inferringincomeriskfromeconomicchoices · 2014. 5. 17. · inferringincomeriskfromeconomicchoices:...

Inferring Income Risk from Economic Choices:An Indirect Inference Approach

Fatih Guvenen Anthony Smith

Minnesota and NBER Yale University

July 29, 2009

Guvenen and Smith (2009) Inferring Income Risk from Choices July 29, 2009 1 / 42

Goals of this Paper:

1 Substantive Question:I To shed light on the nature of labor income risk...

I by exploiting the information contained in the joint dynamics ofhouseholds’ labor earnings and consumption choice decisions.

2 Methodology:I To better understand how “indirect inference” can be used to extract

information from individuals’ economic choices.

Main substantive conclusion: Typical calibrations of incompletemarkets models significantly overstate uninsurable income risk.

A Stochastic Process for Labor Income

y it : log labor earnings of household i at age t.

y it =

[a0 +a1t +a2t2 +a3Educ + ...

]︸︷︷︸common life-cycle component

i + βi t]︸︷︷︸

profile heterogeneity

+[z it + ε

it]︸︷︷︸

stochastic component

where z it = ρz i

t−1 + η it , and η i

t ,εit ∼ iid

y it =

[a0 +a1t +a2t2 +a3Educ + ...

i + βi t]︸︷︷︸

+[z it + ε

it]︸︷︷︸

where z it = ρz i

t ,εit ∼ iid

y it =

[a0 +a1t +a2t2 +a3Educ + ...

i + βi t]︸︷︷︸

+[z it + ε

it]︸︷︷︸

where z it = ρz i

t ,εit ∼ iid

y it =

[a0 +a1t +a2t2 +a3Educ + ...

i + βi t]︸︷︷︸

+[z it + ε

it]︸︷︷︸

where z it = ρz i

t ,εit ∼ iid

Three Questions about Labor Income Risk

1 How persistent and how large are income shocks? i.e., what is ρ andσ2

2 Do individuals differ systematically in their income growth rates? i.e.,is σ2

β� 0?

3 If indeed σ2β� 0, how much do individuals know about their β i at

different points in their life-cycle?

I i.e., what fraction of heterogeneity in β i is uncertainty as opposed toknown heterogeneity?

β� 0?

Outline of the Talk

1 Brief review of existing evidence on labor income risk

2 A life-cycle model of consumption-savings choice

3 Indirect Inference methodology

4 Identification with Consumption Data

5 Results (Monte Carlos and Real data)

6 Conclusions and future agenda

Outline of the Talk

Existing Evidence from Labor Income Data

1 HIP Process (“Heterogenous Income Profiles”):I Early studies estimated the full model and found:

σ2β� 0 and 0.5≤ ρ ≤0.8

I See Hause (1977), Lillard and Weiss (1979), Baker (1997), Haider (2001),Guvenen (2005)

2 RIP Process (“Restricted Income Profiles”):

I MaCurdy (1982) suggested a test for β i≡ 0 and could not reject it.

I Then he and the following literature:F imposed β i≡ 0 and estimated 0.95≤ ρ ≤ 1.0.

I See Abowd and Card (1989), Topel (1991), Moffitt and Gottschalk (1995),Hubbard, Skinner and Zeldes (1994), Storesletten et al. (2004), etc.

σ2β� 0 and 0.5≤ ρ ≤0.8

This Paper

Studies the joint dynamics of consumption and labor income to learn moreabout labor income risk.

Two Difficulties

First, GMM requires strong assumptions.

I “Indirect inference” circumvents many of these difficulties.

Second, long US panel on consumption does not exist.

I We construct a panel of imputed consumption (1968-1992) bycombining CEX and PSID.

This Paper

Two Difficulties

This Paper

Two Difficulties

This Paper

Two Difficulties

This Paper

Two Difficulties

This Paper

Two Difficulties

The Model

Households work for R years. Spend S years in retirement.

Only choice is consumption-savings (inelastic labor supply).

CRRA utility function

Households face the unrestricted labor income process above.

A risk-free asset: Borrowing subject to potentially binding borrowingconstraints.

After retirement: Pension system mimicking the U.S. Social Securitysystem.

The Model

Information Structure

Recall: y it = α i + β i t + z i

t + ε it

At t = 0, each individual observes a signal about her own β i .

In each t > 0, individuals observe (α i ,y it ,ε

it) but must learn about(

β i ,z it).

Individuals use the Kalman filter to update their beliefs about(β i ,z i

as new observatios on labor income arrive.

t + ε it

β i ,z it).

t + ε it

β i ,z it).

t + ε it

β i ,z it).

Prior and Posterior Beliefs

Priors: Bivariate Normal distribution with mean (β i1|0, z

i1|0) and

covariance matrix P1|0 =

β ,0 00 σ2

]Express the prior standard deviation as: σβ ,0 = λσβ . Then:

If λ = 0 → σβ ,0 = 0 and β i1|0 = β i (No prior uncertainty)

If λ = 1 → σβ ,0 = σβ and β i1|0 = β (Full prior uncertainty)

For realistic parameter values learning about β i is very slow (Guvenen2007).

i1|0) and

β ,0 00 σ2

i1|0) and

β ,0 00 σ2

i1|0) and

β ,0 00 σ2

i1|0) and

β ,0 00 σ2

Dynamic Programming Problem

Worker (t = 1, ..,R−1)

Vt(ωt , βt , zt) = maxCt ,at+1

{U(Ct) + δEt

[Vt+1(ωt+1, βt+1, zt+1)

]}s.t. (1)

Ct +at+1 = ωt (2)ωt = (1+ r)at +Yt (3)

at+1 ≥ a,and Kalman recursions

Retiree (t = R, ..,T )

Vt(ωt ,Y ) = maxCt ,at+1

[U(Ct) + δVt+1(ωt+1,Y )] (4)

s.t Y = Φ(YR) , and eq.(2,3)

Dynamic Programming Problem

Worker (t = 1, ..,R−1)

Vt(ωt , βt , zt) = maxCt ,at+1

{U(Ct) + δEt

[Vt+1(ωt+1, βt+1, zt+1)

]}s.t. (1)

Ct +at+1 = ωt (2)ωt = (1+ r)at +Yt (3)

at+1 ≥ a,and Kalman recursions

Retiree (t = R, ..,T )

Vt(ωt ,Y ) = maxCt ,at+1

[U(Ct) + δVt+1(ωt+1,Y )] (4)

s.t Y = Φ(YR) , and eq.(2,3)

Estimation via Indirect Inference

Hallmark of indirect inference: Think in terms of an auxiliary modelthat can be viewed as a reduced form of the structural model.

Specifically, pick an auxiliary model that is:I easy to estimate

I rich enough to provide a good descriptive statistical model of the data.

Mimicking structural equations works well.

Thinking in terms of dynamic relationships often lead to someconditional moment conditions that are not straightforward to seeotherwise.

Why Indirect Inference?

The standard method since Hall and Mishkin (1982) is to derive structuralequations explicitly and estimate them.

For example, with (i) quadratic utility, (ii) no borrowing constraint,(iii) no retirement, and (iv) persistent plus transitory shocks, we have:

∆Ct = φtηt + ψtεt

φ � 0, ψt ∼ 0

Suppose εt ≡ 0:

∆Ct = φtηt = φt∆Yt

Response of consumption growth to income growth reveals persistenceof income shocks.

φ � 0, ψt ∼ 0

Suppose εt ≡ 0:

φ � 0, ψt ∼ 0

Suppose εt ≡ 0:

φ � 0, ψt ∼ 0

Suppose εt ≡ 0:

An Example

0 5 10 15 20 25 30 35 40−1

Labor Income Consumption

An Example: Binding Constraints

0 5 10 15 20 25 30 35 40−1

Labor Income ConsumptionAssets

IDENTIFICATION

Identification

Simplifying assumptions for intuition:(i) quadratic utility, (ii) no borrowing constraints, (iii) εt ≡ 0, and (iv)yt : level of income.

1 Optimal consumption choice:

HIP: ∆Ct = Πt ×(y it −(

αi + β

it−1t + ρ z i

))︸︷︷︸

2 If σ2β≡ 0, learning disappears → we get (certainty equivalent)

permanent income model:

RIP: ∆Ct = Ψt ×ηt

Identification

αi + β

it−1t + ρ z i

))︸︷︷︸

Identification

αi + β

it−1t + ρ z i

))︸︷︷︸

1. Identification from Consumption Changes

1 2 3 4 5 6 7 8

∆C1=∆C2

} η>0

1 2 3 4 5 6 7 8

} η>0

1 2 3 4 5 6 7 8

}ξ1<0

η=ξ2>0

1 2 3 4 5 6 7 8

HIP: ∆C1<0 ∆C2>0

} η>0

η=ξ2>0}

1 2 3 4 5 6 7 8

RIP: ∆C1=∆C

HIP: ∆C1<0, ∆C

} η=ξ2>0

} ξ1<0

2. Identification from Consumption Levels

1 2 3 4 5 6 7 8 9 10

Agent 1: Actual IncomeAgent 2: Actual Income

1 2 3 4 5 6 7 8 9 10

Forecast Income Paths under RIP

Ct=31 =Y

1 2 3 4 5 6 7 8 9 10

Forecast Income Paths under RIP

Ct=31 >C

1 2 3 4 5 6 7 8 9 10

Forecast Income Paths under HIP

Ct=31 >C

Determining the Prior Information in HIP

1 2 3 4 5 6 7 8 9 10

Forecast Income Paths under HIP with λ=1

Forecast Income underHIP with λ<1

If λ=1: Ct=31 =C

If λ<1: Ct=31 >C

A Feasible Auxiliary Model:

∆Ct = Πt ×(Y i

t −(

α + βit−1t + ρ z i

))︸︷︷︸

This regression is not feasible, so approximate with

ct = a0 +a1yt−1 +a2yt−2 +a3yt+1 +a4yt+2

+a5y1,t−3 +a6y t+3,T +a7∆y1,t−3 +a8∆y t+3,T

+a9ct−1 +a10ct−2 +a11ct+1 +a12ct+2 + error

where ct ≡ log (Ct) .

Add a second regression where yt is the dependent variable. Use thesame income regressors above.

∆Ct = Πt ×(Y i

t −(

))︸︷︷︸

+a5y1,t−3 +a6y t+3,T +a7∆y1,t−3 +a8∆y t+3,T

∆Ct = Πt ×(Y i

t −(

))︸︷︷︸

+a5y1,t−3 +a6y t+3,T +a7∆y1,t−3 +a8∆y t+3,T

Monte Carlo Results

RRA = 2

Annual net interest rate is: r = 4%

T = 41, S = 15. Social security system mimicking the U.S.

Use the observed structure of missing data.

2200 individuals with approximately 12 observations per individual.

Starting values assigned randomly 20% away from truth.

Each run takes approximately 3 hours.

Monte Carlo Results

RRA = 2

Monte Carlo Results

RRA = 2

Monte Carlo Results

RRA = 2

Monte Carlo Results

RRA = 2

Monte Carlo Results

RRA = 2

Monte Carlo Results

RRA = 2

Monte Carlo Results

True value Mean estimate Std. dev.ρ 0.670 0.669 0.034ση 0.190 0.191 0.009σε 0.150 0.150 0.015σβ (×100) 2.650 2.652 0.121σα 0.490 0.493 0.031λ 0.500 0.504 0.054σuy 0.200 0.199 0.003σuc 0.200 0.199 0.008Note: Statistics are based on 150 replications

Results

PSID data, 1968-1993.

Labor income data: household non-financial after tax income(adjusted for family size and demographics)

Consumption data: imputed (adjusted for family size anddemographics)

Measurement error:I yobs

t = yt +uyt , uy

t is i.i.d over time.

I cobst = ct +mc +uc

t , mc is an individual fixed effect, uct is i.i.d over

Also estimate δ , γ (RRA coefficient), and a (borrowing constraint).

Results

PSID data, 1968-1993.

t = yt +uyt , uy

Results

PSID data, 1968-1993.

t = yt +uyt , uy

Results

PSID data, 1968-1993.

t = yt +uyt , uy

Results

PSID data, 1968-1993.

t = yt +uyt , uy

Results

PSID data, 1968-1993.

t = yt +uyt , uy

Results

PSID data, 1968-1993.

t = yt +uyt , uy

Estimation of the Full Model

Parameter Estimate Std Error Descriptionρ 0.756 0.029 Persistenceση 0.189 0.005 Std. dev. of perm. shockσε 0.019 0.027 Std. dev. of transit. shockσβ (×100) 1.785 0.392 Growth rate heterogeneityσα 0.383 0.029 Level heterogeneitycorαβ −0.178 0.074 Correlationλ 0.191 0.124 Prior uncertaintyσuy 0.145 0.010 IID meas. error in incomeσuc 0.355 0.002 IID meas. error in cons.σuc 0.434 0.009 Fixed meas. error in cons.δ 0.958 0.003 Time discount factorγ 2.0* — RRA coefficienta -0.25* — Borrowing limit

Quantifying Life Cycle Income Risk

30 35 40 45 50 55 60

Var(log(Income))US Data

30 35 40 45 50 55 60

Var(log(Income))Estimated Model

30 35 40 45 50 55 60

Forecast Variance(This paper)

Conclusion 1: About 1/2 of cross-sectional income dispersion at retirementrepresents risk—the rest is known heterogeneity.

30 35 40 45 50 55 60

Forecast Variance(This paper)

Forecast Variance(Existing Literature)

Conclusion 2: Typical estimates in the literature overstate labor income risksignificantly.

Within-Cohort Evolution of Inequality

25 30 35 40 45 50 55 600.1

Var(log(Income)), US Data

Var(log(Income)), Model

Var(log(Consumption)), Model

Var(log(Consumption)), US Data

Var(log(Consumption)), US Data, Smoothed

Life Cycle Profile of Consumption

25 30 35 40 45 50 55 60 650.8

US DataUS Data, SmoothedModel

Conclusions and Future Directions

Monte Carlo results suggest that the indirect inference method worksvery well.

Estimates indicate

I (a) modest persistence of shocks, (b) large heterogeneity in incomegrowth rates, and (c) significant information about future incomegrowth rate (from 75 to 90% of total variance).

Implies: Typical calibrations of incomplete markets modelssubstantially overstate uninsurable income risk.

Next steps: (a) time-varying shock variances, (b) partial insurance.

Estimates indicate

Imputation of Consumption in PSID

PSID tracks food consumption only. CEX measures non-durableexpenditures but its panel structure is very limited.

Blundell, et al (2006) develop a structural procedure to imputeconsumption in PSID (1980-1992).

We modify and extend this methodology to construct a panel ofconsumption for PSID covering 1968-1992.

In particular, we also make use of CEX 1972-73 and use additionalregressors.

Imputation of Consumption in PSID Results

1965 1970 1975 1980 1985 1990 1995−0.08

−0.06

−0.04

−0.02

PSID Imputed: 1968−1992CE Non−Durable Consumption: 1980−92

CE Non−Durable, Kernel Smoothed

PSID Imputed, Kernel Smoothed

PSID Food Expenditures

PSID Food, Kernel Smoothed

25 30 35 40 45 50 55 60 65−0.06

−0.04

−0.02

PSID Imputed ConsumptionCE Non−Durable Consumption

CE Kernel Smoothed

PSID Kernel Smoothed

25 30 35 40 45 50 55 60 650.8

PSID Imputed ConsumptionCE Non−Durable ConsumptionCE Kernel SmoothedPSID Kernel Smoothed

Imputed=.996*Actual+.277+error

R squared=.67

6 8 10 12Actual

Regression Line 45 degree line

Ignoring profile heterogeneity biases ρ upward

yhit − yh

αi + β

ih+ εhit

αi + βh

βi −β

)h+ ε

Under the (incorrect) assumption that β i ≡ β , a consistent estimatorof ρ is:

ρ =h (h−1)σ2

(h−1)2σ2β

+ σ2ε

−→ 1 for large h

If σ2β

= 0.0004, and σ2ε = 0.03, h = 20, yields ρ = 0.87.

If h = 30 ⇒ ρ = 0.95, when in fact true persistence is zero.If there is a population of individuals uniformly distributed from 25 to65 years of age, ρ = 0.91

A comparison to the existing literature

Body Math: MaCurdy (1982) rejected HIP by testing a simple proposition:If individuals differ systematically in the slope of their income profiles thenincome growth rates should be positively autocorrelated at higher lags.

cov(∆yhi ,∆yh+k

i ) = σ2β−[

ρk−1(1−ρ

)σ2η

], k = 2, ...

Two concerns about the test:1 Low power against the alternative of HIP (even without AR(1) income

shocks):cov(∆yh

i ,∆yh+ki ) = σ

= .00038≈ 0

2 With AR(1) income shocks:

cov(∆yhi ,∆yh+k

i ) = .00038−[.0033∗0.82k−1

for k < 12

⇒ Even with HIP autocovariances should be slightly negative andalways close to zero.

Group ρ σ2α σ2

βcorrαβ σ2

η σ2ε

A .988 .058 — — .015 .061(.024) (.011) (.007) (.010)

A .821 .022 .00038 −.23 .029 .047(.030) (.074) (.00008) (.43) (.008) (.007)

C .979 .031 — — .0099 .047(.055) (.021) (.013) (.020)

C .805 .023 .00049 −.70 .025 .032(.061) (.112) (.00014) (1.22) (.015) (.017)

H .972 .053 — — .011 .052(.023) (.015) (.007) (.008)

H .829 .038 .00020 −.25 .022 .034(.029) (.081) (.00009) (.59) (.008) (.007)

LagSample 0 1 2 3 4 5

Autocovariance(1) Data .0476 −.0176 .00058 −.00166 −.00014 −.00067

(.0019) (.0014) (.0008) (.0007) (.0008) (.0007)(2) Data .1215 −.0385 −.0031 −.0023 −.0025 −.00004

(.0023) (.0011) (.0010) (.0008) (.0007) (.0008)(3) Model .0840 −.0329 −.0014 −.0011 −.0007 −.0007

(.0013) (.0010) (.0008) (.0009) (.0008) (.0007)

Autocorrelation(4) Data 1.00 −.394 .013 −.039 −.003 −.016(5) Data 1.00 −.317 −.026 −.019 −.021 −.001(6) Model 1.00 −.391 −.016 −.012 −.009 −.009

(.000) (.008) (.009) (.010) (.008) (.009)

Imputation of Consumption in PSID: Results

25 30 35 40 45 50 55 60 650.85

CEX all years (before−tax)PSID (before−tax)CEX drop<80PSID (After−tax)

Imputation of Consumption in PSID: Results

30 35 40 45 50 55 60 65−0.1

PSID (before−tax income)PSID (after−tax income)CEX all yearsCEX drop<80

Change in Precision of Beliefs about β i

Missing in the file

inferringincomeriskfromeconomicchoices · 2014. 5. 17. · inferringincomeriskfromeconomicchoices:...

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