Idiosyncratic Risk and the Business Cycle:
A Likelihood Perspective
Alisdair McKay∗
Boston University
April 2014
Abstract
This paper asks whether uninsurable idiosyncratic income risk affects aggregate
consumption over the business cycle. I explore this question using an incomplete markets business cycle model in which the extent of idiosyncratic risk varies over the
business cycle in line with recent empirical findings. I present a method for estimating general equilibrium incomplete markets models with rich household heterogeneity
using likelihood-based methods. I use the estimated model to construct a counterfactual series for aggregate consumption under complete markets. The results show
that time-varying idiosyncratic risk has substantial effects on the dynamics of aggregate consumption. Had markets been complete, the variance of aggregate consumption
growth would have been up to 32% lower and the drop in consumption in the Great
Recession up to 1.9 percentage points smaller.
∗
amckay@bu.edu. I would like to thank Susanto Basu, Simon Gilchrist, Jonathan Parker, Zhongjun
Qu, and Ricardo Reis for helpful discussions and Carlos Ramos for excellent research assistance. I am
also grateful to seminar participants at PSE/IMF Workshop on Advances in Numerical Methods, BU/BC
Greenline Macro Meeting, FRB Chicago, MIT, Brown, ASSA 2014, Columbia, UCL, and the Philadelphia
Workshop on Macroeconomics.
1
Introduction
Are idiosyncratic income risks important to the business cycle dynamics of aggregate quantities? Krusell and Smith (1998) provided a largely negative answer to this question. Their
results show that the business cycle dynamics of an incomplete markets economy are generally very close to those of a representative agent economy.1 I revisit this question with a focus
on cyclical variation in uninsurable risks to an individual’s long-term earnings potential.
Recent empirical work using large panel datasets on individual earnings portray recessions
as times when households face substantially larger downside risks to their earnings prospects.
Moreover, these risks appear to have highly-persistent effects on household earnings. Davis
and von Wachter (2011) show that earnings losses from job-displacement are large, longlasting, and roughly twice as large when the displacement occurs in a recession as opposed
to a expansion. The differential impact of displacement in a recession is evident even twenty
years after the event occurred. Similarly, Guvenen et al. (2013) show that the distribution
of five-year earnings growth rates displays considerable pro-cyclical skewness meaning severe
negative events are more likely in a recession. This type of cyclical risk has previously been
estimated by Storesletten et al. (2004) using PSID data. Those authors find the standard
deviation of idiosyncratic labor income shocks is much larger during recessions than it is
during expansions.
The purpose of this paper is to investigate the implications of cyclical variation in idiosyncratic income risks for the business cycle dynamics of aggregate consumption. To do so,
I develop a general equilibrium business cycle model with uninsurable idiosyncratic shocks
to earnings. The idiosyncratic shock process is calibrated to match cyclical variation in income changes documented by Guvenen et al. (2013). I estimate the model using aggregate
data and recover the structural aggregate shocks that generated the observed time series for
aggregate consumption, investment and labor market conditions. Feeding these shocks into
an alternative model with a full set of insurance markets yields a counterfactual prediction
1
One exception in their paper is that the stochastic-β version of the model displays an increased correlation
of consumption and output.
1
for consumption growth under complete markets.
The results show that the combination of market incompleteness and cyclical idiosyncratic
risks to long-term earnings prospects have a substantial effect on aggregate consumption dynamics. In the complete markets counterfactual, the variance of consumption growth is up
to 32% lower than it is in the incomplete markets economy. The differences in consumption
growth accumulate to substantial differences in the dynamics of aggregate consumption during recessions. For example the cumulative effect of uninsurable idiosyncratic risk during the
Great Recession is such that aggregate consumption would have been up to 1.9 percentage
points higher in the first quarter of 2009 under complete markets.
Methodologically this paper partially integrates two literatures: it presents a method by
which incomplete markets business cycle models in the style of Krusell and Smith (1998) can
be estimated using likelihood-based methods in the style of Sargent (1989) and Schorfheide
(2000) and others. The main idea behind the methodological contribution is to express the
dynamics for aggregate variables implied by the incomplete markets model as a dynamic
system in a small number of aggregate variables. That the aggregate dynamics of the incomplete markets model can very accurately represented in terms of a small number of aggregate
variables was shown by Krusell and Smith (1998). I apply this idea to the statistical analysis of the model. Using this approach, one can apply the tools of statistical filtering to
incomplete markets models to construct the likelihood function for the aggregate data and
conduct inference on unobservable variables.
This methodological approach is useful for the question at hand for three reasons. First, it
allows me to investigate the contribution of idiosyncratic risk to particular episodes, such as
the Great Recession, by backing out the structural shocks that generated the data. Second,
it allows me to decompose the effect of idiosyncratic risk on aggregate dynamics according
to its effect on the impulse response functions for the individual aggregate shocks. Finally,
it allows me to estimate the persistence of the aggregate shock processes. As I explain,
the persistence of the shocks is crucial because it determines the size of the wealth effect
generated by a given innovation and therefore the size of the resulting change in consumption.
2
This is not the first paper to estimate an incomplete markets model. Gourinchas and
Parker (2002) and Guvenen and Smith (2010) estimate versions of the incomplete markets
model focusing on cross-sectional variation in a stationary environment. Heathcote et al.
(2012) estimate an incomplete markets model that allows for time-varying wage and preference distributions. Their focus is on secular trends in inequality as opposed to business cycle
fluctuations in aggregate quantities. The goal here is to estimate an incomplete markets
business cycle model using aggregate time series data.
The contribution of idiosyncratic risk to the dynamics of aggregate quantities has been
studied previously and the most closely related contributions are Krusell and Smith (1998),
Challe and Ragot (2013), and Ravn and Sterk (2013). This paper is primarily concerned
with cyclical variation in highly-persistent earnings shocks as measured in the empirical
papers mentioned above. By contrast, preceding work has incorporated cyclical variation in
idiosyncratic risk through changes in the risk of unemployment, which is generally a short
lived shock to earnings.2 The highly-persistent earning shocks that I study are notable in
that they alter the dynamics of aggregate consumption even when households are relatively
far from the borrowing constraint. This is not true of employment shocks, which only have
aggregate effects when the distribution of wealth is highly skewed (Krusell and Smith, 1998).
Other implications of the cyclical variation in idiosyncratic risks to long-term earnings
potential have been studied. For example, Storesletten et al. (2007) investigate the asset
pricing implications of this type of risk. The welfare cost of business cycles in the presence of
these risks have been analyzed by Storesletten et al. (2001); Krebs (2003, 2007). However, it
does not appear that the consequences for the business cycle dynamics of aggregate quantities
have been studied in the literature.
The paper is organized as follows: Section 2 presents the model. Section 3 discusses
the method to compute the likelihood function for incomplete markets models. Section
4 discusses the parameters that are estimated and the data used to estimate them. The
2
This distinction is not quite as stark as it might seem as Ravn and Sterk (2013) allow for long-term
unemployment spells. The calibration of the shock processes and cyclical variation in the risks is very
different in that approach.
3
baseline results appear in Section 5 and Section 6 extends the model to generate more
realistic heterogeneity in wealth holdings. Section 7 compares estimated parameter values
from different model specifications. The paper concludes with Section 8.
2
Model
I analyze a general equilibrium model with heterogeneous households and aggregate uncertainty. At the aggregate level, the model is similar to that of Krusell and Smith (1998).
At the microeconomic level, I incorporate time-varying idiosyncratic risk with an income
process similar to the one estimated by Guvenen et al. (2013).
2.1
Preferences and endowments
There is a unit continuum of households who survive from one period to the next with
probability 1 − ω. Each period, a mass ω of households is born leaving the total population
unchanged. Households seek to maximize homothetic preferences
E0
X
t
β̂ t
t
Y
τ =0
!
eqτ
c̄t1−γ
,
1−γ
where c̄t is the household’s consumption in period t. The economy grows at rate g and
ct ≡ c̄t /egt will denote consumption relative to trend. qt is a preference shock that follows
qt = ρq qt−1 + q,t .
Throughout the paper, I will assume an aggregate shock z,t , for any z, is normally distributed
with mean zero and standard deviation σz . I use β ≡ β̂(1 − ω) to denote the joint effect of
pure time preference and discounting due to mortality risk.
Households can be either employed (n = 1) or unemployed (n = 0) and transition
between these two states exogenously. Let λ and ζ be the job-finding and -separation rates,
respectively. I assume that aggregate shocks occur at the start of a period and labor market
4
transitions occur under the new transition rates. The job-finding and -separation rates are
exogenous and follow logistic AR(1) processes
λt =
1
1 + e−λ̂t
(1)
¯
λ̂t = (1 − ρλ )λ̂ + ρλ λ̂t−1 + λ,t
1
1 + e−ζ̂−t
¯
ζ̂t = (1 − ρζ )ζ̂ + ρζ ζ̂t−1 + ζ,t .
ζt =
(2)
(3)
(4)
The unemployment rate, u, evolves according to
ut = (1 − λt )ut−1 + ζt (1 − ut−1 ),
(5)
If employed, a household exogenously supplies ey efficiency units of labor, where y is the
households individual efficiency. The cross-sectional dispersion in efficiency units could be
due to differences in wage or due to differences in hours. For lack of a better term, I will
refer to y as skill. This skill evolves according to
y = θ + ξ,
θ0 = θ + η,
where ξ is a transitory income shock distributed N (µξ , σξ ). The permanent shock, η, is drawn
from a time-varying distribution the tails of which vary over the business cycle in such a way
to generate pro-cyclical skewness as shown by Guvenen et al. (2013). Like Guvenen et al., I
assume η is drawn from a mixture of normals. Specifically




N (µ1,t , ση,1 ) with prob. 1 − 2p



η ∼ N (µ2,t , ση,2 ) with prob. p





N (µ3,t , ση,3 ) with prob. p.
5
µ2,t and µ3,t will control the tails of the distribution and I assume they are driven by a single
driving process, xt :
µ2,t = µ02 + µ12 xt
(6)
µ3,t = µ03 + µ13 xt .
(7)
This income process differs from the one estimated by Guvenen et al. in that it mixes three
normals as opposed to two and the extent of idiosyncratic risk is a continuous function of
the driving process xt as opposed to regime switching.
Throughout this paper, I use the job separation rate as the cyclical indicator, xt = ζt .
This choice is motivated in part by the work of Davis and von Wachter (2011) who show
that job loss in mass layoff events is associated with large and long-lasting reductions in
income and it is plausible that times of labor market turbulence lead to both job separations
and more negative outcomes for long-term earnings. I show below that this assumption
does fairly well in accounting for the cyclical variation in measured income risks reported by
Guvenen et al..
The parameter µ1,t controls the center of the distribution for η. I make µ1,t a function
of the other parameters of the distribution such that Eeθ = 1 in all periods.3 Similarly, I
choose the constant parameters of the distribution for ξ, µξ and σξ , such that E[eξ ] = 1. As
employment is independent of y, the aggregate labor input is
Lt = (1 − ut ).
It would be natural to assume that there is a correlation between shocks to skill and shocks
to employment. I have experimented with this and found that it has little impact on the
results because employment shocks are quite transitory.
When a household dies, it is replaced by a household with no assets and θ normalized
3
This requires
2
2
2
(1 − 2p)eµ1,t +ση,1 /2 + peµ2,t +ση,2 /2 + peµ3,t +ση,3 /2 = 1.
6
to 0. The mortality risk allows for a stationary cross-sectional distribution of productivity
along with permanent shocks to θ. The unemployment rate among newborn households is
the same as prevails in the surviving population at that date.
This paper investigates the implications of these cyclical idiosyncratic risks. A related
question concerns the source of these risks. While some progress has been made here (Huckfeldt, 2014), it is still not well understood. The perspective that I adopt here is that the
income shocks are uninsurable and their realizations are unknown before they occur.
2.2
Technology and markets
A composite good is produced out of capital and labor according to
Ȳt = At K̄tα egt Lt
1−α
,
(8)
where g is the trend growth rate. Throughout the paper, I normalize variables by trend
productivity. Defining Kt ≡ K̄t /egt and Yt ≡ Ȳt /egt , Eq. (8) becomes
Yt = At Ktα Lt1−α ,
TFP, At , evolves according to
log At = ρA log At−1 + A,t .
Capital depreciates at rate δ and evolves according to
Ct + eg Kt+1 = Yt + (1 − δ)Kt .
7
The factors of production are rented from the households each period at prices that satisfy
the representative firm’s static profit maximization problem
Wt = (1 − α)At Ktα L−α
t
(9)
R̃t = αAt Ktα−1 Lt1−α + 1 − δ.
(10)
Here R̃ is the return on capital. Households save in the form of annuities and the return to
surviving households is Rt ≡ R̃t /(1 − ω). I assume that savings must be non-negative due
to borrowing constraints.
Other than self-insurance, the only mechanism for risk-sharing is an unemployment insurance scheme. The pre-insurance labor income of a household with characteristics (θ, ξ, n)
is W eθ+ξ n. I assume that all households in the economy share unemployment risk by pooling their incomes and then allocating after-transfer incomes such that a household with
characteristics (θ, ξ, n) receives
`(θ, ξ, n, W, u) ≡ W eθ+ξ [n + bu (1 − n)]
1−u
,
1 − u + bu u
(11)
where the parameter bu controls the degree of insurance against unemployment shocks. The
final term in this expression plays the role of a tax that balances the unemployment insurance
budget period by period.
2.3
The household’s decision problem
The individual state variables of the household’s decision problem are its cash on hand,
call it a, its permanent skill, θ, and its employment status. The aggregate states are S ≡
{A, λ̂, ζ̂, q, Γ}, where Γ is the distribution of households over the state space from which one
can calculate both K and L. The household’s decision variables are consumption and endof-period savings, k 0 . k 0 is normalized by the level of trend productivity in the next period
8
so the current resource cost of saving k 0 is eg k 0 . The household’s decision problem is then4
V (a, n, θ, S) = max eq(S) u(a − eg k 0 ) + eq(S) eg(1−γ) βE0 ,ξ0 ,η0 ,n0 [V (a0 , n0 , θ0 , S 0 )]
k≥0
subject to
a0 = R(S 0 )k 0 + `(θ + η 0 , ξ 0 , n0 , W (S 0 ), u(S 0 )),
the law of motion Γ0 = HΓ (S, λ̂0 , ζ̂ 0 ) and the other equations that govern the dynamics of the
aggregate state.
The household’s Euler equation and budget constraint are
h 0
i
−γ
c−γ = e−γg βE eq R0 c0
c + eg k 0 = Rk + `(θ, ξ, n, W, u).
Analysis of the model can be simplified by normalizing a household’s consumption and
income by it’s permanent income eθ leading to
"
0 −γ #
c −γ
c
0
0
= e−γg βE eq −γη R0 θ0
θ
e
e
0
k
a
c
gk
+
e
= R θ + `(0, ξ, n, W, u) ≡ θ .
θ
θ
e
e
e
e
Making use of this normalization allows one to reduce the size of the state space by only keeping track of a household’s assets relative to its permanent income. A permanent-transitory
income process and finite lifetimes are necessary to achieve this simplicity while maintaining
a finite variance of income in the cross-section and this approach has been used previously
by Carroll et al. (2013). While permanent income can be eliminated as a state of the house4
Due to economic growth, the household’s discount factor is modified by the term eg(1−γ) . If ct is
consumption relative to trend, then total consumption is ct egt . The utility function then becomes
E0
∞
X
t=0
1−γ
βt
(ct egt )
1−γ
= E0
∞ X
t=0
9
βeg(1−γ)
t c1−γ
t
.
1−γ
hold’s problem, the risk to permanent income still affects consumption decisions through the
η 0 term in the Euler equation and through its effect on the denominator of normalized cash
on hand.
2.4
Equilibrium
Let f be the optimal decision rule for k 0 in the household’s problem. Aggregate savings are
0
K =
XZ Z Z
n
k
θ
f (Rk + `(θ, ξ, n, W, u), n, S) Φ(dξ)Γ(dk, dθ, n),
ξ
where Φ is the distribution of ξ. Given a set of exogenous stochastic processes for A, λ̂, ζ̂,
and q, a recursive competitive equilibrium consists of the law of motion for the distribution,
HΓ , household value function, V , and policy rule, f , and pricing functions W and R. In an
equilibrium, V and f are optimal for the household’s problem, R = R̃/(1 − ω) and W satisfy
(9)-(10), and HΓ is induced by f and the idiosyncratic income process.
3
Calculating the likelihood function
The literature has developed several algorithms to solve incomplete markets models with
aggregate uncertainty.5 The difficulty in solving this class of models is that the distribution
of wealth is an infinite-dimensional endogenous state variable. It may appear that the inherit
complexity of the state of this type of model makes computing the likelihood function a
challenging problem. However, the same methods that reduce the state of the system in
solving the model apply to the statistical filtering problem and make it straightforward to
calculate the likelihood.
5
The most widely used algorithm is the Krusell and Smith (1998) algorithm. See also den Haan (2010).
10
3.1
State reduction and filtering
A statistical filtering problem involves recursive application of a prediction step and an
updating step. Let St be the unobserved state at date t, which evolves according to
St = F (St−1 , t )
where F is possibly non-linear and is i.d.d. and possibly non-Gaussian. The data vector
Zt is generated from a possibly non-linear measurement equation
Zt = G(St , νt ),
where ν is an i.i.d. possibly non-Gaussian measurement error. Some prior belief about the
state is given by Pr(S0 ). The prediction step is then
Z
Pr(St |Z1:t−1 ) =
Pr(St |St−1 ) Pr(St−1 |Z1:t−1 )dSt−1 ,
where Z1:t−1 is the data observed up to date t − 1. The updating step is
Pr(St |Z1:t ) =
Pr(Zt |St ) Pr(St |Z1:t−1 )
Pr(Zt |Z1:t−1 )
where
Z
Pr(Zt |Z1:t−1 ) =
Pr(Zt |St ) Pr(St |Z1:t−1 )dSt
is the normalizing constant and also the likelihood of the data at date t conditional on the
preceding data. In these equations, the probability distributions depend on the functions F
and G and on the properties of the shocks t and νt , which in turn depend on the vector of
parameters to be estimated, call it Θ. The log likelihood of the data Z1:T is then given by
T
X
log Pr(Zt |Z1:t−1 ; Θ).
t=1
11
For the model presented in Section 2, the state St includes the distribution of wealth
and so one cannot immediately apply standard filtering methods to the problem at hand.
Under two conditions, however, we can replace the state St with a low-dimensional state
vector S̃t and proceed with standard techniques. The first condition is that the probability
distribution Pr(Zt |St ) can be approximated with Pr(Zt |S̃t ). In the problem at hand, one
does not need to know the distribution of wealth in order to compute aggregate output and
consumption. Suppose Lt , Kt and Kt+1 are given,6 one can then solve for Ct and Yt from
Yt = At Ktα L1−α
t
Ct = Yt + (1 − δ)Kt − eg Kt+1 .
The probability distribution of current-period aggregate data only depends on the evolution
of K and L = 1 − u and not on Γ. Therefore in the updating step it is not necessary
to include Γ in the state as it is sufficient to include K and u. If the observable data
include disaggregated quantities, such as percentiles of the distribution of wealth or the
number of borrowing constrained households, one may need to include more information in
S̃t . Incorporating data of this kind would be a very useful direction for future research.
The second condition that the model must satisfy is that the distribution Pr(S̃t |St−1 )
can be approximated by Pr(S̃t |S̃t−1 ) so it is not needed to keep track of the full state for
the prediction step. This is the same idea that underlies all algorithms to solve incomplete
markets models with aggregate uncertainty: it is impossible to track the entire distribution of
idiosyncratic state variables so these algorithms find ways of replacing that distribution with
some lower-dimensional representation. Most famously, Krusell and Smith (1998) showed
that future values of the capital stock can be forecast very well using only the first moment
of the distribution. So again, S̃ contains K and not Γ.
6
As Kt+1 is determined at date t, it can be included in St and S̃t .
12
3.2
Application
To solve the model I use the Krusell-Smith algorithm in which the distribution of wealth
is summarized by a small number of moments, in particular the first moment, K. The
dynamics of the capital stock can then be expressed in terms of the law of motion h
log K 0 = h(A, λ̂, ζ̂, q, û, log K),
where the distribution of wealth is replaced by log K and the inverse-logistic transformation
of the unemployment rate, û. As usual, the solution algorithm solves for h by making a guess
for h, solving the household’s problem, simulating a population of households, and updating
the guess for h. I assume that h is linear in its arguments, which in this application yields
highly accurate forecasts of the future capital stock. Where the algorithm deviates from the
approach taken in Krusell and Smith (1998) is in the approach to solving the household’s
decision problem for a given h. I approximate the household’s decision problem with a
function that is non-linear with respect to idiosyncratic states, but linear in aggregate states.7
The reason for adopting this approach as opposed to a fully non-linear solution method as in
Krusell and Smith (1998) is to improve the speed of the computations as I will need to solve
the model repeatedly during estimation. Empirical DSGE models typically include a large
number of persistent shocks that result in a considerable number of aggregate state variables
making fully non-linear solutions computationally costly. As the resulting approximate policy
rule is non-linear in the individual’s state variables I am able to capture the precautionary
savings motive that arises out of idiosyncratic risk. Precaution against aggregate shocks,
however, will not be captured by this solution method.8
7
This part of the method is closely related to the work of Reiter (2009) and Ravn and Sterk (2012).
Reiter’s methods has two components: linearizing the household decision rules with respect to aggregate
shocks and summarizing the distribution of wealth with a histogram. I only make use of the first of these
components. Ravn and Sterk (2012) use a version of the Krusell and Smith (1998) algorithm in which the
household decision rules are linear with respect to aggregate states.
8
For a given function h, the household’s decision rule f displays the certainty equivalence property with
respect to the variance of aggregate shocks. However, these variances affect the dynamics of the distribution
of wealth and lead to different solutions for h. Therefore, the full solution of the model does not display
certainty equivalence.
13
To calculate the likelihood function for aggregate data, suppose one has solved for h as
described above. We can then summarize the aggregate dynamics of the economy with the
following system of equations
log K 0 = h(A, λ̂, ζ̂, q, û, log K)
C + K 0 eg = Y + (1 − δ)K
Y = AK α L1−α
L=1−u
u0 = (1 − λ0 )u + ζ 0 (1 − u)
and the exogenous processes for λ̂, ζ̂, and A. The system can be reduced to a small number
of variables and equations because the function h(·) summarizes the aggregate effect of
household heterogeneity. I then take a linear approximation to this system and express the
result in state-space form. Under the assumption of Gaussian aggregate shocks, I can then
use the Kalman filter to compute the likelihood. Non-linear or non-Gaussian models could
be analyzed using the methods of Fernández-Villaverde and Rubio-Ramı́rez (2007) but I do
not pursue this here.
Appendix B discusses the implementation of the methods in more detail and discusses the
accuracy of the solution including the quality of the approximation K 0 = h(A, λ̂, ζ̂, q, û, log K).
4
Parameterization
The model parameters can be grouped into three categories (see Table 1). First, there are
parameters that relate to preferences and technologies, which I calibrate to values on the
balanced growth path. Second, there are the parameters of the income process. I select
these parameters to match the data moments reported by Guvenen et al. (2013) using a
procedure described in Section 4.2. Finally, there are the parameters of the shock processes,
which I estimate from the time series data described in Section 4.3. Prior distributions for
14
these parameters are listed in Table 1 although these do not exert a strong influence on the
results.
The persistence of aggregate shocks is important to the cyclical behavior of aggregate
consumption. More persistent shocks generate larger wealth effects and therefore larger
consumption responses on impact. More persistent shocks also generate more persistent
fluctuations in endogenous variables. Both of these effects are important to assessing the
contribution of a given shock to aggregate consumption dynamics.9 Therefore it is important
to choose these persistence parameters carefully.
4.1
Calibrated parameters
Some of the parameter values are standard in line with Cooley and Prescott (1995): I set
the coefficient of relative risk aversion to 2, and set the household’s discount factor to match
an annual capital-output ratio of 3.32. The capital share is set to 0.36 and the quarterly
depreciation rate is set to 0.02. The average labor market transition rates are set so that the
steady state rates of short-term and long-term unemployment match the average values in
the data described in Section 4.3. Similarly, I set the trend growth rate to the mean growth
of income and consumption in the data to be described. Lastly, I set the mortality rate to
0.01 quarterly to generate an expected working of life of 25 years. I set the unemployment
insurance parameter bu to 0.3 for a 30% replacement rate, which is in line with replacement
rates for the United States reported by Martin (1996).
4.2
Parameters for the income process
The income process I use is motivated by the work of Guvenen et al. (2013). Those authors
estimate a related income process with a mixture of two normals the coefficients of which take
different values in expansions and recessions. To choose parameters of the income process, I
use a simulated method of moments approach similar to Guvenen et al.. I simulate quarterly
9
Cogley and Nason (1995) and King and Rebelo (1999) discuss the importance of the persistence parameters in more detail.
15
Symbol
Panel
γ
λ̄
ζ̄
bu
ω
β
α
δ
g
Panel
µξ
σξ
µ02
µ12
µ03
µ13
ση,1
ση,2
ση,3
p
Panel
ρA
ρλ
ρζ
ρq
σA
σλ
σζ
σq
Parameter
A. Calibrated
Risk aversion
Avg. job finding rate
Avg. high-skill job separation rate
Unemployment insurance
Mortality risk
Discount factor
Capital share
Depreciation rate
Growth rate
B. Income process parameters
Mean transitory skill
Std. dev. transitory skill
Mean of mixture component
Cyclicality of mixture component
Mean of mixture component
Cyclicality of mixture component
Std. dev. of mixture component
Std. dev. of mixture component
Std. dev. of mixture component
Weight on mixture components
C. Estimated
Autoregressive coefficient of A
Autoregressive coefficient of λ
Autoregressive coefficient of ζ
Autoregressive coefficient of q
Standard deviation of A
Standard deviation of λ
Standard deviation of ζ
Standard deviation of q
Value
2
0.679
0.037
0.3
0.01
0.985
0.36
0.02
4.35 × 103
−0.0283
16.82
0.2963
0.1460
−0.3153
0.1534
1.429
10.45
10.45
0.0526
0.9797
0.8955
0.8467
0.9714
0.71
11.60
6.56
0.041
Target/Prior
Mean long-term unemployment.
Mean short-term unemployment.
30% replacement rate.
25-year expected working life.
Capital-output ratio = 3.32.
Mean (∆Y + ∆C)/2
See
See
See
See
See
See
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See
See
text
text
text
text
text
text
text
text
text
text
Beta: mn. = 0.5, var. = 0.04.
Beta: mn. = 0.5, var. = 0.04.
Beta: mn. = 0.5, var. = 0.04.
Beta: mn. = 0.5, var. = 0.04.
Inv. Gam.: mn. = 1, var. = 4.
Inv. Gam.: mn. = 1, var. = 4.
Inv. Gam.: mn. = 1, var. = 4.
Inv. Gam.: mn. = 1, var. = 4.
Table 1: Parameter values, targets and priors. Estimated parameter values correspond to
the general equilibrium economy. Standard deviations are ×100.
16
data from the model, including unemployment spells and mortality risk. I then aggregate
the quarterly data to annual observations and compute 1-year, 3-year, and 5-year log income
changes. I then choose the parameters of the income process to minimize the distance
between the moments of simulated income change distributions to the annual data reported
in Guvenen et al..10 Among the moments that I ask the model to match are the widths of
the right and left tails as measured by the difference between the 90th percentile and the
median and the 10th percentile and the median. These differences can be combined to form
Kelley’s skewness
Kelley’s skewness =
(P 90 − P 50) − (P 50 − P 10)
,
P 90 − P 10
which is Guvenen et al.’s preferred measure of skewness because it is less sensitive to extreme
observations than the third moment. Figure 1 plots Kelley’s skewness for 5-year income
changes for the model and the Guvenen et al. data. The model does a good job of replicating
the amplitude of the fluctuations in skewness but the timing of these fluctuations appears
to lag the data. Appendix A further explains the procedure.
The left panel Figure 2 shows the density of η with x = 0. Most of the mass is concentrated near zero, but there is some probability of both large positive and negative values.
The right panel of the figure omits the mass in the center to focus on the behavior of the
tails. It shows a hypothetical recession with ζ one standard deviation below its mean and an
expansion with ζ one standard deviation above its mean. While the tails are quite volatile
over the cycle, the central mass is relatively stable, which results in the pro-cyclical skewness.
4.3
Time series data and estimation
After fixing the parameters in the first two panels of Table tab:Params, I estimate the
aggregate shock processes by searching for the posterior mode. I use data on short-term
unemployment, long-term unemployment, consumption of non-durables and services, consumption of durables and investment in fixed capital. I relate the data series to model
10
I impose the restriction ση,2 = ση,3 .
17
Kelley’s skewness
0.2
0.15
0.1
0.05
0
−0.05
−0.1
−0.15
−0.2
1975
1980
1985
1990
1995
2000
2005
2010
Figure 1: Keylley’s skewness of 5-year income changes: model (solid line) and data (dashed
line). Data are from Guvenen et al. (2013). The value for year t shows the skewness of
income changes from year t to t + 5.
Full distribution
Tails
0.25
20
0.2
15
0.15
10
0.1
5
0.05
Recession
Boom
density
25
0
−1
−0.5
0
η
0.5
0
−1
1
−0.5
0
η
0.5
1
Figure 2: Density of distribution for η. The right-hand panel omits the central component
of the mixture for clarity.
18
variables as follows
consumption of non-durables and services = Ct
investment plus consumption of durables = Yt − Ct
short-term unemployment = ζt (1 − ut−1 )
long-term unemployment = (1 − λt )ut−1
I will refer to the first two data series as “consumption” and “investment,” respectively, and
each is transformed as 100 × ∆ log(·). I use quarterly data from 1966:I to 2013:IV.
In the context of incomplete markets, the choice of data on labor market conditions
is more complicated than it is under complete markets because the distribution of labor
income is now relevant to the dynamics of aggregate consumption. I ask the model to match
the contribution of aggregate hours to fluctuations in aggregate labor income, but to avoid
complicating the model I abstract from labor force participation and fluctuations in hours
per worker.
I use 1 − Ht /H̄ as the empirical observation of the unemployment rate, where Ht is
aggregate hours per capita and H̄ is aggregate hours at full employment. To define a fullemployment level of aggregate hours per capita I look to 1999:IV as the quarter since 1960
with the largest value of aggregate hours per capita. I then define full employment hours
per capita as the labor force participation rate in 1999:IV times the index of average weekly
hours in that quarter.
The unemployment rate alone does not convey information about the duration of unemployment spells. I therefore use data on the composition of the unemployed pool by splitting
the pool into those with fewer than 15 weeks and at least 15 weeks of unemployment. As the
model period is one quarter, I take less than 15 weeks to be individuals who are unemployed,
but were employed in the previous quarter. These data define the shares of short-term and
long-term unemployment in total unemployment. I use these shares to split the constructed
unemployment rate into short-term and long-term unemployment pools. Figure 3 shows the
19
0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 1966 1971 1976 1981 Short-­‐term 1986 1991 Long-­‐term Total 1996 2001 2006 2011 Reported by BLS Figure 3: Constructed short-term and long-term unemployment data.
constructed unemployment rate and its decomposition by duration. From this figure, one can
see there are some low-frequency movements in the unemployment rates, which are partly
attributable to demographic factors. To remove these low-frequency trends, I de-trend the
unemployment series using an HP filter with smoothing parameter 100,000 following Shimer
(2005).
5
The aggregate effects of time-varying risk
I assess the contribution of time-varying idiosyncratic risk through counterfactual experiments. The first counterfactual is a representative agent economy in which a full set of
insurance markets allow idiosyncratic risks to be fully shared among households. In addition, I assume a dynastic structure such that upon death a household is replaced by a child
20
−3
5
λ
−4
A
x 10
7
x 10
6
4.5
5
4
4
3.5
3
3
0
20
2
60
ζ
−3
0
40
x 10
0
20
−3
2
−0.5
0
−1
−2
−1.5
−4
−2
−6
40
q
x 10
IM
0
20
40
60
0
60
20
CM
40
60
Figure 4: Impulse responses of aggregate consumption to aggregate shocks. IM = incomplete
markets, CM = complete markets. Shocks are scaled to one standard deviation. Units are
log deviation from steady state.
who inherits the parent’s capital holdings and the parent values the child as an extension
of themselves. Finally, I assume a capital subsidy financed by a lump sum tax such that
the steady state capital stock is the same as in the incomplete markets economy.11 In this
counterfactual, aggregate consumption is determined by a relatively standard Euler equation
h 0
i
−γ
C −γ = e−γg β̂(1 + τ K )E eq R0 C 0
.
I call this counterfactual the “complete markets” case. The second counterfactual is simply
the incomplete markets economy with a time-invariant idiosyncratic risk process (µ12 = µ13 =
0). I call this counterfactual the “constant risk” case.
Figure 4 shows the impulse response of aggregate consumption to the four aggregate
shocks for both the incomplete markets and complete markets cases. In the top two panels,
11
This requires β̂(1 + τ K ) = 0.9916.
21
ζ with constant risk case
−3
−0.2
x 10
−0.4
−0.6
−0.8
−1
−1.2
−1.4
−1.6
−1.8
−2
−10
0
10
20
IM
30
CM
40
50
60
Constant Risk
Figure 5: Impulse responses of aggregate consumption to ζ shock. IM = incomplete markets,
CM = complete markets, CR = constant risk. Shocks are scaled to one standard deviation.
Units are log deviation from steady state.
the response of consumption to shocks to TFP (A) and the job-finding rate (λ) is slightly
larger in the incomplete markets economy than in the complete markets economy. The key
difference between the two economies comes in the response to the shock to ζ, which leads
to both an increase in the job-separation rate and negative skewness in the distribution of
skill shocks. Finally, the response to a preference shock (q) is very similar in the two models.
Figure 5 shows that the differences in the response to a ζ shock in the incomplete and
complete markets economies are primarily driven by the time-varying idiosyncratic risk process. Specifically, the figure shows the constant risk case in addition to the two others. One
can see that the the incomplete markets model with a constant level of idiosyncratic risk is
quite similar to the complete markets case.
Table 2 shows the variance of aggregate consumption growth and its correlation with
aggregate income growth, short-term unemployment and long-term unemployment. For the
three model economies, these moments are generated without preference shocks. One can
see that time-varying idiosyncratic risk makes consumption growth more volatile and more
correlated with short-term unemployment.
The estimated model can be used to recover the structural shocks that generated the
data. To recover the structural shocks I use the Kalman smoother. Feeding these shocks
22
Variance
Corr. w/ ∆Y
Corr. w/ short-term u.
Corr. w/ long-term u.
Data
IM
0.27
0.75
-0.33
0.07
0.14
0.97
-0.17
0.00
CM
CR
0.12 0.12
0.96 0.96
-0.09 -0.10
-0.04 -0.04
Table 2: Moments of consumption growth. IM = incomplete markets model. CM = complete
markets model. CR = IM with constant idiosyncratic risk.
λ
A
0.5
0.5
0
0
−0.5
1960
0.01
1980
−0.5
2000
2020
1960
0.01
1980
ζ
0.5
0
0
1960
0.13
1980
2020
2000
2020
q
0.5
−0.5
2000
−0.5
2000
2020
1960
0.01
1980
Figure 6: Decomposition of contribution of idiosyncratic risk to aggregate consumption
growth. Each series is the contribution of the given shock to aggregate consumption growth
in the incomplete markets model less the complete markets model. Standard deviations
listed in the figures.
23
one at a time into the incomplete markets model produces a time series for the contribution
of the given shock’s contribution to aggregate consumption. Given the linear structure of
the empirical model, summing across the four shocks nearly12 reproduces the data. The
same shocks can be fed into the complete markets model to generate a counterfactual contribution of each shock. Figure 6 shows the difference between the generated series for the
incomplete and complete markets economies. Each panel shows the standard deviation of
the difference in the predicted series from that two models. Three of the four panels show
almost no difference between the complete markets and incomplete markets economies. The
exception is the ζ shock where the impact of idiosyncratic risk on aggregate consumption
has a standard deviation of 0.13. The contribution of incomplete markets to consumption
growth is particularly evident during the recessions in 1975, the early 1980’s and the Great
Recession where the incomplete markets model generates larger drops in consumption.
To find a counterfactual prediction for total consumption growth, I could feed the smoothed
structural shocks that were recovered from the incomplete markets model into the complete
markets model. As the model is linear, an equivalent calculation is to sum the series in the
four panels of Figure 6 and then subtract the result from the consumption growth data. This
alternative method for arriving at the same counterfactual is useful because it makes clear
which shocks are contributing to the difference between the counterfactual and the data. As
there is not a clear structural interpretation of the preference shock and I choose to omit it’s
contribution altogether. Therefore, I sum over the remaining three panels of Figure 6 and
subtract the result from the observed data to arrive at the counterfactual.
The variance of the counterfactual consumption growth series under complete markets
is 0.235 as opposed to the observed variance of 0.271 so consumption growth would have
been 13% less volatile had markets been complete. The real benefit of this counterfactual
time series is that it can be used to investigate the contribution of idiosyncratic risk to
particular episodes. The left panel of Figure 7 shows the cumulative consumption growth
12
The data also depends on the past shocks through the state variables. The procedure described sets
these to zero before the sample begins so there is a small discrepancy at the start of the sample that fades
quickly.
24
4
6
Complete markets
5
4
Data
2
percentage points
percentage points
3
1
0
3
2
1
−1
−2
2007
Complete markets
Data
0
2008
2009
2010
2011
2012
2013
−1
1979 1979.5 1980 1980.5 1981 1981.5 1982 1982.5 1983 1983.5 1984
2014
Figure 7: Cumulative consumption growth.
Fraction of wealth held by
Quintile
1
Data
Baseline
80/20
2
3
4
Gini
5
0.0 1.1 4.5 11.2 83.4 0.816
3.2 7.9 12.3 19.3 57.2 0.527
0.9 2.1 3.4 9.8 83.7 0.701
Table 3: Distribution of wealth. The 80/20 model includes preference heterogeneity. Data
are from the 2007 Survey of Consumer Finances as reported by Diaz-Gimenez et al. (2011).
Model distributions refer to the stationary equilibrium around which the economy fluctuates.
during the Great Recession starting from 2007:III. By 2009:I, the difference in aggregate
consumption is 0.71 percentage points. The right panel of the figure shows the cumulative
consumption growth in the early 1980’s starting from 1979:IV. By 1982:I the difference in
aggregate consumption is 1.04 percentage points. The differences in Figure 6 appear small,
but they accumulate to more substantial differences in Figure 7.
6
More wealth inequality
The first two rows of Table 6 show the distribution of wealth for the data and the baseline
incomplete markets economy above. The baseline model does not generate the high concentration of wealth holdings found in the data. In this section, I modify the model to generate
25
additional concentration in wealth holdings. To do so, I incorporate preference heterogeneity
in the spirit of Krusell and Smith’s stochastic-β economy. I assume 80% of the population
of households are impatient and face incomplete markets as described above. The remaining
20% are more patient, more skilled and pool their idiosyncratic income shocks as in the
complete markets counterfactual. Specifically, the budget constraint of a patient household
is
c + eg k 0 = Rk +
χ
W L,
ν
where ν = 0.2 is the mass of patient households and χ is the earnings share of patient
households. The assumption that these households do not face idiosyncratic risk does not
have a large effect on the results because if they did face idiosyncratic labor income risks they
would be very well self insured by virtue of being patient. At the same time, this assumption
is very useful from a computational standpoint because it does not increase the number of
non-linear household decision rules that I need to compute. The budget constraint of an
impatient household is
c + eg k 0 = Rk +
1−χ
`(θ, ξ, n, W, u).
1−ν
These assumptions introduce two new parameters: the relative patience of the two groups
and the relative skills, χ. I choose these parameters to match the wealth share and earnings
share of the patient households to the top 20% of the distribution of wealth. The resulting
distribution of wealth appears in the bottom row of Table 6. The 80/20 model generates
more wealth in the bottom quintiles than we observe in the data, which is partly due to the
tight borrowing limit k 0 ≥ 0.
I re-estimate the model and perform the counterfactual decomposition from the previous section.13 As the 80/20 model includes preference heterogeneity, the complete markets
counterfactual is somewhat different from the one above. In this section the counterfactual
is that all households are patient and fully share risks. The steady state interest rate of
the 80/20 model is determined by the Euler equation of the patient households so it is not
13
The estimated parameter values appear in Table 4.
26
λ
A
0.5
0.5
0
0
−0.5
1960
0.08
1980
−0.5
2000
2020
1960
0.02
1980
ζ
0.5
0
0
1960
0.21
1980
2020
2000
2020
q
0.5
−0.5
2000
−0.5
2000
2020
1960
0.02
1980
Figure 8: Decomposition of contribution of idiosyncratic risk to aggregate consumption
growth for the 80/20 model. Each series is the contribution of the given shock to aggregate
consumption growth in the incomplete markets model less the complete markets model.
Standard deviations listed in the figures.
affected by changes in the share of these households in the population.14
Figure 8 shows the contribution of idiosyncratic risk associated with the four aggregate
shocks. With more agents near the borrowing constraint, consumption becomes more sensitive to TFP shocks and to the ζ shock. In total, the variance of consumption growth is 32%
lower in the counterfactual complete market series.
Precautionary saving behavior generates a consumption function that is concave in wealth
(Carroll and Kimball, 1996). An increase in ζ reduces the expected lifetime income of a
household by changing the probability of unemployment in the future. Due to the concavity
of the consumption function, this wealth effect has larger effects on the consumption of lowwealth households. However, it is only at very low asset levels that this effect is quantitatively
important. This can be seen in Figure 9 which shows the change in consumption in response
to a one standard deviation positive shock to ζ across different levels of wealth. The lightcolored line corresponds to the constant risk case. Notice that is only very near zero cash
14
See McKay and Reis (2013) for further discussion of the determination of the steady state interest rate
in a related model.
27
−3
2
x 10
0
units of consumption
−2
−4
−6
−8
−10
−12
−10
0
10
20
30
40
50
60
normalized cash on hand
70
80
90
100
Figure 9: Percentage change in consumption on impact in response to a standard deviation
increase in ζ for an employed household with the given level of cash on hand relative to
permanent income. 80/20 model (dark line) and 80/20 model with constant risk (light line).
on hand that the drop in consumption is amplified. At higher levels of assets the concavity
of the consumption function is obscured by general equilibrium effects. There is, however,
an additional source of curvature when risk is time-varying. When the increase in ζ brings
about additional uninsurable risk the precautionary savings motive changes and more so at
low wealth levels. Households with low wealth are poorly self insured and so the change
in risk has a particularly strong effect on their consumption. This can be seen in the dark
line in Figure 9 which corresponds to the baseline 80/20 model. The consumption response
to this shock now depends strongly on the household’s asset position. As a result of this
curvature, the distribution of wealth makes a substantial difference to the overall impact of
the shock. Notice that at high wealth levels consumption actually increases due to general
equilibrium effects.
The contributions of idiosyncratic risk to aggregate consumption growth shown in Figure
8 accumulate to substantial differences in consumption during recessions. The left panel of
Figure 10 shows the cumulative effect during the Great Recession. By 2009:I, aggregate consumption would have been 1.93 percentage points higher under complete markets. Similarly,
28
5
7
6
Complete markets
4
5
percentage points
percentage points
3
Data
2
1
0
3
2
Complete markets
1
−1
−2
2007
4
0
2008
2009
2010
2011
2012
2013
Data
−1
1979 1979.5 1980 1980.5 1981 1981.5 1982 1982.5 1983 1983.5 1984
2014
Figure 10: Cumulative consumption growth for the 80/20 model.
during the early 1980s shown in the right panel, aggregate consumption would have been
2.24 percentage points higher by 1982:I.
7
Idiosyncratic risk and parameter estimates
The previous two sections treat the complete markets model as a counterfactual and the
structural parameters are the same in the two models. I now switch to a perspective where
the complete markets model is a potential data generating process and re-estimate the parameters of the model to provide the best fit to the data. I document the extent to which
the estimated structural parameters are sensitive to the inclusion of microeconomic heterogeneity in the model. The value of doing so is nicely illustrated by the work of Chang et al.
(2013) who simulate data from an incomplete markets model and estimate an approximating representative agent model using that data. They find that the estimated structural
parameters of the representative agent model are not stable across policy regimes due to the
misspecification of the micro-foundations. The methods presented here make it possible to
estimate the incomplete markets model itself and therefore check sensitivity of the estimates
to household heterogeneity directly using actual data.
Table 4 shows the estimated parameter values from the posterior mode for both the
complete and incomplete markets models. The primary difference between the two sets of
parameters is the persistence of the job-separation shock, ρζ . With a larger ρζ , a given
29
ρA
ρλ
ρζ
ρq
σA
σλ
σζ
σq
IM
CM
80/20
0.979
0.888
0.885
0.971
0.713
11.585
6.490
0.041
0.981
0.914
0.948
0.972
0.712
11.589
6.555
0.039
0.970
0.896
0.847
0.970
0.711
11.601
6.552
0.040
Table 4: Estimated parameter values. Standard deviations are ×100.
innovation to ζ will have a larger wealth effect on consumption. For a given value of ρζ , the
incomplete markets model makes consumption more sensitive to innovations to ζ because
of the extra effect coming from the increase in idiosyncratic risk, which leads to an extra
precautionary motive for saving. Suppose that fitting the data well requires a certain level of
co-movement between short-term unemployment and consumption growth. To achieve that
co-movement, the incomplete markets model requires less persistence in the shock processes.
Figure 11 illustrates the effect of the persistence parameters on the magnitude of the consumption response to a ζ shock by plotting the impulse responses implied by the baseline
parameters and that implied by the complete markets posterior mode.
Table 4 also shows the estimated parameter values from the posterior mode of the 80/20
model where a similar pattern emerges. In the 80/20 model, the TFP shock and ζ shocks
are less persistent than in the estimates from the incomplete markets model. This fits with
the intuition above as aggregate consumption is more sensitive to these shocks when there
is more heterogeneity in wealth holdings.
8
Conclusion
This paper began by asking whether idiosyncratic risks are relevant for the dynamics of aggregate quantities. The results show that time-varying risks to long-term earnings potential
can generate substantial movements in aggregate consumption. In a version of the model
30
−3
−0.2
x 10
−0.4
−0.6
−0.8
−1
−1.2
−1.4
−1.6
IM
−1.8
−2
−10
0
10
CM
20
quarter
CM − posterior mode
30
40
50
60
Figure 11: Impulse response of consumption to a one standard deviation shock to ζ. Shocks
are scaled to one standard deviation. Units are log deviation from steady state.
with a realistic distribution of wealth the variance of aggregate consumption growth would
have been 32% lower under complete markets.
Several issues remain to be investigated in further detail. Throughout the paper, I have
assumed that the cyclical variation in idiosyncratic risks is perfectly correlated with the jobseparation rate. An important topic for future work is to relax this assumption and instead
estimate the time-varying income risks from data on the distribution of income. Doing so
requires a high-quality panel data set at a frequency suitable for business cycle analysis
and also new methods to incorporate the cross-sectional implications of the model into the
estimation. Additionally, while the model studied here is a general equilibrium model, it
does not include the various frictions that are commonly included in empirical analyses of
business cycles. The interaction between uninsurable idiosyncratic risk and these frictions
has only recently started to receive attention from the literature (Ravn and Sterk, 2013;
McKay and Reis, 2013; Gornemann et al., 2012). The methods presented here can be used
to study this interaction empirically.
31
Appendix
A
Estimating the idiosyncratic income process
This appendix provides additional information on the simulated method of moments procedure used to select the parameters of the idiosyncratic income process, which is a variant of
the procedure used by Guvenen et al. (2013).
Step 1. Calculate λt and ζt implied by the data. To do so, use the data on short-term
unemployment and long-term unemployment described in Section 4.3 and solve for λt and
ζt from
short-term unemployment = ζt (1 − ut−1 )
long-term unemployment = (1 − λt )ut−1
where u is the sum of short-term and long-term unemployment.
Step 2. Guess a vector of parameters
Θ ≡ [µξ , σξ , µ02 , µ12 , µ03 , µ13 , ση,1 , ση,2 , ση,3 , p]
and a sequence {m}2011
t=1978 . mt is the quarterly growth rate of average income in year t, which
shifts the entire distribution from which η is drawn. While I simulate quarterly data, I
assume the mean growth rate is constant in each year as the observed data are at an annual
frequency.
Step 3. Calculate µ2,t and µ3,t from Eq.s 6 and 7 with xt = ζt .
Step 4. Simulate employment, skill, and mortality shocks for a panel of households. The
mortality rate is 1% and the employment transition probabilities are the values for λt and
32
ζt computed in step 1. I simulate 10,000 individuals from 1977 through 2011. The results
are not very sensitive to the way the distribution is initialized because the objects of interest
are all concerned with income changes not levels. I initialize to a 5% unemployment rate
and I draw initial values for θ from a normal distribution that has mean and variance set to
the values to which the distribution of θ would converge if µi , t is constant at its time series
average for 1977 to 2011 for i = 1, 2, 3.15
Step 5. Compute the moments: aggregate the quarterly income observations to annual
observations, take 1-year, 3-year, and 5-year changes in log income. I use the following
moment for each year and for each of the 1-year, 3-year and 5-year changes: the mean, the
difference between the 50th and 10th percentile, the difference between the 90th and 50th
percentile. In addition I use the time-series average of the 10th and 90th percentiles.
Step 6. Compute the objective function: I take the difference between the simulated
moment and the empirical moment from Table A13 in Guvenen et al. (2013). The differences
are expressed as squared percentage differences except for the difference in means, which is
expressed relative to the 90th percentile as in Guvenen et al. (2013).
Step 7. Adjust guess in step 2 and repeat to minimize the objective function from step 6.
The moments of the resulting distribution of income changes is plotted in Figure 12 along
with the empirical counterparts from Guvenen et al. (2013). One can see that estimated
15
This is only an approximation used to initialize the distribution and the distribution of θ will not converge
to a normal distribution. Define
µ̄ = (1 − 2p)µ̄1 + pµ̄2 + pµ̄3 ,
where µ̄i is the time-series average of µi over 1977 to 2011. Define
σ̄ 2 = (1 − 2p)σ12 + pσ22 + pσ32 + (1 − 2p)µ̄21 + pµ̄22 + pµ̄23 − µ̄2 ,
which is the unconditional variance of a shock to η from the law of total variance. The mean is then
1−ω
µ̄
ω
and variance is
1−ω 2
1−ω 2
(µ̄ + σ̄ 2 +
µ̄ ).
ω
ω
33
10
1−year income change
mean
5
3−year income change
0
−10
30
5−year income change
20
10
10
0
−5
1980 1990 2000 2010
−30
−40
P10
20
−10
0
1980 1990
2000 2010
−10
−20
−40
−40
−60
−60
−80
1980 1990 2000 2010
−50
−60
−70
1980 1990 2000 2010
−80
20
5
10
0
0
2000 2010
−100
1980 1990 2000 2010
30
20
P50
10
1980 1990
10
−5
1980 1990 2000 2010
60
−10
0
1980 1990
2000 2010
90
P90
1980 1990 2000 2010
120
80
50
−10
100
70
40
30
80
60
1980 1990 2000 2010
50
1980 1990
2000 2010
60
1980 1990 2000 2010
Figure 12: Simulated (solid line) and empirical (dashed line) moments of the income process.
income process fits well. These moments are to various degrees targetted by the estimation
procedure. As an additional check, I computed the standard deviations of the income changes
and compared those to the results in Guvenen et al. (2013). Figure 13 shows that the
simulated standard deviations are only slightly cyclical while those in the data are more or
less acyclical. The simulated standard deviations are somewhat below the observed values.
B
B.1
Numerical methods and accuracy checks
Numerical methods
To be written.
34
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
Data 5yr
Model 5yr
Data 1yr
Model 1yr
0.2
0.1
0
1975
1980
1985
1990
1995
2000
2005
2010
Figure 13: Simulated and empirical standard deviations of the income process.
B.2
B.2.1
Accuracy checks
The law of motion for capital
This subsection discusses the accuracy of the approximate law of motion for capital K 0 =
h(A, K, λ, ζ, u) for the general equilibrium economy. The accuracy of this approximation
is important to both the solution of the model and calculating the likelihood function.
Den Haan (2010) discusses methods for checking the accuracy of the approximate law of motion for capital and suggests using the law of motion to forecast the capital stock and prices
many periods in the future. Figure 14 shows a plot of the “true” capital stock generated from
simulating the model and the “approximate” capital stock generated by repeatedly applying
the approximate law of motion for capital.16 One can see that the discrepancy is small even
at forecast horizons of 1000 quarters. The maximum absolute log difference between the two
series is 0.0065 and the mean absolute log difference is 0.0017. Another commonly-reported
16
As Den Haan (2010) suggests, the sequence of shocks used to simulate the model for the accuracy shock
differ from those used to calculate the approximate law of motion.
35
58
56
54
K
52
50
48
46
44
0
simulated
forecasting rule
100
200
300
400
500
600
forecast horizon
700
800
900
1000
Figure 14: Simulated aggregate capital stock with implied values from K 0 = h(A, K, λ, ζ, u).
accuracy check is the R2 of the one-step ahead forecast, which is 0.999972.
B.2.2
Euler equation errors
As a check on the accuracy of the solution for the consumer’s decision rule, I calculate unitfree Euler equation errors in the style of Judd (1992). Errors arise for three reasons: (i) in
the stationary equilibrium without aggregate shocks there are errors between the collocation
nodes on which the household’s Euler equation holds exactly, (ii) away from the stationary
equilibrium there are additional errors that arise due to non-linearities just as is normally
the case with linearization methods, (iii) the forecasting rule does not perfectly predict K 0 .
The Euler equation errors I present here reflect all three of these sources of error.
To construct the Euler equation errors, I start with a randomly generated point in the
state space, call it S. I generate these points by simulating the model for 1000 periods starting
from the stationary equilibrium and taking the final position of the economy. This simulation
procedure generates a distribution of wealth Γ(S). Given S, I can find the household policy
rule f (a, n, S) and prices R(S) and W (S). I construct a fine grid on cash on hand, a, that
36
Unemployed
Employed
−1
−1
−1.5
−2
−2.5
−3
−3.5
0
Mean
Max.
Absolute error (log base 10)
Absolute error (log base 10)
Mean
Max.
500
1000
Normalized cash on hand
−1.5
−2
−2.5
−3
−3.5
0
500
1000
Normalized cash on hand
Figure 15: Unit-free Euler equation errors (log base 10). Mean and maximum across 100
points in the state space.
is finer than the one used to solve the model and so it reveals errors from source (i). I can
then compute c(a, n, S) = a − f (a, n, S) and compute K 0 from aggregating the savings across
households. To calculate the right-hand side of the Euler equation, I integrate of aggregate
shocks using Gaussian quadrature with 11 nodes in each dimension. For a given realization
of the aggregate shocks, I can compute λ0 , ζ 0 , R0 , and W 0 . I can also compute the savings
decision rule for that value of S 0 and form c(a0 , n0 , S 0 ). For each set of individual states
(a, n), I integrate over possible idiosyncratic shocks and for each realization of individual
and aggregate states I evaluate c(a0 , n0 , S 0 ). This gives expected marginal utility E [u0 (c0 )].
The unit-free Euler equation error is then
n
h
io−1/γ
−g
0
0 −γ 0
e βE R (c ) η
/c − 1.
I simulate the model 100 times and calculate the Euler equation errors for 100 values of S. I
then take the maximum and mean error across these 100 draws. Figure 15 plots the unit-free
Euler equation errors against cash on hand for both employed and unemployed households.
As one can see, the errors decline with cash on hand. Integrating these functions against the
37
distribution of wealth from the stationary equilibrium gives values of 0.0043 for the mean
and 0.0133 for the maximum.
38
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