NBER Summer Institute Econometrics
Methods Lecture:
GMM and Consumption-Based Asset Pricing
Sydney C. Ludvigson, NYU and NBER
July 14, 2010
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspecified, and macro
variables often measured with error. Therefore:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspecified, and macro
variables often measured with error. Therefore:
1
move away from specification tests of perfect fit (given
sampling error),
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspecified, and macro
variables often measured with error. Therefore:
1
move away from specification tests of perfect fit (given
sampling error),
2
toward estimation and testing that recognize all models are
misspecified,
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspecified, and macro
variables often measured with error. Therefore:
1
move away from specification tests of perfect fit (given
sampling error),
2
toward estimation and testing that recognize all models are
misspecified,
3
toward methods permit comparison of magnitude of
misspecification among multiple, competing macro models.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspecified, and macro
variables often measured with error. Therefore:
1
move away from specification tests of perfect fit (given
sampling error),
2
toward estimation and testing that recognize all models are
misspecified,
3
toward methods permit comparison of magnitude of
misspecification among multiple, competing macro models.
Themes are important in choosing which methods to use.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Generalizations of CRRA utility: recursive utility
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Generalizations of CRRA utility: recursive utility
Semi-nonparametric minimum distance estimators:
unrestricted LOM for data
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Generalizations of CRRA utility: recursive utility
Semi-nonparametric minimum distance estimators:
unrestricted LOM for data
Simulation methods: restricted LOM
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Generalizations of CRRA utility: recursive utility
Semi-nonparametric minimum distance estimators:
unrestricted LOM for data
Simulation methods: restricted LOM
Consumption-based asset pricing: concluding thoughts
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Economic model implies set of r population moment
restrictions
E{h (θ, wt )} = 0
| {z }
(r × 1 )
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Economic model implies set of r population moment
restrictions
E{h (θ, wt )} = 0
| {z }
(r × 1 )
wt is an h × 1 vector of variables known at t
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Economic model implies set of r population moment
restrictions
E{h (θ, wt )} = 0
| {z }
(r × 1 )
wt is an h × 1 vector of variables known at t
θ is an a × 1 vector of coefficients
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Economic model implies set of r population moment
restrictions
E{h (θ, wt )} = 0
| {z }
(r × 1 )
wt is an h × 1 vector of variables known at t
θ is an a × 1 vector of coefficients
Idea: choose θ to make the sample moment as close as
possible to the population moment.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
| {z }
t= 1
(r × 1 )
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
| {z }
t= 1
(r × 1 )
yT ≡ wT′ , wT′ −1 , ...w1′
Sydney C. Ludvigson
′
is a T · h × 1 vector of observations.
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
| {z }
t= 1
(r × 1 )
yT ≡ wT′ , wT′ −1 , ...w1′
′
is a T · h × 1 vector of observations.
b minimizes the scalar
The GMM estimator θ
′
Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )],
( 1× r )
(r×r)
(1)
( r × 1)
{WT }T∞=1 a sequence of r × r positive definite matrices
which may be a function of the data, yT .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
| {z }
t= 1
(r × 1 )
yT ≡ wT′ , wT′ −1 , ...w1′
′
is a T · h × 1 vector of observations.
b minimizes the scalar
The GMM estimator θ
′
Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )],
( 1× r )
(r×r)
(1)
( r × 1)
{WT }T∞=1 a sequence of r × r positive definite matrices
which may be a function of the data, yT .
If r = a, θ estimated by setting each g(θ; yT ) to zero.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
| {z }
t= 1
(r × 1 )
yT ≡ wT′ , wT′ −1 , ...w1′
′
is a T · h × 1 vector of observations.
b minimizes the scalar
The GMM estimator θ
′
Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )],
( 1× r )
(r×r)
(1)
( r × 1)
{WT }T∞=1 a sequence of r × r positive definite matrices
which may be a function of the data, yT .
If r = a, θ estimated by setting each g(θ; yT ) to zero.
GMM refers to use of (1) to estimate θ when r > a.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Asym. properties (Hansen 1982): b
θ consistent, asym.
normal.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Asym. properties (Hansen 1982): b
θ consistent, asym.
normal.
Optimal weighting WT = S−1
∞
S =
r×r
∑
E
j=− ∞
Sydney C. Ludvigson






′
.
[h (θo , wt )] h θo , wt−j

{z
}
| {z } |

r×1
1× r
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
Asym. properties (Hansen 1982): b
θ consistent, asym.
normal.
Optimal weighting WT = S−1
∞
S =
r×r
∑
E
j=− ∞






′
.
[h (θo , wt )] h θo , wt−j

{z
}
| {z } |

r×1
1× r
In many asset pricing applications, it is inappropriate to
use WT = S−1 (see below).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
b T . Employ an
b T depends on b
θT which depends on S
S
iterative procedure:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Review (Hansen, 1982)
b T . Employ an
b T depends on b
θT which depends on S
S
iterative procedure:
1
2
3
4
(1)
Obtain an initial estimate of θ = b
θT , by minimizing
Q (θ; yT ) subject to arbitrary weighting matrix, e.g., W = I.
(1)
(1)
b .
Use b
θT to obtain initial estimate of S = S
T
(1 )
b ; obtain
Re-minimize Q (θ; yT ) using initial estimate S
T
(2)
b
new estimate θT .
Continue iterating until convergence, or stop. (Estimators
have same asym. dist. but finite sample properties differ.)
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Classic Example: Hansen and Singleton (1982)
Investors maximize utility
max Et
Ct
Sydney C. Ludvigson
"
∞
∑ β i u (C t + i )
i= 0
#
Methods Lecture: GMM and Consumption-Based Models
Classic Example: Hansen and Singleton (1982)
Investors maximize utility
max Et
Ct
"
∞
∑ β i u (C t + i )
i= 0
#
Power (isoelastic) utility


 u (C t ) =


1− γ
Ct
1− γ
γ>0
(2)
u (Ct ) = ln(Ct ) γ = 1
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Classic Example: Hansen and Singleton (1982)
Investors maximize utility
max Et
Ct
"
∞
∑ β i u (C t + i )
i= 0
#
Power (isoelastic) utility


 u (C t ) =


1− γ
Ct
1− γ
γ>0
(2)
u (Ct ) = ln(Ct ) γ = 1
N assets => N first-order conditions
n
o
−γ
−γ
Ct = βEt (1 + ℜi,t+1 ) Ct+1
Sydney C. Ludvigson
i = 1, ..., N.
(3)
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Re-write moment conditions
(
"
0 = Et
1 − β (1 + ℜi,t+1 )
Sydney C. Ludvigson
−γ
Ct + 1
−γ
Ct
#)
.
(4)
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Re-write moment conditions
(
"
0 = Et
1 − β (1 + ℜi,t+1 )
−γ
Ct + 1
−γ
Ct
#)
.
(4)
2 params to estimate: β and γ, so θ = ( β, γ) ′ .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Re-write moment conditions
(
"
0 = Et
1 − β (1 + ℜi,t+1 )
−γ
Ct + 1
−γ
Ct
#)
.
(4)
2 params to estimate: β and γ, so θ = ( β, γ) ′ .
xt∗ denotes info set of investors
0=E
nh
n
oi o
−γ
−γ
1 − β (1 + ℜi,t+1 ) Ct+1 /Ct
|xt∗
Sydney C. Ludvigson
i = 1, ...N
(5)
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Re-write moment conditions
(
"
0 = Et
1 − β (1 + ℜi,t+1 )
−γ
Ct + 1
−γ
Ct
#)
.
(4)
2 params to estimate: β and γ, so θ = ( β, γ) ′ .
xt∗ denotes info set of investors
0=E
nh
n
oi o
−γ
−γ
1 − β (1 + ℜi,t+1 ) Ct+1 /Ct
|xt∗
i = 1, ...N
(5)
∗
xt ⊂ xt . Conditional model (5) => unconditional model:
0=E
("
1−
(
β (1 + ℜi,t+1 )
−γ
Ct + 1
−γ
Ct
)# )
xt
i = 1, ...N
(6)
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Let xt be M × 1. Then r = N · M and,
 −γ
Ct+1
xt
 1 − β (1 + ℜ1,t+1 ) Ct−γ
 −
γ

 1 − β (1 + ℜ2,t+1 ) Ct−+γ1
xt

Ct

·
h (θ, wt ) = 

r×1

·


·
 −γ

Ct+1
1 − β (1 + ℜN,t+1 ) −γ
xt
Ct
Sydney C. Ludvigson














(7)
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Let xt be M × 1. Then r = N · M and,
 −γ
Ct+1
xt
 1 − β (1 + ℜ1,t+1 ) Ct−γ
 −
γ

 1 − β (1 + ℜ2,t+1 ) Ct−+γ1
xt

Ct

·
h (θ, wt ) = 

r×1

·


·
 −γ

Ct+1
1 − β (1 + ℜN,t+1 ) −γ
xt
Ct














(7)
Model can be estimated, tested as long as r ≥ 2.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Let xt be M × 1. Then r = N · M and,
 −γ
Ct+1
xt
 1 − β (1 + ℜ1,t+1 ) Ct−γ
 −
γ

 1 − β (1 + ℜ2,t+1 ) Ct−+γ1
xt

Ct

·
h (θ, wt ) = 

r×1

·


·
 −γ

Ct+1
1 − β (1 + ℜN,t+1 ) −γ
xt
Ct














(7)
Model can be estimated, tested as long as r ≥ 2.
Take sample mean of (7) to get g(θ; yT ), minimize
′
min Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )]
θ
| {z } r×r | {z }
1× r
Sydney C. Ludvigson
r×1
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
a
TQ b
θ; yT ∼ χ2 (r − a)
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
a
TQ b
θ; yT ∼ χ2 (r − a)
HS use lags of cons growth and returns in xt ; index and
industry returns, NDS expenditures.
Estimates of β ≈ .99, RRA low = .35 to .999. No equity
premium puzzle! But....
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
a
TQ b
θ; yT ∼ χ2 (r − a)
HS use lags of cons growth and returns in xt ; index and
industry returns, NDS expenditures.
Estimates of β ≈ .99, RRA low = .35 to .999. No equity
premium puzzle! But....
...model is strongly rejected according to OID test.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
a
TQ b
θ; yT ∼ χ2 (r − a)
HS use lags of cons growth and returns in xt ; index and
industry returns, NDS expenditures.
Estimates of β ≈ .99, RRA low = .35 to .999. No equity
premium puzzle! But....
...model is strongly rejected according to OID test.
Campbell, Lo, MacKinlay (1997): OID rejections stronger
whenever stock returns and commercial paper are
included as test returns. Why?
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
a
TQ b
θ; yT ∼ χ2 (r − a)
HS use lags of cons growth and returns in xt ; index and
industry returns, NDS expenditures.
Estimates of β ≈ .99, RRA low = .35 to .999. No equity
premium puzzle! But....
...model is strongly rejected according to OID test.
Campbell, Lo, MacKinlay (1997): OID rejections stronger
whenever stock returns and commercial paper are
included as test returns. Why?
Model cannot capture predictable variation in excess
returns over commercial paper ⇒
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
a
TQ b
θ; yT ∼ χ2 (r − a)
HS use lags of cons growth and returns in xt ; index and
industry returns, NDS expenditures.
Estimates of β ≈ .99, RRA low = .35 to .999. No equity
premium puzzle! But....
...model is strongly rejected according to OID test.
Campbell, Lo, MacKinlay (1997): OID rejections stronger
whenever stock returns and commercial paper are
included as test returns. Why?
Model cannot capture predictable variation in excess
returns over commercial paper ⇒
Researchers have turned to other models of preferences.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
More GMM results: Euler Equation Errors
Results in HS use conditioning info xt –scaled returns.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
More GMM results: Euler Equation Errors
Results in HS use conditioning info xt –scaled returns.
Another limitation with classic CCAPM: large
unconditional Euler equation (pricing) errors even when
params freely chosen.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
More GMM results: Euler Equation Errors
Results in HS use conditioning info xt –scaled returns.
Another limitation with classic CCAPM: large
unconditional Euler equation (pricing) errors even when
params freely chosen.
Let Mt+1 = β(Ct+1 /Ct )−γ . Define Euler equation errors:
j
j
eR ≡ E[Mt+1 Rt+1 ] − 1
j
j
f
eX ≡ E[Mt+1 (Rt+1 − Rt+1 )]
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
More GMM results: Euler Equation Errors
Results in HS use conditioning info xt –scaled returns.
Another limitation with classic CCAPM: large
unconditional Euler equation (pricing) errors even when
params freely chosen.
Let Mt+1 = β(Ct+1 /Ct )−γ . Define Euler equation errors:
j
j
eR ≡ E[Mt+1 Rt+1 ] − 1
j
j
f
eX ≡ E[Mt+1 (Rt+1 − Rt+1 )]
Choose params: min β,γ gT′ WT gT where jth element of gT
gj,t (γ, β) =
gj,t (γ) =
Sydney C. Ludvigson
1
T
1
T
j
∑Tt=1 eR,t
j
∑Tt=1 eX,t
Methods Lecture: GMM and Consumption-Based Models
Unconditional Euler Equation Errors, Excess Returns
j
j
f
eX ≡ E[ β(Ct+1 /Ct )−γ (Rt+1 − Rt+1 )]
RMSE =
q
1
N
j
2
∑N
j=1 [eX ] ,
RMSR =
q
j = 1, ..., N
1
N
j
f
2
∑N
j=1 [E(Rt+1 − Rt+1 )]
Source: Lettau and Ludvigson (2009). Rs is the excess return on CRSP-VW index over 3-Mo T-bill rate. Rs & 6 FF
refers to this return plus 6 size and book-market sorted portfolios provided by Fama and French. For each value of
γ, β is chosen to minimize the Euler equation error for the T-bill rate. U.S. quarterly data, 1954:1-2002:1.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
More GMM results: Euler Equation Errors
Magnitude of errors large, even when parameters are
freely chosen to minimize errors.
Unlike the equity premium puzzle of Mehra and Prescott
(1985), large Euler eq. errors cannot be resolved with high
risk aversion.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
More GMM results: Euler Equation Errors
Magnitude of errors large, even when parameters are
freely chosen to minimize errors.
Unlike the equity premium puzzle of Mehra and Prescott
(1985), large Euler eq. errors cannot be resolved with high
risk aversion.
Lettau and Ludvigson (2009): Leading consumption-based
asset pricing theories fail to explain the mispricing of
classic CCAPM.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
More GMM results: Euler Equation Errors
Magnitude of errors large, even when parameters are
freely chosen to minimize errors.
Unlike the equity premium puzzle of Mehra and Prescott
(1985), large Euler eq. errors cannot be resolved with high
risk aversion.
Lettau and Ludvigson (2009): Leading consumption-based
asset pricing theories fail to explain the mispricing of
classic CCAPM.
Anomaly is striking b/c early evidence (e.g., Hansen &
Singleton) that the classic model’s Euler equations were
violated provided the impetus for developing these newer
models.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
More GMM results: Euler Equation Errors
Magnitude of errors large, even when parameters are
freely chosen to minimize errors.
Unlike the equity premium puzzle of Mehra and Prescott
(1985), large Euler eq. errors cannot be resolved with high
risk aversion.
Lettau and Ludvigson (2009): Leading consumption-based
asset pricing theories fail to explain the mispricing of
classic CCAPM.
Anomaly is striking b/c early evidence (e.g., Hansen &
Singleton) that the classic model’s Euler equations were
violated provided the impetus for developing these newer
models.
Results imply data on consumption and asset returns not
jointly lognormal!
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT , S−1 . Why?
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT , S−1 . Why?
One reason: assessing specification error, comparing
models.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT , S−1 . Why?
One reason: assessing specification error, comparing
models.
Consider two estimated models of SDF, e.g.,
(1)
1
CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected
2
CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected
(2)
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT , S−1 . Why?
One reason: assessing specification error, comparing
models.
Consider two estimated models of SDF, e.g.,
(1)
1
CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected
2
CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected
(2)
May we conclude Model 1 is superior?
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT , S−1 . Why?
One reason: assessing specification error, comparing
models.
Consider two estimated models of SDF, e.g.,
(1)
1
CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected
2
CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected
(2)
May we conclude Model 1 is superior?
No. Hansen’s J-test of OID restricts depends on model
specific S: J = gT′ S−1 gT .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT , S−1 . Why?
One reason: assessing specification error, comparing
models.
Consider two estimated models of SDF, e.g.,
(1)
1
CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected
2
CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected
(2)
May we conclude Model 1 is superior?
No. Hansen’s J-test of OID restricts depends on model
specific S: J = gT′ S−1 gT .
Model 1 can look better simply b/c the SDF and pricing
errors gT are more volatile, not b/c pricing errors are lower.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
HJ: compare models Mt (θ) using distance metric:
DistT (θ) =
r
mingT (θ)′ GT−1 gT (θ),
θ
1
gT (θ) ≡
T
Sydney C. Ludvigson
T
GT ≡
∑ [Mt (θ)Rt − 1N ]
1
T
T
′
t Rt
∑ R|{z}
t= 1
N ×N
t= 1
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
HJ: compare models Mt (θ) using distance metric:
DistT (θ) =
r
mingT (θ)′ GT−1 gT (θ),
θ
1
gT (θ) ≡
T
T
GT ≡
∑ [Mt (θ)Rt − 1N ]
1
T
T
′
t Rt
∑ R|{z}
t= 1
N ×N
t= 1
DistT does not reward SDF volatility => suitable for
model comparison.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
HJ: compare models Mt (θ) using distance metric:
DistT (θ) =
r
mingT (θ)′ GT−1 gT (θ),
θ
1
gT (θ) ≡
T
GT ≡
T
∑ [Mt (θ)Rt − 1N ]
1
T
T
′
t Rt
∑ R|{z}
t= 1
N ×N
t= 1
DistT does not reward SDF volatility => suitable for
model comparison.
DistT is a measure of model misspecification:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
HJ: compare models Mt (θ) using distance metric:
DistT (θ) =
r
mingT (θ)′ GT−1 gT (θ),
θ
1
gT (θ) ≡
T
GT ≡
T
∑ [Mt (θ)Rt − 1N ]
1
T
T
′
t Rt
∑ R|{z}
t= 1
N ×N
t= 1
DistT does not reward SDF volatility => suitable for
model comparison.
DistT is a measure of model misspecification:
Gives distance between Mt (θ) and nearest point in space of
all SDFs that price assets correctly.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
HJ: compare models Mt (θ) using distance metric:
DistT (θ) =
r
mingT (θ)′ GT−1 gT (θ),
θ
1
gT (θ) ≡
T
GT ≡
T
∑ [Mt (θ)Rt − 1N ]
1
T
T
′
t Rt
∑ R|{z}
t= 1
N ×N
t= 1
DistT does not reward SDF volatility => suitable for
model comparison.
DistT is a measure of model misspecification:
Gives distance between Mt (θ) and nearest point in space of
all SDFs that price assets correctly.
Gives maximum pricing error of any portfolio formed from
the N assets.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Appeal of HJ Distance metric:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Appeal of HJ Distance metric:
Recognizes all models are misspecified.
Provides method for comparing models by assessing which
is least misspecified.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Appeal of HJ Distance metric:
Recognizes all models are misspecified.
Provides method for comparing models by assessing which
is least misspecified.
Important problem: how to compare HJ distances
statistically?
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Comparing specification error: Hansen and Jagannathan, 1997
Appeal of HJ Distance metric:
Recognizes all models are misspecified.
Provides method for comparing models by assessing which
is least misspecified.
Important problem: how to compare HJ distances
statistically?
One possibility developed in Chen and Ludvigson (2009):
White’s reality check method.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Chen and Ludvigson (2009) compare HJ distances among
K competing models using White’s reality check method.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Chen and Ludvigson (2009) compare HJ distances among
K competing models using White’s reality check method.
1
2
3
4
5
Take benchmark model, e.g., model with smallest squared
distance d21,T ≡ min{d2j,T }K
j =1 .
Null: d21,T − d22,T ≤ 0, where d22,T is competing model with
the next smallest squared distance.
√
Test statistic T W = T (d21,T − d22,T ).
If null is true, test statistic should not be unusually large,
given sampling error.
Given distribution for T W , reject null if historical value Tb W
is > 95th percentile.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Chen and Ludvigson (2009) compare HJ distances among
K competing models using White’s reality check method.
1
2
3
4
5
Take benchmark model, e.g., model with smallest squared
distance d21,T ≡ min{d2j,T }K
j =1 .
Null: d21,T − d22,T ≤ 0, where d22,T is competing model with
the next smallest squared distance.
√
Test statistic T W = T (d21,T − d22,T ).
If null is true, test statistic should not be unusually large,
given sampling error.
Given distribution for T W , reject null if historical value Tb W
is > 95th percentile.
Method applies generally to any stationary law of motion
for data, multiple competing possibly nonlinear, SDF
models.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Distribution of T W is computed via block bootstrap.
T W has complicated limiting distribution.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Distribution of T W is computed via block bootstrap.
T W has complicated limiting distribution.
Bootstrap works only if have a multivariate, joint,
continuous, limiting distribution under null.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Distribution of T W is computed via block bootstrap.
T W has complicated limiting distribution.
Bootstrap works only if have a multivariate, joint,
continuous, limiting distribution under null.
Proof of limiting distributions exists for applications to
most asset pricing models:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Distribution of T W is computed via block bootstrap.
T W has complicated limiting distribution.
Bootstrap works only if have a multivariate, joint,
continuous, limiting distribution under null.
Proof of limiting distributions exists for applications to
most asset pricing models:
For parametric models (Hansen, Heaton, Luttmer ’95)
For semiparametric models (Ai and Chen ’07).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: econometric problems
Econometric problems: near singular ST−1 or GT−1 .
Asset returns are highly correlated.
We have large N and modest T.
If T < N covariance matrix for N asset returns is singular.
Unless T >> N, matrix can be near-singular.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: economically interesting portfolios
Original test assets may have economically meaningful
characteristics (e.g., size, value).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: economically interesting portfolios
Original test assets may have economically meaningful
characteristics (e.g., size, value).
Using WT = ST−1 or GT−1 same as using WT = I and doing
GMM on re-weighted portfolios of original test assets.
Triangular factorization S−1 = (P′ P), P lower triangular
min gT′ S−1 gT ⇔ (gT′ P′ )I(PgT )
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: economically interesting portfolios
Original test assets may have economically meaningful
characteristics (e.g., size, value).
Using WT = ST−1 or GT−1 same as using WT = I and doing
GMM on re-weighted portfolios of original test assets.
Triangular factorization S−1 = (P′ P), P lower triangular
min gT′ S−1 gT ⇔ (gT′ P′ )I(PgT )
Re-weighted portfolios may not provide large spread in
average returns.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: economically interesting portfolios
Original test assets may have economically meaningful
characteristics (e.g., size, value).
Using WT = ST−1 or GT−1 same as using WT = I and doing
GMM on re-weighted portfolios of original test assets.
Triangular factorization S−1 = (P′ P), P lower triangular
min gT′ S−1 gT ⇔ (gT′ P′ )I(PgT )
Re-weighted portfolios may not provide large spread in
average returns.
May imply implausible long and short positions in test
assets.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Reasons not to use WT = I: objective function dependence on test asset choice
Using WT = [ET (R′ R)]−1 , GMM objective function is
invariant to initial choice of test assets.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Reasons not to use WT = I: objective function dependence on test asset choice
Using WT = [ET (R′ R)]−1 , GMM objective function is
invariant to initial choice of test assets.
Form a portfolio, AR from initial returns R. (Note,
portfolio weights sum to 1 so A1N = 1N ).
[E (MR) − 1N ]′ E RR′
−1
[E (MR − 1N )]
−1
= [E (MAR) − A1N ] E ARR′ A
[E (MAR − A1N )] .
′
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
GMM Asset Pricing With Non-Optimal Weighting
Reasons not to use WT = I: objective function dependence on test asset choice
Using WT = [ET (R′ R)]−1 , GMM objective function is
invariant to initial choice of test assets.
Form a portfolio, AR from initial returns R. (Note,
portfolio weights sum to 1 so A1N = 1N ).
[E (MR) − 1N ]′ E RR′
−1
[E (MR − 1N )]
−1
= [E (MAR) − A1N ] E ARR′ A
[E (MAR − A1N )] .
′
With WT = I or other fixed weighting, GMM objective
depends on choice of test assets.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
More Complex Preferences: Scaled
Consumption-Based Models
Consumption-based models may be approximated:
Mt+1 ≈ at + bt ∆ct+1 ,
Sydney C. Ludvigson
ct+1 ≡ ln(Ct+1 )
Methods Lecture: GMM and Consumption-Based Models
More Complex Preferences: Scaled
Consumption-Based Models
Consumption-based models may be approximated:
Mt+1 ≈ at + bt ∆ct+1 ,
ct+1 ≡ ln(Ct+1 )
Example: Classic CCAPM with CRRA utility
u ( Ct ) =
1− γ
Ct
⇒ Mt+1 ≈ β − βγ ∆ct+1
1−γ
|{z} |{z}
at = a0
Sydney C. Ludvigson
bt = b0
Methods Lecture: GMM and Consumption-Based Models
More Complex Preferences: Scaled
Consumption-Based Models
Consumption-based models may be approximated:
Mt+1 ≈ at + bt ∆ct+1 ,
ct+1 ≡ ln(Ct+1 )
Example: Classic CCAPM with CRRA utility
u ( Ct ) =
1− γ
Ct
⇒ Mt+1 ≈ β − βγ ∆ct+1
1−γ
|{z} |{z}
at = a0
bt = b0
Model with habit and time-varying risk aversion: Campbell and
Cochrane ’99, Menzly et. al ’04
u ( Ct , S t ) =
( Ct S t ) 1 − γ
,
1−γ
St + 1 ≡
C t − Xt
Ct
⇒ Mt+1 ≈ β 1 − γ (φ − 1)(st − s) − γ (1 + ψ (st )) ∆ct+1
|
{z
} |
{z
}
at
Sydney C. Ludvigson
bt
Methods Lecture: GMM and Consumption-Based Models
More Complex Preferences: Scaled
Consumption-Based Models
Consumption-based models may be approximated:
Mt+1 ≈ at + bt ∆ct+1 ,
ct+1 ≡ ln(Ct+1 )
Example: Classic CCAPM with CRRA utility
u ( Ct ) =
1− γ
Ct
⇒ Mt+1 ≈ β − βγ ∆ct+1
1−γ
|{z} |{z}
at = a0
bt = b0
Model with habit and time-varying risk aversion: Campbell and
Cochrane ’99, Menzly et. al ’04
u ( Ct , S t ) =
( Ct S t ) 1 − γ
,
1−γ
St + 1 ≡
C t − Xt
Ct
⇒ Mt+1 ≈ β 1 − γ (φ − 1)(st − s) − γ (1 + ψ (st )) ∆ct+1
|
{z
} |
{z
}
at
bt
Proxies for time-varying risk-premia should be good proxies for
time-variation in at and bt .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Mt+1 ≈ at + bt ∆ct+1
Empirical specification: Lettau and Ludvigson (2001a, 2001b):
at = a0 + a1 zt , bt = b0 + b1 zt
zt = cayt ≡ ct − αa at − αy yt , (cointegrating residual)
cayt related to log consumption-(aggregate) wealth ratio.
cayt strong predictor of excess stock market returns
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Mt+1 ≈ at + bt ∆ct+1
Empirical specification: Lettau and Ludvigson (2001a, 2001b):
at = a0 + a1 zt , bt = b0 + b1 zt
zt = cayt ≡ ct − αa at − αy yt , (cointegrating residual)
cayt related to log consumption-(aggregate) wealth ratio.
cayt strong predictor of excess stock market returns
Other examples: including housing consumption
1
U(Ct , Ht ) =
e 1− σ
C
t
1−
1
σ
ε −1
ε
ε −1
ε −1
ε
ε
e
Ct = χCt + (1 − χ) Ht
,
⇒ ln Mt+1 ≈ at + bt ∆ ln Ct+1 + dt ∆ ln St+1 ,
St + 1 ≡
pC
t Ct
pC
t Ct
+ pH
t Ht
Lustig and Van Nieuwerburgh ’05 (incomplete markets):
at = a0 + a1 zt , bt = b0 + b1 zt , dt = d0 + d1 zt
zt = housing collateral ratio (measures quantity of risk
sharing)
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Euler equation: E Mt+1 Rt+1 |zt = 1
Unconditional version: E Mt+1 Rt+1 = 1
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Euler equation: E Mt+1 Rt+1 |zt = 1
Unconditional version: E Mt+1 Rt+1 = 1
Two forms of conditionality:
1
2
scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1
scaling factors ft+1 , e.g., ft+1 = ∆ ln Ct+1 :
Mt+1
Sydney C. Ludvigson
= bt′ ft+1 with bt = b0 + b1 zt
= b′ ft+1 ⊗ (1 zt )′
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Euler equation: E Mt+1 Rt+1 |zt = 1
Unconditional version: E Mt+1 Rt+1 = 1
Two forms of conditionality:
1
2
scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1
scaling factors ft+1 , e.g., ft+1 = ∆ ln Ct+1 :
Mt+1
= bt′ ft+1 with bt = b0 + b1 zt
= b′ ft+1 ⊗ (1 zt )′
Scaled consumption-based models are conditional in sense
that Mt+1 is a state-dependent function of ∆ ln Ct+1
⇒ scaled factors
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Euler equation: E Mt+1 Rt+1 |zt = 1
Unconditional version: E Mt+1 Rt+1 = 1
Two forms of conditionality:
1
2
scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1
scaling factors ft+1 , e.g., ft+1 = ∆ ln Ct+1 :
Mt+1
= bt′ ft+1 with bt = b0 + b1 zt
= b′ ft+1 ⊗ (1 zt )′
Scaled consumption-based models are conditional in sense
that Mt+1 is a state-dependent function of ∆ ln Ct+1
⇒ scaled factors
Scaled consumption-basedmodels have
been tested on
unconditional moments, E Mt+1 Rt+1 = 1
⇒ NO scaled returns.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled CCAPM turns a single factor model with
state-dependent weights into multi-factor model ft with
constant weights:
Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1
= a0 + a1 zt +b0 ∆ ln Ct+1 +b1 (zt ∆ ln Ct+1 )
|{z}
| {z }
| {z }
f1,t+1
Sydney C. Ludvigson
f2,t+1
f3,t+1
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled CCAPM turns a single factor model with
state-dependent weights into multi-factor model ft with
constant weights:
Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1
= a0 + a1 zt +b0 ∆ ln Ct+1 +b1 (zt ∆ ln Ct+1 )
|{z}
| {z }
| {z }
f1,t+1
f2,t+1
f3,t+1
Multiple risk factors ft′ ≡ (zt , ∆ ln Ct+1 , zt ∆ ln Ct+1 ).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled CCAPM turns a single factor model with
state-dependent weights into multi-factor model ft with
constant weights:
Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1
= a0 + a1 zt +b0 ∆ ln Ct+1 +b1 (zt ∆ ln Ct+1 )
|{z}
| {z }
| {z }
f1,t+1
f2,t+1
f3,t+1
Multiple risk factors ft′ ≡ (zt , ∆ ln Ct+1 , zt ∆ ln Ct+1 ).
Scaled consumption models have multiple, constant betas
for each factor, rather than a single time-varying beta for
∆ ln Ct+1 .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Deriving the “beta”-representation
Let F = (1 f′ )′ , M = b′ F, ignore time indices
= E[MRi ]
1
= E[Ri F′ ]b ⇒ unconditional moments
= E[Ri ]E[F′ ]b + Cov(Ri , F′ )b ⇒
E[Ri ]
=
1 − Cov(Ri , F′ )b
E [F′ ]b
=
1 − Cov(Ri , f′ )b
E [F′ ]b
=
1 − Cov(Ri , f′ )Cov(f, f′ )−1 Cov(f, f′ )b
E [F′ ]b
= R0 − R0 β′ Cov(f, f′ )b
= R0 − β′ λ ⇒ multiple, constant betas
Estimate cross-sectional model using Fama-MacBeth (see Brandt
lecture).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Fama-MacBeth Methodology: Preview–See Brandt
Step 1: Estimate β’s in time-series regression for each
portfolio i:
βi ≡ Cov(ft+1 , ft′ +1 )−1 Cov(ft+1 , Ri,t+1 )
Step 2: Cross-sectional regressions (T of them):
Ri,t+1 − R0,t = αi,t + βi′ λt
T
λ = 1/T ∑ λt ;
t= 1
T
σ2 (λ) = 1/T ∑ (λt − λ)2
t= 1
Note: report Shanken t-statistics (corrected for estimation error
of betas in first stage)
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled models: conditioning done in SDF:
Mt+1 = at + bt ∆ ln Ct+1 ,
not in Euler equation: E(MR) = 1N .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled models: conditioning done in SDF:
Mt+1 = at + bt ∆ ln Ct+1 ,
not in Euler equation: E(MR) = 1N .
Gives rise to a restricted conditional consumption beta
model:
Rit = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled models: conditioning done in SDF:
Mt+1 = at + bt ∆ ln Ct+1 ,
not in Euler equation: E(MR) = 1N .
Gives rise to a restricted conditional consumption beta
model:
Rit = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1
Rewrite as
Rit = a + ( βc + βc,z zt−1 ) ∆ct + βz zt−1
|
{z
}
time-varying beta
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled models: conditioning done in SDF:
Mt+1 = at + bt ∆ ln Ct+1 ,
not in Euler equation: E(MR) = 1N .
Gives rise to a restricted conditional consumption beta
model:
Rit = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1
Rewrite as
Rit = a + ( βc + βc,z zt−1 ) ∆ct + βz zt−1
|
{z
}
time-varying beta
Unlikely the same time-varying beta as obtained from
modeling conditional mean Et (Mt+1 Rt+1 ) = 1.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ).
Latter may require variables beyond a few that capture
risk-premia.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ).
Latter may require variables beyond a few that capture
risk-premia.
Approximating condition mean well requires large
number of instruments (misspecified information sets)
Results sensitive to chosen conditioning variables, may fail
to span information sets of market participants.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ).
Latter may require variables beyond a few that capture
risk-premia.
Approximating condition mean well requires large
number of instruments (misspecified information sets)
Results sensitive to chosen conditioning variables, may fail
to span information sets of market participants.
Partial solution: summarize information in large number
of time-series with few estimated dynamic factors (e.g.,
Ludvigson and Ng ’07, ’09).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Bottom lines:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Bottom lines:
1
Conditional moments of Mt+1 Rt+1 difficult to model ⇒
reason to focus on unconditional moments
E[Mt+1 Rt+1 ] = 1.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Bottom lines:
1
Conditional moments of Mt+1 Rt+1 difficult to model ⇒
reason to focus on unconditional moments
E[Mt+1 Rt+1 ] = 1.
2
Models are misspecified: interesting question is whether
state-dependence of Mt+1 on consumption growth ⇒ less
misspecification than standard, fixed-weight CCAPM.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Bottom lines:
1
Conditional moments of Mt+1 Rt+1 difficult to model ⇒
reason to focus on unconditional moments
E[Mt+1 Rt+1 ] = 1.
2
Models are misspecified: interesting question is whether
state-dependence of Mt+1 on consumption growth ⇒ less
misspecification than standard, fixed-weight CCAPM.
3
As before, can compare models on basis of HJ distances,
using White ”reality check” method to compare statistically.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Two reasons recursive utility is of interest:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Two reasons recursive utility is of interest:
More flexibility as regards attitudes toward risk and
intertemporal substitution.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Two reasons recursive utility is of interest:
More flexibility as regards attitudes toward risk and
intertemporal substitution.
Preferences deliver an added risk factor for explaining asset
returns.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Two reasons recursive utility is of interest:
More flexibility as regards attitudes toward risk and
intertemporal substitution.
Preferences deliver an added risk factor for explaining asset
returns.
But, only a small amount of econometric work on
recursive preferences ⇒ gap in the literature.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
1
For general stationary, consumption growth and cash flow
dynamics: Chen, Favilukis, Ludvigson ’07.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
1
For general stationary, consumption growth and cash flow
dynamics: Chen, Favilukis, Ludvigson ’07.
2
When restricting cash flow dynamics (e.g., “long-run risk”):
Bansal, Gallant, Tauchen ’07.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
1
For general stationary, consumption growth and cash flow
dynamics: Chen, Favilukis, Ludvigson ’07.
2
When restricting cash flow dynamics (e.g., “long-run risk”):
Bansal, Gallant, Tauchen ’07.
In (1) DGP is left unrestricted, as is joint distribution of
consumption and returns (distribution-free estimation
procedure).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
1
For general stationary, consumption growth and cash flow
dynamics: Chen, Favilukis, Ludvigson ’07.
2
When restricting cash flow dynamics (e.g., “long-run risk”):
Bansal, Gallant, Tauchen ’07.
In (1) DGP is left unrestricted, as is joint distribution of
consumption and returns (distribution-free estimation
procedure).
In (2) DGP and distribution of shocks explicitly modeled.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstein-Zin-Weil basics
Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)):
h
i 1−1 ρ
1− ρ
Vt = ( 1 − β ) Ct + β R t ( Vt + 1 ) 1 − ρ
h
i 1−1 θ
Rt (Vt+1 ) = E Vt1+−1θ |Ft
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstein-Zin-Weil basics
Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)):
h
i 1−1 ρ
1− ρ
Vt = ( 1 − β ) Ct + β R t ( Vt + 1 ) 1 − ρ
h
i 1−1 θ
Rt (Vt+1 ) = E Vt1+−1θ |Ft
Vt+1 is continuation value, θ is RRA, 1/ρ is EIS.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstein-Zin-Weil basics
Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)):
h
i 1−1 ρ
1− ρ
Vt = ( 1 − β ) Ct + β R t ( Vt + 1 ) 1 − ρ
h
i 1−1 θ
Rt (Vt+1 ) = E Vt1+−1θ |Ft
Vt+1 is continuation value, θ is RRA, 1/ρ is EIS.
Rescale utility function (Hansen, Heaton, Li ’05):
"
# 1−1 ρ
Vt
Vt + 1 Ct + 1 1 − ρ
= ( 1 − β ) + β Rt
Ct
Ct + 1 Ct
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstein-Zin-Weil basics
Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)):
h
i 1−1 ρ
1− ρ
Vt = ( 1 − β ) Ct + β R t ( Vt + 1 ) 1 − ρ
h
i 1−1 θ
Rt (Vt+1 ) = E Vt1+−1θ |Ft
Vt+1 is continuation value, θ is RRA, 1/ρ is EIS.
Rescale utility function (Hansen, Heaton, Li ’05):
"
# 1−1 ρ
Vt
Vt + 1 Ct + 1 1 − ρ
= ( 1 − β ) + β Rt
Ct
Ct + 1 Ct
C1−θ
t
Special case: ρ = θ ⇒ CRRA separable utility Vt = β 1−
θ.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
The MRS is pricing kernel (SDF) with added risk factor:
 ρ−θ

Vt+1 Ct+1
Ct + 1 − ρ 
C
C
t+1 t 
Mt + 1 = β
Vt+1 Ct+1
Ct
R
t
Sydney C. Ludvigson
Ct+1 Ct
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
The MRS is pricing kernel (SDF) with added risk factor:
 ρ−θ

Vt+1 Ct+1
Ct + 1 − ρ 
C
C
t+1 t 
Mt + 1 = β
Vt+1 Ct+1
Ct
R
t
Ct+1 Ct
Difficulty: MRS a function of V/C, unobservable, embeds
Rt (·).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
The MRS is pricing kernel (SDF) with added risk factor:
 ρ−θ

Vt+1 Ct+1
Ct + 1 − ρ 
C
C
t+1 t 
Mt + 1 = β
Vt+1 Ct+1
Ct
R
t
Ct+1 Ct
Difficulty: MRS a function of V/C, unobservable, embeds
Rt (·).
Epstein-Zin ’91 use alt. rep. of SDF, uses agg. wealth
return Rw,t :
( ) 11−−ρθ 1θ −−ρρ
1
Ct + 1 − ρ
Mt + 1 = β
Ct
Rw,t+1
where Rw,t proxied by stock market return.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
The MRS is pricing kernel (SDF) with added risk factor:
 ρ−θ

Vt+1 Ct+1
Ct + 1 − ρ 
C
C
t+1 t 
Mt + 1 = β
Vt+1 Ct+1
Ct
R
t
Ct+1 Ct
Difficulty: MRS a function of V/C, unobservable, embeds
Rt (·).
Epstein-Zin ’91 use alt. rep. of SDF, uses agg. wealth
return Rw,t :
( ) 11−−ρθ 1θ −−ρρ
1
Ct + 1 − ρ
Mt + 1 = β
Ct
Rw,t+1
where Rw,t proxied by stock market return.
Problem: Rw,t+1 represents a claim to future Ct , itself
unobservable.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
1
If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
1
If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2
If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
1
If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2
If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
But....
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
1
If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2
If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
But....
EIS=1 ⇒ consumption-wealth ratio is constant,
contradicting statistical evidence.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
1
If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2
If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
But....
EIS=1 ⇒ consumption-wealth ratio is constant,
contradicting statistical evidence.
Joint lognormality strongly rejected in quarterly data.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Epstien-Zin-Weil basics
1
If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2
If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
But....
EIS=1 ⇒ consumption-wealth ratio is constant,
contradicting statistical evidence.
Joint lognormality strongly rejected in quarterly data.
Points to need for estimation method feasible under less
restrictive assumptions.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
CFL: semiparametric approach to estimate EZW model
without:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
CFL: semiparametric approach to estimate EZW model
without:
Need to proxy Rw,t+1 with observable returns.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
CFL: semiparametric approach to estimate EZW model
without:
Need to proxy Rw,t+1 with observable returns.
Loglinearizing the model.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
CFL: semiparametric approach to estimate EZW model
without:
Need to proxy Rw,t+1 with observable returns.
Loglinearizing the model.
Parametric restrictions on law of motion or joint dist. of Ct
and Ri,t , or on value of key preference parameters.
Obtain estimates of β, RRA θ, EIS ρ−1
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
CFL: semiparametric approach to estimate EZW model
without:
Need to proxy Rw,t+1 with observable returns.
Loglinearizing the model.
Parametric restrictions on law of motion or joint dist. of Ct
and Ri,t , or on value of key preference parameters.
Obtain estimates of β, RRA θ, EIS ρ−1
Evaluate EZW model’s ability to fit asset return data
relative to competing model specifications.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
CFL: semiparametric approach to estimate EZW model
without:
Need to proxy Rw,t+1 with observable returns.
Loglinearizing the model.
Parametric restrictions on law of motion or joint dist. of Ct
and Ri,t , or on value of key preference parameters.
Obtain estimates of β, RRA θ, EIS ρ−1
Evaluate EZW model’s ability to fit asset return data
relative to competing model specifications.
Investigate implications for Rw,t+1 and return to human
wealth.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
CFL: semiparametric approach to estimate EZW model
without:
Need to proxy Rw,t+1 with observable returns.
Loglinearizing the model.
Parametric restrictions on law of motion or joint dist. of Ct
and Ri,t , or on value of key preference parameters.
Obtain estimates of β, RRA θ, EIS ρ−1
Evaluate EZW model’s ability to fit asset return data
relative to competing model specifications.
Investigate implications for Rw,t+1 and return to human
wealth.
Semiparametric approach is sieve minimum distance
(SMD) procedure.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
First order conditions for optimal consumption choice:



ρ−θ
−ρ
Vt + 1 C t + 1
Ct + 1
Ct + 1 Ct




Et  β
Ri,t+1 − 1 = 0
Vt + 1 C t + 1
Ct
R t Ct + 1 Ct
Sydney C. Ludvigson
(8)
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
First order conditions for optimal consumption choice:



ρ−θ
−ρ
Vt + 1 C t + 1
Ct + 1
Ct + 1 Ct




Et  β
Ri,t+1 − 1 = 0
Vt + 1 C t + 1
Ct
R t Ct + 1 Ct
CFL: plug


Et 
β
Ct+1
Ct
Vt
Ct
(8)
1−ρ 1−1 ρ
t + 1 Ct + 1
= ( 1 − β ) + β Rt V
into (8):
Ct + 1 Ct
− ρ



n h
1
β
ρ−θ
Vt+1 Ct+1
Ct+1 Ct
Vt 1 − ρ
Ct
Sydney C. Ludvigson
− (1 − β )


io 1−1 ρ 


Ri,t+1 − 1
=0
i = 1, ..., N.
(9)
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
First order conditions for optimal consumption choice:



ρ−θ
−ρ
Vt + 1 C t + 1
Ct + 1
Ct + 1 Ct




Et  β
Ri,t+1 − 1 = 0
Vt + 1 C t + 1
Ct
R t Ct + 1 Ct
CFL: plug


Et 
β
Ct+1
Ct
Vt
Ct
(8)
1−ρ 1−1 ρ
t + 1 Ct + 1
= ( 1 − β ) + β Rt V
into (8):
Ct + 1 Ct
− ρ



n h
1
β
ρ−θ
Vt+1 Ct+1
Ct+1 Ct
Vt 1 − ρ
Ct
− (1 − β )


io 1−1 ρ 


Ri,t+1 − 1
=0
i = 1, ..., N.
(9)
N test asset returns, {Ri,t+1 }N
i=1 . (9) is a x-sect asset pricing model.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
First order conditions for optimal consumption choice:



ρ−θ
−ρ
Vt + 1 C t + 1
Ct + 1
Ct + 1 Ct




Et  β
Ri,t+1 − 1 = 0
Vt + 1 C t + 1
Ct
R t Ct + 1 Ct
CFL: plug


Et 
β
Ct+1
Ct
Vt
Ct
(8)
1−ρ 1−1 ρ
t + 1 Ct + 1
= ( 1 − β ) + β Rt V
into (8):
Ct + 1 Ct
− ρ



n h
1
β
ρ−θ
Vt+1 Ct+1
Ct+1 Ct
Vt 1 − ρ
Ct
− (1 − β )


io 1−1 ρ 


Ri,t+1 − 1
=0
i = 1, ..., N.
(9)
N test asset returns, {Ri,t+1 }N
i=1 . (9) is a x-sect asset pricing model.
Moment restrictions (9) form the basis of empirical investigation.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
First order conditions for optimal consumption choice:



ρ−θ
−ρ
Vt + 1 C t + 1
Ct + 1
Ct + 1 Ct




Et  β
Ri,t+1 − 1 = 0
Vt + 1 C t + 1
Ct
R t Ct + 1 Ct
CFL: plug


Et 
β
Ct+1
Ct
Vt
Ct
(8)
1−ρ 1−1 ρ
t + 1 Ct + 1
= ( 1 − β ) + β Rt V
into (8):
Ct + 1 Ct
− ρ



n h
1
β
ρ−θ
Vt+1 Ct+1
Ct+1 Ct
Vt 1 − ρ
Ct
− (1 − β )


io 1−1 ρ 


Ri,t+1 − 1
=0
i = 1, ..., N.
(9)
N test asset returns, {Ri,t+1 }N
i=1 . (9) is a x-sect asset pricing model.
Moment restrictions (9) form the basis of empirical investigation.
Empirical model is semiparametric: δ ≡ ( β, θ, ρ)′ denote finite
dimensional parameter vector; Vt /Ct unknown function.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Assume
Vt
Ct
an unknown function F: R2 → R of form
Vt
Vt − 1 Ct
=F
,
,
Ct
Ct − 1 Ct − 1
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Assume
Vt
Ct
an unknown function F: R2 → R of form
Vt
Vt − 1 Ct
=F
,
,
Ct
Ct − 1 Ct − 1
Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic;
and F(·) is such that the process{Vt /Ct : t = 1, ...} is
asymptotically stationary ergodic.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Assume
Vt
Ct
an unknown function F: R2 → R of form
Vt
Vt − 1 Ct
=F
,
,
Ct
Ct − 1 Ct − 1
Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic;
and F(·) is such that the process{Vt /Ct : t = 1, ...} is
asymptotically stationary ergodic.
Justified if, for example,
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Assume
Vt
Ct
an unknown function F: R2 → R of form
Vt
Vt − 1 Ct
=F
,
,
Ct
Ct − 1 Ct − 1
Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic;
and F(·) is such that the process{Vt /Ct : t = 1, ...} is
asymptotically stationary ergodic.
Justified if, for example,
∆ log(Ct+1 ) is (possibly nonlinear) function of a hidden
first-order Markov process xt .
Under general assumptions, information in xt is
summarized by Vt−1 /Ct−1 and Ct /Ct−1 .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Assume
Vt
Ct
an unknown function F: R2 → R of form
Vt
Vt − 1 Ct
=F
,
,
Ct
Ct − 1 Ct − 1
Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic;
and F(·) is such that the process{Vt /Ct : t = 1, ...} is
asymptotically stationary ergodic.
Justified if, for example,
∆ log(Ct+1 ) is (possibly nonlinear) function of a hidden
first-order Markov process xt .
Under general assumptions, information in xt is
summarized by Vt−1 /Ct−1 and Ct /Ct−1 .
With a nonlinear Markov process for xt , F(·) can display
nonmonotonicities in both arguments.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Assume
Vt
Ct
an unknown function F: R2 → R of form
Vt
Vt − 1 Ct
=F
,
,
Ct
Ct − 1 Ct − 1
Assume {Ct /Ct−1 : t = 1, ...} is strictly stationary ergodic;
and F(·) is such that the process{Vt /Ct : t = 1, ...} is
asymptotically stationary ergodic.
Justified if, for example,
∆ log(Ct+1 ) is (possibly nonlinear) function of a hidden
first-order Markov process xt .
Under general assumptions, information in xt is
summarized by Vt−1 /Ct−1 and Ct /Ct−1 .
Note: Markov assumption only a motivation for arguments
of F(·). Econometric methodology itself leaves LOM for
∆ ln Ct unspecified.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Ft denotes agents information set at time t.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Ft denotes agents information set at time t.
zt+1 contains all observations at t + 1 and
γi (zt+1 , δ, F) ≡ β

Ct+1
− ρ 
t Ct+ 1
F V
Ct , Ct+1
Ct
Ct+1

 1
n o1−ρ
Ct

1− ρ
Vt − 1
Ct
1
F
,
−
1
−
β
)
(
β
C
C
t−1
Sydney C. Ludvigson
t−1
ρ−θ




Ri,t+1 − 1
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Ft denotes agents information set at time t.
zt+1 contains all observations at t + 1 and
γi (zt+1 , δ, F) ≡ β

Ct+1
− ρ 
t Ct+ 1
F V
Ct , Ct+1
Ct
Ct+1

 1
n o1−ρ
Ct

1− ρ
Vt − 1
Ct
1
F
,
−
1
−
β
)
(
β
C
C
t−1
t−1
ρ−θ




Ri,t+1 − 1
δo ≡ ( βo , θo , ρo )′ , Fo ≡ Fo (zt , δo ) denote true parameters that
uniquely solve the conditional moment restrictions (Euler
equations):
E {γi (zt+1 , δo , Fo (·, δo ))|Ft } = 0
Sydney C. Ludvigson
i = 1, ..., N, (10)
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Ft denotes agents information set at time t.
zt+1 contains all observations at t + 1 and
γi (zt+1 , δ, F) ≡ β

Ct+1
− ρ 
t Ct+ 1
F V
Ct , Ct+1
Ct
Ct+1

 1
n o1−ρ
Ct

1− ρ
Vt − 1
Ct
1
F
,
−
1
−
β
)
(
β
C
C
t−1
t−1
ρ−θ




Ri,t+1 − 1
δo ≡ ( βo , θo , ρo )′ , Fo ≡ Fo (zt , δo ) denote true parameters that
uniquely solve the conditional moment restrictions (Euler
equations):
E {γi (zt+1 , δo , Fo (·, δo ))|Ft } = 0
i = 1, ..., N, (10)
Let wt ⊆ Ft . Equation (10) ⇒
E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0.
Sydney C. Ludvigson
i = 1, ..., N.
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Intuition behind minimum distance procedure:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Intuition behind minimum distance procedure:
Theory ⇒
mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0.
Sydney C. Ludvigson
i = 1, ..., N.
(11)
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Intuition behind minimum distance procedure:
Theory ⇒
mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0.
i = 1, ..., N.
(11)
Since mt = 0, mt must have zero variance, mean.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Intuition behind minimum distance procedure:
Theory ⇒
mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0.
i = 1, ..., N.
(11)
Since mt = 0, mt must have zero variance, mean.
Thus can find params by minimizing variance or quadratic
b t.
norm: min E[(mt )2 ]. Don’t observe mt ⇒ need estimate m
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Intuition behind minimum distance procedure:
Theory ⇒
mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0.
i = 1, ..., N.
(11)
Since mt = 0, mt must have zero variance, mean.
Thus can find params by minimizing variance or quadratic
b t.
norm: min E[(mt )2 ]. Don’t observe mt ⇒ need estimate m
Since (11) is cond. mean, must hold for each observation, t.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Intuition behind minimum distance procedure:
Theory ⇒
mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0.
i = 1, ..., N.
(11)
Since mt = 0, mt must have zero variance, mean.
Thus can find params by minimizing variance or quadratic
b t.
norm: min E[(mt )2 ]. Don’t observe mt ⇒ need estimate m
Since (11) is cond. mean, must hold for each observation, t.
Obs > params, need way to weight each obs; using sample
mean is one way: min ET [(mt )2 ].
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Intuition behind minimum distance procedure:
Theory ⇒
mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0.
i = 1, ..., N.
(11)
Since mt = 0, mt must have zero variance, mean.
Thus can find params by minimizing variance or quadratic
b t.
norm: min E[(mt )2 ]. Don’t observe mt ⇒ need estimate m
Since (11) is cond. mean, must hold for each observation, t.
Obs > params, need way to weight each obs; using sample
mean is one way: min ET [(mt )2 ].
Minimum distance procedure useful for distribution-free
estimation involving conditional moments.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Minimum distance procedure useful for distribution-free
estimation involving conditional moments: min ET [(mt )2 ].
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Minimum distance procedure useful for distribution-free
estimation involving conditional moments: min ET [(mt )2 ].
Contrast with GMM, used for unconditional moments:
E[f (xt , α)] = 0.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Minimum distance procedure useful for distribution-free
estimation involving conditional moments: min ET [(mt )2 ].
Contrast with GMM, used for unconditional moments:
E[f (xt , α)] = 0.
With GMM take sample counterpart to population mean:
gT = ∑Tt=1 f (xt , α) = 0.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Minimum distance procedure useful for distribution-free
estimation involving conditional moments: min ET [(mt )2 ].
Contrast with GMM, used for unconditional moments:
E[f (xt , α)] = 0.
With GMM take sample counterpart to population mean:
gT = ∑Tt=1 f (xt , α) = 0.
Then choose parameters α to min gT′ WgT .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Minimum distance procedure useful for distribution-free
estimation involving conditional moments: min ET [(mt )2 ].
Contrast with GMM, used for unconditional moments:
E[f (xt , α)] = 0.
With GMM take sample counterpart to population mean:
gT = ∑Tt=1 f (xt , α) = 0.
Then choose parameters α to min gT′ WgT .
With GMM we average and then square.
With SMD, we square and then average.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
True parameters δo and Fo (·, δo ) solve:
min inf E m(wt , δ, F)′ m(wt , δ, F) ,
δ∈D F∈V
where m(wt , δ, F) = E{γ(zt+1 , δ, F)|wt }
γ(zt+1 , δ, F) = (γ1 (zt+1 , δ, F), ..., γN (zt+1 , δ, F))
Sydney C. Ludvigson
′
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
True parameters δo and Fo (·, δo ) solve:
min inf E m(wt , δ, F)′ m(wt , δ, F) ,
δ∈D F∈V
where m(wt , δ, F) = E{γ(zt+1 , δ, F)|wt }
γ(zt+1 , δ, F) = (γ1 (zt+1 , δ, F), ..., γN (zt+1 , δ, F))
′
For any candidate δ ≡ ( β, θ, ρ) ′ ∈ D , define
V ∗ ≡ F∗ (zt , δ) ≡ F∗ (·, δ) as:
F∗ (·, δ) = arg infE m(wt , δ, F)′ m(wt , δ, F)
F∈V
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
True parameters δo and Fo (·, δo ) solve:
min inf E m(wt , δ, F)′ m(wt , δ, F) ,
δ∈D F∈V
where m(wt , δ, F) = E{γ(zt+1 , δ, F)|wt }
γ(zt+1 , δ, F) = (γ1 (zt+1 , δ, F), ..., γN (zt+1 , δ, F))
′
For any candidate δ ≡ ( β, θ, ρ) ′ ∈ D , define
V ∗ ≡ F∗ (zt , δ) ≡ F∗ (·, δ) as:
F∗ (·, δ) = arg infE m(wt , δ, F)′ m(wt , δ, F)
F∈V
It is clear that Fo (zt , δo ) = F∗ (zt , δo )
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two-Step Procedure
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
First step: For any candidate δ ∈ D , an initial estimate of
F∗ (·, δ) obtained using SMD that consists of two parts:
(Newey-Powell ’03, Ai-Chen ’03, Ai-Chen ’07).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two-Step Procedure
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
First step: For any candidate δ ∈ D , an initial estimate of
F∗ (·, δ) obtained using SMD that consists of two parts:
(Newey-Powell ’03, Ai-Chen ’03, Ai-Chen ’07).
1
Replace the conditional expectation with a consistent,
nonparametric estimator (specified later).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two-Step Procedure
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
First step: For any candidate δ ∈ D , an initial estimate of
F∗ (·, δ) obtained using SMD that consists of two parts:
(Newey-Powell ’03, Ai-Chen ’03, Ai-Chen ’07).
1
Replace the conditional expectation with a consistent,
nonparametric estimator (specified later).
2
Approximate the unknown function F by a sequence of
finite dimensional unknown parameters (sieves) FKT .
Approximation error decreases as KT increases with T.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two-Step Procedure
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
First step: For any candidate δ ∈ D , an initial estimate of
F∗ (·, δ) obtained using SMD that consists of two parts:
(Newey-Powell ’03, Ai-Chen ’03, Ai-Chen ’07).
1
Replace the conditional expectation with a consistent,
nonparametric estimator (specified later).
2
Approximate the unknown function F by a sequence of
finite dimensional unknown parameters (sieves) FKT .
Approximation error decreases as KT increases with T.
Second step: estimates of δo is obtained by solving a
sample minimum distance problem such as GMM.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: First Step SMD Est of F∗
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Approximate
F
Vt
Ct
Vt − 1 Ct
,
;δ
Ct − 1 Ct − 1
=F
Vt−1 Ct
Ct−1 , Ct−1 ; δ
with a bivariate sieve:
KT
≈ FKT (·, δ) = a0 (δ) + ∑ aj (δ)Bj
j= 1
Vt − 1 Ct
,
Ct − 1 Ct − 1
Sieve coefficients {a0 , a1 , ..., aKT } depend on δ
Basis functions {Bj (·, ·) : j = 1, ..., KT } have known
functional forms independent of δ
V0
t
Initial value for V
Ct at time t = 0, denoted C0 , taken as a
unknown scalar parameter to be estimated.
KT KT
0
Given V
,
aj j=1 , Bj j=1 and data on consumption
C0
oT
n
n oT
Vi
Ct
,
use
F
to
generate
a
sequence
.
KT
Ct−1
Ci
t= 1
Sydney C. Ludvigson
i= 1
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: First Step SMD Est of F∗
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Recall m(wt , δo , F∗ (·, δo )) ≡ E {γ(zt+1 , δo , F∗ (·, δo ))|wt } = 0.
First-step SMD estimate b
F (·) for F∗ (·) based on
1
b
F (·, δ) = arg min
T
FK
T
T
∑ mb (wt , δ, FK
T
t= 1
b (wt , δ, FKT (·, δ)),
(·, δ))′ m
b (wt , δ, FKT (·, δ)) any nonpara. estimator of m.
m
Do this for a three dimensional grid of values of
δ = ( β, θ, ρ)′ .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: First Step SMD Est of F∗
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Example of nonparametric estimator of m:
Let p0j (wt ), j = 1, 2, ..., JT , Rdw → R be instruments.
pJT (·) ≡ (p01 (·) , ..., p0JT (·))′
′
Define T × JT matrix P ≡ pJT (w1 ) , ..., pJT (wT ) . Then:
!
T
b (w, δ, F) =
m
∑ γ(zt+1, δ, F)pJ
T
(wt )′ (P′ P)−1 pJT (w)
t= 1
b (·) a sieve LS estimator of m(w, δ, F).
m
Procedure equivalent to regressing each γi on instruments
and taking fitted values as estimate of conditional mean.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: First Step SMD Est of F∗
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
b (·) a sieve LS estimator of m(w, δ, F).
m
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: First Step SMD Est of F∗
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
b (·) a sieve LS estimator of m(w, δ, F).
m
Attractive feature of this estimator of F∗ : implemented as GMM
h
i′
i
−1 h
b
FT (·, δ) = arg min gT (δ,FT ; yT ) {IN ⊗ P′ P
} gT (δ,FT ; yT ) ,
|
{z
}
FT ∈VT
W
where yT = zT′ +1, ...z2′ , wT′ , ...w1′
gT (δ,FT ; yT ) =
Sydney C. Ludvigson
1
T
T
′
(12)
denotes vector of all obs and
∑ γ(zt+1, δ,FT )⊗pJT (wt )
(13)
t=1
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: First Step SMD Est of F∗
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
b (·) a sieve LS estimator of m(w, δ, F).
m
Attractive feature of this estimator of F∗ : implemented as GMM
h
i′
i
−1 h
b
FT (·, δ) = arg min gT (δ,FT ; yT ) {IN ⊗ P′ P
} gT (δ,FT ; yT ) ,
|
{z
}
FT ∈VT
W
where yT = zT′ +1, ...z2′ , wT′ , ...w1′
gT (δ,FT ; yT ) =
1
T
T
′
(12)
denotes vector of all obs and
∑ γ(zt+1, δ,FT )⊗pJT (wt )
(13)
t=1
Weighting gives greater weight to moments more highly
correlated with instruments pJT (·).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: First Step SMD Est of F∗
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
b (·) a sieve LS estimator of m(w, δ, F).
m
Attractive feature of this estimator of F∗ : implemented as GMM
h
i′
i
−1 h
b
FT (·, δ) = arg min gT (δ,FT ; yT ) {IN ⊗ P′ P
} gT (δ,FT ; yT ) ,
|
{z
}
FT ∈VT
W
where yT = zT′ +1, ...z2′ , wT′ , ...w1′
gT (δ,FT ; yT ) =
1
T
T
′
(12)
denotes vector of all obs and
∑ γ(zt+1, δ,FT )⊗pJT (wt )
(13)
t=1
Weighting gives greater weight to moments more highly
correlated with instruments pJT (·).
Weighting can be understood intuitively by noting that variation
in conditional mean m(wt , δ, F) is what identifies F∗ (·, δ).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: Second Step GMM Est of δo
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Under correct specification, δo satisfies :
E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0,
Sydney C. Ludvigson
i = 1, ..., N.
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: Second Step GMM Est of δo
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Under correct specification, δo satisfies :
E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0,
Sample moments:
b (·, δ); yT ) ≡
gT (δ, F
1
T
i = 1, ..., N.
b (·, δ)) ⊗ xt .
∑Tt=1 γ(zt+1 , δ, F
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: Second Step GMM Est of δo
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Under correct specification, δo satisfies :
E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0,
Sample moments:
b (·, δ); yT ) ≡
gT (δ, F
1
T
i = 1, ..., N.
b (·, δ)) ⊗ xt .
∑Tt=1 γ(zt+1 , δ, F
Regardless the model is correctly or incorrectly specified,
estimate δ by minimizing GMM objective:
i′ h
i
h
F (·, δ) ; yT ) W gT (δ, b
F (·, δ) ; yT )
δb = arg min gT (δ, b
δ ∈D
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: Second Step GMM Est of δo
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Under correct specification, δo satisfies :
E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0,
Sample moments:
b (·, δ); yT ) ≡
gT (δ, F
1
T
i = 1, ..., N.
b (·, δ)) ⊗ xt .
∑Tt=1 γ(zt+1 , δ, F
Regardless the model is correctly or incorrectly specified,
estimate δ by minimizing GMM objective:
i′ h
i
h
F (·, δ) ; yT ) W gT (δ, b
F (·, δ) ; yT )
δb = arg min gT (δ, b
δ ∈D
Examples: W = I, W = GT−1 .
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: Second Step GMM Est of δo
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Under correct specification, δo satisfies :
E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0,
Sample moments:
b (·, δ); yT ) ≡
gT (δ, F
1
T
i = 1, ..., N.
b (·, δ)) ⊗ xt .
∑Tt=1 γ(zt+1 , δ, F
Regardless the model is correctly or incorrectly specified,
estimate δ by minimizing GMM objective:
i′ h
i
h
F (·, δ) ; yT ) W gT (δ, b
F (·, δ) ; yT )
δb = arg min gT (δ, b
δ ∈D
Examples: W = I, W = GT−1 .
b
F (·, δ) not held fixed in this step: depends on δ!
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: Second Step GMM Est of δo
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Under correct specification, δo satisfies :
E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0,
Sample moments:
b (·, δ); yT ) ≡
gT (δ, F
1
T
i = 1, ..., N.
b (·, δ)) ⊗ xt .
∑Tt=1 γ(zt+1 , δ, F
Regardless the model is correctly or incorrectly specified,
estimate δ by minimizing GMM objective:
i′ h
i
h
F (·, δ) ; yT ) W gT (δ, b
F (·, δ) ; yT )
δb = arg min gT (δ, b
δ ∈D
Examples: W = I, W = GT−1 .
b
F (·, δ) not held fixed in this step: depends on δ!
b (·, δ) obtained using min. dist over a grid of values δ.
Estimator F
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences: Second Step GMM Est of δo
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Under correct specification, δo satisfies :
E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0,
Sample moments:
b (·, δ); yT ) ≡
gT (δ, F
1
T
i = 1, ..., N.
b (·, δ)) ⊗ xt .
∑Tt=1 γ(zt+1 , δ, F
Regardless the model is correctly or incorrectly specified,
estimate δ by minimizing GMM objective:
i′ h
i
h
F (·, δ) ; yT ) W gT (δ, b
F (·, δ) ; yT )
δb = arg min gT (δ, b
δ ∈D
Examples: W = I, W = GT−1 .
b
F (·, δ) not held fixed in this step: depends on δ!
b (·, δ) obtained using min. dist over a grid of values δ.
Estimator F
Choose the δ and corresponding b
F (·, δ) that minimizes GMM
criterion.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Why estimate in two steps? All params could be estimated
in one step by minimizing the SMD criterion.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Why estimate in two steps? All params could be estimated
in one step by minimizing the SMD criterion.
Less desirable for asset pricing:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Why estimate in two steps? All params could be estimated
in one step by minimizing the SMD criterion.
Less desirable for asset pricing:
1
Want estimates of RRA and EIS to reflect values required to
match unconditional risk premia. Not possible using SMD
which emphasizes conditional moments.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Why estimate in two steps? All params could be estimated
in one step by minimizing the SMD criterion.
Less desirable for asset pricing:
1
Want estimates of RRA and EIS to reflect values required to
match unconditional risk premia. Not possible using SMD
which emphasizes conditional moments.
2
SMD procedure effectively changes set of test assets–linear
combinations of original portfolio returns. But we may be
interested in explaining original returns!
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Why estimate in two steps? All params could be estimated
in one step by minimizing the SMD criterion.
Less desirable for asset pricing:
1
Want estimates of RRA and EIS to reflect values required to
match unconditional risk premia. Not possible using SMD
which emphasizes conditional moments.
2
SMD procedure effectively changes set of test assets–linear
combinations of original portfolio returns. But we may be
interested in explaining original returns!
3
Linear combinations may imply implausible long and short
positions, do not necessarily deliver a large spread in
unconditional mean returns.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Procedure allows for model misspecification:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Procedure allows for model misspecification:
Euler equation need not hold with equality.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Procedure allows for model misspecification:
Euler equation need not hold with equality.
As before, compare models by relative magnitude of
misspecification, rather than...
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Procedure allows for model misspecification:
Euler equation need not hold with equality.
As before, compare models by relative magnitude of
misspecification, rather than...
...asking whether each model individually fits data
perfectly (given sampling error).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Procedure allows for model misspecification:
Euler equation need not hold with equality.
As before, compare models by relative magnitude of
misspecification, rather than...
...asking whether each model individually fits data
perfectly (given sampling error).
Use W = G−1 in second step, compute HJ distance.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Recursive Preferences: Two Step Estimation
Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07
Procedure allows for model misspecification:
Euler equation need not hold with equality.
As before, compare models by relative magnitude of
misspecification, rather than...
...asking whether each model individually fits data
perfectly (given sampling error).
Use W = G−1 in second step, compute HJ distance.
Test whether HJ distances of competing models are
statistically different (White reality check–Chen and
Ludvigson ’09).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Bansal, Gallant, Tauchen ’07: SMM estimation of LRR
model: Bansal & Yaron ’04.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Bansal, Gallant, Tauchen ’07: SMM estimation of LRR
model: Bansal & Yaron ’04.
Structural estimation of EZW utility, restricting to specific
law of motion for cash flows (“long-run risk”LRR).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Bansal, Gallant, Tauchen ’07: SMM estimation of LRR
model: Bansal & Yaron ’04.
Structural estimation of EZW utility, restricting to specific
law of motion for cash flows (“long-run risk”LRR).
Cash flow dynamics in BGT version of LRR model:
∆ct+1 = µc + xc,t + σt ε c,t+1
∆dt+1 = µd + φx xc,t + φs st + σε d σt ε d,t+1
|{z}
LR risk
xc,t = φxc,t−1 + σε x σε xc,t
σt2 = σ2 + ν(σt2−1 − σ2 ) + σw wt
st = (µd − µc ) + dt − ct
ε c,t+1 , ε d,t+1 , ε xc,t , wt ∼ N.i.i.d (0, 1)
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh,
Tauchen ’97, Tauchen ’97.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh,
Tauchen ’97, Tauchen ’97.
Outline of SMM steps
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh,
Tauchen ’97, Tauchen ’97.
Outline of SMM steps
1
Solve the model over grid of values of deep parameters:
ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh,
Tauchen ’97, Tauchen ’97.
Outline of SMM steps
1
Solve the model over grid of values of deep parameters:
ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′
2
For each value of ρd on the grid, combine solutions with
long simulation of length N of model.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh,
Tauchen ’97, Tauchen ’97.
Outline of SMM steps
1
Solve the model over grid of values of deep parameters:
ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′
2
For each value of ρd on the grid, combine solutions with
long simulation of length N of model.
3
Simulation: Monte Carlo draws from the Normal
distribution for primitive shocks.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh,
Tauchen ’97, Tauchen ’97.
Outline of SMM steps
1
Solve the model over grid of values of deep parameters:
ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′
2
For each value of ρd on the grid, combine solutions with
long simulation of length N of model.
3
Simulation: Monte Carlo draws from the Normal
distribution for primitive shocks.
4
Form obs eqn for simulated and historical data, e.g.,
yt = (dt − ct , ct − ct−1 , pt − dt , rd,t )′
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
SMM methodology: Gallant & Tauchen ’96; Gallant, Hsieh,
Tauchen ’97, Tauchen ’97.
Outline of SMM steps
1
Solve the model over grid of values of deep parameters:
ρd = ( β, θ, ρ, φ, φx, µc , µd , σ, σǫd , σǫx ν, φs , σw )′
2
For each value of ρd on the grid, combine solutions with
long simulation of length N of model.
3
Simulation: Monte Carlo draws from the Normal
distribution for primitive shocks.
4
Form obs eqn for simulated and historical data, e.g.,
yt = (dt − ct , ct − ct−1 , pt − dt , rd,t )′
5
Choose value ρd that most closely “matches” moments
between dist of simulated and historical data ( “match”
made precise below.)
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Let {b
yt }N
t=1 denote simulated data (in obs eqn).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Let {b
yt }N
t=1 denote simulated data (in obs eqn).
Let {ỹt }Tt=1 denote historical data on same variables.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Let {b
yt }N
t=1 denote simulated data (in obs eqn).
Let {ỹt }Tt=1 denote historical data on same variables.
Auxiliary model of hist. data: e.g., VAR, with density
f (yt |yt−L , ...yt−1 , α), good LOM for data–f -model.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Let {b
yt }N
t=1 denote simulated data (in obs eqn).
Let {ỹt }Tt=1 denote historical data on same variables.
Auxiliary model of hist. data: e.g., VAR, with density
f (yt |yt−L , ...yt−1 , α), good LOM for data–f -model.
Score function of f -model:
sf (yt |yt−L , ...yt−1 , α) =
Sydney C. Ludvigson
∂
ln[f (yt |yt−L , ..., yt−1 , α)]
∂α
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Let {b
yt }N
t=1 denote simulated data (in obs eqn).
Let {ỹt }Tt=1 denote historical data on same variables.
Auxiliary model of hist. data: e.g., VAR, with density
f (yt |yt−L , ...yt−1 , α), good LOM for data–f -model.
Score function of f -model:
sf (yt |yt−L , ...yt−1 , α) =
∂
ln[f (yt |yt−L , ..., yt−1 , α)]
∂α
QMLE estimator of auxiliary model on historical data
α̃ = arg maxLT (α, {ỹt }Tt=1 )
α
LT (α, {ỹt }Tt=1 ) =
Sydney C. Ludvigson
1
T
T
∑
t= L+ 1
ln f (ỹt |ỹt−L , ..., ỹt−1 , α)
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
First-order-condition:
∂
LT (α̃, {ỹt }Tt=1 ) = 0
∂α
Sydney C. Ludvigson
or,
1
T
T
∑
t= L+ 1
sf (ỹt |ỹt−L , ..., ỹt−1 , α̃) = 0.
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
First-order-condition:
∂
LT (α̃, {ỹt }Tt=1 ) = 0
∂α
or,
1
T
T
∑
t= L+ 1
sf (ỹt |ỹt−L , ..., ỹt−1 , α̃) = 0.
Idea: since above, good estimator for ρd is one that sets
1 N
sf (b
yt (ρd )|b
yt−L (ρd ), ..., b
yt−1 (ρd ), α̃) ≈ 0.
N t=∑
L+1
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
First-order-condition:
∂
LT (α̃, {ỹt }Tt=1 ) = 0
∂α
or,
1
T
T
∑
t= L+ 1
sf (ỹt |ỹt−L , ..., ỹt−1 , α̃) = 0.
Idea: since above, good estimator for ρd is one that sets
1 N
sf (b
yt (ρd )|b
yt−L (ρd ), ..., b
yt−1 (ρd ), α̃) ≈ 0.
N t=∑
L+1
If dim(α) >dim(ρd ), use GMM:
1 N
b T ( ρd , α ) =
m
sf (b
yt (ρd )|b
yt−L (ρd ), ..., b
yt−1 (ρd ), α̃)
| {z }
N t=∑
L+1
dim( α)×1
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
First-order-condition:
∂
LT (α̃, {ỹt }Tt=1 ) = 0
∂α
or,
1
T
T
∑
t= L+ 1
sf (ỹt |ỹt−L , ..., ỹt−1 , α̃) = 0.
Idea: since above, good estimator for ρd is one that sets
1 N
sf (b
yt (ρd )|b
yt−L (ρd ), ..., b
yt−1 (ρd ), α̃) ≈ 0.
N t=∑
L+1
If dim(α) >dim(ρd ), use GMM:
1 N
b T ( ρd , α ) =
m
sf (b
yt (ρd )|b
yt−L (ρd ), ..., b
yt−1 (ρd ), α̃)
| {z }
N t=∑
L+1
dim( α)×1
The GMM estimator is
b T (ρd , α̃ )′ Ĩ −1 m
b T (ρd , α̃)
ρbd = arg min{m
ρd
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
b T (ρd , α̃)′ Ĩ −1 m
b T (ρd , α̃)}.
GMM: ρbd = arg min{m
ρd
Ĩ −1 is inv. of var. of score, data determined from f -model
T
Ĩ =
∑
t= 1
∂
ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃ )]
∂α̃
Sydney C. Ludvigson
′
∂
ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃)]
∂α̃
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
b T (ρd , α̃)′ Ĩ −1 m
b T (ρd , α̃)}.
GMM: ρbd = arg min{m
ρd
Ĩ −1 is inv. of var. of score, data determined from f -model
T
Ĩ =
∑
t= 1
∂
ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃ )]
∂α̃
′
∂
ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃)]
∂α̃
y }N
Sims {b
t=1 follow stationary dens. p(yt−L , ..., yt |ρd ). Note:
no closed-form for p(·|ρd ).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
b T (ρd , α̃)′ Ĩ −1 m
b T (ρd , α̃)}.
GMM: ρbd = arg min{m
ρd
Ĩ −1 is inv. of var. of score, data determined from f -model
T
Ĩ =
∑
t= 1
∂
ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃ )]
∂α̃
′
∂
ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃)]
∂α̃
y }N
Sims {b
t=1 follow stationary dens. p(yt−L , ..., yt |ρd ). Note:
no closed-form for p(·|ρd ).
as
b T (ρd , α) → m(ρd , α) as N → ∞, where
Intuition: m
m ( ρd , α ) =
Z
···
Z
s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
b T (ρd , α̃)′ Ĩ −1 m
b T (ρd , α̃)}.
GMM: ρbd = arg min{m
ρd
Ĩ −1 is inv. of var. of score, data determined from f -model
T
Ĩ =
∑
t= 1
∂
ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃ )]
∂α̃
′
∂
ln[f (ỹt |ỹt−L , ..., ỹt−1 , α̃)]
∂α̃
y }N
Sims {b
t=1 follow stationary dens. p(yt−L , ..., yt |ρd ). Note:
no closed-form for p(·|ρd ).
as
b T (ρd , α) → m(ρd , α) as N → ∞, where
Intuition: m
m ( ρd , α ) =
Z
···
Z
s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt
⇒ use Monte Carlo compute expect. of s(·) under p(·|ρd ).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
as
b T (ρd , α) → m(ρd , α) as N → ∞, where
Intuition: m
m ( ρd , α ) =
Z
···
Z
s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
as
b T (ρd , α) → m(ρd , α) as N → ∞, where
Intuition: m
m ( ρd , α ) =
Z
···
Z
s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt
If f = p above is mean of scores of likelihood. Should be
zero, given f.o.c for MLE estimator.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
as
b T (ρd , α) → m(ρd , α) as N → ∞, where
Intuition: m
m ( ρd , α ) =
Z
···
Z
s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt
If f = p above is mean of scores of likelihood. Should be
zero, given f.o.c for MLE estimator.
Thus, if data do follow the structural model p(·|ρd ), then
m(ρod , αo ) = 0, forms basis of a specification test.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
as
b T (ρd , α) → m(ρd , α) as N → ∞, where
Intuition: m
m ( ρd , α ) =
Z
···
Z
s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt
If f = p above is mean of scores of likelihood. Should be
zero, given f.o.c for MLE estimator.
Thus, if data do follow the structural model p(·|ρd ), then
m(ρod , αo ) = 0, forms basis of a specification test.
Summary: solve model for many values of ρd , store long
simulations of model each time, do one-time estimation of
auxiliary f -model. Choose ρd to minimize GMM criterion
above.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Advantages of using score functions as moments:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Advantages of using score functions as moments:
1
Computational: one-time estimation of structural model;
useful if f -model is nonlinear.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Advantages of using score functions as moments:
1
Computational: one-time estimation of structural model;
useful if f -model is nonlinear.
2
If f -model good description of data, under null, MLE
efficiency is obtained.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Advantages of using score functions as moments:
1
Computational: one-time estimation of structural model;
useful if f -model is nonlinear.
2
If f -model good description of data, under null, MLE
efficiency is obtained.
If dim(α) >dim(ρd ), score-based SMM is consistent,
asymptotically normal, assuming:
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Advantages of using score functions as moments:
1
Computational: one-time estimation of structural model;
useful if f -model is nonlinear.
2
If f -model good description of data, under null, MLE
efficiency is obtained.
If dim(α) >dim(ρd ), score-based SMM is consistent,
asymptotically normal, assuming:
That the auxiliary model is rich enough to identify
non-linear structural model. Sufficient conditions for
identification unknown.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
EZW Preferences With Restricted Dynamics:
Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07
Advantages of using score functions as moments:
1
Computational: one-time estimation of structural model;
useful if f -model is nonlinear.
2
If f -model good description of data, under null, MLE
efficiency is obtained.
If dim(α) >dim(ρd ), score-based SMM is consistent,
asymptotically normal, assuming:
That the auxiliary model is rich enough to identify
non-linear structural model. Sufficient conditions for
identification unknown.
Big issue: are these the economically interesting moments?
Regards both choice of moments, and weighting function.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Little work linking financial markets to macroeconomic
risks, given by primitives in the IMRS over consumption.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Little work linking financial markets to macroeconomic
risks, given by primitives in the IMRS over consumption.
No model that relates returns to other returns can explain
asset prices in terms of primitive economic shocks. Such
models of SDF only describe asset prices.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Little work linking financial markets to macroeconomic
risks, given by primitives in the IMRS over consumption.
No model that relates returns to other returns can explain
asset prices in terms of primitive economic shocks. Such
models of SDF only describe asset prices.
So far many consumption-based models have been
evaluated using calibration exercises ⇒
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Little work linking financial markets to macroeconomic
risks, given by primitives in the IMRS over consumption.
No model that relates returns to other returns can explain
asset prices in terms of primitive economic shocks. Such
models of SDF only describe asset prices.
So far many consumption-based models have been
evaluated using calibration exercises ⇒
A crucial next step in evaluating consumption-based
models is structural econometric estimation. But...
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Little work linking financial markets to macroeconomic
risks, given by primitives in the IMRS over consumption.
No model that relates returns to other returns can explain
asset prices in terms of primitive economic shocks. Such
models of SDF only describe asset prices.
So far many consumption-based models have been
evaluated using calibration exercises ⇒
A crucial next step in evaluating consumption-based
models is structural econometric estimation. But...
...models are imperfect and will never fit data infallibly.
Argue here for need to move away from testing if models
are true, towards comparison of models based on magnitude
of misspecification.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Example: scaled consumption models.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Example: scaled consumption models.
Rather than ask whether scaled models are true...
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Example: scaled consumption models.
Rather than ask whether scaled models are true...
...ask whether allowing for state-dependence of SDF on
consumption growth reduces misspecification over the
analogous non-state-dependent model.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Macroeconomic data, unlike financial measured with error.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Macroeconomic data, unlike financial measured with error.
⇒ Can’t expect such models to perform as well as financial
factor models of SDF.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Macroeconomic data, unlike financial measured with error.
⇒ Can’t expect such models to perform as well as financial
factor models of SDF.
True systematic risk factors are macroeconomic in nature;
asset prices derived endogenously from these.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Macroeconomic data, unlike financial measured with error.
⇒ Can’t expect such models to perform as well as financial
factor models of SDF.
True systematic risk factors are macroeconomic in nature;
asset prices derived endogenously from these.
Financial factors could represent projection of true Mt on
portfolios (i.e., mimicking portfolios).
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Macroeconomic data, unlike financial measured with error.
⇒ Can’t expect such models to perform as well as financial
factor models of SDF.
True systematic risk factors are macroeconomic in nature;
asset prices derived endogenously from these.
Financial factors could represent projection of true Mt on
portfolios (i.e., mimicking portfolios).
In which case, they will always perform at least as well, or
better than, mismeasured macro factors from true Mt ⇒
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Macroeconomic data, unlike financial measured with error.
⇒ Can’t expect such models to perform as well as financial
factor models of SDF.
True systematic risk factors are macroeconomic in nature;
asset prices derived endogenously from these.
Financial factors could represent projection of true Mt on
portfolios (i.e., mimicking portfolios).
In which case, they will always perform at least as well, or
better than, mismeasured macro factors from true Mt ⇒
Not sensible to run horse races between financial factor
models and macro models.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models
Consumption-Based Asset Pricing: Final Thoughts
Goal: not to find better factors, but rather to explain
financial factors from deeper economic models.
Sydney C. Ludvigson
Methods Lecture: GMM and Consumption-Based Models