Estimation of Life-Cycle Consumption

Zhe Li (PhD Student)

Stony Brook University

Introduction

• A CLASSICAL METHOD of moments estimator

• Instead using analytically form, replace the expected response function by a simulation result ---- the method of simulated moments (MSM).

• An application of MSM to life-cycle consumption model (Gourinchas and Parker (2002)).

mmnsobservatio atsponse

Expected

sponse

Observed

Vector

Instrument

..ReRe

25 30 35 40 45 50 55 60 6517

18

19

20

21

22

23

24

25

26

27Figure 1. Life Cycle Consumption and Income Profile

Age

Ave

rage

Con

sum

ptio

n (T

hous

ands

in 1

987

dolla

rs)

Consumption (Raw Data)

Income Profile

Fitted Consumption

Model

• Live t= 0----N, and work for periods T<N. T and N are exogenous.

• The households maximize

• Utility is of CRRA form, and multiplicatively separable in Z.

N

tNN

Ntt

t WBZCuE0

111 )(),(

1)(),(

1CZvZCu

Model

• When working

• Income

• Transitory shock: takes 0 with probability • and otherwise.

• Permanent shock:

)(1 tttt CYWRW

ttt UPY tttt NPGP 1

tU 10 p

),0(ln 2ut NU

),0(ln 2nt NN

1111 )( tttttt YWYCXRX

Model

• After retirement, no uncertainty.

• Illiquid wealth in the first year of retirement

• Retirement value function

• Consumption Rule (Merton (1971))

TTT hPhPH 11

11111111 ))((),,( TTTTTTT HXZvZHXV

)( 1111 TTT HXC

TNR

R

)(1

11/1/1

1/1/1

1

Solution

• Normalization

• At retirement

• When working

ttt PCc / ttt PXx /

1101 TT xc h10

111

1 )(

ttt

ttt UNG

Rcxx

Solution

• In the last period of working

• In periods

))(()(

)(),(max))(( 11

1

TTT

TTTT xcu

Zv

ZvRxuxcu

Tt

)])(()(

)([))(( 1111

1

ttttt

ttt NGxcu

Zv

ZvRExcu

Numerical method

• Intertemporal budget constraint

• Two-dimensional Gauss-Hermite quadrature

0)()1()(

)(

)())((

11

111

1

1

UNGUNG

RcxcuEpNG

NG

RcxcupE

Zv

ZvRxcu

tt

ttttt

ttt

t

ttt

jiijjit

unt

tttttt

wunf

dudneeunf

UdFNdFNGxcuNGxcuE

,

111111

),(

),(

)()())(()])(([

22

Simulation

0 1h

1

2n

2u

26w

Parameters Description Value

Rate of time preference 0.96

Rate of risk aversion 0.514

, where h is the ratio of illiquid wealth to the permanent component of income at retirement

0.001

Marginal propensity to consume at retirement 0.071

R Interest rate 1.0344

p Probability of unemployment 0.00302

Variance of permanent shock 0.0212

Variance of transitory shock 0.0440

Initial log normalized wealth at age 26 -2.794 (1.784)

25 30 35 40 45 50 55 60 6510

15

20

25

30

35Figure 2. Life Cycle Consumption--Different Values for Beta

Age

Ave

rage

Con

sum

ptio

n (T

hous

ands

in 1

987

dolla

rs) 0.96

0.98

0.94

25 30 35 40 45 50 55 60 6516

17

18

19

20

21

22

23

24Figure 4. Life Cycle Consumption--Different Values for Risk Aversion

Age

Ave

rage

Con

sum

ptio

n (T

hous

ands

in 1

987

dolla

rs) 0.5

0.4

0.6

25 30 35 40 45 50 55 60 6516

17

18

19

20

21

22

23

24Figure 6. Life Cycle Consumption-Different values of aphar1

Age

Ave

rage

Con

sum

ptio

n (T

hous

ands

in 1

987

dolla

rs) 0.07

0.06

0.08

25 30 35 40 45 50 55 60 6517

18

19

20

21

22

23

24

25Figure 7. Life Cycle Consumption-Different values of aphar0

Age

Ave

rage

Con

sum

ptio

n (T

hous

ands

in 1

987

dolla

rs) 0.001

0.05

0.5

25 30 35 40 45 50 55 60 6516

17

18

19

20

21

22

23

24Figure 8. Life Cycle Consumption-Lower Income Variance

Age

Ave

rage

Con

sum

ptio

n (T

hous

ands

in 1

987

dolla

rs)

lower income variance

Estimation

• Objective

• Two step MSM:• The first subset:• The second subset:

• Expectation of log consumption,

• Approximation (Monte-Carlo)

0)](ln[ln 0 tit CCE

)ˆ,;,()ˆ,;,(ln)ˆ,(ln PxdFPxCC ttt

)ˆ,(ln)ˆ,(ln1

)ˆ,(ˆln1

t

L

ltt CC

LC

Estimation

• Find that minimize

• Where

• W is a T*T weighting matrix:– Inverse of the sample counterpart of

– Corrected by the variance-covariance matrix for the first-stage

estimation

)ˆ,()'ˆ,( Wgg

)ˆ,(ˆlnln)ˆ,(ˆlnln1

))ˆ,(ˆln(ln1

)ˆ,(11

ttt

I

iit

t

I

itit

tt CCCC

ICC

Ig

tt

]))'(ln))(ln(ln[(ln iiii CECCECE

Estimation Method

• Start at a point x in N-dimensional space, and proceed from there in some vector direction p

• Any function of N variables f(x) can be minimized along the line p, say finding the scalar a that minimizes f(x+ap)

• Replace x by x+ap, and start a new iteration until convergence occurs

• Example: Newton method

pxfDxx ii )(1

Estimation method

• This study, – x is the set of parameters– Dimension is T (time periods)– Objective function is

– Gradient

– Hessian matrix

2

11

2 )]ˆ,(ˆln[ln)]ˆ,([)ˆ,()'ˆ,()( tt

T

t

T

tt CCggWgf

T

t k

ttt

k

CCC

f

1

)ˆ,(ˆln)]ˆ,(ˆln[ln2

lk

ttt

l

t

k

tT

tlk

CCC

CCf

)ˆ,(ˆln

)]ˆ,(ˆln[ln)ˆ,(ˆln)ˆ,(ˆln

22

1

2

Estimation Results

0

1

2

ParametersTrust-Region

NewtonL-M Quasi-Newton

Global convergence

0.9597 0.9595 0.9511 0.9637

(0.0390) (0.0385) (0.0313) (0.0556)

0.4981 0.5236 0.9411 0.2008

(3.9409) (2.6876) (3.6892) (5.0803)

Retirement Rule:

0.0116 0.0001 0.1069 0.0511

(7.3221) (6.7996) (6.5953) (1.8346)

0.0765 0.0822 0.0650 0.0895

(0.6604) (0.6086) (0.3639) (0.4741)

fmin 0.0686 0.0652 0.0772 0.0924

177.8519 169.1142 200.1481 239.5556

25 30 35 40 45 50 55 60 6517

18

19

20

21

22

23

24

25

Figure 9. Life Cycle Consumption (Thousands in 1987 dollars)

Age

Raw data

Trust-region Newton

L-M

Quasi-Newton

Global convergence