lecture viii income process: facts, estimation, and discretization · 2015-03-10 · estimation of...

Lecture VIII

Income Process:

Facts, Estimation, and Discretization

Gianluca Violante

New York University

Quantitative Macroeconomics

G. Violante, ”Income Process” p. 1 /26

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.

• Important to distinguish between wages (endogenous laborsupply), individual earnings (two-earner hh, inelastic laborsupply), hh earnings (one earner hh, inelastic labor supply), hhincome (endowment economy)


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 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 (yi,0)

• 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

• Typically, zit estimated to be a lot less persistent

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


Step 2: Estimation

• You have an 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

yi =

yi1

...

yiA

since the panel is unbalanced, must set missing elements to zeroG. Violante, ”Income Process” p. 9 /26

Step 2: Estimation

• The covariance of earnings is then a (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)


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 comforming (A (A+ 1) /2, 1) vector of empiricalmoments.

• 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 that 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) vector ofparameters.


Weighting matrix

• Define, comformably 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) find, by MC simulations, that there issubstantial small sample bias in the estimates of θ andrecommend using (i) Ω = I (equally weighted estimator) orΩ = 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.


Higher moments of income distribution

Three approaches to think about skewness and excess kurtosis with acontinuous process:

1. Assume innovations drawn from skewed and leptokurticdistributions and add higher moments to estimation

2. Clark (ECMA, 1973): subordinated stochastic processes

• Y (T (t)), where T (t) is the directing process that sets numberof times Y changes in period t

• Both Y (t) and T (t) have independent increments

• If ∆Y is normal and ∆T lognormal, Y (T (t)) displays kurtosis.

3. Kou (Management Science, 2002): combination of diffusion plusjump process, where jump is drawn from an asymmetricdouble-exponential distribution


Step 4: Discretization

• 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]

[

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)

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

a =m−∆

n− 1

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, important 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 give up either on the skewness or on the kurtosis...

• Homework: estimate parameters to fit those moments. Simulateprocess. Can you replicate the var, skewness and kurtosis of theone year changes? Can you derive the moments in firstdifferences in closed form like the ones in level?


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 constarint 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 viii income process: facts, estimation, and discretization · 2015-03-10 · estimation of...

Documents