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What Market Believes
Filippo Massari
School of Banking and Finance, Australian School of Business
Preliminary and incomplete, please do circulate
February 22, 2014
Abstract
This paper examines the implications of the market selection hypothesis on equilibrium
prices. The main focus is the accuracy of the probabilities implied by equilibrium prices, market
beliefs henceforth, and on the “learning” mechanism of markets. In economies populated by
finitely many traders, I show that if there is a unique most accurate trader market beliefs define
a class of probabilities that aggregate information in a way that is, in term of likelihood, asymptotically as good as the probabilities obtained via Bayes’ rule. This class of probabilities is rich
and produces forecasts that can capture observed deviations from standard learning behavior,
such as backward induction, over-reaction and time inconsistency. In economies populated by
finitely many traders in which there is more than one most accurate trader, depending on risk
attitudes, the next period probabilities implicit in equilibrium prices can be more, equally or
less efficient than the probability obtained via Bayes’s rule. In economies populated by a continuum of traders, the asymptotic efficiency of market beliefs can also fail due to the effect of
risk attitudes on the rate at which wealth-shares move. I provide examples in which market
beliefs do not aggregate information efficiently: a Bayesian, given the same information, would
produce forecasts that are more accurate than the market’s.
1
Introduction
According to the market selection hypothesis (Alchian (1950) and Friedman (1953)), henceforth
MSH, the market selects for the traders with the most accurate beliefs. In this paper I investigate
the implications of the MSH on the accuracy of the probabilities implicit in equilibrium prices.
1
The validity of the MSH in general settings has been extensively studied. For general equilibrium
models with finitely many traders, henceforth small economies, Sandroni (2000), Blume-Easley
(2006, 2009) and Massari (2014) have shown that, under general assumptions, markets select for
the traders with the most accurate beliefs. Surprisingly, Massari (2014) shows that this result is not
necessarily true for economies with a continuum of traders, large economies. These papers discuss
the implications of the selection mechanism on wealth-shares, but provide only indirect results
about the accuracy of the probabilities implicit in equilibrium prices: the asymptotic prices reflect
the beliefs of the traders that survive. This intuitive argument is, nevertheless, not sufficient to
properly address the question of the accuracy of market beliefs. For example, in small economies,
equilibrium prices might not converges and, in large economies, the failure of the MSH identified
by Massari (2014) does not translate to equilibrium prices that are asymptotically incorrect or
inefficient.
In this paper, I use the characterization of equilibrium prices of Massari (2014) to directly
discuss market beliefs’ accuracy. The information structure adopted is standard, in the market
selection literature, and is a key component of the model. Each trader has a personal probabilistic
view of the world, his beliefs, and “agrees to disagree” with the other traders. Unlike the models
in the information exchange literature (Grossmann-Stiglits 1980), each trader believes that there is
nothing to learn from prices and is engaged in speculative trading. I define the sparse information
of an economy to be the set of all traders’ beliefs, the initial wealth-share distribution, henceforth
market’s prior, and the sequence of realized states. In this framework, this is all of the available
information. Sparse information and traders’ preferences are the key elements in the determination
of equilibrium prices. The key question I will address is if it is possible to use sparse information
to construct forecasts that are more accurate than the market’s.
I find that, if the economy is small and there is a unique most accurate trader, independent
of risk attitudes, the market selects for the most accurate trader efficiently: market beliefs are,
in terms of asymptotic likelihood, as accurate as the probability obtained via Bayes’ rule from
the market prior. If the economy is small and there is more than one most accurate trader then,
depending on risk attitudes, it is possible to obtain probabilities that are more, equally or less
accurate than the probabilities obtained via Bayes’ rule from the market prior. The results I find
for small economies generalize to the large case with an important caveat: it is possible to construct
2
large economies in which the market fails to select for the traders with the most accurate beliefs.
In some of these cases, also as a function of traders’ risk attitudes, market beliefs’ are less accurate
than the probabilities obtained via Bayes’ rule from the market prior.
2
The model
The model is an infinite horizon Arrow-Debreu exchange economy with complete markets. Time
is discrete and begins at date 0. At each date there is a finite set of states S ≡ {1, ..., S} with
cardinality |S|=S. The set of all infinite sequences of states is S ∞ with representative sequence of
realizations σ = (σ1 , ...). Let σ t = (σ1 , ..., σt ) denote the partial history through date t of path σ,
S t be the set containing all of the different sequences of length t and let Σt be the algebra that
t
consists of all the finite unions of sequences of length t. Σ is the smallest σ-algebra on ∪∞
t=1 Σ . Let
P be the true probability measure on Σ. For any probability measure p on Σ, p(σ t ) is the marginal
probability of the partial history σ t ; that is, p(σ t ) = p({σ1 × ... × σt } × S × S × ...). The information
set at time t is Ft . For simplicity, I assume that traders can only learn from past realizations:
Ft ≡ σ t . The conditional probability of σt , given the information set Ft−1 can therefore be defined
as p(σt |Ft−1 ) = p(σt |σ t−1 ) =
p(σ t )
.
p(σ t−1 )
The expectation operator written without a subscript is the
expectation with respect to the true measure P .
The economy contains a set of traders I. Each trader, i, has consumption set R+ . A conQ
∞
sumption plan c : S ∞ → ∞
t=0 R+ is a sequence of R+ - valued functions {ct (σ)}t=0 in which each
ct (σ) is Ft -measurable. Each trader i is endowed with a particular consumption plan, called the
endowment stream and denoted by eit (σ). Each trader i is characterized by a payoff function
ui : R+ → [−∞, +∞] over consumption, a discount factor βi ∈ (0, 1) and a subjective probability
pi on Σ, his beliefs. There are almost no restrictions on the true probability, nor on the way traders’
beliefs evolve. The only requirement is that the traders’ beliefs define a probability on Σ. Trader
i’s utility for a consumption plan c is:
U i (c) = Epi
∞
X
βit ui (ct (σ)).
t=0
Most of the results of the paper are derived using the time 0 trading setting. Formally, the price of
a claim that pays a unit of consumption at the end of σ t in terms of consumption at time 0 is given
by q(σ t ). Nevertheless, in order to discuss price dynamics, I use the known equivalence between
3
time 0 trading and sequential trading (see Ljungqvist-Sargent (2004, ch.8)). In equilibrium, the
price of an Arrow’s security that pays a unit of consumption in state σ, period t, after history σ t−1
is q(σt |σ t−1 ) =
2.1
q(σ t )
.
q(σ t−1 )
Which, given Assumption 3, is always well-defined.
Assumptions
A competitive equilibrium is a sequence of prices and, for each trader, a consumption plan that
is affordable, preference maximal on the budget set and mutually feasible. Depending on the
cardinality of the set of traders, counted in terms of different beliefs, an economy can be either
small or large.
Definition 1. An economy is small if the union of traders’ beliefs, P =
S
i∈I
pi , has finitely many
elements.
Definition 2. An economy is large if the measure space of agents’ beliefs is (A, P, λ) where A is
the unit interval, P the Borel subsets of A, and λ is the Lebesgue measure.
Following Ostroy’s (1984) approach, a large economy is a pair (, W0 )) where describes preferences and W0 defines an initial allocation of commodities to a group of traders.
In small economies, the existence of the competitive equilibrium is ensured by this set of standard assumptions (see Peleg-Yaari (1970)):
• A1: The payoff functions ui : R+ → [−∞, +∞] are C 1 , concave and strictly increasing and
0
satisfy the Inada condition at 0; that is, ui (c) → ∞ as c & 0.
• A2: The aggregate endowment is fixed.
• A3: For all traders i, all dates t and all paths σ, pi (σ t ) > 0 ⇔ P (σ t ) > 0.
As discussed in Massari (2014), in large economies, the existence of the competitive equilibrium
is ensured by Olsroy’s (1984) existence result if we integrate A1-A3 with these two assumptions on
the commodity space and on the endowment space.
• A4: The beliefs of each trader are exchangeable (i.e., by De Finetti’s Theorem, either iid or
a (Bayesian) mixture of iid probabilities).
4
• A5: The market prior distribution is a function w0 (i):
R
I
w0 (i)dθ(i) = 1.
The integral has to be interpreted in the Lebesgue sense. For small economies, θ(i) is the counting
measure on I. For large economies, θ(i) is the Lebesgue measure on I and w0 (i) is assumed to be
3 times differentiable. This function will be interpreted as a prior distribution on the set of traders’
beliefs.
For tractability reason,in large economies I will need to impose stronger assumptions on traders’
preferences and discount factors:
Definition 3. An economy is identically CRRA (iCRRA) and homogeneous discount factor (HODF)
if all of the traders have identical CRRA payoff function and discount factor: ∀i ∈ I, ui (c) =
c1−γ −1
1−γ
and βi = β.
3
Equilibrium prices and market beliefs
In order to highlight the similarities between price evolution and probabilistic learning, I focus
on the prices of Arrow’s securities in an economy with no aggregate risk and in which all traders
have the same discount factor. For the degenerate case of a representative agent economy, it
is an easy exercise to show that the price of a contingent claim on a certain period-state coincides with the discounted probability of that period-state (∀σ t , q(σ t ) = β t p(σ t )). In economies
populated by traders with log-utility, henceforth log-economy, it is known (Rubinstein (1974),
Blume-Easley (2009), Massari (2014)) that equilibrium prices define, in each period, a discounted
probability measure that coincides with the probabilities obtained via Bayes’ rule from the market
P
prior (∀σ t , q(σ t ) = i∈I w0i pi (σ t )).
The analysis of economies that are not log-economies is complicated by the fact that even
in homogeneous discount factor economies, equilibrium prices are not discounted probabilities.
Specifically, the sum of the prices of contingent claims on next-period consumption differs from
the common discount factor. In order to obtain the probabilities implicit in equilibrium prices, a
further normalization is needed. For economies that are not log, different choices of the horizon of
the normalization deliver different probabilities. I focus on two horizons: the normalized prices on
sequences of length t, representing the probabilities obtained from time 0 trading, market beliefs
(MB); and the conditional one period ahead normalized prices, representing the evolution in the
5
market beliefs over time, market forward forecasts.
Definition 4. Given the sequences of equilibrium prices {q(σ t )}∞
t=1 , market beliefs on sequences of
length t are defined as:
q(σ t )
t
σ̂ t ∈S t q(σ̂ )
pq (σ t ) = P
Which is to say that the market beliefs on σ t is the ratio of the price to move a unit of
consumption from time 0 to σ t and the price of a unit of consumption for sure at period t. It
will be evident that, in economies that are not log, this choice of normalization does not provide
an unambiguous set of one period ahead probabilities. One way to obviate this problem, given a
finite horizons t, is to proceed via backward induction. Operationally, given market beliefs on a
finite horizon t, the probabilities on sequences σ τ of length τ < t can be obtained by summing the
probabilities of all of the histories of length t that depart from it. The per period forecasts are
constructed in a way that is consistent with this construction. I denote this probabilistic scheme
Market Backward Forecasts (MBF).
Definition 5. Given an horizon t and a vector of equilibrium prices q(σ t ), the market backward
forecasts are defined as:
pb (σ t−i |t) =
X
pq (σ t );
pb (στ |t) =
σ̂ t ∈S t :σ̂ t−i =σ t−i
pb (σ τ |t)
pb (σ τ −1 |t)
The market forward forecasts (MFF) are defined as follows:
Definition 6. Given the sequence of next period equilibrium prices q(σt |σ t−1 ), the market forward
forecasts are defined as:
f
p (σt |σ
t−1
q(σt |σ t−1 )
)= P
;
t−1 )
σ̂t ∈S q(σ̂t |σ
f
t
p (σ ) =
t
Y
pf (στ |σ τ −1 )
τ =1
MFF are defined as the price of moving one unit on consumption from σ t−1 to state time σ t
divided the price of moving a unit of consumption from σ t−1 to period t for sure.
Since all of these probabilities preserve the relative likelihoods implicit in equilibrium prices,
these are equally valid candidates to represent the probabilities implicit in equilibrium prices. If
we focus on economies (either small or large) in which all traders have identical CRRA utility
functions, we find that the parameter γ and the type of normalization chosen, determines a class of
probabilities on Σ. This class contains, as special cases, some well known and seemingly unrelated
probabilistic models.
6
Proposition 1. In a iCRRA-HODF economy that satisfies A1-A3 (A4-A5 if large)
(i) If γ = 1, Market beliefs coincide with a Bayesian mixture with market prior.
(ii) If γ → 0 (linear utility), market beliefs coincide with the NML distribution1 .
(iii) If γ → 0 (linear utility), market forward forecasts coincide the Bayesian Factor model.
An advantage of deriving these different models from equilibrium prices is that convergence
of equilibrium prices can be used to derive results on the convergence of all of these probabilistic
models offering a unified framework to discuss the asymptotic efficiency of forecasting schemes that
where previously treated in isolation.
3.1
Asymptotic efficiency of market beliefs
In an economy that is not a log-economy, normalized equilibrium prices do not evolve according to
Bayes’ rule. Nevertheless, via the equilibrium conditions, normalized prices map sparse information
into probabilities. I define a function with this property to be a forecasting scheme. It should be
noted that parametric forecasting schemes depend on the sequence of realizations and on a model
class (a prior support, in Bayesian terminology). A comparison between two different parametric
forecasting schemes can therefore be meaningful only if both schemes use the same model class and
sequence of realizations.
In order to discuss the accuracy of market beliefs I need to provide a definition of asymptotic
efficiency that is independent of the true probability (which is unknown) and to define a benchmark
against which to compare market beliefs. The criterion I adopt is the asymptotic likelihood. The
benchmark to which equilibriums prices are compared is Bayesian updating. This choice depends
on the fact that Bayesian updating produces forecasts that converge to the model with the highest
likelihood in the prior support with the optimal rate of convergence.
Definition 7. Let P be a set of probabilistic models on Σ and pB (σ t ) be the likelihood attached by
a Bayesian model obtained with uniform prior on the models in P.
(i) A probability p on Σ is asymptotically efficient (with respect to P), if:
∀σ ∈ Σ, lim log
t→∞
1
Shtarkov (1987), Rissanen (1986), Grünwald (2007)
7
pB (σ t )
< ∞.
p(σ t )
(ii) A probability p on Σ is asymptotically super-efficient (with respect to P), if
∀σ ∈ Σ, lim log
t→∞
pB (σ t )
pB (σ t ) P̂ -a.s.
>
0,
and
∃
P̂
:
lim
log
→
∞
t→∞
p(σ t )
p(σ t )
(iii) A probability p on Σ is asymptotically sub-efficient (with respect to P) if
∀P ∈ P, lim log
t→∞
pB (σ t ) P̂ -a.s.
pB (σ t ) P -a.s.
>
0,
and
∃
P̂
:
lim
log
→
0
t→∞
p(σ t )
p(σ t )
Definition 7 states that a probability measure p on Σ is efficient if, in every sequence of realizations, the log-likelihood ratio between pB and p is bounded above. In other words, on every
path, σ, there is no statistical test based on the log likelihood principle that can claim that pB is a
better description of the sequence of realizations than p. A super efficient forecasting scheme does
as well as a Bayesian in every sequence and there are some probabilities for which it does better.
A sub-efficient forecasting scheme does as well as the Bayesian in its prior support, but there are
some probabilities (not in the prior support) in which it does worse.
3.2
Equilibrium prices and Bayesian asymptotic
In this section I compare the asymptotic of probabilities consistent with Bayes’ rule and equilibrium
prices (which are not probabilities). The first comparison is between equilibrium prices in small
economies and Bayesian updating from a finite set of models: P = {pi , i ∈ I} and |I| < ∞. It shows
that, even if in non log economies prices are not probabilities, equilibrium prices are asymptotically
equivalent to the probability obtained via Bayes’ rule from an arbitrary prior on the prior on P.
Bayesian with small support VS prices in small economies. Given a Bayesian mixture pB
with prior W0 on a finite set of probabilistic models P = {p1 , ..., pI },
∀σ ∈ S ∞ , ∀t,
pB (σ t ) =
X
w0i pi (σ t )
i∈I
Given a small-HODF-economy that satisfies A1-A3,
q(σ t ) X i i t
$
w0 p (σ )
βt
∀σ ∈ S ∞ , ∀t,
i∈I
Where $ abbreviates ∀x, ∃0 < a < b < ∞ :
f (x)
g(x)
∈ [a, b].
Proof. See Massari (2014)
8
The second comparison is on the large setting. The probabilistic result I present is the BIC approximation (Schwarz (1978),Clarke-Barron (1990), Phillips-Ploberger(2003), Grünwald (2007))).
This result is compared with the characterization of equilibrium prices of Massari (2014). The
comparison reveals that, in the large setting, risk attitudes have an asymptotic effect on equilibrium prices that has the same order of magnitude of an extra dimension in the parameter space to
be estimated. These effect makes equilibrium prices qualitatively different from probabilities even
asymptotically.
Bayesian with large support VS prices in large economies. Given a Bayesian model pB with
smooth prior on a k dimensional non empty open subset of the parameter space of an exponential
family (P):
∀σ ∈ Σ,
t
ln pB (σ t ) ≈ ln pî(σ ) (σ t ) −
k
ln t
2
Given a HODF-iCRRA-large economy that satisfies A1-A5, in which all traders have iid beliefs on
a k-dimensional simplex:
∀σ ∈ Σ,
Where pî(σ
t)
ln
γk
q(σ t )
t
≈ ln pî(σ ) (σ t ) −
ln t
t
β
2
is the parameter/trader with the highest likelihood on σ t .
It is important to stress the generality of these results. These approximations hold in every
sequence, hence independently from the true probability P .
The BIC criterion captures the intuitive idea that there is a “cost” in using models with redundant parameters. To more parameters correspond slower convergence to the best model in the
prior support. Hence lower likelihood. A classical example is the following.
Suppose the true model is Bernoulli with parameter P and that there are two Bayesians (B 1 , B 2 )
that are trying to learn the true model. B 1 has a smooth prior on the Bernoulli family (1 parameter:
k 1 = 1). B 2 has a smooth prior on the Markov (1) family (2 parameters: k 2 = 2). Clearly, both
forecasts will converge to the true model. Nevertheless, substituting the number of parameters in
the BIC formula, it follows that the beliefs of B 1 are more accurate than the beliefs of B 2 .
4
Main result: Small economies
Theorem 1.
In a HODF-iCRRA-small-economy that satisfies A1-A3, MB is asymptotically efficient. Moreover,
9
(i) if γ > 1 MFF is asymptotically super-efficient
(ii) if γ = 1 MFF is asymptotically efficient
(iii) if γ < 1 MFF is asymptotically sub-efficient.
Proof. See Appendix
The first implication of Theorem 1 confirms the intuition that equilibrium prices (averages of the
beliefs of the traders) are as accurate as the probabilities of a Bayesian with market prior support.
Implication i, ii, iii are more surprising. It tells us that the degree of efficiency of MFF depends
on the degree of risk aversion. Moreover, it is possible that MFF are more (less) accurate than the
beliefs of all of the traders in the economy. The probability that verifies the (sub) super efficiency
condition is easily found to be any probability such that equilibrium prices do not converge (for
i
example, every P such that ∃i, j ∈ I : EP ln pi = EP ln pj and ∀z 6= i, j, E ln ppz → ∞).
An example of sub-efficiency is the following:
Example:
Suppose there are two states S = {a, b} and the true probability is iid Bernoulli with P = [ 12 , 12 ].
Consider the MFF obtained from a limit linear economy with two traders with equal Pareto weight
and discount factor and whose beliefs are p1 (a) =
1
3
= p2 (b). Clearly these beliefs are “equally
wrong”, hence equilibrium prices do not converge. The MFF for this economy are

 pi (a), i : pi (σ t−1 ) = arg max{p1 (σ t−1 ), p2 (σ t−1 )}
pf (at |σ t−1 ) =
 1,
if ties occur.
2
We want to compare these forecasts with the forecasts of a Bayesian mixture model with market
prior. This is to say with a model that, to every sequence σ t , attaches probability.
1
1
pB (σ t ) = p1 (σ t ) + p2 (σ t )
2
2
Theorem 1 implies that2
pf (σ t )
pB (σ t )
→P -a.s. 0.
The intuition goes as follows: BFF changes his model discontinuously every time the model
that performed best in the past changes and uses
1
2
when the two models perform equally well. By
∗
∗
construction , the two models are equally wrong hence the event t∗ = {p1 (σ t ) = p2 (σ t )} occurs
2
a proof that is specific to this example can be found in Massari (2014)
10
infinitely often. The result follows, noticing that, BFF always uses the“wrong” model to predict the
∗
realizations at t∗ pM F F (σt∗ |σ t −1 ) = 13 , and verifying that the gains he gets by using the correct
probabilities ( 12 ) in every t∗ +1 period are not enough to compensate for these losses.
4.1
Probabilistic characterization
In this section, I discuss common properties of a forecasting scheme (on iid sequences) and whether
these properties are satisfied by market beliefs, market forward forecasts and market backward
forecasts. Each one of this property has positive appeal, but has been shown to be consistently violated in experiments. These properties are prequentiality (Grünwald (2007)), future independency
and exchangeability (De Finetti (1974)).
Definition 8. A forecasting scheme p is prequential if:
X
p(σt σ t−1 ) = p(σ t−1 )
σt ∈S
An agent with non prequential beliefs believes that the sum of the probabilities of disjoint event
differs from the probability of the union of these events. In the Behavioral literature this type of
errors are well documented and are known as the conjunction fallacy (Kahneman 2011).
Definition 9. A forecasting scheme p is future independent if p(σ t ) only depends on past realizations:
∀t0 , t00
p(σ τ |t0 ) = p(σ τ |t00 )
Where p(σ τ |t) means the probability that p attaches to the sequence σ τ assuming that he only cares
about sequences of t periods.
An individual relying on backward induction typically violates this property. Classical examples
are the probability used to solve “the chain store” game or the probability calibrated in structural
models when a finite horizon is assumed.
Definition 10. A forecasting scheme p is exchangeable if p(σ t ) only depends on the realized frequency:
p(E) = p σ1 , ..., σt : σπ(1) , ..., σπ(t) ∈ E
for every measurable set E ∈ Σt and any permutation π of the indices.
11
Exchangeability capture the idea that the probability of a sequence of events does not depend on
the order of the realizations. It is an appealing criterion whenever the forecasting scheme assumes
an iid model class. For example, we expect a rational agent facing repeated tosses of an iid coin
with unknown bias to attach the same probability to the sequences of realizations {H, H, T } and
{T, H, H}. In terms of conditional forecasts, an agent that attaches less probability to {H, H, T }
than {T, H, H} will appear as overweighting (relatively to a Bayesian) the first two realizations.
The relation between these properties and the different type of probabilities that equilibrium
prices generated depending on the risk attitudes and the choice of normalization is the following:
Proposition 2. In a iid iCRRA-HODF economy with γ = 1 (log), MB, MBB and MFF produce
identical probabilities.
In a iid economy that is not iCRRA-HODF with γ = 1, then:
(i) MB are future independent, exchangeable but not prequential.
(ii) MFF are future independent, prequential but not exchangeable.
(iii) MBF are exchangeable, prequential but not future independent.
For the case of log economies, it is known that market beliefs coincide with the probability of a
Bayesian with market prior, therefore they are prequential, future independent and exchangeable
(in the iid model class). In the next Sections I present an example to show that the forecasting
schemes implicit in equilibrium prices in a non log economy violate at least one of these properties.
Nevertheless, Theorem 1 tells us that the violation of these properties is not sufficient to condemn
a forecasting scheme according to the efficiency criterium proposed (asymptotic likelihoods ratio)
and that in some case it is even better to be not Bayesian.
5
The Leading Example
Consider an Arrow’s security economy with two states, S = {L, R} distributed in every period
according to a Bernoulli distribution with P (L) = 1 − P (R) ∈ (0, 1). There are I traders with
1−γ
identical CRRA utility ( c 1−γ−1 ), identical discount factors (β), iid beliefs pi and positive Pareto
weights ( w01(i) ). There is a trader with correct beliefs. Each trader maximization problem is:
max Epi
X
t=0
β t ui (cit (σ)) s.t.
X X
t=0 σ t ∈S t
12
q(σ t ) cit (σ) − eit (σ) ≤ 0.
The FOC can be used to obtain an explicit equation for equilibrium prices:
P
γ
1
!γ
i (σ t ) γ w (i)
t−1
t
p
X
0
1
i∈I
qγ (σt |σ )
qγ (σ )
γ
=
pi (σ t ) γ w0 (i)
;
= P
1
βt
β
pi (σ t−1 ) γ w (i)
i∈I
(1)
0
i∈I
The following trees of probabilities illustrate the differences between market beliefs (pq ), market
forward forecasts (pf ) and market backward forecasts (pb ) on sequences of leangth 3.
In a log economy (γ = 1) prices are discounted probabilities and coincide with the probabilities of a Bayesian with market prior. Consequently, the three normalizations produces the same
probabilistic tree and these probabilities are prequential, future independent and exchangeable:
LOGpq (σt ),pf (σt ),pb (σt )
1
2
2
9
1
9
1
2
1
2
5
18
1
6
LOGpq (σt |σt−1 ),pf (σt |σt−1 ),pb (σt |σt−1 )
1
9
2
9
1
9
1
9
5
18
1
9
1
9
1
2
5
9
1
6
3
5
4
9
2
5
1
2
4
9
1
2
1
2
5
9
1
2
2
5
3
5
In a limit linear economy (γ → 0), using as equilibrium prices the limit of equation 3) prices
are not discounted probabilities, therefore the three normalizations provides different probabilities:
Market beliefs are future independent, exchangeable but not prequential. Which can be verified
noticing that pq (R, L, L) + pq (R, L, R) =
1
5
>
1
6
= pq (R, L)
LINpq (σt |σt−1 ) not defined
LINpq (σt )
1
2
1
2
1
3
1
5
1
6
1
10
1
10
1
6
1
10
1
10
1
3
1
10
1
10
1
5
Market forward forecasts are future independent, prequential but not exchangeable.Which can
be verified noticing that pf (L, L, R) =
1
9
6=
1
6
= pf (L, R, L)
13
LINpf (σt )
LINpf (σt |σt−1 )
1
2
1
3
2
9
1
6
1
9
1
2
1
2
1
12
1
6
1
12
1
12
2
3
1
3
1
12
1
9
1
2
1
3
2
3
2
9
1
3
1
3
1
2
1
2
2
3
1
2
1
2
2
3
1
3
Market Backward forecasts are prequential, exchangeable but not future independent.Which
can be verified noticing that pb (L, L|T = 3) =
LINt=3
pb (σ t )
1
5
1
10
1
10
1
10
= pb (L, L|T = 2)
SQRTt=2
SQRTt=2
pb (σ t )
pb (σ t |σ t−1 )
1
10
1
10
2
5
1 2
5 3
1
3
1
2
1
2
1
3
3
5
2
5
1
2
1
2
1
2
1
2
1
2
3
5
3
10
1
5
1
10
1
3
1
2
1
2
1
5
6=
LINt=3
pb (σ t |σ t−1 )
1
2
3
10
3
10
1
3
1
6
1
6
1
2
1
2
1
3
2
3
1
3
1
3
2
3
2
3
This example shows that, depending on the utility function of the traders in the economy and
on the choice of the normalization, the probability implicit in equilibrium prices generate different
probabilities in small sample. A natural question to ask is if we can say that one model is better
than the others. Theorem 1 shows that, if there is a unique most accurate model, irrespectively
from the normalization used and from the utility functions of the traders these probabilities are
asymptotically equivalent in terms of likelihood:
∀γ, γ 0 ∈ [0, ∞) lim
pfγ 0 (σ t )
t→∞
6
pqγ (σ)
∈ (0, +∞)
Large economies
Theorem 2.
(i) In a iCRRA-large-HODF-economy that satisfies A1-A5, in which all traders have iid beliefs,
MB and MFF are asymptotically efficient with respect to P = {pi : i ∈ I}.
14
(ii) In a iCRRA-large-HODF-economy that satisfies A1-A5, MB and MFF can be asymptotically
sub-efficient with respect to P = {pi : i ∈ I}.
Proof. See Appendix and Example 1 and 2.
The first implication, shows that the effect of risk attitudes on the rate at which wealth-shares
move does not affect the accuracy of market beliefs: even if the wealth-shares move faster (slower)
in iCRRA economies with γ < (>)1 than in log economies, the accuracy of market beliefs does not
increase (decrease). The second line that there are cases in which the price formation mechanism
destroys some information. The “loss of information” occurs because of the effect that risk attitudes
have on survival. In particular, if γ < 1, it is possible that a positive mass of Bayesian traders
with beliefs that are more accurate than the market’s vanishes (Example 1). If γ > 1 it is possible
that are positive mass of Bayesian traders with beliefs that are less accurate than a Bayesian with
market prior becomes the only survivor (Example 2).
Example 1: Consider a iCRRA-HODF-large-economy with three states S = {a, b, c} with
iid true probabilities P = [ 31 , 31 , 31 ] and γ <
1
2.
The economy contains a positive measure of
traders with iid beliefs whose union covers the 3-dimensional simplex and a positive mass (w0B ) of
Bayesian traders B which know the probability of state a and have a uniform prior on (0, 23 ) for
the probabilities of the other two states. Application of Theorem 2 in Massari (2014) shows that
the equilibrium prices can be asymptotically approximated as follows:

γ
!1 Z Z
Z 2
γ
1
3
q(σ t ) = β t w0B
p(σ t |θ2 )dθ2
+
w0 (θ1, θ2 )p(σ t |θ1 , θ2 ) γ dθ1 dθ2 
0
1
ln P (σ t )− 12 ln t)
≈ β t w0B e γ (
1
+ eγ (
ln P (σ t )− 2γ
ln t)
2
γ
P -a.s.
An application of Massari’s (2014) necessary and sufficient condition to vanish shows that trader
B vanish P − a.s.:
B
t
c (σ ) =
w0B
β t pB (σ t )
q(σ t )
γ1
1
≈
ln P (σ t )− 21 ln t)
w0B e γ (
1
ln P (σ t )− 12 ln t)
ln P (σ t )− 2γ
ln t)
2
w0B e (
+ eγ (
1
γ
→P -a.s. 0.
Consequently, equilibrium prices are asymptotically unaffected by the presence of trader B, which
t
is to say that ln pb (σ t ) ≈ ln pî(σ ) (σ t ) − 22 ln t which is slower than the convergence that a Bayesian
t
with market prior can achieve: ln pB (σ t ) ≈ ln pî(σ ) (σ t ) − 12 ln t.
15
Example 2: Consider an iCRRA-HODF-large-economy with two states S = {a, b} with iid
true probabilities P = [ 12 , 12 ] and IES parameter γ > 2. The economy contains a positive measure
of traders with iid beliefs whose union covers the 2-dimensional simplex and a positive mass (w0B )
of Bayesian traders B with uniform prior on the Markov (1) model (2 parameters). Following the
same logic of the previous example we have that
B
t
c (σ ) =
w0B
β t pB (σ t )
q(σ t )
γ1
w0B
≈
w0B +
1 (ln P (σ t )− γ ln t)
2
eγ
1 ln P (σ t )− 2 ln t
2
eγ
(
→P -a.s. 1.
)
Hence the Bayesian traders dominates and equilibrium prices reflects their beliefs. The accuracy of
t
market beliefs is therefore: ln pb (σ t ) ≈ ln pî(σ ) (σ t )− 22 ln t which are less accurate than the beliefs of a
t
Bayesian with market prior, which, as an application of the BIC is : ln pB (σ t ) ≈ ln pî(σ ) (σ t )− 12 ln t.
These examples suggest that the reason behind the inefficiency in market beliefs lies exclusively
in a failure of the MSH. Indeed, in the two examples the market is selecting against the group
of traders with more accurate beliefs causing an inefficiency in market’s beliefs. Nevertheless, a
failure of the MSH is only necessary to cause market beliefs inefficiency. For example, the failure
of the MSH presented by Massari (2014) constitute an example in which the market selects against
a group of traders with correct beliefs and yet market’s beliefs are asymptotically efficient. The
reason is that the market prior support remains the same even if we remove from the economy the
traders with rational expectations.
7
Conclusion
This paper describes the asymptotic properties of the probabilities implicit in the equilibrium prices
of a complete market exchange economy in which the traders have heterogeneous beliefs. I showed
that, given a set of beliefs and an initial set of weights, the probabilities implicit in equilibrium
prices differ depending on the risk attitudes of the traders and on the normalization chosen to
“extract” probabilities from equilibrium prices. If the economy is small and contains a unique
most accurate trader, risk attitudes and normalization choices do not have an asymptotic effect on
the accuracy of these probabilities. In this case, even if there are small sample differences, both
MB and MFF are asymptotically as good as the probabilities obtained according to Bayes’ rule
from the market prior. If the economy is small and there is more than one most accurate trader
16
then, depending on risk attitudes, MFF can become more (less) accurate than the probabilities
obtained according to Bayes’ rule. These results allow to use prices to define a class of forecasting
scheme that, in terms of asymptotic likelihood, it is as good as (in some cases even better) than
Bayesian updating. This class of probabilities is rich and includes models which are consistent with
behavioral biases commonly observed in experiments such as backward induction, over-reaction
and time inconsistency.
The results I find for small economies generalize to the large case with an important caveat: it
is possible to construct large economies in which the market fails to select for the traders with the
most accurate beliefs. In a subset of these cases, there are values of the CRRA parameter γ for
which MB and MFF are sub-efficient. The existence of cases in which the market is super-efficient
in the large setting is to my knowledge still an interesting open question.
APPENDIX
In this appendices I make use of the notation O(.),o(.) and $ with the following meanings. The bigO notation, f (x) = O(g(x)), means lim supx→∞
(x)
limx→∞ fg(x)
abbreviates
|f (x)|
|g(x)|
< ∞. The little-o notation, f (x) = o(g(x)),
→ 0. The $ notation abbreviates ∀x, ∃0 < a < b < ∞ :
f (x)
g(x)
∈ [a, b].
SMALL ECONOMIES
Lemma 1. In a small economy that satisfies A1-A3: ∃a, b ∈ (0, ∞) : ∀t, ∀σ ∈ Σ :
a<
Proof. i)
1
i∈I u0 (cit (σ)) < ∞:
i
∈ I, u0i (cit (σ)) > 0
1
X
u0 (ci (σ))
i∈I i t
<b
(2)
P
∀σ ∈ Σ, ∀i
because the total endowment is finite (A2) and the payoff functions
are monotone and strictly concave with positive derivative at 0 (A1).
P
ii) 0 < i∈I u0 (c1i (σ)) :
i t
P
1
=
0 ⇔ ∀i ∈ I, u0i (cit (σ) = ∞ which is true iif all the traders have 0 consumption
i∈I u0 (ci (σ))
i
t
and satisfy the Inada condition at 0. The first requirements is impossible as it violates the market
P
P
i
i
clearing condition:
i∈I ct =
i∈I et > 0.
17
Lemma 2. In a homogeneous discount factors CRRA economy with constant aggregate endowment:
P
γ
1
!γ
i (σ t ) γ w (i)
t
t−1
p
X
0
1
i∈I
qγ (σ )
qγ (σt |σ )
γ
=
;
= P
(3)
pi (σ t ) γ w0 (i)
1
t
β
β
pi (σ t−1 ) γ w (i)
i∈I
0
i∈I
Proof. Standard calculations
Proof of Theorem 1
Proof. Asymptotically efficiency of BM: Rearranging the FOCs, normalized equilibrium prices
are:
βt
q(σ t )
pq (σ t ) = P
=
t
σ t q(σ )
P
pi (σ t )w0 (i)
Pi∈I
1
i∈I u0 (c (σ))
i t
β
P
i t
i∈I p (σ̂ )w0 (i)
P
1
P
t
σ̂ t ∈S t
(4)
!
i∈I u0 (c (σ̂))
i t
By Lemma 1 ∃a, b ∈ (0, ∞) : ∀t, ∀σ ∈ Σ,
βt
q
t
p (σ ) ∈
pi (σ t )w0 (i)
Pb i t
,
[ P
i∈I p (σ )w0 (i)
t
β
t
σ∈S
a
P
βt
i∈I
βt
P
P
i∈I
pi (σ t )
Pa
i∈I
σ∈S t
pi (σ t )w0 (i)
b
]
aX i t
bX i t
p (σ )w0 (i),
p (σ )w0 (i)]
b
a
i∈I
i∈I
X
=> pq (σ t ) $
pi (σ t )w0 (i)
=> pq (σ t ) ∈ [
i∈I
Asymptotics of BFF:
Proof. By Lemma 1, independently from the risk attitudes,
q(σ t )
βt
$ pB (σ t ). Therefore:
t
XX
q(σ t )
q(σ τ |σ τ −1 )
ln t B t =
Iσt ln
= k ∈ (−∞, +∞)
β p (σ )
p(σ τ |σ τ −1 )
σ
τ =1
t
Note that:
q(σ τ |σ τ −1 )
P
t X
t X
τ τ −1 )
X
X
X
q(σ t )
pq (σ t )
σt q(σ |σ
ln B t =
Iσt ln
=
ln
−
I
ln
q(σ τ |σ τ −1 )
σ
t
τ |σ τ −1 )
t pB (σ t )
p (σ )
p(σ
β
σ
σ
σ
τ =1
τ =1
t
t
!
(5)
t
Let qγ denote the equilibrium prices in an iCRRA economy with parameter γ. By Proposition 1 in
Massari (2014):
0 < γ 0 ≤ 1 ≤ γ 00 < ∞ ⇒
1 X
1X
qγ 0 (σ t |σ t−1 ) ≥ 1 ≥
qγ 00 (s|σ t−1 )
β
β
σt ∈S
s∈S
18
(6)
Let focus on the case γ > 1 (the proofs of the other cases follow the same logic). To prove that
BFF are super-efficient. I have to verify two conditions:
B
t
(i) ∀σ ∈ Σ, limt→∞ log pp(σ(σt )) < ∞ the condition follows by noticing that ∀σ t , ln
P
σt
q(σ τ |σ τ −1 ) <
0.
B
t
(ii) ∃P̂ : limt→∞ log pp(σ(σt )) →P -a.s. 0.
Let P̂ be such that ∃i, j ∈ I : EP̂ ln pi (σ) $ EP̂ ln pj (σ) and ∀z ∈ IEP̂ ln pz (σ) = o(EP̂ ln pi (σ)).
Note that P always exists. Standard argument can be used to show that as t → ∞ the aggregate
wealth will oscillate infinitely often between trader i and j. Moreover, every time the wealth-shares
P
τ τ −1 ) <
of these two traders are comparable, the inequality 6 is strict, implying that (ln
σt q(σ |σ
0). As a consequence equation 5 diverges; as desired.
LARGE ECONOMIES
Using Lemma 2 and letting the number of traders go to infinity we obtain the following equation
for the normalized equilibrium prices in a CRRA economy with a continuum of traders:
R
γ
1
i (σ t ) γ w(i)dθ(i)
p
I
γ
pqγ (σ t ) = P R
1
i (σ̂ t ) γ w(i)dθ(i)
p
t
σ̂
I
The proof of Theorem 2 relies on a modification of standard proofs on the asymptotic convergence
rate of Bayesian posterior (Clarke-Barron (1990), Phillips-Ploberger (2003)) (γ = 1)with smooth
positive Lebesgue prior together with the proof of the asymptotic minimum regret property of the
Normalized Maximum Likelihood distribution (γ = 0) in Grünwald (2007).
Proof of Theorem 2
For all γ ∈ (0, +∞), if the economy is iid,
R
I
detI(pi )−
pqγ (σ t ) ≈ eln p
3
1+γ
2
î(σ t ) (σ t )− 1
2
ln
< ∞3 and the market prior is smooth:
t
+O(1)
2π
This assumption comport no loss of generality because it is always true for the multinomial model with proba-
bilities strictly bounded away from the boundaries of the simplex.
19
Where I(pi ) be the Fisher information, î(σ t ) be the model in I with Maximum likelihood on σ t ,
t
which is to say the model such that pî(σ ) (a) =
Pt
ν=1 Iσn =a
t
.
Proof. I focus on the case k = 1, which is to say on the Bernoulli family: S = {a, b}. The
generalization to the multinomial family is straightforward. The priors is smooth, therefore satisfy
the standard regularity conditions which make the error in the third and higher order Taylor
expansion around î be o(1).
γ
1
i (σ t ) γ w(i)dθ(i)
p
I
γ
pqγ (σ t ) = P R
1
i (σ̂ t ) γ w(i)dθ(i)
p
t
σ̂
I
R
≈a,b
eln p
î(σ t ) (σ t )+γ
ln
√
Pt
γ+γ ln w(î)−γ 12 ln
t
τ =0 τ
t
−γ
2π
î(σ t ) (σ t )+γ
Pt
τ =0 e
î(σ t ) (σ t )+γ
t
−γ
2π
ln w(î)−γ 12 ln
γ ln w(îτt )−(1+γ) ln
îτt ={0, 1t ,...,1} e
t
≈
ln pî(σ ) (σ t )+γ ln w(î)−γ 12 ln
e
R1
t
√
(2π)
= eln p
√
√
t
γ+γ ln w(î)−γ 12 ln 2π
−γ ln det I(pî )+o(1)
“
”
√
τ
− ln pît (στt )− 12 ln 2πt−ln det I(îτt )+o(1) ln pîτt (σ t )+γ ln √γ+γ ln w(îτ )−γ 1 ln
τ
t
2
eln p
≈P
det I(pî )+o(1)
γ
1
i (σ t ) γ w(i)dθ(i)
p
τ
I
R
eln p
≈c
√
ln
1−γ 1+γ
t
î(σ t ) (σ t )− 1
2
ln
0
t
−γ
2π
detI(îτt )−
√
ln
√
ln
e
t
−γ
2π
√
ln
det I(îτt )+o(1)
det I(pî )+o(1)
ln 2π+o(1)
t det I(îτt )+ 1−γ
2
√
ln
1+γ
2
det I(pî )+o(1)
w(îτt )dîτt
t
+O(1)
2π
a : numerator: by Lemma 4 in Massari (2014)
b : With στt denoting the sequences of length t in which Maximum likelihood estimator for the
P
τ
probability of state a is given by pît (a) = τt (which is to say, such that tν=1 Iσν =a = τ ).
c : Using Stirling’s approximation of the factorial and Lemma ??
20
A
References
Alchian A., (1950): “ Uncertainty, Evolution and Economic Theory”, Journal of Political
Economy, 58, 211221
Ash R., (1999):“ Probability and Measure Theory” Academic Press
Blume L, Easley D. (1992): “Evolution and market behavior” J. Econ. Theory 58:940
Blume L, Easley D. (2006): “If you’re so smart, why aren’t you rich? Belief selection in complete and incomplete markets” Econometrica 74:92966
Blume L, Easley D. (2009a): “The market organism: long-run survival in markets with heterogeneous traders” J. Econ. Dyn. Control 5:102335
Clarke B.S, Barron A.R. (1990): “Information theoretic asymptotics of Bayes methods”. IEEE
Trans. Inform. Theory 36:45371
Cover T. M.,J. A.Thomas. (1991):“ Elements of Information Theory”. Wiley and Son,Inc
Cvitanic, J., Jouini, E., Malamud, S., Napp, C. (2012). “Financial markets equilibrium with
heterogeneous agents. Review of Finance”, 16(1), 285-321.
De Finetti, B. 1974. Theory of Probability. A critical introductory treatment. London:John
Wiley and Sons.
Folland G.B. 1999, Real Analysis: Modern Techniques and Their Applications. Wiley New
York ISBN-10: 0471317160, 2nd edition.
Friedman, M. (1953): Essays in Positive Economics. Chicago: University of Chicago Press.[930,932]
Grossman S. and Stiglitz, J. 1980. On the impossibility of informationally efficient markets.
21
American Economic Review 70, 393408
Grünwald P. 2007, The Minimum Description Length Principle, MIT Press ISBN-10: 0262072815
Hutter M. 2004 Universal Artificial Intelligence: Sequential Decisions Based On Algorithmic
Probability ISBN 3-540-22139-5
Jouini, E., Napp, C. 2006 . “Heterogeneous beliefs and asset pricing in discrete time: An analysis of pessimism and doubt.” Journal of Economic Dynamics and Control 30.7 : 1233-1260.
Kahneman, Daniel. Thinking, fast and slow. Macmillan, 2011.
Ljungqvist, L., Sargent, T.J. 2004. Recursive Macroeconomic Theory, MIT press.
Massari F. 2012. Comment on If you’re so smart, why aren’t you rich? Belief selection in
complete and incomplete markets. Forthcoming, Econometrica
Massari F. 2014 Price Dynamic and Market Selection in Small and Large Economies. Working
paper.
Phillips P.C.B., Ploberger W. 2003. Empirical limits for time series econometric models. Econometrica 71:62773
Rissanen, J. J. 1986. Stochastic Complexity and Modeling, Annals of Statistics, 14, 1080-1100
Rissanen, J. J.1987 “Stochastic Complexity (with discussion)”, Journal of the Royal Statistical
Society, 49,223-239, and 252-265
Rubinstein, M. 1974 “An aggregation theorem for securities markets.” Journal of Financial
Economics 1.3
22
Sandroni A. 2000 “Do markets favor agents able to make accurate predictions?” Econometrica
68:130342
Schwarz, G. E. 1978 “Estimating the dimension of a model”. Annals of Statistics 6 (2): 461464
Shtar’kov, Yurii Mikhailovich. ”Universal sequential coding of single messages.” Problemy
Peredachi Informatsii 23.3 (1987): 3-17.
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