# Consumption Partial Insurance
in the Presence of Tail Income Risk††thanks: We thank Manuel Arellano, Richard
Blundell, Jeff Campbell, George Constantinides, Mariacristina De Nardi, Dmytro
Hryshko, Christian Julliard, Chris Telmer, and seminar participants in the
Structural Econometrics Group at Tilburg University and the BSE Summer Forum
2023 (Macroeconomics and Social Insurance) for many helpful comments. We also
thank Ties Schalij for excellent research assistance.

Anisha Ghosh Desautels Faculty of Management, McGill University; email:
<EMAIL_ADDRESS>Alexandros Theloudis Department of Econometrics &
Operations Research, Tilburg University; email<EMAIL_ADDRESS>
###### Abstract

We measure the extent of consumption insurance to income shocks accounting for
high-order moments of the income distribution. We derive a nonlinear
consumption function, in which the extent of insurance varies with the sign
and magnitude of income shocks. Using PSID data, we estimate an asymmetric
pass-through of bad versus good permanent shocks – 17% of a $3\sigma$ negative
shock transmits to consumption compared to 9% of an equal-sized positive shock
– and the pass-through increases as the shock worsens. Our results are
consistent with surveys of consumption responses to hypothetical events and
suggest that tail income risk matters substantially for consumption.

Keywords: Income risk, skewness, kurtosis, consumption, partial insurance,
PSID.
JEL Classification: D12, D15, D31, E21

##  1 Introduction

How does consumption respond to income shocks? The answer to this question is
important for determining the welfare effects of shifts in the income
distribution, i.e., the link between income and consumption inequality, for
understanding how families cope after adverse events, for designing government
insurance, and for the dynamics of business cycles. A consistent empirical
finding is that consumption is partially insured against income shocks, so
income fluctuations translate into consumption fluctuations less than one-to-
one (e.g., Blundell, Pistaferri, and Preston, 2008; Heathcote, Storesletten,
and Violante, 2014).111The first empirical studies of consumption insurance
date back to tests of the permanent income hypothesis (e.g., Deaton and
Paxson, 1994) or tests (and rejections) of complete markets (e.g., Cochrane,
1991; Attanasio and Davis, 1996; Hayashi, Altonji, and Kotlikoff, 1996; Brav,
Constantinides, and Geczy, 2002). Meghir and Pistaferri (2011) offer a review
of this literature. With a few exceptions discussed subsequently, most
measures of consumption insurance do not distinguish between good and bad
shocks, or between small and large ones. However, recent evidence in multiple
countries shows that the distribution of income risk has a _long and fat left_
tail; that is, there is an asymmetrically large probability that households
face some really bad income shocks. This paper measures the consumption
response to income shocks precisely when the distribution of said shocks may
exhibit long and fat tails.

In a seminal paper, Blundell, Pistaferri, and Preston (2008) –henceforth, BPP–
introduce a methodology to measure the pass-through of income shocks of
varying persistence into consumption. Using Panel Study of Income Dynamics
(PSID) and Consumer Expenditure Survey (CEX) data over 1980–1992, BPP find
evidence of partial insurance to permanent shocks and full insurance to
transitory ones. BPP rely on a workhorse covariance between income and
consumption growth, namely a _second_ moment of their joint distribution. The
idea is that the extent to which income growth (driven by income shocks)
varies with consumption growth reflects the strength of the transmission of
income shocks into consumption.

In a separate strand of literature, Guvenen, Karahan, Ozkan, and Song (2021)
use data from the U.S. Social Security Administration to characterize the
distribution of unexplained income growth, namely the distribution of income
shocks. They establish two important facts. First, income growth is very
negatively skewed, i.e., it has a long left tail. This means there are far
more people in the data who experience large negative than large positive
shocks. Second, income growth exhibits excess kurtosis, i.e., it has fat
tails. This means there are far more people in the data who experience either
small or large income shocks, than people who experience moderate ones,
relative to a Gaussian density. These are now benchmark results in the income
dynamics literature, replicated in many countries (e.g., De Nardi, Fella,
Knoef, Paz-Pardo, and Van Ooijen, 2021, compare the U.S. and the Netherlands).

If most shocks away from the mean are really bad shocks, their impact on
consumption would, intuitively, differ from that of small/moderate shocks. In
other words, left skewness and excess kurtosis, which are statements about the
third and fourth moments of the income distribution, should matter for the
pass-through of shocks into consumption. Yet, by revolving around the second
moments of the joint income-consumption distribution, the second moment being
a poor summary of the tails, the benchmark methodology in BPP ignores these
salient features of income dynamics. This is precisely what we address in this
paper.

The paper measures the degree of consumption partial insurance to income
shocks of varying persistence, accounting for higher-order moments of the
distribution of income and consumption. We offer three main contributions.

First, we replicate BPP (1980–1992) using recent PSID data (1999–2019). The
replication targets the original income and consumption second moments.
However, while BPP imputed consumption from the CEX (the PSID previously
lacked comprehensive consumption data), such data are now internally available
in the PSID. The replication is thus interesting not only because of the
different time period, but also because we no longer need to rely on imputed
consumption from the CEX.

Second, we introduce into BPP the third and fourth moments of the joint
distribution of income and consumption, in addition to the second moments that
BPP originally used. We thus go beyond the workhorse income-consumption
covariance to measure consumption insurance targeting additional information
available in the tails of the distribution.

Third, we generalize the method of measuring consumption partial insurance to
explicitly permit a role for tail income risk. BPP log-linearize the
optimality conditions and intertemporal budget constraint of a lifecycle
consumption/savings problem to obtain a consumption function that depends
_linearly_ on income shocks. We argue (see also Carroll, 2001) that this
linear function is a poor approximation to behavior when income is subject to
tail shocks. We then derive a higher-order (quadratic) consumption function
that nests BPP’s linear specification. We characterize the consumption pass-
through of income shocks in the generalized model and we establish
identification of its parameters. In this environment, the degree of partial
insurance depends on the magnitude and sign of shocks, so bad or extreme
shocks can potentially shift consumption differently than good or small shocks
do.

Our baseline empirical implementation uses income and consumption panel data
from the PSID over the years 1999–2019. We focus on a representative sample of
stable married couples that is similar to the sample in BPP. We show that
unexplained income growth in this sample features large negative skewness and
excess kurtosis, qualitatively and numerically close to the skewness and
kurtosis in the administrative data of Guvenen, Karahan, Ozkan, and Song
(2021). We then obtain four main findings.

First, permanent and transitory shocks to disposable household income are
negatively skewed, with skewness coefficients at $-0.95$ and $-1.31$
respectively, and highly leptokurtic, with kurtosis coefficients at $44.39$
and $49.23$ respectively. Left skewness implies that a negative shock is
typically more unsettling, more hurtful than a positive shock, because it is
further away from the zero mean. Excess kurtosis implies that more shocks are
concentrated either in the middle of the distribution or far out in the tails,
relative to the shoulders. Taken together, large left skewness and excess
kurtosis suggest that the realizations of income shocks tend to be either
close to their zero mean or far out in the left tail; that is, income shocks
for most households are either small/average or quite bad.

Second, in replicating BPP over the recent years, we find full insurance to
transitory shocks and a very small pass-through rate of permanent shocks at
0.15 versus 0.64 in BPP. A 10% permanent income cut thus reduces consumption
by only 1.5%, vis-à-vis 6.4% in BPP. In other words, we find much larger
partial insurance to permanent shocks than originally found in BPP.222Using
the recent PSID and an alternative model, Arellano, Blundell, Bonhomme, and
Light (2023) find a similarly low pass-through rate of permanent shocks,
precisely as we do. We attribute one third of the difference from BPP to the
consumption imputation from the CEX that BPP use. The imputation brings in
large amounts of measurement error, inflating the covariance between
contemporary income and consumption growth. Another third of the difference
arises because the modern PSID provides data biennially (versus annually
previously), which tends to mute higher frequency shocks and their
transmission into consumption. The last third reflects the different calendar
times.

Third, introducing the third and fourth moments of income and consumption
growth into BPP’s linear consumption setting does not change our estimates of
partial insurance. This result is counterintuitive because a big literature
highlights the large welfare implications of tail risk (e.g., De Nardi, Fella,
and Paz-Pardo, 2019; Busch and Ludwig, 2021). One would thus expect that
targeting higher-order moments would affect the degree of partial insurance,
which is not what we find. This result is suggestive of a potential
misspecification in the linear consumption function, which motivates our
generalization of the methodology through the introduction of a higher-order
consumption specification.

Fourth, in estimating our quadratic consumption function, we find an asymmetry
in the pass-through of bad versus good shocks, and of small versus large ones.
With a quadratic consumption function, the pass-through depends on the
magnitude and sign of the shocks; and targeting higher-order income moments
becomes essential for the identification of the generalized transmission
parameters. We draw three main conclusions. First, bad _permanent_ shocks are
more hurtful than good ones because they have larger transmission rates into
consumption – a three standard deviations negative shock (a 50% permanent
income cut) has a pass-through of $0.17$ (17% of the shock transmits into
consumption), whereas a positive shock of similar magnitude and persistence
has a pass-through of $0.09$. Second, the pass-through of bad shocks increases
with the severity of the shock. So, not only are bad permanent shocks more
extreme than good ones (they lie further away from the zero mean in the left
tail), they also have increasingly larger transmission rates as their
magnitude worsens. Third, _transitory_ shocks provide mixed evidence. They are
fully insured in the full sample, at least from a statistical point of view,
even though the point estimates of their transmission parameters suggest that
large transitory windfalls shift consumption considerably, consistent with
quasi-experimental evidence of large responses to transitory gains (e.g.,
Fagereng, Holm, and Natvik, 2021). Among low wealth households, however,
transitory shocks shift consumption statistically significantly, with negative
shocks passing through at an increasingly larger rate than positive ones.
Overall, the results are consistent with an environment in which households
save a good income shock but they dissave in the presence of a bad one. As
there is clearly a limit to how much one can dissave or borrow, a large
negative shock would typically affect consumption by relatively more.

The asymmetric transmission of bad versus good permanent shocks intensifies
with the age of the household. While older households can typically insure
average permanent shocks better than younger households, the former are
associated with larger transmission rates of extreme shocks than the latter.
In other words, severely bad permanent shocks cause more harm among older
households than among younger ones. This agrees with Guvenen, Karahan, Ozkan,
and Song (2021), who show that income skewness becomes more negative with the
age of the earner – there is more room for income to fall as older workers
have higher incomes; we find a similar pattern in our sample too. We further
document that there is also more room for consumption to fall, so bad income
draws are doubly hurtful among the elderly. Overall, the results from our
quadratic specification are indicative of limited insurance to left tail
income risk. Moreover, if the econometrician assumed a linear consumption
function, she would overestimate the true extent of insurance to it, i.e., she
would infer excessive insurance.

The paper is primarily related to the extensive literature that studies the
transmission of income shocks into consumption, i.e., the link between income
and consumption inequality. There are two distinct strands in this literature:
a quasi-reduced form branch that _measures_ the degree of partial insurance in
the data without fully specifying the underlying insurance mechanisms, and a
structural branch that fully _models_ specific insurance channels.

The seminal paper in the first branch, namely BPP, sparked a large volume of
subsequent research.333Hall and Mishkin (1982), Deaton and Paxson (1994) and
Blundell and Preston (1998) characterize the empirical distribution of
consumption and income, so they are early precursors in this literature.
Blundell, Pistaferri, and Saporta-Eksten (2016) extend BPP to an environment
with wage risk to measure the contribution of family labor supply to partial
insurance. Theloudis (2021) extends this further to allow for household
heterogeneity, which he estimates using information in the third income-
consumption moments. Commault (2022) allows consumption to respond to past
transitory shocks and finds a strong response to them, similar to what quasi-
experimental studies find. Hryshko and Manovskii (2022) carry out an
interesting subsample analysis of BPP, identifying sets of households in the
PSID with different degrees of insurance based on certain features of their
incomes. Crawley and Kuchler (2023) offer another empirical extension,
addressing neglected time aggregation in income and consumption in the PSID.
Arellano, Blundell, and Bonhomme (2017) and Arellano, Blundell, Bonhomme, and
Light (2023) generalize the income process by letting shocks of different
size/sign to feature varying persistence.444The latter paper advances the
former by allowing for flexible heterogeneity and unbalanced panels.
Abstracting from income skewness and kurtosis, they measure the consumption
response to income in the modern PSID at about 0.2 on average (specification
with household heterogeneity in the latter paper), though the response varies
over the distribution of shocks.

In the present paper, by contrast, we generalize both the income process, by
allowing shocks to be non-Gaussian, and the consumption function, by allowing
for non-linearities in the transmission of income shocks. The value added is
our finding of asymmetric pass-through of shocks depending on their size and
sign – a recurrent finding in recent surveys of responses to hypothetical
scenarios (e.g., Fuster, Kaplan, and Zafar, 2020). To measure this asymmetry,
information in higher-order income moments becomes indispensable. We thus
characterize carefully how our measures of partial insurance change and what
can be learned as we gradually introduce higher-order information.

The second branch studies specific insurance mechanisms through calibrating
lifecycle models of consumption/savings.555Gourinchas and Parker (2002) and
Krueger and Perri (2006) are early precursors in this strand. Much of the
early research focused on self-insurance, providing mixed evidence for how
much insurance there is. Kaplan and Violante (2010) find less insurance to
permanent shocks in the model compared to the data, while Guvenen and Smith
(2014) estimate income risk and the degree of self-insurance jointly and find
that about one half of income shocks are smoothed, i.e., they find more
insurance than BPP. More recent works study the insurance consequences of
labor supply (Heathcote, Storesletten, and Violante, 2014), durable purchases
(Madera, 2019), health (Blundell, Borella, Commault, and Nardi, 2020), or
added workers (Wu and Krueger, 2021). Except Madera (2019), these papers
abstract from higher-order moments of income, partly due to the computational
difficulties in modelling those. De Nardi, Fella, and Paz-Pardo (2019) embed a
non-Gaussian income process into a model of consumption/savings and show that
tail income risk increases the degree of partial insurance to permanent shocks
due to stronger precautionary motives. Busch and Ludwig (2021) estimate a
similar model of self-insurance targeting income skewness and kurtosis. They
confirm that higher-order moments generate stronger precautionary savings;
they distinguish, however, between good and bad shocks and they show that the
latter are in fact worse insured than the former. Unlike this vast
quantitative literature, we do not take a stance on the insurance mechanisms
available to households; this enables us to _measure_ the overall degree of
insurance from all mechanisms together, keeping our model partially
unspecified while also accounting for the higher-order income moments.

The paper is also related to the modern income dynamics literature that
investigates the non-Gaussian features of the income distribution. Geweke and
Keane (2000) document that earnings growth exhibits left skewness, while
Bonhomme and Robin (2010) document its excess kurtosis, both using PSID data.
Guvenen, Karahan, Ozkan, and Song (2021) provide evidence of left skewness and
excess kurtosis in earnings growth in the population of working males in the
U.S., and show that these features get accentuated with age and the level of
earnings. Guvenen, Ozkan, and Song (2014) and Busch, Domeij, Guvenen, and
Madera (2018) study the cyclical nature of income skewness in several
countries. While these papers are mostly descriptive of the higher-order
features of the income distribution, we go a step further to investigate what
these features imply for consumption partial insurance. We thus contribute to
the growing literature in economics and finance that studies the various
implications of tail risk (e.g., Constantinides and Ghosh, 2017; McKay, 2017;
De Nardi, Fella, and Paz-Pardo, 2019; Ai and Bhandari, 2021, for the equity
premium and asset prices, business cycles, consumer welfare, and labor market
dynamics, respectively).

The asymmetric response to bad versus good shocks has previously been seen in
the literature that uses surveys to elicit responses to various hypothetical
scenarios. For example, Bunn, Le Roux, Reinold, and Surico (2018) find that
the marginal propensity to consume from bad shocks is larger than from good
shocks; Christelis, Georgarakos, Jappelli, Pistaferri, and van Rooij (2019)
and Fuster, Kaplan, and Zafar (2020) find that responses to losses are larger
than to gains with magnitudes increasing with the severity of the
loss.666Given our finding of heterogeneous responses to shocks, our paper is
also broadly related to the literature that studies MPC heterogeneity (e.g.,
Crawley and Kuchler, 2023; Lewis, Melcangi, and Pilossoph, 2022). We show that
these patterns are also present in survey data, such as the PSID.

In the remainder of the paper, section 2 presents the theoretical framework
through which we study the pass-through of income shocks into consumption and
3 establishes identification of the relevant parameters. Section 4 discusses
the empirical implementation, section 5 presents the results, and section 6
concludes.

##  2 Lifecycle model with tail income risk

The framework through which we study the extent of partial insurance to income
shocks is a consumption lifecycle model with idiosyncratic income risk.

Household $i$ chooses lifetime consumption $\\{C_{it}\\}_{t=0}^{T}$ and
savings $\\{A_{it+1}\\}_{t=0}^{T}$ to maximize its expected discounted utility
over its lifecycle. The problem is formulated as

$\max_{\\{C_{it},A_{it+1}\\}_{t=0}^{T}}{\operatorname*{\mathbb{E}}}_{0}\sum_{t=0}^{T}\beta^{t}U(C_{it};\mathbf{Z}_{it}),$
(1)

subject to the lifetime budget constraint

$A_{i0}+{\operatorname*{\mathbb{E}}}_{0}\sum_{t=0}^{T}\frac{Y_{it}}{(1+r)^{t}}={\operatorname*{\mathbb{E}}}_{0}\sum_{t=0}^{T}\frac{C_{it}}{(1+r)^{t}},$
(2)

where $U(\cdot;\mathbf{Z}_{it})$ is utility over consumption,
$\mathbf{Z}_{it}$ is a vector of observed taste shifters (e.g., age,
education), $\beta$ is the discount factor, $r$ is the deterministic interest
rate, $A_{i0}$ is beginning-of-time financial wealth,
${\operatorname*{\mathbb{E}}}_{0}$ denotes expectations over uncertain future
income $Y$, and $T$ is the end of the horizon. We assume that utility
satisfies standard regularity conditions, in particular it is twice
differentiable, but we do not otherwise parameterize it.

###  2.1 Income process

Income is the only source of (idiosyncratic) uncertainty that the household
faces. Its logarithm is given by

$\ln Y_{it}=\mathbf{X}_{it}^{\prime}\boldsymbol{\delta}+P_{it}+v_{it},$

where $\mathbf{X}_{it}$ is a vector of characteristics in the $t=0$
information set of the household, such as age, education, race, that drive the
deterministic profile of income over the lifecycle; $\boldsymbol{\delta}$ is
the loading factor of those variables on the logarithm of income. The
remaining terms make up the stochastic component of income, consisting of
permanent income $P_{it}$ and transitory shock $v_{it}$. We assume that
permanent income has a unit root, $P_{it}=P_{it-1}+\zeta_{it}$, where
$\zeta_{it}$ is the permanent income shock. These modeling choices give rise
to the familiar permanent-transitory formulation777See Meghir and Pistaferri
(2011) for a comprehensive review of modern income processes.

$\Delta y_{it}=\zeta_{it}+\Delta v_{it},$ (3)

where $\Delta y_{it}=\Delta\ln
Y_{it}-\Delta\mathbf{X}_{it}^{\prime}\boldsymbol{\delta}$ is income growth net
of the deterministic profile, i.e., unexplained income growth, and $\Delta$ is
the first difference operator.

A first point of departure from BPP lies in the specification of the permanent
and transitory income shocks, $\zeta_{it}$ and $v_{it}$, respectively. Unlike
BPP, we allow these shocks to be non-Gaussian, with first four central moments
given by

${\operatorname*{\mathbb{E}}}(\zeta_{it}^{m})=\begin{cases}0&\text{for }m=1\\\
\sigma_{\zeta_{t}}^{2}&\text{for }m=2\\\ \gamma_{\zeta_{t}}&\text{for }m=3\\\
\kappa_{\zeta_{t}}&\text{for }m=4\\\
\end{cases}\qquad\text{and}\qquad{\operatorname*{\mathbb{E}}}(v_{it}^{m})=\begin{cases}0&\text{for
}m=1\\\ \sigma_{v_{t}}^{2}&\text{for }m=2\\\ \gamma_{v_{t}}&\text{for }m=3\\\
\kappa_{v_{t}}&\text{for }m=4.\\\ \end{cases}$

$\sigma_{\zeta_{t}}^{2}$ and $\sigma_{v_{t}}^{2}$ describe the dispersion of
the distributions of, respectively, the permanent and transitory shocks about
their mean; the mean is zero by construction. $\gamma_{\zeta_{t}}$ and
$\gamma_{v_{t}}$ denote the skewness/asymmetry of the distributions, i.e., the
relative lengths of the upper and lower tails. Finally, $\kappa_{\zeta_{t}}$
and $\kappa_{v_{t}}$ denote the kurtosis, i.e., the tendency of the
distributions to amass away from the middle $[-\sigma,\sigma]$, and therefore
characterize the thickness of the tails.

We allow the central moments of shocks to depend on $t$ to reflect that
different stages of the lifecycle may exhibit different amounts of dispersion,
skewness, or thickness of the tails of income risk; Guvenen, Karahan, Ozkan,
and Song (2021) provide some evidence for that. Otherwise, shocks across
households are identically distributed within $t$. Permanent and transitory
shocks are mutually independent and independent over time.

As discussed in the introduction, several recent studies document that the
cross-sectional distribution of income growth exhibits left skewness and
excess kurtosis, i.e., a long and fat left tail. Our specification of the
moments of income shocks enables our income process to be consistent with
these facts, allowing us to subsequently decompose its second- and higher-
order moments into the relative contributions of permanent and transitory
shocks.

###  2.2 Linear consumption function

In line with BPP, we first consider a linear specification of the household
consumption function, given by

$\Delta c_{it}=\xi_{it}+\phi_{it}^{(1)}\zeta_{it}+\psi_{it}^{(1)}v_{it},$ (4)

where $\Delta c_{it}$ is the residual from a regression of real consumption
growth, $\Delta\ln C_{it}$, on the taste observables $\mathbf{Z}_{it}$ (i.e.,
consumption growth net of its predictable profile). The linear function (4)
stems from a log-linearization (i.e., a first-order Taylor series expansion)
of the problem’s first-order conditions and the household budget constraint,
as we show in appendix A. These approximations, which are independent of the
distributions of income shocks, mimic similar derivations in BPP.

Permanent and transitory income shocks drive unexplained consumption growth in
(4). The transmission parameters $\phi_{it}^{(1)}=\partial\Delta
c_{it}/\partial\zeta_{it}$ and $\psi_{it}^{(1)}=\partial\Delta c_{it}/\partial
v_{it}$ reflect the pass-through of permanent and transitory shocks,
respectively, into consumption. Adopting BPP’s terminology,
$1-\phi_{it}^{(1)}$ is then the extent of insurance to permanent shocks (i.e.,
the fraction of a permanent shock that does _not_ translate into a consumption
fluctuation), while $1-\psi_{it}^{(1)}$ is the extent of insurance to
transitory shocks. Therefore, $\phi_{it}^{(1)}=\psi_{it}^{(1)}=0$ implies that
consumption is _fully_ insured to income shocks, while
$\phi_{it}^{(1)}=\psi_{it}^{(1)}=1$ implies _no insurance_. By contrast,
values $\phi_{it}^{(1)},\psi_{it}^{(1)}\in(0,1)$ reflect _partial_ insurance:
a given income shock translates into a consumption fluctuation, albeit less
than one-to-one.

The transmission parameters vary, in principle, with $i$ (household) and $t$
(age), reflecting heterogeneity in financial wealth, among other reasons. The
idea is that a household can use wealth to smooth consumption when income
shocks hit; and the more wealth one holds the better they can self-
insure.888In addition, households in Blundell, Pistaferri, and Saporta-Eksten
(2016) have access to insurance through family labor supply, while households
in Theloudis (2021) differ in their preferences; both are reasons why the
transmission parameters may vary with $t$ or $i$. We establish this link from
wealth to $\phi_{it}^{(1)}$, $\psi_{it}^{(1)}$ in appendix A. So while the
pass-through of shocks depends on wealth, it does not depend on the magnitude
or sign of the shock itself. Function (4) thus implies that a household can
insure big or small, good or bad shocks similarly, a point to which we return
subsequently.

In addition to income shocks, unexplained consumption growth is also driven by
$\xi_{it}$, which reflects unobserved (but non-stochastic) consumption taste
heterogeneity across households, independent of the income shocks. We let the
distribution of $\xi_{it}$ across households to have zero mean and variance
$\sigma_{\xi_{t}}^{2}$, as in BPP. We also let its third and fourth central
moments to be $\gamma_{\xi_{t}}$ and $\kappa_{\xi_{t}}$ respectively; these
will appear subsequently in expressions for the skewness and kurtosis of
consumption growth.

###  2.3 Quadratic consumption function

The linear consumption function offers an approximation to the true
consumption response to income shocks when such shocks do not move the
household too far away from its consumption ‘steady-state’ – i.e., its
deterministic consumption profile. Function (4) relies on a linearization of
marginal utility (the problem’s first-order conditions) around such ‘steady-
state’; but such linearization will be inaccurate in the presence of tail
income risk, e.g., when extreme shocks may induce large shifts in consumption
as a consequence of concave utility.999This echoes Carroll (2001)’s critique
of the log-linearized Euler equation. Kaplan and Violante (2010) and Blundell,
Low, and Preston (2013) assess the approximations in BPP across alternative
scenarios, though not in one with respect to the severity of income shocks.

Our second and main point of departure from BPP is the generalization of the
consumption function to a nonlinear (quadratic) specification, given by

$\Delta
c_{it}=\xi_{it}+\phi_{it}^{(1)}\zeta_{it}+\psi_{it}^{(1)}v_{it}+\phi_{it}^{(2)}\zeta_{it}^{2}+\psi_{it}^{(2)}v_{it}^{2}+\omega_{it}^{(22)}\zeta_{it}v_{it},$
(5)

where $\Delta c_{it}$ and $\xi_{it}$ are defined as previously. The quadratic
function (5) stems from a second-order Taylor approximation to the problem’s
first-order conditions and the household budget constraint, as we show in
appendix A. These operations and resulting expressions generalize the original
approximations in BPP and in several papers thereafter.

The quadratic function is characterized by several parameters. As in the
linear case, $\phi_{it}^{(1)}$ and $\psi_{it}^{(1)}$ are the transmission
parameters into consumption of the linear part of a polynomial in the income
shocks. By extension, $\phi_{it}^{(2)}$, $\psi_{it}^{(2)}$, and
$\omega_{it}^{(22)}$ are the transmission parameters of the quadratic part of
the same polynomial. In the context of long left and fat tails in income risk,
$\phi_{it}^{(1)}$ and $\psi_{it}^{(1)}$ can be thought of as the transmission
parameters of the average shock (a shock close to the mean of the
distribution), while $\phi_{it}^{(2)}$, $\psi_{it}^{(2)}$,
$\omega_{it}^{(22)}$ can be seen as the transmission parameters of the typical
shock away from the middle of the distribution. The parameters may again vary
with $i$ (household) and $t$ (age), reflecting heterogeneity in financial
wealth among other things; we establish this link in appendix A.

In this generalized framework, $\phi_{it}^{(1)}$ no longer represents the
pass-through of permanent shocks to consumption. Instead, the pass-through is
given by

$\frac{\partial\Delta
c_{it}}{\partial\zeta_{it}}=\phi_{it}^{(1)}+2\phi_{it}^{(2)}\zeta_{it}+\omega_{it}^{(22)}v_{it},$
(6)

which depends on multiple parameters _and_ on the magnitude and sign of the
income shocks. As such, there is an asymmetric pass-through of good versus bad
shocks into consumption. Suppose, for example, that $\phi^{(1)}>0$ and
$\phi^{(2)}<0$, as our results subsequently suggest. At $v_{it}=0$ (the
average transitory shock), a negative _permanent_ shock has a higher pass-
through to consumption relative to a positive shock of the same magnitude. The
pass-through of negative shocks would thus be underestimated (i.e., excessive
insurance concluded) if the econometrician assessed the extent of partial
insurance on the basis of $\phi^{(1)}$ alone, ignoring the non-linearity in
the consumption response. The size of this bias depends on the magnitude of
$\phi^{(2)}$ (a more negative $\phi^{(2)}$ implies bigger bias) and on the
size of the permanent shock itself (a more severe bad shock implies bigger
bias). When income risk is non-Gaussian (e.g., it is left tailed), the
realization of shocks can be far from their mean, thereby causing large errors
in the inference of the degree of partial insurance based on $\phi^{(1)}$
alone.

A similar asymmetry exists in the transmission of good versus bad transitory
income shocks. In this case, the pass-through into consumption is given by

$\frac{\partial\Delta c_{it}}{\partial
v_{it}}=\psi_{it}^{(1)}+2\psi_{it}^{(2)}v_{it}+\omega_{it}^{(22)}\zeta_{it},$
(7)

which again depends on multiple parameters _and_ on the magnitude and sign of
the income shocks. Importantly, this generalized pass-through does not rule
out the possibility of full insurance to transitory shocks, a recurrent
empirical finding in the literature. Suppose, for example, that
$\psi^{(1)}\approx\psi^{(2)}\approx 0$; then $\partial\Delta c_{it}/\partial
v_{it}\approx 0$ at $\zeta_{it}=0$ (the average permanent shock), indicating
full insurance to transitory shocks.

##  3 Identification of income and consumption parameters

Our specification of the income and consumption processes imposes restrictions
on the second and higher-order moments of income and consumption that can be
used to identify the various parameters. The set of parameters consists of the
variance ($\sigma_{\zeta_{t}}^{2}$, $\sigma_{v_{t}}^{2}$), skewness
($\gamma_{\zeta_{t}}$, $\gamma_{v_{t}}$), and kurtosis ($\kappa_{\zeta_{t}}$,
$\kappa_{v_{t}}$) of the distributions of permanent and transitory
shocks,101010With a slight abuse of terminology, we use the terms skewness and
kurtosis to refer to the third and fourth central moments of shocks. We will
be clear whenever those terms refer to _standardized_ moments. the partial
insurance parameters in the linear ($\phi^{(1)}_{t}$, $\psi^{(1)}_{t}$) and
quadratic consumption functions ($\phi^{(1)}_{t}$, $\psi^{(1)}_{t}$,
$\phi^{(2)}_{t}$, $\psi^{(2)}_{t}$, $\omega^{(22)}_{t}$), and the moments of
the unobserved consumption taste heterogeneity ($\sigma_{\xi_{t}}^{2}$,
$\gamma_{\xi_{t}}$, $\kappa_{\xi_{t}}$). For simplicity of the illustration,
we restrict the partial insurance parameters to not vary across households;
this is _not_ an identifying restriction, and in fact we explore subsequently
their variation across certain demographic groups.

Two important aspects of the data must be addressed before proceeding to
identification: the biennial nature of the modern PSID and measurement error.

As we describe in section 4, the PSID provides data only every two years
within our time period. This necessitates to recast the income and consumption
growth processes of section 2 to this lower frequency, a straightforward
operation that we carry out in appendix B. Nevertheless, for consistency with
the earlier literature, we provide the identifying statements subsequently _as
if_ the data come at a yearly frequency. The complete identifying statements
at the lower frequency of the modern PSID appear in appendix B.

Measurement error in income and consumption is a more challenging issue. In
the permanent-transitory specification of income, the moments of classical
income error are not separately identifiable from the moments of the
economically relevant transitory shock. This necessitates that we restrict the
income error to be Gaussian and retrieve its variance from validation studies
of the PSID, as in Meghir and Pistaferri (2004) and Blundell, Pistaferri, and
Saporta-Eksten (2016). We return to this point in section 4. By contrast, the
moments of classical consumption error are identifiable; this introduces three
additional parameters for the variance, skewness, and kurtosis of consumption
error ($\sigma_{u^{c}_{t}}^{2}$, $\gamma_{u^{c}_{t}}$, $\kappa_{u^{c}_{t}}$).
For simplicity, however, we subsequently show identification _as if_ income
and consumption errors are not present; we provide our full identifying
statements with measurement error in appendix B.

###  3.1 Income process parameters

Given the income process in (3) and the properties of shocks, the second and
higher-order moments of the cross-sectional distribution of income growth are
given by

$\displaystyle{\operatorname*{\mathbb{E}}}((\Delta y_{it})^{2})$
$\displaystyle=\sigma_{\zeta_{t}}^{2}+\sigma_{v_{t}}^{2}+\sigma_{v_{t-1}}^{2},$
$\displaystyle{\operatorname*{\mathbb{E}}}((\Delta y_{it})^{3})$
$\displaystyle=\gamma_{\zeta_{t}}+\gamma_{v_{t}}-\gamma_{v_{t-1}},$
$\displaystyle{\operatorname*{\mathbb{E}}}((\Delta y_{it})^{4})$
$\displaystyle=\kappa_{\zeta_{t}}+\kappa_{v_{t}}+\kappa_{v_{t-1}}+6\sigma_{\zeta_{t}}^{2}(\sigma_{v_{t}}^{2}+\sigma_{v_{t-1}}^{2})+6\sigma_{v_{t}}^{2}\sigma_{v_{t-1}}^{2}.$

The second moment depends on the variance of permanent shocks at $t$ and the
sum of variances of transitory shocks at $t$ and $t-1$; it thus reflects the
dispersion of one permanent and two transitory shocks. The fourth moment
reflects, similarly, contributions from the dispersion and kurtosis of both
types of shocks. By contrast, the third moment is driven primarily by skewness
in the permanent shock. Barring strong non stationarities in the distribution
of transitory shocks, the difference between $\gamma_{v_{t}}$ and
$\gamma_{v_{t-1}}$ will generally be small (and precisely zero if the
distributions are time-invariant), leaving $\gamma_{\zeta_{t}}$, the skewness
in permanent shocks, to drive entirely the third moment of income.

While the above expressions are already partly informative of the parameters
of the income process, the variance of shocks is formally identified as

$\displaystyle\sigma_{\zeta_{t}}^{2}$
$\displaystyle={\operatorname*{\mathbb{E}}}(\Delta y_{it}\times(\Delta
y_{it-1}+\Delta y_{it}+\Delta y_{it+1})),$ $\displaystyle\sigma_{v_{t}}^{2}$
$\displaystyle=-{\operatorname*{\mathbb{E}}}(\Delta y_{it}\times\Delta
y_{it+1}).$

The long sum in the first line strips income growth at $t$ of its contemporary
transitory shock; its covariance with $\Delta y_{it}$ thus picks up the
variance of the permanent shock. The covariance of two consecutive income
growths in the second line identifies the variance of the transitory shocks
due to mean reversion in the shock.

In a similar way, the skewness of shocks is identified as111111In the presence
of income measurement error, the moments that identify the variance and
skewness of transitory shocks also pick up the variance and skewness of the
error. Retrieving its variance from validation studies (see section 4) while
assuming error is Guassian (thus has zero skewness) allows us to identify the
parameters of the economically relevant transitory shock. Appendix B reports
the details of identification in the presence of measurement error.

$\displaystyle\gamma_{\zeta_{t}}$
$\displaystyle={\operatorname*{\mathbb{E}}}((\Delta y_{it})^{2}\times(\Delta
y_{it-1}+\Delta y_{it}+\Delta y_{it+1})),$ $\displaystyle\gamma_{v_{t}}$
$\displaystyle=-{\operatorname*{\mathbb{E}}}((\Delta y_{it})^{2}\times\Delta
y_{it+1}),$

while their kurtosis is identified from

$\displaystyle\kappa_{\zeta_{t}}$
$\displaystyle={\operatorname*{\mathbb{E}}}((\Delta
y_{it})^{4})-\kappa_{v_{t}}-\kappa_{v_{t-1}}-6\sigma_{\zeta_{t}}^{2}(\sigma_{v_{t}}^{2}+\sigma_{v_{t-1}}^{2})-6\sigma_{v_{t}}^{2}\sigma_{v_{t-1}}^{2},$
$\displaystyle\kappa_{v_{t}}$
$\displaystyle={\operatorname*{\mathbb{E}}}((\Delta y_{it})^{2}\times(\Delta
y_{it+1})^{2})-\sigma_{\zeta_{t}}^{2}\sigma_{\zeta_{t+1}}^{2}-\sigma_{\zeta_{t}}^{2}\sum_{\tau=0}^{1}\sigma_{v_{t+\tau}}^{2}-\sigma_{\zeta_{t+1}}^{2}\sum_{\tau=-1}^{0}\sigma_{v_{t+\tau}}^{2}$
$\displaystyle-\sigma_{v_{t-1}}^{2}\sigma_{v_{t}}^{2}-\sigma_{v_{t-1}}^{2}\sigma_{v_{t+1}}^{2}-\sigma_{v_{t}}^{2}\sigma_{v_{t+1}}^{2}.$

There are several over-identifying restrictions for both skewness and
kurtosis.121212Identification of the fourth moments requires prior
identification of the second moments; identification of kurtosis of the
permanent shock also requires prior identification of kurtosis of the
transitory shock.

###  3.2 Linear consumption function parameters

Given the linear consumption function in (4) and the properties of shocks, the
second and higher-order moments of the distribution of consumption growth are
given by

$\displaystyle{\operatorname*{\mathbb{E}}}((\Delta c_{it})^{2})$
$\displaystyle=(\phi_{t}^{(1)})^{2}\sigma_{\zeta_{t}}^{2}+(\psi_{t}^{(1)})^{2}\sigma_{v_{t}}^{2}+\sigma_{\xi_{t}}^{2},$
$\displaystyle{\operatorname*{\mathbb{E}}}((\Delta c_{it})^{3})$
$\displaystyle=(\phi_{t}^{(1)})^{3}\gamma_{\zeta_{t}}+(\psi_{t}^{(1)})^{3}\gamma_{v_{t}}+\gamma_{\xi_{t}},$
$\displaystyle{\operatorname*{\mathbb{E}}}((\Delta c_{it})^{4})$
$\displaystyle=(\phi_{t}^{(1)})^{4}\kappa_{\zeta_{t}}+(\psi_{t}^{(1)})^{4}\kappa_{v_{t}}+\kappa_{\xi_{t}}+6(\phi_{t}^{(1)})^{2}(\psi_{t}^{(1)})^{2}\sigma_{\zeta_{t}}^{2}\sigma_{v_{t}}^{2}$
$\displaystyle+6(\phi_{t}^{(1)})^{2}\sigma_{\zeta_{t}}^{2}\sigma_{\xi_{t}}^{2}+6(\psi_{t}^{(1)})^{2}\sigma_{v_{t}}^{2}\sigma_{\xi_{t}}^{2}.$

The moments of the distribution of consumption growth reflect the underlying
distributions of income shocks, as well as the distribution of unobserved
taste heterogeneity. In general, larger dispersion, skewness, or kurtosis in
the distributions of income shocks implies larger dispersion, skewness, or
kurtosis in the distribution of consumption growth. The link between the two
depends on the degree of insurance with respect to these shocks, so these
moments provide restrictions that help identify the parameters of partial
insurance.

Formally, the transmission parameter of permanent shocks is identified as

$\phi^{(1)}_{t}={\operatorname*{\mathbb{E}}}(\Delta c_{it}\times(\Delta
y_{it-1}+\Delta y_{it}+\Delta y_{it+1}))/\sigma_{\zeta_{t}}^{2},$

while the transmission parameter of transitory shocks is identified as

$\psi^{(1)}_{t}=-{\operatorname*{\mathbb{E}}}(\Delta c_{it}\times\Delta
y_{it+1})/\sigma_{v_{t}}^{2};$

in both cases, the variance of shocks is given by the earlier
expressions.131313The variance of consumption taste heterogeneity is further
given by $\sigma_{\xi_{t}}^{2}={\operatorname*{\mathbb{E}}}((\Delta
c_{it})^{2})-(\phi_{t}^{(1)})^{2}\sigma_{\zeta_{t}}^{2}-(\psi_{t}^{(1)})^{2}\sigma_{v_{t}}^{2}$
while its skewness by $\gamma_{\xi_{t}}={\operatorname*{\mathbb{E}}}((\Delta
c_{it})^{3})-(\phi_{t}^{(1)})^{3}\gamma_{\zeta_{t}}-(\psi_{t}^{(1)})^{3}\gamma_{v_{t}}$.
The kurtosis $\kappa_{\xi_{t}}$ is similarly identified from the fourth moment
of consumption growth. These statements reflect the intuition behind
identification in BPP: the _covariance_ between consumption and (appropriately
defined) income growth captures the strength of the relation between the two
series and, consequently, the pass-through of income shocks into
consumption.141414An equivalent view is that the partial insurance parameters
are identified from a regression of consumption growth on income growth, using
permanent or future income as instruments.

This workhorse covariance, however, is a _second_ moment of the joint
distribution of consumption and income and, as such, neglects information in
the tails. If most income shocks are either zero or rather bad, as Guvenen,
Karahan, Ozkan, and Song (2021) find, identifying partial insurance based on
second moments alone may be misleading. However, the partial insurance
parameters are heavily (over-)identified by higher-order moments of the
marginal and joint distributions of income and consumption; inspect, for
example, the expressions for ${\operatorname*{\mathbb{E}}}((\Delta
c_{it})^{3})$ or ${\operatorname*{\mathbb{E}}}((\Delta c_{it})^{4})$. We will
thus subsequently exploit these higher-order moments in the context of the
linear consumption function in our first attempt to understand the
implications of non-Gaussianity for partial insurance.

###  3.3 Quadratic consumption function parameters

As per the quadratic specification in (5) and the properties of shocks, the
second moment of the distribution of consumption growth is given by

$\displaystyle{\operatorname*{\mathbb{E}}}((\Delta c_{it})^{2})$
$\displaystyle=(\phi_{t}^{(1)})^{2}\sigma_{\zeta_{t}}^{2}+(\psi_{t}^{(1)})^{2}\sigma_{v_{t}}^{2}+2\phi_{t}^{(1)}\phi_{t}^{(2)}\gamma_{\zeta_{t}}+2\psi_{t}^{(1)}\psi_{t}^{(2)}\gamma_{v_{t}}+(\phi_{t}^{(2)})^{2}\kappa_{\zeta_{t}}+(\psi_{t}^{(2)})^{2}\kappa_{v_{t}}$
$\displaystyle+(\omega_{t}^{(22)})^{2}\sigma_{\zeta_{t}}^{2}\sigma_{v_{t}}^{2}+2\phi_{t}^{(2)}\psi_{t}^{(2)}\sigma_{\zeta_{t}}^{2}\sigma_{v_{t}}^{2}+\sigma_{\xi_{t}}^{2}.$

This depends not only on the variance of permanent and transitory income
shocks, as in the linear case, but also on their skewness and kurtosis.
Contrary to the linear case, however, the higher-order moments of consumption
growth depend on higher than fourth-order moments of the shocks, which we do
not model.

Identification of the partial insurance parameters comes from the joint
moments of consumption and income, conceptually similar to identification in
the linear case. Given the non-linearity, however, multiple moments contribute
jointly to the identification of any single parameter, and the identifying
statements are inevitably more involved. Formally, the transmission parameters
are given by

$\left(\begin{array}[]{l}\phi^{(1)}_{t}\\\ \psi^{(1)}_{t}\\\ \phi^{(2)}_{t}\\\
\psi^{(2)}_{t}\\\
\omega^{(22)}_{t}\end{array}\right)=\mathbf{A}^{-1}\left(\begin{array}[]{l}{\operatorname*{\mathbb{E}}}\left(\Delta
c_{it}\times\Delta y_{it}\right)\\\ {\operatorname*{\mathbb{E}}}\left(\Delta
c_{it}\times(\Delta y_{it})^{2}\right)\\\
{\operatorname*{\mathbb{E}}}\left(\Delta c_{it}\times\Delta y_{it+1}\right)\\\
{\operatorname*{\mathbb{E}}}\left(\Delta c_{it}\times(\Delta
y_{it+1})^{2}\right)\\\ {\operatorname*{\mathbb{E}}}\left(\Delta
c_{it}\times\Delta y_{it}\times\Delta y_{it+1}\right)\\\ \end{array}\right),$
(8)

where square matrix $\mathbf{A}$ depends exclusively on the second, third, and
fourth moments of income shocks, and it is generally full rank and
invertible.151515The matrix of coefficients is given by
$\mathbf{A}=\left(\begin{array}[]{lllll}\sigma_{\zeta_{t}}^{2}&\sigma_{v_{t}}^{2}&\gamma_{\zeta_{t}}&\gamma_{v_{t}}&0\\\
\gamma_{\zeta_{t}}&\gamma_{v_{t}}&\kappa_{\zeta_{t}}+\sigma_{\zeta_{t}}^{2}(\sigma_{v_{t}}^{2}+\sigma_{v_{t-1}}^{2})&\kappa_{v_{t}}+\sigma_{v_{t}}^{2}(\sigma_{\zeta_{t}}^{2}+\sigma_{v_{t-1}}^{2})&2\sigma_{\zeta_{t}}^{2}\sigma_{v_{t}}^{2}\\\
0&-\sigma_{v_{t}}^{2}&0&-\gamma_{v_{t}}&0\\\
0&\gamma_{v_{t}}&\sigma_{\zeta_{t}}^{2}(\sigma_{\zeta_{t+1}}^{2}+\sigma_{v_{t}}^{2}+\sigma_{v_{t+1}}^{2})&\kappa_{v_{t}}+\sigma_{v_{t}}^{2}(\sigma_{\zeta_{t+1}}^{2}+\sigma_{v_{t+1}}^{2})&0\\\
0&-\gamma_{v_{t}}&-\sigma_{\zeta_{t}}^{2}\sigma_{v_{t}}^{2}&-\kappa_{v_{t}}&-\sigma_{\zeta_{t}}^{2}\sigma_{v_{t}}^{2}\\\
\end{array}\right).$ There are no obvious linear dependencies across columns
so the matrix is generally full rank and invertible. This is true also if the
underlying distributions of shocks are Gaussian.,161616The variance of
consumption taste heterogeneity is identified from
${\operatorname*{\mathbb{E}}}((\Delta c_{it})^{2})$. Its skewness and kurtosis
enter the third and fourth moments of consumption growth. Those depend on
higher than fourth-order moments of income, which we do not model, so they are
generally not identified in the current setting.

As shown in section 3.1, the higher-order moments of income shocks are
identified through the third and fourth moments of income. Therefore, the
higher than second-order moments of income play an essential role in enabling
the identification of the transmission parameters of the quadratic consumption
function. Unlike the linear case, in other words, the transmission parameters
in the generalized specification cannot be identified with second-order income
moments alone. By contrast, no higher-order moments of consumption are needed;
the identifying moments in (8) are covariances between consumption growth and
various powers of income growth. This is an advantage because, while in
principle we may estimate higher than second-order moments of _consumption_
growth in the PSID, we cannot be certain how good these estimates are. This is
in contrast to higher-order _income_ moments, for which Guvenen, Karahan,
Ozkan, and Song (2021) serves as a natural benchmark given their use of
administrative income data. We return to this point in section 4.

What is the bias in the transmission parameters if one assumes a linear
specification when the true consumption function is quadratic? The identifying
condition for $\phi_{t}^{(1)}$ in the linear case identifies

${\operatorname*{\mathbb{E}}}(\Delta c_{it}\times(\Delta y_{it-1}+\Delta
y_{it}+\Delta
y_{it+1}))/\sigma_{\zeta_{t}}^{2}=\phi_{t}^{(1)}+\phi_{t}^{(2)}(\gamma_{\zeta_{t}}/\sigma_{\zeta_{t}}^{2}),$

namely a linear combination of $\phi_{t}^{(1)}$ and $\phi_{t}^{(2)}$, the
latter weighed by the skewness of permanent shocks. If $\gamma_{\zeta_{t}}<0$,
i.e., permanent shocks are left skewed as we find subsequently, and
$\phi_{t}^{(2)}<0$, as our empirical estimates suggest, linearity biases
$\phi_{t}^{(1)}$ upwards; the magnitude of this bias increases with the extent
of income skewness. Similarly, the identifying condition for $\psi_{t}^{(1)}$
in the linear case identifies

$-{\operatorname*{\mathbb{E}}}(\Delta c_{it}\times\Delta
y_{it+1})/\sigma_{v_{t}}^{2}=\psi_{t}^{(1)}+\psi_{t}^{(2)}(\gamma_{v_{t}}/\sigma_{v_{t}}^{2}).$

If $\gamma_{v_{t}}<0$, i.e., transitory shocks are left skewed as we find
subsequently, and $\psi_{t}^{(2)}>0$, as our baseline estimate suggests,
linearity biases $\psi_{t}^{(1)}$ downwards; the bias is stronger the larger
left skewness is. This may thus help explain the discrepancy between BPP (who
find little pass-through of transitory shocks) and studies based on natural
experiments (who find large consumption responses to said shocks).171717The
discrepancy is between studies that use covariance restrictions in the joint
distribution of income and consumption (they find small pass-through of
transitory shocks) and studies that employ natural experiments (e.g., tax
refunds or lotteries; they find large pass-through). Commault (2022) explains
the discrepancy on the basis of neglected effects from _past_ transitory
shocks on consumption, while Crawley and Kuchler (2023) explain it on the
basis of neglected time aggregation in the data. Both are alternative types of
misspecification in the linear consumption function/empirical implementation
vis-à-vis our focus on nonlinearities here.

##  4 Empirical implementation

###  4.1 Data

As is evident from the previous section, panel data in income and consumption
are needed to identify the moments of the permanent and transitory shocks and
the degrees of insurance to them. The Panel Study of Income Dynamics (PSID)
provides such data for a representative and continuously evolving sample of
U.S. households since $1968$. For most of this period, however, the PSID only
collected consumption expenditures on food items. It is well known that food
is an imperfect measure of consumption of nondurable goods, representing a
declining fraction of the consumer’s basket over time. Since $1999$, however,
the PSID was drastically redesigned in that it started collecting information
over a much broader set of consumption items, currently comprising over 70% of
consumption reported in the National Income and Product Accounts (Blundell,
Pistaferri, and Saporta-Eksten, 2016). We, therefore, use the PSID over the
period between $1999$ and $2019$, the last available wave when writing this
paper; the data are available only biennially. This is in contrast to BPP, who
use PSID data over $1980$–$1992$ and rely on consumption imputations from the
Consumer Expenditure Survey (CEX). We return to this point below.

Our baseline sample consists of married couples in the PSID with a male spouse
aged $30$ to $65$. We focus on a sample that is representative of the U.S.
population, eliminating the supplementary low-income subsample (SEO).
Households are also removed in the event of a break-up or remarriage; our
analysis thus focuses on income risk rather than on divorce or other household
dissolution factors. Naturally, we require non-missing data on income,
expenditure, and basic demographics.181818We also remove observations with
extreme jumps in income or consumption in consecutive years, where a jump is a
$\geq$10-fold increase or a $\geq$90% reduction, or the bottom 0.25% of the
distribution of products of growth rates at $t$ and $t-1$, i.e., large
increases followed by large drops, likely reflecting measurement error. Our
final sample consists of 20,866 household-year observations; on average, a
household is observed for $5.6$ periods, which corresponds to about $10$
calendar years given the biennial nature of the data.

Table 1: Descriptive statistics | Baseline sample |  Low wealth |  High wealth |  No college |  Some college
---|---|---|---|---|---
|  Mean |  Median |  Mean |  Mean |  Mean |  Mean
|  (1) |  (2) |  (3) |  (4) |  (5) |  (6)
_Consumption_ |  |  |  |  |  |
Nondurable+services |  26,766 |  22,895 |  23,106 |  29,852 |  22,137 |  28,908
Food at home |  8,256 |  7,609 |  7,861 |  8,580 |  7,462 |  8,630
Utilities |  3,515 |  3,238 |  3,270 |  3,721 |  3,394 |  3,571
Food out |  3,318 |  2,511 |  2,683 |  3,840 |  2,649 |  3,634
Health |  1,614 |  863 |  1,390 |  1,803 |  1,429 |  1,699
Transportation |  5,866 |  4,900 |  5,398 |  6,261 |  5,436 |  6,065
Education |  3,202 |  0 |  1,529 |  4,614 |  1,262 |  4,100
Childcare |  977 |  0 |  937 |  1,009 |  543 |  1,181
Rent (or rent eq.) |  18,105 |  13,821 |  10,949 |  23,991 |  11,647 |  21,149
Home insurance |  894 |  747 |  654 |  1,092 |  702 |  985
_Income_ |  |  |  |  |  |
Disposable income |  142,910 |  113,581 |  98,527 |  179,416 |  97,675 |  164,239
Taxable+transfer inc. |  137,277 |  109,003 |  93,358 |  173,401 |  90,810 |  159,187
Earnings |  118,423 |  97,154 |  87,005 |  144,266 |  78,565 |  137,217
Male |  81,857 |  62,185 |  56,817 |  102,452 |  50,199 |  96,783
Female |  36,567 |  29,183 |  30,187 |  41,814 |  28,366 |  40,433
Total assets |  547,761 |  191,193 |  69,731 |  950,823 |  284,036 |  669,788
_Demographics_ |  |  |  |  |  |
% working (male) |  0.89 |  1.00 |  0.92 |  0.88 |  0.86 |  0.91
% working (female) |  0.79 |  1.00 |  0.81 |  0.78 |  0.79 |  0.79
Age (male) |  46 |  45 |  43 |  49 |  46 |  46
% some college (male) |  0.68 |  1.00 |  0.55 |  0.79 |  0.00 |  1.00
% some college (female) |  0.71 |  1.00 |  0.62 |  0.78 |  0.44 |  0.84
Periods per household |  5.63 |  5.00 |  4.89 |  6.45 |  5.21 |  5.86
Observations | 20,866 |  9,417 |  11,449 |  6,686 |  14,180
_Notes:_ The table reports summary statistics for consumption, income, wealth,
and basic demographics in our baseline sample from the PSID, and in subsamples
sorted by wealth and education, over $1999$–$2019$. See text for variable
definitions. Rent for owners is calculated as 6% of the self-reported value of
the house.

Table 1 presents summary statistics over the sample period $1999$–$2019$.
About $89\%$ of men and $79\%$ of women work in the labor market (but some
households have intermittent employment); $68\%$ of men have had some college
education, while women fare slightly higher at $71\%$. Our measure of income
is household disposable income, defined as the sum of taxable income (e.g.,
earnings and asset income), transfers, and social security income of the male
and female spouses and of other family unit members, minus taxes paid. Average
disposable income is $142,910 in 2018 prices.191919We express all monetary
figures in 2018 prices because earnings are 1-year retrospective and our
latest data are from 2019. We deflate using the Consumer Price Index from the
Bureau of Labor Statistics. Our measure of consumption is real expenditure on
nondurable goods and services, comprising food (at home and outside),
utilities, out-of-pocket health expenses (doctors, prescriptions), public
transport and own vehicle expenses (gasoline, parking, insurance), education,
and child care. Average real consumption is $26,766, with food at home
accounting for about $30\%$ of this. Housing, including rent and home
insurance, is the main expenditure we exclude; we do so for comparability with
BPP who also exclude it. Including housing does not change any of the
subsequent results meaningfully.202020Blundell, Pistaferri, and Saporta-Eksten
(2016) use consumption data from the modern PSID, including housing, to study
family labor supply as an insurance mechanism. House prices, however,
fluctuated dramatically in 2007–2009, artificially affecting housing
expenditure especially for homeowners. Finally, table 1 reports summary
statistics also for samples sorted by wealth and education; we postpone the
discussion of those samples to section 5.4.

Aside from the PSID, we use a similarly selected sample drawn from the CEX to
facilitate comparison with BPP. We defer the discussion of this secondary data
to section 4.3.

###  4.2 Estimation

Estimation follows a three-step approach. In the first step, we compute income
and consumption growth net of their predictable components, $\mathbf{X}_{it}$
and $\mathbf{Z}_{it}$, respectively, in model notation, by regressing real
income and consumption growth on a rich set of time effects and household
characteristics.212121The full set of controls includes year and year-of-birth
dummies, dummies for education, race, region of residence, family size and
number of children (level and change from last period), employment status
(current and past), presence of outside dependents in the family, and presence
of income recipients other than the husband and wife (both current and past).
The dummies for education, race, and region are interacted with the year
dummies to allow the effect of these characteristics to vary over time. The
residuals from these regressions constitute the empirical counterparts to
unexplained income and consumption growth, $\Delta y_{it}$ and $\Delta
c_{it}$, respectively. In the second step, we estimate the parameters of the
income process – i.e., the second and higher moments of permanent and
transitory shocks – using income data alone. In the third step, we estimate
the transmission parameters of shocks and the moments of taste heterogeneity
and consumption measurement error; we do so using income and consumption data
conditional on the parameters of the income process from the second stage. We
estimate the second and third step parameters using GMM and an identity
weighting matrix.222222Our baseline results come from equally weighted GMM. A
diagonal weighting matrix, where the diagonal entries are the $t$-statistics
of the sample moments, delivers similar results (not reported here but
available upon request), although we did experience some convergence issues in
that case.

We present estimates of the transmission parameters separately for the linear
and quadratic specifications of the consumption function. For the linear
function, we first present results when only second moments of income and
consumption growth are targeted, exactly as in BPP. This purports to replicate
BPP over the more recent sample period. We next report results for the linear
function when, in addition to the second moments, third and fourth moments are
also targeted. This helps assess whether and how, through the lenses of the
linear specification, information in the tails of the income distribution
affects the measurement of partial insurance. We then move on to the results
from the quadratic specification. Our choice to estimate the income parameters
in an earlier step enables those parameters to remain unchanged regardless of
the specification of the consumption function fitted. Appendix B.4 lists the
moments targeted in each case.

Given the multi-step approach, we conduct inference using the block bootstrap
jointly over all stages. We thus account for serial correlation and
heteroskedasticity of arbitrary form, as well as for our use of pre-estimated
parameters in the later stages of the multi-step approach. To avoid the
standard errors being affected by extreme draws, we apply a normal
approximation to the inter-quartile range of bootstrap
replications.232323Instead of estimating standard errors as the standard
deviation of the parameters’ point estimates across replications, we compute
them as the normalized interquartile range of the replications, i.e., the
difference between the $75^{\text{th}}$ and $25^{\text{th}}$ percentiles
divided by the interquartile range of the standard normal cdf. This is
equivalent to applying a normal approximation to the distribution of bootstrap
replications, thereby shielding the standard errors from extreme resampling
draws. See Blundell, Pistaferri, and Saporta-Eksten (2016) and Theloudis
(2021) for details. Standard errors without the normal approximation are very
similar.

As we established in section 3, we must use an external estimate for the
variance of income measurement error. We cannot otherwise separate the moments
of the transitory shock from those of the measurement error. Following Meghir
and Pistaferri (2004) and Blundell, Pistaferri, and Saporta-Eksten (2016), we
retrieve the error variance from Bound, Brown, Duncan, and Rodgers (1994), a
popular validation study of the PSID, setting it equal to $4\%$ of the
variance of log income in any given year.242424Bound, Brown, Duncan, and
Rodgers (1994) survey workers from a single manufacturing firm in 1983 and
1987. They obtain information on earnings in a way that mimics the PSID
questionnaire and coding practices and compare them to administrative data
from the firm. There are two caveats in this approach: first, the sample of
workers in the validation study comes from two decades prior to the data in
this paper; second, we use estimates of error in male earnings to correct for
error in household income.

###  4.3 Empirical moments of income and consumption

Table 2 reports the empirical second, third, and fourth moments of the cross-
sectional distributions of unexplained income and consumption growth in the
sample.

Income moments. Panel A presents the moments of income growth. The variance is
$0.153$, while the first-order autocovariance is $-0.049$ and highly
significant.252525In contrast to the identifying statements of section 3, the
data are biennial so we report moments for income/consumption growth at a
biennial frequency. We maintain the notation $\Delta x_{t}$ for the first
difference of variable $x$ over time, noting that this corresponds to a
difference over two years. See appendix B for details. The income process of
section 2.1 implies that income growth is negatively first-order
autocorrelated due to mean reversion in the transitory shock, and this is
supported by the data.262626The second-order autocovariance is an order of
magnitude smaller and only marginally significant. This feature and the
biennial nature of the data, whereby the second-order autocovariance reflects
correlations 4 calendar years apart, motivates our white noise specification
for the transitory shock.

Table 2: Empirical moments of income and consumption growth | Baseline sample
---|---
_Income data:_ | PSID | PSID
_Consumption data:_ | PSID | imputed from CEX
| (1) | (2)
Panel A. Moments of income growth
${\operatorname*{\mathrm{Var}}}(\Delta y_{it})$ |  0.153 |  (0.005) |  0.153 |  (0.005)
${\operatorname*{\mathrm{Cov}}}(\Delta y_{it},\Delta y_{it+1})$ |  -0.049 |  (0.002) |  -0.049 |  (0.002)
${\operatorname*{\mathrm{Skew}}}(\Delta y_{it})$ |  -0.143 |  (0.098) |  -0.143 |  (0.098)
${\operatorname*{\mathrm{Cov}}}((\Delta y_{it})^{2},\Delta y_{it+1})$ |  0.009 |  (0.003) |  0.009 |  (0.003)
${\operatorname*{\mathrm{Kurt}}}(\Delta y_{it})$ |  10.032 |  (0.509) |  10.032 |  (0.509)
${\operatorname*{\mathrm{Cov}}}((\Delta y_{it})^{2},(\Delta y_{it+1})^{2})$ |  0.080 |  (0.007) |  0.080 |  (0.007)
Panel B. Moments of consumption growth
${\operatorname*{\mathrm{Var}}}(\Delta c_{it})$ |  0.126 |  (0.002) |  0.323 |  (0.015)
${\operatorname*{\mathrm{Cov}}}(\Delta c_{it},\Delta c_{it+1})$ |  -0.044 |  (0.001) |  -0.140 |  (0.009)
${\operatorname*{\mathrm{Skew}}}(\Delta c_{it})$ |  -0.039 |  (0.034) |  -0.177 |  (0.083)
${\operatorname*{\mathrm{Cov}}}((\Delta c_{it})^{2},\Delta c_{it+1})$ |  -0.003 |  (0.001) |  0.190 |  (0.036)
${\operatorname*{\mathrm{Kurt}}}(\Delta c_{it})$ |  4.445 |  (0.100) |  17.109 |  (2.111)
${\operatorname*{\mathrm{Cov}}}((\Delta c_{it})^{2},(\Delta c_{it+1})^{2})$ |  0.027 |  (0.001) |  0.909 |  (0.198)
Panel C. Joint moments
${\operatorname*{\mathrm{Cov}}}(\Delta y_{it},\Delta c_{it})$ |  0.008 |  (0.001) |  0.012 |  (0.002)
_Notes:_ The table presents the second, third, and fourth moments of income
and consumption growth in the baseline sample. See text for the definitions of
income and consumption. Skewness and kurtosis correspond to the third and
fourth standardized moments respectively. Column 1 reports moments of
consumption internally in the PSID, while column 2 reports moments of
consumption imputed from the CEX into the PSID. See appendix C.2 for details
on the imputation. We maintain the notation $\Delta x_{t}$ for the first
difference of variable $x$ over time, noting that, given the biennial nature
of the data, this corresponds to a difference over two calendar years. Block
bootstrap standard errors are in parentheses.

The third _standardized_ moment of income growth is $-0.143$. Despite the
small size of the cross-section and the inherent difficulties in estimating
higher moments, the point estimate misses statistical significance at the 10%
level only by a small margin. The point estimate suggests that income growth
exhibits left skewness, i.e., a long left tail. In simple terms, the left tail
of the distribution of income growth (i.e., of income shocks) is longer than
the right tail, so a negative shock is on average more severe, more
unsettling, than a positive one. This feature of the data is in line with
Guvenen, Karahan, Ozkan, and Song (2021), who document strong left skewness in
the distribution of male earnings growth using large administrative data from
the U.S. Social Security Administration.

The fourth _standardized_ moment of income growth is $10.032$ and, as
expected, highly significant. Income growth is strongly leptokurtic, i.e., it
exhibits very high kurtosis. In simple terms, the distribution of income
growth (i.e., of income shocks) has thicker tails but less mass in the
shoulders, relative to a Gaussian density. Shocks are thus either close to the
center of the distribution, i.e., zero, or out in the tails, i.e., rather
extreme. This is again in line with Guvenen, Karahan, Ozkan, and Song (2021),
who also document strong excess kurtosis in male earnings growth.

Guvenen, Karahan, Ozkan, and Song (2021) focus on male earnings while our
model and, therefore, our statistics for income growth herein, revolve around
household disposable income. To facilitate the comparison, appendix C.1
reports moments of male _earnings_ growth in our baseline sample. The variance
is very similar between the PSID and the U.S. Social Security Administration
data, at $0.271$ and $0.323$ respectively. Skewness in our sample is $-0.569$
(and highly statistically significant) versus $-1.039$ in the administrative
data. While the nature of skewness is similar in both cases, the population
data feature substantially more left skewness than the PSID; Guvenen, Karahan,
Ozkan, and Song (2021) confirm this. Kurtosis, finally, is identical between
the two, at $13.711$ and $13.494$ respectively.

Left skewness and excess kurtosis are thus not just features of the growth
rates in disposable household income (table 2), but they are also observed in
alternative measures of income, such as male earnings (table C.1, panel A) and
household earnings (table C.1, panel B). Together, left skewness and excess
kurtosis suggest that income changes for most households are either close to
their mean value of zero or far out in the left tail (large income
_reductions_). These results corroborate (and, in fact, are very close
numerically with) findings by De Nardi, Fella, and Paz-Pardo (2019) and De
Nardi, Fella, Knoef, Paz-Pardo, and Van Ooijen (2021) over the early period of
the PSID.

Consumption moments. Panel B of table 2 presents the moments of consumption
growth; column 1 specifically reports moments from our baseline sample in the
PSID. The variance of consumption growth is very similar to that of income
growth ($0.126$ versus $0.153$), reflecting either a large pass-through of
income shocks or large taste heterogeneity/measurement error. The first-order
autocovariance is $-0.044$ and highly significant. In the context of our
model, this autocovariance reflects the variance of consumption error
$\sigma_{u^{c}_{t}}^{2}$ (see appendix B).

The third _standardized_ moment of consumption growth is $-0.039$. Albeit not
statistically significant, it is in line with Constantinides and Ghosh (2017),
who use the CEX and document negative skewness in (quarterly) consumption
growth. Similar to income growth, the distribution of consumption growth in
the PSID exhibits left skewness, i.e., it has a long left tail, albeit at a
lesser extent than income.272727The covariance between squared consumption
growth and one-period-ahead consumption growth identifies, in the context of
the linear model, the skewness of consumption error. The covariance is small,
so measurement error does not contribute much to the negative skewness in
consumption.

The fourth _standardized_ moment of consumption growth is $4.445$ and highly
significant. There is excess kurtosis and the distribution is thus
leptokurtic. In contrast to the distribution of income, however, excess
kurtosis is substantially less pronounced here.

The covariance between income and consumption growth is $0.008$. As we
established in section 3, this moment is crucial for the identification of the
pass-through of income shocks into consumption. While we cannot immediately
gauge its magnitude, the strong statistical significance reflects that income
shocks do transmit into consumption.

Unlike income, there are neither administrative data on consumption, nor has
there been a validation study for consumption in the PSID.282828Nevertheless,
Andreski, Li, Samancioglu, and Schoeni (2014) and Li, Schoeni, Danziger, and
Charles (2010) compare consumption in the PSID and in the CEX; Meyer and
Sullivan (2023) offer a critical discussion of the CEX data against the PSID.
As a consequence, we cannot benchmark our estimates of the consumption higher
moments against the literature. The CEX, however, provides high quality
consumption data for a representative sample of the U.S. population, although
it lacks a panel dimension. We can thus use the CEX to select a sample of
households as close to our baseline as possible, and estimate similar cross-
sectional moments across the surveys. The CEX is useful also for another
reason. Since the PSID collected little consumption data prior to $1999$, the
early literature on consumption insurance, notably BPP, resorted to imputing
expenditure from the CEX into the PSID. To facilitate the comparison with BPP,
we can use the CEX to replicate the consumption imputation into the PSID. This
is not strictly needed in our case (the PSID provides comprehensive data over
our sample period) but the imputation is nevertheless informative, as we
illustrate later.

Consumption in the CEX. We use CEX interview survey data over 1999-2019 and
select a sample of stable married households that mimics closely our baseline
selection in the PSID. We report details on the sample selection in appendix
C.2.

Appendix table C.2 offers a comparison between the PSID and CEX samples over
time. The two samples are very close with respect to age, family size, number
of children, education, or labor market participation of the wife and the
husband. Household income is higher in the PSID; BPP noted this and argued
that it may be due to the more comprehensive definition of income in the PSID.
Food expenditure in the PSID is also higher, which is in contrast to BPP who
find that food expenditure over the earlier period 1980–1992 is similar across
the two surveys. Consequently, overall consumption in the PSID is, on average,
higher by about 25% than a similarly selected sample in the CEX.

Appendix figure C.1 plots the variance, skewness, and kurtosis of _log_
consumption in the PSID and CEX samples. The patterns for the variance and
kurtosis are quite similar in the two surveys over time, though the variance
in the PSID is lower and the kurtosis is slightly higher. Skewness, however,
is markedly different. Skewness of log consumption is strongly positive in the
PSID, while it is mostly negative in the CEX; the patterns also diverge over
time. Overall, these statistics suggest that consumption across otherwise
similar samples in the PSID and CEX exhibits notable differences over our
sample period.

This comparison, so far, is not informative about the moments of consumption
_growth_ , which is the relevant variable in our case. The CEX lacks the panel
dimension needed for our analysis, so it cannot be used to obtain measures of
annual (or biennial) consumption growth. This is why BPP imputed consumption
from the CEX into the PSID: the CEX lacked the necessary panel dimension while
the PSID lacked comprehensive consumption data before 1999. We repeat BPP’s
imputation over the _recent_ years (i.e, in our sample period 1999–2019). The
imputation consists of the estimation of food demand in the CEX (food
expenditure expressed as a function of prices and nondurable consumption), the
inversion of demand under standard assumptions, and the use of the inverted
equation to impute nondurable consumption in the PSID based on food that is
available in both surveys. This allows us to obtain an alternative, external
measure of consumption in the PSID – and thus of consumption growth. Appendix
C.2 provides details about the imputation.

The moments of imputed consumption growth are in column 2 of table 2. The
variance at $0.323$ is two-and-a-half times higher than the variance in the
true series (column 1). In fact, the variance of imputed consumption growth is
more than double the variance of income growth. The higher-order moments also
differ markedly between the imputed and true series. Skewness in imputed
consumption growth is $-0.177$ (versus $-0.039$ in the true series), while
kurtosis is $17.109$, almost four times higher than kurtosis in the true
series.292929Appendix C.2 reports additional summary statistics for imputed
consumption.

The main qualitative features of the distribution remain, however, similar.
Consumption growth is non-Gaussian no matter which angle one looks at it.
Nevertheless, the imputation likely imparts substantial measurement/imputation
error to consumption, which likely explains why the second and higher-order
moments are several-fold accentuated relative to the true series. Since the
imputation is not strictly needed over our sample period, we use consumption
_internally_ available in the PSID as our baseline measure of consumption when
estimating the model. Yet, the present analysis suggests that we may not want
to rely too much on the higher-order moments of consumption, especially given
the discrepancy in skewness of _log_ consumption between PSID and CEX (this
discrepancy is independent of the imputation). Fortunately, identification in
the quadratic specification does not use such higher-order consumption
moments, as we illustrated in section 3.3.

##  5 Results

We present our baseline results assuming stationarity of the distributions of
income shocks, taste heterogeneity, and consumption measurement error. We also
assume constant transmission parameters of income shocks over the lifecycle.
We relax the stationarity/lifecycle-invariance assumptions in section 5.4.

###  5.1 Income process

We present estimates of the income process in three specifications. First, we
target only second moments of income growth, which enables us to estimate the
second moments of income shocks. This is similar to what BPP do over the
earlier period 1980–1992. Second, we target second _and_ third moments of
income growth, which enables us to estimate also the skewness of income
shocks. Finally, we target _up to_ the fourth moments of income growth, which
allows us to estimate all the parameters of the income process as presented in
section 2.1. The results from each specification appear in table 3, columns
1-3, respectively. The full set of moments targeted in each case is shown in
appendix B.4.1.

When only the second moments of income growth are targeted, we estimate the
variance of permanent and transitory shocks at $0.027$ and $0.033$
respectively. These are very close to BPP and Meghir and Pistaferri (2004),
who estimate these parameters over an earlier period. As the variance of
income growth is driven by the variance of the contemporaneous permanent shock
and the sum of the variances of transitory shocks in the current and previous
periods, the contribution of the transitory shock to the variance of income
growth is almost $2.5$-fold that of the permanent shock (as in Gottschalk and
Moffitt, 2009).

Table 3: Estimates of the income process, baseline sample _Income moments:_ |  2nd |  2nd, 3rd |  2nd, 3rd, 4th
---|---|---|---
_Income data:_ |  PSID |  PSID |  PSID
|  |  |  (1) |  (2) |  (3)
Panel A. Permanent shocks
Variance | $\sigma_{\zeta}^{2}$ |  |  0.027 |  0.027 |  0.030
|  |  |  (0.002) |  (0.002) |  (0.002)
Skewness | $\gamma_{\zeta}$ |  central |  |  -0.004 |  -0.004
|  |  |  |  (0.003) |  (0.003)
|  |  standardized |  |  -0.947 |  -0.831
|  |  |  |  (0.649) |  (0.571)
Kurtosis | $\kappa_{\zeta}$ |  central |  |  |  0.039
|  |  |  |  |  (0.007)
|  |  standardized |  |  |  44.393
|  |  |  |  |  (11.080)
Panel B. Transitory shocks
Variance | $\sigma_{v}^{2}$ |  |  0.033 |  0.033 |  0.031
|  |  |  (0.002) |  (0.002) |  (0.003)
Skewness | $\gamma_{v}$ |  central |  |  -0.008 |  -0.008
|  |  |  |  (0.003) |  (0.003)
|  |  standardized |  |  -1.310 |  -1.431
|  |  |  |  (0.510) |  (0.566)
Kurtosis | $\kappa_{v}$ |  central |  |  |  0.048
|  |  |  |  |  (0.005)
|  |  standardized |  |  |  49.226
|  |  |  |  |  (10.119)
_Notes:_ The table presents the estimates of the parameters of the income
process, assuming stationarity over time/lifecycle. Column 1 presents
parameter estimates when only the second moments of income growth are
targeted; column 2 targets also third moments; column 3 targets all moments up
to fourth-order. Estimation is via equally weighted GMM; block bootstrap
standard errors are in parentheses.

Targeting the third moments of income growth (in addition to the second
moments) leaves the variances of shocks unchanged. The third moments convey
additional information about the skewness of shocks, which we estimate at
$-0.947$ and $-1.310$ for the permanent and transitory shock respectively
(third standardized moment). Both shocks thus exhibit substantial negative
skewness, i.e., their distribution has a much longer left than right tail.
This is consistent with Geweke and Keane (2000) who find left skewness of
earnings shocks in the early PSID, and Theloudis (2021) who finds left
skewness of wage shocks in the recent period. Assuming for now that good and
bad shocks transmit similarly into consumption, the long left tail implies
that bad shocks are more damaging than good ones because they are typically
further away from the zero mean – they are more extreme. This is true for both
permanent and transitory shocks. Skewness of income growth, however, is driven
mostly by permanent shocks (fully so under stationarity). Therefore, if
targeting the third moments of income growth subsequently changes the degree
of consumption insurance in the household, this must reflect the implications
of permanent rather than of transitory shocks.

Targeting all moments of income growth up to fourth-order leaves the third
central moments of permanent and transitory shocks unchanged and modifies the
variances only by little ($0.030$ versus $0.027$ and $0.031$ versus $0.033$,
respectively). The fourth moments convey additional information about the
kurtosis of shocks, which we estimate at $44.393$ and $49.226$ respectively
(fourth standardized moment). In both cases, the distributions are highly
leptokurtic: they are sharp pointed around the mean, have relatively little
mass in the shoulders and very fat tails. This is consistent with Busch and
Ludwig (2021), who use the PSID until 2012 and estimate numerically similar
kurtosis coefficients for both types of shocks. Together with left skewness,
large excess kurtosis implies that most households experience either
negligible shocks (shocks close to the zero mean) or shocks that are quite
bad. Ignoring these stark features of income shocks may lead to erroneous
inference of the degree of consumption partial insurance, to which we now
turn.

###  5.2 Linear consumption function

We present estimates of the linear consumption function, first, targeting only
second moments of the joint income-consumption distribution and, subsequently,
targeting second as well as higher-order moments of the joint distribution.
The full set of moments targeted in each case is shown in appendix B.4.2.

Targeting second-order moments only. The estimation of the linear consumption
function targeting second moments alone is effectively a replication of BPP
over the recent years (1999-2019) while using consumption data that are
internally available in the PSID.303030We use the pre-estimated second moments
of income shocks obtained while targeting only the second moments of income
growth, i.e., column 1 of table 3. This replication is important in its own
right. The recent years offer superior consumption data (in the sense that we
do not need to rely on a consumption imputation from the CEX), thus reducing
the scope of measurement/imputation error. The data, however, come at a
biennial rather than annual frequency. Moreover, there have been many changes
in taxation and redistribution policies since 1980–1992, the original period
studied in BPP, which may matter for the extent of partial insurance available
to households today (see Borella, De Nardi, Pak, Russo, and Yang, 2022, for a
discussion of such changes).313131The nature and size of families and family
networks may have also changed over the last 5 decades, thus affecting various
informal insurance channels available to households.

Table 4: Estimates of the consumption function, baseline sample _Consumption fn.:_ | Linear |  |  Quadratic
---|---|---|---
_Income moments:_ |  2nd |  2nd |  2nd, 3rd |  2nd, 3rd, 4th |  |  2nd, 3rd, 4th
_Consumption data:_ |  PSID |  CEX |  PSID |  PSID |  |  PSID
|  (1) |  (2) |  (3) |  (4) |  |  (5)
$\phi^{(1)}$ |  0.152 |  0.288 |  0.160 |  0.112 |  |  0.134
|  (0.028) |  (0.044) |  (0.029) |  (0.041) |  |  (0.026)
$\psi^{(1)}$ |  -0.006 |  -0.109 |  -0.012 |  0.007 |  |  -0.003
|  (0.045) |  (0.078) |  (0.038) |  (0.056) |  |  (0.052)
$\phi^{(2)}$ |  |  |  |  |  |  -0.040
|  |  |  |  |  |  (0.026)
$\psi^{(2)}$ |  |  |  |  |  |  0.015
|  |  |  |  |  |  (0.034)
$\omega^{(22)}$ |  |  |  |  |  |  0.320
|  |  |  |  |  |  (0.770)
$\sigma^{2}_{\xi}$ |  0.019 |  0.019 |  0.019 |  0.020 |  |  0.019
|  (0.001) |  (0.004) |  (0.001) |  (0.001) |  |  (0.001)
$\gamma_{\xi}$ |  |  |  -0.001 |  -0.001 |  |
|  |  |  (0.001) |  (0.001) |  |
$\kappa_{\xi}$ |  |  |  |  0.005 |  |
|  |  |  |  (0.001) |  |
$\sigma^{2}_{u_{c}}$ |  0.044 |  0.140 |  0.044 |  0.044 |  |  0.044
|  (0.002) |  (0.009) |  (0.002) |  (0.001) |  |  (0.002)
$\gamma_{u_{c}}$ |  |  |  0.002 |  0.002 |  |
|  |  |  (0.001) |  (0.001) |  |
$\kappa_{u_{c}}$ |  |  |  |  0.013 |  |
|  |  |  |  (0.001) |  |
_Notes:_ The table presents the estimates of the parameters of the consumption
function, assuming homogeneous transmission parameters over the lifecycle/in
the cross-section and stationarity of taste heterogeneity and consumption
measurement error. Columns 1-4 present parameter estimates in the linear
function, while column 5 presents estimates in the quadratic case; the order
of moments targeted in each case is shown at the top of the table. Except for
column 2 where we use imputed data from the CEX, all other columns use
consumption data internally available in the PSID. Appendix table C.4 shows
additional results from the CEX. Estimation is via equally weighted GMM; block
bootstrap standard errors are in parentheses.

The results are in column 1 of table 4. We estimate the transmission parameter
of permanent shocks at $\phi^{(1)}=0.152$. A $10\%$ permanent change in income
thus results in only $1.52\%$ shift in consumption. This reflects a very low
pass-through of permanent shocks and a substantial degree of consumption
insurance. While highly significant, the estimate for $\phi^{(1)}$ is
strikingly different from BPP, whose baseline estimate of $0.64$, four times
bigger, indicates a much larger pass-through of permanent shocks and a much
lower degree of insurance. By contrast, we find evidence of full insurance to
transitory shocks, similarly to BPP. We estimate the pass-through of said
shocks at $\psi^{(1)}=-0.006$, statistically indistinguishable from zero.
Therefore, transitory shocks do not pass through into
consumption.323232Commault (2022) generalizes BPP’s linear consumption
function by allowing _past_ transitory shocks to enter the specification in
(4). She then finds a strong consumption response to transitory income,
similar to the response estimated from quasi-experimental data. We abstract
from this generalization, as our focus here is on higher-order income moments
and on non-linearities in the consumption response.

What explains the discrepancy in $\phi^{(1)}$ with BPP? As suggested earlier,
there are a few important differences between our replication and the original
exercise in BPP: new consumption data and a different time period. To help
decipher their role, we impute consumption from the CEX into the PSID in the
same way BPP did for the earlier period, and we repeat the estimation; see
appendix C.2 for details on the imputation. The results are in column 2 of
table 4. The transmission parameter of permanent shocks doubles to
$\phi^{(1)}=0.288$, while the transmission parameter of transitory shocks
remains statistically zero (though at a larger magnitude). Using imputed
consumption data thus inflates the pass-through of permanent shocks,
substantially lowering the degree of partial insurance. This is most likely
the byproduct of measurement/imputation error, evident in the variance
$\sigma^{2}_{u_{c}}$ of consumption error that almost quadruples between the
true series (column 1) and the imputed data from the CEX (column 2).333333The
large extent of consumption measurement/imputation error is also evident in
the comparison of the consumption moments of the true vs. imputed series in
table 2, as well as in appendix figure C.1. By contrast, the variance of taste
heterogeneity $\sigma^{2}_{\xi}$ remains similar in the two cases, and also
very close to BPP. The imputation, therefore, seems to bring in large amounts
of measurement error, explaining about 1/3 of the difference between our lower
$\phi^{(1)}$ and BPP’s higher estimate. We report additional insights in
appendix C.2.

Although the imputation inflates $\phi^{(1)}$, the inflated value in column 2
is still substantially lower than BPP. The PSID switched in 1997 from annual
to biennial frequency. It is thus possible that at the lower frequency certain
higher frequency income shocks are smoothed out or undone by subsequent
shocks. To assess this, we would need to re-estimate our model on annual
growth rates, which is impossible after 1997. However, we can use the original
BPP data over 1980–1992 to estimate the model _annually_ versus _biennially_.
We do this in appendix C.3. While we obtain $\phi^{(1)}=0.64$ using annual
growth rates (exactly as BPP), this drops to $\phi^{(1)}=0.4$–0.5 using
biennial rates. The biennial nature of the modern data may thus explain
another 1/3 of the difference between our lower $\phi^{(1)}$ and BPP’s higher
estimate.343434In both annual and biennial cases, $\phi^{(1)}$ measures the
pass-through of the permanent shock at an _annual_ rate. This becomes clear in
appendix B that shows the estimating equations with biennial data. The
difference in the estimates of $\phi^{(1)}$ between the two cases arises
because with biennial data we only measure an ‘aggregate’ pass-through within
a given biennial period. Consequently, some higher-frequency shocks will be
inevitably smoothed out at the observed lower frequency and their transmission
will be muted. We attribute the remaining gap to calendar time effects, e.g.,
changes in taxes/benefits or other insurance channels between 1980–1992 and
1999–2019. It is worth noting that we are not the first paper to find such a
low pass-through of permanent shocks in the recent years; Arellano, Blundell,
and Bonhomme (2017) and Arellano, Blundell, Bonhomme, and Light (2023) use the
modern PSID and find (in the specification closest to ours) a pass-through of
about $0.2$, numerically close to us.

Introducing higher-order moments. The estimation of the linear consumption
function targeting higher-order moments is our first attempt to uncover how
the tails of income risk affect the pass-through of shocks into consumption.
In other words, we want to measure the degree of partial insurance going
beyond the workhorse income-consumption covariance to account for the non-
Gaussian features of modern income dynamics. We introduce the higher-order
moments incrementally; we first target third moments (in addition to second);
we then introduce fourth moments (in addition to second and third).353535We
use pre-estimated income parameters targeting, respectively, up to 3rd moments
of income growth (column 2, table 3) and up to 4the moments of income growth
(column 3, table 3).

Column 3 of table 4 presents the results from targeting third- in addition to
second-order moments of consumption and income. The partial insurance
parameters remain largely unchanged upon inclusion of the third moments. We
estimate $\phi^{(1)}=0.160$ (as opposed to $0.152$ with second moments only)
and $\psi^{(1)}=-0.012$ (again indistinguishable from zero). We thus again
find large partial insurance to permanent shocks (a $10\%$ permanent change in
income results in only $1.6\%$ shift in consumption) and full insurance to
transitory shocks. Skewness in taste heterogeneity and consumption error is
small (much smaller than skewness in income shocks), while their variances
remain unchanged from the specification with second moments only. Therefore,
left skewness in consumption is driven by negatively skewed income shocks
rather than by taste heterogeneity or consumption measurement error.

Column 4 of table 4 presents the results from targeting all income and
consumption moments up to fourth-order. The earlier findings hold up. We
estimate $\phi^{(1)}=0.112$ (versus $0.152$ and $0.160$ in the lower-order
cases) and $\psi^{(1)}=0.007$ (indistinguishable from zero). A $10\%$
permanent change in income thus results in $1.12\%$ shift in consumption.
Therefore, if anything, we find evidence for slightly _more_ insurance to
permanent shocks upon inclusion of the fourth-order moments. The dispersion
and skewness of taste heterogeneity and consumption error do not change much
from the lower-order cases; we find evidence for excess kurtosis in both
cases, albeit at a much lesser extent than for income shocks.363636Appendix
table C.4 uses imputed consumption from the CEX to target higher-order moments
as in columns 3-4, table 4. We do see slightly higher pass-through of
permanent shocks upon inclusion of the higher-order moments, but we also find
a strong, significant, and negative response to transitory shocks, which is
hard to explain. We do not think those results are very credible; the
imputation brings in substantial measurement/imputation error, which becomes
worse the higher the order of the target moments is.

Clearly, targeting higher-order moments in the context of the linear
consumption function has not altered the degree of partial insurance vis-à-vis
targeting second-order moments alone. This seems counterintuitive given that
tail income risk has been shown to matter for various phenomena in finance and
macro. For example, tail risk explains asset pricing puzzles in Constantinides
and Ghosh (2017) and plays a crucial role in business cycles in McKay (2017).
Thematically closer to us, Ai and Bhandari (2021) show that tail risk is
uninsured in realistically parameterized labor markets, while Busch and Ludwig
(2021) find sizable welfare implications of higher-order risk. One would thus
expect that tail risk _increases_ the pass-through of shocks, as there must be
a limit beyond which households cannot further insure consumption against
extreme events.373737Similar to our results with fourth moments, De Nardi,
Fella, and Paz-Pardo (2019) and Busch and Ludwig (2021) find slightly higher
insurance with non-normal vs. normal income shocks, which they attribute to
stronger precautionary motives. Busch and Ludwig (2021) caution against BPP’s
linear framework, showing evidence of bias in the presence of tail risk. Our
quadratic specification addresses their concern.

Yet, the linear consumption function leaves no room to such events. BPP rely
on a log-linearization of the optimal consumption path around a lifecycle
‘steady-state’ (appendix A). But tail events move the agent far from such
‘steady-state’: if utility is sufficiently concave, a large shock would
command a large consumption response. A linearized marginal utility, however,
will fail to pick up such large response (echoing Carroll, 2001), which in
turn likely explains why we do not see much action upon including higher-order
moments. In other words, tail risk has, by design, no role in the linear
consumption function, which in turn motivates and justifies our higher-order
specification subsequently.

To summarize our findings from the linear consumption function: 1.) we
replicate BPP over the recent years and find much lower pass-through of
permanent shocks (larger degree of partial insurance); the imputation of
consumption from the CEX and the biennial nature of the modern PSID explain
the bulk of the difference from BPP; 2.) targeting higher-order moments does
not alter the degree of consumption partial insurance, which is not entirely
surprising as tail risk has, by design, no role in the linear framework.

###  5.3 Quadratic consumption function

We now turn to the estimation of the quadratic consumption function. The
results appear in column 5 of table 4; the full set of targeted moments is
shown in appendix B.4.3.383838Estimation of the quadratic consumption function
requires up to 4th-order moments of the income shocks, so we use the pre-
estimated income parameters from column 3, table 3.,393939The consumption-
income moments targeted in the quadratic case differ from the moments targeted
in the most general linear case, i.e., in column 4, table 4. This is because
in the quadratic case certain 3rd\- and 4th-order consumption moments require
up to 8th moments of income shocks, which we do not model. This raises an
issue of comparability between the linear and quadratic specifications, as the
target moments differ (appendix table B.2). However, we can estimate the
linear function targeting the exact same moments that we use in the quadratic
case, as the linear function is identified by a smaller set of lower-order
moments anyway (section 3). Our estimates of the linear function in that case
do not differ from those in column 4, table 4, so estimation is robust to the
choice of moments. These results are available upon request.

We estimate the transmission parameter of the average _permanent_ shock (the
coefficient on the _linear_ part of a polynomial in the permanent shock) at
$\phi^{(1)}=0.134$, highly significant and close to the values $0.11$–$0.16$
obtained from the linear specification. We estimate the transmission parameter
of the average _transitory_ shock (the coefficient on the _linear_ part of a
polynomial in the transitory shock) at $\psi^{(1)}=-0.003$, statistically
indistinguishable from zero. As $\phi^{(1)}$ and $\psi^{(1)}$ remain largely
unaltered relative to the linear case, interest lies in the transmission
parameters of the typical shocks _away_ from the middle of the distribution,
i.e., the coefficients on the _quadratic_ part of a polynomial in the shocks.
We estimate a negative $\phi^{(2)}=-0.04$ for permanent shocks (statistically
significant at the margin for now), while we estimate a positive
$\psi^{(2)}=0.015$ for transitory shocks (statistically insignificant for
now). The coefficient $\omega^{(22)}$ on the product of permanent and
transitory shocks is large but imprecisely estimated. The variance of taste
heterogeneity $\sigma_{\xi}^{2}$ and consumption measurement error
$\sigma_{u_{c}}^{2}$ remain effectively unchanged from the linear
case.404040Skewness and kurtosis of taste heterogeneity and consumption error
are not identified because they require higher-order moments of consumption,
which depend on higher than fourth-order moments of income.

The values of $\phi^{(2)}$ and $\psi^{(2)}$ have important implications for
the pass-through of shocks into consumption. To help interpret the results,
recall that the pass-through of permanent shocks is given by (6), that is, it
depends on $\phi^{(1)}$, $\phi^{(2)}$, $\omega^{(22)}$, and on the magnitudes
and signs of both income shocks. If, for simplicity, we set the transitory
shock to its mean value ($v_{it}=0$) and plug in the parameter estimates from
table 4, we get

$\partial\Delta c_{it}/\partial\zeta_{it}=0.134-0.08\times\zeta_{it}.$

Two main results follow.

First, bad (negative) permanent shocks are more hurtful than good (positive)
ones because they are associated with a much larger transmission parameter
into consumption. Consider, for example, a large permanent shock to household
income, $\zeta_{it}=\pm 0.5$, approximately three standard deviations away
from the mean. If the shock is negative ($\zeta_{it}=-0.5$), then
$\partial\Delta c_{it}/\partial\zeta_{it}=0.174$; by contrast, if the shock is
positive ($\zeta_{it}=0.5$), then $\partial\Delta
c_{it}/\partial\zeta_{it}=0.094$. We can generalize this to bad and good
shocks of any magnitude and conclude that

$\left.\frac{\partial\Delta
c_{it}}{\partial\zeta_{it}}\right|_{\zeta<0}~{}>~{}\left.\frac{\partial\Delta
c_{it}}{\partial\zeta_{it}}\right|_{\zeta>0}.$

Therefore, bad permanent shocks are not just more extreme than good shocks,
i.e., further away from the zero mean – see left skewness in table 3, but they
also have much larger pass-through rates into consumption. Bad permanent
shocks are thus double hurtful.

Second, severe bad shocks are more hurtful than moderate ones because they too
are associated with larger transmission parameters into consumption. Contrast,
for example, a very large permanent bad shock, $\zeta_{it}=-0.7$, about four
standard deviations away from the mean, with a moderate one,
$\zeta_{it}=-0.07$, half standard deviation from the mean. In the first case,
$\partial\Delta c_{it}/\partial\zeta_{it}=0.19$ while in the latter case
$\partial\Delta c_{it}/\partial\zeta_{it}=0.139$. We can generalize this to
bad shocks of any magnitude and conclude that

$\left.\frac{\partial\Delta
c_{it}}{\partial\zeta_{it}}\right|_{\zeta<<<0}~{}>~{}\left.\frac{\partial\Delta
c_{it}}{\partial\zeta_{it}}\right|_{\zeta<0}.$

This result is not hard to comprehend. When a bad shock hits, households can
insure consumption by running down savings or by using other formal or
informal insurance arrangements. But there is clearly a limit to this. As the
magnitude of the bad shock increases, these insurance arrangements will be
eventually exhausted, thus resulting in a larger fraction of the shock
eventually transmitting into consumption.414141The parameter estimates also
imply that the pass-through of a _good_ shock decreases with its magnitude.
While this seems counterintuitive, recall that _extreme_ good shocks are rare
in our data, which follows from left skewness and excess kurtosis in table 3.
We should thus not extrapolate the implications of the parameter estimates for
the case of _extreme_ good shocks, as such shocks are effectively not present
in our sample.

Why would a positive or small shock be better insured? While we cannot assess
candidate mechanisms without a structural model, our best explanation is that
people save a good income shock (thus consumption does not respond much) but
they dissave in the presence of a bad one. As there is clearly a limit to how
much households can dissave or borrow, especially in the presence of borrowing
constraints, a large negative shock would typically affect consumption by
relatively more.

Our finding of asymmetric consumption response to bad versus good income news
has recently been documented in multiple survey questions in which subjects
directly report how they would react in different hypothetical scenarios.
Based on the 2011-2014 Bank of England/NMG Consulting Survey, Bunn, Le Roux,
Reinold, and Surico (2018) find that the marginal propensity to consume from
negative income shocks is larger than from positive shocks. Similarly,
Christelis, Georgarakos, Jappelli, Pistaferri, and van Rooij (2019) and
Fuster, Kaplan, and Zafar (2020) find that responses to losses are much larger
than responses to gains, and the magnitude of the response increases with the
magnitude of the shock. While their hypothetical shocks are more transitory in
nature, they argue that these responses are consistent with lifecycle models
with concave utility and borrowing constraints.424242This is precisely the
type of model that Busch and Ludwig (2021) simulate; they find larger
consumption pass-through of negative permanent shocks versus positive ones.
While we do not model borrowing constraints here, a strength of our approach
is that it enables us to measure partial insurance in the data without taking
a stance on the precise underlying structural model/economic mechanism at
play.

Moving on to transitory shocks, their pass-through is given by (7); this
depends on $\psi^{(1)}$, $\psi^{(2)}$, $\omega^{(22)}$, and on the magnitudes
and signs of both income shocks. If we set the permanent shock to its mean
value ($\zeta_{it}=0$) and plug in the parameter estimates from table 4, we
get

$\partial\Delta c_{it}/\partial v_{it}=-0.003+0.03\times v_{it}.$

Unlike the transmission parameters of permanent shocks, neither $\psi^{(1)}$
nor $\psi^{(2)}$ are statistically significant in the full sample. This
implies that transitory shocks, be they good or bad, moderate or extreme, do
not pass through into consumption. This is similar to our findings for the
linear case as well as to BPP. By contrast, this is not the case for certain
well-defined subsamples, but we postpone the discussion of the subsamples for
section 5.4.

Despite the lack of statistical significance, the partial derivative with
respect to transitory shocks has, at face value, an interesting implication:
large good transitory shocks have a numerically larger pass-through than
smaller ones, whose pass-through is economically zero. While we should be
cautious in drawing conclusions about _extreme_ good transitory shocks (given
left skewness and excess kurtosis, large good transitory shocks are not
present in our sample), this implication helps reconcile the large pass-
through found in quasi-experimental studies with the negligible pass-through
found in BPP and other semi-structural studies that use covariance
restrictions on survey data. The former studies typically exploit large and
infrequent transitory episodes (e.g., tax rebates or lotteries as in Parker,
Souleles, Johnson, and McClelland, 2013; Misra and Surico, 2014; Fagereng,
Holm, and Natvik, 2021), which are unlikely to be present in the pooled survey
data that the latter studies exploit (e.g., BPP, Blundell, Pistaferri, and
Saporta-Eksten, 2016; Theloudis, 2021). In an environment where the
transmission of shocks depends on their magnitude, these two types of settings
will lead to opposite conclusions. Our nonlinear consumption function
reconciles the two by permitting transitory shocks to pass through to
consumption at a varying rate that depends on the shocks’ magnitude and sign.

To summarize our findings from the quadratic consumption function: 1.) bad
permanent shocks reduce consumption by more than good permanent shocks of the
same magnitude increase it; 2.) severe bad permanent shocks reduce consumption
by more than moderate ones do. Tail risk from permanent income shocks thus
matters for consumption. For a given value of $\phi^{(1)}$, the econometrician
will overestimate the true degree of partial insurance against bad permanent
shocks if she assumes a linear consumption function instead of the nonlinear
specification herein. By contrast, transitory shocks, be they good or bad,
small or large, do to not impact consumption, at least not in a statistically
significant way.434343Appendix table C.4 shows that similar conclusions are
obtained if we use consumption imputed from the CEX. In particular,
$\phi^{(2)}<0$ (albeit not statistically significant), with a magnitude
similar to that obtained directly from the PSID ($-0.036$ versus $-0.04$), and
$\psi^{(2)}>0$.

###  5.4 Sensitivity to age, wealth, and education

If bad permanent shocks impact consumption more severely than good ones, one
would expect this asymmetry to be influenced by household age, wealth, or
education, all of which seem to matter for the insurance means available to
households. To assess this claim, we repeat the estimation over subsamples
sorted on the basis of these household characteristics.

We first consider how partial insurance changes with age, by repeating the
estimation over three, almost equally sized, subsamples of households with the
male spouse between 30–40, 41–50, and 51–65 years old respectively.444444We
target the same moments as earlier, broken down by the age of the male spouse.
The moments of income shocks are also age-specific but we do not report those
results for brevity. Intuitively, the transmission parameters of income shocks
should exhibit some age dependence since wealth accumulates over the life
cycle, which in turn influences the household’s ability to insure against
income fluctuations. In fact, the derivation of the consumption function in
appendix A shows that the transmission parameters are linked to the share of
financial wealth in the household’s total wealth (comprising financial assets
and lifetime income); for example, $\phi^{(1)}_{it}=1-\pi_{it}$, where
$\pi_{it}$ is the aforementioned share of financial wealth. Therefore,
$\phi^{(1)}_{it}$ should decrease monotonically over the lifecycle as
households gradually accumulate more wealth.

The results appear in table 5. Column 1 presents results from the linear
specification when only second-order income and consumption moments are
targeted. The transmission parameter $\phi^{(1)}$ exhibits a non-monotonic
pattern over the three age brackets – decreasing from $0.154$ over ages 30–40
to $0.126$ over 41–50 before increasing to $0.180$ over ages 51–65. A similar
non-monotonic pattern is observed in column 2, where second- and third-order
moments are targeted, and is further accentuated in column 3 where all moments
up to fourth-order are included in the estimation. This is again indicative of
potential misspecification in the linear function, in the sense that its
theoretical restrictions are not supported by the empirical results.454545This
conclusion seems to be consistent with BPP who find a decreasing pattern with
age only when they impose a linear age trend on the transmission parameters.
By contrast, when they generalize this to a quadratic trend, they report a
worsening of their results without including further details. By contrast, the
transmission parameter $\psi^{(1)}$ is statistically indistinguishable from
zero in all three age brackets.

Table 5: Estimates of the consumption function, by age

_Consumption fn.:_ | Linear |  | Quadratic
---|---|---|---
_Income moments:_ | 2nd | 2nd, 3rd | 2nd, 3rd, 4th |  | 2nd, 3rd, 4th
_Consumption data:_ | PSID | PSID | PSID |  | PSID
|  | (1) | (2) | (3) |  | (4)
$\phi^{(1)}$ | 30-40 | 0.154 | (0.045) | 0.164 | (0.051) | 0.089 | (0.058) |  | 0.149 | (0.050)
| 41-50 | 0.126 | (0.051) | 0.146 | (0.055) | 0.079 | (0.089) |  | 0.115 | (0.049)
| 51-65 | 0.180 | (0.049) | 0.175 | (0.051) | 0.136 | (0.061) |  | 0.113 | (0.057)
$\psi^{(1)}$ | 30-40 | 0.094 | (0.079) | 0.083 | (0.073) | 0.095 | (0.107) |  | 0.105 | (0.090)
| 41-50 | 0.010 | (0.079) | -0.016 | (0.072) | -0.075 | (0.061) |  | 0.019 | (0.090)
| 51-65 | -0.046 | (0.070) | -0.044 | (0.066) | 0.037 | (0.101) |  | -0.063 | (0.088)
$\phi^{(2)}$ | 30-40 |  |  |  |  |  |  |  | 0.006 | (0.040)
| 41-50 |  |  |  |  |  |  |  | -0.008 | (0.063)
| 51-65 |  |  |  |  |  |  |  | -0.105 | (0.054)
$\psi^{(2)}$ | 30-40 |  |  |  |  |  |  |  | 0.036 | (0.061)
| 41-50 |  |  |  |  |  |  |  | 0.048 | (0.046)
| 51-65 |  |  |  |  |  |  |  | -0.052 | (0.079)
$\omega^{(22)}$ | 30-40 |  |  |  |  |  |  |  | -1.090 | (1.465)
| 41-50 |  |  |  |  |  |  |  | -0.290 | (1.195)
| 51-65 |  |  |  |  |  |  |  | 3.003 | (1.306)
$\sigma^{2}_{\xi}$ | 30-40 | 0.014 | (0.002) | 0.014 | (0.002) | 0.016 | (0.002) |  | 0.013 | (0.004)
| 41-50 | 0.022 | (0.002) | 0.022 | (0.002) | 0.022 | (0.002) |  | 0.022 | (0.002)
| 51-65 | 0.021 | (0.002) | 0.021 | (0.002) | 0.023 | (0.002) |  | 0.010 | (0.008)
$\gamma_{\xi}$ | 30-40 |  |  | -0.001 | (0.001) | -0.001 | (0.001) |  |  |
| 41-50 |  |  | 0.006 | (0.001) | 0.006 | (0.002) |  |  |
| 51-65 |  |  | -0.006 | (0.002) | -0.006 | (0.002) |  |  |
$\kappa_{\xi}$ | 30-40 |  |  |  |  | 0.004 | (0.002) |  |  |
| 41-50 |  |  |  |  | 0.007 | (0.002) |  |  |
| 51-65 |  |  |  |  | 0.006 | (0.002) |  |  |
$\sigma^{2}_{u_{c}}$ | 30-40 | 0.046 | (0.003) | 0.046 | (0.003) | 0.046 | (0.003) |  | 0.046 | (0.003)
| 41-50 | 0.040 | (0.002) | 0.040 | (0.002) | 0.040 | (0.002) |  | 0.040 | (0.002)
| 51-65 | 0.043 | (0.002) | 0.043 | (0.002) | 0.043 | (0.002) |  | 0.043 | (0.002)
$\gamma_{u_{c}}$ | 30-40 |  |  | 0.000 | (0.002) | 0.000 | (0.002) |  |  |
| 41-50 |  |  | 0.002 | (0.002) | 0.002 | (0.002) |  |  |
| 51-65 |  |  | 0.004 | (0.002) | 0.004 | (0.002) |  |  |
$\kappa_{u_{c}}$ | 30-40 |  |  |  |  | 0.014 | (0.003) |  |  |
| 41-50 |  |  |  |  | 0.012 | (0.002) |  |  |
| 51-65 |  |  |  |  | 0.012 | (0.002) |  |  |

_Notes:_ The table presents the estimates of the parameters of the consumption
function, allowing them to vary over ages 30–40, 41–50, and 51–65 of the male
spouse. Columns 1-3 present parameter estimates in the linear function, while
column 4 presents estimates in the quadratic case; the order of moments
targeted in each case is shown at the top of the table. All columns use
consumption data internally available in the PSID. Estimation is via equally
weighted GMM; block bootstrap standard errors are in parentheses.

Column 4 presents the results from the quadratic specification. In this case,
$\phi^{(1)}$ decreases monotonically with age, so the average permanent shock
(the shock close to the mean of the distribution) impacts consumption
gradually _less_ as households age. In line with theory, this pattern reflects
the age-increasing role of financial wealth for self-insurance. Perhaps more
importantly, $\phi^{(2)}$, the transmission parameter of the typical shock
_away_ from the mean, also decreases monotonically with age. $\phi^{(2)}$
drops from $0.006$ to $-0.008$ as we move from ages 30–40 to ages 41–50
(neither statistically significant) and becomes large and statistically
significantly negative at $\phi^{(2)}=-0.105$ in the 51–65 age bracket. As a
result, large negative permanent shocks impact consumption more severely at
older than at younger ages.

The last finding echoes Guvenen, Karahan, Ozkan, and Song (2021), who show
that skewness of earnings growth becomes more negative with the level of
earnings and, consequently, with the earner’s age on average. There is more
room for income to fall sharply in this case than among the low-earners (and
therefore the young). In fact, we do find larger left skewness in income
growth as households age in the PSID and a more negative coefficient of
skewness of permanent shocks among older households.464646The coefficient of
skewness of permanent shocks more than triples in magnitude from $-0.65$ to
$-2.39$ as we move from ages 30–40 to 51–65. The coefficient of skewness of
transitory shocks, by contrast, does not change much with the age – its
estimate is $-1.43$ and $-1.45$, respectively, over ages 30–40 and 51–65. So,
not only do permanent shocks become more negative as households age, but they
also transmit into consumption at an increasingly larger rate. In other words,
not only is there more room for income to fall among older earners, but this
also causes even greater harm to consumption as households age. By contrast,
despite a similar pattern for transitory shocks, their transmission parameters
remain statistically indistinguishable from zero over the entire lifecycle.

We next consider two subsamples formed on the basis of household wealth. We
define a household as ‘low wealth’ (‘high wealth’) if its average wealth over
its entire observation window is less (more) than median real wealth in the
sample over $1999$–$2019$. Wealth is defined as the sum of home equity (value
of main house minus mortgage), other estates, vehicles, value of any business,
investment, pension wealth, savings, and other assets, net of medical,
student, and credit card debt. Table 1 provides summary statistics for the
wealth subsamples. Average wealth in the first group is $69,731 in 2018
prices, while the second group has 14 times more wealth, an average of
$950,823. Income and earnings are almost twice as much in the second group,
while consumption is only 30% higher. Labor force participation is slightly
higher in the low-wealth sample.

Table 6, columns 1-2, presents the results in the quadratic
model.474747Appendix table C.5, columns 1-4, presents the parameter estimates
in the linear model. $\phi^{(1)}$ is consistently smaller among the high
wealth households, regardless of which set of moments are targeted.
$\psi^{(1)}$ is statistically indistinguishable from zero for both groups.
Three points emerge. First, $\phi^{(1)}$ is more than twice as large among the
low vis-à-vis high wealth households ($0.212$ versus $0.091$). As $\phi^{(1)}$
measures the transmission of the _average_ permanent shock, low wealth
households exhibit a larger pass-through of average shocks, i.e., they are
less able to insure against them, compared to high wealth households. Second,
$\phi^{(2)}<0$ for both groups, albeit statistically insignificant. With the
caveat of insignificance, the magnitude of $\phi^{(2)}$ is bigger among the
wealthy. Large negative permanent shocks thus impact consumption of the
wealthy more severely than they do among the less wealthy. As in the case of
older households previously, there is more room for consumption to fall
sharply among the wealthier. Third, $\psi^{(2)}$, the transmission parameter
of the typical _transitory_ shock _away_ from the mean, is negative, large,
and statistically significant among the less wealthy. Transitory shocks thus
have an asymmetric pass-through among lower wealth households, with large bad
shocks reducing consumption substantially more than good shocks increase it
(expression (7)). So while _average_ transitory shocks are fully insured among
low wealth households, larger negative transitory shocks are _not_ : they
reduce consumption quite substantially as low wealth households lack the self-
insurance means to smooth out even transitory fluctuations.

Table 6: Estimates of the consumption function, by wealth and education _Consumption fn.:_ | Quadratic
---|---
| By wealth |  | By education
_Subsample:_ |  Low wealth |  High wealth |  |  No college |  Some college
_Income moments:_ |  2nd, 3rd, 4th |  2nd, 3rd, 4th |  |  2nd, 3rd, 4th |  2nd, 3rd, 4th
_Consumption data:_ |  PSID |  PSID |  |  PSID |  PSID
|  (1) |  (2) |  |  (3) |  (4)
$\phi^{(1)}$ |  0.212 |  0.091 |  |  0.219 |  0.105
|  (0.041) |  (0.033) |  |  (0.057) |  (0.032)
$\psi^{(1)}$ |  -0.081 |  0.017 |  |  -0.071 |  0.030
|  (0.090) |  (0.059) |  |  (0.079) |  (0.063)
$\phi^{(2)}$ |  -0.018 |  -0.039 |  |  -0.095 |  -0.023
|  (0.052) |  (0.035) |  |  (0.048) |  (0.035)
$\psi^{(2)}$ |  -0.108 |  0.031 |  |  0.050 |  0.012
|  (0.065) |  (0.041) |  |  (0.063) |  (0.040)
$\omega^{(22)}$ |  1.116 |  -0.049 |  |  0.668 |  0.095
|  (0.874) |  (0.897) |  |  (1.067) |  (0.971)
$\sigma^{2}_{\xi}$ |  0.013 |  0.022 |  |  0.013 |  0.021
|  (0.002) |  (0.002) |  |  (0.003) |  (0.002)
$\sigma^{2}_{u_{c}}$ |  0.048 |  0.042 |  |  0.043 |  0.044
|  (0.002) |  (0.002) |  |  (0.003) |  (0.002)
_Notes:_ The table presents the estimates of the parameters of the quadratic
consumption function, allowing them to vary across wealth and education groups
(linear specification estimates appear in appendix table C.5). Low (high)
wealth is defined on the basis of household wealth being less (more) than
median real wealth in the sample over $1999$–$2019$. No versus some college is
defined on the basis of the highest level of education attained by the male
spouse. All columns use consumption data internally available in the PSID.
Estimation is via equally weighted GMM; block bootstrap standard errors are in
parentheses.

We finally turn to the role of education. We define a household as ‘no
college’ (‘some college’) if the male spouse has not gone to college (has some
college education). Table 1 provides summary statistics. Those with some
college education earn 70% more income, they have more than twice as much
wealth, they consume about 30% more, while they also work slightly more than
those without college education.

Table 6, columns 3-4, presents the results in the quadratic
model.484848Appendix table C.5, columns 5-8, presents the parameter estimates
in the linear model. The results do not present a clear pattern across
education groups. When only second moments of income and consumption are
targeted, $\phi^{(1)}$ is larger for households without college education –
$0.225$ versus $0.123$. However, when all moments up to fourth-order are
included, $\phi^{(1)}$ is almost identical in the two groups. Similar to the
patterns observed by wealth, we find that $\phi^{(1)}$ is higher (double) for
households without college education ($0.219$ versus $0.105$). The pass-
through of the _average_ permanent shock is therefore larger among the lower
educated, presumably because of the lower wealth that these households hold.
$\phi^{(2)}$ is negative in both groups, suggesting that large bad permanent
shocks transmit into consumption more than equal-sized good shocks. However,
the magnitude of the estimate in the lower education group is over four-fold
larger (and statistically significant) compared to the higher education group.
Permanent shocks _away_ from the mean are therefore way more hurtful among the
lower educated. By contrast, the transmission parameters of all transitory
shocks are statistically indistinguishable from zero in both groups.

##  6 Conclusion

We measure the degree of consumption partial insurance to income shocks,
accounting for higher-order moments of the distribution of income. Such
moments, in particular left skewness and excess kurtosis, are important
features of unexplained income growth, highlighted in several recent papers.
Our focus is on _measuring_ partial insurance without taking a stance on
specific mechanisms at the nexus of income and consumption; as such, we relate
closely to the seminal work of Blundell, Pistaferri, and Preston (2008) – BPP
as we called it throughout – who introduced a methodology to measure partial
insurance directly in the data.

Tail income risk, however, plays little role in their framework because they
linearize the consumption rule (a poor approximation when extreme shocks
command a large consumption response due to concave utility) and they ignore
moments higher than second-order. These are all features that the present
paper improves on. We use recent income and consumption data from the PSID
(1999–2019) and we establish three main new insights.

First, in replicating BPP over the recent years, we find a much lower pass-
through of permanent shocks (larger partial insurance) and full insurance to
transitory ones. We attribute one third of the difference from BPP to the
consumption imputation from the CEX, another third to the biennial nature of
our data, and the remainder to time effects.

Second, in introducing third- and fourth-order income and consumption moments
into the linear consumption model does not alter the above results.

Third, in estimating a generalized higher-order consumption function, we find
that bad permanent shocks transmit at larger rates than good shocks, and their
pass-through increases with the severity of the shock. By contrast, transitory
shocks are fully insured in the full sample, at least from a statistical
point-of-view, but they are not insured among the less wealthy. Among them, in
particular, bad transitory shocks transmit into consumption at increasingly
larger rates than good ones, exactly as permanent shocks do.

In summary, we contribute an analytical framework to measure the transmission
of income shocks into consumption, allowing such transmission to vary with the
sign and severity of the shock. To measure the transmission, we exploit
information in higher-order income moments and we characterize how our
measures of partial insurance change as we gradually introduce those moments.
Our results help reconcile the traditional partial insurance measurement
literature, in which BPP is the default methodology, with both the
quantitative literature (e.g., De Nardi, Fella, and Paz-Pardo, 2019) and the
more empirical one based on surveys or quasi-natural experiments (e.g.,
Fuster, Kaplan, and Zafar, 2020), which document asymmetries in the pass-
through of good and bad, or small and large shocks. Overall, our results
suggest that tail income risk matters substantially for consumption.

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