lecture iii income process: facts, estimation, and discretization · 2015-09-10 · lecture iii...

Lecture III

Income Process:

Facts, Estimation, and Discretization

Gianluca Violante

New York University

Quantitative Macroeconomics

G. Violante, ”Income Process” p. 1 /30

Estimation of income process

• Competitive model: unique market price per efficiency unit →wage distribution reflects heterogeneity in efficiency units:

Yiat = Pt · exp(

Yiat

)

• Compute residuals yiat from regression in logs

log Yiat = Dt + β′Xiat + yiat

time dummies Dt = [logP1, logP2, ...], and Xiat are fixedobservables that capture predictable part of Yiat: age, edu, race

• Gender: be aware of of selection. Typically use data on men.


Which definition of income should one use?

• Models with endogenous labor supply: hourly wages

• Models with exogenous labor supply and saving: labor earnings(wages × hours)

• Is individual or household the unit of analysis?

• Is fiscal policy important? If yes, choose earnings beforetax/transfers, if not choose directly disposable income

• Endowment economy: total (labor + capital) disposable income


Step 1: choose statistical model

• Assume process is stationary: when you compute momentconditions, do it t by t, but then average across t.

• Identification/estimation in nonstationary process is morecomplicated, see Heathcote-Storesletten-Violante (2010).

• We choose the following specification for log earnings process:

yia = zia + εia

zia = ρzi,a−1 + uia

uiaiid∼ (0, σu)

zi0iid∼ (0, σz0)

εiaiid∼ (0, σε)

• Why persistent + transitory?


20 22 24 26 28 30 32 34 360

0.05

0.1

0.15

0.2

0.25NLSY 79: Covariance of wages for college educated males

Age (a)

Cov

aria

nce

COV (age 22, a)

COV( age 23,a)

COV( age 24, a)


Identification in levels

yia = zia + εia


var (yi0) = σz0 + σε for a = 0

var (yia) = var (zia) + σε for a > 0

var (zia) = ρ2var (zi,a−1) + σu

cov (yia, yi,a−j) = cov (zia, zi,a−j) for j > 0

cov (zia, zi,a−j) = ρjvar (zi,a−j)

• Identification of ρ from the slope of the covariance at lags > 0:

cov (yia, yi,a−2)

cov (yi,a−1, yi,a−2)=

ρ2var (zi,a−2)

ρvar (zi,a−2)= ρ


Identification in levels

• Identification of σε from the difference between variance andcovariance

var (yi,a−1)−ρ−1cov (yia, yi,a−1) = var (zi,a−1)+σε−var (zi,a−1)

• Given σε, obtain σz0 residually from var (yi0)

• Finally, identification of σu:

var (yi,a−1)− cov (yia, yi,a−2)− σε =

ρ2var (zi,a−2) + σu + σε − ρ2var (zi,a−2)− σε = σu

• Identification achieved with two lags: (a− 2, a− 1, a) .

• Typically, with long panels (T ≃ 10− 15 ) model is largelyoveridentified, and parameters tightly estimated because N islarge (thousands of observations) but if nonstationary...


Identification in first differences (for ρ = 1)

∆yia = uia + εia − εi,a−1

∆yi,a−1 = ui,a−1 + εi,a−1 − εi,a−2

var (∆yia) = σu + 2σε

cov (∆yia,∆yi,a−1) = −σε

• Of course, you can use moments in first differences as additionalmoments, together with those in level, to estimate the parametersof AR(1)

• Not-so-much-fun fact: if you estimate a permanent-transitoryprocess in levels and first-differences, you obtain very differentresults. Why?


Heterogeneous Income Profiles

• A common alternative specification to the persistent-transitoryrepresentation of idiosyncratic income risk is:

yia = αi + βia+ zia


uiaiid∼ (0, σu)

(αi, βi)iid∼ (0,Σ)

• (αi, βi) are, respectively, constant and slope of earning profile,heterogeneous across individuals

• Persistence of zit estimated to be a lot lower

• HIP difficult to distinguish, empirically, from persistent-transitory


Step 2: Estimation

• Start from unbalanced panel of individuals aged a = 1, ..., A.

• For every individual i = 1, ..., I define a (A× 1) vector

di =

di1

...

diA

where dia ∈ 0, 1 is an indicator variable for whether individual iis present at age a in the sample.

• Analogously to dia define an (A× 1) vector of wage observations

yi =

yi1

...

yiA

since the panel is unbalanced, must set missing elements to zeroG. Violante, ”Income Process” p. 10 /30

Step 2: Estimation

• The covariance of earnings is then an (A×A) symmetric matrixcomputed as

C =Y

D

This is an element by element division, where Y and D are(A×A) matrices given by

Y =

I∑

i=1

yiy′i D =

I∑

i=1

did′i

• Each element of yiy′i is product of earnings at ages (p, q) for i

• Each element of did′i is 1 only if i is present at both ages (p, q) .

• Each entry of C is the cross-sectional covariance of earnings atages (p, q) → C symmetric


Step 2: Estimation

• Take the upper triangular portion of C and vectorize it into an(A (A+ 1) /2, 1) vector.

m =vech(

CUT)

• Let f (Θ) be the conforming (A (A+ 1) /2, 1) vector of empiricalcovariances

• Estimation based on a minimum distance estimator that minimizesthe distance btw the model covariance matrix and the empiricalcovariance matrix (Chamberlain, 1984) — family of GMM.

• The estimator solves the following minimization problem

minΘ

[m− f (θ)]′Ω [m− f (θ)]

where Ω is a weighting matrix and θ is a (n, 1) parameter vector,e.g. (ρ, σu, σz0, σε)


Weighting matrix

• Define, conformably with m, the vector mi that contains thedistinct elements of the cross-product matrix yiy

′i.

• Similarly, define the vector s = vech(

DUT)

. Chamberlain provedthat m has asymptotic variance

V =

∑I

i=1 (m−mi) (m−mi)′

ss′

where V is a (A (A+ 1) /2, A (A+ 1) /2) matrix

• Chamberlain shows that the optimal weighting matrix Ω is V−1

• Altonji and Segal (1996): MC simulations show there issubstantial small sample bias in θ based on Ω and recommendusing (i) Ω = I (equally weighted estimator) or Ω = diag

(

V−1)

(diagonally weighted estimator)


Step 3: S.E. and Testing

• Standard errors of θ are then computed as

(G′ΩG)−1

G′ΩVΩG (G′ΩG)−1

where G is the (A (A+ 1) /2, n) gradient matrix ∂f (θ) /∂θevaluated at θ∗

• Finally, the chi-squared goodness of fit statistic is

[m− f (θ∗)]R− [m− f (θ∗)]′∼ χ2

q

where R− is the generalized inverse of R = WVW′ whereW = I−G (G′ΩG)

−1G′Ω, and q = A(A+ 1)/2− n, are the the

degrees of freedom


Step 4: Discretization (focus on AR1)

• Two dominant methodologies:

1. Tauchen method (Tauchen EL, 1986)

2. Rowenhorst method (article in “Frontiers of business cycle”,Tom Cooley, editor, 1995)

Rowenhorst method better as ρ → 1, plus one can derivefirst four moments in closed form

Two approaches to derive moments in closed form:Kopecki-Suen (RED, 2010) and Lkhagvasuren (2012)

Multivariate version: Terry-Knotek (EL, 2011),Galindev-Lkhagvasuren (JEDC, 2010)


Tauchen vs Rowenhorst (9 points, σy = 1)

2000 4000 6000 8000 10000

−2

0

2

Continuous, ρ = 0.99

2000 4000 6000 8000 10000

−2

0

2

Continuous, ρ = 0.999

2000 4000 6000 8000 10000

−2

0

2

Tauchen, ρ = 0.99

2000 4000 6000 8000 10000

−2

0

2

Tauchen, ρ = 0.999

2000 4000 6000 8000 10000

−2

0

2

Rouwenhorst, ρ = 0.99

2000 4000 6000 8000 10000

−2

0

2

Rouwenhorst, ρ = 0.999


Rowenhorst method to discretize AR(1)

• Consider the following two-state Markov chain x ∈ 0, 1 where

Pr (x′ = 0|x = 0) = p Pr (x′ = 1|x = 0) = 1− p

Pr (x′ = 0|x = 1) = 1− q Pr (x′ = 1|x = 1) = q

• Compute the invariant distribution (1− α, α):

[

1− α α]

[

p 1− p

1− q q

]

=[

1− α α]

A system of 2 equations in 2 unknowns with with solution

α ≡ Pr(x = 1) =1− p

2− p− q


Moments in closed form

• Bernoulli invariant distr. → we can compute moments analytically:

E (x) = α =1− p

2− p− q

V ar (x) = α (1− α) =(1− p) (1− q)

(2− p− q)2

Skew (x) =1− 2α

√

α (1− α)=

p− q√

(1− p) (1− q)

Ekurt (x) = −6 +1

α (1− α)= −2 +

(p− q)2

(1− p) (1− q)

Corr (x, x′) = p+ q − 1


Generalization to n-state process

• Consider the auxiliary process

Xn =n−1∑

i=1

xi

where each x is an independent 2-state Markov chain

• Therefore Xn ∈ 0, 1, ..., k, ..., n− 1 . Define the process:

y =

n−1∑

i=1

(a+ bxi) = (n− 1)a+ bXn

and choose a and b so that y takes values in [m−∆,m+∆]where m is the mean and 2∆ the range of the state space.


Generalization to n-state process

• By setting ymin = m−∆ and ymax = m+∆, it is easy to see that

ymin = (n− 1)a → a =m−∆

n− 1

ymax = (n− 1)(a+ b) → b =2∆

n− 1

• It follows that the kth point of the grid (0 < k < n− 1) is:

yk = m−∆+2∆

n− 1k

• We have the grid for y as a function of three parameters (n,m,∆)


Moments in closed form for n-state process

• Then, it is easy to see that:

E (y) = (n− 1) [a+ bE (x)] = m−∆+ 2∆1− p

2− p− q= m+∆

q − p

2− p− q

V ar (y) = (n− 1) b2V ar (x) =4∆2

n− 1

(1− p) (1− q)

(2− p− q)2

Corr(

y, y′)

= Corr(

x, x′)

= p+ q − 1

• Note: don’t need to set q = p for the process to have mean zero.

• Less easy to show that:

Skew (y) =Skew (x)√

(n− 1)=

p− q√

(n− 1) (1− p) (1− q)

Ekurt (y) =Ekurt (x)

n− 1=

−2

n− 1+

(p− q)2

(n− 1) (1− p) (1− q)

therefore, implied constraint: Ekurt (y) = −2/(n− 1) + Skew (y)2


Exact moment matching

• 5 moments E (y) , V ar (y) , Corr (y, y′) , Skew (y) , Ekurt (y)and 5 parameters m,∆, n, p, q

• Moment matching:

(p, q, n) → Corr (y, y′) , Skew (y) , Ekurt (y)

(m,∆) → E (y) , V ar (y)

• In the data (SCF log household earning residuals, ages 22-60):

E (y) = 0, V ar (y) = 0.8, Corr (y, y′) = 0.95, Skew (y) = −1.1, Ekurt = 4

so one must either give up on the skewness or on the kurtosis...

• Additional problem: kurtosis of the implied (discretized) innovationshould be 3 but it is much higher — the fewer the grid points themore serious the problem


Introduction New Empirical Facts Estimation Conclusions

What Do Data on Millions of U.S. Workers

Reveal About Life Cycle Income Risk?1

Fatih Guvenen Fatih KarahanMinnesota and NBER New York Fed

Serdar Ozkan Jae SongToronto SSA

September 23, 2014

1The findings and conclusions expressed are solely those of the authors and do not represent the views of

Federal Reserve Board, Federal Reserve Bank of New York or SSA.


Data: SSA Master Earnings File

• Representative sample of US males covering 34 years:1978 to 2011

• Salary and wage workers (from W-2 forms)

• Individuals aged 25–60

• Key Advantages:

• Very large sample size (200+ million observations)

• No survey response error

• No sample attrition

• No top-coding


II.a Standard Deviation of y

t+1 y

t

0 20 40 60 80 100

0.4

0.5

0.6

0.7

0.8

0.9

1

Percentiles of Recent Earnings (RE) Distribution

Standard

Deviationof(y

t+1−

yt)

25-2930-3435-3940-4445-4950-54


II.b Skewness of y

t+1 y

t

0 20 40 60 80 100−3

−2.5

−2

−1.5

−1

−0.5

0

0.5


Skewness

of(y

t+1−

yt)

25-2930-3435-3940-4445-4950-54


II.c Histogram of y

t+1 y

t

−3 −2 −1 0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

yt+1− yt

Den

sit

y

US Data

Normal (0,0.48)

Std. Dev. = 0.48Skewness = –1.35Kurtosis = 17.80


II.c Center of Histogram: [-0.12, 0.19]

−3 −2 −1 0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

yt+1− yt

Den

sit

y

US Data

Normal (0,0.48)

US Data: 65%


II.c Center of Histogram: [-0.12, 0.19]

−3 −2 −1 0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

yt+1− yt

Den

sit

y

US Data

Normal (0,0.48)

Normal: 25%


II.c Shoulders of Histogram

−3 −2 −1 0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

yt+1− yt

Den

sit

y

US Data

Normal (0,0.48)

US Data: 31%


II.c Shoulders of Histogram

−3 −2 −1 0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

yt+1− yt

Den

sit

y

US Data

Normal (0,0.48)

Normal: 71%


II.c Distribution of Income Changes

Prob(|y i

t+1 y

i

t

| < x)x : Data N (0, 0.482) Ratio

0.05 0.35 0.08 4.380.10 0.54 0.16 3.380.20 0.71 0.32 2.230.50 0.86 0.70 1.221.00 0.94 0.96 0.98


II.c Kurtosis of y

t+1 y

t

0 20 40 60 80 1000

5

10

15

20

25

30

35


Kurtosisof(y

t+1−

yt)

25-2930-3435-3940-4445-4950-54

Business Cycle Variation in Shocks

Myth #2:

The variance of idiosyncratic income shocks

rises substantially during recessions.

Fatih Guvenen (Myths vs. Facts) Myths vs. Facts 15 / 41

Myth #2: Countercyclical Shock Variances

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

Density

yt+k

− yt

Recession

Expansion


Countercyclical Variance

Constantinides and Duffie (1996): countercyclical variance cangenerate interesting and plausible asset pricing behavior.

Existing indirect parametric estimates find a tripling of the varianceof persistent innovations during recessions (e.g., Storesletten et al(2004)).

Our direct and non-parametric estimates show no change invariance over the cycle. See the next figure.

The following figures on Myths 2 to 4 are from Guvenen et al.(2014b).


Fact #2: No Change in Variance

0 10 20 30 40 50 60 70 80 90 100

0.8

1

1.2

1.4

1.6

1.8

2

Percentiles of 5-Year Average Income Distribution (Y t−1)

Dispersionin

Recession/Dispersionin

Expansion

Std. dev. ratio

L90−10 ratio

Storesletten et al (2004)’s

benchmark estimate: 1.75


Fact #2: Countercyclical Left-Skewness

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5

Density

yt+k

− yt

Expansion

Recession


Fact #2: Countercyclical Skewness

0 10 20 30 40 50 60 70 80 90 100−0.4

−0.3

−0.2

−0.1

0

0.1

Percentiles of 5-Year Average Income Distribution (Y t−1)

Kelley’s

Skewness

Measu

reofyt+k−

yt,k=

1,5

Expansion

Recession


Modelling a process with high skewness and kurtosis

• The way to do it is through a mixture of Normals (Kon 1984)

• Normal Mixture innovation AR + Transitory process (NMART)

yt = zt + εt

zt = ρzt−1 + ut

where

ut ∼

N (µ1, σ1) with probability p1

N (µ1, σ2) with probability p2

εt ∼ N (0, σε)

• Civale, Diez-Catalan, and Fazilet (2015) derive first to fourthmoments in closed form for levels and differences.

• Given data counterpart, you can estimateθ = (µ1, µ2, σ1, σ2, p1, p2, σε) by MDE


Discretization of NMAR

• They argue, you really need the transitory shock to matchGuvenen’s facts

• But let’s pretend you don’t and focus discussion on NMAR

• C-DC-F extend Tauchen’s method (and call it the ET method):

1. Choose the number of gridpoints n

2. Guess a vector z = z1, z2, ..., zn

3. Define midpoints di = (zi + zi+1) /2 with d1 = −∞ anddn+1 = ∞ and the implied transition function:

Tij = F (dj+1 − ρzi)− F (dj − ρzi)

where F is the CDF of the persistent innovation ut


Discretization of NMAR

1. Choose the number of gridpoints n

2. Guess a vector z = z1, z2, ..., zn

3. Define midpoints di = (zi + zi+1) /2 with d1 = −∞ and dn+1 = ∞and the implied transition function:

Tij = F (dj+1 − ρzi)− F (dj − ρzi)

where F is the CDF of the persistent innovation ut

4. Compute the distance between the model moments generated bythe continuous process m (θ) and its discretized counterpart m (z)

5. Repeat steps by choosing until

|m (θ)− m (z, T ) | < δ

• Difference from Tauchen: with Normal innovation, you can chooseequidistant gridpoints


Application to Aiyagari model

Figure 11: Level curves of the general equilibrium interest rateFor each combination of skewness and kurtosis of the shocks to labor endowment displayed in

this figure, we compute the interest rate at the general equilibrium of an Aiyagari economy.

In this figure we report the level curves of such interest rate function. Notice that since

we use the ET method to calibrate the 15 states Markov chain for labor endowment, only

the combinations northeast of the blue line can be calibrated exactly: for more details see

Section 4.2.

-

-

-

-

-


Stationarizing the stochastic growth model

• The representative household solves:

maxIt,Ct,Nt

E0

∞∑

t=0

βt

[

Cφt (1−Nt)

1−φ]1−σ

1− σ

s.t.

Ct + It = (1− τt)WtNt + rtKt +Φt

Kt+1 = (1− δ)Kt + It

Zt = Gtzt, zt stationary, G = 1 + g

• The representative firm solves (F is CRS):

maxNt,Kt

F (Kt, ZtNt)−WtNt − rtKt

• The government balances its budget: Φt = τtWtNt



• Define a blue new set of stationary variables:

ct ≡ Ct/Gt, kt ≡ Kt/G

t, it ≡ It/Gt, φt ≡ Φt/G

t, wt ≡ Wt/Gt, zt ≡ Zt/G

t

• Rewrite the household problem as:

maxit,ct,Nt

E0

∞∑

t=0

βt

[

cφt (1−Nt)1−φ

]1−σ

1− σ

s.t.

ct + it = (1− τt)wtNt + rtkt + φt

kt+1G = (1− δ) kt + it

where βt =(

βGφ(1−σ)

)t



• By CRS, the firm problems becomes

maxNt,kt

F (kt, ztNt)− wtNt − rtkt

• The government budget constraint becomes

φt = τtwtNt

• Thus all the stationarized variables grow at rate (G− 1)

• Labor (btw 0 and 1) and interest rate (Y/K) do not grow

• The model has a balanced growth path

• Key: preferences s.t. inc. effect = subst. effect on labor supply


Stationarized Euler equation

cφ(1−σ)−1t (1−Nt)

(1−φ)(1−σ) = λt

βtλtG = βt+1λt+1 [1 + Fk (kt+1, zt+1Nt+1)− δ]

which implies:

βtGφ(1−σ)t+1

λt = βt+1Gφ(1−σ)(t+1)

λt+1 (1 + rt+1 − δ)

G1−φ(1−σ)

(

ct+1

ct

)1−φ(1−σ)

·

(

1−Nt

1−Nt+1

)(1−φ)(1−σ)

= β [1 + Fk (kt+1, zt+1Nt+1)− δ]

• We could have written the FOC from the model with trends andstationarized it (note that Fk is homogenous of degree zero).What is preferable? It depends how you solve the model!


lecture iii income process: facts, estimation, and discretization · 2015-09-10 · lecture iii...

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