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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. 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