CAPÍTULO II: APUNTES REFERENCIALES PARA LA INVESTIGACIÓN EN
2.1 De las Escuelas de Periodismo a la investigación en comunicación
The probability distribution νa conditional on each ambiguity state a will
generate a distribution of return for each asset. We say one asset is
ambiguity-free if the its conditional distributions are equal across ambi- guity states.
Definition 3. Ambiguity-free asset
An asset is ambiguity-free if its return distributions conditional on each ambiguity state are the same across states of ambiguity.
Example 1. Suppose there are 3 states of risk, i.e. S = 3, and 2 states of uncertainty, i.e. A = 2. Assume the asset pays (1, a, a) contingent on risk states. The probability distributions conditional on two uncertainty states are (12,0,12) and (12,12,0). Then this asset is ambiguity-free.
Since the existence of an ambiguity-free asset is important for our iden- tification argument, it deserves a bit more discussion. Does there exist an ambiguity-free asset given any conditional probabilities? The risk-free asset is one, but it’s trivial.1 The existence of a risky ambiguity-free asset is not
guaranteed for arbitrary conditional probabilities. The reasoning is this: if A conditional probabilities over S-dimension return vector generate the same return distribution, then they must have the same mean; however, if
A > S, the existence of such S-dimension return vector is not generic. The existence of a risky ambiguity-free asset relies on underlying con- ditional probabilities. If we restrict the space of conditional probabilities, the existence of such asset is not a problem. One restriction which gen- erates an ambiguity-free asset is that we focus on such probability space:
P={νa:ν11 =...=νa1 =...=νA1}, that is, all conditional probabilities
in this probability space put the same probability on the first state. Then any return vector (a, b, ..., b) is ambiguity-free and risky for a 6= b. The restricted probability spacePis not generic in (non-restricted) probability space, however, such space is big enough for us to work on, and it is widely
used in experimental work. For example, in Ahn, Choi, Gale, and Kariv
(2014), the composition of one color balls is publicly announced, and the composition of other two color balls is unknown. This falls in our setting, and one ambiguity-free asset can be traded.
Proposition 4. Suppose that
(1) the objective function satisfies Assumption 1, and asset return satis- fies Assumption 2;
1In the remaining of this chapter, when we refer to an ambiguity-free asset, we will implicitly mean it is ambiguity-free and risky, even though we do not explicitly emphasize its riskiness property.
(2) there is an asset j = 1 that is risk-free: r1 = 1 across states of the
world;
(3) there is an asset j = 2 that is ambiguity-free: its payoff distribution is invariant to the states of ambiguity;
(4) the family of conditional probability measures over states of risk, ν :
A→∆(S) is known;
(5) the matrix of expected marginal utility,
..., Eνau 0 (Ry)R u0 u−1 E νau(Ry) , ... has full row rank A at each portfolio y.
Then, the demand for assets identifies the cardinal index for risk, u, up to a positive affine transformation, as well as the ordinal utility function Φ, up to a monotonically increasing transformation.
Proof. Step 1−recovering risk index:
Restrict attention to portfolios y= (y1, y2,0, ...,0), and let ˜y= (y1, y2)
be the associated truncated portfolio.
Since the distribution of payoffs of assets 1 and 2 are invariant across states of ambiguity, there exists a probability measure, ˜ν ∈ ∆(S), and a matrix of payoffs of assets over states of risk ˜R = (1#S,r˜2), such that,
the distribution of payoffs of assets generated by (νa,Ry), for any state of
ambiguity, coincides with the distribution generated by (˜ν,R˜y˜).
As a consequence, m12(˜y) = Eν˜u 0 ( ˜Ry˜) Eν˜u 0 ( ˜Ry˜) ˜r2 >0. (3.12)
Identification of the cardinal risk index, u, then follows as in Lemma 3.
The first order conditions for an optimum, X a ∂Φ ∂wa Eνau 0 (Ry)R u0 u−1 E νau(Ry) =λp, λ >0, (3.13)
can be written in matrix form,
[Φ1, ...,Φa, ...ΦA] caj = [λp1, ..., λpj, ...λpJ], (3.14) where Φa = ∂w∂Φa, caj = Eνau 0 (Ry)rj u0 u−1 E νau(Ry)
, and matrix C has dimension A
times J.
Since we have recovered index u and conditional distribution of asset
return is known, the matrix C is computable. If matrix C has full row
rank, then
[Φ1, ...,Φa, ...ΦA] = [λp1, ..., λpj, ...λpJ]CT[CCT]−1. (3.15)
So we can trace out the marginal rate of substitution Φa
Φ0a
. Under as- sumption 1, Φ is strictly quasi-concave, continuously differentiable, and has strictly positive gradient everywhere on RA+, following Mas-Colell (1977), knowing the marginal rate of substitution Φa
Φ0a will identify function Φ up to monotonically increasing transformation.
In Klibanoff, Marinacci, and Mukerji (2005), the smooth ambiguity utility functional form isU =Eµφ
u−1 Eνau(xs)
, which is a special case of the above utility functional form, so the identification follows directly from Proposition 4.
Corollary 2. Suppose that
(1) the objective function satisfies Assumption 10, and asset return satis- fies Assumption 2;
(2) there is an asset j = 1 that is risk-free: r1 = 1 across states of the
(3) there is an asset j = 2 that is ambiguity-free: its payoff distribution is invariant to the states of ambiguity;
(4) the family of conditional probability measures over states of risk, ν :
A→∆(S) is known; (5) the payoffs EµEνa(r3)
2 and E
µ(Eνar3)
2 of an ambiguous assetj = 3
are known and satisfy (EµEνar3)
2 6=E
µ(Eνar3)
2.
Then, the demand for assets identifies the risk index u and uncertainty index φ up to positive affine transformation.
Proof. The identification of index u follows the same argument in Propo- sition 4, we sketch the recovery of indexφ.
The marginal rate of substitution between risk-free asset 1 and ambigu- ous asset 3 gives
Eµφ 0 u−1 Eνau(Ry) Eνau 0 (Ry)r1 u0 u−1 E νau(Ry) = m13(y)Eµφ 0 u−1 Eνau(Ry) Eνau 0 (Ry)r3 u0 u−1 E νau(Ry) . (3.16)
Take derivative on both sides with respect to y3, and evaluate at y =
(x,0, ...,0), we get [(EµEνar3) 2−E µ(Eνar3) 2]φ 00 (x) φ0(x) = [EµEνa(r3) 2−E µ(Eνar3) 2]u 00 (x) u0(x) + (EµEνar3) 2∂m13(x,0, ...,0) ∂y3 . (3.17)
Given risk index u recovered, this will identify uncertainty index φ
uniquely up to a positive affine transformation.
Remark 14. The coefficient of φ 00
(x)
φ0(x) is the variance of random variable
Eνar3 evaluated by ambiguity probability measure µ, and nonzero of the coefficient requires ambiguity on the mean. The full row rank condition implies the non-vanishing of the coefficient, but this condition is much weaker than full row rank condition.
Remark 15. The above recovery argument entails knowing some moments of assets evaluated by bothµandν: EµEνar3, EµEνa(r3)
2, andE
µ(Eνar3)
2.
If the domain of preference is compound objective lotteries, such moments can be computed directly from the known objective probabilities. However, if the domain is subjective-objective two stage lotteries within Anscombe and Aumann(1963) framework, these payoff moments are not directly ob- servable, and should be elicited from subjects.
In the above recovery arguments, we only observe individual portfolio choice; however, in reality, individual makes joint decision on consumption and portfolio. In this case, individual optimization problem can be written as max y∈RJ U = Φ c0, φ−1 Eµφ(u−1(Eνau(Ry))) s.t. p0c0+py = 1. Assumption 100
(1) u isC2 on R++, is strictly concave and satisfies ∀x∈R++,u
0
>0; (2) Φ is C1 onR2
++, is strictly concave, and satisfies ∀c ∈ R2++, Φi >0
for all i= 1,2; (3) φ−1 E
µφ(u−1(Eνau(x)))
is strictly concave on R++.
The demand for assets satisfies necessary and sufficient first order condi- tions Φ1 =λp0, (3.18) Φ2 Eµφ 0 u−1 E νau(Ry) Eνau 0 (Ry)R u0(u−1(E νau(Ry))) φ0 φ−1 E µφ(u−1(Eνau(Ry))) =λp, λ >0, (3.19) p0c0+py = 1. (3.20)
Corollary 3. Suppose that
(1) the objective function satisfies Assumption 100, and asset return sat- isfies Assumption 2;
(2) there is an asset j = 1 that is risk-free: r1 = 1 across states of the
world;
(3) there is an asset j = 2 that is ambiguity-free: its payoff distribution is invariant to the states of ambiguity;
(4) the family of conditional probability measures over states of risk, ν :
A→∆(S) is known; (5) the payoffs EµEνa(r3)
2 and E
µ(Eνar3)
2 of an ambiguous assetj = 3
are known and satisfy (EµEνar3)
2 6=E
µ(Eνar3)
2.
Then, the demand for consumption and assets identifies the cardinal index for risk, u, as well as the cardinal index for uncertainty φ, up to a positive affine transformation, the ordinal index for time preference, Φ, up to a monotonically increasing transformation.
Proof. Step 1−recovering risk index:
First order conditions trace out the marginal rate of substitution be- tween asset 1 and asset 2,
Eµφ 0 u−1 E νau(Ry) Eνau 0 (Ry)r1 u0 u−1 E νau(Ry) Eµφ 0 u−1 E νau(Ry) Eνau0(Ry)r2 u0 u−1 E νau(Ry) =m12(c0,y). (3.21)
Since the return distribution of asset 1 and 2 is ambiguity free, at ˜y= (y1, y2,0, ...,0), we have m12(c0,y˜) = Eν˜u 0 ( ˜Ry˜) E˜νu 0 ( ˜Ry˜) ˜r2 >0. (3.22)
Then recovery of u follows from Lemma 3.
The marginal rate of substitution between risk-free asset 1 and ambigu- ous asset 3 gives
Eµφ 0 u−1 Eνau(Ry) Eνau 0 (Ry)r1 u0 u−1 E νau(Ry) = m13(c0,y)Eµφ 0 u−1 Eνau(Ry) Eνau 0 (Ry)r3 u0 u−1 E νau(Ry) . (3.23)
Take derivative on both sides with respect to y3, and evaluate at y =
(x,0, ...,0), we get [(EµEνar3) 2− Eµ(Eνar3) 2 ]φ 00 (x) φ0(x) = [EµEνa(r3) 2−E µ(Eνar3) 2]u 00 (x) u0(x) + (EµEνar3) 2∂m13(c0, x,0, ...,0) ∂y3 . (3.24) Given risk index u recovered, this will identify uncertainty index φ
uniquely up to a positive affine transformation.
Step 3−recovering time index:
The marginal rate of substitution between risk-free asset 1 and con- sumption c0 gives Φ1 Φ2Eµφ 0 u−1 E νau(Ry) Eνau 0 (Ry)r2 u0 u−1 E νau(Ry) =m01(c0,y). (3.25) Evaluate at y= (x,0, ...,0), we have Φ1(c0, x) Φ2(c0, x) =m01(c0, x,0, ...,0)EµEνar1. (3.26) It will recover Φ up to monotonically increasing transformation.
Remark 16. The functional form Φ
c0, φ−1 Eµφ(u−1(Eνau(xs)))
con- tains one important special case where the three preference parameters−
elasticity of inter-temporal substitution, risk aversion and ambiguity aver- sion are separated as in Hayashi and Miao (2011), which shows that such
separation will explain the historical data better.
Remark 17. As in Lemma3and Lemma4, knowledge of the second moment of the distribution of payoffs of an asset invariant across states of ambiguity permits identification of the risk index u, as well as identification of the payoffs of assets invariant over states of ambiguity.
Example 2. Identification of risk and uncertainty aversion
Suppose there are one riskless asset with payoff r, one ambiguity free as- set with payoff r1, r1 ∼ N(µ1, σ12) and one ambiguous asset with payoff
r2, r2 ∼ N(µ2, σ22), where individual has ambiguity on the mean of r2 ,
and µ2 ∼ N(θ, σ20). It is assumed that payoffs of assets are independent. Individual is endowed with risk preference u(x) = −e−ρx
ρ , and ambiguity
preference φ = −e−Au
A . Individual will demand the ambiguity free asset
α1 = µρσ1−2r 1
, and the ambiguous asset α2 = ρσ2θ−r 2+Aσ02
. It follows that risk aversion index u can be recovered from ambiguity-free asset demand α1
if we know its return distribution; and ambiguity aversion index φ can
be recovered from ambiguous asset demand α2 if we know its conditional
distribution and ambiguity.
Remark 18. The above argument identifies cardinal risk index u and un-
certainty index φ once knowing certain moments of asset returns under
ambiguity probability measure µ. One question is: suppose we know con- ditional distribution of asset return, can we recover without reference to
ambiguity probability measure µ? The above example shows that this is
not the case: even we know payoffs of the riskless asset r, and payoffs of the ambiguity-free risky asset µ1, σ1, and all conditional distribution of
ambiguous asset θ, σ2, we can not identify risk index u and uncertainty
index A uniquely.
The above identification argument requires observing individual asset demand. An equivalent way to establish recoverability of risk and uncer- tainty preference is to know individual portfolio indifference correspondence
I(y) = x∈RJ :E µφ u−1 Eνau(Rx) =Eµφ u−1 Eνau(Ry) .2
Under the same assumption as above on asset return and individual
2We illustrate the argument using the this special functional form, but the argument here can recover more general utility function.
belief, risk indexuand uncertainty indexφcan be identified from individual portfolio indifference correspondence.
Corollary 4. Suppose that
(1) the objective function satisfies Assumption 10, and asset return satis- fies Assumption 2;
(2) there is an asset j = 1 that is risk-free: r1 = 1 across states of the
world;
(3) there is an asset j = 2 that is ambiguity-free: its payoff distribution is invariant to the states of ambiguity;
(4) the family of conditional probability measures over states of risk, ν :
A→∆(S) is known; (5) the payoffs EµEνa(r3)
2 and E
µ(Eνar3)
2 of an ambiguous assetj = 3
are known and satisfy (EµEνar3)
2 6=E
µ(Eνar3)
2.
Then, the portfolio indifference correspondence identifies the cardinal in- dex for risk u,and cardinal index for uncertainty φ, up to a positive affine transformation.
Proof. See the proof in the appendix.
Remark 19. Also, the coefficient of φ 00
φ0 is the variance of random variable
Eνar3 under ambiguity probability measure µ, and the required informa- tion on distribution of asset return is the same as in Corollary2.
Remark 20. Under the same assumptions on asset return and individual be- lief, both observing individual asset demand and knowing individual port- folio indifference correspondence give the identification result. This should not be surprising, since we can trace out individual asset demand from knowledge of his indifference correspondence.