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