Sets of Models and
Prices of Uncertainty
Lars Peter Hansen (University of Chicago)
Thomas J. Sargent (New York University)
1/1
Main ideas
Aim:
• Use dynamic max-min utility as a tool to study consequences of
fears of model misspecification
Approach:
• Surround a baseline model with sets that include particular
statistically similar time-varying parameter models
• Use Chernoff entropy to calibrate statistically plausible sets
• Study how concerns about misspecification alter market prices of
uncertainty
2/1
Application
Ingredients:
• Baseline model for log consumption growth with a predictable
growth state variable
• Set of alternative parametric and other less structured models
• Tractable robust decision problems for planner and representative
investor
Outcome:
• Log stochastic discount factor is a quadratic function of the growth
state
• Uncertainty prices high when growth state is low
3/1
Cast of characters
• Baseline model and parametric alternatives
• Positive martingales that represent alternative probabilities
• Convex set of martingales
• Statistical discrimination via Chernoff entropy
4/1
References
• Anderson, Hansen, and Sargent, 2003. “A Quartet of Semigroups for
Model Specification, Robustness, Prices of Risk, and Model Detection.”
JEEA.
• Gilboa and Schmeidler, 1989. “Maxmin Expected Utility with
Non-unique Prior.” Journal of Mathematical Economics.
• Petersen, James, and Dupuis, 2000. “Minimax Optimal Control of
Stochastic Uncertain Systems with Relative Entropy Constraints,”
Automatic Control, IEEE Trans. Aut. Control
• Newman and Stuck, 1979. “Chernoff Bounds for Discriminating between
Two Markov Processes,” Stochastics.
5/1
Baseline model
• Baseline model parameterized by µ̂, ϕ̂, β̂, κ̂, α, σ
(
)
d log Ct = (.01) µ̂ + β̂Xt dt + (.01)α · dWt
dXt = ϕ̂dt − κ̂Xt dt + σ · dWt
• {Wt } a Brownian motion
• Think of C as consumption
• {Xt } generates “long-run risk”
6/1
Other models
• Generate an alternative model with a nonnegative martingale
(∫ t
)
∫
1 t
h
2
Zt = exp
ht · dWt −
|hu | du
2 0
0
• ∫Z is set of all martingales Zh for some adapted process h with
t
2
0 |hu | du < ∞.
7/1
Alternative models
• Implied perturbed probabilities
[
]
Eh [Bt |F0 ] = E Zht Bt |F0
• Implied perturbed evolution of W:
dWt = ht dt + dWht
where dWht is a standard Brownian increment
8/1
A family of parametric alternative
models
•
d log Ct = .01 (µ + Xt ) dt + .01α · dWht
dXt = ϕdt − κXt dt + σ · dWht
• Represent by setting
ht = η(Xt ) ≡ η0 + η1 Xt
where
[
]
[ ′]
µ − µ̂
α
η =
σ′ 0
ϕ − ϕ̂
[
]
[ ′]
α
β − β̂
η =
σ′ 1
κ̂ − κ
9/1
Instantaneous vs. intertemporal
Alternative ways to restrict |ht |2
• Instant by instant (Chen and Epstein, 2000)
• Intertemporal based on statistical discrepancy (Hansen and Sargent)
We include intertemporal neighborhoods of parametric alternatives
using relative entropy and Chernoff entropy as measures of statistical
discrepancy
10 / 1
Statistical discrepancy: Step 1
• Form a convex function
ef(z, ζ) =
( r+1
)
1
z
− 1 + (1 − z)ζ
r(r + 1)
where ef(1, ζ) = 1
• r = 0 is particularly convenient and implies
ef(z, ζ) = z log z + (1 − z)ζ
11 / 1
Statistical discrepancy: Step 2
For martingales Zh construct measures of statistical discrepancy using
subjective rate of discount δ
e t to use as ζ in ef(z, ζ):
• Construct a process Ξ
∫ t
˜ u )du
e
Ξt =
ξ(X
0
e h |x) equal to
• Form a discrepancy measure ∆(Z
∫
[ (
)
]
e t X0 = x dt
exp(−δt)E ef Zht , Ξ
0
( )∫ ∞
[( )
)
r+1
1
h
2
=
exp(−δt)E Zt
|ht | X0 = x dt
2
( ) ∫ 0∞
[
(
)
)
1
˜ t ) 1 − Zh X0 = x dt
+
exp(−δt)E ξ(X
t
2
0
∞
δ
12 / 1
Statistical discrepancy: Step 3
Look at martingales that satisfy:
∫ ∞
[
)
˜ t )X0 = x dt
e h |x) ≤ 1
exp(−δt)E ξ(X
∆(Z
2 0
• Includes parametric models satisfying
˜ t)
|ht | ≤ ξ(X
• Allows time-varying parameter models
• Allows other statistically similar models
13 / 1
Convex set Z ∗
• Construct set Z ∗ of martingales Zh satisfying
∫
[
]
1 ∞
h
˜ t )X0 = x dt
e
∆(Z , x) ≤
exp(−δt)E ξ(X
2 0
• Use probability measure Q to average over x
• Use Z ∗ to pose a recursive robust decision problem
14 / 1
Robust planner’s problem
• h∗ (x, θ) = − θ1 [.01α − συ2 (θ)x − συ1 (θ)] attains HJB equation
1
0 = min − δυ(x, θ) + (.01)(µ̂ + x) − υ ′ (x, θ)κ̂x + |σ|2 υ ′′ (x, θ)
h
2
θ
θ
˜
+ (.01)α · h + υ ′ (x, θ)σ · h + |h|2 − ξ(x)
2
2
• Implied stochastic discount factor
1
d log St = −δdt − .01 (µ̂ + Xt ) dt − .01α · dWt + h∗t · dWt − |h∗t |2 dt
2
• h∗ = η0∗ + η1∗ x
“local risk price” =
.01α
+
(−h∗t )
risk price
uncertainty price
15 / 1
SDF carpentry
˜ shapes uncertainty
• By affecting h∗t , the functional form of ξ(x)
prices
˜ gives rise to a quadratic log stochastic discount
• A quadratic ξ(x)
factor
16 / 1
A stylized model selection problem
• To discriminate between two models:
(i) a baseline model
(ii) an alternative within the parametric family
• Given historical data, choose a model as either a Bayesian or
max-min decision maker
• Compute ex ante probabilities of making Type I and Type II errors
as sample size increases
• Equate decay rates in error probabilities
17 / 1
A large deviation approximation
Chernoff entropy measures statistical distance between models
•
{
}
Zb = Zh ∈ Z : χ(Zh ; x) ≤ χ
where Chernoff entropy is
χ(Zh |x) = − inf lim sup
0≤r≤1
t→∞
[( )r ]
1
log E Zht
t
• Use χ to compute half-life of decay in Type I and Type II errors
18 / 1
Calibrating robustness concerns
e up to scale.
• Specify quadratic ξ(x)
e to match a given half-life implied by baseline and
• Scale ξ(x)
worst-case models.
19 / 1
Two statistical calibrations
• Bansal-Yaron model without stochastic volatility matched to post
WWII quarterly consumption data
▷ no correlation between shock to consumption and shock to
the growth rate in consumption (α · σ = 0)
▷ state variable hidden to econometricians but not to investors
• Hansen-Heaton-Li VAR style model matched to quarterly
consumption data and data on business related earnings projected
onto our parametric class
▷ correlation between shock to consumption and shock to the
growth rate in consumption (α · σ ̸= 0)
▷ state variable inferred by econometrics
20 / 1
Worst-case models for specification
one
Half-Life
∞
120
80
40
µ
0.499
0.477
0.475
0.475
ϕ
0
-0.0348
-0.0407
-0.0517
β
1
1
1
1
κ
0.169
0.134
0.122
0.095
µ + βϕ
κ
0.499
0.218
0.142
- 0.070
˜ ∝ x2
ξ(x)
21 / 1
Figure : Expected values and interdecile ranges of log Ct for the baseline
model and the 90 half-life worst case model. The shaded black and red areas
show the .1 and .9 interdecile ranges. The black line is the mean growth for
the baseline model, and the red circle line is the mean growth for the
worst-case model.
22 / 1
Figure : Shock-price elasticities. Top to bottom: half-lives 120, 80, and 40,
respectively. Each column displays elasticities for the corresponding shock.
Shaded regions show the .1 to .9 interdecile ranges. Blue circles are
elasticities when ξ˜ ∝ 1.
23 / 1
Worst-case models for specification
two
Half-Life
∞
120
80
40
µ
0.464
0.394
0.386
0.374
ϕ
0
-0.0095
-0.0107
-0.0128
β
1
1.173
1.221
1.316
κ
0.065
0.039
0.032
0.018
µ + βϕ
κ
0.464
0.150
0.049
-0.355
˜ ∝ x2
ξ(x)
24 / 1
Figure : Shock-price elasticities. Top to bottom: half-lives 120, 80, and 40,
respectively. Each column displays elasticities for the corresponding shock.
Shaded regions show .1 to .9 interdecile ranges. Blue circles are elasticities
when ξ˜ ∝ 1.
25 / 1
Concluding remarks (I)
• Use tilted discounted relative entropy to pose robust decision
problem
• Use Chernoff entropy as a calibration device
• Allow for time-varying parameter models and other statistically
similar ones
• Induce larger uncertainty prices when growth prospects are weaker
26 / 1
Concluding remarks (II)
More complicated ways of inducing variation in “risk-prices”
• Add stochastic volatility shocks
• Impose reduced-form linear variation in risk prices as in Ang and
Piazzesi (2003)
• Impose social externalities in the form of a habit state as in
Cochrane and Campbell (1999)
• Incorporate robust learning of a hidden Markov state as in Hansen
and Sargent (2010)
27 / 1