Pages from ludvigson methodslecture 2

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


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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