Carlo Benetti

Christian Bidard

Edith Klimovsky

Antoine Rebeyrol
email: antoine.rebeyrol@u-paris10.fr

Reproduction and temporary disequilibrium: a Classical approach

Keywords: Classical Reproduction, Market prices, Disequilibrium, Growth, Cycle Classification: E11, E30, E32, O41

We build a bisector reproduction model with classical features in which the capitalists aim at maximizing accumulation of their profits. At variance with gravitation models, it is assumed that they invest their profits in their own industry. Their plans are based on actual productions and expected prices. Effective prices and effective allocations of resources are determined by a market-clearing mechanism. A simple law on the formation of expectations allows us to define the dynamics of disequilibria, which let appear endogenous self-sustained fluctuations, around a long-run path. The long-run rate of growth and the amplitude of the fluctuations depend on the initial conditions.

I. Introduction

It is often considered that the study of equilibria is the only solid scope for economic theory. Our approach is different: we study the reproduction of a bisector economy conceived as a simplified representation of an economy in which individual decisions are irreversible and taken independently from each other. As a consequence the analysis is centered on temporary market disequilibrium. In our model, the agents are capitalists who aim at accumulating capital. Given their expectations at a given date, they make up plans which, in general, are globally inconsistent.

The market is not a tâtonnement process ensuring the instantaneous adjustment of individual plans, but a device which reveals their incompatibility. Such private economic decisions are thus submitted to a social evaluation which shows itself in differences between expected and effective profits and allocation. The model proposes a formalization of this feature of a market economy by introducing a market mechanism which determines market prices, the actual rates of profit and the final allocations of resources among the sectors, as a result of a set of individual decisions.

The independent producers must adapt their productions to circumstances they had not foreseen at the beginning of the period. Then a new period opens and the economy moves from a temporary disequilibrium state to another one. A simple hypothesis on the formation of expectations allows us to link a period to the next and gives its dynamic structure to the model. It is only through a sequence of temporary or market disequilibria that a full or long-period equilibrium may be reached.

Note that in our model profits are accumulated in their own sector instead of moving from a sector to another according to their relative profitability, as in the Classical tradition of gravitation. A historical root of that idea is found in some parts of Torrens"s and Marx"s works, such as Torrens"s "Principles of Demand and Supply" (1821, chap.VI, section VI) and Marx"s extended reproduction scheme (1885, Book II, chap. XXI). Torrens is a little-known but important author who emphasized the physical constraints on the reproduction of capital if the whole profits are accumulated in their original own sectors. His main ideas on value and distribution have been set out by De Vivo (1986). More recently, Benetti et alii (2007) studied the dynamics of a classical disequilibrium model when a part of profits is unproductively used. In the present paper, we propose a new approach centered on a market mechanism determining the relevant disequilibrium variables on which our notion of temporary disequilibrium is based.

In section II, we introduce the model, study its properties and define the equilibrium and disequilibrium concepts. Section III examines the dynamics by decomposing them into a long-run tendency and short-run cycles. In section IV, we discuss the market mechanism we have retained and compare our results with those obtained when the price rule is a Walrasian tâtonnement.

II. Temporary disequilibrium in a reproduction model

We consider an economy where capitalists are the only economic agents: workers do not appear explicitly. The economy is made of two sectors. In each sector, there is a unique method with constant returns, which makes use of both types of inputs (the inputs include an exogenously given wage basket). All capital is circulating. When the accumulation of goods is economically impossible, capitalists may get rid of them without cost, for instance by consumption. Since we assume that all capitalists have the same price expectations, the producers in the same sector can be aggregated and everything proceeds as if there were only one capitalist producing good 1 and another good 2.

The model has a sequential three-step structure. Production takes one period. Depending on price expectations at date t (end of period t-1 and beginning of period t), a part of the quantities produced during the previous period is brought to the market. Exchanges occur at t and determine market prices and the allocation of inputs. The effective production of period t is then determined.

II.1. Producers' plans

Let us start with the determination of the ith producer"s plan (i = 1, 2), when he knows the quantity i q obtained as a result of his previous investment and, before the opening of the market, expects a relative price / e e e ij i j p p p for his product at that date (for the moment, the price expectations are considered as exogenously given). His objective being maximum accumulation, the producer retains a part of his product in order to invest it and brings the remaining part to the market in order to exchange it for the complementary input j. He thus aims at converting his product into a basket of inputs which fits the proportions required by his method of production, the expected level of production e i q being determined by his technique and his budget constraint at the expected prices. Let ii a and ij a the quantities of inputs i and j necessary to produce one unit of good i (these production coefficients are assumed to be positive). The expected budget constraint being written

e i i e j ij e i ii e i p q p a p a q ) (
, the production plan can be expressed as a function of the previous production and the expected relative price:

i e i e ii ij ji q q a a p (1)
The quantity i s of commodity i brought to the market is the difference between the previous production and the quantity of input i required by the production plan:

e i i ii i
s q a q . The producers of the other sector have a similar behaviour. Thanks to the aggregation of all producers belonging to the same sector, the economy can be represented in an Edgeworth box (Figure 1). The origin of the axes is the south-west corner for sector 1 and the north-east corner for sector 2, and the sides of the box are the quantities 1 q and 2 q . The initial endowments are represented by point I at the south-east corner. The production methods are respectively represented by the line OE with slope / aa (for that of good 2). For an expected price vector e p , the expected budget constraint is orthogonal to e p . The production plans are represented by point B for the first producer and point C for the second. Supplies

II.2. Market mechanism and production

Flukes apart, the capitalists' plans are not compatible (the points B and C are distinct) and cannot be both implemented. According to the market clearing mechanism we introduce, inspired from [START_REF] Cantillon | Essai sur la nature du commerce en général[END_REF]see Benetti and Cartelier, 2001), the market price is established through the exchange of the supplied quantities:

e j j jj j ij e i i ii i s q a q p s q a q
(2)

In Figure 1, the relative price is the slope of segment IA and the effective exchange is represented by the move from point I to point A. A comparison of the expected and market prices leads to the property:

e ij ij j j i i p p s d d s (3)
The first equivalence stems from the equality / ij j i p s s and the expected budget constraint e j i ij d s p , and the second from the Walras law ( ) ( ) 0

e i i ij j j d s p d s .
Let us dub 'scarce' the commodity in excess demand (good 1 in the case of Figure 1) and 'superabundant' that in excess supply. The scarce commodity (denoted by r for "rare") is therefore the one that the producers underevaluated. That commodity is entirely accumulated. The producer of the scarce good gets windfall profits, since its effective price is higher than expected:

he obtains more of the other good than expected. Nevertheless he cannot accumulate it because of the lack of his own input. After having got rid of the quantity AB of the superabundant good (for instance by consuming it), the producer of the scarce commodity implements his investment plan at B, therefore the effective rate of growth (or accumulation) in that sector is equal to both the expected rate of growth and the expected rate of profit ( . Note that both producers get rid of the same good, so that the definitions of scarcity and superabundancy are consistent with the effective uses of the goods. In this economy in temporary disequilibrium, each agent must withdraw a part of the same superabundant good from accumulation: were the market opened again, both producers would demand the same input and no exchange would take place.

Nevertheless the outcome of the market is an efficient input allocation (A in Figure 1):2 the production of a sector could only be improved by obtaining some quantity of the scarce commodity at the expense of the other sector.

The effectively produced quantities are determined by the inputs owned by the producers after the exchange: min / , / e i i ii ii j ij q q a a s a , or:

min , e j jj j e ii ij q a q qq a (4)

In that equality, let us replace the expected production by its expression as a function of past production and expected prices, as given by (1). The effective production is then defined as a function of initial quantities and expected prices:

min , ji j i i ee ii ij ji ij ji jj ji aq q q a a p a a a p (5)
Similarly, starting from given quantities and price expectations, the effective market prices are

1 The rate of profit π is calculated as a value ratio between net production and inputs at current prices, i.e. at their

replacement cost: 1 (1 )( ). i ii ij ji a a p
This rate measures the capital reproduction capability.

obtained by eliminating e i q and e j q between relationships (1) and ( 2), namely:

e ji j ii ij ij ij e ij i
jj ji ji a q a p a p a q a p a (6)

II.3. A remarkable property

We know that the producer of the scarce commodity implements his production plan and that this commodity is entirely accumulated. The first condition implies that the producer of the scarce commodity brings to the market a quantity r s which has the same expected value as the quantity of input a he plans to invest and does invest: ee r r r ra a s p q a p ; the second condition implies that the quantity r s is actually invested by the producer of the superabundant good: r a ar s q a . The equality ee a ar r r ra a q a p q a p follows. In Figure 1, the segment IB makes visible that the (partial) use of the superabundant commodity available in sector r (i.e., the quantity 1 12 2 q a d ) has the same expected value as the (full) use of the scarce commodity in sector a (i.e., the quantity 2 21 1 q a s ). As the scarce and the superabundant goods play symmetric roles in this equality, it can also be written:

ji i e ij j ij a q p qa (7)
This equality might also have been found by direct computation from (5). It shows that the proportion between the effective productions only depend on expected prices and not on the past activity levels: the economic intuition is that the share of the existing quantity of the scarce commodity between the sectors is determined by its producer only on the basis of the expected price, and determines the ongoing productions. 3 The relationship between the relative activity level and the expected price is linear and increasing, therefore a rise of the expected price leads to a proportional rise of the relative production of the corresponding commodity. The study of the dynamics will shed light on the strength and the limits of this market regulation.

II.4. Equilibrium

Let a full (or long-period) equilibrium of the economy be defined by the implementation of two conditions: (i) market temporary equilibrium, that is the equality between demand and supply at the expected price before the opening of the current market; and (ii) reproduction equilibrium, that is the equality between the two rates of accumulation.

Market equilibrium presumes the equality e ij ij pp , as shown by relations (3). No commodity is then under-or over-evaluated, there is no scarce or superabundant commodity, and all plans for exchange and production are implemented. Both arguments of the min functions are equal. In Figure 1, this condition occurs if e p is orthogonal to segment IT. Note that market equilibrium holds only by a sheer fluke. At variance with the Hicksian temporary equilibrium [START_REF] Hicks | Value and Capital, 2d edition[END_REF], it is not the result of an underlying stable adjustment process, and it is not maintained from a period to the next. The dynamics of our model are those of a sequence of temporary disequilibria.

Reproduction equilibrium is defined by // i j i j q q q q . Let us set 1 2 12 / p p p p and 12 / q q q . At full equilibrium, both equalities e pp and qq prevail. Then, by eliminating p between ( 6) and ( 7), it turns out that the full equilibrium proportion q is defined by: 11 21 12 22 a q a q a q a (8)

This formula shows that the full equilibrium row vector ( ,1) q , or 12 ( , ) qq as well, is the positive Perron-Frobenius eigenvector of the matrix ij a

A

. Let 1 be the dominant root of A, which is smaller than 1 if the economy is productive. In equilibrium both goods are entirely accumulated and the uniform rate of growth and accumulation is 1 1 1: the economy is on a von Neumann maximum growth path. In that case the rate of profit is uniform and equal to the maximum growth rate of the economy, and the market pricesthe column positive Perron-Frobenius eigenvector of Acoincide with the classical prices of production.

Point T in Figure 1 is named after Torrens: the historical reference to our notion of equilibrium is found in Torrens"s numerical model, in which all profits are accumulated and enduring reproduction goes on only when Torrens"s "good proportions", which are those defined by equality (8), are met.

III. Dynamics of disequilibria

A hypothesis on the making of the expectations will allow us to study the dynamics of disequilibria. For simplicity, we assume static expectations: the expected price at some date is the last known market price e pp (9)

Then the market prices depend on the produced quantities and the past prices (formulas ( 6) and ( 9)) and the same for the quantities produced (formulas ( 5) and ( 9)). Equality ( 7), which is written as:

p a a q ) / (
) 12 21 (10) shows that the dynamics of the relative price and the relative quantity are basically identical up to a one-period lag. It therefore suffices to study the evolution of quantities. We shall examine in the first place the long-term tendency of the quantities, and then the fluctuations of their proportion.

III.1. Long-run growth

Out of equilibrium the evolution of quantities is irregular. We proceed to computer simulations to study their long-run dynamics. These experiments show that the average rate of growth is roughly the same for both sectors in the medium run (let us say over 10 periods) and is approximately constant afterwards. When the initial distance to the equilibrium proportion increases, the longrun rate of growth decreases (in the long-run, no sector can grow faster than the other because its expansion would ultimately require to absorb more than the gross product obtained by the other).

Figure 2 plots the long-run ratecomputed over 1000 periodsagainst the initial value of relative quantity (0) q (for simplicity, we have chosen (0) pp and 1 q

).

Figure 2

The rate of growth reaches its (von Neumann) maximum value for q q ) 0 ( and decreases with the distance between (0) q and q . It may become negative, though the economy is productive, because the superabundant commodity is partly 'spoilt' at each date: the dynamics of disequilibrium imply the exclusion of some commodities from accumulation.

A noteworthy property of the model is that the hysteresis effects concern the long-run rates and not only the levels of the growth paths: the transitory shocks of productivity, what the Classicals named "a good or bad crop", have permanent effects on the long-run rates of growth.

III.2. Cycles

The study of cycles is based on the observation of the relative quantity. The elimination of the relative price from the one-step induction formulas ( 6), ( 9) and ( 10) allows us to obtain the twostep induction formula governing the evolution of the relative quantity:

1 11 1 21

2 12 1 22 tt t tt q a q a q q a q a (11)

In case of uniform composition of capital ( xx : the sequence of relative quantities is periodic of period six. So is the sequence of relative prices, and the long-run rate of growth is then equal to the average rate of growth over six consecutive periods. Outside this simple case, we study first the fluctuations in a neighborhood of the equilibrium, then in the large.

III.2.1. Local fluctuations

The fixed point of the dynamics ( 11) is the unique positive solution q of (8). The standard method to identify the nature of the dynamics in a neighbourhood of the fixed point consists in linearizing the dynamical equation 12 ( , ) t t t q h q q and writing it as 1 (det )/( ) h a q a A . Let 1 and 2 be the dominant and second eigenvalue of matrix A. Since ( ,1) q is the positive row-Perron-Frobenius eigenvector, we have

1 1 2 2 t t t X h X h X ,
1 12 22
a q a , hence

2 1 1 2 1 1/ h
. As 21 , it is convenient to introduce the scalar defined by:

2 1 2 cos 1 ( 0, ) 2 (12)
The linearized dynamics obey the law:

12 2 cos 0 t t t X X X ( 13
)
The characteristic polynomial 2 2 cos 1 XX has two complex roots i e of modulus one, which is unfortunately the case in which linearization cannot conclude to the local convergence or divergence.

Let 1 ( , )

t t t
M q q and ( , ) M q q . A more precise study of the local dynamics makes us introduce the quadratic form 2 defined by: t MM . This means that the distance to M remains constant from one point to the next, therefore, up to first order approximations, the trajectory of the point

t M is such that 0 ( , ) t MM
, where 0 is the initial distance. The curve defined by this equation is an ellipse: its centre is the fixed point M and its main axis the 45-degree line. The economic interpretation is that the relative quantities t q fluctuate around a central value q . Appendix I shows that the period only depends on the technical coefficients (of course, the amplitude depends also on initial conditions). In general, the period is not an integer. The case of uniform composition of capital serves as a benchmark and a divide between quicker and smaller fluctuations on the one hand, or slower and larger fluctuations on the other hand.

Because these calculations rely on a first order approximation, they do not prove the cyclical nature of the relative quantities: the points t M might actually move on a spiral with a slight inwards or outwards deviation at each loop. Then the approximation by an ellipse holds for a certain time in a neighbourhood of equilibrium but is not admissible globally.

III.2.2. Global fluctuations

Apart the case of uniform composition of capital, the study of the global dynamics is complex because the dynamics are not linear. Let us plot the points 1 ( , )

t t t
M q q and proceed to computer simulations for an arbitrary starting point 0 M . It turns out that these points move on a closed curve. Figure 3 represents three such curves, which depend on the initial condition (500 points plotted on each curve). These curves are fitted into each other and gradually lose their initial ellipse shape as they wander away from equilibrium. In Figure 4, we have drawn the segments 1 tt MM , and the fact their envelope is smooth shows that the movement of t M on its curve is regular. The observed regularities are the expression of cyclical movements in the relative activity level, and similarly for the relative price. An important property of the formula ( 11) is its temporal symmetry, as it is written 21 () t t t q q f q , or 11 ()

t t t
q q f q :

1 t q and 1 t q play symmetric roles for a given value of t q . Time can be reversed in a strong sense as the future and the past are defined by the same formulas. This temporal symmetry implies the geometrical symmetry of the curves with respect to the 45-degree line, a property illustrated by Figure 3. That property excludes the slightly inwards and outwards spirals as possible outcomes, the existence of which has been suggested in the previous subsection. Therefore there are strong reasons to think that the dynamics are those of a self-sustained cycle. Appendix II goes into some working tracks concerning the property that the points t M move on a closed curve.

The economic interpretation of the clockwise move of the point t M on its curve in Figure 3 is as follows. By relation (10), 1 t q is equal to t p up to a constant factor so that, up to a linear deformation, Figure 3 can be seen as plotted in the plane ( , tt qp ). To draw the phase diagram in this plane, we consider the curves 0 t q and 0 t p . Since

1 21 12 ( / ) t t t t t
q q q a a p q , the locus 0 t q is a straight line from the origin with slope 12 21 ( / ) aa (from an economic point of view, 0 t q means that market equilibrium prevails at t (because Its slope depends on the sign of the determinant. The arrows in Figure 6 result from the signs of t q and t p .

Figure 6

Above the line 0 t q , q is increasing because commodity 1 is scarce: it had been underevaluated before the opening of the market at date t ( 1 e t t t p p p ) and is completely invested, whereas good 2 is partly excluded from accumulation. The market regulation, expressed by the increasing relationship (6) between the proportion q and the expected price, is at work. The price may keep rising for a while, but sooner or later it will fall because high price at t means high expectation at 1 t , high activity plan in sector 1, high demand for good 2 and low supply of good 1 (because the producer 1 keeps a great quantity of his own product as input); and mutatis mutandis for the low activity plan in sector 2. As soon as the price has declined once, commodity 2 becomes the scarce good at the next period and is in excess demand. Then the same story holds the other way around, with commodity 2 being the scarce good during the other half of the cycle.

IV. Comparisons

IV.1. Disequilibrium prices

In our model, the agents are in disequilibrium and the baskets they buy differ from those they expected. The reason is that the mechanism retained for the market price formation presumes that the agents take irrevocable decisions before knowing the market prices. An obvious objection is that the customer knows the price of meat before he pays it to the butcher, and that is why he does not get more of it than the quantity he decided. At first sight, the opposite situation, which occurs in our model for the producer of the scarce commodity, might seem in contradiction with the idea of voluntary exchange. But the demand for meat is not meaningful: going out of the butcher's, the consumer may discover that chocolates are sold off and that he brings home more meat than he would have bought, had he been aware of that sale. [START_REF] Walras | Eléments d'économie politique pure[END_REF] taught us that the demand for meat depends as much on its price as on those of the other goods and, therefore, assumed that the decisions are taken on the basis of a perfect knowledge of the entire price vector. In modelling the working of the market, some aspects of reality are ignored while others are stressed: our basic hypothesis that the expenditure decisions are taken on the basis of (generally erroneous) expectations of prices is less strong than the one retained by the general equilibrium theory. It allows us to deal with disequilibrium, which is not the case for the Walrasian theory. Suppose indeed that the price e p is announced by an auctioneer and that, as in a Hahn process [START_REF] Hahn | A theorem of non-tatonnement stability[END_REF], transactions are effective at these prices. The producers of the scarce commodity would be in equilibrium and the quantity of the superabundant commodity not used in production (the segment BD in Figure 1) would remain in the hands of his producers. Then the produced quantities coincide with those we have determined but the prices at which the next transactions will take place are unknown. That is why the Walrasian theory must call on for an auctioneer who decides the price changes whereas, in our decentralized and perfect competition model, the price is determined (by equation ( 2)) as a function of the agents' decisions and independently of their wills. Other price formation rules might be taken into account, but the one we retain is simple and its origin can be traced back to Cantillon's (1734) analysis of market prices.

IV.2. A comparison with Walrasian tâtonnement

The differences between the dynamics generated by our price formation rule and the Walrasian tâtonnement can be put into light by comparing their consequences in the same situation. Since the treatment of complementarity is at stake here, let us first recall how the Walrasian theory deals with it and consider a two-good exchange economy with 2000 price-taking agents, all of them desiring one lump of sugar for one cup of coffee. The initial endowments are: 1000 agents hold 2 cups of coffee, 999 hold 2 sugar lumps, whereas the last agent has 3 sugar lumps. At the Walrasian equilibrium sugar is a free good, the 1000 agents who hold it do not get anything while the 1000 other agents can drink 2 cups of coffee.

Let us return to the economy examined in section II but replace our market mechanism by a Walrasian tâtonnement (future markets are not included in the model). e p is now interpreted as a price called by the auctioneer and taken by producers as a parameter for their calculations. In order to reach maximum accumulation, each producer maximizes his future production under the given constraints. Let i q be the available endowments (previously noted as kk , suppose for a moment that:

1 2 1 2 min , max , k k q k k (17)
where 12 / q q q . Then: ), and is also stable.

This case is represented in the Figure 1: equilibrium T is unstable but E and F are two more equilibria, with a zero price for commodity 2 at E and for commodity 1 at F (for instance at E, there is an excess supply for good 2 whereas good 1 is neither demanded nor supplied). As can be seen on the Figure, the tâtonnement process during the current period converges towards an equilibrium with one free commodity, which is either 1 or 2 according as the first called price is lower or higher than T p . At the next date, the sector producing the zero-price good cannot buy any amount of the other input and its production will be nil. During the following period, the other sector will be in the same situation, therefore the economy collapses in two periods.

A case in which a cyclical movement can be proved occurs when the starting point belongs to the 45° line ) ( 1 0 X X and the level of 0 X is chosen in order that two consecutive values t X and 1 t X are also equal for some t. Then the reversibility property implies that t periods later the initial point will be reached again, therefore the dynamics admit period 2t. tt ) the first derivative is positive and increasing as long as () x t x , therefore the function () xt reaches value

x and goes beyond it. A point such that 3 () x t

x and 3 ( ) 0 xt is reached. That position is qualitatively symmetric from the one considered in the first case we have examined and a similar argument applies: x(t) increases, goes up to a local maximum, then decreases to x and below.

The trajectory x(t) thus admits incessant fluctuations around x . Moreover, let us take the origin of time a date when a local extremum is reached. Then, if x(t) is a solution of the differential equation () x f x with the initial conditions (0) x and (0) 0 x , the same for () xt , therefore ( ) ( ) x t x t . The graph of x(t) is therefore symmetric with respect to any vertical axis corresponding to a local extremum. Since there are several local extrema and that two consecutive symmetries amount to a translation, the graph is also invariant some translation.

That translation defines a period of the solution x(t). The overall conclusion is that q(t) oscillates and admits a periodic movement around the long-run level q .

  Figure1

  the production of good 1) and the line O'F with slope 22 21

  superabundant good (denoted by a) receives less of the other commodity than expected and therefore cannot invest the amount of his own commodity he had put aside for production. He must revise his plan and downsize production from C to D by eliminating the quantity AD of the superabundant good, therefore e aaa g

Figure 3 Figure 4

 34 Figure 3

Figure 5 ,

 5 Figure5, which represents a simulation of t q over 1000 periods, is another significant aesthetical contribution to the study of the reproduction model, in that it suggests the existence of hidden regularities underlying complex moves.

Figure 5

 5 Figure 5

In Figure1the set of the efficient allocations is made of the two triangles OFT and O'ET.

The same idea can be found in Marx"s schemas of enlarged reproduction in which the total accumulation of means of production determines the growth path of the economy (1885, Book II, chap. XXI).

0,

is an equilibrium price, at which the resources are fully employed (in Figure 1, this price supports the equilibrium point T). For any other price p , the excess demand The economic interpretation of the condition 12 kk is that the inputs of each industry come more from the product of that industry itself than from the other: production is 'introverted'. The equilibrium at price T p is unstable, but the excess demand formula (19) also shows the existence of two 'hidden' equilibria with a free good. One of them corresponds to 0 p , i.e. If 12 k q k there is no equilibrium with a zero price, because the excess demand for such a commodity would be positive. The equilibrium at T is unique and globally stable. The tâtonnement within the period leads to the full utilization of the resources.

Let us study the dynamics of quantities across the sequence of successive equilibria. Let A be the matrix of technical coefficients and q the row-vector of the produced quantities. The activity levels q of the next period are such that q A q , i.e.

, where 1 y and 2 y are row eigenvectors of A. Since the root 2 with maximum modulus of 1 A is the inverse of the non dominant root of A, the vector

takes the direction of 2 y in the long run, and therefore has a negative component. There exists a minimum integer τ such that () q admits a negative component. In economic terms, the full utilization of resources at this date is impossible. At this date, the tâtonnement converges towards the only equilibrium, with a zero price for either good 1 (if

). The production of the free good vanishes at date 1 and the big crunch of the economy occurs at date 2 .

Summing up, the technical complementarities imply that, sooner or later, some commodities are excluded from accumulation: a planner who would aim at the full utilization of resources and would therefore select the allocation T at any date would meet the same difficulty. The Walrasian tâtonnement allows for the disposal of some good only if its price is zero: this will occur either immediately (

12 kk ) or after some time ( 12kk ), but the economy will then disappear. By contrast, the price formation rule we have considered allows for the partial disposal of the superabundant commodity even if its price is positive and the reproduction of the economy is possible at any date.

V. Conclusion

In the temporary disequilibrium model with Classical features we have dealt with, the conditions of production, the present quantities and the price expectations lead to a determination of market prices, quantities, rates of accumulation and profit. In disequilibrium no agent can accumulate the whole of his effective profit and some agents cannot implement their initial plans. The dynamics of our model exhibit endogenous self-sustained cycles around a long-term path. The level of the long-term rate of growth depends on the gap between the initial proportions and the equilibrium (or von Neumann) proportions. If the gap is significant, the amount of commodities discarded from accumulation is great, a phenomenon which generates lower long-term rates of growth and cycles with larger amplitudes. Since, in our formalization, no substitution is possible between inputs, the survival of the economy in disequilibrium requires a not-too-tough price mechanism.

By contrast with the Walrasian tâtonnement, the one we have even if its formulation is too simple and other market mechanisms are conceivable, allows for the determination of effective prices and quantities outside equilibrium. A natural extension of the model would consist in examining if the introduction of money into the market mechanism would help to improve the economic performances.

Appendix I: Local fluctuations

In a neighbourhood of the equilibrium, the local fluctuations are defined by the dynamic equation ( 13) where the scalar depends on the ratio between the first and the second eigenvalues of matrix A by formula (12). The solution of the equation ( 13) being a linear combination of i e and i e , the movement is periodic of period

The amplitude A is obtained as the difference between the extreme abscissas of the points of the ellipse (these points are those for which the tangent is vertical). Calculation gives: a a a a ), then detA is negative, 21 21 det / A is negative and defined by ( 12) is greater than /3: then the period is smaller than 6 and the amplitude is small. In intuitive terms, when the interindustrial relationships are strong, the transmission of fluctuations is rapid ( 6 T

) and the overall fluctuations small.

On the contrary, when the interdependencies are weak (

), the period is long and the amplitude large: the economy drifts far from its regular state, comes back slowly and drifts again in the opposite direction.

Appendix II: Global fluctuations

As an application of the temporal symmetry established in section III.2.2, an invariant expression 1 ( , ) , where C is a constant. This suggests that the invariant is symmetric:

An idea to prove the existence of a cycle would be to generalize the local calculation and look for an invariant of order n , for any n . That is, we set tt X q q and consider a polynomial of the type ( , ) bb . There are two difficulties with that method: first, the linear system defining the ij b coefficients is not always of the Cramer type; second, once a sequence ( , ) n P X Y of invariants has been identified, its convergence towards some XX any magnitude such as the average value of the whole sequence t X (or its minimum value, its period, etc). Such a magnitude X remains the same if the starting point was 1 ( , )

nn XX , and therefore X is itself invariant on the curve. This means that the average value X depends on the 01 ( , ) XX only through the value 01 ( , ) I X X . In that the expression I can be viewed as a 'universal formula'. Expressed the other way round, the property becomes: the knowledge of the function f which gives the centre of the curve as a function of its starting point ( 01 ( , ) X f X X ) suffices to know the equation of the curve itself, which is


Georges Prat
email: georges.prat@u-paris10.fr

EQUITY RISK PREMIUM AND TIME HORIZON : WHAT DO THE U.S. SECULAR DATA SAY?

Keywords: JEL classification : D81, D84, E44, G11, G12 equity risk premium, time horizon

An ex-ante equity risk premium is the difference between the expected return of a risky asset at time t for a given future time horizon and an equivalent maturity risk-free interest rate. Using annual US secular data from 1871 to 2008, this study aims to model simultaneously the measures and the explanations of ex-ante equity risk premia for two polar horizons: the one period ahead horizon (i.e. the "short term" premium) and the infinite time horizon (i.e. the "long term" premium). Expectations being represented by traditional adaptive processes, large disparities in the dynamics of the two premia are evidenced. According to the conditional CAPM, each premium is at time t explained by the product of the price of risk by the expected variance of returns, these two magnitudes being horizon dependant. The expected variances depend on the past values of the centered squared returns (we found 5 and 8 years for the one year and the infinite horizon, respectively). For each horizon, the price of risk is determined by a spread of interest rates capturing economic factors of uncertainty and by an unobservable variable determined according to the kalman filter methodology (i.e. a state variable). The state variables are supposed to capture the influence of hidden variables and of non directly measurable psychological effects. The model gives a valuable representation of the "short term" and "long term" premia.

Résumé -Une prime de risque ex-ante est définie par la différence entre la rentabilité anticipée à la date t d'un actif risqué pour un horizon de placement donné, et le taux d'intérêt d'un titre de créance sans risque ayant une maturité égale à cet horizon. En utilisant les données séculaires annuelles américaines sur la période 1871-2008, cet article a pour objectif de modéliser simultanément les mesures et les explications des primes de risque ex-ante des actions pour deux horizons pôlaires : l'horizon d'une période (« prime « de court terme ») et horizon infini (prime « de long terme »). Les anticipations étant représentées par des processus adaptatifs traditionnels, on constate des différences importantes entre ces deux primes au cours de la période. Suivant le MEDAF conditionnel, à la date t, chaque prime est expliquée par le prix du risque multiplié par la variance anticipée des rentabilités, ces grandeurs étant paramétrées par l'horizon. Les variances anticipées sont supposées dépendre des valeurs passées du carré des rentabilités centrées (on trouve une durée de 5 ans et 8 ans pour la prime de court terme et la prime de long terme, respectivement). Pour chaque horizon, le prix du risque est déterminé par un « spread » de taux d'intérêt capturant les facteurs économiques de l'incertitude et par une variable inobservable déterminée suivant la méthode du filtre de Kalman (i.e. une variable d'état). Les variables d'état sont supposées capturer à la fois les influences de variables cachées et d'effets psychologiques non directement mesurables. Le modèle donne une représentation valable des primes de court et de long terme.

3 the price that would prevail on a virtual market characterized by a given single horizon, the weigh for each virtual price in the effective market price being represented by the share of equities that investors wish to held according to the considered horizon. 4 [START_REF] Anufriev | Asset pricing model with heterogeneous investment horizons[END_REF] discussed the conditions of existence of a market equilibrium price in a multiple horizon framework and show that, under a suitable parameterization, the no-arbitrage market condition leads the fundamental value to be a stable fixed point. 5Using annual US secular data from 1871 to 2008, the present study aims to model exante equity risk premia for two polar horizons: the one period (year) ahead horizon (i.e. the "short term" premium) and the infinite time horizon (i.e. the "long term" premium). Regarding the literature and in addition to the length of the analyzed period, the new contribution of this paper is in several ways. First, the two polar horizons are modeled simultaneously in the CAPM framework. Second, the question of the measurement of ex-ante equity risk premia -implying the knowledge of how expectations are formed -and the question of their explanation are solved in the same model. Third, since the price of risk is not directly observable, it is represented using the Kalman filter methodology, while taking into account information contained in a spread of interest rates. The rest of this paper is organized as it follows. Part 2 presents a brief survey of the literature on market equity risk premia and shows why it seems relevant to consider ex-ante premia in place of ex-post premia, and why premia can be considered both as time varying and horizon-dependant phenomena. Part 3 presents the general theoretical framework of our approach based on the conditional CAPM, according to which each market premium is at time t the product of the price of risk by the expected According to any stock valuation model, three factors determine an equity value: the expected return, the riskless rate and the required risk premium measured by the difference between these two last magnitudes. The intuitive assumption underlying our work is that each horizon of investment is characterized by a particular combination of these factors, the stock market price being of course the starting common knowledge information for all agents whatever the horizon of investment. 4 Let * τ P the virtual price related to investors with an horizon τ and τ n the number of equities held by this class of investors. With a number h of independent horizons, the market clearing condition is for an equity priced P may be written as ∑ = = - variance of returns. Part 4 presents assumptions concerning the expected variances and the prices of risk determinations, which allow empirical analyses for the two horizons considered.

Part 5 presents the Kalman filter methodology used to estimate jointly the two premia, and gives the empirical results. Part 6 gives concluding remarks.

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The equity risk premia in the literature: surveying concepts,

approaches and empirical results

Any equity premium is defined by the difference between a given representation of the expected return of the risky asset at time t for a given time horizon and an equivalent horizon risk-free rate. Two kinds of premia are distinguished: the ex-post premium and the ex-ante one.

Unlike the ex-ante premium, the ex-post premium is deduced from the return observed over the future horizon considered, and not from the return expected over this horizon. Since investors cannot consider the ex-post premium to make their financial choices at time t, this magnitude cannot be regarded as a decision-making concept, unless the perfect foresight hypothesis holds, in which case it is clear that there is no risk premium, so that the ex-post excess return could not be viewed as a risk premium. Considering now the rational expectation hypothesis (REH), the so-called ex-post premium equals in fact the rational ex-ante premium plus a white noise representing the forecasting error which is of course unknown at time t. In this instance, the rational expected return and then the ex-ante premium remain unknown variables. Empirical analyses evidence that, because of excessively large error terms, the values of ex-post premia are almost often as negative as positive (among others, see [START_REF] Mpacko-Priso | La prime de risque des actions, Théories et Applications[END_REF]) and this is somewhat disconcerting and likely to generate severe econometric biases, in particular when errors are not white noises. Moreover, many studies in the literature use lagged predictors to forecast the excess equity returns (i.e. the ex-post premium): dividend yield, earnings price ratio, short-term interest rate, payout ratio, term and default spreads of interest rates, inflation rate, book-to-market ratio, consumption, wealth, etc. As a result, no robust predictors are found (Goyal andWelch 2003, 2007), confirming that the ex-post premium is probably more a countable observation than an operational concept. In fact, experts' expected returns derived from surveys convey systematic forecast errors [START_REF] Abou | A propos de la rationalité des anticipations boursières : quel niveau d'agrégation des opinions?[END_REF]) and are mainly driven by autoregressive processes [START_REF] Abou | Modelling stock price expectations: lessons from micro data[END_REF]), which confirm that modeling ex-ante or ex_post premia are two different subjects. [START_REF] Merton | Lifetime portfolio selection under uncertainty: the continuous-time case[END_REF], [START_REF] Samuelson | Lifetime portfolio selection by dynamic stochastic programming[END_REF] and Py (1973) early analysed the relation between the time life horizon and the portfolio risk, and showed that, if the financial market is efficient (i.e.

returns are not predictable), rational investors can behave myopically if their utility function exhibits constant relative risk aversion. 6 This implies that the required risk premium is not horizon dependant. However, if returns are predictable, this result does not prevail any more.

To illustrate this, [START_REF] Barberis | Investing for the long run when returns are predictable[END_REF] interestingly builds optimal portfolios made up of stocks and bonds quoted on the US market. The author shows that, taking into account predictable features of stock returns (actual returns depend on past values of dividend yield), the optimum is reached by 40% of stocks for a one-month time horizon and by 100% of stocks for a 10-year time horizon. When return are supposed to be unpredictable, the share of stocks is near 35% whatever the horizon. This result suggests that the horizon of investment intervenes strongly to determine the required equity risk premium. In fact, when returns are partially predictable on the basis of their past values and/or macroeconomic variables, agents do not require a unique risk premium but a set of premia scaled by the time horizon.7

So, it is not astonishing to find the existence of a term structure for ex-ante equity premia deduced from survey data revealing experts' stock price expectations. Such studies largely confirm that, despite common trends, substantial discrepancies characterize risk premia according to the time horizon of the investment.8

Concerning the studies relating to the long term view, the usual method to analyse the equity risk premium is to observe historical averages of the difference between stock market returns and a risk-free interest rate. At the theoretical level, this approach refers to the well-Following papers by [START_REF] Welch | Views of financial Economists on the equity premium and on professional controversies[END_REF][START_REF] Welch | The equity premium consensus forecast revisited[END_REF] and [START_REF] Prat | Une analyse de la dynamique des primes de risque des actions suivant l'horizon de placement[END_REF], Graham andHarvey (2001-2007) corporations. They found that, in contrast with the 10-year expected risk premium, the oneyear risk premium is highly erratic through time, ranging between 1.3 and 6.6% depending on the quarter surveyed. As a result, these studies strongly confirm that ex-ante premia appear to be both time-varying and horizon-dependent.

6 known debate about the so-called "equity premium puzzle": with reasonable preference parameters values (i.e. the risk aversion coefficient and the subjective discount rate), the theoretical risk premium inferred from the consumption asset-based general equilibrium model is far too low (about 1-2% a year) as against the observed market premium, which stand about 6-7% a year on average [START_REF] Mehra | The equity premium: a puzzle[END_REF]).9

Another approach to measure the long term premium is to base the value of the premium on the dividends discount model (DDM). Study by [START_REF] Harris | The market risk premium: expectational estimates using analysts' forecasts[END_REF] is based the Gordon formula according to which the long term premium equals the dividend yields plus the expected long term rate of growth in dividends minus the yield on long-term US government bonds as proxy of the risk-free rate. The five years ahead expected growth in earnings per share issued from surveys is supposed to approximate the expected growth in dividends, allowing to evaluate an ex-ante long term risk premium for US stocks (S&P 500) over the period 1982-98. The authors show strong evidence that this long term risk premium change over time and that a significant part of its dynamics may be explained by readily available forward-looking proxies for risk, as the spread of interest rates, the consumer confidence index reported by the Conference Board, the degree of discrepancy between financial analysts' forecasts, or the implicit volatility issued from options prices. But the average market risk premium is found to be 7.1% which joins the equity premium puzzle.

However, the period was not large enough to allow reliable conclusions. This is not the case of the paper by [START_REF] Fama | The equity premium[END_REF] who still inferred ex-ante premia on the US stock market (S&P index) from the DDM, using alternatively dividends and earnings per share. The authors infer the expected growth rate of dividends (or earnings) per share and the risk-free Study by [START_REF] Siegel | Perspectives on the Equity Risk Premium[END_REF] shows that the premium was substantially lower during the periods 1802-1870 (3.2%) and 1871-1925 (4.00%), while Ibbotson [START_REF] Associates | Stocks, Bonds, Bills, and Inflation, Valuation Edition[END_REF] consider that the relevant historical premium is 7.1% during the period 1926[START_REF] Siegel | Perspectives on the Equity Risk Premium[END_REF][START_REF] Dimson | Global Evidence on the risk premium[END_REF] report that premium was generally higher during the second half of the 20 th century. Overall, we can see that these estimations of equity premium are particularly widespread according to the averaged period, which led [START_REF] Shiller | Irrational exuberance[END_REF] to point out that "the future will not necessarily be like the past " and Fernandez (2006, p.12) (earnings). This hypothesis seems to be confirmed by semi-annual S&P industrial expectations 6 and 12-month ahead carried out by J. Livingston' surveys on a panel of experts. Abou and Prat (2010) calculate indeed risk premia over the period 1952 to 1993 using these data and showed that their mean values are around 2.3% per year. This result joins in another way the Fama and French results, and then confirm the importance in distinguishing the ex-ante premium from ex-post premium.

Concerning the short term view about equity risk premium, as soon as 1987, French et al. showed that monthly risk premia fluctuations on the US stock market are partly driven by conditional variance of returns (ARCH effects), while using daily stock returns for three european countries, paper by [START_REF] Koutmos | Modeling common volatility characteristics and dynamic risk premia in european equity markets[END_REF] gives a recent illustration of this shorttem approach. In another way, De [START_REF] Santis | International asset pricing and portfolio diversification with time varying risk[END_REF] analysed the factors of the shortterm dynamics of premia by using a conditional multivariate CAPM while study by [START_REF] Kryzanowski | Performance attribution using an APT with pre-specified macro factors and time-varying risk premia and betas[END_REF] based on the conditional Arbitrage Pricing Theory put into evidence several macroeconomic factors of the time-varying monthly equity premia for a set of 130 mutual funds equities on the Canadian market. According to these studies, there is an implicit assimilation between the frequency of observation in returns and the time horizon of the investment, which is a simplifying assumption since in fact, the frequency of observation can be larger or smaller than the horizon of the investment. 10 10 For instance, [START_REF] Benartzi | Myopic loss aversion and the equity premium puzzle[END_REF] suggested that long-term investors can adopt a myopic behaviour since they observe returns over periods shorter than the horizon. Conversely, studies modeling high frequency data using GARCH specifications suggests that the one period ahead expected variance refers to the squared returns generally to a much longer past than one period.

-Equity risk premia and time horizon: theoretical framework

-The one-period ahead standard model: recall

Let us consider a closed economy where no international effects exist on the stock market (i.e. a segmented market), and suppose that this market is efficient in the sense that any portfolio (among them, the market portfolio) is nondiversifiable (i.e. the market allowance of equities is optimal). Expectations can be rational or not, and this means that the stock market price does not necessarily convey all the relevant information (i.e. informational efficiency can prevail or not). In this context, according to the conditional CAPM, the equilibrium required risk premium on any risky asset j is given by

) , ( ) ( 1 1 1 t t i j t i t i ot t j t i R R Cov R R E Ω = - Ω + + + γ (1)
where j t i R is the return of equity (or portfolio) j in the country i, t i R the market return in country i, t o R the one period riskless rate and it γ >0 the unit local price of risk at time t in country i. The expected return ) (

1 t j t i R E Ω +
of the risky asset j made at time t for t+1 is conditional to the set of information available at time t t Ω . The covariance between the return of asset j and the market return represents the magnitude of the uncertainty taken into account by holders of asset j to determine their required risk premium. When asset j is the market portfolio itself, the relation (1) leads to the following expression of the ex-ante market premium at time t in the country i :

) ( ) ( 1 1 t t i t i t o t t i t i R Var R R E Ω = - Ω = Φ + + γ (2)
In this case, the market risk premium is measured by the difference between expected one period ahead stock return and the risk-free rate, and is explained at any time by the product of the unit local price of risk 11 11 Consider a representative agent whom wealth is made with a share of riskless asset and a share of a risky asset represented by the non diversifiable market portfolio. The representative agent is supposed to maximise at time t the expected utility of his wealth at time t+1. Put in the expectation/variance form, this program is by the expected variance or the market return. During

Max Θ ) ( 2 ) ( 1 1 t t i it t w t i t R Var R E Ω - Ω + + δ where w t i
R is at time t the return of the total wealth which equals a weighted average of t i R and t o R (the weigh depending on the share 1 0 ≤ Θ ≤ it of the risky asset in the wealth), and where it δ >0 is the absolute risk

aversion coefficient. Given that ) ( ) ( 1 2 1 t t i t i t w t i R Var R Var Ω Θ = Ω + +
, the first order condition of the program gives the expression of the risk premium required by the representative agent, that is

) ( ) ( 1 1 t t i t i t i t o t t i R Var R R E Ω Θ = - Ω + + δ
. Comparing this result with (2) leads to the equality

it it it δ γ Θ = .
the last century, many studies confirm that the influence of international stock markets on US stock market is negligible. This result suggests that the US market is that approaching more the hypothesis of a segmented market, and this is not astonishing because it represents the more important stock market. Since subscript i will always refers to US, we now remove it from the variables and parameters.

-Short term versus long term horizons

The time-horizon over which the expected market return ) 2) ) may equal a priori one hour ahead, one day ahead, one month ahead, one year ahead or more … provided that these variables are conditional to information available at time t. We will now consider the one year return

( 1 t t R E Ω + and the expected variance ) ( 1 t t R Var Ω + (equation (
1 1 - -+ - = t t t t t P D P P R
of the US stock market, where t P and t D are the S&P composite stock market price index at time t and the corresponding dividends distributed during the last year, respectively. To model the ex-ante risk premium we will consider at the same time its measurement (left hand side of (2)) and its explanation (right hand side of (2)). We will consider two traditional polar timehorizons : the one period ahead horizon and the infinite time horizon. We will refer to the "short term" ex-ante risk premium for the first one and to the "long term" ex-ante risk premium for the second.

According to (2), the one period ahead ex-ante risk premium is defined by the difference between the expected one period ahead stock return and the risk-free rate ot R one year to maturity:

t o t t t R R E - Ω = Φ + ) ( 1 1 (3)
this premium being explained by the product of the unit US price of risk by the one period ahead expected variance of the market return :

) ( 1 1 1 t t t t R Var Ω = Φ + γ (4) Given that t t o t t t t t t t t R P D E P E P R E 1 1 1 1 ) ( ) ( ) ( Φ + = Ω + Ω - = Ω + + +
, the forward iterative resolution of this last equation leads to the well known expression of stock price in an infinite time horizon :

∏ ∑ = ∞ = + + = k j t k k t t t R D E P 0 0 ) 1 ( 1 with t ot t R R 1 Φ + =
(5)

According to (5), the price equals the present value of the expected stream of future dividends. 12 t g Supposing that the expected rate of growth in dividends and the actualisation rate are uniform between t and all the future periods t+k , that are and ot R respectively, we obtain the "Gordon-Shapiro" stock price valuation formula which allows to define the infinite time horizon ex-ante risk premium t 2 Φ , this horizon being viewed here as a single long term representative horizon:

t o t t t t t R g P g D - + + = Φ ) 1 ( 2 (6)
In accordance with (2),

t 2 Φ is explained by ) ( 2 , 2 2 t t t t R Var Ω = Φ γ (7) where t 2 γ and ) ( 2 , t t R Var Ω
represent the US price of risk and the expected variance of returns (expressed per year) in an infinite time horizon context, respectively.

-Expected returns, expected variances and prices of risk : assumptions

-The expected returns and expected variances

Concerning the measurement of the two ex-ante premia, it is necessary to make hypotheses about the expected stock return for the one period horizon and the expected growth rate of dividends for the infinite horizon. It is worth noting that expected returns revealed by survey data strongly suggest that experts form their expectations mainly according to an autoregressive process 13

12 When expectations are assumed to be rational, (5) gives the "fundamental value" of equities. Under the transversality condition, there is no bubble and the price equals the fundamental value. 13 Among others, see [START_REF] Abou | Modelling stock price expectations: lessons from micro data[END_REF].

, the adaptive model appearing to be a simplified form -and the most popular -of such a process. Accordingly, it is supposed that expectations can be represented by simple adaptive processes which are :

) ( ) 1 ( ) ( 1 1 1 1 - + Ω - + = Ω t t t t t R E R R E β β , with 1 1 - -+ - = t t t t t P D P P R , 1 0 1 ≤ ≤ β (8) 1 2 2 ) 1 ( - - + = t t t g g g β β , with 1 log - = t t t D D g , 1 0 2 ≤ ≤ β (9)
For given values of coefficient 1 β and 2 β and of initial values of the expected variables, equations ( 8) and (9) determine the time series of the two expected variables.

Provided that indicators of riskless interest rates are given for the two horizons, we get measurements of the one period risk premium

t 1
Φ and of the infinite horizon risk premium t 2 Φ by reporting ( 8) and ( 9) in ( 3) and ( 6), respectively.

We now turn to the conditional expected variances intervening in the explanation of the premia. Appendix 1 illustrates why the expected variance is horizon-dependent when the returns are partially predictable (what is the case in particular for annual returns) and show why, according to the date, the short term premium may be greater or smaller than the long term premium. This leads to consider that the one period ahead expected variance 14 Moreover, annual data are generally not appropriate to implement an ARCH process, this latter being generally adapted to at least monthly data. 15 We also considered the actual value of the instantaneous variance (i=0) in ( 10) and ( 11), but this assumption lowered the criteria of information of the risk premia model compared to the retained specification (i=1) where the expected variance is determined only by the past values of the observed variance. Moreover, we let the possibility to add a constant term in the right hand side of the equations; but, for the two equations, these constants appeared to be insignificant and then were removed.

:

One year horizon :

∑ ∑ = = - + = Ω m i i m i i t i t t R Var 1 1 1 2 1 1 ) ( α σ α , 0 ; 1 1i 11 > = α α (10) Infinite horizon : ∑ ∑ = = - = Ω m i i m i i t i t t R Var 1 2 ' 1 2 2 2 , ) ( α σ α , 0 ; 1 2 21 > = i α α (11) with 2 2 ) ( = R R t t - σ
, where R is the average annual return. 16 i 1 α

The parameters , i 2 α , m and m' and are determined in the course of the estimation of the risk premia model. Note that these two equations represent at the same time how agents observe the variance and how they expect it, and they are in accordance with many other studies. 17

Concerning the unit US price of risk intervening in the explanation of the premia, Using CFO'surveys, Graham andHarvey (2001-2007) asked questions designed to determine expert's assessment of market volatility. Interestingly, this latter appeared to be much lower than usual alternative measures, hence suggesting that the expected variance have a lower magnitude than observed variance, which is in accordance with hypotheses ( 10) and ( 11).

-The prices of risk

t 1 γ and t 2 γ are typically unobservable variables. This lead us to estimate a state-space model where for each horizon a signal (or measurement) equation describes the risk premium and a state (or transition) equation contribute to determine the unobservable US local price of risk together with macroeconomic variables. Accordingly, the two state variables are AR(1) processes augmented possibly by a constant term and by the rate of change in CPIs, of the real consumption per capita, of earnings per share and by various spreads of interest rates. In fact, none of them was found to be significant when added in the sate equation, excepted the rate of change in corporate earnings per share for the long term price of risk. However, when 16 Note that the centered squared return 2 t σ representing the instantaneous variance appeared to be insignificantly autocorrelated over the sample period. 17 In a forecasting view and following [START_REF] Bollerslev | Z conditional heteroskedastic time series model for speculative prices and rates of return[END_REF] and [START_REF] Hansen | A forecast comparison of volatility models: does anything beat a Garch (1,1)?[END_REF] propose to represent the expected variance as an ARMA model for the squared returns plus a constant term. [START_REF] Müller | Volatilities of different time resolutions -analysing the dynamics of market components[END_REF], and for high frequency data, authors refer to an equation analogue with (10) and ( 11) including a constant term. considered separately by adding them to the state equation, the relative spread between the 30 years high grade corporate bonds yield and the 10 years government bonds yield appeared significantly in the determination of the price of risk for the two horizons. The prices of risk t 1 γ and t 2 γ are then determined according to the following relations:

The price of risk for the one year time horizon

1 1 1 1 = κ S sv t t t + + δ γ 0 1 > δ (12) t t t t + B B b + sv sv 1 1 1 1 1 1 = η ρ ∆ - 1 0 1 ≤ ≤ ρ , 0 1 > b ( 13
)
The price of risk for the infinite time horizon

2 2 2 2 = κ S sv t t t + + δ γ 0 2 > δ (14) t t t t B B b + sv sv 2 2 1 2 2 2 = η ρ + ∆ - 1 0 2 ≤ ≤ ρ , 0 2 > b ( 15
)
where κ and 2 κ are a priori undetermined. The spread variable t S captures the risk of default since corporate bonds are more risky assets than government bonds. But t S also captures risk due to the fact that corporate bonds are 30 years to maturities while government bonds are 10 years to maturity. Our intuition is not that this spread of interest rates influences directly the equity risk premia, but rather that economic factors of uncertainty which are imbedded in t S intervene in the equity premia determination. Because of these reasons, coefficients 1

t C t o t C t R R R S - =
δ and 2 δ are expected to be positive. ), it can be expected that an increase of corporate earnings will incite to rise the share of risky asset in the portfolio, thus implying a rise in the price of risk (see note ( 11)). Overall, the state variables t sv 1 and t sv 2 are supposed to capture the total influence on the price of risk of hidden factors and of non directly measurable psychological effects.

-Equity risk premia and time horizon : empirical evidence on US secular data

We R the 10 years government bonds yield and the consumer price index, and the real consumption per capita. These data are in january of each year. We also extended a 30 years high grade corporate bonds yield time series over this period by using [START_REF] Friedman | Monetary trends in the United States and the United Kingdom, Their relation to income, prices and interest rates, 1867-1975[END_REF] data ending in 1975. 19t P Data about , t D and t B were discussed and revised by Wilson and Jones (2002) (WJ) over the period 1870 to 1999. Note that contrary to the Shiller's annual data, the WJ series are for december of each year. Comparing the Shiller's december S&P stock price index, dividends and earnings with the corresponding WJ data, we observed that the series are practically confused from 1957 to 1999. However, during the period 1870 to1956, the WJ values appeared to be at a level slightly larger compared to the Shiller's data, although the two data sets exhibit very similar movements. To appreciate if there is significant bias to use a data set in place of the other, we compared over the period 1871-1957 the annual stock returns (including dividends) calculated by using the december Shiller' S&P data with the december stock returns deduced from the WJ data. The two measures appeared to be closely correlated. Regressing the Shiller' S&P returns on the WJ returns, the coefficient of regression was found to be insignificantly different from one and the intercept insignificantly different from zero, the residuals being insignificantly autocorrelated. As a result, concerning our study, we can indifferently use the Shiller or the WJ data.

For given values of 1 β and of the initial value of the expected stock return, (8) gives the one period ahead expected return time series and (3) the short term risk premium measurement. Reporting in the structural equation ( 4) the expected variance given by ( 10) and the unit US price of risk given by ( 12), and adding an error term in (4), we obtain the signal equation of the one year ahead risk premium ( 16), where the variable t sv 1 is determined by the state equation ( 13) :

t o t t t R R E - Ω = Φ + ) ( 1 1 (3) ) ( ) 1 ( ) ( 1 1 1 1 - + Ω - + = Ω t t t t t R E R R E β β 1 0 1 ≤ ≤ β (8) t m i i m i i t i t t t κ S sv 1 1 1 1 2 1 1 1 1 1 ) ( υ α σ α δ + + + = Φ ∑ ∑ = = - (16) t t t t + B B b + sv sv 1 1 1 1 1 1 = ε ρ ∆ - 1 0 1 ≤ ≤ ρ (13)
In a similar manner, for given values of 2 β and of the initial value of the expected growth in dividends, (9) gives the long run expected rate of growth in dividends and then (6)

gives the long term risk premium measurement. Reporting in the structural equation ( 7) the expected variance given by ( 11) and the unit US price of risk given by ( 14), and adding an error term in (7), we obtain the signal equation of the infinite time horizon risk premium (17),

where the variable t sv 2 is determined by the state equation ( 15):

t o t t t t t R g P g D - + + = Φ ) 1 ( 2 (6) 1 2 2 ) 1 ( - - + = t t t g g g β β 1 0 2 ≤ ≤ β (9) t m i i m i i t i t t t κ S sv 2 1 2 ' 1 2 2 2 2 2 2 ) ( υ α σ α δ + + + = Φ ∑ ∑ = = - (17) t t t t B B b + sv sv 2 2 1 2 2 2 = ε ρ + ∆ - 1 0 2 ≤ ≤ ρ , 0 2 > b (15)
For given values of 2 1 , β β and initial values of expected variables, the 4-equationssystem ( 16), ( 17), ( 13) and ( 15), can be estimated jointly as a system using the Kalman filter methodology, where ( 16) and ( 17) are the signal equations while ( 13) and ( 15) are the state equations (see Appendix 2 for a formal presentation of the state-space model and of the recurrent equations used in the estimation method). The innovations t υ 1 and t υ 2 of the signal equations are supposed Niid and independent of the auxiliary residuals t 1 ε and t 2 ε of the state equations, respectively. However, a significant correlation may exist between the two signal residuals resulting from the interdependences between the two premia and between their factors, which could cause biases in estimates. This is why a covariance term (covar) between t 1 υ and t 2 υ were added to the set of hyperparameters to be estimated. All parameters (including initial values of state variables) are those minimizing the Akaike, Schwarz and Hannan-Quinn criteria of information.

[ Insert Table 1 ]

Table 1 gives descriptive statistics of the two "observed" premia as determined by equation ( 3) and ( 8) for the one year premium and by equations ( 6) and ( 9) for the infinite horizon premium, with optimal values of the expectation coefficients of % 7 1 = β and % 3 2 = β respectively for the adaptive processes ( 8) and ( 9). 20 25 . 0 2 = R These coefficients show a rather long memory for the two horizons, but with longer for the long run horizon, which rather intuitive. It is worth noting that, excepted in the beginning of 1980's, the values of the premia keep positive values. In addition to error measurements, the negative values observed could be connected to the shock due to the Republican Presidential election of Ronald Reagan in 1980 reinforced at the same time by the Republican majority elected at the Senate; if these events had an effect on equity risk premia, it was probably in the direction of a fall. The short tem premium appears to be higher in mean since more volatile than the long term premium and these outcomes are rather intuitive. Interestingly, the mean value of the infinite horizon premium (2.3% per year) joins the one of 2.5% obtained by [START_REF] Fama | The equity premium[END_REF]. Figures 1 and2 exhibit the time pattern of the components contributing to the measure of the premia, for the one year and the infinite time horizon, respectively: interestingly, in both cases, the components appears to be not negligible one compared to others. Figure 3 compares the values of the measurement of two observed premia : although these two magnitudes are stationary at the 10% level of significance, the horizon is a very discriminant parameter conditioning the dynamics of the premia (the value of is not very high), and this result confirms lesson from survey data reflecting experts' opinions (see part 2). 20 The optimal initial values of the expected return in ( 8) and of the expected rate of growth in ( 9) are found to be 4% and 0% per year.

[ Insert figure 1] [ Insert figure 2] [ Insert figure 3]

Table 2 gives the estimates of the structural model of risk premia. Concerning the signal equations, a grid search over the lags m and m' intervening in the expected variances determination led to the optimal values 5 and 8 for the one year and the infinite horizon, respectively. As a result, compared to the short term premium, the long term premium is influenced by the variance over a longer time span, which is rather intuitive since this result contributes to explain why the former is more volatile that the latter, as shown by figure 3.

Although the coefficients of the lagged variances decrease from the third lag for both horizons, the lag-coefficients increase before decreasing for the long term premium, and this outcome was already observed in the literature. 21 67 . 0 2 = R However, excepted two weighs for the long term premium, standard errors of estimates suggest that they are not significantly different from 1 at the 5% level, hence suggesting that the expected variances could be approached with a simple arithmetic average of a set of past values of centered squared returns. Figure 4 compares the two expected variance time patterns and confirms that despite similar trends (

), the short term expected variance exhibits higher volatility of the long term one.

The significant positive value obtained for the covariance between the residuals of the two signal equations (covar) results from the correlations between the two premia, between the two expected variances and between the two state variables, as displayed in Figures 3,4 and 7, respectively. The covariance between the two state residuals is found to be insignificantly different from zero and this is why this parameter has been removed from the estimations. 22 Another point is that, as expected, the coefficients

[ Insert

S

The larger sensibility to the spread found for the short term premium than for the long term one seems intuitive insofar as represents default risk perceived due to economic factors or default risk 21 For instance, see [START_REF] Ederington | How markets process information: news releases and volatility[END_REF]. 22 We also found a zero covariance between the signal residuals and the state residuals for each horizon. This was a condition underlying the updating equations ( B5) and (B6) used and presented in Appendix 3. 23 See note (18).

expected in a rather near future. The constants 1 κ and 2 κ were not found to be significant and therefore have been removed at the final stage of estimation. According to ( 12) and ( 14), the state variables t sv 1 and t sv 2 given by ( 13) and ( 15) contributes with the spread t S to the representation of the prices of risk for each horizon t 1 γ and t 2 γ . Since this paper is concerned by a structural model, the state variables are estimated at each point in time conditional on the whole sample data (smoothed inference) rather than using only the past observations (predicted inference) or actual and past observations (filtered inference). At each point of time, the kalman filter method yields the standard deviations (std) of each state variable. For an accepted 5% level of significance, the std allows to determine a zone of uncertainty defined at each date by the line ranging between the estimated value plus and minus 1.96 std.

In the same manner, the std of coefficients of t S , which are constant over time (although horizon dependant), allow to define a zone of uncertainty associated to to understand why it is not necessary to consider explicitly structural changes over this long period to fit the data. We further checked the goodness of the fits by using the conventional coefficient of determination 2 R and a modified measure, 2 D R , assessing the relevance of our model with respect to the simple random walk plus drift process (benchmark). 24

D R

The values of reach 0.88 and 0.68 for the long and the short term horizons (table 2) indicate that the residual variance of the measurement equation is 0.12 and 0.32 times the one of the benchmark model for the infinite and 1-year horizons, respectively : clearly, the risk premia model strongly outperforms the benchmark model.

[ Insert figure 8] [ Insert figure 9]

We now examine the statistical properties of the residuals of the signal equations (innovations) t υ 1 and t υ 2 . The diagnostic tests we refer to are presented in Appendix 3.25

According to [START_REF] Harvey | Forecasting, structural time series models and the Kalman filter[END_REF] heteroskedasticity test, the null of homoskedasticity of these residuals is rejected for the one year horizon but accepted for the infinite horizon. The appropriate Ljung-Box Q test by [START_REF] Harvey | Forecasting, structural time series models and the Kalman filter[END_REF] based on the first 11 autocorrelations applied to the signal standardized smoothed residuals showed a rather weak26 but significant autocorrelation for the two horizons, which suggest that, beyond a possible specification bias, market frictions such as transaction costs 27 and risky arbitrage opportunities28 t τ Φ could cause delayed adjustments of premia toward their theoretical values. Supposing these adjustments to be represented by an error correction models (ECM), we showed in Appendix 4 that ECM's residuals are independents and homoscedastic at the 5% level of significance.

-Concluding remarks

Any ex-ante equity premium is defined by the difference between a given representation of the expected return of the risky asset at time t for a given future time horizon 24

The two measures of goodness of fit are defined by

∑ = - - = T t t y y SSR R 1 2 2 ) ( / 1 and ∑ = ∆ - ∆ - = T t t D y y SSR R 2 2 2 ) ( / 1
where

t t y τ Φ = ( 2 , 1 = τ
) and SSR is the sum of the squared residuals. A negative 2 D R implies that the estimated model is worse than a simple random walk plus drift (Harvey, 1992).

and an equivalent maturity risk-free interest rate. Using annual US secular date from 1871 to 2008, the challenge of our study with respect to the literature is to model at the same time the measures of ex-ante equity risk premia (which specify how expected returns are formed) and the explanations of these premia in the CAPM framework, by considering simultaneously two polar horizons: the one period (year) ahead horizon (i.e. the "short term" premium) and the infinite time horizon (i.e. the "long term" premium).

Representing expected returns by traditional adaptive processes, large disparities in the dynamics of the two measured premia are evidenced, as expected when the market is not efficient and as shown by ex-ante premia deduced from survey data reflecting experts'opinion. According to the CAPM, each of the two premia is explained by the product of the unit price of risk by the expected variance of observed returns, each of these time varying magnitudes depending on the horizon. The expected variances are found to depend on the past values of variances during 5 and 8 years for the short term and long term premia, respectively. For each horizon, the price of risk is determined both by a spread of interest rates capturing economic factors of uncertainty and by an unobservable variable determined according to the kalman filter methodology (i.e. a state variable), that is supposed to capture the influence of hidden variables and of non directly measurable psychological effects. The model gives a valuable representation of the "short term" and "long term" premia over the period 1881-2008, and then gives an explanation of the large disparities in their dynamics.

Finally, we find a weak but significant autocorrelation in residuals that could be explained by short term adjustments of premia toward their theoretical values, possibly due to transaction costs and risky arbitrage. Overall, these results highlight the existence of a time varying term structure of ex-ante equity risk premia and suggest that it is necessary to solve simultaneously their measurement and their explanation, although, when expectations are not rational, results are conditional to the hypothesis retained for the expected return representation.

Appendix 1 -Why equity risk premia are horizon-dependant when returns are predictable? Illustrations

It is supposed that no dividends are distributed. Let t p denote the logarithm of the price of an equity or a portfolio at time t and ∆ the 1-period change operator. Suppose that the market is efficient in the sense that t p conveys all available information about the future price (i.e. expectations are rational).

( ) ( ) 1 + + t t τ t p τV = p p Var ∆ - , 1 ≥ τ
, that is, the two first moments increase in the same proportion with τ . Since the risk premium depends on the expected variance (equation ( 2)), the premium averaged per period may be time-varying if the variance is so but does not depend on τ , so that there is a single premium. Conversely, when returns are partially predictable on the basis of their past values and/or macroeconomic variables 30 t X ∆

, the stock market is not efficient and agents do not require a unique risk premium but a set of premia scaled by the time horizon. For example, suppose that the one period return is related to the variable according to the simple relation

1 + 1 + t t t + X = p η ∆ ∆
, where

1 + t η is a white noise, with 0 ) ( ) ( = = ∆ t t E X E η , 2 ) ( θ = ∆ t X Var , 2 ) ( ω η = t Var and t Cov X X Cov t t ∀ = ∆ ∆ + ) ; ( 1 . Suppose further that 1 0 ) ; ( > ∀ = ∆ ∆ + τ τ t t X X Cov
, the one period variance is ( )

2 2 1 + = ω + θ p Var t ∆
while the τ period horizon variance is ( )

Cov τ ω + θ p p Var τ t τ t       - + = - 1 1 2 1 2 2 +
. It can be seen that when 0 > Cov , the variance and therefore the required premium increase with the horizon, while when 0 < Cov , the variance and the premium decrease with the horizon. 31

29 Even if we introduce a discount rate with constant variance which is independent of the white noise forecast error, this conclusion is still valid. 30 Lo and MacKinlay (1988) and Cochrane (1999a) give overviews concerning the predictable character of stock returns. Beyond the autocorrelation of returns, economic variables such as spreads of interest rates, change in money supply, production growth, change in corporate earnings, the ratio dividend/earnings and the dividend yield are often shown to be significant predictors. 31 Two examples for the sign of This implies that a sufficient condition to generate an increasing or decreasing term structure of risk premia is the existence of a

Cov are given by Cochrane (1999b) with

t t P a ΔX ∆ = , 0 > a : in this case,
Cov is positive when the actual return is positively autocorrelated but negative when a mean-reversion describes the dynamics of returns. Here, the condition 0 = a corresponds to the efficiency hypothesis according to which returns are a white noise.

1 ) 2 , 2 ( ) 1 , 2 ( / t t t t c B B b sv = sv ε ρ + + ∆ + - , T t ,..., 1 = (B2) where         Φ Φ = Φ t t t 2 1 ,       t t t sv sv = sv 2 1 ,       Σ 2 1 0 0 Var Var = t ,       = 2 1 0 0 ρ ρ ρ ,       = 2 1 κ κ c ,       2 1 0 0 δ δ δ = ,       2 1 . . Var S Var S = X t t t ,       = 2 1 b b b ,       = t t
c B B b v s = v s t t t t t + ∆ + - - - / ˆ1 / 1 1 / ρ (B3)
and it can be shown that the covariance matrix of the forecast error,

] )' v s )(sv v s E[(sv = M t t t t t t t t 1 / 1 / 1 / ˆ- - - - -
, can be written as:

Q + ' M = M t t t t ρ ρ 1 / 1 1 / - - - (B4)
The equations (B3) and (B4) are the prediction equations of the Kalman filter. From (B1) we get the forecast error on t Φ

and its covariance matrix given by

U + M = ] )' )( E[( = H t t t t t t t t t t t ' 1 / 1 / 1 / ˆΣ Σ Φ - Φ Φ - Φ - - -
. The linear projection of t sv on t Ω leads to the following updating equations:

) ( ˆ1 / 1 , , - - Φ - Φ + = t t t t t t t t K v s v s (B5) 1 / 1 / - - Σ - t t
∑ Φ T t= t ) f( = L 1 log
, where

      Φ - Φ Φ - Φ - Φ - - ) ( )' ( 2 1 exp ) 2 ( 1 1/2 - 1 t t t t t t t H H = ) f( π is the pdf of t Φ .
34 Note that ) , ( ' t t E ε υ may be equal to some non-zero matrix G if ' t t = and 0 otherwise, that is, the residuals may be contemporaneously correlated. In this case the prediction equations (B3) and (B4) are unaltered but the updating equations (B5) and (B6) are modified as described in Harvey (1992, sub-section 3.2.4). Var are given by (10) and ( 11)) and the two state equations by 16) and ( 17 estimates are significant at the 1%, 5% or 10% levels, respectively. 2 R and 2 D R are two goodness of fit measures (see footnote 21). Q and hH represent Ljung-Box serial correlation and heteroskedasticity test statistics (see Appendix 3 for a presentation of these statistics). For the Qstatistic, the asymptotic critical values for a 2 χ with 5 d.f. (one year horizon) and 1 d.f. (infinite horizon) are (9.24, 11.07, 15.09) and (2.71, 3.84, 6.61) for (10%, 5%, 1%) levels of significance, respectively. For the hH statistic, the asymptotic critical values for a 2 χ with 43 d.f. (both horizons) are 57.51, 61.66 and 69.96 for 10%, 5% and 1% levels of significance, respectively.

  present a set of four studies about the expected equity premia defined as the difference between the experts' mean expected stock returns and an equivalent horizon bonds yields. These studies are based on quarterly surveys conducted since June 2000 by Duke University and CFO Magazine and concern stock market returns expected by about 270 anonymous Chief Financial Officers (CFOs) of U.S.

  take the same value. An ARCH-M model would not be appropriate since the conditional volatility would concern the residuals of the risk premium equation and not the total variance of the return as required by our model. An ARCH model where the mean equation specifies the stock return as a constant term plus an error term would give a first step estimation of the conditional expected variance, but this estimation would be disconnected from the estimation of the portfolio model.14 As a tentative, for the two horizons, the expected variances are assumed to be represented as an m-order (resp. m') weighted averages of the past values of the centered squared returns, this last one representing a proxy of the instantaneous observed variance15

η

  are Niid error terms, and where the values of the constant drifts 1

  Concerning now the positive influence of change in earnings per share (

γ and t 2 γFigures 8

 28 Figures 8 and 9represent the observed and the fitted values of the premia for the two horizons, respectively: the state-space model fits well the main fluctuations of the long term premium whereas the fit of the 3-month premium is of lower quality because of the higher volatility of the latter. It is noted that the expected variances and the prices of risk can explain sudden changes in the risk premia and periods of high or low premia, what makes it possible

  τ

  of equations using the Kalman filter methodology. covar stands for the covariance between the two signal residuals. The estimates are obtained by setting to zero the insignificant covariance between the two auxiliary (state) residuals and the insignificant values of intercepts τ κ . The variances of τ t, ε and τ t, υ are estimated as ) (c τ exp and ) (c ' exp τ , respectively. AIC, SC and HQC stand for Akaike, Schwarz and Hannan and Quinn information criteria for the system estimation. The initial values of t sv τ have been optimized as 0.01 and 0.05 according to the

  Figure 1 -The two components of the short term equity risk premium 1881-2008

  used the annual secular data over the period 1871 to 2008 as available on the web

	site of Robert Shiller. t P is the Standard and Poor's 500 stock price index, t D and t B are the
	S&P dividends and earnings per share during last year,	t R the one-year interest rate, o

t o

Table 2 ] [ Insert figure 4]

 2

	1 δ and 2 δ of the spread of interest

Table 1 -Short term and long term risk premia : descriptive statistics

 1 Notes. The risk premia are expressed in percent per month. The sample period is1881-2008 (128 observations). The asymptotic critical values for the ADF test statistic is -3.48, -2.88 and -2.58 at the 1% , 5% and 10% levels, respectively.

				Std deviation
		Mean	Maximum	Skewness	Jarque-Bera	Unit root
		Median	Minimum	Kurtosis	(probability)	ADF test
	One year horizon premium	5.54 5.25	12.81 -6.73	4.10 -0.27 2.43	3.24 0.19	t-stat=-2.79
	Infinite horizon premium	2.29 2.35	10.03 -3.88	2.16 0.36 4.01	8.17 0.02	t-stat=-2.74

Table 2 : Estimating the one year and infinite time horizon equity risk premia model using the Kalman filter methodology

 2

	Notes. Estimations cover the period 1880-2008 (128 years). For	τ	=	, 1	2	, the two signal equations
	Φ	τ	t	=	sv τ	t	Var τ	+	δ	τ	S	t	Var τ	+	τ t, υ	(where the expected variances

  ) Numbers in brackets are the t-values. ***, ** and * indicate that

	minimum information criteria.
						0.051**	0.030***
	δ	τ	(3.0)	(4.9)
	α τ	1	1.39*** (4.5)	1.41*** (4.7)
	α	τ	2	1.40*** (4.4)	2.30*** (4.2)
	α	τ	3	1.40*** (4.9)	2.80*** (4.8)
	α	τ	4	098*** (3.4)	1.58*** (2.8)
	α	τ	5	_	1.30** (2.3)
	α	τ	6	_	0.93*** (2.8)
	α	τ	7	_	0.58** (2.3)
	' τ c	-1.17*** (-10.4)	-3.86*** (-34.8)
	covar	0.31**
						(1.98)
	R	2	0.925	0.942
	R	2 D	0.676	0.881
	Q	19.77	19.34
	hH	117.01	54.66
	AIC	8.56
						9.02
						8.74
						30

[START_REF] Subbotin | Multiple Investment Horizons and Stock Prices Dynamics[END_REF] gives a survey of the rare literature about this subject.

More recently see[START_REF] Gollier | Time horizon and portfolio risk[END_REF] who extended this approach.

Appendix 1 gives a formal illustration of this point.

Abou and Prat (2010) give an overview concerning studies about equity premia issued from survey data.

Papers by[START_REF] Kocherlakota | The equity premium: It's still a puzzle?[END_REF],Cochrane (1997) and[START_REF] Siegel | Anomalies: the equity premium puzzle[END_REF] provide comprehensive surveys of the literature related to the equity premium puzzle and tentative to solve this puzzle.

Number of contributions have shown that spreads of interest rates are negatively (positively) correlated with stock prices (returns). Concerning the spread between different ratings US bond yields over a long period, see in particular[START_REF] Prat | La bourse et la conjonctuire économique[END_REF], Chapter IV which is entirely devoted to this relation. Concerning the link between spread between different maturities US bond yields and equity risk premium through the APT, see, among others[START_REF] Chen | Economic forces and the stock market: testing the APT and alternative asset pricing theories[END_REF] and Elton, Gruber and[START_REF] Gruber | Cost of capital using Arbitrage Pricing Theory: a case study of nine New York Utilities[END_REF]) and[START_REF] Kryzanowski | Performance attribution using an APT with pre-specified macro factors and time-varying risk premia and betas[END_REF].

[START_REF] Friedman | Monetary trends in the United States and the United Kingdom, Their relation to income, prices and interest rates, 1867-1975[END_REF], Table6.17, pp. 296-98.

The same tests are implemented byPrat and Uctum (2008) for a similar model concerning exchange rates risk premia.

In fact, no coefficient of correlation R exceeds 0.20.

See Shleifer and Summers (1990).

serial correlation in returns. 32 2 ) ( θ = ∆ t X Var

If the sign or the magnitude of the covariance is time-varying, the slope of the term structure of premia is also time-varying, and this may explain why the short term premium may be greater or smaller than the long term premium. If now we relax the hypothesis and assume that ) ( t X Var ∆ is a conditional AR(n) process, it can be shown that ( )

is also an AR(n) process. When 1 > τ , an autoregressive structure with order greater than n is preserved for

Appendix 2 -The equity risk premia model and the Kalman filter equations

The system formed by the four equations ( 16), ( 17), ( 13) and ( 15) can be written in the following state-space form (see Harvey (1992), Ch. 3;[START_REF] Hamilton | Time series analysis[END_REF], Ch.13):

Appendix 3 -Diagnostic residual tests for the Kalman filter inference

We describe [START_REF] Harvey | Forecasting, structural time series models and the Kalman filter[END_REF] autocorrelation and heteroskedasticity tests for the standardized signal residuals t υ ˆ (that are errors t υ 1 and t υ 2 of ( 16) and ( 17)) resulting from the smoothed inference over our sample size 128 = T

. Let θ γ be the sample autocorrelations in t υ ˆat lag p , = ,... 0 θ . We set 12 T = ≈ p (see Harvey (1992, p.259)). The null of no serial autocorrelation in the residuals can be tested by using the Ljung-Box Q statistic

is the number of non-stationary elements of the state vector that are associated to a signal equation, equal to 0 in our case).

-, where n is the number of hyperparameters to be estimated minus one, equal to 7 and 11 for our 1 year and infinite horizon models, respectively. The author suggests to calculate the test for heteroskedasticity as

, where h is the nearest integer to 3 / * T , equal to 73 with our sample size. The asymptotic distribution of the statistic

Appendix 4 -Testing the adjustments of premia toward their theoretical values according to an ECM

We consider the two following error correction models: Φ are the fitted values of the premia given by the signal equations (here, the targets) and where t 1 ξ and t 2 ξ are error terms. For both horizons, we found that τ ω is not significantly different from zero, so that the lagged endogenous variable has been removed.

Estimating equations ( 18) and ( 19) simultaneously with the SUR method, (