parameter uncertainty and learning in dynamic financial decisions

Parameter Uncertainty and Learning in DynamicFinancial Decisions

Daniel A. Bruggisser

May 15, 2010

Lecture Notes

Financial Econometrics

Keywords: Bayesian portfolio analysis; parameter uncertainty; dynamic multi-period asset allocation; Bayesian learning; informative prior; return predictability; return forecasting; regime-switching; model uncertainty.

Agenda

1. Introduction

2. Bayesian portfolio analysis

3. Multi-period Bayesian asset allocation

4. Bayesian learning

5. Conclusion

2

http://www.amazon.de/Parameter-Uncertainty-Learning-Financial-Decisions/dp/3258077096

http://www.haupt.ch/verlag/oxid.php/cl/details/anid/9783258077093

Introduction

• Parameter uncertainty and learning is ubiquitous in finance.• Parameters (or certain states) of financial models are never known with certainty.• Participants in financial markets acquire substantial information over time andlearn about the true parameters or states of the economy.

• Furthermore, they may want to use non-sample information in their decisionmaking process.

• Bayesian portfolio analysis allows the investor to combine proprietary beliefs,parameter uncertainty and learning in dynamic decision making process.

• Whereas dynamic decision making under conditions of risk is meanwhile quite wellunderstood, the impact of parameter uncertainty and learning is not.

• Bayesian portfolio analysis is extended to a multi-period dynamic setting andconditions for learning dynamic models established.

• The Bayesian learning process for conditionally normal linear models is shown tobe the Bayesian version of the well known Kalman filter.

• The learning process for the unobservable regime of a regime-switching model isshown to be the Bayesian version of Hamilton′s basic filter.

3

Bayesian Portfolio Analysis (1)

• Bayesian portfolio analysis has a long tradition in finance.

• The literature includes:

– Uninformative prior approach.1

– Informative prior approach.2

– Shrinkage models (such as James-Stein estimators and Bayes-Stein).3

– Mixed estimation and the Black-Litterman Model.4

– Prior beliefs in an Asset Pricing Theory.5

– Prior beliefs in no-predictability in forecasting models.6

– Model uncertainty and Bayesian model selection and averaging.7

• All these approaches are deeply rooted in the theory of Bayesian analysis.

• Significant simplification can be achieved for wide class of conditionally normallinear models in application of conjugate informative priors or diffuse (uninforma-tive) Jeffreys′ priors.

4


• The classical portfolio selection problem:8

maxω

ET [U(WT+1)] = maxω

∫

Ω

U(WT+1)p(rT+1|θ)drT+1, (1)

where Ω is the sample space, U(WT+1) is a utility function, WT+1 is the wealthat time T + 1, θ is a set parameters, ω are portfolio weights, and p(rT+1|θ) isthe sample density of returns.

• Bayesian portfolio selection problem:9

maxω

ET [U(WT+1)] = maxω

∫

Ω

U(WT+1)p(rT+1|ΦT )drT+1, (2)

where ΦT is the information available up to time T , p(rT+1|ΦT ) is the Bayesianpredictive distribution (density) of asset returns.

5


• Bayesian decomposition, Bayes′ rule and Fubini′s theorem:10

ET [U(WT+1)] =

∫

Ω

U(WT+1)p(rT+1|ΦT )drT+1 (3)

=

∫

Ω×Θ

U(WT+1)p(rT+1,θ|ΦT )d(drT+1,θ)

=

∫

Ω

∫

Θ

U(WT+1)p(rT+1|θ)p(θ|ΦT )dθdrT+1

=

∫

Ω

U(WT+1)

(∫

Θ

p(rT+1|θ)p(ΦT |θ)p(θ)dθ

)

drT+1,

where Θ is the parameter space, p(rT+1, θ|ΦT ) is the joint density of parametersand realizations, p(θ|ΦT ) is the posterior density, p(ΦT |θ) is the conditionallikelihood, and p(θ) is the prior density of the parameters.

• There are many ways to derive the predictive distribution.

6


• Uninformative prior approach

The model:11

rt = µ+ ut (4)

ut ∼ i.i.d. N (0ι,Σ) (5)

where rt is the m× 1 vector of asset returns at time t, µ is the m× 1 vector ofunknown means, ut is the m× 1 of disturbances, ι is a m× 1 vector of ones, andΣ is a PDS covariance matrix. The model has sample density:

p(rt|µ,Σ) = N(

µ,Σ)

. (6)

With a uninformative prior density p(µ) = ιc and assuming Σ known, the posterioris given by12

p(µ|ΦT ,Σ) = N

(

µ,1

TΣ

)

, (7)

7

where ΦT is the information history and

µ =1

T

T∑

t=1

rt. (8)

The predictive density of the one period ahead returns is13

p(rT+1|ΦT ,Σ) = N

(

µ,Σ+1

TΣ

)

. (9)

Following the variance decomposition, the k-period ahead predictive density is14

p(rT,T+k|ΦT ,Σ) = N (kµ, kΣ+k2

TΣ) = N

(

kµ,k(T + k)

TΣ

)

. (10)

Parameter uncertainty has a large impact on predictive density in the long-termas noted by Barberis (2000).

8


• Informative prior approach

The model is the same as before, but here the prior is given a multivariate normaldensity p(µ) = N (m0,Λ0), where m0 is the m× 1 vector of priors on the means,and Λ0 is a m×m matrix of prior uncertainty. Σ is again assumed to be known.Then, the posterior of µ is15

p(µ|ΦT ,Σ) = N(

mT ,ΛT

)

(11)

mT =(

Λ−1

0 + TΣ−1)−1

(Λ−1

0 m0 + TΣ−1µ) (12)

ΛT =(

Λ−1

0 + TΣ−1)−1

. (13)

The predictive density of one period ahead asset returns is given by

p(rT+1|ΦT ,Σ) = N (mT ,Σ+ΛT ) . (14)

9


• Posterior Shrinkage16

The mean of the posterior in (12) can be written in shrinkage form:

mT = δm0 + (I− δ)µ, (15)

where I is an m×m identity matrix with principal diagonal elements of one andzeros elsewhere. δ is called the posterior shrinkage factor. It can be obtainedusing matrix algebra and can be shown to be17

δ =(

Λ−1

0 + TΣ−1)−1

Λ−1

0 (16)

=(

[prior covariance]−1 + [conditional covariance]−1)−1

[prior covariance]−1

= [posterior covariance][prior covariance]−1.

The shrinkage target m0 and Λ0 can be obtained from a minimum varianceportfolio, reversed optimization or any other economic reasonable information theinvestor might have prior to seeing the data.

10


• Bayes-Stein estimator

The mean of the posterior is the same as in posterior shrinkage above. TheBayes-Stein estimator introduced by Jorion (1986) use the minimum varianceportfolio as a shrinkage target:

p(µ) = N (µ0ι,1

κΣ) (17)

µ0 =ι′Σ−1

ι′Σ−1ιµ.

Jorion (1986) obtains a posterior of the form:

p(µ|ΦT ,Σ) = N(

mT ,ΛT

)

(18)

mT =(

κΣ−1 + TΣ−1)−1

(κΣ−1µ0ι+ TΣ−1µ)

ΛT =(

κΣ−1 + TΣ−1)−1

.

11

and in shrinkage form, this is

mT = δµ0ι+ (1− δ)µ (19)

δ =κ

κ+ T. (20)

The predictive density for one period ahead returns is

p(rT+1|ΦT ,Σ) = N(

mT ,Σ+ΛT

)

(21)

= N

(

mT ,

(

1 +1

κ+ T

)

Σ

)

. (22)

Because the shrinkage target µ0 has minimum variance ιι′

ι′Σ−1ι, Jorion also finds

that

p(rT+1|ΦT ,Σ) = N

(

mT ,

(

1 +1

κ+ T

)

Σ+κ

T (T + 1 + κ)

ιι′

ι′Σ−1ι

)

. (23)

12


• Optimal shrinkage approach (James-Stein estimator)

James & Stein (1961) define a loss function based on the estimate µs such that(Jorion, 1986, p. 283)

L(µ, µs) = (µ− µs)′Σ−1(µ− µs). (24)

They find that for a shrinkage target µ0 and prior variance λ20, such that the prior

is p(µ) = N (µ0ι, λ20I), the posterior is given by

p(µs|δ∗,Σ) = N (ms,Λs) (25)

ms = δ∗µ0ι+ (1− δ∗)µ

Λs = δ∗(λ20I),

13

where δ∗ is called the optimal shrinkage factor

δ∗ = min

1,(m− 2)/T

(µ− µ0ι)′Σ−1(µ− µ0ι)

, (26)

and

µ =1

T

T∑

t=1

rt. (27)

Although the James-Stein estimator is usually used as a point estimate, it can beshown that the predictive density of one period ahead asset returns is

p(rT+1|δ∗,Σ) = N (ms,Σ+Λs) . (28)

14


• Mixed estimation18

Let the sample density of returns be given a multivariate normal density

p(rt|µ,Σ) = N(

µ,Σ)

, (29)

and the prior density for the m× 1 vector µ also have multivariate normal densitywith

p(µ) = N (m0,Λ0) . (30)

The investor expresses views about µ by imposing

p(v|µ) = N (Pµ,Ω) , (31)

where P is an m × m design matrix that selects and combines returns intoportfolios about which the investor is able to express his views. v is a m × 1vector of views and Ω expresses the uncertainty of those views. It emerges that

15

the posterior of µ updated by the views is

p(µ|v) = N (mv,Λv) (32)

mv =(

Λ−1

0 +P′Ω−1P)−1 (

Λ−1

0 m0 +P′Ω−1v)

Λv =(

Λ−1

0 +P′Ω−1P)−1

.

Then, the predictive density of one period ahead returns is obtained by integratingover the unknown parameter µ

p(rT+1|v,Σ) =

∫

Θ

p(rT+1|µ,Σ)p(µ|v)dµ, (33)

which can be shown to result in

p(rT+1|Σ,v) = N(

mv,Σ+Λv

)

. (34)

16


• Black-Litterman model

Black & Litterman (1992) suggest using the market equilibrium model as a prior

µequ = γΣω∗mkt, (35)

where γ is the risk aversion of a power utility investor and ω∗mkt is the market

capitalization. Black & Litterman assume a natural conjugate prior for the vectorof means such that

p(µ) = N(

µequ, λ0Σ)

. (36)

The investor expresses views about the µ by imposing

p(v|µ) = N (Pµ,Ω) . (37)

17

It follows that the posterior of µ updated by the views is

p(µ|v) = N (mv,Λv) (38)

mv =(

(λ0Σ)−1

+P′Ω−1P)−1 (

(λ0Σ)−1

µequ +P′Ω−1v)

Λv =(

(λ0Σ)−1 +P′Ω−1P)−1

.

Then, the predictive density of one period ahead returns is again

p(rT+1|Σ,v) = N(

mv,Σ+Λv

)

. (39)

18


• Bayesian Asset Pricing Model

Pastor (2000) formulated a Bayesian Asset Pricing Model

rt = x′tθ + εt (40)

εt ∼ i.i.d. N (0, σ2), (41)

where xt = [1, z′t]′ denotes a (k + 1)× 1 vector with zt containing a k × 1 vector

of observable factors. Then, θ is a (k + 1) × 1 vector with θ = [α,β′]′ where αis the intercept and β is a k × 1 vector containing the sensitivities (betas) of theassets to the factors zt. Pastor (2000) imposes a prior on θ such that

p(θ) = N (m0,Λ0), (42)

where m0 is a m× 1 vector of prior means and Λ0 is a m×m uncertainty matrix.In a single-factor model such as the CAPM, the benchmark portfolio zt is the

19

market portfolio, and market efficiency implies α = 0. Therefore he assumes thatm0 = 0ι and by sets the first element of the diagonal matrix Λ0 equal to σ2

α andthe remaining to a high but finite value, indicating an uninformative prior for β.He then obtains the posterior of θ by updating the prior with observations suchthat19

p(θ|y,X, σ2) = N (mT ,ΛT ) (43)

mT =(

Λ−1

0 +[

σ2(X′X)−1]−1

)−1 (

Λ−1

0 m0 +[

σ2(X′X)−1]−1

θ)

ΛT =(

Λ−1

0 +[

σ2(X′X)−1]−1

)−1

,

where X = [x1,x2, . . . ,xt]′, y = [r1, r2, . . . , rt]

′and

θ = (X′X)−1X′y. (44)

The predictive density of one period ahead asset returns is then

p(rT+1|y,X, σ2) = N(

x′T+1mT , σ

2 + x′T+1ΛTxT+1

)

. (45)

20


• Bayesian return forecasting with a belief in no-predictability

Kandel & Stambaugh (1996) formulate the following predictive regression model:

rt = x′t−1θ + εt (46)

εt ∼ i.i.d. N (0, σ2). (47)

where xt−1 = [1, z′t−1]′ denotes a (k+ 1)× 1 vector with zt−1 containing a k× 1

vector of exogenous explanatory variables. Then, θ is a (k + 1) × 1 vector withθ = [α,β′]′, where α is the intercept, which is scalar, and β is a k × 1 vectorcontaining the regression coefficients of the explanatory variables zt−1. Kandel &Stambaugh (1996) imposes a prior on θ such that

p(θ) = N (m0,Λ0), (48)

where m0 is a m × 1 vector of prior means and Λ0 is a m × m uncertaintymatrix. Kandel & Stambaugh (1996) and Connor (1997) recommend imposing

21

an informative prior centered on the economic notion of (weak form) marketefficiency, which implies that the slope coefficient should be zero. Specifically,they use the prior p(β) = N (0, σ2

βI) and an uninformative prior for α. This choiceof prior translates into m0 = 0ι where ι is a (k + 1) × 1 vector of ones; the firstelement of the diagonal matrix Λ0 has a high but finite value, and the remainingdiagonal elements are assigned according to σ2

β.

The posterior density is then:

p(θ|y,X, σ2) = N (mT ,ΛT ) , (49)

mT =(

Λ−1

0 +[

σ2(X′X)−1]−1

)−1 (

Λ−1

0 m0 +[

σ2(X′X)−1]−1

θ)

ΛT =(

Λ−1

0 +[

σ2(X′X)−1]−1

)−1

.

where X = [x0,x1, . . . ,xt−1]′, y = [r1, r2, . . . , rt]

′and

θ = (X′X)−1X′y. (50)

22

The predictive density of one period ahead asset returns is then

p(rT+1|y,X, σ2) = N(

x′T+1mT , σ

2 + x′T+1ΛTxT+1

)

. (51)

23


• Bayesian Model Uncertainty20

Each model Mj ∈ M1, . . . ,MJ is given a sample density p(rt|Mj,θj). Eachmodel has then a conditional likelihood p(ΦT |Mj, θj), whereΦT is the observationacquired up to time T and θj is the vector of parameters for model j ∈ 1, . . . , J.Then the posterior model probability is given by

Pr(Mj|ΦT ) =p(ΦT |Mj)Pr(Mj)

∑J

j=1p(ΦT |Mj)Pr(Mj)

, (52)

where

p(ΦT |Mj) =

∫

Θθj

p(ΦT |Mj,θj)p(θj|Mj)dθj (53)

The predictive return density for one period ahead returns is given by predictive

24

Multi-period Bayesian Asset Allocation (1)

• Learning is a result of parameter (or state) uncertainty in dynamic decisionproblems.

• Generally, Bayesian learning leads to a non-Markovian dynamic decision problem.

• However, under certain conditions an equivalent (re-)representation of the statevariables can be found that recaptures the Markovian structure.

• Conditions for the existence of the state (re-)representation are presented.

• Whenever the posterior density exhibits a finite number of sufficient statistics,these sufficient statistics define a compact filter on observations and become statevariables in the dynamic decision problem.

• These state variables can be updated sequentially and exhibit a Markovian struc-ture.

26


• Terminal wealth problem:21

Vt(Wt,Zt) = maxωs

T−1s=t

Et

[

U(WT ) | FI

t

]

(56)

s.t. Ws+1 = Ws

(

ω′s exp(rs+1 + rfs ) + (1− ι′ωs) exp(r

fs ))

, (57)

for all s ∈ t, . . . , T − 1, and where ωs is the optimal allocation, FI

t and is theinformation filtration.

• The finite horizon Bellman equation:

Vt(Wt,Zt) = maxωt

Et

[

Vt+1(Wt+1,Zt+1) | FI

t

]

, (58)

with boundary condition VT+1(WT ,ZT ) = U(WT ).

27


• Bayesian decomposition, Bayes′ rule and Fubini′s theorem:22

Vt(Wt,Zt) = maxωt

∫

Ω×Θ

Vt+1 (Wt+1,Zt+1) p(rt+1, θ|Φt)d(rt+1, θ) (59)

= maxωt

∫

Ω

∫

Θ

Vt+1 (Wt+1,Zt+1) p(rt+1|θ,Zt)p(θ|Φt)dθdrt+1

= maxωt

∫

Ω

Vt+1 (Wt+1,Zt+1)

(∫

Θ

p(rt+1|θ,Zt)p(θ|Φt)dθ

)

drt+1

= maxωt

∫

Ω

Vt+1 (Wt+1,Zt+1) p(rt+1|Φt)drt+1,

with boundary condition Vt+1(WT ,ZT ) = U(WT ).

28


• The predictive distribution breaks the Markovian structure of the multi-perioddynamic allocation problem.

• State (re-)representation theorem:23

A mere change in variables representing the state-space does not affect thesolution, provided that both sets of variables convey equivalent information.The condition for the theorem to hold is that the two sets of variables spanequivalent σ-fields. Whenever a set of sufficient statistics exists, the informationnecessary to characterize p(rt+1|Φt) can be captured by these sufficient statistics.Including these sufficient statistics in the state representation Zt, the Markovianrepresentation of p(rt+1|Φt) is p(rt+1|Zt). In other words, p(rt+1|Zt) andp(rt+1|Φt) convey equivalent information and therefore span equivalent σ-fields.Then, the last line of (59) can be written as

Vt(Wt, Zt) = maxωt

∫

Ω

Vt+1 (Wt+1,Zt+1) p(rt+1|Zt)drt+1. (60)

29


• Implications of the state (re-)representation theorem:24

– Two representations of the state-space convey equivalent information if aset of sufficient statistics exists such that no other statistic calculated fromobservations is needed to uniquely determine the predictive distribution, whichcould eliminate the need to condition the observation history.

– Should such sufficient statistics exist, we might sequentially update the statistics.The resulting path is described by a process that exhibits a Markovian structure.

– Provided it does not depend upon unobservable states, the objective functionis irrelevant in determining the predictive distribution. Therefore, the predictivedistribution and the process of learning can be derived outside the decisioncontext.

– The state (re)-representation does not depend on the form of the probabilitydistribution of the state variables.25 However, the form of the probabilitydistribution and the prior determine the number of sufficient statistics.

– The sufficient statistics become state variables in the multi-period decisionproblem.

30


• Conditions for learning:26

– The learning process must be compact, meaning that additional informationdoes not change the form of the posterior distribution. Instead, the additionalinformation modifies the values of the sufficient statistics that determine thepredictive distribution.

– The observation process must be described by a parametric statistical model,implying a likelihood function of parametric form.

– We must use an informative prior of the conjugate family or an uninforma-tive Jeffreys′ prior. Only this family of priors guarantees that the posteriordistribution of the observations will preserve the distributional form.

– Alternatively, it is possible to use uninformative (diffuse) priors to represent theabsence of prior knowledge.

– Combining the prior with the likelihood function must result in a joint posteriordistribution for which a set of sufficient statistics can summarize all necessaryinformation to determine the predictive density.

31

Bayesian Learning (1)

• A wide class of conditionally normal linear models are discussed and the metho-dology of Bayesian learning presented.

• The following models are analyzed: (1) Learning the risk-premium under un-certainty; (2) Learning in a forecasting model; (3) Learning the parametersof a vector-autoregressive model; (4) Learning the parameters of a generalizedstate-space model; (5) Learning in a regime-switching model under uncertainty.

• The Bayesian learning process of the first four models is shown to be the Bayesianversion of the well known Kalman filter.

• The Bayesian learning process of the regime-switching model under uncertaintycan be seen as the Bayesian equivalent of the well known Hamilton filter.

• The presented models fulfil the conditions for the state (re-)representation suchthat the multi-period dynamic decision problem remains solvable.

32


• Learning in a generalized state-space model27

– The model is riche enough to encompass model (1)-(4), therefore the presenta-tion is restricted to this class.

– The Bayesian learning process is the Bayesian equivalent of the well knownKalman filter.

Measurement equation:

yt = Ft−1θt + vt (61)

vt ∼ i.i.d. N (0ι,V), (62)

where yt is an m× 1 vector of observable random variables, θt is a k× 1 vector ofunobservable time-varying parameters and Ft−1 is an m×k design matrix relatingthe parameters to observations and assuring that the product Ft−1θt is an m× 1vector. V is a m×m PDS covariance matrix.

33

Transition equation:

θt = Gt−1θt−1 +wt (63)

wt ∼ i.i.d. N (0ι,W), (64)

where Gt−1 is a k × k design matrix relating the parameters of the last period toparameters of the current period ensuring that the product Gt−1θt−1 is a k × 1vector. W is a k × k PDS covariance matrix.

Let Φt contain the information acquired up to time t. Imposing the usualconjugate prior on θ0 such that p(θ0) = N

(

m0,Λ0), the posterior density is givenby28

p(θt|Φt) = N(

mt,Λt

)

(65)

Rt−1 = W +Gt−1Λt−1G′t−1

mt = Gt−1mt−1 +Rt−1F′t−1(V + Ft−1Rt−1F

′t−1)

−1(yt − Ft−1Gt−1mt−1)

Λt = Rt−1 −Rt−1F′t−1(V + Ft−1Rt−1F

′t−1)

−1Ft−1Rt−1.

34

The predictive return density for one period ahead returns is29

p(yt+1|Φt) = N(

FtGtmt,V + FtRtF′t

)

. (66)

The state (re-)representation is therefore

Zt = rft ,mt, unique(Λt), unique(Ft), unique(Gt), (67)

where the operator unique(·) determines all unique elements that are not constant.

35


• Learning in regime-switching model with uncertainty30

– The investor faces a generalized state-space models with multiple parametersets θj, one for each regime j = 1, . . . , S.

– The actual regime st is unobservable to the investor, while the other parametersare assumed to be known.

– The regime-switching model can be seen as an application of Bayesian modeluncertainty.31

The model is governed by the following measurement equation

yt = Ft−1θst + vt (68)

vt ∼ i.i.d. N (0ι,Vst). (69)

36

The regimes are governed by the following transition matrix:

πij = Pr(st+1 = j|st = i), Π =

π11 · · · π1S... . . . ...

πS1 · · · πSS

. (70)

Prediction: Let Φt−1 contain all information acquired up to time t − 1. At theend of the (t− 1)-th iteration, the investor is given the posterior state probabilityPr(st−1 = j|Φt−1), for all j ∈ 1, . . . , S. Based on the posterior, the investormust predict the state probability of the next period by

Pr(st = j|Φt−1) =S∑

i=1

Pr(st = j|st−1 = i)Pr(st−1 = i|Φt−1) (71)

where Pr(st = j|st−1 = i), for all i = 1, . . . , S and j = 1, . . . , S, are thetransition probabilities πji in the transition matrix Π.

Update: For each date of the sample the optimal inference and forecast on the

37

active regime can be found by iterating on the equations

Pr(st = j|Φt) =p(yt|st = j,Φt−1)Pr(st = j|Φt−1)

∑S

i=1p(yt|st = i,Φt−1)Pr(st = i|Φt−1)

, (72)

Initialization: The initial state probability Pr(s1 = j|Φ0) must sum up to pro-bability one and Pr(s1 = j|Φ0) ≥ 0. Reasonable choices include the steadystate probability or attributing the same probability to all states. The procedurestarts with the initial state probability followed by iterating on the prediction andupdating steps.

Predictive density for one period ahead returns is then given by

p(yt+1|Φt) =S∑

i=1

p(yt+1|st+1 = i,Φt)Pr(st+1 = i|Φt). (73)

The and state (re-)representation of the Bayesian multi-period investor is

Zt = rft ,Pr(st = j|Φt), unique(Ft), ∀j ∈ 1, . . . , S. (74)

38

Conclusion (1)

• Classical portfolio selection assumes that parameters are known and constant overtime and that there is no other source of information useful to the investor thansample statistics.

• Bayesian portfolio analysis assumes that parameters (or states) are unknown tothe investor, that the investor learns more about quantities of interest as newobservations become available and that there are sources of information otherthan sample evidence that are useful in his decision making process.

• Parameters (or certain states) of economic models are never known with certainty.In fact, these variables are often unknown, partly known or may vary randomly.

• The aim of the work is to extend Bayesian portfolio analysis to a multi-perioddynamic setting.

39

Conclusion (2)

• The Bayesian paradigm allows the investor to treat parameters (or certain states)as uncertain, to include prior beliefs or views in the decision making process andto learn the quantities of interest as new information become available.

• Given that the observation history is rarely a good predictor of the future, it isevident that information other than the sample statistics of past observations maybe very useful in a portfolio selection context.

• Furthermore, portfolio choices are by nature subjective decisions and not objectiveinference problems as the mainstream literature on portfolio choice might suggest.Therefore, there is no need to facilitate comparison.32

• Given that the overall approach is feasible, it remains to show by empirical studiesthat the Bayesian approach to parameter uncertainty and learning in fact leads tobetter allocations in multi-period dynamic decision problems.

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Footnotes1See, e.g., Jeffreys (1961), Zellner (1996), Barry (1974), Klein & Bawa (1976), and Bawa, Brown & Klein (1979).

2See, e.g., Brandt (2010, p. 312), Hamilton (1994, p. 355), Raiffa & Schlaifer (1968).

3Shrinkage estimation relates to either posterior shrinkage, James-Stein optimal shrinkage estimation and Jorion’s(1986) Bayes-Stein estimation.

4Mixed estimation is attributed to Theil & Goldberger (1961); The Black-Litterman model refers to the work of Black& Litterman (1992).

5Prior beliefs in an asset pricing theory has been formulated by Pstor (2000).

6Prior beliefs in no-predictability in a forecasting model has been introduced by Kandel & Stambaugh (1996).

7Bayesian model uncertainty, model selection and model averaging in financial applications can be found in Avramov(2002), Brandt (2010, p. 319), Cremers (2002), Carlin & Louis (2009, pp. 203-204), and Koop, Poirier & Tobias (2007,Ch. 16), among others.

8See, e.g., Campbell & Viceira (2003, p. 22), Barberis (2000).

9See, e.g., Barberis (2000), Kandel & Stambaugh (1996, p. 388), Rachev et al. (2008, p. 96), Wachter (2007, p. 14).

10See Barberis (2000), Brandt (2010, p. 308), Brown (1976, 1978), Kandel & Stambaugh (1996, p. 388), Klein & Bawa(1976), Pstor (2000), Skoulakis (2007, p. 7), and Zellner & Chetty (1965).

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11See, e.g., Barberis (2000), Brandt (2010, p. 310), Hamilton (1994, p. 748), Hoff (2009, p. 108; Rachev et al., 2008,p. 95; Zellner, 1996, p. 379)

12See, e.g., Brandt (2010, p. 310), Zellner (1996), Hamilton (1994, p. 353).

13Brandt (2010, p. 310).

14The variance decomposition and the effect of parameter uncertainty in the long-run predictive density has beendiscussed in Barberis (2000), Pastor & Veronesi (2009, p. 12), and Pastor & Stambaugh (2009).

15See, e.g., Brennan & Xia (2001, p. 918), Hoff (2009, p. 108), Koop, Poirier & Tobias (2007, p. 26).

16Posterior shrinkage is a generalization of the Bayes-Stein estimator (Jorion, 1986) and is a direct result fromreformulating the posterior obtained by an informative prior in shrinkage form.

17See, e.g., Greene (2008, p. 607), Hoff (2009, p. 108), Koop, Poirier & Tobias (2007, p. 26).

18Mixed estimation is attributed to the work of Theil & Goldberg (1961). It is also presented in Brandt (2010, p. 313),Satchell & Scowcroft (2000), Scowcroft & Sefton (2003). As a special case of mixed estimation, Black & Litterman

(1992) present the Black-Litlerman model.

19See, e.g., Greene (2008, p. 607), Hamilton (1994, p. 356), Hoff (2009, p. 155).

20Bayesian model uncertainty, model selection and model averaging in financial applications can be found in Avramov

(2002), Brandt (2010, p. 319), Cremers (2002), Carlin & Louis (2009, pp. 203-204), and Koop, Poirier & Tobias (2007,Ch. 16), among others.

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21Instead of optimizing expected utility over final wealth, the investor could have other objectives, such maximizingexpected utility over interim consumption. The terminal wealth problem has been discussed in Barberis (2000, p. 251),

Brandt (2010, p. 274), Brandt et al. (2005, p. 836), and Brannan & Xia (2001, p. 918). Consumption-based models havebeen presented in Rubinstein (1976), Lucas (1978), Breeden (1979), Campbell & Viceira (2003, p. 122), Cochrane (1989,p. 322), and Mehra & Prescott (1985, 2003) and in other classical papers, such as those of Samuelson (1969, 1970) and

Merton (1969, 1971).

22See, e.g., Barberis (2000, p. 255), Brandt (2010, p. 309), Bauwens, Lubrano & Richard (1999, p. 5). Fubini′s theorem

is presented in Billingsely (1986, p. 236), Chung (1974, p. 59), Capinski & Kopp (2004, p. 171), Duffie (1996, p. 282).

23See Feldman (2007, p. 127).

24See Feldman (2007, p. 127).

25See Feldman (2007, p. 127).

26See Feldman (2007, p. 124).

27The presentation is mainly based on the work of Harvey (1993), Kim & Nelson (1999, pp. 22-57, pp. 189-236), West& Harrison (1997), Meinhold & Singpurwalla (1983), and the references therein.

28The derivation can be found in West & Harrison ( 1997, p. 583, p. 639), and Meinhold & Singpurwalla (1983, p. 125).

29See, e.g., Carlin & Louis (2009, p. 26), West & Harrison (1997, p. 584).

30The derivation is based on Hamilton (1994, pp. 692), Guidolin & Timmermann (2007), Harvey (1993, p. 289), andKim & Nelson (1999, p. 63).

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31Guidolin & Timmermann (2007).

32See Brandt (2010, p. 311).

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parameter uncertainty and learning in dynamic financial decisions

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