Lanouar Charfeddine
email: lanouar_charf@yahoo.fr.

Dominique Guégan
email: dguegan@univ-paris1.fr

Breaks or Long Memory Behaviour: An empirical Investigation

Keywords: JEL classication: C13 -C32 -E3 Structural breaks models, Spurious long memory behavior, Ination series

 show, under specic conditions, that switching models and long memory processes can be easily confused. In this paper, using several generating models like the mean-plus-noise model, the STOchastic Permanent BREAK model, the Markov switching model, The TAR model, the sign model and the Structural CHange model (SCH) and several estimation techniques like the the GPH technique, the Exact Local Whittle (ELW) and the Wavelet methods, we show that, if the answer is quite simple in some cases, it can be mitigate in other cases. Using French and American ination rates we show that these series cannot be characterized by the same class of models. The main result of this study suggests that estimating the long memory parameter without taking account existence of breaks in the data sets may lead to misspecication and to overestimate the true parameter.

Introduction

In recent years, many works have provided a theoretical justication and empirical evidence that several structural change models can create spurious long memory behavior. For instance, short memory models with changing regime in mean can generate identical behaviour compared with the one derived from long memory process looking at the evolution of the autocovariance function. This behavior has been pointed out by a lot of authors: Chen and Tiao (1990) for the mean-plus-noise model, Engle and Smith (1999) for the stochastic permanent break model, [START_REF] Granger | A simple nonlinear time series model with misleading linear properties[END_REF] for the sign model, [START_REF] Hamilton | A new Approach to the Economic Analysis of Non stationarity Times Series and the Business Cycle[END_REF] for the Markov switching model, [START_REF] Lim | Threshold autoregressions, limit cycles, and data[END_REF] for the Threshold Auto-Regressive (TAR) model, [START_REF] Gourieroux | Memory and infrequent breaks[END_REF] for the infrequent breaks model, for instance.

In the literature, until now, no consensus has been reached about the link between presence of structural changes models and detection of long memory behavior. [START_REF] Diebold | Long memory and regime switching[END_REF] argued that switching regimes and long memory behavior can be easily confused. [START_REF] Granger | Occasional structural breaks and long memory with application to the S&P500 absolute stock returns[END_REF] suggested that it is dicult to discriminate between these two classes of models, say structural changes model and long memory process. Empirical studies in testing for long-range dependence show that, when the data are weakly dependent with changes in mean, statistical tests used to detect long memory behavior have a high size distortion, indeed they are not able to reject the alternative hypothesis of long memory with a high power.

This paper focus on the behaviors of the estimations for the long memory parameter under misspecication. We consider as estimation techniques the Geweke-Porter-Hudak (1989) method (GPH), the Exact Local Wittle estimate of Shimotsu and Phillips (2005) and the Wavelet estimator of [START_REF] Lee | Estimating memory parameter in the US ination rate[END_REF]. The generating processes are the mean-plus-noise, the Stopbreak, the Markov-Switching, the Threshold autoregressive and the Structural Changes processes. [START_REF] Diebold | Long memory and regime switching[END_REF] already provide simulation results for the performance of the GPH estimator under some data generating processes such as mean-plus-noise and stopbreak models in order to study this problem of misspecication. Here we perform some similar studies for a wider class of processes and estimators. Then, an application is provided in order to investigate the possible origin of the long range dependence behavior observed in the American and French ination series.

All along this paper we say that a process (Y t ) t has a long memory behavior if its autocovariance function

ρ k ≈ c(k)k 2d-1 , k → ∞,
where d > 0 and c(k) is a slowly varying function at innity. This means, that given a sample (Y 1 , • • • , Y T ), the sample correlation function ρ(k) decays with an hyperbolic rate towards zero. In the next section, we introduce stochastic processes which exhibit this kind of behavior. For more details on long memory behavior, we refer to [START_REF] Guégan | How can we dene the concept of long memory ? An econometric survey[END_REF].

We say that we observe a spurious long memory behavior inside data sets, if knowing that the theoretical generating process is short memory (in this later case the autocorrelation function decreases with an exponential rate towards zero), the sample autocorrelation function decays slowly towards zero with an hyperbolic rate.

The rest of the paper is organized as follows. Section 2 recalls the analytical expressions of the processes used in this study: the long memory process and several structural changes models. Section 3 describes the estimation methods used to estimate the long memory parameter. Section 4 reports Monte Carlo simulation results, which illustrate the problem of misspecication of estimation of long memory behavior when breaks are present in the data sets. In Section 5 we investigate the behavior of French and US ination rates. Section 6 concludes.

Long memory and Structural changes models

In this section, we specify the models that we study all along the paper.

FI(d) process

In 1980 Granger and Joyeux introduced the so-called Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) model, developed independently by [START_REF] Hosking | Fractional dierencing[END_REF]. We say that a process {Y t } t , follows a Fractional Integrated FI(d) process if, ∀t,

(I -L) d (Y t -µ) = u t , (1)
where (u t ) t is assumed to be a Gaussian strong white noise N(0, σ 2 u ), µ is the mean and L is the lag operator. In the following we assume that 0 < d < 1. This means that the autocorrelation function of the process (1) exhibits a slow decay towards zero which characterizes existence of the long memory behavior. The parameter d is allowed to be greater than 1/2 which permits to work with non-stationary processes and, as soon as d < 1 this means that the process (Y t ) t is mean reverting.

Structural changes models

Now, we dene some structural changes models. First we assume that, ∀t

Y t = µ t + t , (2)
where ( t ) t is a Gaussian strong white noise N(0, σ 2 ) and (µ t ) t is an occasional level shift process. In the following, we propose several forms for the processes (µ t ) t and ( t ) t . We can classify these models as follows:

1. We assume that the process (µ t ) t follows a random walk denoted also I(1) process, controlled by two random variables q t and η t .

2. We assume that the process (µ t ) t follows a Markov switching model, a TAR model, or a structural changes model.

All these models are known to have theoretically a short memory behavior (in the autocorrelation sense) as soon as they are stationary. We describe now these classes of models.

1. Model 1: The mean-plus-noise model. We assume that the process (µ t ) t is such that, ∀t

µ t = µ t-1 + q t η t , (3)
where (η t ) t is a Gaussian white noise with zero mean and variance σ 2 η and, q t is a sequence of independent identically distributed random variables with a binomial distribution q t = 0 with probability 1-p 1 with probability p.

The process (2)-( 4) is the so-called mean plus noise model introduced by Chen and Tiao, (1990) and Engle and Smith, (1999), see also [START_REF] Diebold | Long memory and regime switching[END_REF]. Given a sample (Y 1 , • • • , Y T ) we provide on Figure 1, the trajectory of such a process and the sample autocorrelation function ρ(h). The breaks that we observe on the Figure 1(a) are obtained as soon as p × T is a non-zero nite constant with 0 < p < 1 and T the sample size. The breaks arise when the sample size increases and the probability p decreases. On Figure 1(b) we can observe that the autocorrelation function decreases very slowly as the lag k increases.

Model 2:

The Stopbreak model. The occasional level shifts model (µ t ) t is dened by,

µ t = µ t-1 + q t-1 t-1 (5)
where,

q t-1 = 2 t-1 γ + 2 t-1 . ( 6
)
The function q t-1 is a non decreasing function and it is bounded by zero and one. Many other smooth transition functions q t-1 can be proposed, for instance, the logistic function q t-1 = 1 1+exp( t-1 ) and also the functions q t-1 = exp( t-1 ) γ+exp( t-1 ) , q t-1 =

1 γ+ 2 t-1 , or q t-1 = 1 γ+| t-1 | .
The model (2)-( 5)-( 6) is the so-called Stopbreak process introduced by Engle and Smith (1999). On Figure 2(a), we report the trajectory of this model. Looking at the sample autocorrelation function, on Figure 2 (b) we observe a slow decreasing of this function.

3. Model 3. Markov switching model. We assume that the occasional level shifts process, (µ t ) t , depends on an unobserved Markov chain s t , where s t is a random variable which takes values 0 and 1, characterized by a transition probability which is dened by:

µ t = µ st , (7)
p(s t = j|s t-1 = i) = p ij , i, j = 0, 1, with 0 ≤ p ij ≤ 1, 1 j=0 p ij = 1, i = 0, 1. The transition matrix is equal to P = p 00 1 -p 00 1 -p 11 p 11 .
Diebold and Inoue (2001) have shown that for some sets of values p 00 and p 11 , the autocorrelation function decreases slowly towards zero. This behavior is described on Figure 3.

4. Model 4. Threshold Auto-Regressive model. The occasional level shifts process (µ t ) t , depends on an observed variable, a threshold parameter r and a delay parameter

d µ t = µ 0 if Y t-d ≤ r µ 1 if Y t-d > r. ( 8
)
The model dened by ( 2) and ( 8) is a simple form of the Threshold Auto-Regressive model introduced by [START_REF] Lim | Threshold autoregressions, limit cycles, and data[END_REF]. This model ( 2)-( 8) is also a particular example of Markov switching model when the Markov chain I{Y t-1 ≤ r} is not exogenous, see [START_REF] Carrasco | Misspecied Structural Change, Threshold, and Markov-switching models[END_REF]. On Figures 4 (a) and (b), we report the trajectory and the autocorrelation function for this model. Again we observe a slow decreasing of the autocorrelation function.

5. Model 5. The sign model. The occasional level shifts process (µ t ) t , follows a sign model if where the sign(.) function is dened by

µ t = sign(Y t-1 ), (9)
sign(Y t-1 ) =    1 if Y t-1 < 0 0 if Y t-1 = 0 -1 if Y t-1 > 0,. (10)
This model is the so-called sign model introduced by Granger and Teräsvirta (1999). They assume that the noise

( t ) t given in (2) is such that p = Prob( t < -1) = Prob( t > 1)
. In theory the process dened by the relationships (2), [START_REF] Diebold | Long memory and regime switching[END_REF], and (10) still have a short memory behavior. Nevertheless, when p is small, a slow decay of the autocorrelation function can be observed. We illustrate this fact on Figure 5.

6. Model 6. The Structural CHange model. The occasional level shifts process (µ t ) t , follows a process with two regimes and an unknown time break

µ t = µ 0 if t = 1, ..., m µ 1 if t = m + 1, ..., T, (11)
where m = πT and π ∈ (0, 1). This model ( 2)-( 11) has been introduced by [START_REF] Quandt | The Estimation of the Parameter of a linear Regression System Obeying Two Separate Regimes[END_REF]. We illustrate its behavior on Figure 6.

3 Estimation methods to detect long memory behavior

In the last two decades, many procedures have been proposed to estimate the fractional long memory parameter d in [START_REF] Bai | A note on spurious break[END_REF]. In this paper, we focus on the GPH technique introduced by Geweke and Porter-Hudak (1983), the Exact Local Whittle method proposed by Philips and [START_REF] Shimotsu | Exact Local Whittle Estimation of Fractional Integration[END_REF] and the wavelet method introduced by Lee (2005). The testing procedure to detect the presence of long memory behavior is:

H 0 : d = 0 against H 1 : d = 0. ( 12
)
Under the null hypothesis H 0 , the process (Y t ) t has a short memory behavior and under the alternative, it has a long memory behavior. Now we specify dierent methods to estimate the parameter d.

The GPH technique

The GPH technique developed by Geweke and Porter-Hudak (1983) and [START_REF] Robinson | Log-periodogram regression of time series with long range dependence[END_REF] is based on the use of log-periodogram I(w j ) at frequency w j = 2πj/T . For frequency near zero, Geweke and Porter-Hudak (1983) and Robinson (1995) show that the parameter d can be consistently estimated using the following least squares regression

ln{I(w j )} = a -d ln{4sin 2 (w j /2)} + t , j = 1, ..., m. ( 13
)
where ( t ) t is a white noise. If the integer m grows slowly with respect to the sample size as for instance m = √ T , then the ordinate least-square estimator d is asymptotically normal with standard error equal to π(6m) -1/2 .

The Exact Local Whittle (ELW)

The Exact Local Whittle (ELW) method proposed by Shimotsu and Phillips (2005) avoids some approximations in the derivation of the Local Whittle estimator proposed by Künsch (1987) and [START_REF] Yajima | A central limit theorem of Fourier transforms of strongly dependent stationary processes[END_REF]. The estimated value dELW for d obtained from this method is such that:

dELW = Arg min d∈[d 1 ,d 2 ] R(d), (14)
where d 1 and d 2 are the lower and upper bounds of the admissible values of

d such that -∞ < d 1 < d 2 < ∞ and, R(d) = ln G(d) -2d 1 m m j=1 ln(ω j ), ( 15
)
where m is the truncation parameter, G(d) = 1 m m j=1 I(ω j ), and I(ω j ) is the periodogram. Under regular assumptions, the ELW estimator dELW is asymptotically Gaussian

√ m( dELW -d) → d N (0, 1/4), when T → ∞. ( 16
)

The Wavelet method

In recent years, the use of wavelets have been extended to a lot of areas other than physics. In econometrics, [START_REF] Jensen | Time-Varying Long-Memory in Volatility: Detection and Estimation with Wavelets[END_REF] and [START_REF] Jensen | An Approximate Wavelet MLE of Short-and Long-Memory Parameters[END_REF] propose a wavelet estimator for the long memory parameter d. In this paper, we use the wavelet method proposed by [START_REF] Lee | Estimating memory parameter in the US ination rate[END_REF] which permits to estimate d inside the interval (0, 1.5). The estimated value of d is obtained by applying the least squares method using the following regression,

ln(I (j) q ) = c -2(d -1)ln(λ q ) + t for q = 1, 2, ..., m. ( 17
)
where, c is a constant, λ q = 2πq n and

I (j) q = 1 2πT 2 j -1 k=0 |ω j (k)exp(iλ q k)| 2 , q = 1, 2, ..., m. ( 18
)
The number of frequencies m included in the regression is such that m → ∞ and m/n → 0 as n → ∞. ω j (k) is the wavelet transform of a times series Y t , t = 1, ..., T = 2 j -1,

w j (k) = 2 -j/2 t Y t ψ(2 -j t -k), (j, k) ∈ Z 2 . ( 19
)
The wavelet coecient w j (k) is the detail coecient at scale j and position k. Using this approach the dW AV E estimator, for d ∈ (0, 1.5), veries,

m 1/2 ( dW AV E -d) → d N (0, π 2 24 ) as T → ∞. ( 20
)
4 Simulation experiment

In this section, using Monte Carlo simulations, we investigate the performance of the estimation methods recalled in section 3. We use 1000 Monte Carlo realizations, sample size T = 2000. In each Table , from Table 1 to Table 9, we provide the estimation obtained for the parameter d assuming that the generating process is theoretically a FI(d) model ( 1). The standard deviations and the t-statistics are not reported in the paper and can be provided under request. We specify now the sets of parameters for each generating process used in the simulations:

• The mean plus noise model: σ 2 η = (0.05, 0.01, 0.1, 1) and p = (0.001, 0.0025, 0.005, 0.01, 0.025, 0.05, 0.1).

• The Stopbreak model : σ 2 = (0.05, 0.01, 0.1, 1) and γ = (0.001, 0.01, 0.1, 1, 10, 100, 1000).

• The Markov switching model: σ 2 = (0.05, 0.01, 0.1, 1) and (p 00 , p 11 ) = ((0.5, 0.99), (0.95, 0.95), (0.95, 0.99), (0.99, 0.99), (0.999, 0.999)).

• The TAR model : r = 0, d = 1 and σ 2 = (0.05, 0.01, 0.1, 0.2, 0.3, 0.4, 1).

• The sign model : p = (0.001, 0.0025, 0.005, 0.01, 0.025, 0.05, 0.1).

• The SCH model : σ 2 = (0.05, 0.01, 0.1, 1) and π = (0.1, 0.2, 0.35, 0.5, 0.65, 0.8, 0.95).

In all Tables, we report the mean of the estimated value of the long memory parameter d using the GPH, the Exact Local Whittle and the Wavelet methods for the six previous models. We summarize now our results.

1. The mean plus noise model. In Table 1, we observe that the estimation performance is similar for the GPH and the ELW methods. In all cases and for all methods we cannot accept the null hypothesis of short memory. The results show also that the estimated value of the fractional long memory parameter d depends on the two parameters σ 2 η and the product pT . We observe that the increase of pT or σ 2 η have similar eects on the estimated parameter d whose value increases too. Indeed, when pT increases, the number of breaks increases and the estimated value of d tends to be close to one. This makes the mean plus noise process similar to an I(1) process. Thus, we can retain the following rule:

Rule 1: When the data generating process is the mean plus noise model, if pT is a nite non zero number and if pT σ 2 η ∼ 1, then, the estimated value of the fractional long memory parameter d in model ( 1) is dierent of zero. We remark also that for dierent triplets (p, T, σ 2 η ), which gives the same value for the product pT σ 2 η , then the estimated parameter of long memory d is nearly the same. In Table 2, we report results which illustrate this fact. For example, when pT σ 2 η = 1 with (p, T, σ 2 η ) = (0.05, 2000, 0.01) or (0.01, 2000, 0.05) or (0.005, 2000, 0.1), d is equal respectively to 0.479, 0.472 and 0.47 when we use the GPH technique and when we use the ELW technique, we get respectively 0.432, 0.427 and 0.416 for d.

The Stopbreak model. The Table 3 reports the mean estimated values

for the long memory parameter d for this model. In all cases, we cannot accept the null hypothesis, except when the value of σ 2 T /γ → 0.

When the value σ 2 T /γ is far from 0 and does not tend to ∞, all methods fail to reject the null hypothesis of short memory. These results are summarized in the following rule, Rule 2: When data generating process is the Stopbreak model ( 2)-( 5)-( 6) if σ 2 T /γ ∼ R, where R is a nite non zero number, then the estimated value of the fractional long memory parameter d in model ( 1) is signicantly dierent from zero. 4 and5.

3. The Markov switching (MS-AR) model ( 2) and ( 7). In Table 6 we observe that the estimated value for d seem to depend on the values taken by the couple (p 00 , p 11 ). For example, when (p 00 , p 11 ) = (0.5, 0.95), d is close to zero. We know that when p 00 and p 11 are far from

(T σ 2 /γ) (200) (20)
(2) (0.2) (0.02) 0.1 0.987 0.949 0.559 0.090 -0.008

(T σ 2 /γ) (2000) (200) (20)
(2) (0.2) 1 0.992 0.987 0.953 0.564 0.094

(T σ 2 /γ) (20000) (2000) (200) (20)
(2) zero, then switches between the two regimes occur frequently and the behavior of the process seems close to the one of a white noise.

In the other hand, when p 00 and 11 are close to one, (for instance (p 00 , p 11 ) = (0.99, 0.99)), h atd is signicantly dierent from zero. In that case the breaks do not occur frequently inside the data sets. Thus, when the underlying process is a Markov switching process, we have no specic rules with respect of the test we consider. Sometimes we can accept H 0 , sometimes no. 4. The Threshold Auto-Regressive and the Sign models. The results reported in Tables 7 and8 for these two models show that the estimated values for d are close for both models.

More generally, in these Tables we observe that when the value of σ 2 varied between 0.2 and 0.6 the alternative hypothesis of long memory behavior cannot be rejected. Thus, for these models, we have no specic rule to decide when the estimated procedure detect a long memory behavior. 5. The Structural change model. This model is a linear process with one break which occurs at unknown time m. We use dierent values of noise's variance and dierent values for the date of break m. In Table 9 we observe that the performance to estimate d using all methods depends on the location of the break. The results depend on the noise's variance. For example, for large value of σ 2 , the ELW method detects nearly always the long memory behavior whatever the localization of the break. This simulation exercise shows that 1. If the underlying processes is a mean-plus-noise process or a stop break process, then whatever the estimation methods we use to estimate the long memory parameter d of the model (1), we reject the null assumption and retain the presence of persistence inside the data set.

2. If the underlying processes is a Markov switching process, a TAR process, a Sign process or a structural changes model, the results are mitigate. We cannot provide rules permitting to say when these models create spurious long memory behaviors, even if in some cases, they do: the rejection of the null depends strongly on the values of specic parameters of the considered model.

5 Application to the American and the French ination rates

Ination time series are subject to many domestic and external shocks. International like oil price shocks, wars, nancial crisis and monetary policies play an important role on the evolution of this variable. To describe its dynamics, practitioners and researchers use mainly two classes of models. The rst class includes models with changes in mean like the Markov switching model of [START_REF] Hamilton | A new Approach to the Economic Analysis of Non stationarity Times Series and the Business Cycle[END_REF] and the structural change model introduced in the literature by [START_REF] Quandt | The Estimation of the Parameter of a linear Regression System Obeying Two Separate Regimes[END_REF][START_REF] Quandt | Tests of the Hypothesis that a Linear Regression System Obeys Two Separate Regimes[END_REF] and developed recently by [START_REF] Bai | A note on spurious break[END_REF]Perron (1998, 2003). The second class includes long memory models like the FARMA model developed by [START_REF] Granger | An introduction to long memory time series and fractional dierencing[END_REF] and [START_REF] Hosking | Fractional dierencing[END_REF]. If we use this last class of models to explain the evolution of ination, this means that these series are characterized by existence of persistence. This fact has already been pointed out by [START_REF] Hassler | Long memory in ination rates: international evidence[END_REF], Baillie, Chung and Tieslau (1996), Bos, Franses and Ooms (1999) and [START_REF] Lee | Estimating memory parameter in the US ination rate[END_REF] for instance.

We know that presence of breaks in a data set can be cause of misspecication for the underlying process. We will see that the problem can arise with ination rate, even if, in ne, it is very dicult to conclude.

We consider monthly data sets corresponding to the French and the US ination on the period January 1959 -April 2006, sample size T=568. The data set is provided from Datastream Base. Ination rates are introduced computing y t = 1200 * [log(p t )log(p t-1 )], where p t is the Consumer Price Indices. The trajectories and the autocorrelation functions for the two ination rates are provided in Figures 7 and8 existence of long memory inside the data sets. In another hand, the trajectories of the series ( seem to present some breaks that could be modelled uding models with structural changes. Indeed, using the sequential method of [START_REF] Bai | A note on spurious break[END_REF]Perron (1998, 2003), we detect four breaks in the French ination series and three breaks in the US ination series. Dates of breaks coincide exactly with some economics and nancial events. For example, we detect, for the two time series, a break that corresponds to the Vietnam war which exerts a positive eect on the US ination rate (April 1967). This shock is transmitted ten months after to the French economic (February 1968).

The second detected break corresponds to the beginning of the rst oil price crisis (January 1973). Another break is detected at the end of the second oil price shock, July 1981 for the US series and, August 1983 for the French one. These two last shocks correspond to international events that have affected the two economies at the same time even if their magnitudes and their durations are not been the same. Finally the fourth break detected on the French ination series coincides with the crisis of the European Monetary system (September 1991).

Thus, it appears that these two ination rates time series can be modelled using long memory process (study based on the behavior of the autocorrelation functions) and using structural change models (study based on a economic approach). To detect if the long memory behavior is spurious or not, we proceed in the following way:

1. In a rst step, assuming that the data sets can be modelled using a FI(d) process, we estimate the long memory parameter d using GPH method with m = T 0.5 , and m = T /2 for the ELW method, for the whole sample.

2. In a second step, we identify existence of breaks using Bai and Perron method. We get 4 breaks for the French ination rate and 3 breaks for the American ination rate.

3. We divide the whole sample in four subsamples for the French series and in three subsamples for the American series.

4. On each subsample, we estimate the long memory parameter d using a FI(d) process with the same methods as in the rst step.

5. We compare the values obtained for d for each series on the whole sample and on the subsamples.

6. We consider the following rule. If there is a "`true"' long memory behavior, we will nd it on the whole sample and on the subsamples. If this behavior exists only on the whole sample and not on the subsamples, then we conclude that the breaks create a spurious long memory behavior. The results are not identical for the both series, see Tables 10 and11. For the French ination rate, the values of d are signicantly dierent from zero on the whole sample and not on the subsamples. This means that we are in presence of spurious long memory and it seems preferable to model this series with a model taking into account presence of breaks. We can chose it amongst the models proposed in Section 2. For the American ination, the values of d are always far from zero whatever the sample we use. Thus, it seems that persistence is an important feature of this last series. Nevertheless, we observe thet d > 1 on the whole period and d < 1/2 on the subsamples. It seems reasonable, for this series to consider a model which is able to take into account regimes with long memory behavior. This paper focus on the behaviors of the estimation methods for the long memory parameter under misspecication. Using several techniques to estimate the long memory parameter of a FI(d) process, we show, through an experiment study, that it can exist misspecication between some classes of structural changes models including mainly the mean-plus-noise and the Stopbreak and the FI(d) model. For the Markov-Switching, the Threshold autoregressive and the Structural Changes processes the results are mitigate.

In this paper, we have extended the work of [START_REF] Diebold | Long memory and regime switching[END_REF], to a large class of models using dierent techniques of estimation.

Studying French and American ination rates on a specic period, we detect several behaviors inside the data sets and we use a procedure which permits to privilege a model with respect to another one. We show that the French ination rate may be characterized by a spurious long memory behavior if we do not take into account existence of breaks while the US ination series can be characterized using mixing of long memory behavior and switching process in order to take into account its main features. Thus, the structural behavior of these two series appear sligtly dierent from this empirical study.

Figure 1 :

 1 Figure 1: (a) and (b) are respectively the trajectory and the ACF of the Mean-Plus-Noise model (2)-(4) with p = 0.01, T = 2000, σ 2 = σ 2 η = 1.

Figure 2 :

 2 Figure 2: (a) and (b) represent respectively the trajectory and the ACF of the stochastic permanent break model (2)-(5)-(6) with γ = 100, T = 2000, σ 2 = 1.

Figure 3 :

 3 Figure 3: (a) and (b) are respectively the trajectory and the ACF of the Markov switching model dened by (2) and (7) with T = 2000, σ 2 = 0.2, p 00 = 0.95, p 11 = 0.95 and µ0 = -µ1 = -1.

Figure 4 :

 4 Figure 4: (a) and (b) are respectively the trajectory and the ACF of the Threshold Auto-Regressive model dened by (2) and (8) with T = 2000, σ 2 = 0.2, r = 0, d = 1 and µ0 = -µ1 = -1.

Figure 5 :

 5 Figure 5: (a) and (b) are respectively the trajectory and the ACF of the sign model (2)-(9)-(10), T = 2000 and σ 2 = 0.2.

Figure 6 :

 6 Figure 6: (a) and (b) are respectively the trajectory and the ACF of the Structural CHange model (2 )-(11) with T = 2000, σ 2 = 0.19, π = 0.85 and µ 0 = -µ 1 = -1.

  (a) and (b). From these Figures, we observe a slow decaying of the autocorrelation function which indicate

Figure 7 :

 7 Figure 7: (a) and (b) are respectively the trajectory and the ACF of the French ination time series.

Figure 8 :

 8 Figure 8: (a) and (b) are respectively the trajectory and the ACF of the US ination time series.

Table 1 :

 1 Mean estimate of the long memory parameter d when the DGP is the Mean Plus Noise model with T = 2000

	σ 2 η 0.01 0.05 0.1 1	p(pT) dGP H dELW dW AV E dGP H dELW dW AV E dGP H dELW dW AV E dGP H dELW dW AV E	0.001(2) 0.0025(5) 0.005(10) 0.01(20) 0.05(100) 0.1(200) 0.066 0.118 0.174 0.251 0.479 0.589 0.058 0.117 0.176 0.240 0.432 0.524 0.209 0.236 0.237 0.241 0.336 0.373 0.142 0.268 0.359 0.472 0.728 0.819 0.142 0.247 0.334 0.427 0.643 0.734 0.241 0.266 0.273 0.332 0.474 0.577 0.209 0.350 0.471 0.579 0.818 0.887 0.188 0.314 0.416 0.518 0.736 0.819 0.267 0.269 0.321 0.377 0.575 0.670 0.557 0.712 0.786 0.877 0.975 0.988 0.393 0.632 0.713 0.803 0.934 0.961 0.349 0.458 0.557 0.659 0.862 0.925

Table 2 :

 2 Mean estimate of the fractional integration parameter d using GPH method and the corresponding value of pT σ 2

	σ 2 η |p	. 0.001 0.0025 0.005	η 0.01	0.05	0.1
	0.01 (pT σ 2 η ) (0.02) (0.05) 0.06 0.118	0.178 (0.1)	0.25 (0.2)	0.479 0.589 (1) (2)
	0.05 (pT σ 2 η )	0.142 (0.1)	0.268 (0.25)	0.360 0.472 0.727 0.820 (0.5) (1) (5) (10)
	0.1 (pT σ 2 η )	0.209 (0.2)	0.350 (0.5)	0.471 0.579 0.818 0.887 (1) (2) (10) (20)
	1 (pT σ 2 η )	0.557 (2)	0.682 (5)	0.786 (10)	0.877 (20)	0.970 0.988 (100) (200)

Table 3 :

 3 Mean estimate of the long memory parameter d when the DGP is the STOPBREAK model with T = 2000

	σ 2 0.01 0.05 0.1 1	γ dGP H dELW dW AV E 0.997 0.993 0.896 0.407 0.228 0.216 0.001 0.01 0.1 1 10 100 0.997 0.998 0.977 0.634 0.089 0.001 0.991 0.994 0.950 0.563 0.092 -0.009 -0.012 1000 0.004 0.211 dGP H 0.999 0.995 1.001 0.944 0.427 0.097 0.008 dELW 0.994 0.992 0.983 0.898 0.385 0.029 -0.002 dW AV E 1.001 1.002 0.992 0.788 0.302 0.215 0.219 dGP H 0.999 0.998 0.999 0.976 0.634 0.09 0.005 dELW 0.990 0.989 0.987 0.949 0.559 0.090 -0.008 dW AV E 0.996 1.003 1.001 0.901 0.406 0.232 0.198 dGP H 1.004 0.997 1.002 0.998 0.985 0.636 0.089 dELW 0.993 0.992 0.992 0.987 0.953 0.564 0.094 dW AV E 0.992 0.987 0.982 0.964 0.783 0.404 0.211
	We can illustrate this rule with the following example. If we take
	σ 2 T /γ = 20 for dierent triplets (σ 2 , T, γ) = (0.01, 2000, 1) or (0.1, 2000, 10) or (1, 2000, 100), d is respectively equal to 0.634, 0.634 and 0.636, using the GPH technique. Using the ELW, we got d equal to 0.563,
	0.559 and 0.564, and equal to 0.407, 0.406 and 0.404 using the Wavelet
	method, see Tables

Table 4 :

 4 Mean estimate value of the parameter d using wavelet method and its corresponding value of T σ 2 /γ.

	σ 2 |γ	0.1	1	10	100	1000
	0.01	0.950	0.563 0.092 -0.009 -0.012

Table 5 :

 5 Mean estimate of the fractional integration parameter d using wavelet method and its corresponding value of T σ 2 /γ

	σ 2 T |γ d	2 0.087 0.117 0.163 0.202 0.241 0.314 0.432 0.501 0.634 2.5 3.34 4 5 6.67 10 13.34 20

Table 6 :

 6 Mean estimate of the long memory parameter d when the DGP is the MS-AR model with T = 2000

	σ 2 0.01 0.05 0.1 1	(p 00 , p 11 ) (0.5, 0.99) (0.95, 0.95) (0.95, 0.99) (0.99, 0.99) (0.999, 0.999) dGP H 0.001 0.139 0.244 0.555 0.565 dELW 0.003 0.157 0.283 0.685 0.977 dW AV E 0.315 0.395 0.538 0.783 0.974 dGP H -0.002 0.143 0.242 0.556 0.573 dELW 0.004 0.151 0.267 0.684 0.932 dW AV E 0.303 0.386 0.535 0.774 0.900 dGP H 0.006 0.140 0.244 0.561 0.577 dELW 0.004 0.154 0.277 0.675 0.901 dW AV E 0.289 0.377 0.545 0.760 0.840 dGP H 0.0005 0.139 0.238 0.552 0.566 dELW 0.001 0.145 0.244 0.621 0.617 dW AV E 0.233 0.345 0.453 0.649 0.529

Table 7 :

 7 Mean estimate of the long memory parameter d when the DGP is the SETAR model with T = 2000

	σ 2 dGP H dELW dW AV E 0.949 0.745 0.799 0.677 0.472 0.383 0.160 0.01 0.05 0.1 0.2 0.3 0.4 1 0.004 0.001 0.729 0.562 0.263 0.136 0.023 1.024 0.874 0.866 0.602 0.255 0.127 0.009

Table 8 :

 8 Mean estimate of the long memory parameter d when the DGP is the Sign model

	p	0.001 0.0025 0.005	0.01	0.025	0.05	0.1
	(

Table 9 :

 9 Mean estimate of the long memory parameter d when the DGP is the SCH model with T = 2000

	σ 2 0.01 0.05 0.1 1	π dGP H dELW dW AV E 0.883 0.927 0.899 0.557 0.896 0.951 0.943 0.1 0.2 0.35 0.5 0.65 0.8 0.95 0.976 0.957 1.013 0.623 1.01 0.958 0.965 0.991 0.986 0.963 0.953 0.977 0.988 0.970 dGP H 0.910 0.904 0.931 0.604 0.923 0.902 0.867 dELW 0.914 0.900 0.864 0.876 0.888 0.895 0.911 dW AV E 0.757 0.774 0.765 0.540 0.762 0.781 0.793 dGP H 0.848 0.855 0.867 0.592 0.872 0.857 0.786 dELW 0.851 0.794 0.777 0.787 0.805 0.832 0.835 dW AV E 0.679 0.683 0.680 0.493 0.675 0.685 0.693 dGP H 0.515 0.547 0.529 0.433 0.529 0.547 0.399 dELW 0.276 0.315 0.347 0.397 0.434 0.450 0.467 dW AV E 0.379 0.371 0.382 0.317 0.382 0.383 0.388

Table 10 :

 10 Estimate of the number of breaks and the fractional long memory parameter

	d for the French ination series.
	Periods	N. of breaks	GPH dGP H t-stat dELW t-stat ELW
	P1 = 1959:01-1968:02	0	-0.114 -0.262 -0.038 0.916
	P2 = 1968:03-1973:01	0	-0.085 -0.174 0.039 -0.655
	P3 = 1973:02-1983:08	0	0.504	1.211	0.438	5.409
	P4 = 1983:09-1991:09	1	0.696	4.613	0.502	5.567
	P5 = 1991:10-2006:04	0	0.414	1.581	0.141	0.758
	PT = 1959:01-2006:04	4	0.928	6.613	0.338	7.99

Table 11 :

 11 Estimate of the number of breaks and the fractional long memory parameter d for the US ination series.

	Periods	N. of breaks	GPH dGP H t-stat dELW ELW t-stat
	P1 = 1959:01-1967:04	1	0.492 3.231 0.205	1.466
	P2 = 1967:05-1973:01	0	0.408 2.220 0.481	4.637
	P3 = 1973:02-1981:07	2	0.734 2.004 0.540	6.160
	P4 = 1981:08-2006:04	1	0.302 1.704 0.428	7.983
	PT = 1959:01-2006:04	3	1.038 7.564 0.483 12.915
	6 Conclusion


Lionel De Boisde¤re
email: lionel.de-boisde¤re@univ-paris1.fr

Keywords: sequential equilibrium, temporary equilibrium, perfect foresight, expectations, incomplete markets, asymmetric information, arbitrage. JEL Classi…cation: D52

). Yet, when anticipations are private, we argue, any element of a typically uncountable 'minimum uncertainty set' may prevail as an equilibrium price tomorrow. This outcome is inconsistent with perfect foresight and appeals for a broader de…nition of sequential equilibrium, which we propose hereafter.

.

Introduction

In economies subject to uncertainty and asymmetric information, agents are traditionally assumed to have a 'model' of how equilibrium prices are determined, along [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF], and to infer any additional information from comparing actual prices and price expectations to their theoretical values. This so-called 'rational expectations' hypothesis is convenient to deal with sequential equilibrium, but relies on a strong assumption regarding agents' capacities, and leads to standard cases of non-existence of equilibrium. In [START_REF] Cornet | Arbitrage and price revelation with asymmetric information and incomplete markets[END_REF][START_REF] Cornet | Elimination of arbitrage states in asymmetric information models[END_REF]) and de Boisde¤re (2007), we showed that treating asymmetric information without Radner's rational expectations assumption was always possible and improved existence results, namely, it guaranteed the existence of equilibrium on a purely …nancial market under the same conditions as for symmetric information.

Our former model of asymmetric information still retained the standard sequential equilibrium's second hypothesis, namely, [START_REF] Radner | Existence of equilibrium plans, prices and price expectations in a sequence of markets[END_REF] perfect foresight, along which agents anticipate with certainty exactly one price for each commodity (or asset) in each random state, which turns out to be the true price, if that state prevails tomorrow. The rationale for doing this was technical. Indeed, keeping perfect foresight, while dropping rational expectations, might have been di¢cult to justify economically. Technically, however, perfect foresight is the easiest way to insure equilibrium unfolds sequentially, as opposed to 'temporarily'. That is, when all uncertainty is removed, no agent would ever face bankruptcy or a welfare increasing trade opportunity, and agents' equilibrium decisions (ex ante) clear on markets at all time periods. These outcomes no longer hold at a temporary equilibrium, because agents may fail to forecast prices correctly at the outset.

Yet, perfect foresight is a su¢cient but, by no means, a necessary condition to guarantee sequential unfolding. Though it has remained the standard setting so far, the perfect foresight model deals with a restrictive notion of sequential equilibrium, which embeds feasibility and existence problems, akin to the rational expectation model's dealing with asymmetric information. In particular, the perfect foresight model virtually requires the common knowledge of the equilibrium price, leads to standard existence problems, and barely explains speculation or crash phenomena, stemming from agents' beliefs. These shortcomings appeal for an alternative broader de…nition of the sequential …nancial equilibrium concept. The main purpose of this paper is to introduce such a concept and to lay foundations for proving its existence in a companion paper, by presenting, …rst and hereafter, a theory of arbitrage under asymmetric information in an in…nite dimensional setting, where agents are prone to uncertainty between (possibly uncountable) price forecasts.

We do this by extending our earlier model of asymmetric information, so as to let agents be uncertain of future prices on each spot market, whenever required.

We introduce new concepts of beliefs, structures of beliefs and re…nements, noarbitrage prices and the information they reveal. These concepts enhance those of our 2002 model to …t with the new setting, and coincide with the earlier ones under perfect foresight. Via standard, in…nite dimensional analysis techniques, we extend the arbitrage properties of [START_REF] Cornet | Arbitrage and price revelation with asymmetric information and incomplete markets[END_REF][START_REF] Cornet | Elimination of arbitrage states in asymmetric information models[END_REF] to the new model. In particular, we show how agents, with no price model a la Radner, may still update their beliefs from observing prices on current markets and, thereby, free markets from arbitrage opportunities. Along this inference process, agents' beliefs are said to be price-revealed. Besides the new results, this paper embeds all main properties demonstrated in the …nite dimensional case by [START_REF] Cornet | Arbitrage and price revelation with asymmetric information and incomplete markets[END_REF][START_REF] Cornet | Elimination of arbitrage states in asymmetric information models[END_REF] as a particular application -since the two models coincide under perfect foresight.

Formally, the model we propose is a two-period pure exchange economy, where …nitely many agents face an exogenous uncertainty, represented by …nitely many random states of nature (on which they may be asymmetrically informed), exchange goods on spot markets, for the purpose of consumption, and trade, unrestrained, on a (possibly incomplete) …nancial market, so as to transfer wealth across periods and states. At the …rst period, besides the above exogenous uncertainty on the future state of nature, agents may face an 'endogenous uncertainty' on the future price, in each state they expect. Namely, consumers have private sets of anticipations for future spot prices, distributed along idiosyncratic probability laws, called beliefs.

The latter uncertainty on prices is traditionally referred to as 'endogenous', because it concerns and may a¤ect the endogenous variables. The model's equilibrium, called 'correct foresight equilibrium' (C.F.E.), is reached when agents anticipate tomorrow's 'true' price as a possible outcome, and make optimal trade and consumption decisions today, given their preferences, which clear on markets and remain optimal ex post, given prior portfolio choices. In a companion paper, we show how a C.F.E. may be reached and why this equilibrium may solve the classical existence problems, which followed, not only [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF], but [START_REF] Hart | On the existence of equilibrium in a securities model[END_REF]. Whenever required, agents' revised beliefs at the C.F.E. may be revealed by the equilibrium price itself, in the sense de…ned above. This concept is, indeed, a sequential one, that is, di¤ers from the temporary equilibrium's, introduced by Hicks (1939), developed by [START_REF] Grandmont | Temporary general equilibrium theory[END_REF][START_REF] Grandmont | Temporary general equilibrium theory[END_REF], [START_REF] Green | Temporary general equilibrium in a sequential trading model with spot and futures transactions[END_REF], [START_REF] Hammond | Overlapping expectations and Hart's condition for equilibrium in a securities model[END_REF], [START_REF] Balasko | Temporary …nancial equilibrium[END_REF], among others, where agents' forecasts need not be correct.

The paper is organized as follows: we present the model, in Section 2, its arbitrage properties, in Section 3, and inference mechanisms, in Section 4.

We consider a pure-exchange …nancial economy with two time-periods (t 2 f0; 1g) and two markets, a commodity market and a …nancial market. There is an a priori uncertainty at the …rst period (t = 0) about which state s of a given state space S will prevail at the second period (t = 1), when all uncertainty is removed. The state of nature at t = 0 is non random and denoted by s = 0. The sets of agents (or consumers), I := f1; :::; mg, of commodities, L := f1; :::; Lg, of states of nature, S := f1; :::; N g, and …nancial assets, J := f1; :::; Jg, are all …nite subsets of N.

Before presenting the model, we introduce notations, which are used throughout.

The model's notations

Throughout, we denote by the scalar product and by k:k the Euclidean norm on an Euclidean space, by B(K) the Borel sigma-algebra of a topological space, K. We let s = 0 be the non-random state at t = 0 and S 0 := f0g [ S. For all sets S 0 and tuples ("; s; l; x; x 0 ; y; y

0 ) 2 R ++ L R R (R L ) (R L )
, we denote by:

x s 2 R, y s 2 R L the scalar and vector, indexed by s 2 , of (resp.) x and y; y l s the l th component of y s 2 R L ;

x 6 x 0 and y 6 y 0 (resp. x << x 0 and y << y 0 ) the relations x s 6 x 0 s and y l s 6 y 0l s (resp. x s < x 0 s and y l s < y 0l s ) for all (l; s) 2 f1; :::; Lg ;

x < x 0 (resp. y < y 0 ) the joint relations x 6 x 0 , x 6 = x 0 (resp. y 6 y 0 , y 6 = y 0 );

R L := (R L ) , R L + = fx 2 R L : x > 0g and R + := fx 2 R : x > 0g, R L ++ := fx 2 R L : x >> 0g and R ++ := fx 2 R : x >> 0g, M 0 := f(p 0 ; q) 2 R L + R J : kp 0 k + kqk = 1g;
M := [ s2S M s and M " := [ s2S M " s .

The commodity and asset markets

The L commodities, l 2 L, are used for the purpose of consumption and may be exchanged between agents on spot markets. There are #S 0 ex ante possible spot markets, namely one in each state s 2 S 0 . In each state s 2 S, an expectation of a spot price, p 2 R L + , is denoted by the pair ! s := (s; p) 2 S R L + (which will also stand for the spot price p in state s itself). At little cost, we normalize admissible expectations and spot prices, in each state s 2 S, to the above de…ned set M s .

Agents exchange commodities in order to increase their welfare. Ex post, the generic i th agent's welfare is measured by u i (x; y) 2 R + , where x := (x 1 ; :::; x L ) 2 R L + and y := (y 1 ; :::; y L ) 2 R L + are the vectors of consumptions, respectively, at t = 0 and t = 1, and u i : R 2L + ! R + is a utility function, assumed to be continuous.

Trade may take place because each agent, i 2 I, can rely on an endowment,

e i := (e is ) 2 R LS 0
++ , of the L goods, which grants her the commodity bundle e i0 2 R L ++ at t = 0, and e is 2 R L ++ , in each state s 2 S if this state prevails at t = 1. For the sake of simpler notations, we henceforth let e i! := e is , for every triple (i; s; !) 2 I S 0 M s .

The …nancial market permits limited transfers across periods and states, via J assets, also called securities, j 2 J := f1; :::; Jg, which are exchanged at t = 0 and pay o¤ at t = 1. Assets may be nominal or real (i.e., pay o¤ in account units or in commodities). For any spot price, or expectation, ! 2 M, the payo¤s, v j (!) 2 R, of each asset j 2 f1; :::; Jg conditional on the occurence of !, de…ne a row vector, 5 Documents de Travail du Centre d'Economie de la Sorbonne -2012.53 V (!) = (v j (!)) 2 R J , and the mapping ! 2 M 7 ! V (!) is continuous from the de…nition and the continuity of the scalar product (since assets pay in money or commodities).

Provided she can a¤ord, every agent i 2 I may take unrestrained positions, z j i 2 R (positive, if purchased; negative, if sold), in every security j 2 f1; :::; Jg, which de…ne her portfolio, z i := (z j i ) 2 R J . When an asset price, q 2 R J , is observed at t = 0, a portofolio, z 2 R J , is thus a contract, which costs q z units of account at t = 0, and promises to pay V (!) z units tomorrow, for each spot price ! 2 M, if ! obtains.

Similarly, we henceforth normalize …rst period prices, ! 0 := (p 0 ; q), to the set M 0 .

Information and beliefs

During the …rst period (t = 0), each agent receives a private information signal, S i S, which informs her that the true state, which will prevail at t = 1, will be in S i . Henceforth, the collection (S i ) of all agents' signals is set as given and we let S := \ m i=1 S i , referred to as the pooled information set. Agents are assumed to receive no wrong information in the sense that no state of SnS will prevail, hence, S is non-empty. Agents form private anticipations of future spot prices in each state they expect, distributed along idiosyncratic probability laws. Formally:

De…nition 1 For all probability , on (M; B(M)), and pair (! := (s; p); ") 2 M R ++ , we let B(!; ") := f(s 0 ; p 0 ) 2 M : kp 0 pk+js 0 sj < "g and P ( ) := f! 2 M : (B(!; ")) > 0; 8" > 0g be a compact set, whose elements are called anticipations, expectations or forecasts.

A probability, , on (M; B(M)), is called a belief if the following Condition holds: Similarly, a collection of classes,

( o i ) 2 CB m , is called a class structure (of beliefs),
and we denote it by

( o i ) 2 CSB, if \ m i=1 P ( o i ) 6 = ?. Let (( i ); ( 0 i )) 2 SB 2 , ( o i ) 2 CSB and payo¤ mapping, V , be given. The couples, [V; ( i )]
and [V;

( o i )],
are called, respectively, a structure and a class structure (of payo¤s and beliefs). The structure ( 0 i ) is said to re…ne ( i ), and we denote it by ( 0 i ) ( i ), if the relations 0 i i hold for each i 2 I. The two structures are equivalent, and we denote it by

( i ) ( 0 i ), if both relations ( i ) ( 0 i ) and ( 0 i ) ( i ) hold. A re…nement, ( i ) 2 SB, of ( i ) 2 SB
, is said to be self-attainable if the following Condition holds:

(d) \ m i=1 P ( i ) = \ m i=1 P ( i ).
The notions of re…nement and self-attainable re…nement are de…ned alike on CSB.

Remark 1 Without changing the paper's results, a belief could be de…ned as a probability on (M; B(M)), whose support cannot take arbitrary large or low values.

With normalized expectations this is stated by Condition (a). Under perfect foresight, class structures, ( o i ) 2 CSB, and information structures, (S i ), coincide, as well as the above de…nitions of re…nements with Cornet-de Boisde¤re's (2002).

Consumers' behavior and the notion of equilibrium

In this sub-Section, we assume that agents make their trade and consumption plans after having reached a (…nal) structure of beliefs, ( i ) 2 SB, and observed the market price at t = 0, ! 0 := (p 0 ; q) 2 M 0 , which are set as given and referred to throughout. The generic i th agent's consumption set is, then, de…ned as:

X( i ) := C(f0g [ P ( i ); R L + ) ,
where

C(f0g [ P ( i ); R L + ) stands for the set of continuous mappings from f0g [ P ( i ) to R L + . A consumption, x 2 X( i )
, is, thus, a mapping, which relates continuously s = 0 to a (…xed) consumption decision, x 0 := x !0 2 R L + , at t = 0, and every anticipation,

! := (s; p) 2 P ( i ), to a random consumption decision x ! 2 R L + at t = 1,
which is conditional on the occurence of the spot price !, that is, of both state s 2 S and price p 2 R L + , on the spot market, at t = 1.

Each agent i 2 I elects and implements a consumption and investment decision, or strategy, (x; z) 2 X( i ) R J , that she can a¤ord on markets, given her endowment,

e i 2 R LS 0
+ , and her expectation set, P ( i ). This de…nes her budget set as follows:

B i (! 0 ; i ) := f(x; z) 2 X( i ) R J : p 0 (x 0 -e i0 ) 6 q z; p s (x ! -e i! ) 6 V (!) z; 8! := (s; p s ) 2 P ( i )g . An allocation, (x i ) 2 X[( i )] := m i=1 X( i )
, is a collection of consumptions across consumers. We de…ne the following sets of attainable allocations, for every price collection, (! s ) := (! s ) s2S 2 s2S M s , and attainable portfolios, respectively: Hence, only spot markets, whose price is commonly expected (or observed at t = 0) by all agents, need clear, along the above de…nition of an attainable allocation.

A((! s ); ( i )) := f(x i ) 2 X[( i )] : P m i=1 (x i0 -e i0 ) = 0; P m i=1 (x i!s -e i!s ) = 0; 8s 2 S; s:t: ! s 2 \ m i=1 P ( i )g ; Z := f(z i ) 2 (R J ) m : P m i=1 z i = 0g.
Each agent i 2 I has preferences represented by the V.N.M. utility function:

u i i : x 2 X( i ) 7 ! u i i (x) := R !2P ( i) u i (x 0 ; x ! )d i (!) .
The generic i th agent's behavior is, then, to elect a strategy, which maximises this utility function in the buget set, that is, a strategy in

B i (! 0 ; i ) := arg max (x;z)2Bi(!0; i) u i i (x).
The above economy is denoted by E, whose equilibrium is de…ned as follows:

De…nition 2 A collection of prices, (! s ) 2 s2S 0 M s , beliefs, ( i ) 2 SB, and strategies,

(x i ; z i ) 2 B i (! 0 ; i ), de…ned for each i 2 I,
(a) 8s 2 S, ! s 2 \ m i=1 P ( i ); (b) 8i 2 I; (x i ; z i ) 2 B i (! 0 ; i ) := arg max (x;z)2Bi(!0; i) u i i (x); (c) (x i ) 2 A((! s ); ( i )); (d) (z i ) 2 Z.

The arbitrage properties

We de…ne and characterize no-arbitrage prices and the information they reveal.

The model's no-arbitrage prices

We start with a standard application of separation theorems in topological vector spaces, which yields a no-arbitrage characterization used throughout the paper.

Claim 1 Let 2 B and q 2 R J be given. The following statements are equivalent:

(i) there is no portfolio z 2 R J , such that q z > 0 and V (!) z > 0 for every ! 2 P ( ),

with at least one strict inequality;

(ii) there exists a continuous mapping

: P ( ) ! R ++ , such that q = R !2P ( ) (!)V (!)d (!).
Remark 3 It follows from Claim 1, that if a belief, 2 B, meets the conditions of Claim 1 for a given price, q 2 R J , any other equivalent belief, 0 , meets the same conditions. This will make De…nition 3, below, consistent.

Proof Let 2 B and q 2 R J be given.

(ii) ) (i) Assume that assertion (ii) holds, and, let z 2 R J be given such that q z > 0 and V (!) z > 0 for every ! 2 P ( ). Assume, …rst, that V (!) z > 0, for some

! 2 P ( ).
Then, the above inequalities V (!) z > 0, which hold every ! 2 P ( ), and the

continuity of V at ! 2 P ( ) imply q z = R !2P ( ) (!)V (!) zd (!) > 0,
in contradiction with the above relation -q z > 0. Hence, V (!) z = 0, for all ! 2 P ( ), which also yields, from Assertion (ii), q z = R !2P ( ) (!)V (!) zd (!) = 0, that is, assertion (i) holds.

(i) ) (ii) Assume that Assertion (i) holds and let P := fs = 0g [ P ( ) and C(P; R) be the set of continuous (hence, Borel measurable) mappings from P to R, endowed with the (well de…ned) operator (f; g) 2 C(P; R) 2 ), the induced metric and topology. The set C(P; R) is a convex metric space, with the linear sub-spaces:

7 ! < f; g > := f (0)g(0)+ R !2P ( ) f (!)g(!)d (!), the norm f 2 C(P; R) 7 ! kf k := q f (0) 2 + R !2P ( ) f (!) 2 d (!
A := ff 2 C(P; R) : 9z 2 R J ; f (0) = q z and f (!) = V (!) z; 8! 2 P ( )g; A ? := ff 2 C(P; R) : < a; f > = 0; 8a 2 Ag.
Let C(P; R + ) and C(P; R ++ ) be, respectively, the subsets of continuous non-negative and strictly positive valued mappings of C(P; R). Assertion (i) is written A\ C(P; R + ) = f0g. Assume, by contraposition, that A ? \ C(P; R ++ ) = ?, i.e., Assertion (ii) fails.

Then, from the Interior Separating Hyperplane Theorem and the fact that A ? is a linear sub-space (see Aliprantis-Border (1999), pp. 188, 190), there exists a nonzero continuous linear functional, ', which properly separates A ? and C(P; R ++ ), and such that '(a) = 0 6 '(b), for every (a; b) 2 A ? C(P; R ++ ).

From Riesz' Theorem (see Aliprantis-Border (1999), p. 440), there exists f 2 C(P; R), such that '(h) = < f; h >, for every h 2 C(P; R). The linear space A is closed hence, with obvious de…nition, A ?? = A (Aliprantis-Border (1999), p. 215). Then, from the above inequalities, the relations f 2 A ?? \ C(P; R + )nf0g = A \ C(P; R + )nf0g hold and contradict the above formulation, A \ C(P; R + ) = f0g, of assertion (i).

We can now de…ne and characterize arbitrage-free prices, beliefs, and structures. o , and a price, q 2 R J , be given. The couples, (V; o ) or (V; ), are said to to be q-arbitrage-free (hence, arbitrage-free), or q to be a no-arbitrage price of (V; o ), or (V; ), if the following equivalent Conditions hold:

(a) there is no portfolio z 2 R J , such that q z > 0 and V (!) z > 0 for every

( ) ! R ++ , such that q = R !2P ( ) (!)V (!)d (!).
We let Q[V; o ] be the set of no-arbitrage prices of (V; o ) (or

(V; )) and Q c [V; ( o i )] := \ m i=1 Q[V; o i ] be the set of common no-arbitrage prices of [V; ( o i )].
The class structure

[V; ( o i )
] is said to be arbitrage-free (resp., q-arbitrage-free

) if Q c [V; ( o i )] 6 = ? (resp., if q 2 Q c [V; ( o i )]
).We say that q is a no-arbitrage price (resp., a self-attainable no-

arbitrage price) of [V; ( o i )]
if there exists a re…nement (resp., a self-attainable re…nement),

( o i ) ( o i ), such that q 2 Q c [V; ( o i )],
and we denote their set by Q[V;

( o i )].
All above de…nitions and notations can be stated, equivalently from Remark 3, in terms of any representative structure,

( i ) 2 m i=1 o i . We then refer to Q c [V; ( i )] := Q c [V; ( o i )] and Q[V; ( i )] := Q[V; ( o i )]
as, respectively, the sets of common no-arbitrage prices, and no-arbitrage prices, of the structure

[V; ( i )].
When no confusion is possible, the reference to V may be dropped in all above de…nitions and notations.

Remark 4 We notice that a symmetric re…nement of any structure

( i ) 2 SB, that is, a re…nement ( 0 i ) ( i ), such that P ( 0 i ) = P ( 0 1
), for every i 2 I, is always arbitrage-free along De…nition 3. The latter De…nition embeds and extends Cornetde Boisde¤re's (2002) De…nition 2.2 (p. 397). Henceforth, without recalling, we let the reader notice the same conclusion for all subsequent de…nitions and properties.

The following no-arbitrage characterization will be used throughout the paper.

Claim 2 Given ( o i ) 2 CSB, ( i ) 2 m i=1 o
i and q 2 R J , the following Conditions are equivalent, the latter being called absence of future arbitrage opportunity (AFAO):

(i) the class structure ( o i ) (or, equivalently, the structure ( i )) is arbitrage-free;

(ii) there is no portfolio collection (z i ) 2 (R J ) I , such that

P m i=1 z i = 0 and V (! i ) z i > 0 for every couple (i; ! i ) 2 I P ( i ) = I P ( o i ), with at least one strict inequality. Proof Let ( o i ) 2 CSB and ( i ) 2 m i=1 o i be given. (i) ) (ii) Assume that ( i ) 2 SB is arbitrage-free. Set as given q 2 Q c [V; ( i )],
and, for each i 2 I, a continuous mapping, i :

P ( i ) ! R ++ , such that q = R !2P ( i) i (!)V (!)d i (!),
which exist from De…nition 3. Assume that there exists (z i ) 2 (R J ) I , such that V (! i ) z i > 0 for every (i; !) 2 I P ( i ), with at least one strict inequality. Then, summing up the latter relations, wheigthed by i (!), for every (i; !) 2 I P ( i ), yields, from above and the continuity of V ,

P m i=1 q z i = P m i=1 R !2P ( i) i (!)V (!) z i d i (!) > 0,
which implies that P m i=1 z i 6 = 0 and, consequently, that Condition AFAO holds.

(ii) ) (i) Assume Condition AFAO holds. The proof is akin to that of Claim 1.

For each i 2 I, we let P i := fs = 0g [ P ( i ), and C(P i ; R) be the set of continuous mappings from P i to R, endowed with the operator, ), and the induced norm, metric and topology, as in Claim 1. Then, we endow from above the set C := m i=1 C(P i ; R) with the operator, metric and topology of product spaces, let C + and C ++ be the subsets of non-negative and strictly positive valued functions of C and A, A ? be the linear sub-spaces:

(f; g) 2 C(P i ; R) 2 7 ! < f; g > := f (0)g(0) + R !2P ( i) f (!)g(!)d i (!
A := f(f i ) 2 C : (f i (0)) = 0; 9(z i ) 2 R JI : P m i=1 z i = 0; f i (! i ) = V (! i ) z i ; 8(i; ! i ) 2 I P ( i )g; A ? := ff 2 C : < a; f > = 0; 8a 2 Ag.
The AFAO Condition is written:

A \ C + = f0g. If A ? \ C ++ = ?
, the very same arguments as in Claim 1 apply, and, as we let the reader check, yield a contradiction.

Hence, we may set as given ( i ) 2 A ? \ C ++ 6 = ?. Then, by taking (z i ) 2 (R J ) I , such that (z i ; z j ) = ( z 1 ; 0), for every (i; j) 2 I 2 , i 6 = 1, j = 2 f1; ig, the relation ( i ) 2 A ? yields: ), for every i 2 I, which implies, from De…nition 3, that q 2 Q c [V; ( i )], i.e., assertion (i) holds.

R !2P ( i) i (!)V (!) zd i (!) = R !2P ( 1) 1 (!)V (!) zd 1 (!), for every pair (i; z) 2 I R J . Let q := R !2P ( 1) 1 (!)V (!)d 1 (!). From above, q = R !2P ( i) i (!)V (!)d i (!
(i) [V; ( o i )] is arbitrage-free;
(ii) every arbitrage-free re…nement of

( o i ) is a re…nement of ( o i ).
The coarsest arbitrage-free re…nement, denoted

o [V; ( o i )] or o [( o i )], is self-attainable. Proof Along De…nitions 1 & 3, let ( o i ) 2
k; k 0 ) 2 K 2 , either ( o k i ) ( o k 0 i ) or ( o k 0 i ) ( o k i ).
We set as given such a chain, f( o k i )g k2K . Along De…nition 1, we let f(P k i )g := f(P ( o k i ))g k2K be its chain of supports and, for each

o k i ) ( o i ) ( o i ) holds, for every k 2 K.

Assume, by contraposition, that

( o i ) = 2 R ( o i) , i.e.
, from Claim 2, there exists (z i ) 2 (R J ) I , such that P m i=1 z i = 0 and V (! i ) z i > 0 for every couple (i; ! i ) 2 I P i , with at least one strict inequality, say V (! 1 ) z 1 > 0 for ! 1 2 P 1 . Since, for all k 2 K, the class structure ( o k i ) 2 CSB is arbitrage-free and such that (

o k i ) ( o i ), the latter relations imply, from Claim 2, that V (! k i ) z i = 0 for every triple (k; i; ! k i ) 2 K I P k i . Since V is continuous, the relation V (! 1 ) z 1 > 0 for ! 1 2 P 1 := [ k2K P k 1 implies that there exists k 2 K and ! k 1 2 P k 1 , such that V (! k 1 ) z 1 > 0, in
o i ; q) 2 ? [ CB, such that:

(i) either o (V; o i ; q) = ?, or o (V; o i ; q) o i along De…nition 1; (ii) if o (V; o i ; q) o i , then, (V; o (V; o i ; q)
) is q-arbitrage-free along De…nition 3;

(iii) every q-arbitrage-free re…nement of Moreover, for every pair (i; i ) 2 I o (V; o i ; q), there exists a continuous mapping

i : P ( i ) ! R ++ , such that q = R !2P ( i ) i (!)V (!)d i (!).
Proof The proof is similar to that of Claim 3. Let i 2 I, q 2 R J , and a class structure, ( o i ) 2 CSB, along De…nition 1, be given. Let R ( o i;q) be the set of q-arbitrage-free re…nements of Assume, by contraposition, that o i = 2 R ( o i;q) , i.e., from De…nition 3, there exists z 2 R J , such that q z > 0 and V (!) z > 0 for every ! 2 P i , with at least one strict 15 Documents de Travail du Centre d'Economie de la Sorbonne -2012.53 inequality, say (V (!) z q z) > 0 for ! 2 P i . Since, o k i is q-arbitrage-free and P k i P i , for every k 2 K, the above relations imply, from De…nition 3, that q z = 0 and

V (! k ) z = 0, for every (k; ! k ) 2 K P k i . Since V is continuous, the relation V (!) z > 0, which holds for ! 2 P i := [ k2K P k i , implies that there exists k 2 K and ! k i 2 P k i , such that V (! k i ) z > 0, in

Structures of beliefs revealed by prices

The following Claim characterizes no-arbitrage prices along De…nition 3.

Claim 5 Let a price, q 2 R J , a class structure of payo¤s and beliefs, [V; ( o i ; q)) i2I , along De…nition 4, be given. Then, the following statements are equivalent:

(i) q is a no-arbitrage price of [V; ( o i )], i.e., q 2 Q[V; ( o i )]; (ii) ( o (V; o i ; q)) is the coarsest q-arbitrage-free re…nement of ( o i ); (iii) ( o (V; o i ; q)) ( o i ), i.e., ( o (V; o i ; q)) is a re…nement of ( o i ); (iv) \ m i=1 P ( o (V; o i ; q)) 6 = ?, i.e., ( o (V; o i ; q)
) is a class structure of beliefs.

Proof (i) ) (ii) Let q 2 Q[V; ( o i
)] be given. Then, along De…nition 3, there exists a q-arbitrage-free re…nement,

( o i ), of ( o i
), which we set as given, and, for each i 2 I, the set, R ( o i;q) , of q-arbitrage-free re…nements of

o i ; q)) is a re…nement of ( o i )
and, moreover, the (unique) coarsest q-arbitrage-free re…nement of ( o i ).

The relations (ii) ) (iii) ) (iv) are immediate from De…nition 1.

(iv) ) (i) If ( o (V; o i ; q)) 2 CSB, then, ( o (V; o i ; q)) re…nes ( o i ), from Claim 4-(i), and is q-arbitrage-free, from Claim 4-(ii), that is, q 2 Q c [V; ( o (V; o i ; q))] Q[V; ( o i )].
De…nition 5 Let a class structure of payo¤s and beliefs, [V; ( o i )], and no-arbitrage price, q, be given. The coarsest q-arbitrage-free re…nement of ( o i ) along Claim 5,

( o (V; o i ; q)), also denoted by ( o ( o i ; q)), is said to be revealed by price q 2 Q[V; ( o i )]. A re…nement, ( o i ) ( o i ) (or, equivalently, ( i ) ( i ) 2 m i=1 o i
) is said to be pricerevealable, if there exists q 2 R J , such that

( o i ) = ( o (V; o i ; q)) (or ( i ) 2 m i=1 o (V; o i ; q)).
Remark 5 As shown in Cornet-de Boisde¤re (2002), given ((

o i ); q) 2 CSB Q[( o i )],
there may exist other q-arbitrage-free re…nements of ( o i ) than the coasest, (

i ; q)). o ( o
Claim 6 shows that the coarsest arbitrage-free re…nement is always price-revealable.

( o ( o i ; q)), for every q 2 Q[V; ( o i )],
be, respectively, the coarsest arbitrage-free, and coarsest q-arbitrage-free, re…nements of ( o i ). Then, the following assertion holds:

(i) ? 6 = fq 2 R J : o [( o i )] = ( o ( o i ; q))g = Q c [V; o [( o i )]].
Hence

)] can be revealed by any price

q 2 Q c [V; o [( o i )]]. Moreover, if [V; ( o i )] is arbitrage-free, o [( o i )] = ( o i ) and ( o i ) can be revealed by any price q 2 Q c [V; ( o i )] 6 = ?. Proof Let ( o i ) 2 CSB be given and let o [( o i )] and ( o ( o i ; q)), for every q 2 Q[V; ( o i )],
be de…ned as in Claim 6. From Claim 3 and De…nition 3, the set

Q c [V; o [( o i )]] is non-empty. Let q 2 Q c [V; o [( o i )]
] be given. Then, from Claim 5, the relation

i ; q)) holds, since o [( o i o [( o i )] ( o ( o
) is q-arbitrage-free, whereas, from Claim 3, the converse

i ; q)) o [( o i )], also holds, since o [( o i ) is the coarsest arbitrage-free re…nement. Hence, o [( o i ) = ( o ( o i ; q)) ( o i ). relation, ( o ( o
We have thus shown the relations:

? 6 = Q c [V; o [( o i )]] fq 2 R J : o [( o i )] = ( o ( o i ; q))g. Conversely, let q 2 R J be given, such that [( o i )] = ( o ( o i ; q)). Then, from Claim 5, [( o i )] is q-arbitrage-free, i.e., q 2 Q c [V; o [( o i )]]. Hence, fq 2 R J : o [( o i )] = ( o ( o i ; q))g Q c [V; o [( o i )]
] and, from above, assertion (i) holds. The rest is immediate from the de…nitions and assertion (i).

We now characterize prices, which reveal a self-attainable re…nement.

Claim 7 Let ( o i ) 2 CSB & q 2 R
J be given. Let AS be the non-empty set of arbitragefree self-attainable re…nements of ( o i ). The following assertions are equivalent:

(i) q 2 [ ( o i )2AS Q c [V; ( o i )],
i.e., q is a self-attainable no-arbitrage price;

(

i ; q)) is a self-attainable re…nement of ( o i ). ii) ( o ( o
Proof From Claim 3 and Remark 4, the set AS contains both the symmetric selfattainable and the coarsest arbitrage-free re…nements of

( o i ). The relation (i) ) (ii)
is a direct consequence of Claim 5. Moreover, if assertion (ii) holds, (

i ; q)) ( o i ) is q-arbitrage-free, from Claim 5, i.e., q 2 Q c [V; ( o ( o i ; q))], o ( o
and assertion (i) holds.

We now examine how agents, endowed with no price model a la Radner, may still update their beliefs from observing markets, namely, prices, if they can meet agreement, or exchange opportunities, if this is not the case.

This Section generalizes Cornet-de Boisde¤re (2009) to the above economy E.

Sequential re…nement through prices

Throughout, we let a class structure, [V;

( o i )],
of payo¤s and beliefs, an agent, i 2 I, and an asset price, q 2 R J , be given. We study how the generic i th agent, endowed with the initial expectation set, P ( o i ), along De…nition 1, may narrow down in steps her expectation set from observing the current price, q, on …nancial markets, without having any model of how market prices are determined. To that aim, we de…ne, by induction on n 2 N, two sequences, fA n i g n2N and fP n i g n2N , of sub-sets of ? [ M, called, respectively, the arbitrage and (interim) expectation sets:

for n = 0, we let A 0 i = ? and P 0 i := P ( o i );

for n 2 N arbitrary, with A n i and P n i de…ned at step n, we let A n+1 i := P n+1 i := ?, if P n i = ?, and, otherwise:

A n+1 i := f! 2 P n i : 9z 2 R J ; q z > 0; V (!) z > 0 and V (!) z > 0;
( o i ; q)) \ A n+1 i
and z 2 R J , such that q z > 0, V (!) z > 0 and V (!) z > 0, for every ! 2 P ( o ( o i ; q)) P n i , which contradicts Claim 5, along which

(V; P ( o ( o i ; q))) is q-arbitrage-free, if non-empty.
Hence, the relation P ( o ( o i ; q)) P n i holds for every n 2 N.

Assume, …rst, that P i := \ n2N P n i = ?. Since the sequence fP n i g n2N is non-increasing and made of compact or empty sets, there exists N 2 N, such that P n i = A n i = ?, for every n > N , and, from above, assertions (i)-(ii) of Claim 8 hold. Assume, next, that P i 6 = ?, then, P i , a non-empty intersection of compact sets, is compact, and, from above, P ( Second, we assume, by contraposition, that assertion (i) of Claim 8 fails, that is:

8n 2 N; 9(! n ; z n ) 2 P n i Z n i : q z n > 0; V (! n ) z n > 0 and V (!) z n > 0; 8! 2 P n
i , in which we may take Z n i := fz 2 R J : kzk = 1; z 2 hZ on i i

? g orthogonal to Since fP n i g n2N is decreasing, the sequence of vector spaces, fZ on i g, is non-decreasing in R J , hence, stationary. We let N 2 N be such that Z n i = Z N i , for every n > N , and denote simply Z i := Z N i and Z o i := Z oN i . Since P 0 i Z i is compact, we may assume there exists (! ; z ) = lim n!1 (! n ; z n ) 2 P 0 i Z i . We recall that Z i and Z o i are orthogonal and notice that Z o i = fz 2 R J : q z = 0; V (!) z = 0; 8! 2 P i g.

Z on i := fz 2 R J : q z = 0; V (!) z = 0; 8! 2 P n i g.
The above relations on (! n ; z n ), for each n 2 N, and the continuity of V and of the scalar product imply that q z > 0 and V (!) z > 0, for every ! 2 P i , with one strict inequality, since z 2 Z i , is orthogonal to Z o i and such that kz k = 1. Hence, there exists ! 2 P i , such that V (! ) z > 0. By construction,

P i := \ n2N P n i is disjoint from A i := [ n2N A n i . Hence, ! = 2 A i .
From the continuity of the scalar product and above, there exists n > N , such that V (! ) z n > 0, which implies, from the above relations on

(! n ; z n ), that ! 2 A n+1 i
, in contradiction with the fact that ! = 2 A i . This contradiction proves that assertion (i) of Claim 8 holds, and completes the proof.

The joint resuls of Claims 4, 5 and 8 yield the following De…nition.

De…nition 6 Given a class structure of payo¤s and beliefs, [V; ( o i )], and price, q 2 R J , we let, for each i 2 I, fP n i g n2N and P i := lim n!1 P n i , be, respectively, the set sequence and the …nal expectation set, de…ned above by induction. The information revealed by price q to the generic i th agent along the no-arbitrage principle is the (possibly empty) set of beliefs, denoted by o (V; o i ; q) 2 ? [ CB, whose support is P i . This set coincides with the information set, o (V; o i ; q), revealed by price q of De…nition 4. If q 2 R J is a no-arbitrage price, the re…nement revealed by price q along the no-arbitrage principle is the class structure, o i ; q)), revealed by price q of De…nition 5.

The meaning of the no-arbitrage principle of De…nition 6 is now clear. Agents having no clue of how market prices are determined can still update their beliefs from observing current prices so as to rule out arbitrage. They will, in any case, reach their …nal update after a …nite number of inference steps. As long as markets have not reached a no-arbitrage price, traders cannnot agree on prices and a sequential equilibrium may not exist. Claims 5 and 8 above show, jointly, that agents have common expectations -a necessary condition for sequential equilibrium to existif, and only if, the asset price is a no-arbitrage price (and such prices exist).

Along the same Claims, once markets have reached a no-arbitrage price, decentralized agents, who update their beliefs from the no-arbitrage principle, may infer a re…nement, agree on current prices, and share common forecasts. From Claim 5, 7 and 8, whenever the no-arbitrage price is self-attainable, the re…nement inferred is self-attainable. We now show that a similar re…nement process is possible without observing such a price. Agents may infer information from trade opportunities on markets, in a way which leads to preclude arbitrage, after …nitely many inference steps. Then, agents infer the coarsest arbitrage-free re…nement of Claim 3, which may also be revealed, along Claim 6, by any of its common no-arbitrage prices. Let fA n g n2N and fP n g n2N , be de…ned from above. Then, the following assertions hold:

A n+1 i 0 := f! 2 P n i 0 : 9(z i ) 2 (R J ) m ; P m i=1 z i =0; -q z i >0; V (!) z i 0 >0; V (! i ) z i >0; 8(i; ! i ) 2 I P n i g 22
(i) 9N 2 N : 8n > N; A n = ? and P n = P N ;

(ii) P N = lim n!1 P n = P , along assertion (i). We show, …rst, by induction on n 2 N, that P i P n i , for every pair (i; n) 2 I N. Let i 0 2 I be given. From Claim 3, the relation P i 0 P 0 i 0 holds for n = 0. Assume, now, by contraposition, that, for some n 2 N, P i 0 P n i 0 and P i 0 * P n+1 i 0

Proof Let (

. Then, there exist

! 2 P i 0 \ A n+1 i 0
and (z i ) 2 (R J ) m , such that P m i=1 z i = 0, V (!) z i 0 > 0 and V (! i ) z i > 0, for every (i; ! i ) 2 I P i I P n i , which (from Claim 2) contradicts Claim 3, along which [V; ( First, assume that assertion (i) of Claim 10 holds and let N 2 N satisfy that condition. Then, P N = P , and the re…nement, ( o i ), is q-arbitrage-free (since is, for each i 2 I, the orthogonal complement of Z on i := fz 2 R J : V (!) z = 0; 8! 2 P n i g.

Since fP n g n2N is decreasing, the sequence of vector spaces, f m i=1 Z on i g, is nondecreasing in (R J ) m , hence, stationary. We let N 2 N be such that Z n = Z N , for every n > N , and denote simply Z := Z N and Z o := m i=1 Z oN i . To simplify, we will assume, non restrictively, that i n = 1, for every n 2 N. Since P 0 1 Z is compact, we may assume there exists (! ; (z i )) = lim n!1 (! n 1 ; (z n i )) 2 P 0 1 Z. We recall that Z and Z o are orthogonal and notice that Z o = m i=1 fz 2 R J : V (!) z = 0; 8! 2 P i g.

The above relations on (! n 1 ; (z n i )), for each n 2 N, and the continuity of V and of the scalar product imply that V (!) z i > 0, for every pair (i; !) 2 I P i , with one strict inequality, since (z i ) 2 Z, is orthogonal to Z o and such that k(z i )k = 1. Hence, there exists (i; ! ) 2 I P i , such that V (! ) z i > 0. By construction, P i := \ n2N P n i is disjoint from

A i := [ n2N A n i . Hence, ! = 2 A i .
From the de…nition of P m i=1 Z on i and above, for each n 2 N, there exist (attainable) portfolios, (z n i ) 2 R JI , such that P m i=1 z i = 0 and V (! i ) z n i = V (! i ) z n i > 0, for every (i; ! i ) 2 I P n i . From the continuity of the scalar product, the above relations, V (! ) z i > 0 and z i = lim z n i , imply, for n 2 N large enough, V (! ) z n i = V (! ) z n i > 0, with ! 2 P n i , hence, ! 2 A n+1 i , from the latter relations on (z n i ). This contradicts the fact that ! =

(a) 9 " 2 R

 92 ++ : P ( ) M " . We denote by B the set of all beliefs. A belief 0 2 B is said to re…ne 2 B and we denote it by 0 , if the following Condition holds:(b) P ( 0 ) P ( ).

8

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.53

  is a sequential equilibrium (respectively, a temporary equilibrium) of the economy E, or correct foresight equilibrium (C.F.E.), if the following Conditions (a)-(b)-(c)-(d) (resp., Conditions (b)-(c)-(d)) hold:

Remark 2

 2 Along De…nition 2, a C.F.E. is reached when agents forecast prices correctly (Condition (a)), make optimal decisions at t = 0 (Condition (b)), which clear on all markets at all dates (Conditions (c)-(d)), and is, indeed, a sequential equilibrium. At t = 1, agents never face bankruptcy and have no incentive to exchange on the spot market. Under perfect foresight, the above sequential equilibrium concept coincides with Cornet-de Boisde¤re's (2002, De…nition 2.3, p. 399).

De…nition 3

 3 Let a class structure of payo¤s and beliefs, [V; ( o i )], a class of beliefs, o 2 CB, a representative belief,

2

 2

! 2 P

 2 ( ) = P ( o ), with at least one strict inequality; 11 Documents de Travail du Centre d'Economie de la Sorbonne -2012.53 (b) there exists a continuous mapping : P

13Claim 3

 3 Documents de Travail du Centre d'Economie de la Sorbonne -2012.53 3.2 Individual beliefs revealed by prices First, we show a structure of beliefs admits a coarsest arbitrage-free re…nement. Let a class structure of payo¤s and beliefs, [(V; ( o i )], be given. Then, there exists a unique coarsest arbitrage-free re…nement of ( o i ) 2 CSB, namely, a re…newhich satis…es the two following Conditions:

i 2 I

 2 , P i := [ k2K P k i P ( o i ) be a compact set, and o i 2 CB be the class of beliefs with support P ( o i ) = P i . Then, by construction, (

o i 2

 2 CB, (partially) ordered by the relation of De…nition 1. Then, either R ( o i;q) = ?, and we let o (V; o i ; q) = ?, or R ( o i;q) 6 = ?, which we henceforth assume. Along De…nition 1, let f o k i g k2K be a chain in R ( o i;q) , fP k i g := fP ( o k i )g k2K be the related chain of supports, P i := [ k2K P k i P ( o i ) be compact, and o i 2 CB be the class of beliefs with support P ( o i ) = P i . By construction, all k 2 K.

De…nition 4

 4 meets Conditions (ii) and (iii) of Claim 4. The last part of Claim 4 results from De…nition 3 and from above. Let a class structure of payo¤s and beliefs, [V; ( o i )], and a price, q 2 R J , be given. The information (set) revealed by price q to the generic i th agent is the (possibly empty) set, o (V; o i ; q) 2 ? [ CB, of Claim 4, also denoted by o

Claim 6

 6 Let ( o i ) 2 CSB be given. Along Claim 3 and De…nition 5, let o

  )) P i . Along De…nition 1, we let o i 2 CB be the (unique) re…nement ofo i , de…ned by P ( o i ) = P i , and consider two cases.First, we assume that assertion (i) of Claim 8 holds and let N 2 N be such thatA N +1 i = ?.Then, by construction, P N i = P i , and o i o i is q-arbitrage-free (sinceA N +1 i = ?), which implies, from Claim 4, )), and, from above, assertions (i)-(ii) of Claim 8 hold again.

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  (

4. 2

 2 Sequential re…nement through trade Throughout, we let a class structure, [V; ( o i )], be given. We study how agents may narrow down in steps their expectation sets from observing exchange opportunities on …nancial markets. As above, we de…ne, by induction on n 2 N, two sequences, fA n g n2N := f m i=1 A n i g n2N and fP n g n2N := f m i=1 P n i g n2N , of sub-sets of (? [ M) m : we let A 0 i = ? and P 0 i := P ( o i ), for each i 2 I, and A 0 := m i=1 A 0 i and P 0 := m i=1 P 0 i ; with A n := m i=1 A n i and P n := m i=1 P n i de…ned at step n 2 N, we let, for each i 0 2 I:

n+1 i 0 := P n i 0 n A n+1 i 0 ,) 2 CSB

 002 Documents de Travail du Centre d'Economie de la Sorbonne -2012.53 P i.e., agents rule out expectations, granting an arbitrage, be given. We will denote simply ( the coarsest arbitrage-free re…nement of ( o i ) along Claim 3, and let, for each i 2 I, P i := P ( o i ) be the support of o i 2 CB, and P := m i=1 P i .

o i ) 2

 2 CSB be given, and let( o i ), P := m i=1 P i := m i=1 P ( o i ), fA n g n2N := f m i=1 A n i g n2Nand fP n g n2N := f m i=1 P n i g n2N be de…ned as in Claim 10 and denote P := m i=1 P i := \ m i=1 P n = lim n!1 & P n .

  o i )] is arbitrage-free. Hence, the relation P i P n i holds for every pair (i; n) 2 I N, which implies that P := m i=1 P i := \ m i=1 P n is a compact set, such that P := m i=1 P i P . Let

  be the re…nement of ( o i ), de…ned by P ( o i ) = P i , for each i 2 I. From above, now consider two cases.

  Two beliefs, ( ; 0) 2 B 2 , are said to be equivalent, and we denote it by 0

								, if both
	relations 0	and	0 hold, and we let	o := f 2 B :	g be their equivalence
	class. We denote by CB := f	o	: 2 Bg the set of classes, forming a partition, of B,
	and by P (	o ) the expectation support of any class	o 2 CB, namely, the set of anticipa-
	tions, P (	o ) := P ( ), which is common to all beliefs 2	o , and which characterizes	o .
	We say that a class,	o 0 2 CB, re…nes	o 2 CB, and denote it by	o 0	o , if P (	o 0) P (	o ).
	A collection of beliefs, ( i ) 2 B m , is called a structure (of beliefs), and we denote
	it by ( i ) 2 SB, if the following Condition holds:
	(c) \ m i=1 P (

i ) 6 = ?:

  CSB be a given class structure, and let R ( o i) be the set of arbitrage-free re…nements of ( k2K , where K is a non-empty set, such that for every pair (

		o	i ), along Remark 4. A chain in R ( o	i) is a totally ordered subset,
	say f(	o k i )g

o i ), (partially) ordered by the relation . This set, R ( o i)

, is non-empty, for it contains the symmetric self-attainable re…nement of (

  contradiction with the above equalities. This contradiction yields

		o	i 2 R ( o	i;q) , hence, from above,	o	i is an upper bound of the chain f	o k i g k2K in
	R ( o	i;q) . From Zorn's Lemma (Aliprantis-Border, 1999, p. 14), R ( o	i;q) has a maximal
	element, i.e., a re…nement,

o (V;

  CSB, the latter assertions, which hold and are written for each i 2 I, imply that the collection (

	the proof of Claim 4, at this stage (R ( o	i;q)	6 = ?) and in all subsequent steps, we let
	the reader check that the set	o	(V;	o	i ; q) (of De…nition 4) is non-empty and meets
	Conditions (i)-(ii)-(iii) of Claim 4. The latter Conditions imply that, for each i 2 I,
	o	(V;	o	i ; q) is the unique coarsest q-arbitrage-free re…nement of	o	i , hence, from above,
	that		o	i	o	(V;	o	i ; q)	o	i . Since (	o	i ) 2
												o	i is non-empty. Resuming exactly

o (V;

  With a non-restrictive convention that the empty set is included in any other set, we show, …rst, by induction on n 2 N, that P (

											o	(	o	i ; q))	P n i , for every n 2 N.
	Indeed, from Claim 4, the relation P (	o	(	o	i ; q)) P 0 i holds for n = 0. Assume, now, by
	contraposition, that, for some n 2 N, P (	o	(	o	i ; q)) P n i and P (	o	(	o	i ; q)) * P n+1 i	. Then,
	there exist ! 2 P (
											8! 2 P n i g;
	P n+1 i	:= P n i n A n+1 i	, i.e., the agent rules out expectations, granting an arbitrage.
	Claim 8 Let (	o	i ) 2 CSB and q 2 R J be given. Let	o	(	o	i ; q) be the set of De…nition 4
	and fA n i g n2N and fP n i g n2N , be de…ned from above. Let P (	o	(	o	i ; q)) be empty, if	o	(	o	i ; q)
	is empty, and the support of	o	(	o	i ; q) 2 CB otherwise. The following assertions hold:
	(i) 9N 2 N : 8n > N; A n i = ? and P n i = P N i ;
	(ii) P N i = lim P n i = P (	o	(	o	i ; q)), along assertion (i).
	Proof Let (								o	i ; q)), fA n i g n2N
	and fP n i g n2N be de…ned as in Claim 8 and denote P i := \ m i=1 P n i = lim n!1 & P n i .

o i ) 2 CSB and q 2 R J be given, and let o ( o i ; q), P ( o ( 19 Documents de Travail du Centre d'Economie de la Sorbonne -2012.53 o

  Documents de Travail du Centre d'Economie de la Sorbonne -2012.53for each i 2 I), which implies, from Claim 3, that , that is, assertions (i) and (ii) of Claim 10 hold.Second, assume, by contraposition, that assertion (i) of Claim 10 fails, that is:8n 2 N; 9(! n in ; (z n i )) 2 P n in Z n : V (! n in ) z n in > 0 and V (! i ) z n i > 0; 8(i; ! i ) 2 I P n i ,whereZ n := f(z i ) 2 (R J ) m :

					o	i	o	i , hence, from above, that
	o	i =	o	i P m i=1 z i 2	P m i=1 Z on i ; k(z i )k = 1; (z i ) 2 m i=1 hZ on i i ? g, in which hZ on i i ?

A N +1 i = ?,

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Documents de Travail du Centre d'Economie de la Sorbonne -2012.53

Documents de Travail du Centre d'Economie de la Sorbonne -2012.53

A i . Hence, assertion (i) holds, which completes the proof. 24 Documents de Travail du Centre d'Economie de la Sorbonne -2012.53

We have now extended to the in…nite dimensional economy, E, the basic arbitrage properties of the Cornet-de Boisde¤re model (2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009). Proving the extended properties was a perequisite to study existence of the C.F.E. in the general setting, where agents are, both, asymmetrically informed about the future, and endowed with private idiosyncratic beliefs on forthcoming prices.

In the traditional perfect foresight model, we are used to deal with locally isolated predictable equilibria, whose price is common knowledge. When consumers are prone to uncertainty between several private forecasts, a single prediction is no longer possible. Then, the set of sequential equilibria is easily shown to be typically uncountable, with di¤erent equilibria related to di¤erent possible private beliefs.

Such an indeterminacy leads to de…ne a minimum uncertainty set, as the set of all prices, which might prevail as a sequential equilibrium price tomorrow, for some structure of beliefs today. When beliefs are private or prone to change, any such price is, a priori, possible. We already know that this set is non-empty in a standard economy with purely …nancial markets, since this economy admits a perfect foresigth (possibly price-revealed) equilibrium (see De Boisde¤re, 2007). In a companion paper, we will extend this result to all types of …nancial markets.

On the one hand, it is clear that agents endowed with private beliefs need include the minimum uncertainty set into their expectation sets, if their forecasts are to be correct (along De…nition 2 above). On the other hand, we will show a converse result. Namely, that a standard economy, where agents' expectations embed that set, always admits a correct foresight (possibly price-revealed) equilibrium, whatever the structure of payo¤s and beliefs of the economy. In particular, a correct foresight equilibrium exists in the standard cases, following [START_REF] Hart | On the existence of equilibrium in a securities model[END_REF] and [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF], where the perfect foresight and the rational expectation equilibria fail to exist.