1.
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
2.
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
3.
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
4.
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
5.
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspeciﬁed, and macro
variables often measured with error. Therefore:
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
6.
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspeciﬁed, and macro
variables often measured with error. Therefore:
1 move away from speciﬁcation tests of perfect ﬁt (given
sampling error),
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
7.
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspeciﬁed, and macro
variables often measured with error. Therefore:
1 move away from speciﬁcation tests of perfect ﬁt (given
sampling error),
2 toward estimation and testing that recognize all models are
misspeciﬁed,
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
8.
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspeciﬁed, and macro
variables often measured with error. Therefore:
1 move away from speciﬁcation tests of perfect ﬁt (given
sampling error),
2 toward estimation and testing that recognize all models are
misspeciﬁed,
3 toward methods permit comparison of magnitude of
misspeciﬁcation among multiple, competing macro models.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
9.
GMM and Consumption-Based Models: Themes
Why care about consumption-based models?
True systematic risk factors are macroeconomic in
nature–derived from IMRS over consumption–asset
prices are derived endogenously from these.
Some cons-based models work better than others, but...
...emphasize here: all models are misspeciﬁed, and macro
variables often measured with error. Therefore:
1 move away from speciﬁcation tests of perfect ﬁt (given
sampling error),
2 toward estimation and testing that recognize all models are
misspeciﬁed,
3 toward methods permit comparison of magnitude of
misspeciﬁcation among multiple, competing macro models.
Themes are important in choosing which methods to use.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
10.
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
11.
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
12.
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Generalizations of CRRA utility: recursive utility
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
13.
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Generalizations of CRRA utility: recursive utility
Semi-nonparametric minimum distance estimators:
unrestricted LOM for data
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
14.
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Generalizations of CRRA utility: recursive utility
Semi-nonparametric minimum distance estimators:
unrestricted LOM for data
Simulation methods: restricted LOM
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
15.
GMM and Consumption-Based Models: Outline
GMM estimation of classic representative agent, CRRA
utility model
Incorporating conditioning information: scaled
consumption-based models
Generalizations of CRRA utility: recursive utility
Semi-nonparametric minimum distance estimators:
unrestricted LOM for data
Simulation methods: restricted LOM
Consumption-based asset pricing: concluding thoughts
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
16.
GMM Review (Hansen, 1982)
Economic model implies set of r population moment
restrictions
E{h (θ, wt )} = 0
(r × 1 )
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
17.
GMM Review (Hansen, 1982)
Economic model implies set of r population moment
restrictions
E{h (θ, wt )} = 0
(r × 1 )
wt is an h × 1 vector of variables known at t
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
18.
GMM Review (Hansen, 1982)
Economic model implies set of r population moment
restrictions
E{h (θ, wt )} = 0
(r × 1 )
wt is an h × 1 vector of variables known at t
θ is an a × 1 vector of coefﬁcients
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
19.
GMM Review (Hansen, 1982)
Economic model implies set of r population moment
restrictions
E{h (θ, wt )} = 0
(r × 1 )
wt is an h × 1 vector of variables known at t
θ is an a × 1 vector of coefﬁcients
Idea: choose θ to make the sample moment as close as
possible to the population moment.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
20.
GMM Review (Hansen, 1982)
Sample moments:
T
g(θ; yT )≡ (1/T ) ∑ h (θ, wt ) ,
t= 1
(r × 1 )
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
21.
GMM Review (Hansen, 1982)
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.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
22.
GMM Review (Hansen, 1982)
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.
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 .
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
23.
GMM Review (Hansen, 1982)
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.
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 .
If r = a, θ estimated by setting each g(θ; yT ) to zero.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
24.
GMM Review (Hansen, 1982)
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.
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 .
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
25.
GMM Review (Hansen, 1982)
Asym. properties (Hansen 1982): θ consistent, asym.
normal.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
26.
GMM Review (Hansen, 1982)
Asym. properties (Hansen 1982): θ consistent, asym.
normal.
Optimal weighting WT = S−1
∞
′
S =
r×r
∑ E
[h (θo , wt )] h θo , wt−j
.
j=− ∞
r×1 1× r
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
27.
GMM Review (Hansen, 1982)
Asym. properties (Hansen 1982): θ consistent, asym.
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).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
28.
GMM Review (Hansen, 1982)
ST depends on θT which depends on ST . Employ an
iterative procedure:
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
29.
GMM Review (Hansen, 1982)
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.)
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
30.
Classic Example: Hansen and Singleton (1982)
Investors maximize utility
∞
max Et
Ct
∑ β i u (C t + i )
i= 0
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
31.
Classic Example: Hansen and Singleton (1982)
Investors maximize utility
∞
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
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
32.
Classic Example: Hansen and Singleton (1982)
Investors maximize utility
∞
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)
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
33.
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
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
34.
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
2 params to estimate: β and γ, so θ = ( β, γ) ′ .
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
35.
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
2 params to estimate: β and γ, so θ = ( β, γ) ′ .
∗
xt denotes info set of investors
−γ −γ ∗
0=E 1 − β (1 + ℜi,t+1 ) Ct+1 /Ct |xt i = 1, ...N
(5)
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
36.
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
2 params to estimate: β and γ, so θ = ( β, γ) ′ .
∗
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)
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
37.
Asset Pricing Example: Hansen and Singleton (1982)
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
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
38.
Asset Pricing Example: Hansen and Singleton (1982)
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
Model can be estimated, tested as long as r ≥ 2.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
39.
Asset Pricing Example: Hansen and Singleton (1982)
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
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
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
40.
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
a
TQ θ; yT ∼ χ2 (r − a)
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
41.
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
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....
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
42.
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
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....
...model is strongly rejected according to OID test.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
43.
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
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....
...model is strongly rejected according to OID test.
Campbell, Lo, MacKinlay (1997): OID rejections stronger
whenever stock returns and commercial paper are
included as test returns. Why?
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
44.
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
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....
...model is strongly rejected according to OID test.
Campbell, Lo, MacKinlay (1997): OID rejections stronger
whenever stock returns and commercial paper are
included as test returns. Why?
Model cannot capture predictable variation in excess
returns over commercial paper ⇒
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
45.
Asset Pricing Example: Hansen and Singleton (1982)
Test of Over-identifying (OID) restrictions:
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....
...model is strongly rejected according to OID test.
Campbell, Lo, MacKinlay (1997): OID rejections stronger
whenever stock returns and commercial paper are
included as test returns. Why?
Model cannot capture predictable variation in excess
returns over commercial paper ⇒
Researchers have turned to other models of preferences.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
46.
More GMM results: Euler Equation Errors
Results in HS use conditioning info xt –scaled returns.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
47.
More GMM results: Euler Equation Errors
Results in HS use conditioning info xt –scaled returns.
Another limitation with classic CCAPM: large
unconditional Euler equation (pricing) errors even when
params freely chosen.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
48.
More GMM results: Euler Equation Errors
Results in HS use conditioning info xt –scaled returns.
Another limitation with classic CCAPM: large
unconditional Euler equation (pricing) errors even when
params freely chosen.
Let Mt+1 = β(Ct+1 /Ct )−γ . Deﬁne Euler equation errors:
j j
eR ≡ E[Mt+1 Rt+1 ] − 1
j j f
eX ≡ E[Mt+1 (Rt+1 − Rt+1 )]
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
49.
More GMM results: Euler Equation Errors
Results in HS use conditioning info xt –scaled returns.
Another limitation with classic CCAPM: large
unconditional Euler equation (pricing) errors even when
params freely chosen.
Let Mt+1 = β(Ct+1 /Ct )−γ . Deﬁne Euler equation errors:
j j
eR ≡ E[Mt+1 Rt+1 ] − 1
j j f
eX ≡ E[Mt+1 (Rt+1 − Rt+1 )]
′
Choose params: min β,γ gT WT gT where jth element of gT
j
gj,t (γ, β) = 1
T ∑T=1 eR,t
t
j
gj,t (γ) = 1
T ∑T=1 eX,t
t
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
50.
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.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
51.
More GMM results: Euler Equation Errors
Magnitude of errors large, even when parameters are
freely chosen to minimize errors.
Unlike the equity premium puzzle of Mehra and Prescott
(1985), large Euler eq. errors cannot be resolved with high
risk aversion.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
52.
More GMM results: Euler Equation Errors
Magnitude of errors large, even when parameters are
freely chosen to minimize errors.
Unlike the equity premium puzzle of Mehra and Prescott
(1985), large Euler eq. errors cannot be resolved with high
risk aversion.
Lettau and Ludvigson (2009): Leading consumption-based
asset pricing theories fail to explain the mispricing of
classic CCAPM.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
53.
More GMM results: Euler Equation Errors
Magnitude of errors large, even when parameters are
freely chosen to minimize errors.
Unlike the equity premium puzzle of Mehra and Prescott
(1985), large Euler eq. errors cannot be resolved with high
risk aversion.
Lettau and Ludvigson (2009): Leading consumption-based
asset pricing theories fail to explain the mispricing of
classic CCAPM.
Anomaly is striking b/c early evidence (e.g., Hansen &
Singleton) that the classic model’s Euler equations were
violated provided the impetus for developing these newer
models.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
54.
More GMM results: Euler Equation Errors
Magnitude of errors large, even when parameters are
freely chosen to minimize errors.
Unlike the equity premium puzzle of Mehra and Prescott
(1985), large Euler eq. errors cannot be resolved with high
risk aversion.
Lettau and Ludvigson (2009): Leading consumption-based
asset pricing theories fail to explain the mispricing of
classic CCAPM.
Anomaly is striking b/c early evidence (e.g., Hansen &
Singleton) that the classic model’s Euler equations were
violated provided the impetus for developing these newer
models.
Results imply data on consumption and asset returns not
jointly lognormal!
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
55.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT S−1 . Why?
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
56.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT S−1 . Why?
One reason: assessing speciﬁcation error, comparing
models.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
57.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT S−1 . Why?
One reason: assessing speciﬁcation error, comparing
models.
Consider two estimated models of SDF, e.g.,
(1)
1 CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected
(2)
2 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
58.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT S−1 . Why?
One reason: assessing speciﬁcation error, comparing
models.
Consider two estimated models of SDF, e.g.,
(1)
1 CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected
(2)
2 CAPM: Mt+1 = a + bRm,t+1 , OID restricts rejected
May we conclude Model 1 is superior?
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
59.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT S−1 . Why?
One reason: assessing speciﬁcation error, comparing
models.
Consider two estimated models of SDF, e.g.,
(1)
1 CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected
(2)
2 CAPM: Mt+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 model
speciﬁc S: J = gT S−1 gT .
′
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
60.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Asset pricing applications often require WT S−1 . Why?
One reason: assessing speciﬁcation error, comparing
models.
Consider two estimated models of SDF, e.g.,
(1)
1 CCAPM: Mt+1 = β(Ct+1 /Ct )−γ , OID restricts not rejected
(2)
2 CAPM: Mt+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 model
speciﬁc S: J = gT S−1 gT .
′
Model 1 can look better simply b/c the SDF and pricing
errors gT are more volatile, not b/c pricing errors are lower.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
61.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
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
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
62.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
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
DistT does not reward SDF volatility => suitable for
model comparison.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
63.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
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
DistT does not reward SDF volatility => suitable for
model comparison.
DistT is a measure of model misspeciﬁcation:
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
64.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
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
DistT does not reward SDF volatility => suitable for
model comparison.
DistT is a measure of model misspeciﬁcation:
Gives distance between Mt (θ) and nearest point in space of
all SDFs that price assets correctly.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
65.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
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
DistT does not reward SDF volatility => suitable for
model comparison.
DistT is a measure of model misspeciﬁcation:
Gives distance between Mt (θ) and nearest point in space of
all SDFs that price assets correctly.
Gives maximum pricing error of any portfolio formed from
the N assets.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
66.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Appeal of HJ Distance metric:
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
67.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Appeal of HJ Distance metric:
Recognizes all models are misspeciﬁed.
Provides method for comparing models by assessing which
is least misspeciﬁed.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
68.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Appeal of HJ Distance metric:
Recognizes all models are misspeciﬁed.
Provides method for comparing models by assessing which
is least misspeciﬁed.
Important problem: how to compare HJ distances
statistically?
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
69.
GMM Asset Pricing With Non-Optimal Weighting
Comparing speciﬁcation error: Hansen and Jagannathan, 1997
Appeal of HJ Distance metric:
Recognizes all models are misspeciﬁed.
Provides method for comparing models by assessing which
is least misspeciﬁed.
Important problem: how to compare HJ distances
statistically?
One possibility developed in Chen and Ludvigson (2009):
White’s reality check method.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
70.
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Chen and Ludvigson (2009) compare HJ distances among
K competing models using White’s reality check method.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
71.
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Chen and Ludvigson (2009) compare HJ distances among
K competing models using White’s reality check method.
1 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.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
72.
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Chen and Ludvigson (2009) compare HJ distances among
K competing models using White’s reality check method.
1 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.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
73.
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Distribution of T W is computed via block bootstrap.
T W has complicated limiting distribution.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
74.
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Distribution of T W is computed via block bootstrap.
T W has complicated limiting distribution.
Bootstrap works only if have a multivariate, joint,
continuous, limiting distribution under null.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
75.
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Distribution of T W is computed via block bootstrap.
T W has complicated limiting distribution.
Bootstrap works only if have a multivariate, joint,
continuous, limiting distribution under null.
Proof of limiting distributions exists for applications to
most asset pricing models:
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
76.
GMM Asset Pricing With Non-Optimal Weighting
Statistical comparison of HJ distance: Chen and Ludvigson, 2009
Distribution of T W is computed via block bootstrap.
T W has complicated limiting distribution.
Bootstrap works only if have a multivariate, joint,
continuous, limiting distribution under null.
Proof of limiting distributions exists for applications to
most asset pricing models:
For parametric models (Hansen, Heaton, Luttmer ’95)
For semiparametric models (Ai and Chen ’07).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
77.
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: econometric problems
− −
Econometric problems: near singular ST 1 or GT 1 .
Asset returns are highly correlated.
We have large N and modest T.
If T < N covariance matrix for N asset returns is singular.
Unless T >> N, matrix can be near-singular.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
78.
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: economically interesting portfolios
Original test assets may have economically meaningful
characteristics (e.g., size, value).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
79.
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: economically interesting portfolios
Original test assets may have economically meaningful
characteristics (e.g., size, value).
− −
Using WT = ST 1 or GT 1 same as using WT = I and doing
GMM on re-weighted portfolios of original test assets.
Triangular factorization S−1 = (P′ P), P lower triangular
min gT S−1 gT ⇔ (gT P′ )I(PgT )
′ ′
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
80.
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: economically interesting portfolios
Original test assets may have economically meaningful
characteristics (e.g., size, value).
− −
Using WT = ST 1 or GT 1 same as using WT = I and doing
GMM on re-weighted portfolios of original test assets.
Triangular factorization S−1 = (P′ P), P lower triangular
min gT S−1 gT ⇔ (gT P′ )I(PgT )
′ ′
Re-weighted portfolios may not provide large spread in
average returns.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
81.
GMM Asset Pricing With Non-Optimal Weighting
Reasons to use identity matrix: economically interesting portfolios
Original test assets may have economically meaningful
characteristics (e.g., size, value).
− −
Using WT = ST 1 or GT 1 same as using WT = I and doing
GMM on re-weighted portfolios of original test assets.
Triangular factorization S−1 = (P′ P), P lower triangular
min gT S−1 gT ⇔ (gT P′ )I(PgT )
′ ′
Re-weighted portfolios may not provide large spread in
average returns.
May imply implausible long and short positions in test
assets.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
82.
GMM Asset Pricing With Non-Optimal Weighting
Reasons not to use WT = I: objective function dependence on test asset choice
Using WT = [ET (R′ R)]−1 , GMM objective function is
invariant to initial choice of test assets.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
83.
GMM Asset Pricing With Non-Optimal Weighting
Reasons not to use WT = I: objective function dependence on test asset choice
Using WT = [ET (R′ R)]−1 , GMM objective function is
invariant to initial choice of test assets.
Form a portfolio, AR from initial returns R. (Note,
portfolio weights sum to 1 so A1N = 1N ).
−1
[E (MR) − 1N ]′ E RR′ [E (MR − 1N )]
′ −1
= [E (MAR) − A1N ] E ARR′ A [E (MAR − A1N )] .
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
84.
GMM Asset Pricing With Non-Optimal Weighting
Reasons not to use WT = I: objective function dependence on test asset choice
Using WT = [ET (R′ R)]−1 , GMM objective function is
invariant to initial choice of test assets.
Form a portfolio, AR from initial returns R. (Note,
portfolio weights sum to 1 so A1N = 1N ).
−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.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
85.
More Complex Preferences: Scaled
Consumption-Based Models
Consumption-based models may be approximated:
Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 )
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
86.
More Complex Preferences: Scaled
Consumption-Based Models
Consumption-based models may be approximated:
Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 )
Example: Classic CCAPM with CRRA utility
1− γ
Ct
u ( Ct ) = ⇒ Mt+1 ≈ β − βγ ∆ct+1
1−γ
at = a0 bt = b0
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
87.
More Complex Preferences: Scaled
Consumption-Based Models
Consumption-based models may be approximated:
Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 )
Example: Classic CCAPM with CRRA utility
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
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
88.
More Complex Preferences: Scaled
Consumption-Based Models
Consumption-based models may be approximated:
Mt+1 ≈ at + bt ∆ct+1 , ct+1 ≡ ln(Ct+1 )
Example: Classic CCAPM with CRRA utility
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 .
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
89.
Scaled Consumption-Based Models
Mt+1 ≈ at + bt ∆ct+1
Empirical speciﬁcation: Lettau and Ludvigson (2001a, 2001b):
at = a0 + a1 zt , bt = b0 + b1 zt
zt = cayt ≡ ct − αa at − αy yt , (cointegrating residual)
cayt related to log consumption-(aggregate) wealth ratio.
cayt strong predictor of excess stock market returns
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
90.
Scaled Consumption-Based Models
Mt+1 ≈ at + bt ∆ct+1
Empirical speciﬁcation: Lettau and Ludvigson (2001a, 2001b):
at = a0 + a1 zt , bt = b0 + b1 zt
zt = cayt ≡ ct − αa at − αy yt , (cointegrating residual)
cayt related to log consumption-(aggregate) wealth ratio.
cayt strong predictor of excess stock market returns
Other examples: including housing consumption
1
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)
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
91.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
92.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Euler equation: E Mt+1 Rt+1 |zt = 1
Unconditional version: E Mt+1 Rt+1 = 1
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
93.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Euler equation: E Mt+1 Rt+1 |zt = 1
Unconditional version: E Mt+1 Rt+1 = 1
Two forms of conditionality:
1 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 )′
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
94.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Euler equation: E Mt+1 Rt+1 |zt = 1
Unconditional version: E Mt+1 Rt+1 = 1
Two forms of conditionality:
1 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 )′
Scaled consumption-based models are conditional in sense
that Mt+1 is a state-dependent function of ∆ ln Ct+1
⇒ scaled factors
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
95.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Two kinds of conditioning are often confused.
Euler equation: E Mt+1 Rt+1 |zt = 1
Unconditional version: E Mt+1 Rt+1 = 1
Two forms of conditionality:
1 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 )′
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.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
96.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled CCAPM turns a single factor model with
state-dependent weights into multi-factor model ft with
constant weights:
Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1
= a0 + a1 zt +b0 ∆ ln Ct+1 +b1 (zt ∆ ln Ct+1 )
f1,t+1 f2,t+1 f3,t+1
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
97.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled CCAPM turns a single factor model with
state-dependent weights into multi-factor model ft with
constant weights:
Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1
= a0 + a1 zt +b0 ∆ ln Ct+1 +b1 (zt ∆ ln Ct+1 )
f1,t+1 f2,t+1 f3,t+1
′
Multiple risk factors ft ≡ (zt , ∆ ln Ct+1 , zt ∆ ln Ct+1 ).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
98.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled CCAPM turns a single factor model with
state-dependent weights into multi-factor model ft with
constant weights:
Mt+1 = (a0 + a1 zt ) + (b0 + b1 zt ) ∆ ln Ct+1
= a0 + a1 zt +b0 ∆ ln Ct+1 +b1 (zt ∆ ln Ct+1 )
f1,t+1 f2,t+1 f3,t+1
′
Multiple risk factors ft ≡ (zt , ∆ ln Ct+1 , zt ∆ ln Ct+1 ).
Scaled consumption models have multiple, constant betas
for each factor, rather than a single time-varying beta for
∆ ln Ct+1 .
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
99.
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).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
100.
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)
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
101.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled models: conditioning done in SDF:
Mt+1 = at + bt ∆ ln Ct+1 ,
not in Euler equation: E(MR) = 1N .
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
102.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled models: conditioning done in SDF:
Mt+1 = at + bt ∆ ln Ct+1 ,
not in Euler equation: E(MR) = 1N .
Gives rise to a restricted conditional consumption beta
model:
Ri = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1
t
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
103.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled models: conditioning done in SDF:
Mt+1 = at + bt ∆ ln Ct+1 ,
not in Euler equation: E(MR) = 1N .
Gives rise to a restricted conditional consumption beta
model:
Ri = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1
t
Rewrite as
Ri = a + ( βc + βc,z zt−1 ) ∆ct + βz zt−1
t
time-varying beta
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
104.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Scaled models: conditioning done in SDF:
Mt+1 = at + bt ∆ ln Ct+1 ,
not in Euler equation: E(MR) = 1N .
Gives rise to a restricted conditional consumption beta
model:
Ri = a + β ∆c ∆ct + β ∆c,z ∆ct zt−1 + βz zt−1
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.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
105.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
106.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
107.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
108.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ).
Latter may require variables beyond a few that capture
risk-premia.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
109.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ).
Latter may require variables beyond a few that capture
risk-premia.
Approximating condition mean well requires large
number of instruments (misspeciﬁed information sets)
Results sensitive to chosen conditioning variables, may fail
to span information sets of market participants.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
110.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Distinction is important.
Conditioning in SDF: theory provides guidance:
typically a few variables that capture risk-premia.
Conditioning in Euler eqn: model joint dist. (Mt+1 Rt+1 ).
Latter may require variables beyond a few that capture
risk-premia.
Approximating condition mean well requires large
number of instruments (misspeciﬁed information sets)
Results sensitive to chosen conditioning variables, may fail
to span information sets of market participants.
Partial solution: summarize information in large number
of time-series with few estimated dynamic factors (e.g.,
Ludvigson and Ng ’07, ’09).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
111.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Bottom lines:
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
112.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Bottom lines:
1 Conditional moments of Mt+1 Rt+1 difﬁcult to model ⇒
reason to focus on unconditional moments
E[Mt+1 Rt+1 ] = 1.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
113.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Bottom lines:
1 Conditional moments of Mt+1 Rt+1 difﬁcult to model ⇒
reason to focus on unconditional moments
E[Mt+1 Rt+1 ] = 1.
2 Models are misspeciﬁed: interesting question is whether
state-dependence of Mt+1 on consumption growth ⇒ less
misspeciﬁcation than standard, ﬁxed-weight CCAPM.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
114.
Scaled Consumption-Based Models
Distinguishing two types of conditioning, or state dependence
Bottom lines:
1 Conditional moments of Mt+1 Rt+1 difﬁcult to model ⇒
reason to focus on unconditional moments
E[Mt+1 Rt+1 ] = 1.
2 Models are misspeciﬁed: interesting question is whether
state-dependence of Mt+1 on consumption growth ⇒ less
misspeciﬁcation than standard, ﬁxed-weight CCAPM.
3 As before, can compare models on basis of HJ distances,
using White ”reality check” method to compare statistically.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
115.
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
116.
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Two reasons recursive utility is of interest:
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
117.
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Two reasons recursive utility is of interest:
More ﬂexibility as regards attitudes toward risk and
intertemporal substitution.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
118.
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Two reasons recursive utility is of interest:
More ﬂexibility as regards attitudes toward risk and
intertemporal substitution.
Preferences deliver an added risk factor for explaining asset
returns.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
119.
Asset Pricing Models With Recursive Preferences
Growing interest in asset pricing models with recursive
preferences, e.g., Epstein and Zin ’89, ’91, Weil ’89, (EZW).
Two reasons recursive utility is of interest:
More ﬂexibility as regards attitudes toward risk and
intertemporal substitution.
Preferences deliver an added risk factor for explaining asset
returns.
But, only a small amount of econometric work on
recursive preferences ⇒ gap in the literature.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
120.
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
121.
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
1 For general stationary, consumption growth and cash ﬂow
dynamics: Chen, Favilukis, Ludvigson ’07.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
122.
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
1 For general stationary, consumption growth and cash ﬂow
dynamics: Chen, Favilukis, Ludvigson ’07.
2 When restricting cash ﬂow dynamics (e.g., “long-run risk”):
Bansal, Gallant, Tauchen ’07.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
123.
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
1 For general stationary, consumption growth and cash ﬂow
dynamics: Chen, Favilukis, Ludvigson ’07.
2 When restricting cash ﬂow dynamics (e.g., “long-run risk”):
Bansal, Gallant, Tauchen ’07.
In (1) DGP is left unrestricted, as is joint distribution of
consumption and returns (distribution-free estimation
procedure).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
124.
Asset Pricing Models With Recursive Preferences
Here discuss two examples of estimating EZW models:
1 For general stationary, consumption growth and cash ﬂow
dynamics: Chen, Favilukis, Ludvigson ’07.
2 When restricting cash ﬂow dynamics (e.g., “long-run risk”):
Bansal, Gallant, Tauchen ’07.
In (1) DGP is left unrestricted, as is joint distribution of
consumption and returns (distribution-free estimation
procedure).
In (2) DGP and distribution of shocks explicitly modeled.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
125.
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
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
126.
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
Vt+1 is continuation value, θ is RRA, 1/ρ is EIS.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
127.
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
Vt+1 is continuation value, θ is RRA, 1/ρ is EIS.
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
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
128.
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
Vt+1 is continuation value, θ is RRA, 1/ρ is EIS.
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
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
129.
EZW Recursive Preferences
Epstien-Zin-Weil basics
The MRS is pricing kernel (SDF) with added risk factor:
ρ−θ
Vt+1 Ct+1
Ct + 1 − ρ Ct+1 Ct
Mt + 1 = β
Ct R Vt+1 Ct+1
t Ct+1 Ct
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
130.
EZW Recursive Preferences
Epstien-Zin-Weil basics
The MRS is pricing kernel (SDF) with added risk factor:
ρ−θ
Vt+1 Ct+1
Ct + 1 − ρ 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 (·).
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
131.
EZW Recursive Preferences
Epstien-Zin-Weil basics
The MRS is pricing kernel (SDF) with added risk factor:
ρ−θ
Vt+1 Ct+1
Ct + 1 − ρ 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 (·).
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.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
132.
EZW Recursive Preferences
Epstien-Zin-Weil basics
The MRS is pricing kernel (SDF) with added risk factor:
ρ−θ
Vt+1 Ct+1
Ct + 1 − ρ 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 (·).
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.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
133.
EZW Recursive Preferences
Epstien-Zin-Weil basics
1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
134.
EZW Recursive Preferences
Epstien-Zin-Weil basics
1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2 If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
135.
EZW Recursive Preferences
Epstien-Zin-Weil basics
1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2 If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
But....
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
136.
EZW Recursive Preferences
Epstien-Zin-Weil basics
1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2 If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
But....
EIS=1 ⇒ consumption-wealth ratio is constant,
contradicting statistical evidence.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
137.
EZW Recursive Preferences
Epstien-Zin-Weil basics
1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2 If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
But....
EIS=1 ⇒ consumption-wealth ratio is constant,
contradicting statistical evidence.
Joint lognormality strongly rejected in quarterly data.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
138.
EZW Recursive Preferences
Epstien-Zin-Weil basics
1 If EIS=1, and ∆ log Ct+1 follows a loglinear time-series
process, log(V/C) has an analytical solution.
2 If returns, Ct are jointly lognormal and homoscedastic, risk
premia are approx. log-linear functions of COV between
returns, and news about current and future Ct growth.
But....
EIS=1 ⇒ consumption-wealth ratio is constant,
contradicting statistical evidence.
Joint lognormality strongly rejected in quarterly data.
Points to need for estimation method feasible under less
restrictive assumptions.
Sydney C. Ludvigson Methods Lecture: GMM and Consumption-Based Models
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