ludvigson methodslecture1

NBER Summer Institute EconometricsMethods 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 innature–derived from IMRS over consumption–assetprices are derived endogenously from these.





Some cons-based models work better than others, but...






...emphasize here: all models are misspecified, and macrovariables often measured with error. Therefore:







1 move away from specification tests of perfect fit (givensampling error),








2 toward estimation and testing that recognize all models aremisspecified,









3 toward methods permit comparison of magnitude ofmisspecification among multiple, competing macro models.









3 toward methods permit comparison of magnitude ofmisspecification 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, CRRAutility model




Incorporating conditioning information: scaledconsumption-based models





Generalizations of CRRA utility: recursive utility






Semi-nonparametric minimum distance estimators:unrestricted LOM for data







Simulation methods: restricted LOM







Simulation methods: restricted LOM

Consumption-based asset pricing: concluding thoughts


GMM Review (Hansen, 1982)

Economic model implies set of r population momentrestrictions

E{h (θ, wt)︸︷︷︸(r×1)

} = 0




E{h (θ, wt)︸︷︷︸(r×1)

} = 0

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




E{h (θ, wt)︸︷︷︸(r×1)

} = 0


θ is an a × 1 vector of coefficients




E{h (θ, wt)︸︷︷︸(r×1)

} = 0


θ is an a × 1 vector of coefficients

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



Sample moments:

g(θ; yT)︸︷︷︸(r×1)

≡ (1/T)T

∑t=1

h (θ, wt) ,



Sample moments:

g(θ; yT)︸︷︷︸(r×1)

≡ (1/T)T

∑t=1

h (θ, wt) ,

yT ≡(w′

T, w′T−1, ...w′

1

)′is a T · h × 1 vector of observations.



Sample moments:

g(θ; yT)︸︷︷︸(r×1)

≡ (1/T)T

∑t=1

h (θ, wt) ,

yT ≡(w′

T, w′T−1, ...w′

1


The GMM estimator θ minimizes the scalar

Q (θ; yT) = [g(θ; yT)]′

(1×r)

WT(r×r)

[g(θ; yT)](r×1)

, (1)

{WT}∞T=1 a sequence of r × r positive definite matrices

which may be a function of the data, yT.



Sample moments:

g(θ; yT)︸︷︷︸(r×1)

≡ (1/T)T

∑t=1

h (θ, wt) ,

yT ≡(w′

T, w′T−1, ...w′

1



Q (θ; yT) = [g(θ; yT)]′

(1×r)

WT(r×r)

[g(θ; yT)](r×1)

, (1)



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



Sample moments:

g(θ; yT)︸︷︷︸(r×1)

≡ (1/T)T

∑t=1

h (θ, wt) ,

yT ≡(w′

T, w′T−1, ...w′

1



Q (θ; yT) = [g(θ; yT)]′

(1×r)

WT(r×r)

[g(θ; yT)](r×1)

, (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.Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models


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




Optimal weighting WT = S−1

Sr×r

=∞

∑j=−∞

E

[h (θo, wt)]︸︷︷︸r×1

[h(θo, wt−j

)]′︸︷︷︸

1×r

.




Optimal weighting WT = S−1

Sr×r

=∞

∑j=−∞

E

[h (θo, wt)]︸︷︷︸r×1

[h(θo, wt−j

)]′︸︷︷︸

1×r

.

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



ST depends on θT which depends on ST. Employ aniterative procedure:



ST depends on θT which depends on ST. Employ aniterative procedure:

1 Obtain an initial estimate of θ = θ(1)T , by minimizing

Q (θ; yT) subject to arbitrary weighting matrix, e.g., W = I.

2 Use θ(1)T to obtain initial estimate of S = S

(1)T .

3 Re-minimize Q (θ; yT) using initial estimate S(1)T ; obtain

new estimate θ(2)T .

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


Classic Example: Hansen and Singleton (1982)

Investors maximize utility

maxCt

Et

[∞

∑i=0

βiu (Ct+i)

]




maxCt

Et

[∞

∑i=0

βiu (Ct+i)

]

Power (isoelastic) utility

u (Ct) =C

1−γt

1−γ γ > 0

u (Ct) = ln(Ct) γ = 1

(2)




maxCt

Et

[∞

∑i=0

βiu (Ct+i)

]

Power (isoelastic) utility

u (Ct) =C

1−γt

1−γ γ > 0

u (Ct) = ln(Ct) γ = 1

(2)

N assets => N first-order conditions

C−γt = βEt

{(1 +ℜi,t+1) C

−γt+1

}i = 1, ..., N. (3)


Asset Pricing Example: Hansen and Singleton (1982)

Re-write moment conditions

0 = Et

{1 − β

[(1 + ℜi,t+1)

C−γt+1

C−γt

]}. (4)




0 = Et

{1 − β

[(1 + ℜi,t+1)

C−γt+1

C−γt

]}. (4)

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




0 = Et

{1 − β

[(1 + ℜi,t+1)

C−γt+1

C−γt

]}. (4)


x∗t denotes info set of investors

0 = E{[

1 −{

β (1 + ℜi,t+1) C−γt+1/C

−γt

}]|x∗t}

i = 1, ...N

(5)




0 = Et

{1 − β

[(1 + ℜi,t+1)

C−γt+1

C−γt

]}. (4)


x∗t denotes info set of investors

0 = E{[

1 −{

β (1 + ℜi,t+1) C−γt+1/C

−γt

}]|x∗t}

i = 1, ...N

(5)

xt ⊂ x∗t . Conditional model (5) => unconditional model:

0 = E

{[1 −

{β (1 +ℜi,t+1)

C−γt+1

C−γt

}]xt

}i = 1, ...N

(6)



Let xt be M × 1. Then r = N · M and,

h (θ, wt)r×1

=

[1 − β

{(1 + ℜ1,t+1)

C−γt+1

C−γt

}]xt

[1 − β

{(1 + ℜ2,t+1)

C−γt+1

C−γt

}]xt

···[

1 − β

{(1 + ℜN,t+1)

C−γt+1

C−γt

}]xt

(7)




h (θ, wt)r×1

=

[1 − β

{(1 + ℜ1,t+1)

C−γt+1

C−γt

}]xt

[1 − β

{(1 + ℜ2,t+1)

C−γt+1

C−γt

}]xt

···[

1 − β

{(1 + ℜN,t+1)

C−γt+1

C−γt

}]xt

(7)

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




h (θ, wt)r×1

=

[1 − β

{(1 + ℜ1,t+1)

C−γt+1

C−γt

}]xt

[1 − β

{(1 + ℜ2,t+1)

C−γt+1

C−γt

}]xt

···[

1 − β

{(1 + ℜN,t+1)

C−γt+1

C−γt

}]xt

(7)

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

Take sample mean of (7) to get g(θ; yT), minimize

minθ

Q (θ; yT) = [g(θ; yT)]︸︷︷︸1×r

′WTr×r

[g(θ; yT)]︸︷︷︸r×1



Test of Over-identifying (OID) restrictions:

TQ(

θ; yT

)a∼ χ2(r − a)




TQ(

θ; yT

)a∼ χ2(r − a)

HS use lags of cons growth and returns in xt; index andindustry returns, NDS expenditures.

Estimates of β ≈ .99, RRA low = .35 to .999. No equitypremium puzzle! But....




TQ(

θ; yT

)a∼ χ2(r − a)



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




TQ(

θ; yT

)a∼ χ2(r − a)




Campbell, Lo, MacKinlay (1997): OID rejections strongerwhenever stock returns and commercial paper areincluded as test returns. Why?




TQ(

θ; yT

)a∼ χ2(r − a)





Model cannot capture predictable variation in excessreturns over commercial paper ⇒




TQ(

θ; yT

)a∼ χ2(r − a)





Model cannot capture predictable variation in excessreturns 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: largeunconditional Euler equation (pricing) errors even whenparams freely chosen.





Let Mt+1 = β(Ct+1/Ct)−γ. Define Euler equation errors:

ejR ≡ E[Mt+1R

jt+1] − 1

ejX ≡ E[Mt+1(R

jt+1 − R

ft+1)]





Let Mt+1 = β(Ct+1/Ct)−γ. Define Euler equation errors:

ejR ≡ E[Mt+1R

jt+1] − 1

ejX ≡ E[Mt+1(R

jt+1 − R

ft+1)]

Choose params: minβ,γ g′TWTgT where jth element of gT

gj,t(γ, β) = 1T ∑

Tt=1 e

jR,t

gj,t(γ) = 1T ∑

Tt=1 e

jX,t


Unconditional Euler Equation Errors, Excess Returns

ejX ≡ E[β(Ct+1/Ct)−γ(R

jt+1 − R

ft+1)] j = 1, ..., N

RMSE =√

1N ∑

Nj=1[e

jX]2, RMSR =

√1N ∑

Nj=1[E(R

jt+1 − R

ft+1)]

2

Source: Lettau and Ludvigson (2009). Rs is the excess return on CRSP-VW index over 3-Mo T-bill rate. Rs & 6 FFrefers to this return plus 6 size and book-market sorted portfolios provided by Fama and French. For each value ofγ, 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 arefreely chosen to minimize errors.

Unlike the equity premium puzzle of Mehra and Prescott(1985), large Euler eq. errors cannot be resolved with highrisk aversion.





Lettau and Ludvigson (2009): Leading consumption-basedasset pricing theories fail to explain the mispricing ofclassic CCAPM.






Anomaly is striking b/c early evidence (e.g., Hansen &Singleton) that the classic model’s Euler equations wereviolated provided the impetus for developing these newermodels.






Anomaly is striking b/c early evidence (e.g., Hansen &Singleton) that the classic model’s Euler equations wereviolated provided the impetus for developing these newermodels.

Results imply data on consumption and asset returns notjointly lognormal!


GMM Asset Pricing With Non-Optimal WeightingComparing specification error: Hansen and Jagannathan, 1997

Asset pricing applications often require WT , S−1. Why?




One reason: assessing specification error, comparingmodels.





Consider two estimated models of SDF, e.g.,

1 CCAPM: M(1)t+1 = β(Ct+1/Ct)−γ, OID restricts not rejected

2 CAPM: M(2)t+1 = a + bRm,t+1, OID restricts rejected








May we conclude Model 1 is superior?









No. Hansen’s J-test of OID restricts depends on modelspecific S: J = g′

TS−1gT.









No. Hansen’s J-test of OID restricts depends on modelspecific S: J = g′

TS−1gT.

Model 1 can look better simply b/c the SDF and pricingerrors gT are more volatile, not b/c pricing errors are lower.



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

DistT(θ) =

√min

θgT (θ)′ G−1

T gT (θ), GT ≡ 1

T

T

∑t=1

RtR′t︸︷︷︸

N×N

gT (θ) ≡ 1

T

T

∑t=1

[Mt(θ)Rt − 1N]




DistT(θ) =

√min

θgT (θ)′ G−1

T gT (θ), GT ≡ 1

T

T

∑t=1

RtR′t︸︷︷︸

N×N

gT (θ) ≡ 1

T

T

∑t=1

[Mt(θ)Rt − 1N]

DistT does not reward SDF volatility => suitable formodel comparison.




DistT(θ) =

√min

θgT (θ)′ G−1

T gT (θ), GT ≡ 1

T

T

∑t=1

RtR′t︸︷︷︸

N×N

gT (θ) ≡ 1

T

T

∑t=1

[Mt(θ)Rt − 1N]


DistT is a measure of model misspecification:




DistT(θ) =

√min

θgT (θ)′ G−1

T gT (θ), GT ≡ 1

T

T

∑t=1

RtR′t︸︷︷︸

N×N

gT (θ) ≡ 1

T

T

∑t=1

[Mt(θ)Rt − 1N]



Gives distance between Mt(θ) and nearest point in space ofall SDFs that price assets correctly.




DistT(θ) =

√min

θgT (θ)′ G−1

T gT (θ), GT ≡ 1

T

T

∑t=1

RtR′t︸︷︷︸

N×N

gT (θ) ≡ 1

T

T

∑t=1

[Mt(θ)Rt − 1N]



Gives distance between Mt(θ) and nearest point in space ofall SDFs that price assets correctly.

Gives maximum pricing error of any portfolio formed fromthe N assets.



Appeal of HJ Distance metric:




Recognizes all models are misspecified.

Provides method for comparing models by assessing whichis least misspecified.






Important problem: how to compare HJ distancesstatistically?






Important problem: how to compare HJ distancesstatistically?

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


GMM Asset Pricing With Non-Optimal WeightingStatistical comparison of HJ distance: Chen and Ludvigson, 2009

Chen and Ludvigson (2009) compare HJ distances amongK competing models using White’s reality check method.




1 Take benchmark model, e.g., model with smallest squareddistance d2

1,T ≡ min{d2j,T}K

j=1.

2 Null: d21,T − d2

2,T ≤ 0, where d22,T is competing model with

the next smallest squared distance.

3 Test statistic T W =√

T(d21,T − d2

2,T).

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 squareddistance d2

1,T ≡ min{d2j,T}K

j=1.

2 Null: d21,T − d2

2,T ≤ 0, where d22,T is competing model with

the next smallest squared distance.

3 Test statistic T W =√

T(d21,T − d2

2,T).

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 motionfor data, multiple competing possibly nonlinear, SDFmodels.



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 tomost asset pricing models:






Proof of limiting distributions exists for applications tomost asset pricing models:

For parametric models (Hansen, Heaton, Luttmer ’95)

For semiparametric models (Ai and Chen ’07).


GMM Asset Pricing With Non-Optimal WeightingReasons to use identity matrix: econometric problems

Econometric problems: near singular S−1T or G−1

T .

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.


GMM Asset Pricing With Non-Optimal WeightingReasons to use identity matrix: economically interesting portfolios

Original test assets may have economically meaningfulcharacteristics (e.g., size, value).




Using WT = S−1T or G−1

T same as using WT = I and doingGMM on re-weighted portfolios of original test assets.

Triangular factorization S−1 = (P′P), P lower triangular

min g′TS−1gT ⇔ (g′

TP′)I(PgT)








TP′)I(PgT)

Re-weighted portfolios may not provide large spread inaverage returns.








TP′)I(PgT)

Re-weighted portfolios may not provide large spread inaverage returns.

May imply implausible long and short positions in testassets.


GMM Asset Pricing With Non-Optimal WeightingReasons not to use WT = I: objective function dependence on test asset choice

Using WT = [ET(R′R)]−1, GMM objective function isinvariant 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)]

= [E (MAR) − A1N]′ E(ARR′A

)−1[E (MAR − A1N)] .




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)]

= [E (MAR) − A1N]′ E(ARR′A

)−1[E (MAR − A1N)] .

With WT = I or other fixed weighting, GMM objectivedepends on choice of test assets.


More Complex Preferences: ScaledConsumption-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

u(Ct) =C

1−γt

1 − γ⇒ Mt+1 ≈ β

︸︷︷︸at=a0

− βγ︸︷︷︸bt=b0

∆ct+1




Mt+1 ≈ at + bt∆ct+1, ct+1 ≡ ln(Ct+1)


u(Ct) =C

1−γt

1 − γ⇒ Mt+1 ≈ β

︸︷︷︸at=a0

− βγ︸︷︷︸bt=b0

∆ct+1

Model with habit and time-varying risk aversion: Campbell andCochrane ’99, Menzly et. al ’04

u(Ct, St) =(CtSt)

1−γ

1 − γ, St+1 ≡ Ct − Xt

Ct

⇒ Mt+1 ≈ β(

1 − γ(φ − 1)(st − s)︸︷︷︸

at

− γ(1 + ψ(st))︸︷︷︸bt

∆ct+1

)




Mt+1 ≈ at + bt∆ct+1, ct+1 ≡ ln(Ct+1)


u(Ct) =C

1−γt

1 − γ⇒ Mt+1 ≈ β

︸︷︷︸at=a0

− βγ︸︷︷︸bt=b0

∆ct+1

Model with habit and time-varying risk aversion: Campbell andCochrane ’99, Menzly et. al ’04

u(Ct, St) =(CtSt)

1−γ

1 − γ, St+1 ≡ Ct − Xt

Ct

⇒ Mt+1 ≈ β(

1 − γ(φ − 1)(st − s)︸︷︷︸

at

− γ(1 + ψ(st))︸︷︷︸bt

∆ct+1

)

Proxies for time-varying risk-premia should be good proxies fortime-variation in at and bt.


Scaled Consumption-Based Models

Mt+1 ≈ at + bt∆ct+1

Empirical specification: Lettau and Ludvigson (2001a, 2001b):

at = a0 + a1zt, bt = b0 + b1zt

zt = cayt ≡ ct − αaat − αyyt, (cointegrating residual)cayt related to log consumption-(aggregate) wealth ratio.cayt strong predictor of excess stock market returns


Scaled Consumption-Based Models

Mt+1 ≈ at + bt∆ct+1

Empirical specification: Lettau and Ludvigson (2001a, 2001b):

at = a0 + a1zt, bt = b0 + b1zt

zt = cayt ≡ ct − αaat − αyyt, (cointegrating residual)cayt related to log consumption-(aggregate) wealth ratio.cayt strong predictor of excess stock market returns

Other examples: including housing consumption

U(Ct, Ht) =C

1− 1σ

t

1 − 1σ

Ct =

[χC

ε−1ε

t + (1 − χ) Hε−1

εt

] εε−1

,

⇒ ln Mt+1 ≈ at + bt∆ ln Ct+1 + dt∆ ln St+1, St+1 ≡ pCt Ct

pCt Ct + pH

t Ht

Lustig and Van Nieuwerburgh ’05 (incomplete markets):

at = a0 + a1zt, bt = b0 + b1zt, dt = d0 + d1zt

zt = housing collateral ratio (measures quantity of risksharing)


Scaled Consumption-Based ModelsDistinguishing two types of conditioning, or state dependence

Two kinds of conditioning are often confused.




Euler equation: E[Mt+1Rt+1|zt

]= 1

Unconditional version: E[Mt+1Rt+1

]= 1





]= 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 = b′t ft+1 with bt = b0 + b1zt

= b′(ft+1 ⊗ (1 zt)′)





]= 1


]= 1



(Ri,t+1 ⊗ (1 zt)′

)]= 1



= b′(ft+1 ⊗ (1 zt)′)

Scaled consumption-based models are conditional in sensethat Mt+1 is a state-dependent function of ∆ ln Ct+1

⇒ scaled factors





]= 1


]= 1



(Ri,t+1 ⊗ (1 zt)′

)]= 1



= b′(ft+1 ⊗ (1 zt)′)

Scaled consumption-based models are conditional in sensethat Mt+1 is a state-dependent function of ∆ ln Ct+1

⇒ scaled factors

Scaled consumption-based models have been tested onunconditional moments, E

[Mt+1Rt+1

]= 1

⇒ NO scaled returns.



Scaled CCAPM turns a single factor model withstate-dependent weights into multi-factor model ft withconstant weights:

Mt+1 = (a0 + a1zt) + (b0 + b1zt) ∆ ln Ct+1

= a0 + a1 zt︸︷︷︸f1,t+1

+b0 ∆ ln Ct+1︸︷︷︸f2,t+1

+b1(zt∆ ln Ct+1︸︷︷︸f3,t+1

)




Mt+1 = (a0 + a1zt) + (b0 + b1zt) ∆ ln Ct+1

= a0 + a1 zt︸︷︷︸f1,t+1

+b0 ∆ ln Ct+1︸︷︷︸f2,t+1

+b1(zt∆ ln Ct+1︸︷︷︸f3,t+1

)

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




Mt+1 = (a0 + a1zt) + (b0 + b1zt) ∆ ln Ct+1

= a0 + a1 zt︸︷︷︸f1,t+1

+b0 ∆ ln Ct+1︸︷︷︸f2,t+1

+b1(zt∆ ln Ct+1︸︷︷︸f3,t+1

)

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

Scaled consumption models have multiple, constant betasfor 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[RiF′]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′)−1Cov(f, f′)b

E[F′]b

= R0 − R0β′Cov(f, f′)b

= R0 − β′λ ⇒ multiple, constant betas

Estimate cross-sectional model using Fama-MacBeth (see Brandtlecture).


Fama-MacBeth Methodology: Preview–See Brandt

Step 1: Estimate β’s in time-series regression for eachportfolio i:

βi ≡ Cov(ft+1, f′t+1)−1Cov(ft+1, Ri,t+1)

Step 2: Cross-sectional regressions (T of them):

Ri,t+1 − R0,t = αi,t + β′iλt

λ = 1/TT

∑t=1

λt; σ2(λ) = 1/TT

∑t=1

(λt − λ)2

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



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 betamodel:

Rit = a + β∆c∆ct + β∆c,z∆ctzt−1 + βzzt−1








Rewrite as

Rit = a + (βc + βc,zzt−1)︸︷︷︸

time-varying beta

∆ct + βzzt−1








Rewrite as

Rit = a + (βc + βc,zzt−1)︸︷︷︸

time-varying beta

∆ct + βzzt−1

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







Latter may require variables beyond a few that capturerisk-premia.








Approximating condition mean well requires largenumber of instruments (misspecified information sets)

Results sensitive to chosen conditioning variables, may failto span information sets of market participants.








Approximating condition mean well requires largenumber of instruments (misspecified information sets)

Results sensitive to chosen conditioning variables, may failto span information sets of market participants.

Partial solution: summarize information in large numberof time-series with few estimated dynamic factors (e.g.,Ludvigson and Ng ’07, ’09).



Bottom lines:



Bottom lines:

1 Conditional moments of Mt+1Rt+1 difficult to model ⇒reason to focus on unconditional momentsE[Mt+1Rt+1] = 1.



Bottom lines:


2 Models are misspecified: interesting question is whetherstate-dependence of Mt+1 on consumption growth ⇒ lessmisspecification than standard, fixed-weight CCAPM.



Bottom lines:


2 Models are misspecified: interesting question is whetherstate-dependence of Mt+1 on consumption growth ⇒ lessmisspecification 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 recursivepreferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).




Two reasons recursive utility is of interest:





More flexibility as regards attitudes toward risk andintertemporal substitution.






Preferences deliver an added risk factor for explaining assetreturns.






Preferences deliver an added risk factor for explaining assetreturns.

But, only a small amount of econometric work onrecursive preferences ⇒ gap in the literature.



Here discuss two examples of estimating EZW models:




1 For general stationary, consumption growth and cash flowdynamics: 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 ofconsumption and returns (distribution-free estimationprocedure).






In (1) DGP is left unrestricted, as is joint distribution ofconsumption and returns (distribution-free estimationprocedure).

In (2) DGP and distribution of shocks explicitly modeled.


EZW Recursive PreferencesEpstein-Zin-Weil basics

Recursive utility (Epstein, Zin (’89, ’91) & Weil (’89)):

Vt =[(1 − β) C

1−ρt + βRt (Vt+1)

1−ρ] 1

1−ρ

Rt (Vt+1) =(

E[V1−θ

t+1 |Ft

]) 11−θ




Vt =[(1 − β) C


1−ρ] 1

1−ρ

Rt (Vt+1) =(

E[V1−θ

t+1 |Ft

]) 11−θ

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




Vt =[(1 − β) C


1−ρ] 1

1−ρ

Rt (Vt+1) =(

E[V1−θ

t+1 |Ft

]) 11−θ


Rescale utility function (Hansen, Heaton, Li ’05):

Vt

Ct=

[(1 − β) + βRt

(Vt+1

Ct+1

Ct+1

Ct

)1−ρ] 1

1−ρ




Vt =[(1 − β) C


1−ρ] 1

1−ρ

Rt (Vt+1) =(

E[V1−θ

t+1 |Ft

]) 11−θ


Rescale utility function (Hansen, Heaton, Li ’05):

Vt

Ct=

[(1 − β) + βRt

(Vt+1

Ct+1

Ct+1

Ct

)1−ρ] 1

1−ρ

Special case: ρ = θ ⇒ CRRA separable utility Vt = βC1−θ

t1−θ .


EZW Recursive PreferencesEpstien-Zin-Weil basics

The MRS is pricing kernel (SDF) with added risk factor:

Mt+1 = β

(Ct+1

Ct

)−ρ

Vt+1

Ct+1

Ct+1

Ct

Rt

(Vt+1

Ct+1

Ct+1

Ct

)

ρ−θ




Mt+1 = β

(Ct+1

Ct

)−ρ

Vt+1

Ct+1

Ct+1

Ct

Rt

(Vt+1

Ct+1

Ct+1

Ct

)

ρ−θ

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




Mt+1 = β

(Ct+1

Ct

)−ρ

Vt+1

Ct+1

Ct+1

Ct

Rt

(Vt+1

Ct+1

Ct+1

Ct

)

ρ−θ


Epstein-Zin ’91 use alt. rep. of SDF, uses agg. wealthreturn Rw,t:

Mt+1 =

{β

(Ct+1

Ct

)−ρ} 1−θ

1−ρ {1

Rw,t+1

} θ−ρ1−ρ

where Rw,t proxied by stock market return.




Mt+1 = β

(Ct+1

Ct

)−ρ

Vt+1

Ct+1

Ct+1

Ct

Rt

(Vt+1

Ct+1

Ct+1

Ct

)

ρ−θ


Epstein-Zin ’91 use alt. rep. of SDF, uses agg. wealthreturn Rw,t:

Mt+1 =

{β

(Ct+1

Ct

)−ρ} 1−θ

1−ρ {1

Rw,t+1

} θ−ρ1−ρ

where Rw,t proxied by stock market return.

Problem: Rw,t+1 represents a claim to future Ct, itselfunobservable.



1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-seriesprocess, log(V/C) has an analytical solution.




2 If returns, Ct are jointly lognormal and homoscedastic, riskpremia are approx. log-linear functions of COV betweenreturns, 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 lessrestrictive assumptions.


EZW Recursive PreferencesUnrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

CFL: semiparametric approach to estimate EZW modelwithout:




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 datarelative to competing model specifications.










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










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

Semiparametric approach is sieve minimum distance

(SMD) procedure.



First order conditions for optimal consumption choice:

Et

β

(Ct+1

Ct

)−ρ

Vt+1

Ct+1

Ct+1

Ct

Rt

(Vt+1

Ct+1

Ct+1

Ct

)

ρ−θ

Ri,t+1 − 1

= 0 (8)




Et

β

(Ct+1

Ct

)−ρ

Vt+1

Ct+1

Ct+1

Ct

Rt

(Vt+1

Ct+1

Ct+1

Ct

)

ρ−θ

Ri,t+1 − 1

= 0 (8)

CFL: plug VtCt

=

[(1 − β) + βRt

(Vt+1

Ct+1

Ct+1

Ct

)1−ρ] 1

1−ρ

into (8):

Et

β

(Ct+1

Ct

)−ρ

Vt+1Ct+1

Ct+1Ct

{1β

[VtCt

1−ρ − (1 − β)]} 1

1−ρ

ρ−θ

Ri,t+1 − 1

= 0 i = 1, ..., N.

(9)




Et

β

(Ct+1

Ct

)−ρ

Vt+1

Ct+1

Ct+1

Ct

Rt

(Vt+1

Ct+1

Ct+1

Ct

)

ρ−θ

Ri,t+1 − 1

= 0 (8)

CFL: plug VtCt

=

[(1 − β) + βRt

(Vt+1

Ct+1

Ct+1

Ct

)1−ρ] 1

1−ρ

into (8):

Et

β

(Ct+1

Ct

)−ρ

Vt+1Ct+1

Ct+1Ct

{1β

[VtCt

1−ρ − (1 − β)]} 1

1−ρ

ρ−θ

Ri,t+1 − 1

= 0 i = 1, ..., N.

(9)

N test asset returns, {Ri,t+1}Ni=1. (9) is a x-sect asset pricing model.




Et

β

(Ct+1

Ct

)−ρ

Vt+1

Ct+1

Ct+1

Ct

Rt

(Vt+1

Ct+1

Ct+1

Ct

)

ρ−θ

Ri,t+1 − 1

= 0 (8)

CFL: plug VtCt

=

[(1 − β) + βRt

(Vt+1

Ct+1

Ct+1

Ct

)1−ρ] 1

1−ρ

into (8):

Et

β

(Ct+1

Ct

)−ρ

Vt+1Ct+1

Ct+1Ct

{1β

[VtCt

1−ρ − (1 − β)]} 1

1−ρ

ρ−θ

Ri,t+1 − 1

= 0 i = 1, ..., N.

(9)


Moment restrictions (9) form the basis of empirical investigation.




Et

β

(Ct+1

Ct

)−ρ

Vt+1

Ct+1

Ct+1

Ct

Rt

(Vt+1

Ct+1

Ct+1

Ct

)

ρ−θ

Ri,t+1 − 1

= 0 (8)

CFL: plug VtCt

=

[(1 − β) + βRt

(Vt+1

Ct+1

Ct+1

Ct

)1−ρ] 1

1−ρ

into (8):

Et

β

(Ct+1

Ct

)−ρ

Vt+1Ct+1

Ct+1Ct

{1β

[VtCt

1−ρ − (1 − β)]} 1

1−ρ

ρ−θ

Ri,t+1 − 1

= 0 i = 1, ..., N.

(9)


Moment restrictions (9) form the basis of empirical investigation.

Empirical model is semiparametric: δ ≡ (β, θ, ρ)′ denote finitedimensional parameter vector; Vt/Ct unknown function.



Assume VtCt

an unknown function F: R2 → R of form

Vt

Ct= F

(Vt−1

Ct−1,

Ct

Ct−1

),



Assume VtCt


Vt

Ct= F

(Vt−1

Ct−1,

Ct

Ct−1

),

Assume {Ct/Ct−1 : t = 1, ...} is strictly stationary ergodic;and F(·) is such that the process{Vt/Ct : t = 1, ...} isasymptotically stationary ergodic.



Assume VtCt


Vt

Ct= F

(Vt−1

Ct−1,

Ct

Ct−1

),


Justified if, for example,



Assume VtCt


Vt

Ct= F

(Vt−1

Ct−1,

Ct

Ct−1

),



∆ log(Ct+1) is (possibly nonlinear) function of a hiddenfirst-order Markov process xt.

Under general assumptions, information in xt issummarized by Vt−1/Ct−1 and Ct/Ct−1.



Assume VtCt


Vt

Ct= F

(Vt−1

Ct−1,

Ct

Ct−1

),





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



Assume VtCt


Vt

Ct= F

(Vt−1

Ct−1,

Ct

Ct−1

),





Note: Markov assumption only a motivation for argumentsof 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

γi(zt+1, δ, F) ≡ β

(Ct+1

Ct

)−ρ

F(

VtCt

,Ct+1Ct+1

)Ct+1

Ct

{1β

[{F(

Vt−1Ct−1

, CtCt−1

)}1−ρ− (1 − β)

]} 11−ρ

ρ−θ

Ri,t+1 − 1





γi(zt+1, δ, F) ≡ β

(Ct+1

Ct

)−ρ

F(

VtCt

,Ct+1Ct+1

)Ct+1

Ct

{1β

[{F(

Vt−1Ct−1

, CtCt−1

)}1−ρ− (1 − β)

]} 11−ρ

ρ−θ

Ri,t+1 − 1

δo ≡ (βo, θo, ρo)′, Fo ≡ Fo(zt, δo) denote true parameters thatuniquely solve the conditional moment restrictions (Eulerequations):

E {γi(zt+1, δo, Fo (·, δo))|Ft} = 0 i = 1, ..., N, (10)





γi(zt+1, δ, F) ≡ β

(Ct+1

Ct

)−ρ

F(

VtCt

,Ct+1Ct+1

)Ct+1

Ct

{1β

[{F(

Vt−1Ct−1

, CtCt−1

)}1−ρ− (1 − β)

]} 11−ρ

ρ−θ

Ri,t+1 − 1

δo ≡ (βo, θo, ρo)′, Fo ≡ Fo(zt, δo) denote true parameters thatuniquely solve the conditional moment restrictions (Eulerequations):

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)





(11)

Since mt = 0, mt must have zero variance, mean.





(11)


Thus can find params by minimizing variance or quadraticnorm: min E[(mt)2]. Don’t observe mt ⇒ need estimate mt.





(11)



Since (11) is cond. mean, must hold for each observation, t.





(11)




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





(11)




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

Minimum distance procedure useful for distribution-freeestimation involving conditional moments.



Minimum distance procedure useful for distribution-freeestimation 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.






Tt=1 f (xt, α) = 0.

Then choose parameters α to min g′TWgT.






Tt=1 f (xt, α) = 0.

Then choose parameters α to min g′TWgT.

With GMM we average and then square.

With SMD, we square and then average.



True parameters δo and Fo (·, δo) solve:

minδ∈D

infF∈V

E[m(wt, δ, F)′m(wt, δ, F)

],

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))′




minδ∈D

infF∈V


],



For any candidate δ ≡ (β, θ, ρ)′ ∈ D, defineV∗ ≡ F∗ (zt, δ) ≡ F∗ (·, δ) as:

F∗ (·, δ) = arg infF∈V


]




minδ∈D

infF∈V


],



For any candidate δ ≡ (β, θ, ρ)′ ∈ D, defineV∗ ≡ F∗ (zt, δ) ≡ F∗ (·, δ) as:

F∗ (·, δ) = arg infF∈V


]

It is clear that Fo (zt, δo) = F∗ (zt, δo)


EZW Recursive Preferences: Two-Step ProcedureUnrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

First step: For any candidate δ ∈ D, an initial estimate ofF∗ (·, 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 offinite dimensional unknown parameters (sieves) FKT

.

Approximation error decreases as KT increases with T.





2 Approximate the unknown function F by a sequence offinite dimensional unknown parameters (sieves) FKT

.

Approximation error decreases as KT increases with T.

Second step: estimates of δo is obtained by solving asample minimum distance problem such as GMM.


EZW Recursive Preferences: First Step SMD Est of F∗Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

Approximate VtCt

= F(

Vt−1

Ct−1, Ct

Ct−1; δ)

with a bivariate sieve:

F

(Vt−1

Ct−1,

Ct

Ct−1; δ

)≈ FKT

(·, δ) = a0(δ)+KT

∑j=1

aj(δ)Bj

(Vt−1

Ct−1,

Ct

Ct−1

)

Sieve coefficients {a0, a1, ..., aKT} depend on δ

Basis functions {Bj(·, ·) : j = 1, ..., KT} have knownfunctional forms independent of δ

Initial value for VtCt

at time t = 0, denoted V0C0

, taken as aunknown scalar parameter to be estimated.

Given V0C0

,{

aj

}KT

j=1,{

Bj

}KT

j=1and data on consumption

{Ct

Ct−1

}T

t=1, use FKT

to generate a sequence{

ViCi

}T

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

F (·, δ) = arg minFKT

1

T

T

∑t=1

m(wt, δ, FKT(·, δ))′m(wt, δ, FKT

(·, δ)),

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

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



Example of nonparametric estimator of m:

Let{

p0j(wt), j = 1, 2, ..., JT

}, R

dw → R be instruments.

pJT (·) ≡ (p01 (·) , ..., p0JT(·))′

Define T × JT matrix P ≡(pJT (w1) , ..., pJT (wT)

)′. Then:

m(w, δ, F) =

(T

∑t=1

γ(zt+1, δ, F)pJT(wt)′(P′P)−1

)pJT (w)

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

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


EZW Preferences: First Step SMD Est of F∗Unrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07





Attractive feature of this estimator of F∗: implemented as GMM

FT (·, δ) = arg minFT∈VT

[gT(δ,FT ; yT)

]′{IN⊗

(P′P)−1}

︸︷︷︸W

[gT(δ,FT; yT)

],

(12)

where yT =(z′T+1, ...z′2, w′

T, ...w′1

)′denotes vector of all obs and

gT(δ,FT ; yT) =1

T

T

∑t=1

γ(zt+1, δ,FT)⊗pJT (wt) (13)






[gT(δ,FT ; yT)

]′{IN⊗

(P′P)−1}

︸︷︷︸W

[gT(δ,FT; yT)

],

(12)

where yT =(z′T+1, ...z′2, w′

T, ...w′1


gT(δ,FT ; yT) =1

T

T

∑t=1


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






[gT(δ,FT ; yT)

]′{IN⊗

(P′P)−1}

︸︷︷︸W

[gT(δ,FT; yT)

],

(12)

where yT =(z′T+1, ...z′2, w′

T, ...w′1


gT(δ,FT ; yT) =1

T

T

∑t=1


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

Weighting can be understood intuitively by noting that variationin conditional mean m(wt, δ, F) is what identifies F∗(·, δ).


EZW Preferences: Second Step GMM Est of δoUnrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

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) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.




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

Sample moments:

gT(δ, F (·, δ); yT) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.

Regardless the model is correctly or incorrectly specified,estimate δ by minimizing GMM objective:

δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

]




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

Sample moments:

gT(δ, F (·, δ); yT) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.


δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

]

Examples: W = I, W = G−1T .




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

Sample moments:

gT(δ, F (·, δ); yT) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.


δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

]


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) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.


δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

]



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) ≡ 1T ∑

Tt=1 γ(zt+1, δ, F (·, δ)) ⊗ xt.


δ = arg minδ∈D

[gT(δ, F (·, δ) ; yT)

]′W[gT(δ, F (·, δ) ; yT)

]



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

Choose the δ and corresponding F (·, δ) that minimizes GMMcriterion.


EZW Recursive Preferences: Two Step EstimationUnrestricted Dynamics, Distribution-Free Estimation: Chen, Favilukis, Ludvigson ’07

Why estimate in two steps? All params could be estimatedin one step by minimizing the SMD criterion.




Less desirable for asset pricing:





1 Want estimates of RRA and EIS to reflect values required tomatch unconditional risk premia. Not possible using SMDwhich emphasizes conditional moments.






2 SMD procedure effectively changes set of test assets–linearcombinations of original portfolio returns. But we may beinterested in explaining original returns!






2 SMD procedure effectively changes set of test assets–linearcombinations of original portfolio returns. But we may beinterested in explaining original returns!

3 Linear combinations may imply implausible long and shortpositions, do not necessarily deliver a large spread inunconditional mean returns.



Procedure allows for model misspecification:




Euler equation need not hold with equality.





As before, compare models by relative magnitude ofmisspecification, rather than...






...asking whether each model individually fits dataperfectly (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 arestatistically different (White reality check–Chen andLudvigson ’09).


EZW Preferences With Restricted Dynamics:Structural Estimation of Long-Run Risk Models: Bansal, Gallant, Tauchen ’07

Bansal, Gallant, Tauchen ’07: SMM estimation of LRRmodel: Bansal & Yaron ’04.




Structural estimation of EZW utility, restricting to specificlaw of motion for cash flows (“long-run risk”LRR).




Structural estimation of EZW utility, restricting to specificlaw of motion for cash flows (“long-run risk”LRR).

Cash flow dynamics in BGT version of LRR model:

∆ct+1 = µc + xc,t + σtεc,t+1

∆dt+1 = µd + φx xc,t︸︷︷︸LR risk

+ φsst + σεdσtεd,t+1

xc,t = φxc,t−1 + σεx σεxc,t

σ2t = σ2 + ν(σ2

t−1 − σ2) + σwwt

st = (µd − µc) + dt − ct

εc,t+1, εd,t+1, εxc,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

, σǫxν, φs, σw)′







2 For each value of ρd on the grid, combine solutions withlong simulation of length N of model.








3 Simulation: Monte Carlo draws from the Normaldistribution 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” momentsbetween dist of simulated and historical data ( “match”made precise below.)



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




Let {yt}Tt=1 denote historical data on same variables.





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






Score function of f -model:

sf (yt|yt−L, ...yt−1, α) =∂

∂αln[f (yt|yt−L, ..., yt−1, α)]






Score function of f -model:

sf (yt|yt−L, ...yt−1, α) =∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

QMLE estimator of auxiliary model on historical data

α = arg maxα

LT(α, {yt}Tt=1)

LT(α, {yt}Tt=1) =

1

T

T

∑t=L+1

ln f (yt|yt−L, ..., yt−1, α)



First-order-condition:

∂

∂αLT(α, {yt}T

t=1) = 0 or,1

T

T

∑t=L+1

sf (yt|yt−L, ..., yt−1, α) = 0.




∂

∂αLT(α, {yt}T

t=1) = 0 or,1

T

T

∑t=L+1

sf (yt|yt−L, ..., yt−1, α) = 0.

Idea: since above, good estimator for ρd is one that sets

1

N

N

∑t=L+1

sf (yt(ρd)|yt−L(ρd), ..., yt−1(ρd), α) ≈ 0.




∂

∂αLT(α, {yt}T

t=1) = 0 or,1

T

T

∑t=L+1

sf (yt|yt−L, ..., yt−1, α) = 0.


1

N

N

∑t=L+1


If dim(α) >dim(ρd), use GMM:

mT(ρd, α)︸︷︷︸dim(α)×1

=1

N

N

∑t=L+1

sf (yt(ρd)|yt−L(ρd), ..., yt−1(ρd), α)




∂

∂αLT(α, {yt}T

t=1) = 0 or,1

T

T

∑t=L+1

sf (yt|yt−L, ..., yt−1, α) = 0.


1

N

N

∑t=L+1


If dim(α) >dim(ρd), use GMM:

mT(ρd, α)︸︷︷︸dim(α)×1

=1

N

N

∑t=L+1

sf (yt(ρd)|yt−L(ρd), ..., yt−1(ρd), α)

The GMM estimator is

ρd = arg minρd

{mT(ρd, α)′I−1mT(ρd, α)



GMM: ρd = arg minρd

{mT(ρd, α)′I−1mT(ρd, α)}.

I−1 is inv. of var. of score, data determined from f -model

I =T

∑t=1

{∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

}{∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

}′






I =T

∑t=1

{∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

}{∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

}′

Sims {y}Nt=1 follow stationary dens. p(yt−L, ..., yt|ρd). Note:

no closed-form for p(·|ρd).






I =T

∑t=1

{∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

}{∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

}′



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

m(ρd, α) =∫

· · ·∫

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






I =T

∑t=1

{∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

}{∂

∂αln[f (yt|yt−L, ..., yt−1, α)]

}′




m(ρd, α) =∫

· · ·∫


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




m(ρd, α) =∫

· · ·∫





m(ρd, α) =∫

· · ·∫


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




m(ρd, α) =∫

· · ·∫



Thus, if data do follow the structural model p(·|ρd), thenm(ρo

d, αo) = 0, forms basis of a specification test.




m(ρd, α) =∫

· · ·∫



Thus, if data do follow the structural model p(·|ρd), thenm(ρo

d, αo) = 0, forms basis of a specification test.

Summary: solve model for many values of ρd, store longsimulations of model each time, do one-time estimation ofauxiliary f -model. Choose ρd to minimize GMM criterionabove.



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, MLEefficiency is obtained.






If dim(α) >dim(ρd), score-based SMM is consistent,asymptotically normal, assuming:







That the auxiliary model is rich enough to identifynon-linear structural model. Sufficient conditions foridentification unknown.







That the auxiliary model is rich enough to identifynon-linear structural model. Sufficient conditions foridentification 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 macroeconomicrisks, given by primitives in the IMRS over consumption.




No model that relates returns to other returns can explainasset prices in terms of primitive economic shocks. Suchmodels of SDF only describe asset prices.





So far many consumption-based models have beenevaluated using calibration exercises ⇒






A crucial next step in evaluating consumption-basedmodels is structural econometric estimation. But...






A crucial next step in evaluating consumption-basedmodels is structural econometric estimation. But...

...models are imperfect and will never fit data infallibly.

Argue here for need to move away from testing if modelsare true, towards comparison of models based on magnitudeof 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 onconsumption growth reduces misspecification over theanalogous non-state-dependent model.



Macroeconomic data, unlike financial measured with error.




⇒ Can’t expect such models to perform as well as financialfactor 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 onportfolios (i.e., mimicking portfolios).







In which case, they will always perform at least as well, orbetter than, mismeasured macro factors from true Mt ⇒







In which case, they will always perform at least as well, orbetter than, mismeasured macro factors from true Mt ⇒

Not sensible to run horse races between financial factormodels and macro models.



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


ludvigson methodslecture1

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