Estimation of Life-Cycle Consumption
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Estimation of Life-Cycle Consumption
Zhe Li (PhD Student)
Stony Brook University
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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
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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
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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
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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
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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
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Solution
• Normalization
• At retirement
• When working
ttt PCc / ttt PXx /
1101 TT xc h10
111
1 )(
ttt
ttt UNG
Rcxx
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Figure 9. Life Cycle Consumption (Thousands in 1987 dollars)
Age
Raw data
Trust-region Newton
L-M
Quasi-Newton
Global convergence