Assessing Specification Errors in Stochastic
Discount Factor Models
Lars Peter Hansen and Ravi Jagannathan
Journal of Finance 1997
How can we compare misspecified asset pricing models?
Environment
I
at date t a set of assets is purchased at price qt
I
these assets have a payoff of xt+τ at date t + τ
I
let F be the conditioning observation observable at t + τ and L2
the space of all random variables with finite second moments in
F
I
inner product and norm: hh1 |h2 i = E(h1 h2 ), khk = hh|hi1/2
I
P, the space of portfolio payoffs used in the econometric analysis
is a closed linear subspace of L2 . For the empirical analysis, let
P = {x · c : c ∈ Rn } with x being an n dimensional vector with
entries in L2
Environment
the expected price π(p)
I
π is continuous and linear on P, and there exists a p ∈ P such that
π(p) = 1
Environment
Stochastic Discount Factors
I
an admissible sdf is a random variable in L2 such that
π(p) = E(pm)∀p ∈ P
(1)
Let M be the space of all admissible stochastic discount factors.
Further, let M++ be the set of all stochastic discount factors that
are positive with probability one and let M+ = closure(M++ )
I
call the extension of π to all h ∈ L2 πm (h) = Ehm. This pricing
functional does not induce arbitrage opportunities if m ∈ M++
Least Squares Approximation of Proxies
I
an asset pricing model gives us a proxy for a stochastic discount
factor:
πa (h) = E(yh)∀h ∈ L2
(2)
Least Squares Approximation of Proxies
Least Squares Problems
I
Problem 1:
δ = min ky − mk
(3)
δ + = min ky − mk
(4)
m∈M
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Problem 2:
m∈M+
Least Squares Approximation of Proxies
Maximum Pricing Errors
I
δ=
max
p∈P,kpk=1
|πa (p) − π(p)|
(5)
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δ + = min
max
m∈M+ h∈L2 ,khk=1
|πm (h) − πa (h)|
(6)
Least Squares Approximation of Proxies
Duality to solve the LS Problems: Problem 2
I
(δ + )2 =
min
m∈L2 ,m≥0
ky − mk2
(7)
subject to E(mx) = Eq
I
(δ + )2 = maxn
min (E(y − m)2 + 2λ0 E(xm) − 2λ0 Eq) (8)
λ∈R m∈L2 ,m≥0
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solving the inner minimization problem we get:
(δ + )2 = maxn E(y2 − [(y − λ0 x)+ ]2 − 2λ0 q)
λ∈R
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this needs to be solved numerically
(9)
Least Squares Approximation of Proxies
Duality to solve the LS Problems: Problem 1
I
calculations similar to the ones for problem 2 give us:
(δ + )2 = maxn E(y2 − [(y − λ0 x)]2 − 2λ0 q)
(10)
E[x(y − λ̃0 x) − q = 0]
(11)
λ̃ = (Exx0 )−1 E(xy − q)
(12)
δ = [E(xy − q)0 (Exx0 )−1 E(xy − q)]1/2
(13)
λ∈R
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FONC:
I
I
Implementation
I
dT = {maxn T −1
λ∈R
T
X
[y2t+τ − (yt+τ − λ0 xt+τ )2 − 2λ0 qt ]}1/2 (14)
t=1
I
dT+ = {maxn T −1
λ∈R
T
X
t=1
[y2t+τ − (yt+τ − λ0 xt+τ )+2 − 2λ0 qt ]}1/2 (15)
Implementation
Hansen, Heaton and Luttmer (1995)
I
T 1/2 (dT − δ) =⇒D N[0, σ
b2 /(4δ 2 )]
(16)
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T 1/2
T
X
[y2t+τ − (yt+τ − λ0 xt+τ )+2 − 2λ0 qt − δ 2 ] =⇒D N[0, σ
b2 ]
t=1
(17)
I
T 1/2 dT
(dT − δ) =⇒D N[0, 1]
2sT
(18)
Empirical Application
Specification Errors for Power Utility
Empirical Application
Lagrange Multipliers for Power Utility
Empirical Application
Specification Errors for Habits
Empirical Application
Specification Errors for Linear Factor Models