Habit, Long Run Risk, Prospect?
Eric Aldrich
Duke University
Department of Economics
Durham NC 27708-0097 USA
A. Ronald Gallant
Duke University
Fuqua School of Business
Durham NC 27708-0120 USA
Robert E. McCulloch
University of Texas
Graduate School of Business
Austin TX 78712-1175 USA
Methodology - 1
• Gallant and McCulloch (2008), “On the Determination General
Statistical Models with Application to Asset Pricing,” Journal of the
American Statistical Association, forthcoming.
• A statistical model determines the likelihood.
• A prior is placed on the statistical model by imposing
1. a scientific model, which restricts its parameters to a lower
dimensional manifold
2. a prior on the parameters of the the scientific model
3. a prior on functionals of the the scientific model
• Relaxed versions of the prior attract the parameters of the
statistical model to the lower dimensional manifold.
Tinker Toy Example
Scientific Model: p(y | θ) = n(y | θ, θ2)
Statistical Model: f (y | η) = n(y | η1, η2)
Implied map: g : θ 7→ η = (η1, η2) = (θ, θ2)
The scientific model states that excess returns yt = rdt − rf t are
iid normal with a Sharpe ratio of one. The statistical model
states that excess returns are iid normal.
1.0
1.5
2.0
2.5
3.0
2 4 6 8 10
eta2
2 4 6 8 10
eta2
Fig 1. The Scientific and Statistical Model
3.5
1.0
1.5
1.5
2.0
2.5
eta1
3.0
3.5
3.0
3.5
3.0
3.5
2 4 6 8 10
eta2
1.0
2.5
eta1
2 4 6 8 10
eta2
eta1
2.0
1.0
1.5
2.0
2.5
eta1
The the dotted lines are contours of the likelihood of the statistical model f (y|x, η) of the
tinker toy example. The line is the prior on η determined by the implied map η = g(θ) from
the parameters θ of scientific model p(y|x, θ) to the parameters η of the statistical model. In
the left panels the scientific model is true, in the right it is false. The thickness of the line is
proportional to the posterior of η. The prior π(η) can be relaxed as indicated by the shading.
The lower panels are more relaxed than the upper. The solid contours show the posterior
under the relaxed prior. Relaxation causes the contours to enlarge in all cases. When the
scientific model is false, the posterior shifts in search of the likelihood.
Determination of Model Adequacy
Ranking Scientific Models
• Put equal prior probability on each model.
• Model comparision is by means of posterior probabilities.
Determination of Model Adequacy
Scientific Model vs. Statistical Model
• Relax the prior determined by the scientific model.
• The parameters of the statistical model are now free to move
off the lower dimensional manifold into their natural higher
dimensional parameter space.
• But the prior retains enough influence to overcome data
sparseness.
• Model adequacy is determined by observing if parameters
or functionals of the statistical model that have scientific
relevance change.
• Model comparision is by means of posterior probabilities.
4
3
2
1
0
0
1
2
3
4
Fig 2. Determination of Model Adequacy
0.5
1.0
1.5
2.0
0.5
1.5
2.0
3
2
1
0
0
1
2
3
4
kappa=5, restriction false
4
kappa=5, restriction true
1.0
0.5
1.0
1.5
2.0
kappa=100, restriction true
0.5
1.0
1.5
2.0
kappa=100, restriction false
The posterior of the Sharpe ratio for the tinker toy example is the solid line; the dashed line
is prior. In the left panels the scientific model is true, in the right it is false. The prior is
more relaxed in the lower panels than it is in the upper panels. The panels correspond to
those of Figure 1
Data
Annual observations 1929–2001, 72 years, on
a
Pdt
end-of-year per capita stock market value
Dta
annual aggregate per capita dividend
Cta
annual per capita consumption
a
rdt
annual real geometric return
Qat
annual quadratic variation
Data are real, i.e. inflation adjusted.
Source: Bansal, Gallant, and Tauchen (2007).
Data for Estimation
Used here
cat − cat−1
a
rdt
!
dat − cat
cat − cat−1
padt − dat
a
rdt

Used by BGT









Habit Persistence Asset Pricing Model
Driving Processes
Consumption: ct − ct−1 = g + vt
Dividends: dt − dt−1 = g + wt
Random Shocks:
vt
wt
!
∼ NID
"
!
σ2
0
ρσσw
,
2
0
ρσσw σw
!#
The time increment is one month. Lower case denotes logarithms of upper
case quantities; i.e. ct = log(Ct), dt = log(Dt). From Campbell and Cochrane
(1999).
Habit Persistence Asset Pricing Model
Utility function

E0
∞
X
t=0

1−γ
(StCt)
−1
t
δ
,
1−γ
Habit persistence
Surplus ratio: st − s̄ = φ (st−1 − s̄) + λ(st−1)vt−1
Sensitivity function: λ(s) =
(
1
S̄
q
1 − 2(s − s̄) − 1 st ≤ smax
0
st > smax
Et is conditional expectation with respect to St, St−1, ... . Lower case denotes
logarithms of upper case quantities: st = log(St). S̄ and smax can be computed
p
from model parameters θ = (g, σ, ρ, σw , φ, δ, γ) as S̄ = σ γ/(1 − φ), smax =
s̄ + (1 − S̄ 2)/2. From Campbell and Cochrane (1999).
Habit Persistence Asset Pricing Model
Return on dividends


St+1Ct+1
V (St) = Et δ

St C t
"
!−γ
!

i
Dt+1 h
1 + V (St+1)

Dt
Dt
1 + V (St)
rdt = log
V (St−1) Dt−1
!#
V (·) is defined as the solution of the Euler condition above. It is the price
dividend ratio; i.e. Pdt /Dt = V (St), where Pdt is the price of the asset that
pays the dividend stream. rdt is the logarithmic real return, i.e. rdt = log(Pdt +
Dt) − log(Pd,t−1 ), where Pdt and Dt are measured in real (inflation adjusted)
dollars. Dividend error can be integrated out analytically. Consumption error
integrated by quadrature. From Campbell and Cochrane (1999).
Habit Persistence Asset Pricing Model
Solution Method
Approximate the log policy function
v(st) = log V (est )
by a piecewise linear function and use policy function iteration.
Campbell and Cochrane used Gauss’s intquad1 and set join
points at s̄, smax, smax −0.01, smax −0.02, smax −0.03, smax −0.04,
and log[iS/(m + 1)] for i = 1, . . . , m = 10. We used GaussHermite quadrature; we added the abscissae of the GaussHermite quadrature formula at the maximum and minimum of
the above join points; we deleted all points less than 0.001 apart.
Figure 3, next slide, plots the approximation at the Campbell
and Cochrane parameter values.
Fig 3. Piecewise Linear Approximation
x
x
x
x x
x
xxxxx
x x
x
2.4
2.0
p−d
2.8
3.2
Annualized Log Policy Function v(s)
x
xx
x
−5.0
x
x
−4.5
x
−4.0
−3.5
−3.0
−2.5
15
x
10
P/D
20
25
Annualized Policy Function V(S)
x
x
x
x x
x
x
x
xxxxxx
x
x x
x
xx
0.02
0.04
0.06
0.08
x’s mark Campbell and Cochrane join points; o’s mark extra join
points from the quadrature rule.
Habit Persistence Asset Pricing Model
Risk Free Rate



rf t = − log Et δ

!−γ 

St+1Ct+1


St C t
rf t is the logarithmic return on an asset that pays one real dollar one month
hence with certainty. From Campbell and Cochrane (1999).
Solution method is similar to the foregoing.
Habit Persistence Asset Pricing Model
Large Model Output
Given model parameters
θ = (g, σ, ρ, σw , φ, δ, γ)
simulate monthly and aggregate to annual:
Cta =
11
X
C12t−k
k=0
cat = log(Cta)
a =
rdt
11
X
rd,12t−k
11
X
rf,12t−k
k=0
rfat =
k=0
Habit Persistence Asset Pricing Model
Prior Distribution
"
p(θ) = N rf | 0.89,
2 # Y
p
1
1.96
i=1

N θi | θi∗,
!2 
0.1θi∗ 
1.96
where the θi∗ are the calibrated values from Campbell and
Cochrane (1999).
Table 1. Habit Model
Prior
Parameter
g
σ
ρ
σw
φ
δ
γ
rf
rd − rf
σr d
Posterior
Mode
Std.Dev.
Mode
Std.Dev.
0.0015640
0.0042114
0.21045
0.033386
0.98691
0.98850
2.17383
0.87084
6.8610
20.560
8.8319e-05
0.00021870
0.010179
0.0016510
0.00033643
0.00062957
0.0861067
0.60467
0.38795
1.1170
0.00157928
0.00494385
0.18604
0.033142
0.98837
0.98934
2.12305
2.0611
5.4221
18.6145
7.2418e-05
0.00016592
0.010208
0.0017159
0.00029442
0.00033657
0.050678
0.55120
0.35970
0.73469
Parameter values are for the monthly frequency. Returns are annualized. Mode is the
mode of the multivariate density. It acutually ocurrs in the MCMC chain whereas other
measures of central tendancy may not even satisfy support conditions. In the data,
rd − rf = 6.02 − 0.89 = 5.13 and σrd = 19.29.
Fig 4. Posterior Returns
0.4
0.0
0.2
Density
0.6
0.8
bond returns
1.0
1.5
2.0
2.5
3.0
equity premium
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Density
0.5
5.0
5.5
6.0
6.5
1.0
0.0
0.5
Density
1.5
2.0
stock returns
7.0
7.5
8.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Density
sdev stock returns
18
19
20
21
Long Run Risk Asset Pricing Model
Driving Processes
Consumption: ct+1 − ct = µc + xt + σtηt+1
Long Run Risk: xt+1 = ρxt + φeσtet+1
2
= σ̄ 2 + ν(σt2 − σ̄ 2) + σw wt+1
Stochastic Volatility: σt+1
Dividends: dt+1 − dt = µd + φdxt + πdσtηt+1 + φuσtut+1


  
1
0
ηt
 0   0
e 
  
 t
Random Shocks:   ∼ NID  , 
 0   0
wt 
0
0
ut
0
1
0
0
0
0
1
0

0

0


0 
1
The time increment is one month. Lower case denotes logarithms of upper
case quantities; i.e. ct = log(Ct), dt = log(Dt). From Bansal, Kiku, and Yaron
(2007).
Long Run Risk Asset Pricing Model
Epstein-Zin utility function
"
Ut = (1 −
1−γ
δ)C θ
t
1−γ θ
+ δ Et Ut+1
where
γ
ψ
1
is the coefficient of risk aversion
is the elasticity of intertemporal
substitution
and
1−γ
θ=
1 − 1/ψ
Et is conditional expectation with respect to xt , σt.
#
θ
1−γ
Long Run Risk Asset Pricing Model
Return on consumption
mrst+1 = δ θ exp[−(θ/ψ)(ct+1 − ct) + (θ − 1)rc,t+1]
(
Ct+1
VC (xt, σt) = Et mrst+1
Ct
"
!
h
1 + VC (xt+1, σt+1)
1 + VC (xt, σt)
Ct
rct = log
VC (xt−1, σt−1) Ct−1
i
)
!#
VC (·) is defined as the solution of the Euler condition above. It is the price
consumption ratio; i.e. Pct /Ct = VC (xt, σt ), where Pct is the price of the asset
that pays the consumption stream.
rct is the logarithmic real return, i.e.
rct = log(Pct + Ct ) − log(Pc,t−1), where Pct and Ct are measured in real (inflation
adjusted) dollars.
Long Run Risk Asset Pricing Model
Solution Method
Use the log linear approximation
.
rc,t+1 = κ0 + κ1zt+1 + ∆ct+1 − zt
κ1 = [exp(z̄)]/[1 + exp(z̄)]
k0 = log[1 + exp(z̄)] − κ1z̄
where zt = log(Pc,t/Ct) and z̄ is its endogenous mean.
To compute z̄, use the approximation
.
zt = A0(z̄) + A1(z̄) xt + A2(z̄) σt2
Ai(z̄) = tedious expressions in model parameters and z̄
and solve the fixed point problem
z̄ = A0(z̄) + A1(z̄) xt + A2(z̄) σt2
Long Run Risk Asset Pricing Model
Return on dividends
mrst+1 = δ θ exp[−(θ/ψ)(ct+1 − ct) + (θ − 1)rc,t+1]
(
Dt+1
VD (xt, σt) = Et mrst+1
Dt
"
!
h
i
1 + VD (xt+1, σt+1)
Dt
1 + VD (xt, σt)
rdt = log
VD (xt−1, σt−1) Dt−1
)
!#
VD (·) is defined as the solution of the Euler condition above. It is the price
dividend ratio; i.e. Pdt /Dt = VD (xt , σt), where Pct is the price of the asset
that pays the dividend stream. rdt is the logarithmic real return, i.e. rdt =
log(Pdt + Dt) − log(Pd,t−1), where Pdt and Dt are measured in real (inflation
adjusted) dollars.
Solution method is similar to the foregoing.
Long Run Risk Asset Pricing Model
Risk Free Rate
h
io
θ
rf t = − log Et δ exp −(θ/ψ)(ct+1 − ct) + (θ − 1)rc,t+1
n
rf t is the logarithmic return on an asset that pays one real dollar one month
hence with certainty.
Solution method is similar to the foregoing.
Long Run Risk Asset Pricing Model
Large Model Output
Given model parameters
θ = (δ, γ, ψ, µc, ρ, φe, σ̄ 2, η, σw , µd, φu)
simulate monthly and aggregate to annual:
Cta =
11
X
C12t−k
k=0
cat = log(Cta)
a =
rdt
11
X
rd,12t−k
11
X
rf,12t−k
k=0
rfat =
k=0
Long Run Risk Asset Pricing Model
Prior Distribution
"
p(θ) = N rf | 0.89,
2 # Y
p
1
1.96
i=1

N θi | θi∗,
!2 
0.1θi∗ 
1.96
where the θi∗ are the calibrated values from Kiku (2006).
Table 2. Long Run Risk Model
Prior
Parameter
δ
γ
ψ
µc
ρ
φe
σ̄ 2
ν
σw
µd
φd
πd
φu
rf
rd − rf
σr d
Mode
Std.Dev.
0.99919
0.00012674
10.191
0.21061
1.4882
0.061835
0.0014458 8.011084e-05
0.98672
0.0024294
0.033661
0.0010018
4.2468e-05
1.3089e-06
0.99295044
0.0016819
1.7397e-06
4.8917e-07
0.0012093
7.2206e-05
2.78515
0.14961
3.9492
0.63146
6.0507
0.66856
1.17504
0.27676
6.24790
1.5816
19.076
1.8840
Posterior
Mode
0.99949
11.269
1.5585
0.0015296
0.98812
0.032684
4.2528e-05
0.98648
1.7844e-06
0.0011482
2.6835
4.1523
5.9648
0.59225
7.1733
18.970
Std.Dev.
0.00010942
0.29135
0.065250
6.6133e-05
0.00088681
0.0010092
1.2106e-06
0.00093868
5.0889e-07
7.7365e-05
0.15192
0.62532
0.51864
0.20659
0.51500
1.1396
Parameter values are for the monthly frequency. Returns are annualized. Mode is the mode
of the multivariate density. It acutually ocurrs in the MCMC chain whereas other measures of
central tendancy may not even satisfy support conditions. In the data, rd − rf = 6.02 − 0.89 =
5.13 and σrd = 19.29.
Fig 5. Posterior Returns
0.0 0.5 1.0 1.5 2.0
Density
bond returns
0.6
5.5
1.0
6.0
6.5
1.2
1.4
7.0
7.5
8.0
stock returns
0.0 0.2 0.4 0.6 0.8 1.0
Density
0.8
equity premium
0.0 0.2 0.4 0.6 0.8
Density
0.4
7.0
7.5
8.0
8.5
0.2
0.0
Density
0.4
sdev stock returns
15
16
17
18
19
Prospect Theory Asset Pricing Model
Driving Processes
Aggregate Consumption: c̄t+1 − c̄t = gC + σC ηt+1
Dividends: dt+1 − dt = gD + σD ǫt+1
Random Shocks:
ηt
ǫt
!
∼ NID
"
!
0
1 ω
,
0
ω 1
!#
C̄t is aggregate, per capita consumption which is exogenous to the agent.
The time increment is one year. Lower case denotes logarithms of upper
case quantities; i.e. c̄t = log(C̄t), dt = log(Dt). All variables are real. From
Barberis, Huang, Santos (2001).
Prospect Theory Asset Pricing Model
Other Model Variables
• Gross Stock Return: Rt
−1
2
2
• Gross Risk Free Rate: Rf = ρ exp γgC − γ σC /2
• Allocation to Risky Asset: St
• Gain or Loss: Xt+1 = St(Rt+1 − Rf )
• Benchmark Level (State Variable): zt+1 = η zt R R̄ +(1−η)
t+1
• Choose R̄ to make Median {zt} = 1
• The Agent’s Consumption: Ct
Prospect Theory Asset Pricing Model
Utility function

E0
∞
X
t=0

1−γ
h
i
C
−1
−γ t+1
 ρt t
+ b0C̄t ρ
St v̂(Rt+1, zt) 

1−γ
h
Utility from Gains and Losses: St v̂(Rt+1, zt)
v̂(Rt+1, zt)
= Rt+1 − Rf
= (ztRf − Rf ) + λ(Rt+1 − ztRf )
= Rt+1 − Rf
= λ(zt)(Rt+1 − Rf )
λ(zt) = λ + k(zt − 1)
i
zt ≤ 1, Rt+1 ≥ ztRf
zt ≤ 1, Rt+1 < ztRf
zt > 1, Rt+1 ≥ Rf
zt > 1, Rt+1 < Rf
0.0
−0.6 −0.4 −0.2
Utility
0.2
0.4
0.6
Fig 6. Utility of Gains and Losses
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
Gain/Loss
The dot-dash line represents the case where the investor has prior
gains (z < 1), the dashed line the case of prior losses (z > 1), and
the solid line the case where the investor has neither prior gains nor
losses (z = 1).
Prospect Theory Asset Pricing Model
Return on dividends
2
2
2
1 = ρ exp gD − γgC + γ σC (1 − ω )/2
"
#
1 + f (zt+1)
× Et
exp[(σD − γωσC )ǫt+1]
f (zt)
"
!#
1 + f (zt+1)
+ b0ρ Et v̂
exp(gD + σD ǫt+1), zt
f (zt)
"
1 + f (zt)
exp(gD + σD ǫt)
rdt = log
f (zt−1)
#
f (·) is defined as the solution of the Euler condition above. It is the price
dividend ratio; i.e. Pdt/Dt = f (zt), where Pct is the price of the asset that pays
the dividend stream. rdt is the logarithmic real return, i.e. rdt = log(Pdt +
Dt) − log(Pd,t−1 ), where Pdt and Dt are measured in real (inflation adjusted)
dollars.
Prospect Theory Asset Pricing Model
Self Referential Equations
zt+1 = η zt
R̄
Rt+1
!
+ (1 − η)
1 + f (zt+1)
exp(gD + σD ǫt+1)
Rt+1 =
f (zt)
1 = Median{zt}
Prospect Theory Asset Pricing Model
Solution Method
Approximate f by a piecewise linear function f (0)(z).
Approximate R̄ by (1 + f (1)) exp(gD )/f (1), which is a departure
from Barberis, Huang, and Santos (2001).
Define h(0) such that zt+1 = h(0)(zt, ǫt+1) solves the self referential equations that define zt+1 and Rt+1 on previous slide. A
root finding problem. We use Brent’s method.
Substitute h(0)(zt, ǫt+1) into the Euler equation. Use GaussHermite quadrature to integrate out ǫt+1. Solve for f (1)(z). A
root finding problem at each join point of f (1) .
Repeat h(i) → f (i+1) until convergence.
20
15
10
P/D
25
30
Fig 7. Piecewise Linear Approximation
0
1
2
z
3
4
Prospect Theory Asset Pricing Model
Risk Free Rate
i
−1
2
2
rf = log ρ exp γgC − γ σC /2
h
rf t is the logarithmic return on an asset that pays one real dollar one year
hence with certainty.
Prospect Theory Asset Pricing Model
Large Model Output
Given model parameters
θ = (gC , gD , σC , σD , ω, γ, ρ, λ, k, b0, η)
simulate annually and set
cat = log(Ct)
a =r
rdt
dt
rfat = rf
Prospect Asset Pricing Model
Prior Distribution
"
p(θ) = N rf | 0.89,
2 # Y
p
1
1.96
i=1

N θi | θi∗,
!2 
0.1θi∗ 
1.96
where the θi∗ are the calibrated values from Barberis, Huang,
Santos (2001).
Table 3. Prospect Model
Prior
Parameter
gC
gD
σC
σD
ω
γ
ρ
λ
k
b0
η
rf
rd − rf
σr d
Mode
Std.Dev.
0.017212 0.00092156
0.018188 0.00089243
0.036621 0.0020330
0.12402 0.0060679
0.14941 0.0081401
0.99219
0.051017
0.99982 0.0010958
2.28125
0.11124
10.09375
0.52937
2.18018
0.033314
0.93909
0.014177
1.6604
0.166210
6.3091
0.385231
30.984
2.074061
Posterior
Mode
Std.Dev.
0.017700 0.00096645
0.019165 0.00087639
0.034668 0.0019947
0.11035 0.0051616
0.14746 0.0076743
1.00781 0.0498657
0.99823 0.0013179
2.28125
0.11559
10.28125
0.60546
1.96436 0.0572277
0.85754 0.0120773
1.8999
0.170305
4.5090
0.347421
21.3325
1.33432
Parameter values are for the annual frequency. Mode is the mode of the multivariate
density. It acutually ocurrs in the MCMC chain whereas other measures of central
tendancy may not even satisfy support conditions. In the data, rd − rf = 6.02 − 0.89 =
5.13 and σrd = 19.29.
Fig 8. Posterior Returns
1.5
1.0
0.0
0.5
Density
2.0
bond returns
1.8
2.0
2.2
2.4
2.6
equity premium
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Density
1.6
4.0
4.5
5.0
5.5
0.0 0.2 0.4 0.6 0.8 1.0
Density
stock returns
5.5
6.0
6.5
7.0
7.5
0.20
0.10
0.00
Density
0.30
sdev stock returns
20
22
24
26
Model Assessment
Posterior Probabilities for Three Models
Long Run Risk
0.34736
Habit Persistence
0.33278
Prospect Theory
0.31986
Model Assessment
Posterior Probabilities for Long Run Risk
κ = 0.01
0.32531
κ = 50.00
0.33692
κ = 75.00
0.33777
Methodology - 2
• Gallant and Hong (2007), “A Statistical Inquiry into the Plausibility
of Recursive Utility,” Journal of Financial Econometrics 5, 523–590.
• Pricing kernel: θ = (θ1, . . . , θn+1)
Qn
• Hierarchical likelihood: L(θ, η) = L(θ) t=1 f (θt+1|θt, . . . , θ1, η)
– L(θ) function of returns derived from the Euler equation
1 = Etθt+1Rit by a conditioning argument.
– f (·|·, η) seminonparametric transition density on θ.
• Composite prior: p(θ, η) = p(η) pT (θ, η)
– p(η) prior on hyperparameter η : habit, long run risk, etc.
– pT (θ, η) imposes technical conditions
• Inference:
– MCMC posterior probabilities P (A) =
RR
IA(θ, η)P (dθ, dη)
Data: Two Data Sets
• First: 551 monthly observations from February 1959 through
December, 2004
– real returns including dividends on 24 Fama-French (1993) portfolios.
– real returns on U.S. Treasury debt of ten year, one year, and thirty
day maturities
– real returns including dividends on the aggregate stock market
– real, per-capita, consumption expenditure and labor income growth
• Second: The second is 75 annual observations from 1930
through 2004 on the same variables except U.S. Treasury
debt of ten and one year maturities.
• Exclusion of one Fama-French portfolio and debt due to
missing values.
Findings
• Monthly data:
– P(Habit) = 0.51
– P(Long Run Risk) = 0.49
• Annual data:
– P(Habit) = 0.50
– P(Long Run Risk) = 0.50
Remainder
• Use long run risk model to illustrate
– Law of motion of pricing kernel f (θt|θt−1, . . . , θ1, η)
– Prior p(η) on hyperparameter η
– Likelihood for observables L(θ)
– Implementation of L(θ)
– Technical prior pT (θ, η)
Consumption Endowment and Cash Flows
Ct consumption endowment
Pct price of an asset that pays the consumption endowment
Rct = (Pct + Ct)/Pc,t−1 gross return on consumption endowment
Dst cash flow S
Pst price of cash flow S
Rst = (Pst + Dst)/Ps,t−1 gross return on cash flow S
Prices are real
LRR Uses Recursive Utility
Marginal rate of substitution
Mt,t+1 = δ β
Ct+1
Ct
!−(β/ψ)
(Rc,t+1)(β−1),
δ time preference parameter
γ coefficient of risk aversion
ψ elasticity of intertemporal substitution
1−γ
β = 1−1/ψ
In General
Euler equation
1 = Et θt+1Rt,t+1
Satisfied by any pricing kernel θt+1 including Mt,t+1 given by
recursive utility
Law of motion of pricing kernel f (θt|θt−1, . . . , θ1, η)
• Fit SNP expansion to simulated data from a calibrated long
run risk economy.
• Use BIC to determine the truncation point.
• The form of the fitted density of θt = Mt−1,t is
f (θt |θt−1, . . . , θ1 , η)
yt
f (yt |yt−1 , . . . , y1 , η)
η
P(yt|a)
µt−t
2
σt−1
=
=
=
=
=
=
=
f (yt |yt−1, . . . , y1 , η)/ exp(yt)
log(θt−1,t )
2
)
P 2(yt|a) n(yt | µt−1, σt−1
(a1, a2 , a3 , a4 , b0, b1, r0, r1, r2)
1 + a1 yt + a2 yt2 + a3 yt3 + a4 yt4
b0 + b1 yt−1
2
r02 + r12σt−2
+ r22 (yt−1 − µt−2)2
Likelihood Derivation – 1 of 4
• Consider a general set of moment conditions
n
1 X
m̄n =
mt(yt, θt)
n t=1
mt ∈ ℜK
that a scientific theory implies ought to have expectation
zero where both yt and θt are regarded as random
• To estimate the variance when the mt(yt, xt, θo) are uncorrelated, use
n
1 X
Wn =
[mt(yt, θt) − m̄n] [mt(yt, θt) − m̄n]′
n t=1
• When correlated, use HAC.
−1/2√
n m̄n
• Rely on CLT for normality of z = Wn
Likelihood Derivation – 2 of 4
• View as a data summary: z = Z(Y, θ)
Z : (Y, θ) 7→
√
nW −1/2n m̄n
where Y = [y1|y2| · · · |yn], θ = [θ1|θ2| · · · |θn]
• Y is all possible Y
• Information loss: Probability can only be assigned to sets A
expressed in terms of the original data (Y, θ) that are preimages A = Z −1(B) of a Borel set B in ℜK .
• Assignment formula:
P (A) = PZ (B) =
Z
···
Z
IB (z) (2π)−K/2 exp(−z ′z/2) dz1 · · · dzK
Likelihood Derivation – 3 of 4
• Let Zθ denote the map Y 7→ Z(Y, θ) for θ fixed.
• The sets C expressed in terms of Y to which we can assign
conditional probability knowing that θ has occurred have the
form C = Zθ−1(Bθ ) for some Borel set Bθ ∈ ℜK .
• The principle of conditioning is that one assigns conditional
probability proportionately to joint probability.
• The constant of proportionality is 1/P (ℜθ ), where ℜθ is the
Borel set for which Y = Zθ−1(ℜθ ).
• The conditional probability of C is computed as
P (C|θ) =
PZ (Bθ )
PZ (ℜθ )
Likelihood Derivation – 4 of 4
• Conditional probability is computed as
1
P (C|θ) =
PZ (ℜθ )
Z
···
Z
IBθ (z) (2π)−K/2 exp(−z ′z/2) dz1 · · · dzK
• The data summary is
√
−1/2
n mn(θ)
z = [Wn(θ)]
• In our application PZ (ℜθ ) = 1
• Conclude that the likelihood is
n
L(θ) ∝ exp − mn(θ) ′[Wn(θ)]−1mn(θ)
2
• Same argument as Fisher (1930) used to define fiducial probability. Is Bayesian inference for continuously updated GMM.
Implementation of L(θ) (monthly data)
• st: Fama-French gross returns S11 − S54
bt: T-debt gross returns t30ret, b1ret, and b10ret.
• Parameters θ = θ1, . . . , θ551
• Euler equation errors
s
et(θ) = et(st+1, bt+1, θt+1) = 1 − θt+1 t+1
bt+1
!
• Instruments: Zt = (st − 1, bt − 1, log(cgt) − 1, log(lgt) − 1, 1)
• Moment conditions: t = 1, . . . , n = 550
mt(θ) = mt(st, bt, st+1, bt+1, θt+1) = Zt ⊗ et(st+1, bt+1, θt+1)
• Length of mt(θ): K = 810
Number of overidentifying restrictions: 260.
Computing the Weighting Matrix W
• Assume factor structure for the random part of (st, bt).
• Σe : one error common for st, one error common for bt, one idiosyncratic
error for each of (st, bt).
o )] = 0 to
• Strengthen zero correlation condition E[Zi,t ej,t(st+1 , bt+1 , θt+1
o
Var[Zt ⊗ et(st+1 , bt+1 , θt+1
)] = Σz ⊗ Σe ,
• Implies Σz ⊗ Σe is diagonalized by Uz ⊗ Ue where Uz and Ue are known,
constant, orthogonal matrices: 1000 fold efficiency gain.
• Mean correction when estimating diagonal elements to avoid acceptance
of an absurd MCMC proposal due to a large variance estimate
!2
n
n
X
X
1
1
si (θ) =
vt,i(θ)
vt,i(θ) −
n t=1
n t=1
Implementation of L(θ) (monthly data)
• The likelihood for the pricing kernel θ is, then,
n
L(θ) ∝ exp − mn(θ) ′(Uz ⊗ Ue)Sn−1(θ)(Uz ⊗ Ue) ′mn(θ)
2
n
1 X
Zt ⊗ et(st+1, bt+1, θt+1),
mn(θ) =
n t=1
• Full likelihood L(η, θ) ∝ L(θ) ×
Qn
t=1 f (θt+1 |θt, θt−1, . . . , θ1 , η).
• dim(η, θ) = 560 implies high correlation in MCMC chain.
• To reduce, sample 30, 000 out of 30 million MCMC draws.
Technical Prior p(θ, η)
• Support condition:
– Impose mean reversion on law of motion f (θt|θt−1, . . . , θ1, η)
• Constrain the size of the Euler equation errors to be about
the same
– Annual data: P (−0.5 < ei,n(θ) < 0.5) = 0.95
– Monthly data: P (−0.05 < ei,n(θ) < 0.05) = 0.95
• Require an approximation PB = n
t=1 θt to the average price
of a risk-free, one period bond to be between 1% and 4%
per annum,
P
f (PB ) ∝ (1 + cos(α + βPB ))
where β = 2π/(b − a), α = π − βb.
a < PB < b
Summary
• Pricing kernel: θ = (θ1, . . . , θn+1)
Q
• Hierarchical likelihood: L(θ, η) = L(θ) n
t=1 f (θt+1 |θt , . . . , θ1 , η)
– L(θ) is a function of returns implied by Euler equations
– f (·|·, η) is the seminonparametric transition density of the
pricing kernel.
• Composite prior: p(θ, η) = p(η)pT (θ, η)
– p(η) prior on hyperparameter η : tight, intermediate, loose
– pT (θ, η) imposes technical conditions
• Inference: MCMC
– Moments of posterior distribution P (θ, η)
RR
– Posterior probabilities P (A) =
IA(θ, η)P (dθ, dη) of events
A that correspond to hypotheses
0.5
1.0
2.0
5.0
10.0
Figure 5. The Posterior Mean of the Monthly Pricing Kernel
1962
1964
1966
1968
1970
0.5
1.0
2.0
5.0
10.0
1960
1975
1980
1985
1990
0.5
1.0
2.0
5.0
10.0
1970
1990
1995
2000
2005
0.1
0.2
0.5
1.0
2.0
5.0
Figure 6. The Posterior Mean of the Annual Pricing Kernel
1940
1960
1980
2000