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Journal of Business & Economic Statistics

ISSN: 0735-0015 (Print) 1537-2707 (Online) Journal homepage: http://www.tandfonline.com/loi/ubes20

The Intergenerational Transmission of Income
Volatility: Is Riskiness Inherited?
Stephen H. Shore
To cite this article: Stephen H. Shore (2011) The Intergenerational Transmission of Income
Volatility: Is Riskiness Inherited?, Journal of Business & Economic Statistics, 29:3, 372-381, DOI:
10.1198/jbes.2011.08091
To link to this article: http://dx.doi.org/10.1198/jbes.2011.08091

Published online: 01 Jan 2012.

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The Intergenerational Transmission of Income
Volatility: Is Riskiness Inherited?
Stephen H. S HORE

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Department of Economics, Johns Hopkins University, 458 Mergenthaler Hall, Baltimore, MD 21218
(shore@jhu.edu)
This article examines the intergenerational transmission of income risk. Do risky parents have risky
kids? Income volatility—a proxy for income risk—is not observed directly; instead, it must be estimated
with substantial error from the time series variability of income. I characterize an income process with
individual-specific volatility parameters and estimate the joint distribution of volatility parameters for fathers and for their adult sons. In data from the Panel Study of Income Dynamics, fathers with higher
income volatility have sons with higher income volatility. This finding is correlated with, but far from
fully explained by, the intergenerational transmission of risk tolerance and of the propensity for selfemployment.

1. INTRODUCTION
Parents transmit human capital to their children. Welleducated parents tend to have well-educated children (Black,
Devereux, and Salvanes 2005). Tall parents tend to have tall
children (Persico, Postlewaite, and Silverman 2005). Healthy
parents tend to have healthy children. Personality—including
propensity for risky behaviors—is inherited as well (Plomin
and Caspi 1998; Plomin, Caspi, and John 1999; Duncan et al.
2005; Dohmen et al. 2006). Of most interest to economists,
high-earning (and high-consuming) parents tend to have highearning (and high-consuming) children (income findings surveyed in Solon 1999; consumption findings in Waldkirch, Ng,
and Cox 2004). This article seeks evidence of intergenerational
transmission of higher moments of human capital. Do parents
with risky—or, more precisely, volatile—income streams have
children with risky income streams?
The very large literature on the intergenerational transmission of income from parents to children acknowledges that there
is no single statistic that describes it completely (Haider and
Solon 2006). To address this concern, researchers have identified the correlation between the incomes of parents and children at multiple points in the life cycle or at the “best” points
(Dahl and DeLeire 2006; Grawe 2006). For example, Haider
and Solon (2006) presented a summary statistic for the intergenerational transmission of permanent income. Here I do the
same for income volatility. Since income paths are not known
initially but are realizations of a stochastic process, I consider

intergenerational transmission of the key parameters of that
process. These parameters are level of income (as in the existing literature) and the variance of income changes (volatility,
as in this article).
This intergenerational transmission of income volatility is
apparent from simple summary statistics: fathers with large
sample variances of income changes tend to have children
with the same. This pattern persists even after removing predictable income changes, heterogeneous income growth rates,
or the predictable component of sample variances. However, a
full characterization of the joint distribution of the fathers’ and
sons’ volatilities is more involved.

I write down an income process (for a single individual)
that is standard in the income dynamics literature (following Hall and Mishkin 1982; Carroll and Samwick 1997), in
which incomes evolve in response to predictable changes, permanent shocks, and transitory shocks. The variances of these
shocks are individual-specific volatility parameters. The object of interest is the joint distribution from which fathers’ and
sons’ volatility parameters are drawn, particularly the correlation of fathers’ and sons’ volatility parameters. Unfortunately,
individual-specific volatility parameters are estimated with substantial error, and this error increases with volatility. Because
fathers’ volatility parameters are estimated with error, regressions using a father’s volatility on the right-hand side to predict a son’s volatility are prone to standard attenuation bias
(Spearman 1904). In addition, the magnitude of estimation error in sons’ volatility parameters will vary, because sons differ in the number of years of income data used to estimate
volatility; as a result, reduced-form estimates will be prone to

heteroscedasticity. These two problems interact because in the
data used here, fathers with longer income series (and therefore
smaller right-hand side measurement error) tend to have sons
with shorter income series (and therefore larger left-hand side
measurement error).
To overcome these problems, I look explicitly at estimates of
the correlation of volatility parameters, not at the correlation of
the estimates of volatility parameters. I use a hierarchical model
in which each individual’s unobserved volatility parameters are
drawn from a joint distribution of fathers’ and sons’ volatility
parameters. That distribution has hyperparameters characterizing the means, variances, and covariances of fathers’ and sons’
volatility parameters. Observed income data are drawn conditional on each individual’s unobserved volatility. Hyperparameters are estimated by maximum likelihood. By integrating
explicitly over all possible values that volatility could take for
any father–son pair, this approach tackles the problem that any
individual’s observed data are consistent with a wide variety of
possible income volatility parameters.

372

© 2011 American Statistical Association

Journal of Business & Economic Statistics
July 2011, Vol. 29, No. 3
DOI: 10.1198/jbes.2011.08091

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Shore: The Intergenerational Transmission of Income Volatility

In principle, this estimation strategy could accommodate a
very general income process with multiple dimensions of heterogeneity. In practice, this is computationally infeasible. As a
result, I focus on the case in which permanent and transitory
shocks are iid. I estimate the model under the assumption that
each individual’s (log) permanent and transitory volatilities are
either perfectly (positively) correlated or completely uncorrelated. I show how results on intergenerational persistence are
qualitatively similar under these two opposite assumptions (perfect correlation or zero correlation) and discuss the relative merits of these two assumptions, as well as the problems with explicitly estimating the correlation of permanent and transitory
volatility.
The sample of 4304 male household heads with sufficient
data from the Panel Study of Income Dynamics (PSID) includes 841 adult sons whose fathers are also included in the
sample. In this sample, fathers with volatile incomes have sons
with volatile incomes, regardless of whether or not education,

income, and demographic controls are included. When allowing only one dimension of variation in individuals’ volatilities,
the correlation of fathers’ and sons’ log volatility parameters is
approximately 28 percent. As would be expected given attenuation bias, these estimates are modestly larger than the 14 percent correlation of log sample variances. The intergenerational
correlation is somewhat lower when individuals’ permanent and
transitory volatilities are assumed to be uncorrelated.
Income volatility could be inherited because preferences relating to income volatility (e.g., risk-tolerance) are inherited, or
because the technology that generates a volatile income (e.g.,
occupation, skills) is inherited. I find clear evidence for inherited preferences related to income volatility: measures of
self-described risk tolerance are inherited. I also find evidence
for inherited self-employment related to income volatility: selfemployment is correlated with income volatility, and parents
who are self-employed for more of their lives have children with
higher levels of income volatility. This is true primarily because
their children are more likely to be self-employed. Whether inherited self-employment reflects inherited preference for selfemployment or inherited skills needed for self-employment is
not clear.
The intergenerational transmission of volatility has important
implications for economic mobility. When income is volatile,
individuals tend to move through the earnings distribution faster
during their own work lives (Haider 2001; Solon 2001). If children inherit income volatility from their parents, then they also
inherit the ability to move up or down the economic ladder during their work lives. Previous research has shown that where
people begin on the economic ladder is inherited; this article

shows that the rate at which people subsequently move up and
down this ladder over their lives is also inherited. Put another
way, I document the intergenerational transmission of the parameter (volatility) that governs intragenerational mobility. This
inheritance of unconditional volatility is of interest regardless
of the covariates (observable to the econometrician or not) that
may drive it.
2.

DATA

2.1 Sample
The data are from the core samples of the PSID, a nationally
representative panel of U.S. households. They include annual

373

observations on education, income, employment status, and a
host of other variables. The PSID surveys not only families
from the original sample, but also families formed from those
original families. As a result, the sample includes income histories both for fathers (from the original sample) and for their

adult sons (after they move out and form their own households).
I use PSID sample weights throughout; in regressions where the
unit of observation is the father–son pair, I use the product of
these PSID sample weights with the number of observations for
the father.
Data are from 1968 through 2001. I use information on the
log labor incomes of male household heads who are at least 25
and at most 65 years of age. I drop any observations in which
the head’s labor income is missing or top-coded and include
only men with at least 10 years of data. One empirical challenge is how to handle observations where labor income is zero
or close to zero. When annual income is below (federal minimum hourly wage) × (1000 hours)—the income that would be
earned if the head worked half-time for minimum wage—I replace income with this amount. To ensure that time-variation in
PSID top-coding cannot drive results, I replace real incomes
above $200,000 in 2001 dollars with the real equivalent of
$200,000 in 2001 dollars. Because actual income is a poor measure of potential income for those not in the labor force (e.g.,
retired people, students), I drop observations in which the head
is out of the labor force. These adjustments lead to substantial
gaps in data for some individuals. Therefore, I drop from the
sample anyone for whom data is missing or dropped in more
than four intervals or with gaps in the data of longer than 4

years.
To remove predictable income changes from income dynamics, I run an ordinary least squares (OLS) regression of log income on the following right-hand side variables: a quartic in
age; the interaction of age and the square of age with education dummies, occupation dummies, and race dummies; year
dummies; and dummies for number of children and for marital
status. When specified but not otherwise, robustness checks remove individual-specific trends in log income. I use the residuals from this regression to capture excess log incomes, and
examine the dynamics of excess log income. Of the 4304 individuals with sufficient data, 841 are the adult sons of fathers
also included in this subsample.
The econometric procedure outlined in Section 3 is not set
up to accommodate gaps in the data. Most of these gaps are
random for our purposes; no data were collected in 1998 and
2000. To overcome this shortcoming in the data, I use a singleimputation hot-deck procedure to fill holes in the data (Rao and
Shao 1992; Reilly 1993; Little and Rubin 2002). For each gap
in the data, I examine the change in excess log income around
that gap. I choose income data from another person in the sample without a gap and with the closest change in excess log income. I fill in the gap in income with excess log income values
to match the bootstrapped changes. This procedure is conservative in the sense that it will bias estimates of income volatility
parameters toward the mean as gaps are filled with bootstrapped
values pulled from the full distribution.

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Journal of Business & Economic Statistics, July 2011

Table 1. Summary statistics for adult children and their parents
Adult sons

Initial age (years)
# of observations
Education (years)
Labor income (′ 000s 2001 $s)
Ever self-employed? (1 or 0)
Fraction of time self-emp.
Cigarettes (#, 2001 $s/year)
Risk tolerance (1/CRRA)
Risk tolerant? (CRRA ≤ 3?)
v
ar(yit − yit−4 )
log(
var(yit − yit−4 ))

Mean

St. dev.

Mean

St. dev.

Corr.

25.4
14.7
14.1
$47.7
0.401
0.138
4.96
1.36
0.60
0.43
−1.65

1.3
5.1
2.1
$28.0
0.490
0.257
11.94
2.35
0.49
0.66
1.32

38.1
20.6
13.0
$55.7
0.506
0.202
602.25
1.14
0.43
0.27
−2.03

6.5
6.3
2.8
$33.0
0.500
0.319
721.81
2.34
0.50
0.34
1.28

14.9%
−43.2%
42.1%
33.0%
17.0%
29.3%
16.7%
13.5%
7.0%
18.5%
14.5%

# of father–son pairs

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Their fathers

841

NOTE: This table presents summary statistics on the 841 parent–adult child pairs with volatility estimates in the sample. Fathers with multiple
sons in the sample will show up repeatedly. Observations with age 65 are dropped, and at least 10 observations must be present. Labor
income is the median (for that individual) real labor income (in thousands of 2001 dollars), and education is the maximum number of years of
education reported for that individual. Cigarettes refers to the number of cigarettes smoked per day (when available) for sons (available for 821 sons)
and to annual cigarette expenditure for fathers (2001 dollars). Risk tolerance refers to 1/CRRA, as calculated from the 1988 PSID risk tolerance
questions. This is available for 663 sons, 192 fathers, and 164 father–son pairs. v
ar(yit − yit−4 ) is the sample variance of 4-year changes in excess
log income.

2.2 Summary Statistics
Table 1 presents summary statistics for father–son pairs.
Each row presents a different variable: initial age, number of
observations, years of education, real (2001 dollars) median
labor income, self-employment (whether ever self-employed
and fraction of work life self-employed), cigarette consumption (number of cigarettes per day for sons; annual expenditure
for fathers), risk tolerance (from survey questions about preferences over hypothetical gambles), and sample variance of excess log income changes. For each, the table provides the mean
and standard deviation for sons and also for their fathers (who
will appear multiple times when a father has more than one
son in the sample), and the correlation between values for these
pairs. Note that except for the number of years in the sample, fathers’ and sons’ attributes are all positively correlated. An initial
taste of the key result of this article is apparent in the last row of
the last column. The crudest measure of income volatility is the
variance of income changes, specifically the sample variance of
changes in excess log income, v
ar(yit − yit−k ). If income volatility is inherited, then fathers with large realized income changes
will tend to have sons with large realized income changes. From
the last row, fathers and sons have positively correlated sample
variances of 4-year changes with a correlation of 0.18 (0.14 in
logs).
The relationship between the income volatility of fathers and
sons is shown graphically in Figure 1. As in the last row of Table 1, Figure 1 uses the sample variance of 4-year excess log income changes, v
ar(yit − yit−4 ). The left column presents sample
variance levels, and the right column presents log sample variances. In each case, circles represent father–son pairs. Fathers’
sample variances are shown on the x-axis, and sons’ sample
variances are shown on the y-axis. Circle size represents the
sample weight, the product of the fathers’ average normalized
PSID sample weight and the number of years of data available
for the father. The figure shows the unpredictable component of

sample variances. This is merely the residual of a regression of
the individual’s sample variance on the following covariates:
initial and average income of the father and son, number of
years of data for each, dummies for the final calendar year of
data for each, and number of years of education for each.
Although there is a great deal of “noisy” variation in these
volatility estimates, the positive relationship between fathers’
and sons’ volatility measures is apparent in both panels. OLS
regression coefficients (with the regression line plotted) are
0.27 in levels (left) and 0.13 in logs (right). OLS standard errors
are all highly significant. Table 2 presents this relationship with
regressions predicting sons’ sample variances with the sample
variances of their fathers. Volatility is computed as the average squared excess log income change where income changes
are taken over 3-, 4-, 5-, and 6-year periods, v
ar(yit − yit−k ). Results are shown both with and without controls (same covariates
as in Figure 1). Coefficient estimates range from 0.20 to 0.36;
again, OLS standard errors imply that these results are highly
significant.
Figure 1 and Table 2 show that fathers with volatile incomes
tend to have sons with volatile incomes. This finding is quite
robust. It is apparent with and without a variety of controls,
for log and level squared changes; for 3-, 4-, 5-, and 6-year
changes, and (in results not shown) when income changes are
and are not adjusted for individual-specific trends in income.
3. MODEL
Although the relationship between fathers’ and sons’ volatilities is shown robustly in Figure 1 and in Table 2, there are
three major problems of interpretation for these reduced-form
results. First, without a model of income dynamics, it is not
clear what the variance of income changes means. Second, the
sample variances of fathers’ income changes are estimated with
error, so the regressions shown in Table 2 suffer from attenuation bias. Third, the sample variances of sons’ income changes

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Shore: The Intergenerational Transmission of Income Volatility

375

Figure 1. Fathers’ and sons’ average squared excess log income changes. Each panel plots a measure of fathers’ income volatilities against
that same measure for their sons. Each circle represents a father–son pair, with size indicating their weight. Weights are the fathers’ sample
weight times the number of observations. Volatility is measured as the average residual squared change in excess log income over all 4-year
periods for which there is data, as described in the text. The left column of panels presents these averages, and the right column presents the
log of these averages. The residual is from a regression of the volatility measure on the initial and average incomes of the father and son, the
number of years of data for each, dummies for the final calendar year of data for each, and the number of years of education for each. Slopes
(and standard errors) for the trend lines in each figure are 0.266 (0.070) and 0.127 (0.038), respectively.

are estimated with error; the magnitude of that error is increasing in the (unobserved) variance, so that heteroscedasticity is
also a problem. The latter two concerns imply that whereas
reduced-form regressions generally yield the right sign, there
may be systematic errors in magnitude and significance.
To address the first problem, I write an income process for
each individual governed by individual-specific income volatility parameters. To address the latter two problems, I estimate
a hierarchical model in which each individual’s volatility parameters are drawn from a joint distribution of fathers’ and
sons’ volatility parameters. This distribution is characterized
by hyperparameters (e.g., the mean and variance of fathers’ and
sons’ volatility parameters, the covariance between fathers’ and
sons’ volatility parameters), which I estimate using maximum
likelihood. Each individual’s observed income data are drawn

conditional on his unobserved volatility parameters. This approach explicitly acknowledges uncertainty about each individual’s volatility parameters, integrating over each possible value
that volatility could take for each person.
3.1 Income Dynamics
The log income for individual i in year t is assumed to take
the following form:
ln(Yit ) = f (Zi,t ) + yit .

Log income is a function of covariates Z (e.g., age, education,
and their interaction; see Section 2 for a list of covariates) and a
stochastic component, yit . The model seeks to capture the evolution of the stochastic term, yi,t , excess log income, composed

Table 2. Intergenerational transmission of total volatility: reduced-form regression of adult children’s
total volatility on their parents’ total volatility
Dependent var.:
Independent var.:
Interval (years), k:
bˆ OLS
Time controls
Educ. controls
Income controls
Observations
R2

0.292
(4.85)
No
No
No
841
0.027

Sample variance of son’s income changes, v
ar(yit − yit−k )
Sample variance of fathers’s income changes, v
ar(yit − yit−k )
3
4
5
6
0.195
(2.93)
Yes
Yes
Yes
841
0.098

0.359
(5.44)
No
No
No
841
0.034

(1)

0.266
(3.68)
Yes
Yes
Yes
841
0.105

0.291
(4.35)
No
No
No
841
0.022

0.201
(2.70)
Yes
Yes
Yes
841
0.096

0.336
(5.04)
No
No
No
841
0.029

0.261
(3.54)
Yes
Yes
Yes
841
0.093

NOTE: Regressions use the sample variance of income changes v
ar(yit − yit−k ), for k = 3, 4, 5, and 6, as both dependent and independent variables. The sample consists of the 841 adult sons with sufficient data in the sample who also have a father with sufficient
data in the sample. Note that fathers with more than one adult son in the sample may appear more than once. When noted, regressions
include as covariates the number of years that the father (and son) are in the sample, dummy variables for the last year the father (and
adult son) were in the sample, the number of years of education for the father (and adult son), and the median and initial income (in
thousands) for the father (and adult son). t-statistics are given in parentheses.

376

Journal of Business & Economic Statistics, July 2011

of permanent income, pi,t , and transitory income. Permanent
income evolves in response to mean 0 permanent shocks, ωi,t ,
and transitory income evolves in response to mean 0 transitory
shocks, εi,t :
yit = pit +

t


φεt−k εik ;

3.2 Econometric Model

k=t−qε +1

pit = pi0 +

t−q
ω

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k=1

ωik +

t


the restriction that φε1 = 0 (at considerable computational cost)
accommodates the slight autocorrelation estimated at 2-year
lags, but the restriction has little impact on the substance of the
results on intergenerational transmission.

(2)
φωt−k ωik .

k=t−qω +1

Permanent shocks (ω) persist forever, although they may enter into income gradually over qω periods as governed by φω .
Transitory shocks (ε) may damp out gradually over qε periods, as governed by φε . The variance of permanent shocks is
viω ≡ E[ωit2 ], the permanent variance or permanent volatility.
The variance of transitory shocks is viε ≡ E[εit2 ], the transitory
variance or transitory volatility.
I assume that the shocks to permanent and transitory income
{ωit , εit } are uncorrelated and normally distributed conditional
on vi .
This income process is standard within the income dynamics literature, following Hall and Mishkin (1982), Carroll and
Samwick (1997), Meghir and Pistaferri (2004). The assumptions underlying this class of models have been studied at length
(e.g., Abowd and Card 1989). Chief among these assumptions
is that one-period (in this case, 1 year) changes in yit are uncorrelated at lags longer than max(qω , qε + 1). Whereas Abowd
and Card found evidence for positive autocorrelation at lags of
1 year (as allowed by these restrictions) and (to a far lesser
degree) 2 years, they found no evidence of autocorrelation at
longer lags. This parsimonious framework is not the only existing model of earnings dynamics, however. In particular, it
rules out the possibility of an AR structure in income levels
(Gottschalk and Moffitt 2002) and does not allow for heterogeneous rates of income growth (Baker 1997; Baker and Solon
2003). Baker (1997) argued that the absence of autocorrelation in income changes in Abowd and Card (1989) used to
defend homogeneous rates of income growth in equation (2)
merely reflects sample and power problems. Note that reducedform robustness checks in the previous section include heterogeneous rates of income growth as a robustness check, without
substantial differences in the rate of intergenerational transmission. With this in mind, the simplifying assumption that income
growth rates do not vary seems not to be problematic for this application. Heterogeneous growth rates may affect the estimated
level of income volatility without changing the correlation of
fathers’ and sons’ volatility.
I estimate the simplest model of the form in equation (2),
in which φωq = 1 for all q and φεq = 0 for all q > 0. In other
words, permanent and transitory shocks are iid. This simplification is made for computational tractability, although reducedform results—coupled with the fact that this simplifying assumption is not wildly at odds with reality—suggest that the
main results will be robust to relaxing this restriction. This
restriction implies that 1-year changes in excess log income,
dyit ≡ yit − yit−1 , are not autocorrelated at lags greater than 1
year (E[dyit dyit−q ] = 0 for q > 1). After 2 years, shocks will
have disappeared completely or will persist forever. Relaxing

This article considers the joint distribution of permanent and
transitory volatility parameters for adult sons and their fathers,
vi ≡ {viωp , viεp , viωk , viεk } ≡ {vip , vik }. p subscripts refer to fathers (parents), k subscrips refer to sons (kids), y refers to the
set of all data on excess log income, yi refers to data for father–
son pair i, and yip and yik refer to the income series for fathers
and sons in pair i, respectively. I assume that the rates at which
shocks enter into permanent income and damp out of transitory income, φ ≡ {φω , φε }, do not differ across fathers or across
sons, although the parameters for fathers could differ from those
for sons.
I assume that fathers’ and sons’ income changes are conditionally independent, each following the income process described in Section 3.1. As a result, the likelihood of observing a
father’s and son’s income data given their volatility parameters
is the product of the conditional likelihood functions:
f (yi |vi ) = fp (yip |vip )fk (yik |vik ),

(3)

f (yi |vi ) is the probability of observing the data for father–son
pair i, yi , given a particular set of volatility values, vi ; fp (yip |vip )
is the probability of observing father i’s data, yip , given a particular set of volatility values, vip ; sons are the same, with k subscripts replacing p subscripts. Furthermore, the income changes
of father–son pairs are assumed to be independent of one another, so that

f (y|v) =
f (yi |vi ).
(4)
i∈N

I consider a hierarchical model in which each father–son
pair’s four volatility parameters have a joint lognormal distribution,
ln vi ∼ N(ln v, θ) ≡ g(ln vi |),

(5)

governed by (4 × 1) mean vector ln v and (4 × 4) variance–
covariance matrix θ ;  ≡ {ln v, θ, φ}. Draws are independent
across father–son pairs. The lognormal distribution is chosen because its long right tail is consistent with the empirical
volatility distribution (Jensen and Shore 2008). The variance–
covariance matrix of fathers’ and sons’ log volatility parameters, θ , is
⎡
⎤
θωp,ωp θωp,εp θωp,ωk θωp,εk


θpp θkp
θεp,εp θεp,ωk θεp,εk ⎥
⎢θ
θ ≡ ⎣ εp,ωp
. (6)
⎦≡
θωk,ωp θωk,εp θωk,ωk θωk,εk
θpk θkk
θεk,ωp θεk,εp θεk,ωk θεk,εk
As an example of notation, θωp,εk is the covariance between
fathers’ permanent volatility parameters and sons’ transitory
volatility parameters. We are interested in the (2 × 2) off′ );
diagonal blocks of covariance hyperparameters (θpk = θkp
these identify the correlation of fathers’ and sons’ volatility parameters.

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Shore: The Intergenerational Transmission of Income Volatility

377

Unfortunately, the four-dimension integrals required to solve
this problem are computationally intensive. To make the problem tractable, I make simplifying assumptions in two areas.
First, I restrict the income process, requiring iid income shocks
as discussed in the last paragraph of Section 3.1. Second, I restrict the structure of within-person volatility heterogeneity.
I propose two opposite assumptions about within-person heterogeneity and show that results are robust across these disparate assumptions.
At one extreme, I assume that permanent volatility and transitory volatility are uncorrelated. Nonetheless, the permanent
and transitory volatility of parents may be correlated with the
permanent and transitory volatility of children. At the other extreme, I can assume that log permanent and transitory volatility
are perfectly (positively) correlated within an individual. Naturally, the preferred alternative would be to estimate the correlation directly. I avoid estimating this correlation, because estimates are highly sensitive to violations of the iid assumption
for the shocks required to estimate the model.
When permanent and transitory volatility are uncorrelated,
there are four free covariance hyperparameters in θpk : the covariance of fathers’ log permanent and transitory volatilities
with sons’ log permanent and transitory volatilities. When log
permanent and transitory volatility are perfectly correlated,
there is only one free covariance hyperparameter. Restrictions
on θ implied by perfect and zero correlation are given in the Appendix.
It is now computationally feasible to use maximum likeliˆ k,
ˆ p and 
hood to find 
p ≡ {ln vωp , ln vεp , θωp,ωp , θεp,εp , φp };

(7)

k ≡ {ln vωk , ln vεk , θωk,ωk , θεk,εk , φk },
the hyperparameters that govern fathers’ and sons’ marginal
volatility distributions. Given these estimates, I estimate the
covariance(s) of fathers’ and sons’ log volatility parameters,
θpk , again by maximum likelihood. The Appendix discusses the
computation of these estimates.
4. RESULTS
4.1 Distribution of Volatility Parameters
Table 3 presents the distribution of volatility parameters implied by maximum likelihood point estimates. The table shows
the distribution of the standard deviation of permanent (excess
log) income changes in the population, and also the standard

deviation of transitory (excess log) income changes in the population. This table shows the distribution under the assumption
that permanent and transitory log volatility are perfectly positively correlated (top panel) or uncorrelated (bottom panel). The
2.5th, median, mean, and 97.5th percentile of the distribution
for fathers and sons are shown. Under perfect correlation for
fathers, the median permanent and transitory (excess log) income changes have standard deviations of 22 percent and 10
percent (in log points) per year, respectively. For sons, the income changes are modestly larger, with standard deviations of
29 and 12 percent (in log points) per year, respectively. Under the assumption that permanent and transitory variance are
uncorrelated, the marginal distributions for permanent and transitory volatility are similar.
Volatility parameters vary substantially across individuals,
and likelihood ratio tests overwhelmingly reject the hypothesis
that all fathers (and separately all sons) have the same volatility parameter. When permanent and transitory volatility are assumed to be perfectly correlated, 95 percent of fathers’ permanent and transitory shocks have standard deviations ranging
from 7 percent and 2 percent at the low end to 66 percent and 66
percent at the high end, respectively. Some 95 percent of sons
have permanent and transitory shocks ranging from 9 percent
and 2 percent to 92 percent and 63 percent, respectively.
Fathers’ and sons’ volatility parameters are highly correlated.
This is apparent from Figure 2, which graphs the estimated joint
distribution of fathers’ and sons’ volatility parameters, assuming that permanent and transitory volatility for an individual are
perfectly correlated. The covariance between fathers’ and sons’
log permanent volatility parameters is 0.37, which implies a
correlation of 28 percent. The hypothesis that the correlation is
below 22 percent or above 35 percent can be rejected at the 5
percent level. This covariance implies a regression coefficient
of 0.33 (when fathers’ log volatility is used to predict sons’
log volatility). This is larger than 0.14 correlation of log sample variances in the right panel of Figure 1, consistent with the
attenuation bias expected from the reduced-form measure.
Table 4 presents estimates of the off-diagonal elements from
equation (6), θpk , the correlation between fathers’ and sons’
volatility parameters (assuming that permanent and transitory
volatility parameters are uncorrelated for fathers and uncorrelated for sons). Correlations are presented in the top row, with
covariances of logs (the primitive estimated by the model and
used to construct estimates of correlation) and their standard
errors. The first column shows the correlation between fathers’
and sons’ permanent volatility parameters (17.5%); the second
column shows the correlation between fathers’ permanent and

Table 3. Distribution of volatility parameters
Standard
deviation

Father’s volatility
2.5th

Median

Mean

Son’s volatility
97.5th

2.5th

Median

Mean

97.5th

Permanent and transitory volatility perfectly correlated
Permanent
7.1%
21.6%
29.4%
65.5%
Transitory
1.5%
10.1%
24.5%
66.4%

9.3%
2.4%

29.2%
12.5%

40.5%
24.1%

91.6%
63.4%

Permanent and transitory volatility uncorrelated
Permanent
4.7%
18.0%
25.3%
Transitory
2.3%
15.5%
24.9%

5.7%
2.5%

22.4%
20.1%

31.6%
33.9%

88.5%
161.0%

69.7%
102.7%

378

Journal of Business & Economic Statistics, July 2011

the inheritance of income volatility are robust to the assumed
correlation between an individual’s permanent and transitory
volatilities.

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5. CHANNELS OF TRANSMISSION

Figure 2. Joint distribution of (log, permanent) volatility parameters. This figure presents graphically the estimate of the joint distribution of fathers’ and sons’ (permanent) volatility parameters (under
the assumption that transitory volatility is a deterministic function of
permanent volatility for any individual.

sons’ transitory volatility parameters (11.1%); the third column
shows the correlation between fathers’ transitory and sons’ permanent volatility parameters (6.7%); the fourth column shows
the correlation between fathers’ and sons’ transitory volatility
parameters (11.7%).
The correlation between fathers’ and sons’ permanent and
transitory volatility parameters is nontrivial. Each of these correlations is estimated with substantial error, so that the hypothesis that any one of them is zero cannot be rejected. However, the
correlation of estimation errors is such that the hypothesis that
all four correlations are zero can be strongly rejected. In particular, I can reject the hypothesis that fathers’ permanent volatility
has no impact on their sons’ (permanent or transitory) volatility; furthermore, I can reject the hypothesis that sons’ transitory
volatility is not affected by their fathers’ (permanent or transitory volatility).
Results under perfect correlation and no correlation imply
similar explanatory power of fathers’ volatility in predicting
sons’ volatility. When I assume that log permanent and transitory volatility are perfectly correlated, the univariate regression
has an R2 of 0.078, and the correlation is 28%. When I assume
no correlation, regressions to predict sons’ permanent and transitory volatility (each with fathers’ two volatilities) have R2 values of 0.035 and 0.026, respectively. This shows that results on
Table 4. Estimated correlation of fathers’ and sons’ volatilities
Fathers:
Sons:
Corr.
Cov. (logs)
(s.e.)

Permanent

Transitory

Perm.
θωp,ωk

Trans.
θωp,εk

Perm.
θεp,ωk

Trans.
θεp,εk

0.175
0.412
(0.550)

0.111
0.405
(1.014)

0.067
0.242
(0.325)

0.117
0.552
(0.642)

NOTE: This table presents estimates of the off-diagonal elements from equation (6), θpk ,
the correlation between fathers’ and sons’ volatility parameters (assuming that permanent
and transitory volatility parameters are uncorrelated for fathers and uncorrelated for sons).
Correlations are presented in the top row, with covariances of logs and their standard errors
below. Standard errors are calculated from the estimate of the information matrix.

The last section documented the intergenerational transmission of income volatility. What explains this inherited volatility? Parents pass along a variety of traits to their children. Many
of these are correlated with income volatility. The two most
natural channels of such transmission (as it related to income
volatility) are the inheritance of a preference (or tolerance) for
a volatile income and the inheritance of a technology (human
capital) that generates a volatile income.
Dohmen et al. (2006) provided an extensive treatment of the
risk-related behaviors that parents pass along to their children:
“Children of parents who smoke, take drugs, commit crimes,
and engage in early sex are more likely to do the same compared
with children whose parents do not engage in these activities.”
Although most of these measures of risky behavior are absent
from the PSID, the PSID does contain information about smoking, as shown in Table 1. Smoking is measured with cigarette
expenditure in the first 5 years of the sample; because fathers
are generally present early in the sample, I use median smoking
expenditure to capture fathers’ smoking behavior.
Smoking is measured by the number of cigarettes per day in
1986 and after 1998; because sons are generally present later
in the sample, I use the median number of cigarettes per day
to capture sons’ smoking behavior. Fathers’ and sons’ smoking
measures have a correlation of 0.17 in the PSID, a result that is
significant at the 1 percent level. However, smoking behavior is
uncorrelated with income volatility in the PSID. Although this
evidence is far from conclusive, there is no evidence that the
inheritance of risky behaviors more generally is linked to the
intergenerational transmission of income volatility.
The PSID provides another risk measure that might be more
salient for income volatility: risk tolerance. In 1988, the PSID
asked respondents whether they would be willing to accept hypothetical income gambles. Answers to these questions were
used to back out a coefficient of relative risk aversion (CRRA).
These estimates are summarized in Table 1. Because this question was asked in the latter half of the PSID, responses are
available for most sons but not for most fathers, with data on
both in father and son in only 164 pairs. Table 1 shows that the
correlation of fathers’ and sons’ risk aversion parameter estimates is positive. Fathers’ and sons’ 1/CRRA estimates have
a correlation of 0.13, significantly different from 0 at the 10
percent level; fathers’ and sons’ incidence of CRRA > 3 (1 if
yes, 0 if no) have a correlation of 0.09, which is significantly
different from 0 at the 5 percent level. Sons’ volatility parameter estimates are correlated with these estimates of risk aversion. The correlation between sons’ volatility parameter estimates and 1/CRRA is 0.07 (significant at the 10 percent level);
the correlation between volatility parameter estimates and incidence of CRRA > 3 (1 if yes, 0 if no) is 0.08 (significant at
the 5 percent level).
The facts that fathers and sons have correlated risk tolerance,
and that risk tolerance is correlated with income volatility, provide suggestive evidence that inheritance of income volatility

Shore: The Intergenerational Transmission of Income Volatility

379

Table 5. Determinants of son’s volatility
Volatility
measure
Son selfemployed
Father Selfemployed
Father’s
volatility
Obs.
R2

Model

Reduced
form

0.424
(6.79)*

0.359
(5.44)*

841
0.052

841
0.034

Model

Reduced
form

0.329
(21.14)*

1.411
(18.97)*

841
0.348

841
0.300

Model

Reduced
form

Model

Reduced
form

Model
0.326
(19.97)*
−0.039
(2.70)*
0.215
(3.60)*

1.429
(18.26)*
−0.206
(3.02)*
0.163
(2.58)*

841
0.359

841
0.310

0.063
(4.10)*

0.210
(2.95)*

0.017
(0.99)
0.390
(5.45)*

0.047
(0.60)
0.339
(4.59)*

841
0.020

841
0.010

841
0.053

841
0.034

Reduced
form

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NOTE: Dependent variable is the son’s income volatility. Model volatility refers to the estimates of permanent volatility from the econometric model, under the assumption that log
permanent and transitory volatility are perfectly correlated; reduced-form volatility refers to the sample variance of four-year changes in excess log income. Self-employment refers to
the fraction of work life the individual spent self-employed. Standard errors are from an OLS regression.

might be related to the inheritance of risk preference. Riskaverse fathers have low income volatility and tend to have riskaverse children, who in turn have low income volatility. However, neither the relationship between risk tolerance and income
volatility nor the relationship between risk tolerance of fathers
and sons is strong enough to explain much of the intergenerational transmission of income volatility. Self-employment is a
more promising candidate explanation.
Self-employment is strongly inherited (Dunn and HoltzEakin 2000), and also strongly correlated with volatility. In
these data, I measure self-employment as the fraction of time
in the sample that the individual (father or son) identifies as
self-employed. A regression to predict sons’ self-employment
with fathers’ self-employment yields a highly significant OLS
coefficient of 0.24.
Table 5 shows how self-employment affects inherited volatility. Odd-numbered columns use model estimates of individualspecific permanent volatility, under the assumption that log
permanent and transitory volatility are perfectly correlated;
even-numbered columns use reduced-form ones (same variance
of 4-year excess log income changes). Columns 1 and 2 repeat earlier regressions showing inheritance of volatility. Subsequent columns add measures of self-employment.
Columns 3 and 4 of Table 5 show that sons who spend
more time self-employed have much higher income volatility.
Columns 5 and 6 show that fathers who spend more time selfemployed tend to have children with higher income volatility. Columns 7 and 8 show that this effect goes away once
fathers’ volatility is included. Self-employed fathers tend to
have sons with volatile incomes, but only to the degree that
self-employment makes their incomes volatile. The last two
columns (columns 9 and 10) show that accounting for sons’
self-employment substantially reduces the direct effect of fathers’ volatility on sons’ volatility. In other words, one reason why fathers with volatile incomes tend to have sons with
volatile incomes is that fathers with volatile incomes are more
likely to have self-employed sons, and the self-employment of
these sons leads to higher income volatility.
Whereas the intergenerational transmission of self-employment and income volatility are clearly linked, the interpretation of this linkage is less clear. Self-employment is a choice
that may reflect inherited preference (e.g., inherited risk tolerance) or inherited skills (e.g., working independently). This

leaves unanswered the question of whether it is a preference
for a volatile income or a volatile income-producing technology that is inherited. The inheritance of risk tolerance and its
link with volatility suggests that inherited preferences account
for at least some of the inheritance of income volatility.
6. CONCLUSION
This article provides evidence of intergenerational transmission of higher moments of human capital. Parents with more
volatile incomes tend to have children with more volatile incomes. This pattern is visible in sample moments and figures,
but is quantified with an econometric structure to explicitly account for measurement error in individual-specific estimates of
income volatility. Just as previous research has found intergenerational transmission of the level of income, I show that there
is intergenerational transmission of income volatility, the parameter that governs the rate of intragenerational mobility. This
inheritance of volatility is linked to the inheritance of financial
risk tolerance and self-employment, providing suggestive evidence of an inherited taste or tolerance for volatile incomes.
The inheritance of income volatility may inform our view of
social programs aimed at entrenched poverty (e.g., Head Start)
or wealth (e.g., the estate tax). Children will inherit their parents’ place in the lifetime income distribution to the extent that
both (a) they have an expected lifetime income or initial income similar to their parents and (b) their place in the income
distribution does not change substantially from that expectation or starting point. Such policies tackle (a); they are most
needed (as a tool to dislodge dynasties) under (b). Because income volatility is inherited, such policies are most needed in
families with low-volatility parents; they have children who
will likely become low-volatility adults. Absent an intervention
to change their expected lifetime income, these low-volatility
individuals will not stray far from their inherited trajectories.
High-volatility dynasties will not remain stuck at a point in the
income distribution even without policy intervention. An example of such a targeted intervention would be an estate tax with
an exclusion amount that increases with the decedent’s income
variability. Such a policy would dislodge wealthy low-volatility
dynasties (who would not otherwise be dislodged) more aggressively than high-volatility dynasties (whose intragenerational

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380

Journal of Business & Economic Statistics, July 2011

mobility means that they are unlikely to remain at the same
place in the income distribution).
Of course, income volatility can lead incomes to move both
up and down. Volatility may dislodge a poor dynasty from its
place in the income distribution by making it even poorer or
may dislodge a rich one by making it richer. Although the word
“volatility” evokes a “bad” and “mobility” a “good,” volatility
will be welcomed ex post only if shocks turn out to be positive.
This suggests that the optimal timing of policy interventions
may differ by dynastic volatility. Early interventions (e.g., Head
Start) may be preferred for low-volatility families, for whom the
children are less likely to move from their expected low level of
income; for high-volatility dynasties, expensive interventions
might be more effective later, targeting only those whose incomes fail to increase. Among initially poor, high-volatility
dynasties who receive positive shocks, policy interventions to
ameliorate poverty will be unnecessary.
APPENDIX: ESTIMATION DETAILS
Because individuals’ volatility parameters, v ≡ {v1 , . . . , vN },
are unobserved, finding a solution requires integrating over all
possible volatility values vi for all father–son pairs i (Gelman et
al. 2004):
arg max L(y|),

where

ln v,θ

L(y|) =



···



f (y|v)g(ln v|) dv,

(A.1)

v

where f (y|v) is the probability of observing all of the data, y,
given a particular set of volatility values for all individuals,
v, and g(ln v|) is the probability of observing those volatility values, v, given the distribution from which volatility values are drawn, . Recall that for i = j, yi is independent of yj
co