) Linear regressions for each subsample with standard error
clustered by parental family identifiers.
(b) l, unconstrained; 0, constrained.
(c) l, received actual inheritances/gifts greater than $25,000
(165 cases); 0, received actual inheritances/gifts less than
$25,000 (1,616 cases).
(d) Consumption is the logarithm of multi-year average of Skinner
(1987) consumption measure.
(e) Parent's age is the household head's age as of 1967; child's
age is the child's age as of 1987.
(f) Probability of being "unconstrained" or "constrained"
conditional on the value of proxy indicator [D.sub.a] calculated
from [PHI]([gamma]O + [gamma]l[D.sub.a]), where [PHI](.) is the
CDF of standard normal distribution.
*** Statistical significance of a coefficient at .01; **
statistical significance of a coefficient at .05; * statistical
significance of a coefficient at .10.
TABLE 5
Estimated Consumption Persistence by Using Alternative Cutoffs of
Actual Inheritances/Gifts for Constructing the Classification
Indicator: Simple Sample Splitting and Switching Regressions
(Online)
Threshold Value $0 $5 k $10 k
Sample Size (D = 1) 372 329 265
Simple sample splitting regression estimates
[[beta].sub.1] (unconstrained) 0.5712 0.5449 0.5717
(0.0591) (0.0640) (0.0706)
[[beta].sub.0] (constrained) 0.5067 0.5126 0.5157
(0.0360) (0.0351) (0.0338)
Switching regression
[[beta].sub.1] (unconstrained) 0.4401 0.4439 0.4426
(0.0236) (0.0237) (0.0237)
[[beta].sub.0] (constrained) 0.9889 0.9936 1.0061
(0.1408) (0.1417) (0.1390)
Pr([DELTA] = 1|D = 1) 0.9246 0.9220 0.9274
Pr([DELTA] = 0|D = 1) 0.0754 0.0780 0.0726
Pr([DELTA] = 1|D = 0) 0.8329 0.8411 0.8415
Pr([DELTA] = 0|D = 0) 0.1671 0.1589 0.1585
Threshold Value $25 k $30 k
Sample Size (D = 1) 165 141
Simple sample splitting regression estimates
[[beta].sub.1] (unconstrained) 0.6254 0.6514
(0.0893) (0.0978)
[[beta].sub.0] (constrained) 0.5229 0.5239
(0.0324) (0.0322)
Switching regression
[[beta].sub.1] (unconstrained) 0.4398 0.4396
(0.0236) (0.0236)
[[beta].sub.0] (constrained) 1.0176 1.0269
(0.1376) (0.1310)
Pr([DELTA] = 1|D = 1) 0.9284 0.9057
Pr([DELTA] = 0|D = 1) 0.0716 0.0943
Pr([DELTA] = 1|D = 0) 0.8399 0.8435
Pr([DELTA] = 0|D = 0) 0.1627 0.1565
Threshold Value $40 k $50 k
Sample Size (D = 1) 121 102
Simple sample splitting regression estimates
[[beta].sub.1] (unconstrained) 0.6905 0.6421
(0.1104) (0.1214)
[[beta].sub.0] (constrained) 0.5278 0.5328
(0.0318) (0.0316)
Switching regression
[[beta].sub.1] (unconstrained) 0.4387 0.4353
(0.0237) (0.0236)
[[beta].sub.0] (constrained) 0.9994 1.0461
(0.1368) (0.1597)
Pr([DELTA] = 1|D = 1) 0.8876 0.9077
Pr([DELTA] = 0|D = 1) 0.1124 0.0923
Pr([DELTA] = 1|D = 0) 0.8419 0.8373
Pr([DELTA] = 0|D = 0) 0.1581 0.1366
Notes: Standard error in parentheses. D = 1 if received actual
inheritances/gifts are greater than the specified cutoff value,
and D = 0 if otherwise.
TABLE 6
Switching Regression Estimates of Intergenerational Consumption
Persistence: Log of Actual Inheritances/Gifts in the Switching
Equation
Consumption Persistence Switching Regression: Log of Actual
Inheritances/Gifts
Switching
Regime Equation Equation
[DELTA] = [DELTA] = Prob([DELTA] =
Variable 1 (a) 0 (a) l|D) (b)
Parental consumption (c) 0.4442 *** 0.9832 ***
(0.0266) (0.2824)
Daughter dummy -0.0378 * 0.3421 ***
(0.0207) (0.1188)
Parental marital status -0.0215 *** 0.0204
(0.0072) (0.0370)
Child's marital status 0.4493 *** 1.3048 ***
(0.0254) (0.1469)
Parent's age (d) -0.0108 -0.0224
(0.0126) (0.0517)
Parent's age squared 0.0001 0.0003
(0.0001) (0.0006)
Child's age (d) -0.1223 0.5812
(0.0877) (0.5609)
Child's age squared 0.0024 * -0.0091
(0.0014) (0.0090)
Log of actual -- -- -0.0838 **
inheritances/gifts (e) (0.0426)
(Constant) 6.6932 *** -10.556 -1.6380 ***
(1.3469) (8.6517) (0.1920)
[[sigma].sub.U] 0.3565 *** 0.6606 ***
(0.0118) (0.0413)
Sample average of:
Predicted Prob([DELTA] 0.8552
= 1) (f)
Predicted Prob([DELTA] 0.9231
= 1| actual
inheritances > 0) (f)
Predicted Prob([DELTA] 0.8373
= 1| actual
inheritances = 0) (f)
Maximized log-likelihood -1106.67
Notes: Sample size 1,781; dependent variable: adult child's
logarithm of consumption; standard error in parentheses.
(a) l, unconstrained; 0, constrained.
(b) Probability of being truly "unconstrained" or "constrained"
conditional on the value of a vector of variables D, in the form
of F(D[gamma]) = 1/(1 + exp(D[gamma])) in accordance with the
identification of unconstrained group.
(c) Consumption is the logarithm of multi-year average of Skinner
(1987) consumption measure for a household.
(d) Parent's age is the household head's age as of 1967; child's
age is the child's age as of 1987.
(e) Continuous variable, computed as log of amount of actually
received inheritances/gifts.
(f) Computed as the predicted value based on the estimated
coefficients of switching equation.
*** Statistical significance of a coefficient at .01; **
statistical significance of a coefficient at .05; * statistical
significance of a coefficient at .10.
TABLE 7
Switching Regression Estimates of Intergenerational Consumption
Persistence: Alternative Sets of Switching Variables
Alternative Specifications of Switching Equation
(1) (2)
Regime equation
[DELTA] = 1 (borrowing unconstrained
group) Parental consumption 0.4436 *** 0.4351 ***
(0.0257) (0.0262)
[DELTA] = 0 (borrowing constrained
group) Parental consumption 0.9009 *** 0.8209 ***
(0.1641) (0.1551)
Switching equation (a)
Parental savings (b) -0.9073 ***
(0.2616)
Number of children in school 0.1585 **
(0.0617)
Parental homeownership -0.0765
(0.2534)
Parental car ownership -0.6504 **
(0.2832)
Parental head is nonwhite 1.2903 ***
(0.2704)
Father's age above 50 0.7855 **
(0.3355)
Mother's education level college and -0.5175
above (0.3264)
Lots of reading at parental home (c)
Parental home in rural area (d)
Sample average of predicted 0.8556 0.8478
Prob([DELTA] = l) (e)
Maximized log-likelihood value -1089.9510 -1090.2499
(3) (4)
Regime equation
[DELTA] = 1 (borrowing unconstrained
group) Parental consumption 0.4407 *** 0.4381 ***
(0.0273) (0.0256)
[DELTA] = 0 (borrowing constrained
group) Parental consumption 0.9513 *** 0.7630 ***
(0.1615) (0.1664)
Switching equation (a)
Parental savings (b) -0.7628 ***
(0.2616)
Number of children in school 0.1524 **
(0.0640)
Parental homeownership
Parental car ownership -0.4246
(0.2792)
Parental head is nonwhite 0.8158 ***
(0.2958)
Father's age above 50 1.0232 ***
(0.3636)
Mother's education level college and -0.7574 * -0.4501
above (0.3873) (0.3364)
Lots of reading at parental home (c) -0.3275
(0.3161)
Parental home in rural area (d) 0.1266
(0.3118)
Sample average of predicted 0.8566 0.8515
Prob([DELTA] = l) (e)
Maximized log-likelihood value -1106.4757 -1079.5978
Notes: These regressions are conducted on the full sample of
1,781 parent--child pairs. The dependent variable is the adult
child's logarithm of consumption. The independent variables in
regime equations, other than the logarithm of parental
consumption, are identical to those in previous tables, and their
estimated coefficients are omitted. Standard error in
parentheses.
(a) All the variables in the switching equation are measured in
1968/1972 when a child resided with his/her parents.
(b) l if parents ever saved more than 2 months' income in
1968-1972; 0 if otherwise.
(c) 1 if a lot of reading material was visible in the house; 0 if
otherwise.
(d) 1 if lived 50 miles or more from the center of a city in each
year of 1968-1972; 0 if otherwise.
(e) The predicted value of Prob([DELTA] = 1) is computed based on
the estimated coefficients of switching equation.
*** Statistical significance of a coefficient at .01; **
statistical significance of a coefficient at .05; * statistical
significance of a coefficient at .10.
TABLE 8
Prediction Regression of Actual
Inheritances/Gifts Received Conditional on
Expected Inheritances
Prediction of Actual Inheritance/
Gift Conditional on Expected Inheritance
Variable Tobit (a,b)
Expected inheritances 8.6245 ***
> 25,000 (c) (1.044)
Log of parental income in 5.2678 ***
1967 - 1971 (d) (0.969)
Log of adult children's 0.3467
income in 1984-1988 (d) (0.877)
Log of adult children's 0.4847 ***
wealth in 1984 (e) (0.118)
Adult children's 0.6585 ***
education attainment (0.246)
(measured in years)
Adult children's 2.9156 **
homeownership in (1.246)
1984-1988
Adult children's car 0.8955
ownership in 1984-1988 (2.672)
Average number of -1.2842 ***
adult children's kids (0.485)
(underage of 18 in
1984-1988)
Maximized log- -1937.2367
likelihood value
SHENG GUO, I am grateful to Casey Mulligan for his guidance and
support, as well as generously sharing his data. 1 would like to thank
Gary Becker, Susanne Schennache, Jeffery Smith, and three anonymous
referees for helpful comments and suggestions. This article is a
substantially revised version of Chapter 3 of my dissertation thesis
finished at the University of Chicago. I gratefully acknowledge a
dissertation fellowship from the Chicago Center of Excellence in Health
Promotion Economics at the University of Chicago. All remaining errors
are my own.
Guo: Assistant Professor, Department of Economics, Florida
International University, 11200 SW 8th Street, DM 318A, Miami, FL 33199.
Phone 1-305-348-2735, Fax 1-305 348-1524, E-mail Sheng.Guo@fiu.edu
