Ludvigson methodslecture1

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?


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),


sampling error),
2 toward estimation and testing that recognize all models are
misspeciﬁed,


sampling error),
misspeciﬁed,
3 toward methods permit comparison of magnitude of
misspeciﬁcation among multiple, competing macro models.


sampling error),
misspeciﬁed,
3 toward methods permit comparison of magnitude of
misspeciﬁcation among multiple, competing macro models.
Themes are important in choosing which methods to use.

GMM and Consumption-Based Models: Outline

GMM estimation of classic representative agent, CRRA
utility model



utility model

Incorporating conditioning information: scaled
consumption-based models



utility model


Generalizations of CRRA utility: recursive utility



utility model



Semi-nonparametric minimum distance estimators:
unrestricted LOM for data



utility model



Simulation methods: restricted LOM



utility model



Simulation methods: restricted LOM

Consumption-based asset pricing: concluding thoughts


GMM Review (Hansen, 1982)

Economic model implies set of r population moment
restrictions

E{h (θ, wt )} = 0
(r × 1 )



restrictions

E{h (θ, wt )} = 0
(r × 1 )

wt is an h × 1 vector of variables known at t



restrictions

E{h (θ, wt )} = 0
(r × 1 )


θ is an a × 1 vector of coefﬁcients



restrictions

E{h (θ, wt )} = 0
(r × 1 )


θ is an a × 1 vector of coefﬁcients

Idea: choose θ to make the sample moment as close as
possible to the population moment.


Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
t= 1
(r × 1 )


Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
t= 1
(r × 1 )

′ ′ ′ ′
yT ≡ wT , wT −1 , ...w1 is a T · h × 1 vector of observations.


Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
t= 1
(r × 1 )

′ ′ ′ ′

The GMM estimator θ minimizes the scalar
′
Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )], (1)
( 1× r ) (r×r) ( r × 1)

∞
{WT }T =1 a sequence of r × r positive deﬁnite matrices
which may be a function of the data, yT .


Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
t= 1
(r × 1 )

′ ′ ′ ′

′
Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )], (1)
( 1× r ) (r×r) ( r × 1)

∞
If r = a, θ estimated by setting each g(θ; yT ) to zero.


Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
t= 1
(r × 1 )

′ ′ ′ ′

′
Q (θ; yT ) = [g(θ; yT )] WT [g(θ; yT )], (1)
( 1× r ) (r×r) ( r × 1)

∞
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.


Asym. properties (Hansen 1982): θ consistent, asym.
normal.



normal.

Optimal weighting WT = S−1

 
∞

 

′
S =
r×r
∑ E

[h (θo , wt )] h θo , wt−j

.
j=− ∞  
r×1 1× r



normal.

Optimal weighting WT = S−1

 
∞

 

′
S =
r×r
∑ E

[h (θo , wt )] h θo , wt−j

.
j=− ∞  
r×1 1× r

In many asset pricing applications, it is inappropriate to
use WT = S−1 (see below).



ST depends on θT which depends on ST . Employ an
iterative procedure:



ST depends on θT which depends on ST . Employ an
iterative procedure:

(1)
1 Obtain an initial estimate of θ = θT , by minimizing
Q (θ; yT ) subject to arbitrary weighting matrix, e.g., W = I.

(1) (1)
2 Use θT to obtain initial estimate of S = ST .

(1 )
3 Re-minimize Q (θ; yT ) using initial estimate ST ; obtain
(2)
new estimate θT .

4 Continue iterating until convergence, or stop. (Estimators
have same asym. dist. but ﬁnite sample properties differ.)


Classic Example: Hansen and Singleton (1982)
Investors maximize utility

∞
max Et
Ct
∑ β i u (C t + i )
i= 0



∞
max Et
Ct
∑ β i u (C t + i )
i= 0

Power (isoelastic) utility
 1− γ
 u (C ) =
 Ct
γ>0
t 1− γ
(2)


u (Ct ) = ln(Ct ) γ = 1



∞
max Et
Ct
∑ β i u (C t + i )
i= 0

Power (isoelastic) utility
 1− γ
 u (C ) =
 Ct
γ>0
t 1− γ
(2)


u (Ct ) = ln(Ct ) γ = 1

N assets => N ﬁrst-order conditions
−γ −γ
Ct = βEt (1 + ℜi,t+1 ) Ct+1 i = 1, ..., N. (3)


Asset Pricing Example: Hansen and Singleton (1982)
Re-write moment conditions
−γ
Ct + 1
0 = Et 1 − β (1 + ℜi,t+1 ) −γ . (4)
Ct


−γ
Ct + 1
0 = Et 1 − β (1 + ℜi,t+1 ) −γ . (4)
Ct

2 params to estimate: β and γ, so θ = ( β, γ) ′ .


−γ
Ct + 1
0 = Et 1 − β (1 + ℜi,t+1 ) −γ . (4)
Ct

∗
xt denotes info set of investors

−γ −γ ∗
0=E 1 − β (1 + ℜi,t+1 ) Ct+1 /Ct |xt i = 1, ...N
(5)


−γ
Ct + 1
0 = Et 1 − β (1 + ℜi,t+1 ) −γ . (4)
Ct

∗
xt denotes info set of investors

−γ −γ ∗
0=E 1 − β (1 + ℜi,t+1 ) Ct+1 /Ct |xt
i = 1, ...N
(5)
∗ . Conditional model (5) => unconditional model:
xt ⊂ xt
−γ
Ct + 1
0=E 1− β (1 + ℜi,t+1 ) −γ xt i = 1, ...N
Ct
(6)

Let xt be M × 1. Then r = N · M and,
 −γ

C +1
 1 − β (1 + ℜ1,t+1 ) t−γ xt 
Ct
 
 −γ
Ct+1 
 1 − β (1 + ℜ2,t+1 ) −γ xt 
 Ct 
 

h (θ, wt ) =  ·  (7)

r×1  · 
 
 · 
 
 −γ
C +1 
1 − β (1 + ℜN,t+1 ) t−γ xt
Ct


 −γ

C +1
 1 − β (1 + ℜ1,t+1 ) t−γ xt 
Ct
 
 −γ
Ct+1 
 1 − β (1 + ℜ2,t+1 ) −γ xt 
 Ct 
 

h (θ, wt ) =  ·  (7)

r×1  · 
 
 · 
 
 −γ
C +1 
1 − β (1 + ℜN,t+1 ) t−γ xt
Ct

Model can be estimated, tested as long as r ≥ 2.


 −γ

C +1
 1 − β (1 + ℜ1,t+1 ) t−γ xt 
Ct
 
 −γ
Ct+1 
 1 − β (1 + ℜ2,t+1 ) −γ xt 
 Ct 
 

h (θ, wt ) =  ·  (7)

r×1  · 
 
 · 
 
 −γ
C +1 
1 − β (1 + ℜN,t+1 ) t−γ xt
Ct

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 r×r
1× r r×1

Test of Over-identifying (OID) restrictions:
a
TQ θ; yT ∼ χ2 (r − a)


a
TQ 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....


a
TQ θ; yT ∼ χ2 (r − a)

...model is strongly rejected according to OID test.


a
TQ θ; yT ∼ χ2 (r − a)

Campbell, Lo, MacKinlay (1997): OID rejections stronger
whenever stock returns and commercial paper are
included as test returns. Why?


a
TQ θ; yT ∼ χ2 (r − a)

Model cannot capture predictable variation in excess
returns over commercial paper ⇒


a
TQ θ; yT ∼ χ2 (r − a)

Model cannot capture predictable variation in excess
returns over commercial paper ⇒
Researchers have turned to other models of preferences.

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 )]


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
j
gj,t (γ, β) = 1
T ∑T=1 eR,t
t

j
gj,t (γ) = 1
T ∑T=1 eX,t
t


Unconditional Euler Equation Errors, Excess Returns

j j f
eX ≡ E[ β(Ct+1 /Ct )−γ (Rt+1 − Rt+1 )] j = 1, ..., N
j j f
RMSE = 1
N ∑N 1 [eX ]2 ,
j= RMSR = 1
N
N
∑j=1 [E(Rt+1 − Rt+1 )]2

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.


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.


risk aversion.
Lettau and Ludvigson (2009): Leading consumption-based
asset pricing theories fail to explain the mispricing of
classic CCAPM.


risk aversion.
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.


risk aversion.
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!

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.




models.
Consider two estimated models of SDF, e.g.,
(1)
1 CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected
(2)
2 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected




models.
(1)
(2)

May we conclude Model 1 is superior?




models.
(1)
(2)


No. Hansen’s J-test of OID restricts depends on model
speciﬁc S: J = gT S−1 gT .
′




models.
(1)
(2)


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.


HJ: compare models Mt (θ) using distance metric:

T
1
DistT (θ) = mingT (θ)′ GT 1 gT (θ),
−
GT ≡ ∑ Rt Rt′
θ T t= 1
N ×N
T
1
gT (θ) ≡
T ∑ [Mt (θ)Rt − 1N ]
t= 1




T
1
−
GT ≡ ∑ Rt Rt′
θ T t= 1
N ×N
T
1
gT (θ) ≡
T ∑ [Mt (θ)Rt − 1N ]
t= 1

DistT does not reward SDF volatility => suitable for
model comparison.




T
1
−
GT ≡ ∑ Rt Rt′
θ T t= 1
N ×N
T
1
gT (θ) ≡
T ∑ [Mt (θ)Rt − 1N ]
t= 1

model comparison.
DistT is a measure of model misspeciﬁcation:




T
1
−
GT ≡ ∑ Rt Rt′
θ T t= 1
N ×N
T
1
gT (θ) ≡
T ∑ [Mt (θ)Rt − 1N ]
t= 1

model comparison.
Gives distance between Mt (θ) and nearest point in space of
all SDFs that price assets correctly.




T
1
−
GT ≡ ∑ Rt Rt′
θ T t= 1
N ×N
T
1
gT (θ) ≡
T ∑ [Mt (θ)Rt − 1N ]
t= 1

model comparison.
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.


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?






Important problem: how to compare HJ distances
statistically?

One possibility developed in Chen and Ludvigson (2009):
White’s reality check method.


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 Take benchmark model, e.g., model with smallest squared
distance d1,T ≡ min{d2 }K 1 .
2
j,T j=

2 Null: d2 − d2 ≤ 0, where d2 is competing model with
1,T 2,T 2,T
the next smallest squared distance.
√
3 Test statistic T W = T (d2 − d2,T ).
1,T
2

4 If null is true, test statistic should not be unusually large,
given sampling error.
5 Given distribution for T W , reject null if historical value T W
is > 95th percentile.



1 Take benchmark model, e.g., model with smallest squared
distance d1,T ≡ min{d2 }K 1 .
2
j,T j=

2 Null: d2 − d2 ≤ 0, where d2 is competing model with
1,T 2,T 2,T
the next smallest squared distance.
√
3 Test statistic T W = T (d2 − d2,T ).
1,T
2

4 If null is true, test statistic should not be unusually large,
given sampling error.
5 Given distribution for T W , reject null if historical value T W
is > 95th percentile.

Method applies generally to any stationary law of motion
for data, multiple competing possibly nonlinear, SDF
models.


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:






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).


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.


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.




− −


′ ′

Re-weighted portfolios may not provide large spread in
average returns.

May imply implausible long and short positions in test
assets.


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 ).

−1
[E (MR) − 1N ]′ E RR′ [E (MR − 1N )]
′ −1
= [E (MAR) − A1N ] E ARR′ A [E (MAR − A1N )] .




Form a portfolio, AR from initial returns R. (Note,
portfolio weights sum to 1 so A1N = 1N ).

−1
[E (MR) − 1N ]′ E RR′ [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.


More Complex Preferences: Scaled
Consumption-Based Models
Consumption-based models may be approximated:
Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 )


Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 )
Example: Classic CCAPM with CRRA utility
1− γ
Ct
u ( Ct ) = ⇒ Mt+1 ≈ β − βγ ∆ct+1
1−γ
at = a0 bt = b0


Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 )
1− γ
Ct
u ( Ct ) = ⇒ Mt+1 ≈ β − βγ ∆ct+1
1−γ
at = a0 bt = b0

Model with habit and time-varying risk aversion: Campbell and
Cochrane ’99, Menzly et. al ’04
( Ct S t ) 1 − γ C t − Xt
u ( Ct , S t ) = , St + 1 ≡
1−γ Ct

⇒ Mt+1 ≈ β 1 − γ (φ − 1)(st − s) − γ (1 + ψ (st )) ∆ct+1
at bt


Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 )
1− γ
Ct
u ( Ct ) = ⇒ Mt+1 ≈ β − βγ ∆ct+1
1−γ
at = a0 bt = b0

Model with habit and time-varying risk aversion: Campbell and
Cochrane ’99, Menzly et. al ’04
( Ct S t ) 1 − γ C t − Xt
u ( Ct , S t ) = , St + 1 ≡
1−γ Ct

⇒ Mt+1 ≈ β 1 − γ (φ − 1)(st − s) − γ (1 + ψ (st )) ∆ct+1
at bt
Proxies for time-varying risk-premia should be good proxies for
time-variation in at and bt .

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


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
1− σ ε
Ct ε −1 ε −1 ε −1
U(Ct , Ht ) = 1
Ct = χCt ε
+ (1 − χ) Ht ε
,
1− σ

pC Ct
t
⇒ ln Mt+1 ≈ at + bt ∆ ln Ct+1 + dt ∆ ln St+1 , St + 1 ≡
pC Ct
t + pH Ht
t
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)


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 scaling returns: E Mt+1 Ri,t+1 ⊗ (1 zt )′ = 1
2 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 )′



′
= 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



′
= 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-based models have been tested on
unconditional moments, E Mt+1 Rt+1 = 1
⇒ NO scaled returns.


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 )
f1,t+1 f2,t+1 f3,t+1



constant weights:

Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1
f1,t+1 f2,t+1 f3,t+1

′
Multiple risk factors ft ≡ (zt , ∆ ln Ct+1 , zt ∆ ln Ct+1 ).



constant weights:

Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1
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 .


Deriving the “beta”-representation
Let F = (1 f′ )′ , M = b′ F, ignore time indices
1 = E[MRi ]
= E[Ri F′ ]b ⇒ unconditional moments
= E[Ri ]E[F′ ]b + Cov(Ri , F′ )b ⇒

1 − Cov(Ri , F′ )b
E[Ri ] =
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).

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 T
λ = 1/T ∑ λt ; σ2 (λ) = 1/T ∑ (λt − λ)2
t= 1 t= 1

Note: report Shanken t-statistics (corrected for estimation error
of betas in ﬁrst stage)



Scaled models: conditioning done in SDF:
Mt+1 = at + bt ∆ ln Ct+1 ,
not in Euler equation: E(MR) = 1N .



Mt+1 = at + bt ∆ ln Ct+1 ,
Gives rise to a restricted conditional consumption beta
model:
Ri = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1
t



Mt+1 = at + bt ∆ ln Ct+1 ,
model:
t

Rewrite as
Ri = a + ( βc + βc,z zt−1 ) ∆ct + βz zt−1
t
time-varying beta



Mt+1 = at + bt ∆ ln Ct+1 ,
model:
t

Rewrite as
Ri = a + ( βc + βc,z zt−1 ) ∆ct + βz zt−1
t
time-varying beta

Unlikely the same time-varying beta as obtained from
modeling conditional mean Et (Mt+1 Rt+1 ) = 1.


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.




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.




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).


Bottom lines:



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.



Bottom lines:

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.



Bottom lines:

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.


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.






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.



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 (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.


EZW Recursive Preferences
Epstein-Zin-Weil basics

Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)):
1
1− ρ
+ β R t ( Vt + 1 ) 1 − ρ
1− ρ
Vt = ( 1 − β ) Ct
1
1− 1− θ
Rt (Vt+1 ) = E Vt+1θ |Ft



1
1− ρ
+ β R t ( Vt + 1 ) 1 − ρ
1− ρ
Vt = ( 1 − β ) Ct
1
1− 1− θ

Vt+1 is continuation value, θ is RRA, 1/ρ is EIS.



1
1− ρ
+ β R t ( Vt + 1 ) 1 − ρ
1− ρ
Vt = ( 1 − β ) Ct
1
1− 1− θ

Rescale utility function (Hansen, Heaton, Li ’05):
1
1− ρ 1− ρ
Vt Vt + 1 Ct + 1
= ( 1 − β ) + β Rt
Ct Ct + 1 Ct



1
1− ρ
+ β R t ( Vt + 1 ) 1 − ρ
1− ρ
Vt = ( 1 − β ) Ct
1
1− 1− θ

Rescale utility function (Hansen, Heaton, Li ’05):
1
1− ρ 1− ρ
Vt Vt + 1 Ct + 1
= ( 1 − β ) + β Rt
Ct Ct + 1 Ct

C1−θ
Special case: ρ = θ ⇒ CRRA separable utility Vt = β 1−θ .
t


Epstien-Zin-Weil basics

The MRS is pricing kernel (SDF) with added risk factor:
  ρ−θ
Vt+1 Ct+1
Ct + 1 − ρ  Ct+1 Ct 
Mt + 1 = β
Ct R Vt+1 Ct+1
t Ct+1 Ct



  ρ−θ
Vt+1 Ct+1
Ct + 1 − ρ  Ct+1 Ct 
Mt + 1 = β
Ct R Vt+1 Ct+1
t Ct+1 Ct

Difﬁculty: MRS a function of V/C, unobservable, embeds
Rt (·).



  ρ−θ
Vt+1 Ct+1
Ct + 1 − ρ  Ct+1 Ct 
Mt + 1 = β
Ct R Vt+1 Ct+1 t Ct+1 Ct

Rt (·).
Epstein-Zin ’91 use alt. rep. of SDF, uses agg. wealth
return Rw,t :
1− θ θ −ρ
−ρ 1− ρ
Ct + 1 1 1− ρ
Mt + 1 = β
Ct Rw,t+1

where Rw,t proxied by stock market return.



  ρ−θ
Vt+1 Ct+1
Ct + 1 − ρ  Ct+1 Ct 
Mt + 1 = β
Ct R Vt+1 Ct+1 t Ct+1 Ct

Rt (·).
Epstein-Zin ’91 use alt. rep. of SDF, uses agg. wealth
return Rw,t :
1− θ θ −ρ
−ρ 1− ρ
Ct + 1 1 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.


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....




But....
EIS=1 ⇒ consumption-wealth ratio is constant,
contradicting statistical evidence.




But....
Joint lognormality strongly rejected in quarterly data.




But....
Joint lognormality strongly rejected in quarterly data.

Points to need for estimation method feasible under less
restrictive assumptions.


Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

CFL: semiparametric approach to estimate EZW model
without:



without:
Need to proxy Rw,t+1 with observable returns.



without:
Loglinearizing the model.



without:
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



without:

Evaluate EZW model’s ability to ﬁt asset return data
relative to competing model speciﬁcations.



without:

Investigate implications for Rw,t+1 and return to human
wealth.



without:

Investigate implications for Rw,t+1 and return to human
wealth.
Semiparametric approach is sieve minimum distance
(SMD) procedure.


First order conditions for optimal consumption choice:
   
Vt + 1 C t + 1 ρ−θ
−ρ
 Ct + 1  Ct + 1 Ct  
Et  β Ri,t+1 − 1 = 0 (8)
Ct Rt Vtt+1 CC+1
+1 t
C t



   
Vt + 1 C t + 1 ρ−θ
−ρ
 Ct + 1  Ct + 1 Ct  
Et  β Ri,t+1 − 1 = 0 (8)
Ct Rt Vtt+1 CC+1
+1 t
C t

1
Vt + 1 C t + 1 1− ρ 1− ρ
Vt
CFL: plug Ct = ( 1 − β ) + β Rt Ct + 1 Ct into (8):

  ρ−θ 
−ρ Vt+1 Ct+1
 Ct+1  Ct+1 Ct  
Et  β


 1

 Ri,t+1 − 1 = 0
 i = 1, ..., N.
Ct 1 Vt 1 − ρ 1− ρ
β Ct − (1 − β )
(9)



   
Vt + 1 C t + 1 ρ−θ
−ρ
 Ct + 1  Ct + 1 Ct  
Et  β Ri,t+1 − 1 = 0 (8)
Ct Rt Vtt+1 CC+1
+1 t
C t

1
Vt + 1 C t + 1 1− ρ 1− ρ
Vt

  ρ−θ 
−ρ Vt+1 Ct+1
 Ct+1  Ct+1 Ct  
Et  β


 1

 Ri,t+1 − 1 = 0
 i = 1, ..., N.
Ct 1 Vt 1 − ρ 1− ρ
β Ct − (1 − β )
(9)
N test asset returns, {Ri,t+1 }N 1 . (9) is a x-sect asset pricing model.
i=



   
Vt + 1 C t + 1 ρ−θ
−ρ
 Ct + 1  Ct + 1 Ct  
Et  β Ri,t+1 − 1 = 0 (8)
Ct Rt Vtt+1 CC+1
+1 t
C t

1
Vt + 1 C t + 1 1− ρ 1− ρ
Vt

  ρ−θ 
−ρ Vt+1 Ct+1
 Ct+1  Ct+1 Ct  
Et  β


 1

 Ri,t+1 − 1 = 0
 i = 1, ..., N.
Ct 1 Vt 1 − ρ 1− ρ
β Ct − (1 − β )
(9)
i=
Moment restrictions (9) form the basis of empirical investigation.



   
Vt + 1 C t + 1 ρ−θ
−ρ
 Ct + 1  Ct + 1 Ct  
Et  β Ri,t+1 − 1 = 0 (8)
Ct Rt Vtt+1 CC+1
+1 t
C t

1
Vt + 1 C t + 1 1− ρ 1− ρ
Vt

  ρ−θ 
−ρ Vt+1 Ct+1
 Ct+1  Ct+1 Ct  
Et  β


 1

 Ri,t+1 − 1 = 0
 i = 1, ..., N.
Ct 1 Vt 1 − ρ 1− ρ
β Ct − (1 − β )
(9)
i=
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.



Vt
Assume Ct an unknown function F: R2 → R of form
Vt Vt − 1 Ct
=F , ,
Ct Ct − 1 Ct − 1



Vt
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.



Vt
Vt Vt − 1 Ct
=F , ,
Ct Ct − 1 Ct − 1

Justiﬁed if, for example,



Vt
Vt Vt − 1 Ct
=F , ,
Ct Ct − 1 Ct − 1

∆ 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 .



Vt
Vt Vt − 1 Ct
=F , ,
Ct Ct − 1 Ct − 1

With a nonlinear Markov process for xt , F(·) can display
nonmonotonicities in both arguments.



Vt
Vt Vt − 1 Ct
=F , ,
Ct Ct − 1 Ct − 1

Note: Markov assumption only a motivation for arguments
of F(·). Econometric methodology itself leaves LOM for
∆ ln Ct unspeciﬁed.


Ft denotes agents information set at time t.



zt+1 contains all observations at t + 1 and
 ρ−θ
 Vt Ct+1 Ct+1

Ct+1 −ρ
 F Ct , Ct+1 Ct 
γi (zt+1 , δ, F) ≡ β  1  Ri,t+1 − 1
Ct  1− ρ 1− ρ 
1 Vt − 1 Ct
β F Ct−1 , Ct−1 − (1 − β )



 ρ−θ
 Vt Ct+1 Ct+1

Ct+1 −ρ
 F Ct , Ct+1 Ct 
γi (zt+1 , δ, F) ≡ β  1  Ri,t+1 − 1
Ct  1− ρ 1− ρ 
1 Vt − 1 Ct
β F Ct−1 , Ct−1 − (1 − β )

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)



 ρ−θ
 Vt Ct+1 Ct+1

Ct+1 −ρ
 F Ct , Ct+1 Ct 
γi (zt+1 , δ, F) ≡ β  1  Ri,t+1 − 1
Ct  1− ρ 1− ρ 
1 Vt − 1 Ct
β F Ct−1 , Ct−1 − (1 − β )

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. i = 1, ..., N.



Intuition behind minimum distance procedure:



Theory ⇒
mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0. i = 1, ..., N.
(11)



Theory ⇒
mt ≡ E {γi (zt+1 , δo , Fo (·, δo ))|wt } = 0.
i = 1, ..., N.
(11)
Since mt = 0, mt must have zero variance, mean.



Theory ⇒
i = 1, ..., N.
(11)
Thus can ﬁnd params by minimizing variance or quadratic
norm: min E[(mt )2 ]. Don’t observe mt ⇒ need estimate mt .



Theory ⇒
i = 1, ..., N.
(11)
Since (11) is cond. mean, must hold for each observation, t.



Theory ⇒
i = 1, ..., N.
(11)

Obs > params, need way to weight each obs; using sample
mean is one way: min ET [(mt )2 ].



Theory ⇒
i = 1, ..., N.
(11)

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.


estimation involving conditional moments: min ET [(mt )2 ].




Contrast with GMM, used for unconditional moments:
E[f (xt , α)] = 0.




E[f (xt , α)] = 0.

With GMM take sample counterpart to population mean:
gT = ∑T=1 f (xt , α) = 0.
t




E[f (xt , α)] = 0.

gT = ∑T=1 f (xt , α) = 0.
t
′
Then choose parameters α to min gT WgT .




E[f (xt , α)] = 0.

gT = ∑T=1 f (xt , α) = 0.
t
′
Then choose parameters α to min gT WgT .

With GMM we average and then square.

With SMD, we square and then average.



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))




δ∈D F∈V

′

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




δ∈D F∈V

′

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 )


EZW Recursive Preferences: Two-Step Procedure

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.





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.


EZW Recursive Preferences: First Step SMD Est of F∗

Vt Vt−1 Ct
Approximate Ct =F Ct−1 , Ct−1 ; δ with a bivariate sieve:

KT
Vt − 1 Ct Vt − 1 Ct
F , ;δ ≈ FKT (·, δ) = a0 (δ) + ∑ aj (δ)Bj ,
Ct − 1 Ct − 1 j= 1
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
Initial value for Vtt at time t = 0, denoted V0 , taken as a
C C
0

unknown scalar parameter to be estimated.
V0 KT KT
Given C0 , aj j= 1
, Bj j= 1
and data on consumption
T T
Ct Vi
Ct−1 , use FKT to generate a sequence Ci .
t= 1 i= 1



Recall m(wt , δo , F∗ (·, δo )) ≡ E {γ(zt+1 , δo , F∗ (·, δo ))|wt } = 0.

First-step SMD estimate F (·) for F∗ (·) based on
T
1
F (·, δ) = arg min ∑ m(wt , δ, FK T
(·, δ))′ m(wt , δ, FKT (·, δ)),
FKT T t= 1

m(wt , δ, FKT (·, δ)) any nonpara. estimator of m.

Do this for a three dimensional grid of values of
δ = ( β, θ, ρ)′ .



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
m(w, δ, F) = ∑ γ(zt+1, δ, F)pJ T
(wt )′ (P′ P)−1 pJT (w)
t= 1

m(·) a sieve LS estimator of m(w, δ, F).

Procedure equivalent to regressing each γi on instruments
and taking ﬁtted values as estimate of conditional mean.

EZW Preferences: First Step SMD Est of F∗





Attractive feature of this estimator of F∗ : implemented as GMM
′ −1
FT (·, δ) = arg min gT (δ,FT ; yT ) {IN ⊗ P′ P } gT (δ,FT ; yT ) ,
FT ∈VT
W
(12)
′ ′ ′ ′ ′
where yT = zT +1, ...z2, wT , ...w1 denotes vector of all obs and
T
1
gT (δ,FT ; yT ) = ∑ γ(zt+1, δ,FT )⊗pJT (wt ) (13)
T t=1




′ −1
FT ∈VT
W
(12)
′ ′ ′ ′ ′
T
1
T t=1

Weighting gives greater weight to moments more highly
correlated with instruments pJT (·).




′ −1
FT ∈VT
W
(12)
′ ′ ′ ′ ′
T
1
T 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).


EZW Preferences: Second Step GMM Est of δo

Under correct speciﬁcation, δo satisﬁes :
E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, i = 1, ..., N.



E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, i = 1, ..., N.

Sample moments:
gT (δ, F (·, δ); yT ) ≡ 1
T ∑T=1 γ(zt+1 , δ, F (·, δ)) ⊗ xt .
t



E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, i = 1, ..., N.

Sample moments:
gT (δ, F (·, δ); yT ) ≡ 1
T ∑T=1 γ(zt+1 , δ, F (·, δ)) ⊗ xt .
t
Regardless the model is correctly or incorrectly speciﬁed,
estimate δ by minimizing GMM objective:
′
δ = arg min gT (δ, F (·, δ) ; yT ) W gT (δ, F (·, δ) ; yT )
δ ∈D



E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, i = 1, ..., N.

Sample moments:
gT (δ, F (·, δ); yT ) ≡ 1
T ∑T=1 γ(zt+1 , δ, F (·, δ)) ⊗ xt .
t
′
δ ∈D

−
Examples: W = I, W = GT 1 .



E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, i = 1, ..., N.

Sample moments:
gT (δ, F (·, δ); yT ) ≡ 1
T ∑T=1 γ(zt+1 , δ, F (·, δ)) ⊗ xt .
t
′
δ ∈D

−
F (·, δ) not held ﬁxed in this step: depends on δ!



E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, i = 1, ..., N.

Sample moments:
gT (δ, F (·, δ); yT ) ≡ 1
T ∑T=1 γ(zt+1 , δ, F (·, δ)) ⊗ xt .
t
′
δ ∈D

−
Estimator F (·, δ) obtained using min. dist over a grid of values δ.



E {γi (zt+1, δo , F∗ (·, δo )) ⊗ xt } = 0, i = 1, ..., N.

Sample moments:
gT (δ, F (·, δ); yT ) ≡ 1
T ∑T=1 γ(zt+1 , δ, F (·, δ)) ⊗ xt .
t
′
δ ∈D

−
Estimator F (·, δ) obtained using min. dist over a grid of values δ.
Choose the δ and corresponding F (·, δ) that minimizes GMM
criterion.

EZW Recursive Preferences: Two Step Estimation

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!






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.



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.







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).

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).



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
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)



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 )′



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.)


Let {yt }N 1 denote simulated data (in obs eqn).
t=



t=

Let {yt }T=1 denote historical data on same variables.
˜ t



t=

˜ t
Auxiliary model of hist. data: e.g., VAR, with density
f (yt |yt−L , ...yt−1 , α), good LOM for data–f -model.



t=

˜ t
Score function of f -model:
∂
sf (yt |yt−L , ...yt−1 , α) = ln[f (yt |yt−L , ..., yt−1 , α)]
∂α



t=

˜ t
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
α = arg maxLT (α, {yt }T=1 )
˜ ˜ t
α

T
1
LT (α, {yt }T=1 ) =
˜ t ∑ ln f (yt |yt−L , ..., yt−1 , α)
˜ ˜ ˜
T t= L+ 1



First-order-condition:
T
∂ 1
LT (α, {yt }T=1 ) = 0
˜ ˜ t or, ∑ sf (yt |yt−L , ..., yt−1 , α) = 0.
˜ ˜ ˜ ˜
∂α T t= L+ 1



T
∂ 1
LT (α, {yt }T=1 ) = 0
˜ ˜ t or, ∑ sf (yt |yt−L , ..., yt−1 , α) = 0.
˜ ˜ ˜ ˜
∂α T t= L+ 1
Idea: since above, good estimator for ρd is one that sets
1 N
∑ s (yt (ρd )|yt−L (ρd ), ..., yt−1(ρd ), α) ≈ 0.
N t= L+ 1 f
˜



T
∂ 1
LT (α, {yt }T=1 ) = 0
˜ ˜ t or, ∑ sf (yt |yt−L , ..., yt−1 , α) = 0.
˜ ˜ ˜ ˜
∂α T t= L+ 1
1 N
N t= L+ 1 f
˜

If dim(α) >dim(ρd ), use GMM:
1 N
mT ( ρ d , α ) = ∑ s (yt (ρd )|yt−L (ρd ), ..., yt−1 (ρd ), α)
N t= L+ 1 f
˜
dim( α)×1



T
∂ 1
LT (α, {yt }T=1 ) = 0
˜ ˜ t or, ∑ sf (yt |yt−L , ..., yt−1 , α) = 0.
˜ ˜ ˜ ˜
∂α T t= L+ 1
1 N
N t= L+ 1 f
˜

If dim(α) >dim(ρd ), use GMM:
1 N
mT ( ρ d , α ) = ∑ s (yt (ρd )|yt−L (ρd ), ..., yt−1 (ρd ), α)
N t= L+ 1 f
˜
dim( α)×1

The GMM estimator is
ρd = arg min{mT (ρd , α )′ I −1 mT (ρd , α)
˜ ˜ ˜
ρd



˜ ˜
GMM: ρd = arg min{mT (ρd , α)′ I −1 mT (ρd , α)}.
˜
ρd

˜
I −1 is inv. of var. of score, data determined from f -model
T ′
˜ ∂ ∂
I= ∑ ∂α˜
ln[f (yt |yt−L , ..., yt−1 , α )]
˜ ˜ ˜ ˜
∂α˜
ln[f (yt |yt−L , ..., yt−1 , α)]
˜ ˜ ˜ ˜
t= 1



˜ ˜
˜
ρd

˜
T ′
˜ ∂ ∂
I= ∑ ∂α˜
ln[f (yt |yt−L , ..., yt−1 , α )]
˜ ˜ ˜ ˜
∂α˜
ln[f (yt |yt−L , ..., yt−1 , α)]
˜ ˜ ˜ ˜
t= 1

Sims {y}N 1 follow stationary dens. p(yt−L , ..., yt |ρd ). Note:
t=
no closed-form for p(·|ρd ).



˜ ˜
˜
ρd

˜
T ′
˜ ∂ ∂
I= ∑ ∂α˜
ln[f (yt |yt−L , ..., yt−1 , α )]
˜ ˜ ˜ ˜
∂α˜
ln[f (yt |yt−L , ..., yt−1 , α)]
˜ ˜ ˜ ˜
t= 1

t=
as
Intuition: mT (ρd , α) → m(ρd , α) as N → ∞, where

m ( ρd , α ) = ··· s(yt−L , ..., yt , α)p(yt−L , ..., yt |ρd )dyt−L · · · dyt



˜ ˜
˜
ρd

˜
T ′
˜ ∂ ∂
I= ∑ ∂α˜
ln[f (yt |yt−L , ..., yt−1 , α )]
˜ ˜ ˜ ˜
∂α˜
ln[f (yt |yt−L , ..., yt−1 , α)]
˜ ˜ ˜ ˜
t= 1

t=
as


⇒ use Monte Carlo compute expect. of s(·) under p(·|ρd ).



as




as


If f = p above is mean of scores of likelihood. Should be
zero, given f.o.c for MLE estimator.



as



Thus, if data do follow the structural model p(·|ρd ), then
m(ρo , αo ) = 0, forms basis of a speciﬁcation test.
d



as



Thus, if data do follow the structural model p(·|ρd ), then
m(ρo , αo ) = 0, forms basis of a speciﬁcation test.
d

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.



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.





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.


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...





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.


Example: scaled consumption models.




Rather than ask whether scaled models are true...




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.


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 ⇒






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.


Goal: not to ﬁnd better factors, but rather to explain
ﬁnancial factors from deeper economic models.


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