business cycle: a parametric approach · higher-order income risk over the business cycle: a...

HIGHER-ORDER INCOME RISK OVER THE

BUSINESS CYCLE: A PARAMETRIC APPROACH

Christopher Buscha Alexander Ludwigb

aMOVE-UAB and Barcelona GSE

bSAFE, Goethe University Frankfurt

V Winter Macroeconomics Workshop

Bellaterra, March 2019

Motivation

I Idiosyncratic income risk important for

I Welfare costs of fluctuations

I Benefits from insurance

I Design of tax and transfer policies

I Recent empirical work documents rich deviations from

“Normality”

I Distribution of shocks

I Persistence of shocks

I Question: does it matter from a macroeconomic perspective?

Busch & Ludwig (MOVE-UAB & Goethe U): Higher-Order Income Risk 1 / 38

This paper

1. Characterize idiosyncratic income risk over business cycle

I Counter-cyclical variance; Pro-cyclical skewnessI High (excess) kurtosisI Post government: weaker cyclicality

2. Welfare implications of remaining higher-order income risk?

I Welfare losses (gains) for high (low) risk aversion

3. Does higher-order risk matter for costs of cycles?

I Yes! With high RA, larger welfare losses from fluctuations

4. Does higher-order risk matter for self-insurance?

I Yes! With high RA, better consumption smoothing (moreprecautionary savings)


This paper

1. Characterize idiosyncratic income risk over business cycleI Counter-cyclical variance; Pro-cyclical skewnessI High (excess) kurtosisI Post government: weaker cyclicality

2. Welfare implications of remaining higher-order income risk?

I Welfare losses (gains) for high (low) risk aversion






This paper


2. Welfare implications of remaining higher-order income risk?I Welfare losses (gains) for high (low) risk aversion






This paper



3. Does higher-order risk matter for costs of cycles?I Yes! With high RA, larger welfare losses from fluctuations




This paper



3. Does higher-order risk matter for costs of cycles?I Yes! With high RA, larger welfare losses from fluctuations

4. Does higher-order risk matter for self-insurance?I Yes! With high RA, better consumption smoothing (more

precautionary savings)


Approach

1. Individual income risk over the business cycle

I Residual income (pre- and post-government)

I Parametric approach: GMM estimation

2. Evaluate macroeconomic implications

I Tool: Life-cycle model with exogenous income

I First: Analytical illustration in two-period model

3. Intermediate step:

I Transparent shock discretization method


Literature

I Earnings dynamics over business cycle

I Early Evidence: 2nd Moment Shock (Storesletten, Telmer & Yaron,

JPE, 2004; Bayer & Juessen, EL, 2012)I Newer Evidence: 3rd Moment Shock (Guvenen, Ozkan & Song,

JPE, 2014; Busch, Domeij, Guvenen & Madera, 2018, Huggett & Kaplan,

RED, 2016)

I Rich earnings dynamics

I Arellano, Blundell & Bonhomme, Ecta, 2017; Guvenen, Karahan, Ozkan,

Song, 2018; Holter, Halvorsen, Ozkan & Storesletten, 2018;...

I Macroeconomic implications

I Consumption insurance (De Nardi, Fella & Paz-Pardo, JEEA, 2018)I Aggregate savings (Civale, Díez-Catalán & Fazilet, 2016)I Optimal fiscal policy (Golosov, Troshkin & Tsyvinski, AER, 206)I Portfolio choice & equity premium (Catherine, 2019)


LiteratureI Earnings dynamics over business cycle

I Early Evidence: 2nd Moment Shock (Storesletten, Telmer & Yaron,

JPE, 2004; Bayer & Juessen, EL, 2012)I Newer Evidence: 3rd Moment Shock (Guvenen, Ozkan & Song,

JPE, 2014; Busch, Domeij, Guvenen & Madera, 2018, Huggett & Kaplan,

RED, 2016)

I Rich earnings dynamics

I Arellano, Blundell & Bonhomme, Ecta, 2017; Guvenen, Karahan, Ozkan,

Song, 2018; Holter, Halvorsen, Ozkan & Storesletten, 2018;...

I Macroeconomic implications

I Consumption insurance (De Nardi, Fella & Paz-Pardo, JEEA, 2018)I Aggregate savings (Civale, Díez-Catalán & Fazilet, 2016)I Optimal fiscal policy (Golosov, Troshkin & Tsyvinski, AER, 206)I Portfolio choice & equity premium (Catherine, 2019)


2nd Moment Shock: Symmetric Risk?

yt+1 − yt

-1 -0.5 0 0.5 1

Den

sity

0

0.5

1

1.5

2

Expansion


2nd Moment Shock: Symmetric Risk?

yt+1 − yt

-1 -0.5 0 0.5 1

Den

sity

0

0.5

1

1.5

2

ExpansionRecession


3rd Moment Shock: Asymmetric Risk?

yt+1 − yt

-1 -0.5 0 0.5 1

Den

sity

0

0.5

1

1.5

2

Expansion


3rd Moment Shock: Asymmetric Risk?

yt+1 − yt

-1 -0.5 0 0.5 1

Den

sity

0

0.5

1

1.5

2

ExpansionRecession


Outline

Intro

Higher-Order Risk in Two-Period Model

Estimation of Idiosyncratic Income Process

Macroeconomic Implications

Conclusion

Two-Period Model: Setup

I EndowmentsI Agents live for two periodsI Non-risky first-period income y0

I Risky second-period income y1 = exp(ε1), ε ∼ Ψ:

a1 = a0 + y0 − c0, a0 given

c1 = a1 + y1

I Preferences: Epstein-Zin-Weil


Two-Period Model: Preferences

U =

1

1−ρ

(c

1− 1ρ

0 + v (c1, θ,Ψ)1− 1ρ

)for ρ 6= 1

ln (c0) + ln (v (c1, θ,Ψ)) for ρ = 1.

with

v(c1, θ,Ψ) =

(∫

c1(ε)1−θdΨ(ε)) 1

1−θ =(E[c1(ε)1−θ]) 1

1−θ for θ 6= 1

exp(∫

ln(c(ε))dΨ(ε))

= exp (E [ln(c(ε))]) for θ = 1.

I ρ Intertemporal Elasticity of Substitution

I θ Risk Aversion (and “higher risk attitudes”)


Two-Period Model: Hand-to-Mouth Consumers

I Shut down risk-free savings technology: a1 = 0

I Fourth-Order Taylor approximation around µc1 = E0[c1]:

U ≈c

1− 1ρ

0

1− 1ρ

+

(1

1− 1ρ

− θ

2µc

2 +θ(θ + 1)

6µc

3 −θ(θ + 1)(θ + 2)

24µc

4

)(1)

I Increase in risk: welfare losses for θ > 1(µ

exp(ε)2 ↑, µexp(ε)

3 ↓, µexp(ε)4 ↑

)I Losses ↑ in θ for θ > 1.

I Relative weight ↓ in order of risk

I Equation (1) is for ρ = 1θ ; extends readily to ρ 6= 1

θ


Two-Period Model: Precautionary Savings Reaction

I Allow for savings. Approximation of First Order Condition:

(y0 − a1)− 1ρ = E0

[(exp(ε) + a1)

−θ]

(2)

≈ (1 + a1)−θ +

θ(1 + θ)

2(1 + a1)

−(2+θ) µexp(ε)2

− θ(1 + θ)(2 + θ)

6(1 + a1)

−(3+θ) µexp(ε)3

+θ(1 + θ)(2 + θ)(3 + θ)

24(1 + a1)

−(4+θ) µexp(ε)4 .

I LHS increasing in a1, RHS decreasing in a1

I Increase in risk→ More savings:(µ

exp(ε)2 ↑, µexp(ε)

3 ↓, µexp(ε)4 ↑

)→ RHS ↑→ a1 ↑

I Reaction depends on risk attitudes (pinned down by θ)


Note on Logs vs. Levels

I Moments in previous analysis are “in levels”

I Income process characterized “in logs”

I How do moments of logs translate into moments of levels?I In logs: µ3 ↓⇒ In levels: µ2 ↓, µ3 ↓, µ4 ↓

I In logs: µ4 ↑⇒ In levels: µ2 ↑, µ3 ↑, µ4 ↑

⇒ Quantitative question whether change in µ2 & µ4 or change in µ3

dominate

Answer depends on magnitude of change and on risk attitudes

Numerical Example Numerical Illustration


Outline

Intro




Conclusion

Overview of Estimation

1. Log income:

yith = f (Xith,Yt ) + yith

f (Xith,Yt ) = β0t + fh (h) + Ieit =c fEP(EP) + βsize log (hhsizeith)

fh (h) = βage1 h + β

age2 h2 + β

age3 h3

fEP(EP) = βEP0 + βEP

1 t + βEP2 t2

2. Fit stochastic process to residual income yith

3. Inference: Bootstrap


GMM Estimation Approach

I Residual income process:

yith = χi + zith + εith

zith = ρzit−1h−1 + ηith

I Distributions of stochastic components:

εith ∼ Fε, χi ∼ Fχ, ηith ∼ Fη (s (t))

I Interested in central moments µ2, µ3, and µ4


GMM Estimation ApproachIdentification

I Persistent shocks accumulate

⇒ Cross-sectional moments identify distribution of shocks:Fη (s (t)) varies with aggregate state s(t): cross-sectionaldistribution at age h differs across cohorts t − h

I Identifies F s(t)=EXPη and F s(t)=CON

η

I Extend Storesletten, Telmer & Yaron (2004) to 3rd central

moment


Variance as Share of Recessions

I More recessions: higher varianceBusch & Ludwig (MOVE-UAB & Goethe U): Higher-Order Income Risk 15 / 38

Skewness as Share of Recessions

I More recessions: stronger left skewness


Approach: Theoretical Moments

Conditional on history St = [s0, . . . , st ]:

µ2 (yith; θ) = E[(yith − E [yith])2 | St

]µ11 (yith, yit+1h+1; θ) = E

[(yith − E [yith]) (yit+1h+1 − E [yit+1h+1]) | St]

µ3 (yith; θ) = E[(yith − E [yith])3 | St

]µ21 (yith, yit+1h+1; θ) = E

[(yith − E [yith])2 (yit+1h+1 − E [yit+1h+1]) | St

]µ4 (yith; θ) = E

[(yith − E [yith])4 | St

]µ31 (yith, yit+1h+1; θ) = E

[(yith − E [yith])3 (yit+1h+1 − E [yit+1h+1]) | St

]


Approach: Theoretical Moments

µ2 (yith; θ) = µχ2 + µε2 +∑h−1

j=0ρ2jµη2 (s (t − j))

µ11 (yith, yit+1h+1; θ) = µχ2 + ρ∑h−1

j=0ρ2jµη2 (s (t − j))

µ3 (yith; θ) = µχ3 + µε3 +∑h−1

j=0ρ3jµη3 (s (t − j))

µ21 (yith, yit+1h+1; θ) = µχ3 + ρ∑h−1

j=0ρ3jµη3 (s (t − j))

µ4 (yith; θ) = µχ4 + µε4 +∑h−1

j=0ρ4jµη4 (s (t − j)) + g1 (ρ,−→µ 2)

µ31 (yith, yit+1h+1; θ)) = µχ4 + ρ2∑h−1

j=0ρ4jµη4 (s (t − j)) + g2 (ρ,−→µ 2)

More details on 4th moment


Moment Conditions

I Parameters to estimate:

θ =(ρ, µχ2 , µ

ε2, µ

η,E2 , µη,C2 , µχ3 , µ

ε3, µ

η,E3 , µη,C3 , µχ4 , µ

ε4, µ

η,E4 , µη,C4

)

I Stack moment conditions into F (θ) and minimize (weighted with

#obs):

minθF (θ)′WF (θ)


Data and Sample Selection

I Germany (GSOEP) and US (PSID)

I GSOEP: about 5,000 households in 1984 / 10,500 in 2013

I PSID: about 2,000 households each wave (annual 1968 –1997,

biannual afterwards)

I Sample Selection:I 2 adults present in householdI Head of household age 25-60I Annual household income ≥ 520 hours at half minimum wage

I Pre-sample aggregate information: expansions, contractions


Estimation Results PSID

HH Pre HH Post

ρ .9873 .9815µχ2 .2150 .1926µε2 .1038 .0936µη,C2 .0132 .015µη,E2 .0076 .0067µχ3 −.1248 −.0966µε3 −.1489 −.1219µη,C3 −.0121 −.0145µη,E3 −.003 −.0004µχ4 .1386 .1113µε4 .4393 .3125µη,C4 .0413 .0584µη,E4 .0137 .0117


Implied Standardized Moments

HH Pre HH Post

αχ3 −1.25 −1.14

αε3 −4.45 −4.26

αη,C3 −7.96 −7.90

αη,E3 −4.58 −0.82

αχ4 3 3

αε4 40.76 35.69

αη4 237.31 260.03

αk = µk/µk/22 : standardized k th moment

I Ratio of Std. Dev. Contraction/Expansion = 1.5


Implied Age Profiles: Post Government

I Age profiles, data and model, post government

Pre Government


Outline

Intro




Conclusion

Questions & Approach

I 3 Questions:

1. What’s the welfare cost of higher-order income risk?

2. Does it matter for welfare costs of business cycles?

3. Does it matter for consumption self-insurance?

I Tool: Life-cycle model with exogenous income

I No analysis of

I nature of realized income changes

I direct effect of aggregate risk


Welfare Criterion

I Ex-ante welfare: Into which world would you prefer to be born

into as the average individual?

I Welfare measure: CEV gc in (a) to make households indifferent

to (b)

I gc < 0: welfare losses fromI higher order riskI aggregate risk

I Decomposition into mean and distribution effects


Insurance Coefficients

I Model-version of Blundell, Pistaferri & Preston (2008) a la

Kaplan & Violante, 2010

I Share of variance of shock x ∈ {ε, η} which does not translate

into consumption growth

I By age h and aggregate state s ∈ {C,E}:

φxh(s) = 1−

cov(∆ ln ch(s), xh(s))

var(xh(s))


Quantitative Model: Overview

I H-period lived households

I Working and retirement phase, retirement at age hr

I Endowments:

I exogenous

I discretization of estimated income process

I zero borrowing constraint

I Preferences: Epstein-Zin-Weil: RA θ and IES ρ

More details


Quantitative Model: Shock Distributions

I Model shock distributions Fε, Fη: discrete

I Fit to estimated central moments: continuous Fε, Fη (s (t))

I Approach: Flexible Generalized Lambda Distribtion (FGLD):

1. Target estimated µ2 − µ4

2. Discretize FGLD distribution on grids Gε,Gη,

I Lowest income level unchanged over scenarios


Quantitative Model: The FGLD

I Quantile function of FGLD (Freimer et al. 1988):

Q(p) = λ1 +1λ2

(pλ3 − 1λ3

− (1− p)λ4 − 1λ4

)

I Closed form expressions for moments µ1 − µ4

I Numerical solution for λ3, λ4 to match µ3, µ4

I Analytical solution for λ1, λ2 to match µ1, µ2

I Then discretization

More details


Quantitative Model: Fit of Discretized Distribution

Moment µ2 µ3 µ4

Transitory Shock

estimate 0.094 -0.122 0.312fitted 0.094 -0.122 0.313discrete 0.091 -0.112 0.297

Persistent Shock

Contraction

estimate 0.015 -0.015 0.058fitted 0.015 -0.014 0.059discrete 0.015 -0.014 0.058

Expansion

estimate 0.007 0 0.012fitted 0.007 0 0.012discrete 0.007 0 0.012

I For US post government earnings


Quantitative Model: Persistent Income ShocksIncome Profiles

(a) NORM (b) LK

(c) LKSW


Quantitative Model: Other Parameters

Working period 25 (h = 0) to 60

Maximum age 80

IES ρ = 1

RA θ ∈ {1,2,3,4}Discount factor (2nd stage) β ∈ {0.971,0.969,0.967,0.964}Interest rate r = 0.042

Borrowing constraint a = 0

Pension contribution rate τ = 0.124

Aggregate shocks π(s′ = c | s = c) = 0.43

π(s′ = e | s = e) = 0.79

Initial ass. / perm. inc. 1.54

I estimate β to match assets / permanent income


Welfare Costs of Higher-Order Risk, IES ρ = 1CEV gc gmean

c gdistrc g lcd

c gcsdc

Risk Aversion, θ = 1LK -0.99 1.141 -2.131 -2.107 -0.024LKSW 0.609 0.66 -0.052 -0.064 0.012Risk Aversion, θ = 2LK -1.754 1.267 -3.021 -3.006 -0.016LKSW -0.097 0.661 -0.757 -0.793 0.036Risk Aversion, θ = 3LK -4.867 1.62 -6.487 -6.456 -0.031LKSW -4.247 1.137 -5.383 -5.379 -0.005Risk Aversion, θ = 4LK -11.097 2.286 -13.383 -13.264 -0.119LKSW -11.98 2.019 -13.999 -13.857 -0.142

I Positive mean effects

I Substantial welfare losses from higher-order risk in distribution effects

I Losses increase in measure of risk aversion


Welfare Costs of Aggregate Risk, IES ρ = 1CEV gc gmean

c gdistrc g lcd

c gcsdc %-change

Risk Aversion, θ = 1NORM -1.334 0.329 -1.663 -1.621 -0.042 0LK -1.7 0.389 -2.089 -2.044 -0.046 27.42LKSW -0.984 0.277 -1.261 -1.227 -0.034 -26.259Risk Aversion, θ = 2NORM -2.512 0.638 -3.149 -3.049 -0.101 0LK -2.794 0.601 -3.395 -3.305 -0.09 11.234LKSW -2.686 0.61 -3.296 -3.211 -0.085 6.933Risk Aversion, θ = 3NORM -3.553 0.903 -4.455 -4.298 -0.157 0LK -4.641 0.851 -5.493 -5.355 -0.138 30.645LKSW -5.961 1.109 -7.07 -6.899 -0.171 67.785Risk Aversion, θ = 4NORM -4.442 1.138 -5.58 -5.374 -0.205 0LK -7.252 1.13 -8.382 -8.188 -0.194 63.276LKSW -10.236 1.63 -11.866 -11.587 -0.28 130.443


Life-Cycle Consumption1st and 2nd Moment (Log), ρ = 1, θ = 4

(a) Mean (b) Variance

I Higher-order income risk: lower consumption when young

I LK: higher variance, LKSW: lower variance

I Precautionary savings reaction


Insurance CoefficientsTransitory & Persistent Shock, ρ = 1, θ = 4

(a) Transitory Shock (b) Persistent Shock

I LK: less insurance against transitory shocks

I Total effect: more insurance b/c of higher precautionary savings

3rd and 4th


Outline

Intro




Conclusion

SummaryEmpirical Analysis

I GMM estimator of 2nd–4th moments of transitory and (cyclical)

persistent component of income process

I German (SOEP) and US (PSID) households:

I Variance of persistent shocks countercyclical

I Skewness of persistent shocks procyclical

I Highly leptokurtic shocks

I Role of government taxes and transfers?

I Similar transitory shocks

I Dampen cyclicality of persistent shocks


SummaryMacroeconomic Implications

I Household life-cycle model

I (Cyclical) higher-order risk welfare implications:

I θ = 1: welfare gain of approx. .6%

I θ = 4: welfare losses of approx. 12.0%

I Quantitatively relevant for welfare costs of business cycles:

I For θ = 1, welfare cost of fluctuations lower by 26%

I For θ = 4, welfare cost of fluctuations larger by 130%

I Better consumption insurance against shocksBusch & Ludwig (MOVE-UAB & Goethe U): Higher-Order Income Risk 38 / 38

Fourth Central Moment of Income Processµ4 (yith ; θ) = E

[(yith − E [yith ])4

]= µ

χ4 + µ

ε4 +

∑h−1j=0

ρ4jµη4 (s (t − j)) + 6

(µχ2 µ

ε2 + . . .(

µχ2 + µ

ε2

)∑h−1j=0

ρ2jµη2 (s (t − j)) + . . .

∑h−2j=0

∑h−1k=j+1

ρ2jρ

2kµη2 (s (t − j))µ

η2 (s (t − k))

)

µ31(yith, yit+1h+1) = E[

(yith − E [yith ])3 (yit+1h+1 − E[yit+1h+1

])|St]

= E[µχ4 + ρ

h−1∑j=0

ρ4jµη4 (s(t − j)) + 3

(µχ2 µ

ε2 + µ

χ2

h−1∑j=0

ρ2jµη2 (s(t − j)) + . . .

(µχ2 + µ

ε2

)ρ

h−1∑j=0

ρ2jµη2 (s(t − j)) + . . .

ρ

h−1∑j=0

ρ2jµη2 (s(t − j))

h−1∑k 6=j

ρ2kµη2 (s(t − k))

︸︷︷︸∑j−1k=0 ρ

2kµη2 (s(t−k))+∑h−1

k=j+1 ρ2kµη2 (s(t−k))

)]

Back


Implied Age Profiles: Pre Government

I Age profiles, data and model, pre government

Go Back


Note on Logs vs. Levels: Numerical Example

I Numerical example with 3-point distribution Details

Moments of Innovation in Logs, ε

µε1 µε2 µε3 µε4

NORM –.25 0.5 0 0.75

LK –.57 0.5 0 7.5

LKSW –.11 0.5 -1.7678 7.5

Moments of Innovation in Levels, exp(ε)

µexp(ε)1 µ

exp(ε)2 µ

exp(ε)3 µ

exp(ε)4

NORM 1 0.5868 0.6691 1.3045

LK 1 11.6316 299.3406 7842.5727

LKSW 1 0.1039 0.0190 0.0523

I Welfare gain of LKSW with log-utility (hand-to mouth)Back


Two-Period Model: Illustration

I Effects of higher-order risk

I Take 3-point distribution for yt Details

I Three distributions: “normal” (NORM), leptokurtic (LK),

leptokurtic and skewed, (LKSW)

I Two normalizations: E [ε] = 0 (LN), E [exp ε] = 1 (EN)

I Two institutional setups: No capital markets (NCM), natural debt

limit (NDL)

I Two degrees of “risk aversion”: low (θ = 1), high (θ = 4)


Simple Model: Numerical Example

CEV relative to NORM:

(a) LN (b) EN

Scen / CEV LK LKSW LK LKSW

θ = 1

NCM 0 0 -0.1859 0.1096

θ = 4

NCM -0.7888 -0.8554 -0.8281 -0.8396

CM -0.6978 -0.8004 -0.7540 -0.7785

I (N)CM: (No) capital markets

I Important to distinguish mean and distribution effects

I Distribution effects dominate with increasing risk aversion

Back


Two Period Model: DiscretizationLet x ∈ {xl , x0, xh}, drawn with respectiveprobabilities {(1− p) · q,p, (1− p) · (1− q)}. Then,

1. if and only if α(3) = 0 or2. if α(3) 6= 0 and α(4) ≥ 1 + α(3)2

we match E [exp(x)] = 1 and µ(2), α(3), α(4) by determining

q =12

+12

√1−

4 α(4)

α(3)2−4

4 α(4)

α(3)2−3if α(3) > 0

−12

√1−

4 α(4)

α(3)2−4

4 α(4)

α(3)2−3if α(3) < 0

0.5 if α(3) = 0

p =

1− (2q−1)2

q(1−q)α(3)2 if α(3) 6= 0

1− 1α(4) if α(3) = 0


Two Period Model: Discretization (cont’d)

∆x =

√µ(2)α(3)

2q−1 if α(3) 6= 0

2√µ(2)

√α(4) if α(3) = 0,

and

xl = − ln [p exp ((1− q)∆x ) + (1− p) (q + (1− q) exp(∆x ))]

x0 = xl + (1− q)∆x

xh = xl + ∆x .

Back to Illustration Back to Numerical Example


Quantitative ModelEndowments

I 2 aggregate states of economy s ∈ c,eI Transition: time-invariant Markov

I Households:I live for H periodsI 1 unit of labor at h ∈ 0, ...,hr − 1; retired hr , . . . ,HI Household i ’s (post-government) log income:

yith = eh + zith + εith

zith = ρzit−1h−1 + ηith

I eh : fh and fEP from first stageI zit+1h+1 = ρzith + ηit+1h+1

I εith ∼ Fε, ηih ∼ Fη (s (t))


Quantitative ModelEndowments (cont’d)

I Contributions to pension system: τp

I Retirement income: risk-free, US bend-point formula

I Savings: risk-free 1-period bonds, rate of return r

I Dynamic budget constraint with borrowing constraint:

ait+1h+1 = aith (1 + r) + (1− τp)yith − cith ≥ −a

I Only idiosyncratic effects of aggregate risk→ lower bound of

welfare losses


Quantitative ModelPreferences

I Epstein-Zin-Weil recursive utility: separate IES ρ from RA θ

Uh (x , z; s) =

((1− β) c

1− 1ρ

h + β(

ce (Uh+1 (·))1− 1ρ

))1/(

1− 1ρ

)if ρ 6= 1

exp {(1− β) ln ch + β ln (ce (Uh+1 (·)))} otherwise

with

ce (Uh+1 (·)) =

(

Eh

[Uh+1 (x ′, z′; s′)

1−θ])1/(1−θ)

if ρ 6= 1

exp (Eh [lnUh+1 (x ′, z′; s′)]) otherwise

I x = a(1 + r) + y : cash on hand

Go Back


Details on FGLD

I 1st moment:

µ1 = E [X ] = a + bE [Y ] = a + bν1

= λ1 −1

λ2λ3+

1λ2λ4

+1λ2ν1

I Moments of Y by binomial expansion:

E[(Y − E [Y ])k

]= E

k∑j=0

(kj

)(−1)j (Y )k−j ν(1)j

=

k∑j=0

(kj

)(−1)j E

[(Y )k−j

]ν(1)j


Details on FGLD (cont’d)

I Higher moments of X :

µ2 =1λ2

2(ν2 − ν2

1)

µ3 =1λ3

2

(ν3 − 3ν1ν2 + 2ν3

1

]µ4 =

1λ4

2

(ν4 − 4ν1ν3 + 6ν2

1ν2 − 3ν41

).

I Jointly estimate λ3, λ4 with more weight on skewness:

min{λi∈(− 1

4 ,λi ]}4i=3

4∑i=3

(pi

(µi(~λ)− µi

)2)

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Life-Cycle Earnings1st Moment and 2nd Moment of Log

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Life-Cycle Earnings3rd and 4th Moment (of Logs)

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Life-Cycle Consumption3rd and 4th Moment (of Logs), θ = 4

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business cycle: a parametric approach · higher-order income risk over the business cycle: a...

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