Christian Bidard

The Ricardian Dynamics Revisited

Keywords: JEL Classi…cation. B12, D24, D33 Dynamics, Lemke, Linear Complementarity Problem, Rent, Ricardo

The Ricardian dynamics describe the substitution of a new marginal method for an outgoing marginal method when demand increases. The process of extension or intensi…cation of cultivation allows for spasmodic changes in prices and rents but is smooth on the physical side. We criticize the notion of extension of cultivation and show the existence of non-Ricardian equilibria, when some good is not produced by a marginal method. The working of the dynamics requires that the incoming method satis…es a productivity condition which is expressed in algebraic terms. A parallel is drawn between Ricardo's views on dynamics and the working of a Lemke algorithm for linear complementarity problems.

Introduction

The 1 Ricardian dynamics are among the most classical pieces of economic analysis: Ricardo (1817) combined parsimonious assumptions with a deep analysis of the extension of cultivation, from which he drew dramatic conclusions on the evolution of distribution between rents and pro…ts and a plea for free trade. It is often admitted that, under the retained hypotheses, which ignore technical progress, the description of the extension or intensi…cation of cultivation is correct and general (the argument is used today to explain the rise of the prices of raw materials). We intend to show that the dynamics are less obvious than it may appear and that their working requires a productivity condition which does not hold automatically.

Let us start from a long-term equilibrium for a given demand. Somewhat paradoxically, Ricardo's strategy consists in eliminating rents in the …rst step of his analysis. This is done by considering the operated industrial methods and the marginal agricultural methods, which pay a zero rent. For a given distribution variable (which is the real wage in Ricardo's analysis), these methods determine the set of prices, as if we were contemplating an economic system with no scarce resource. Ricardo eventually reintroduces rents as a re ‡ection of di¤erential costs between lands or agricultural methods. The increase in the demand for agricultural products leads to a change of equilibrium and triggers the dynamics. As long as lands of the same grade are available, the extension of cultivation requires an adjustment of activity levels but no changes in prices, wages and rents. When some land becomes fully cultivated, a physical limit is reached and the price of the scarce agricultural good rises suddenly up to the level where the cultivation of a more costly type of land becomes pro…table. Then the previous marginal land yields a positive rent and all prices are modi…ed. A new equilibrium is found and the transformation opens another quiet period with adaptations of activity levels, followed again by a new rupture. We consider as an essential feature of Ricardo's views that the spasmodic upheavals in value and distribution do not prevent a smooth physical transition from the previous to the new long-term equilibrium: in normal times, some methods of cultivation are progressively extended; from time to time, the introduction of cultivation on a new land, that the economists consider as a break, is hardly noticeable for a geographer.

The paper studies the Ricardian dynamics in a general framework with several agricultural goods. Curiously enough, the topic has rarely been examined in the analytical literature. It will turn out that the traditional de-scription, which argues that the demand for corn triggers the introduction of a new corn method on another …eld (extension of cultivation) or that of more productive corn method on the same …eld (intensi…cation of cultivation), is a too much simpli…ed story. Section 2 shows that the equilibria may be of an unexpected (non-Ricardian) type. Section 3 analyzes the transition between consecutive equilibria and establishes a criterion for the working of the dynamics. Section 4 draws a parallel between the Ricardian dynamics and the [START_REF] Lemke | Bimatrix equilibrium points and mathematical programming[END_REF] algorithm, initially conceived to …nd a solution to bimatrix games. Some references are given in Section 5.

The Ricardian dynamics result from a comparison of long-term equilibria: each equilibrium is associated with a given demand for industrial and agricultural goods and the rates of pro…ts are uniform. We shall not consider time explicitly and, when demand changes, disequilibria will only be referred to as the reason for a move from an equilibrium to another. Following [START_REF] Sra¤a | Production of Commodities by Means of Commodities[END_REF] proposal, we assume that the independent distribution variable is the given rate of pro…ts rather than the real wage: this inessential change with regard to Ricardo's own hypothesis allows for analytical simpli…cations. Finally, since our basic purpose is to clarify the Ricardian dynamics, we discard any type of degeneracy and reduce calculations to minimum.

Non-Ricardian equilibria

Let there be a multisector reproduction model with n industrial or agricultural commodities. All methods are of the single-product type with constant returns (land is not treated here as a joint product of agriculture), but each agricultural method admits an upper activity level due to the scarcity of lands. In a steady state, the industrial methods yield the ruling rate of profits r (r is nonnegative and economically feasible) if they are operated, less if they are non operated. As for the agricultural methods, the rent is zero on partially cultivated lands (marginal land) and nonnegative (in fact positive, ‡ukes apart) on fully cultivated lands: the positivity of rent stems from the scarcity of a given grade of land. Its level re ‡ects the di¤erences in the costs of production for two methods producing the same good and compensates for it: this di¤erential cost may concern two methods of cultivation on different lands (extensive rent) or two methods of cultivation on the same land, one of them being more productive per acre but more expensive per quarter of product than the other (intensive rent). Once rent is taken into account, each good has a unique price and the rates of pro…t are uniform among all industries. Some lands are not cultivated at all, because this would be too costly.

Let us start from an initial long term-equilibrium I sustained by the price vector p I (labor is chosen as numéraire) and assume that the demand for corn increases. An adaptation of activity levels, with no change in prices and rents, is a su¢ cient response of the productive system until some marginal land becomes fully cultivated. When a limit is reached, the price of corn rises, and Ricardo assumes that it increases up to the level where a new corn method becomes pro…table, be it on a new type of land (extension of cultivation) or an already cultivated land (intensi…cation of cultivation): a new equilibrium II with higher prices p II and higher rents succeeds equilibrium I and provides a solution to the scarcity problem.

The point missed by Ricardo is that the rise in the price of corn triggers a rise in the prices of all commodities, be they industrial (iron) or agricultural (rice), in the production of which corn enters as a direct or indirect input. It may well be the case that a new iron method or a new rice method becomes pro…table before any new corn method. Imagine for simplicity that, when the price of corn changes, the adjustment of the other prices to the changing costs is immediate, so that the market has only to …nd the new price of corn by means of a sequence of small successive rises. In that process, if an iron method which was not pro…table at the initial price vector becomes pro…table before any new corn method, that iron method would pay extra-pro…ts if the rise continued, and this would be incompatible for the notion of long-equilibrium for an industrial method. If it is a rice method on a new land which becomes …rst pro…table, the extra-pro…ts might be absorbed and transformed into a rent paid to the landowner, but a comparison of the physical sides of both equilibria shows that a land which was not exploited at all at the …rst equilibrium would be fully cultivated at the second equilibrium, this being a condition for reaping a positive rent. The notion of Ricardian dynamics excludes such a discontinuity in activity levels. To sum up, the answer of the market to a scarcity of corn may consist in introducing a new iron method or a new rice method instead of a new corn method, as this may be the only way to ensure a smooth physical transition from the old to the new equilibrium.

A numerical example illustrates the point. For simplicity the rate of pro…ts is zero (the phenomenon we point at occurs even under the golden rule hypothesis), but little would be changed for a small positive rate of pro…ts.

Example 1. The rate of pro…ts is zero. Let there be n = 2 agricultural goods, corn and rice, and l = 4 types of lands. Lands 1 and 2 are specialized in the production of corn, lands 3 and 4 in that of rice, with a unique method on each type of land (extensive cultivation). The available area of land i is denoted by i . Per unit of land, the constant returns methods are described as follows:

Corn methods: (method 1:) Let the …nal demand basket be d(t) = (t; 4). When t is small (0 t 4), the corn method 1 and the rice method 3 are used, the price vector with labor as numeraire being p I = (p corn = p rice = 1): For t = 4, the corn land 1 is fully cultivated. The next equilibrium (4 t 4:5), however, does not consist in extending the cultivation of corn to land 2 but in introducing cultivation of rice on land 4 jointly with land 3, while land 1 remains fully cultivated. This non-Ricardian equilibrium with two marginal rice methods is sustained by prices p II = (p corn = 4; p rice = 2): land 1 yields a positive rent and demand is met for activity levels y 1 = 1 ; y 3 (t) = 9 2t and y 4 (t) = 3t 12. For t = 4:5, land 3 is no longer cultivated and, for t > 4:5; a new Ricardian equilibrium is made of the marginal methods 2 and 4, land 1 being fully cultivated and land 3 being left fallow. An ultimate limit due to the scarcity of lands is eventually reached.

Some lessons are:

-In a non-Ricardian equilibrium, some good (corn in the above example) is produced only on fully cultivated lands and the methods which produce it do not matter for the determination of prices.

-The number of methods playing an active role in the determination of prices is however su¢ cient since some other industrial or agricultural good is produced by two methods simultaneously.

-Ricardo's general strategy is to discard rent from the analysis of value by considering the marginal methods only, because they pay no rent and determine the prices. The idea is to reduce the study of a system with scarce resources to that of a single-product system, in which each commodity is produced by a unique method. Long after Ricardo, such single-product systems have indeed been studied by Leontief and Sra¤a and their formal properties elucidated by means of the Perron-Frobenius theorem. But that strategy fails because the standard properties of single-product systems do not hold: even if each method produces a unique commodity, the set M of marginal meth-ods may contain two methods producing the same good and no method for another good and, therefore, is not a usual single-product system.

-When demand increases, the rice land 3 is …rst cultivated up to a certain maximum level reached for t = 4. Then, because of an external phenomenon (the corn land 1 is fully cultivated), the level of cultivation on land 3 decreases and drops to zero for t = 4:5. A comparison of these changes with those occurring on land 4, which is cultivated up from the level t = 4, shows that Ricardo's notion of 'order of cultivation', which presumes that a land cultivated for some level of demand remains cultivated for higher levels, does not apply.

-A number of conditions must be met for the occurrence of a non-Ricardian equilibrium. On the value side, the price vector sustained by both rice methods must be positive and such that some corn method yields a positive rent (otherwise, corn will not be produced at all). On the physical side, the activity levels of the second equilibrium describe the progressive transfer of rice from land 3 to land 4, which in the present case is an adequate answer to the demand for more corn because the rice method 4 is corn-saving with regard to the rice method 3.

Local and global dynamics

In this section, we look at the conditions for a smooth physical transition between consecutive equilibria. Due to the possible occurrence of non-Ricardian equilibria in the dynamics of the economy, there is no harm in assuming that the industrial or agricultural methods are of the multiple-product type, i.e. the productive method i (i = 1; :::; m) is represented by a input vector a i 2 R n + of reproducible commodities, an amount l i 2 R + of homogenous labour and the area(s) i 2 R l + of land(s) used by the method at unit activity level (that unit is arbitrary, and i = 0 for industrial methods), while the output vector is a basket b i 2 R n + . We assume that all goods can be disposed of freely and that lands can be left fallow: fallowing is considered as a speci…c agricultural method with no inputs and outputs, the rent on the corresponding land being zero. (That formalization may look rather ar-ti…cial, but it facilitates the counting of equations and the identi…cation of the bounds to equilibrium, since a limit is then reached when some positive variable becomes zero.) At equilibrium, and ‡ukes apart, the number of operated processes, including fallowing and free disposal, is equal to the number n of produced commodities plus the number l of lands: otherwise, either the degrees of freedom on the positive activity levels would not su¢ ce to adapt the net product to the given demand basket, as described by the conditions

(B A)y = d (1) y = (2)
or the prices-and-rents vector (p; ) 2 R n + R l + would be overdetermined, since a price equality

(1 + r)p T a i + T i + l i = p T b i (3)
stating that the method yields the ruling rate of pro…ts is associated with each operated process i. In this equality, following [START_REF] Sra¤a | Production of Commodities by Means of Commodities[END_REF], we assume that the wage is paid post factum, but nothing is changed if it is paid at the beginning of the period. By setting c i (r) = (b i (1 + r)a i ; i ) 2 R n+l and = (p; ) 2 R n+l + , the same price equality is written more simply

T c i (r) = l i (4)
De…nition. Given a set M of n + l operated methods, C M (r) is the square matrix obtained by stacking the n + l vectors c i (r) of the operated methods, and

C M = C M (0).
With this notation, the price-and-wage vector is the solution of

T C M (r) = l T (5)
Let us start from a given equilibrium for a certain demand d = d(t) and change that demand slightly. In general, the activity levels y(t) adapt themselves to the new demand vector. With the convention that fallowing is a method of production, lands are fully cultivated (possibly partly or totally with weeds) and the limit to the adaptation of activity levels is always reached when the activity level y j (t) of some method j, which was positive at t = t 0 ", drops to zero at t = t 0 and would become negative at t 0 + " ( ‡ukes apart, that method is unique). The existing equilibrium I thus fails for a physical reason, and we have to build the physical side and the value side of the next equilibrium II. The outgoing method j is well identi…ed. In order to ensure a continuous physical transition between the previous and the next equilibrium II, the other n + l 1 operated methods must continue to work at the same level, whereas some incoming method k is introduced at a zero level. How is that method selected? The point is that the choice of method k is uniquely determined by the value side of the problem: since the previous n + l 1 marginal methods other than j satisfy condition (3) or (4) in both equilibria, the change = II I in the price-and-rent vector satis…es the n + l 1 linear equalities T c i (r) = 0. The new price-and-rent vector II is therefore written as II = I + , where is a given solution, unique up to a factor, of these equations, and is an unknown scalar. Moreover, the sign of is determined by the property that the outgoing method j pays extracosts at the new price-and-rent vector: for a choice of such that T c j < 0, is positive. On the whole, the incoming method is determined as follows: starting from = 0, let us increase and 'pivot'the prices I + until some method k which was not operated becomes pro…table. If the new price vector were not the one corresponding to the minimum value of satisfying that condition, method k would either yield extra-pro…ts at the new equilibrium if it is an industrial method, or its activity level would jump from zero to maximum if it is an agricultural method. This is excluded in the Ricardian dynamics.

Note that equilibrium I and the candidate II have a speci…c relationship. All but one of their methods are identical and, by construction, the method j which is operated in I pays overcosts j > 0 at prices II , while the method k operated in II pays overcosts k > 0 at prices I . An algebraic expression of this property is obtained when, in the matrix C II (r), the incoming method k takes the column previously occupied by the outgoing method j. Then: Clearly enough, as the unique candidate for the incoming method is selected on pro…tability considerations, that method does not necessarily meet the change in demand which caused the loss of the previous equilibrium (an exception is the golden rule hypothesis which ensures a strict duality between the value and the quantity sides). The condition for the local working of the Ricardian dynamics is that the new system is able to meet the demand vector d(t 0 + "), in which case the candidate II is indeed an equilibrium for that demand.

The matrix C(r = 0) being denoted by C, the physical constraints (1)-(2) on activity levels are written as Cy = (d(t); ). For t = t 0 ", all activity levels for equilibrium I are positive but the activity level y j (t) vanishes at t = t 0 and would become negative at t = t 0 + ". When method j is replaced by another method, say method k, an algebraic decomposition of the vector d(t) leads to the formal equality

d(t) = X i2M nfjg y i (t)c i + y j (t)c j = X i2M nfjg y 0 i (t)c i + y 0 k (t)c k (6)
At t = t 0 both decompositions coincide, with y i (t 0 ) = y 0 i (t 0 ) > 0 and y j (t 0 ) = y 0 k (t 0 ) = 0: At t = t 0 + ", y 0 i (t) remains positive by continuity and y 0 k (t) has a small nonzero value. The new set of methods including method k sustains the demand vector d(t 0 + ") if and only if y 0 k (t) is positive, whereas y j (t) has become negative. Proof. Equality [START_REF] Freni | Capitale tenico nei modelli dinamici ricardiani[END_REF] shows that the n + l 1 vectors c i for i 2 M n fjg and the vector y j (t)c j y 0 k (t)c k are linearly dependent, therefore y j (t) det C I y 0 k (t) det C II = 0. y 0 k (t) and y j (t) have opposite signs if and only if it is the same for the two determinants.

The condition on the sign of the two determinants can be considered as a productivity condition, that productivity being relative to the local variations of the demand vector. It follows from Lemmas 1 and 2 that: Proposition. Let the rate of pro…ts be the given distribution variable. The Ricardian dynamics when demand changes work locally if and only if the sign of det C I (r)= det C I is preserved when the outgoing method is replaced by the incoming method.

As a consequence, the Ricardian dynamics always work when the rate of pro…t is zero and, by continuity, when it is small enough. They also hold in the case of single-product methods with extension of cultivation on specialized lands: simpli…cations then occur in the calculation of the determinants, and the signs of det C I (r) and det C I are deduced from those of det(I (1 + r)A) and det(I A), which are both positive by the Hawkins-Simon criterion. On the contrary, the productivity condition matters in the case of pure intensive cultivation, as shown by Example 2 in the next Section.

If the condition stated in the Proposition holds for any pair of consecutive sets, the Ricardian dynamics work globally. Let us link a basket d 0 to another d 1 by a path d t . An equilibrium at d 0 is progressively transferred, by means of successive transforms along the path, to another at d 1 . If d 0 is sustained by a unique equilibrium, could another path joining d 0 to d 1 lead to a di¤erent equilibrium at d 1 ? Along the path, an equilibrium I is transformed into a uniquely de…ned equilibrium II when some activity level reaches its bound. The process is reversible in the sense that, if one starts from equilibrium II and moves in the opposite direction on the same path, its successor is equilibrium I (because, by construction, the equilibrium which succeeds equilibrium II is sustained by the price vector II , which is equal to I ). Imagine that demand d 1 is sustained by several equilibria. By following the path d t , each of them is transferred to d 0 and two equilibria never merge during the successive transforms: otherwise, when moving in the opposite direction, an equilibrium would have two successors. As multiple equilibria at d 1 would give birth to multiple equilibria at d 0 , the uniqueness property at d 0 ensures uniqueness for any demand basket.

Ricardo and Lemke

Lemke and Howson (1964) and [START_REF] Lemke | Bimatrix equilibrium points and mathematical programming[END_REF] introduced an algorithm to …nd a solution of a bimatrix game. The algorithm was soon adapted to a variety of problems, including linear and nonlinear complementarity problems (that class includes general equilibrium). The linear complementarity problem LCP(q; M ), where q is a given vector in R s and M an s s matrix, consists in …nding two nonnegative and orthogonal vectors z and w linked by the relationship w = q + M z (the orthogonality condition amounts to stating that w i or z i is zero for any component i). When free disposal and fallowing are not explicitly taken into account as methods of production, a long-term equilibrium can alternatively be de…ned as -a solution to a set of inequalities with complementarity relationships:

(B A)y d [p] (7) y [ ] (8) p
T (B (1 + r)A) T l [y] (9)
The …rst inequality means that demand is met, the overproduced commodities having a zero price. The second inequality expresses the scarcity of lands, with a zero rent on non fully cultivated lands. The third inequality means that the operated methods yield the ruling rate of pro…ts, and the non-operated methods do not yield more. With z obtained by stacking the price vector p, the rent vector and the activity levels y; and q by stacking d; and l, the system ( 7)-( 8)-( 9) is written as w = M z +q 0 where the vector w, obtained by stacking the excess supply, the areas of non cultivated lands and the overcosts, is complementary to z. An equilibrium is therefore a solution of the complementarity problem LCP(q; M ) of dimension s = n+l+m: (Conversely, the equalities (1)-( 2) are nothing but the inequalities ( 7)-( 8) rewritten with slack variables.) The speci…c structure of matrix M is inessential for the present purpose. The idea of the parametric LCP algorithm (Cottle et al., 1993) is to …nd a solution to LCP(q 1 ; M ) by transferring a given solution to LCP(q 0 ; M ) along the path q(t) = q t = (1 t)q 0 + tq 1 . A solution at q 0 , more generally at q t , is characterized by a set S 0 of s zero components of vector (z; w), the other s components being strictly positive in normal times. Slight changes in q t are met by adjusting these positive components. A limit is reached when some positive component, say w j (t), vanishes at t = t 0 . Then the next set of solutions is obtained by solving the a¢ ne system w = M z + q and imposing the same zero components to (z; w), except that the constraint w j = 0 is substituted for the previous one z j = 0. In a neighborhood of t 0 , the s 1 positive components of (z(t 0 ); w(t 0 )) remain positive by continuity whereas z j (t), which is zero for t = t 0 , is close to zero at t 0 + ":

It turns out that the parametric LCP algorithm is nothing but a version of the Ricardian dynamical model in which vector q plays the role of an augmented demand vector. In general, the transfer from q t to q t+" sets no problem, with a rupture from time to time when a positive variable drops to zero. Then the new system is a neighbor of the previous, in the sense that only one condition is changed. The search for an incoming method as described in Section 3 is also similar to that of a new facet in the LCP problem.

A signi…cant di¤erence, however, occurs when z j (t 0 +"), which we know to be close to zero, is negative. Then, the Ricardian dynamics stop because the incoming method is not productive enough. By contrast, the Lemke algorithm goes on by making a U-turn on the path: since z j (t 0 ") is then positive, a second solution to LCP(q t ; M ) for t = t 0 " is found. (The relative sign of two determinants is reversed with regard to that of the initial sequence of equilibria, and this sign plays a role similar to that of the index in the di¤erentiable version of general equilibrium theory [START_REF] Mas-Colell | The Theory of General Economic Equilibrium: A Di¤erentiable Approach[END_REF].) The algorithm continues for lower and lower values of t ('antitone'move, in the terminology of the LCP literature). Eventually, however, another reverse may occur (or must occur, if the conditions ensuring the working of the parametric algorithm are met) and after the change of several methods of production, a new long-term equilibrium sustains increasing levels of demand. A third solution sustaining q(t 0 ") is …rst reached and, eventually, a solution for level q 1 , as an example shows: Example 2. Let corn be the unique good and consider three methods of cultivation on a homogenous land with total area = 20 acres. The methods are: method 1 : 3 corn + 4 labor + 4 acres ! 10 corn method 2: 2 corn + 7 labor + 5 acres ! 10 corn method 3: 3 corn + 21 labor + 1 acre ! 10 corn The given rate of pro…t is r = 1. For low levels of demand, method 1 is the cheapest (the price of corn in terms of wage is p 1 = 1, and the rent 1 is zero). At level d = 35, the land is fully cultivated and the Ricardian dynamics stop because the next cheapest method is method 2, which is less productive per acre than method 1. On the contrary, the Lemke algorithm goes on: at price p 1;2 = 2 and rent 1;2 = 1, method 2 is progressively substituted for method 1 and the net product drops to 32. Eventually, at price p 2;3 = 7 and rent 2;3 = 7, method 3 is introduced jointly with method 2 and, after a progressive substitution, the net product increases up to 140. Two comments are in order. First, the phenomenon we point at remains if the number of methods was a continuum. Second, the solutions to Example 2 give the set of equilibria for any level of demand, because there is only one good. In a more general framework, the powerfulness of the Lemke algorithm is illustrated by the following observation. Assume that d 0 is sustained by a unique equilibrium (which is the case for single-product systems if d 0 is small), and the same for d 1 (for instance, d 1 is high). For another demand vector d, let us draw a path from d 0 to d 1 with d 0:5 = d. Crucial properties of the Lemke algorithm are that the sequence of equilibria generated along the path is made of distinct techniques (the algorithm does not cycle) and that the passage from a technique to the next is uniquely determined and reversible. The algorithm which starts from the unique equilibrium at d 0 will eventually reach the unique equilibrium at d 1 . The algorithm which starts from a given equilibrium sustaining d will not stop before reaching the unique equilibrium sustaining either d 0 or d 1 . Therefore the second path is a part of the …rst. This shows that any equilibrium at d is reached on the path d t (exhaustivity property).

References to the literature

The revival of analytical Ricardian studies is due to [START_REF] Sra¤a | Production of Commodities by Means of Commodities[END_REF]. Sra¤a rejected Ricardo's idea of a natural order of cultivation because the decision to cultivate some land rather than another depends on the relative costs, which are sensitive to the distribution of income. [START_REF] Montani | Scarce natural resources and income distribution[END_REF] showed that, for a given rate of pro…ts and in the case of pure extensive rent, the order of cultivation coincides with that of the decreasing wages paid by the agricultural methods for a zero rent. By contrast, the Ricardian dynamics do not work automatically in the case of pure intensive rent [START_REF] Bidard | The dynamics of intensive cultivation[END_REF].

Probably under the in ‡uence of Sra¤a's preface which stresses that no change in quantities is ever considered (though Chapter 11 on lands makes a clear reference to the Ricardian dynamics), most post-Sra¢ an studies assume a given level of demand. From this abundant and intricate literature, we retain for our purpose the recognition of multiple equilibria (D' Agata, 1982;[START_REF] Freni | Capitale tenico nei modelli dinamici ricardiani[END_REF] and of non-Ricardian equilibria [START_REF] Erreygers | Terre, Rente et Choix de Techniques[END_REF]), a general existence result based on the identi…cation with a linear complementarity problem [START_REF] Salvadori | Land and choice of techniques within the Sra¤a framework[END_REF] and the studies on uniqueness [START_REF] Erreygers | Terre, Rente et Choix de Techniques[END_REF][START_REF] Erreygers | On the uniqueness of square cost-minimizing techniques[END_REF]: starting from geometrical considerations, Erreygers elaborates a criterion for local uniqueness identical to that mentioned in the Proposition and states a global uniqueness result (which, in our views, is not totally convincing).

Conclusion

Ricardo's idea was to get rid of lands and rents by reducing the core of the productive system to the marginal methods. The existence of non-Ricardian equilibria, for which some good is not produced by a marginal method, shows the limit of the approach. But the dynamics can only be facilitated by the presence of such equilibria, since the introduction of a corn-saving method in some industry may be an adequate answer to a global lack of corn. Even then, the working of the Ricardian dynamics comes up against a productivity condition: when some long-term equilibrium fails because it cannot meet the increase of demand, the reconstruction of another equilibrium starts from pro…tability considerations, so that the incoming method may be unable to follow the evolution of demand. That productivity condition is expressed as a simple algebraic criterion.

The lessons of the study go well beyond the case of lands: little would be changed if the scarcity constraints concerned consumption or production goods. A formal reason for that generality is that the Ricardian dynamics turn out to be very close to an algorithm used to solve a class of problems often encountered in economics. The main di¤erence is that the Ricardian dynamics, which allow for discontinuities in prices and rents, pays attention to the smoothness of the physical transition between equilibria, and there is no doubt that this condition is economically sensible. The domain of validity of the Ricardian dynamics can be enlarged by taking into account the in ‡uence of relative prices on demand: allowing for substitution in …nal demand alleviates the physical constraints on the scarce goods and facilitates the working of the dynamics.

Lemma 1

 1 det C I (r) and det C II (r) have opposite signs. Proof. Since T II c j (r) = l j j = T I c j (r) j and T I c k (r) = l k k = T II c k (r) k , we have ( II I ) T ( k c j (r) + j c k (r)) = 0. As ( II I ) T c i (r) = 0 for the other operated methods i, the matrix with the columns c i and k c j + j c k has a zero determinant. Therefore k det C I (r)+ j det C II (r) = 0:

Lemma 2

 2 The set of methods fkg [ M n fjg can sustain the demand vector d(t 0 + ") if and only if det C I and det C II have opposite signs.

  1 qr. corn + 2 qr. rice + 4 labour + 1 land 1 ! 7 qr. corn ( 1 = 1) (method 2:) 1 qr. corn + 2 qr. rice + 4 labour + 1 land 2 ! 2 qr. corn Rice methods: (method 3:) 2 qr. corn + 1 qr. rice + 4 labour + 1 land 3 ! 7 qr. rice ( 3 = 2) (method 4:) 1 qr.corn + 3 qr. rice + 4 labour + 1 land 4 ! 7 qr. rice ( 4 > 1:5)

With acknowledgements to Guido Erreygers for many discussions on the topic.


Sylvain Prado
email: sylvain.prado@gmail.com

Free lunch in the oil market: a note on Long Memory

Keywords: oil market, long memory, ARFIMA-FIGARCH. JEL Classi…cation: C22, F47, G17, Q47

In the crude oil market the phenomenon of Long Memory can be easily identi…ed with the help of the simple (but e¤ective) methodology of Katsumi Shimotsu. The Exact Local Whittle estimator and two testing strategies provide a strong assessment of the phenomenon. We present evidences and we suggest a pro…t opportunity. Furthermore, the existence of Long Memory discloses an ine¢ cient oil market.

1 Evidences of Long Memory reveal a pro…t opportunity.

The phenomenon of Long Memory implies that the prices of yesterday will impact the prices of tomorrow on a long term horizon. In other words, an autocorrelation function decaying slowly characterizes a Long Memory process.

The standard model of stock market variations implies that (h), the autocorrelation function of realized returns, must decrease geometrically ( (h) 6 cr h with c > 0). [START_REF] Adelman | Long cycles: facts or artefacts?[END_REF] and [START_REF] Granger | The typical spectral shape of an economic variable[END_REF], however, observed a phenomenon with a hyperbolic decay called Long Dependance or Long Memory ( (h) ch 2H2 with c > 0 and 0 < H < 1). The phenomenon lead [START_REF] Mandelbrot | Une classe de processus stochastiques homothétiques à soi; application à la loi climatologique de H.E. Hurst[END_REF], during his research on fractal, to rediscover [START_REF] Hurst | Long term storage capacity of reservoirs[END_REF]'s law and to introduce the concept of self-similar process which later became [START_REF] Mandelbrot | Fractional Brownian motion, fractional noises, and applications[END_REF] fractional Brownian motion. The Fractal approach moved to econometrics with the discrete time perspective. While the discrete time analog of Brownian motion is the random walk, the discrete time version of fractional Brownian motion is a fractionally di¤erenced white noise.

They are many explanations for the phenomenon in …nancials markets, some of which are related to the various characteristics of the agents and, as a consequence, to the heterogeneity of the trading behaviors.

Econometrical models for Long Memory in mean were …rst introduced by [START_REF] Granger | An Introduction to Long Memory Time Series Models and Fractional Di¤erencing[END_REF] and [START_REF] Hosking | Fractional Di¤erencing[END_REF]. The Autoregressive Fractionally Integrated Moving Average (ARFIMA) manifests a hyperbolic decay rate and constitutes an extension of the ARMA model (with a geometric rate). The ARFIMA framework was itself extended towards volatility models. Baillie et al. (1996) and [START_REF] Bollerslev | Modeling and pricing long memory in stock market volatility[END_REF] introduced the Fractionally Integrated Generalized Autoregressive Conditionally Heteroskedastic (FIGARCH) models re ‡ecting the …ndings that, even when the return series has no serial correlation, the autocorrelation function of the squared, log-squared, or the absolute value series of an asset return could decays slowly. [START_REF] Chung | Estimating the fractionally integrated garch model[END_REF] proposed a signi…cant improvement of the ARFIMA-FIGARCH models to …x a problem of speci…cation from Baillie et al. (1996) 2 .

Regarding economic theory, the existence of Long Memory in …nancial returns goes against the e¢ cient market hypothesis stating that the prices of assets contain all past information and so the expected pro…t for the speculator should be zero. [START_REF] Fama | E¢ cient Capital Markets: A Review of Theory and Empirical Work[END_REF] said that "a market in which prices always fully re ‡ect available information is called e¢ cient". In the weak-form of informationally e¢ cient market, all past information is incorporated in the prices of assets. As a consequence, under the e¢ cient market hypothesis, the oil market should be a faire game with rare opportunities of pro…t.

The academic literature has been highlighting Long Memory phenomenons in markets expected to be ine¢ cient. Numerous papers have identi…ed Long Memory in small …nancial markets of developing countries 3 . Intuitively, ine¢ cient markets are expected to be small, new, or partially aggregated. With more than 86.6 million barrels per day sold in 20104 , the crude oil market should be one of the most e¢ cient with very few opportunity of gains. Surprisingly, although "there is more trade internationally in oil than in anything else"5 , a growing literature is showing evidences of Long Memory in the oil market, especially on volatility 6 .

We demonstrate that, with the help of the simple (but e¤ective) methodology of Shimotsu, the existence of Long Memory in the crude oil market can be easily identi…ed.

A straightforward approach prevents spurious results

By spending enough time, …ddling observations and experimenting statistical tools, an econometrician can reveal spurious patterns correlations from any dataset.

In order to reduce the problem of induction7 embedded in the econometric approach and to prevent spurious …ndings from data mining, we apply a straightforward strategy: initially, we collect the last 20 years volatility of two oil spot prices major index (Brent and WTI8 ). Secondly, we strictly replicate the methodology of [START_REF] Shimotsu | Exact Local Whittle Estimation of Fractional Integration with Unknown Mean and Time Trend[END_REF] to estimate the Long Memory parameter, d. Finally, we run the tests of [START_REF] Shimotsu | Simple (but E¤ective) Tests of Long Memory versus Structural Breaks[END_REF] to assess the quality of d.

To compute the volatility, we transpose the calculation of [START_REF] Shimotsu | Simple (but E¤ective) Tests of Long Memory versus Structural Breaks[END_REF] from a daily basis to a weekly basis. The weekly realized volatility is the sum of square of the daily returns in a week. The …rst return is computed using the closing price of the previous week. To avoid the disruptions created by the days without quotation, we divide the total by the number of open days in the week. Let p de…ne the price of crude oil index at time and the h-period return at time as r +h;h = p +h;h p . Then, the realized weekly volatility of week t is calculated as

x t = v 2 with v 2 = X 1= j=1 r 2 t+j ;
and the number of open days in the week. In our study, we analyze X t = log p x t . At every step, we do not change the values and we do not remove possible outliers. In addition, we do not adjust break points: an alternative methodology would be to identify break points using the algorithm of [START_REF] Sanso | Testing for change in the unconditional variance of …nancial time series[END_REF] and to adjust them using the …lter of [START_REF] Nouira | Long-memory and shifts in the unconditional variance in the exchange rate Euro/US Dollar returns[END_REF]. Whereas the alternative methodology produces an adjusted d, our approach simply consist in the identi…cation of a non spurious d. We use samples from 1rst January 1990 until the 31th December of 2010.

3 We con…rm the existence of Long Memory in the crude oil market.

The volatility densities of the Brent and the WTI shows skewness and excess kurtosis, a common characteristic of …nancial returns9 .

The descriptive statistics and the stationary tests suggest the existence of Long Memory for the two time series: the Autocorrelation function, the Ljung-Box and Lagrange Multiplier test exhibit signi…cant evidences of serial correlation; additionally the ADF test rejects the null hypothesis of the series being I(1) and, at the same time, the KPSS tests also rejects the series being I(0). Therefore modelling these series either as I(0) or I(1) processes appears too restrictive. [START_REF] Shimotsu | Exact Local Whittle Estimation of Fractional Integration[END_REF] proposed an consistent estimator of Long Memory, called the Exact Local Whittle estimator (ELW). Later, [START_REF] Shimotsu | Exact Local Whittle Estimation of Fractional Integration with Unknown Mean and Time Trend[END_REF] created a more generalized version under the assumption that the mean and trend of the observations are unknown: the Two Step ELW (TSELW) estimator10 . Applying the TSELW, we …nd d = 0:625 for the Brent and d = 0:672 for the WTI11 .

We have to mention that the estimation of the long memory parameter is subject to a signi…cant amount of research and controversy. [START_REF] Baillie | Semi Parametric Estimation of Long Memory: Comparisons and Some Attractive Alternatives[END_REF] said that: "[Surprisingly...], one of the most heavily researched topics in this area [Long Memory], continues to be the issue of semi-parametric estimation of the long memory parameter [d]". Lopez and Mendes (2006), for instance, focused on a speci…c parameter in the estimation of d, the bandwidth selection. The tuning constant de…nes the bandwidth m with m = n and n the sample size. In our calculation, we apply a common value of = 0:6 and therefore m = 66. The whole question of Long Memory estimation, however, does not constitutes the purpose of this article.

The presence of structural breaks in the observations can introduce spurious e¤ects in the covariance function and, as a consequence, creates spurious evidence of fractional integration. To deal with the problem [START_REF] Shimotsu | Simple (but E¤ective) Tests of Long Memory versus Structural Breaks[END_REF] provided two testing strategies: In the …rst one, we split the sample into subsamples, then we estimate d for each subsample, and …nally we compare them with the estimate of d from the full sample, using a modi…ed Wald test. Basically, the …rst and the second testing strategy tend to overreject the null d = d. In the second strategy, following a demeaning of the series and using an estimated d, we apply the well known KPSS and PP tests to check for stationarity of the di¤erenced series 12 . Assuming the observations being I(d), the dth di¤erence must be I(0). On the contrary, it would not be I(0) with a spurious I(d).

In the end, the two testing strategies validate the estimated d and we conclude in the existence of Long Memory in the crude all market 13 .

3 The results bring two questions.

As a conclusion, the crude oil market appears ine¢ cient. The results highlight pro…ts opportunities and new questions:

Is the magnitude of the estimated e¤ect big enough to matter? The question is inspired by Mc Closkey and Ziliak (2008) 14 . A highly statistically sig-ni…cant relationship may have a very small e¤ect and not be of substantive importance. To be more speci…c, are the Long Memory phenomenons big enough to provide a pro…t relative to the transaction cost required to exploit them? Our results indicate that modeling these series with the usual ARMA-GARCH models is inapropriate. The next step will be to evaluate how much the gain collected from ARFIMA-FIGARCH models exceeds the transaction cost.

Where does Long Memory come from? Henry and Za¤aroni (2003) provided an exogenous explanation of a Long Memory inherited from underlying geophysical in ‡uences: oil consumption may be impacted by climatic factors containing Long Memory. They provided also an endogenous explanation: Long Memory appears because the observations are an aggregation of outputs created by heterogeneous agent with "time varying cost of adjustment". In the same way, [START_REF] Cont | Long range dependence in …nancial markets -Fractals in Engineering[END_REF] said that Long Memory appears because of the heterogeneity in the time horizon of the economic agents. Another possibility could be the agents changing their trading behavior: Cont (2005) also mention evolutionnary models with a natural selection of agents or investor inertia with a threshold behavior as possible cause of Long Memory. Above all, explanations for Long Memory are numerous and are left for future research.

tests (H0 = I(1)) perform relatively poorly in distinguishing between the I(1) null hypothesis and the I(d) alternative meanwhile the KPSS and LM tests (H0 = I(d)) can be used to distinguish short memory from Long Memory stationary processes. [START_REF] Shimotsu | Simple (but E¤ective) Tests of Long Memory versus Structural Breaks[END_REF], however, showed the complementarity of KPSS and PP test and how by using the two tests it minimizes the risk of overaccept the null (d = d). 13 The statistical results are in Appendix 1 and Appendix 2. The …rst window (1.) represents the observations. The second window (2.) illustrates the identi…cation of skewness and kurtosis. Windows (3.), (4.), and (5.) show the Autocorrelation function, the Ljung-Box and the Lagrange multiplier. The ADF and the KPSS tests are in (6.) and (7.). The last windows (7.) contains the results of the TSELW estimator.

14 See [START_REF] Mccloskey | The cult of statistical signi…cance: how the standard error costs us jobs, justice, and lives[END_REF], Chapter 8, p.89, 'How Big is Big in economics'.

See Baillie (1996),Pérez and Ruiz (2001) and[START_REF] Walter | Research of Scaling Law on Stock Market Variations[END_REF] for a review of concepts and literature.

As an illustration, see[START_REF] Tan | Long Memory Features in Return and Volatility of the Malaysian Stock Market[END_REF] for the Malaysian market or[START_REF] Korkmaz | Testing for Long Memory in ISE Using ARFIMA-FIGARCH Model and Structural Break Test[END_REF] for the Turkish market.

The U.S. Energy Information Administration (EIA).

The U.S. Energy Information Administration (EIA).

See Choi and Hammoudeh (2009) and[START_REF] El Hedi Arouri | Forecasting the conditional volatility of oil spot and futures prices with structural breaks and long memory models[END_REF].

The problem of induction discloses the philosophical question of whether inductive reasoning leads to knowledge.

The Brent (produced in the North Sea region) and the West Texas Intermediate (WTI -Cushing, produced in Texas and southern Oklahoma) serve as a reference or 'marker'for pricing a number of other crude streams. Source: http://www.eia.gov/dnav/pet/pet_pri_spt_s1_d.htm

See Cont (2001).

The ELW and the TSELW are semiparametric estimators. Shimostu modi…ed the ELW objective function to estimate the mean by combining two estimators: the sample average, and the …rst observation depending on the value of d. In addition, Shimotsu dealt with the presence of a linear or quadratic time trend by …rst detrending the data. Above all, Shimotsu showed that the TSELW estimator has a N (0; 1 4 ) asymptotic distribution for d 2 ( 1 2 ; 2) (or d 2 ( 1 2 ; 7 4 ) when the data has a polynomial trend).

 11 The ELW and the TSELW produced similar estimations. The complete results are available upon request. The TSELW estimator, as well as the testing strategies, are implemented using the Matlab R code provided by Katsumi Shimotsu. Other calculations are done using common features from Eviews R and Matlab

R .12 In a review of litterature,Baillie (1996, chapter 4.3) mentioned that the Dickey-Fuller and the PP

Appendix

Appendix 1: Brent Volatility.

(1 .

(2 .

(3 . (4 .

(5 .

(6 . (7 .

(8 .

Appendix 2: WTI Volatility.

(1 .

(2 .

(3 . (4 .

(5 .

(6 . (7 .

(8 .


Marie Brière

Bastien Drut

Valérie Mignon
email: valerie.mignon@u-paris10.fr.

Kim Oosterlinck

Ariane Szafarz

Is the Market Portfolio Efficient? A New Test to Revisit the Roll (1977) versus Levy and Roll (2010) Controversy

Keywords: Efficient portfolio, mean-variance efficiency, efficiency test. JEL codes: G11, G12, C12. '

recently revived the debate related to the market portfolio's efficiency, suggesting that it may be mean-variance efficient after all. This paper develops an alternative test of portfolio mean-variance efficiency based on the realistic assumption that all assets are risky. The test is based on the vertical distance of a portfolio from the efficient frontier. Monte Carlo simulations show that our test outperforms the previous mean-variance efficiency tests for large samples since it produces smaller size distortions for comparable power. Our empirical application to the U.S. equity market highlights that the market portfolio is not mean-variance efficient, and so invalidates the zerobeta CAPM.

Introduction

Testing the mean-variance (MV) efficiency of the market portfolio, or equivalently testing the validity of the Capital Asset Pricing Model (CAPM) of [START_REF] Sharpe | Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk[END_REF] and [START_REF] Lintner | Security Prices, Risk, and the Maximal Gains from Diversification[END_REF], is a major task for financial econometricians. The debate on this issue dates back to the breakthrough theoretical contributions of [START_REF] Roll | A Critique of the Asset Pricing Theory's Tests. Part I: On Past and Potential Testability of the Theory[END_REF] and [START_REF] Ross | The Capital Asset Pricing Model (CAPM), Short-Sale Restrictions and Related Issues[END_REF] questioning the efficiency of the market portfolio. In the wake of these contributions, numerous empirical studies [START_REF] Gibbons | Multivariate Tests of Financial Models: A New Approach[END_REF][START_REF] Gibbons | A Test of the Efficiency of a Given Portfolio[END_REF][START_REF] Mackinlay | Using Generalized Method of Moments to Test Mean-Variance Efficiency[END_REF]among others) found that the market portfolio may indeed lie far away from the efficient frontier.

Ironically, this debate was recently fuelled by [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF], who published an article in the Review of Financial Studies entitled "The market portfolio may be mean-variance efficient after all". Based on a new test, we take a fresh look at this issue with the ambition to arbitrate between the contradictory arguments of [START_REF] Roll | A Critique of the Asset Pricing Theory's Tests. Part I: On Past and Potential Testability of the Theory[END_REF] and [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF].

More generally, all portfolio managers are-or should be-faced with the issue of checking whether a given portfolio is optimal within a predefined investment universe. For this purpose, MV efficiency, as defined by [START_REF] Markowitz | Portfolio Selection[END_REF][START_REF] Markowitz | Portfolio Selection: Efficient Diversification of Investments[END_REF], remains the key optimality concept.

Currently, the econometric literature offers a wide variety of tests for MV efficiency. Most are designed for universes that include a riskless asset. 1 This represents a considerable constraint when it comes to practical implementation. By contrast, this paper focuses on MV efficiency tests that allow all assets to be risky. 1 When the investment universe includes a riskless asset, the efficient frontier is a straight line, which makes the derivations far simpler [START_REF] Gourieroux | Econométrie de la Finance : Approches Historiques[END_REF]. Tests falling in this category have been proposed by [START_REF] Gibbons | Multivariate Tests of Financial Models: A New Approach[END_REF], [START_REF] Jobson | Potential Performance and Tests of Portfolio Efficiency[END_REF], and [START_REF] Mackinlay | Using Generalized Method of Moments to Test Mean-Variance Efficiency[END_REF], among others. The test introduced by [START_REF] Gibbons | A Test of the Efficiency of a Given Portfolio[END_REF] has since then become the standard. [START_REF] Michaud | The Markowitz Optimization Enigma: Is 'Optimized' Optimal?[END_REF] and [START_REF] Green | When Will Mean-Variance Efficient Portfolios Be Well Diversified[END_REF] discuss the limitations of this framework. Besides, MV efficiency tests must be distinguished from MV spanning tests, which examine whether the efficient frontier built from a given set of assets intersects the frontier resulting from a larger set (see De [START_REF] De Roon | Testing for Mean-Variance Spanning: A Survey[END_REF] for a survey).

The assumption that all assets are risky is highly relevant given that riskless assets are no longer realistic in modern financial markets. The recent debt crisis has highlighted that even the supposedly safest assets, namely sovereign bonds issued by developed countries, are exposed to default risk. In the same way, the freezing of the money markets and the Lehman Brothers' bankruptcy underlined the counterparty and liquidity risks associated with money market investments (Acharya et al., 2010;[START_REF] Bruche | Deposit Insurance and Money Market Freezes[END_REF][START_REF] Krishnamurthy | The Financial Meltdown: Data and Diagnoses[END_REF].

Investors can thus meet severe restrictions on borrowing [START_REF] Black | Capital Market Equilibrium with Restricted Borrowing[END_REF], and the riskless borrowing rate can largely exceed the Treasury bill rate [START_REF] Brennan | Capital Market Equilibrium with Divergent Borrowing and Lending Rates[END_REF]. For all these reasons, MV efficiency is better tested without assuming the availability of a riskless asset.

Two broad classes of MV efficiency tests for risky-asset universes exist in the literature: likelihood-based tests and geometric tests. The likelihood-based tests are directly inspired by the formulation of the CAPM. While the riskless asset is needed to establish the original CAPM, further refinements by [START_REF] Black | Capital Market Equilibrium with Restricted Borrowing[END_REF] allow the riskless asset to be replaced by the zero-beta portfolio. To address the nonlinearities embedded in the Black CAPM, Gibbons (1982) builds a likelihood-ratio test statistic, for which [START_REF] Kandel | The Likelihood Ratio Test Statistic of Mean-Variance Efficiency without a Riskless Asset[END_REF][START_REF] Kandel | The Geometry of the Likelihood Estimator of the Zero-Beta Return[END_REF] derives the exact asymptotic chi-square distribution. However, because this test uses the Gauss-Newton algorithm, practical implementation turns out to be complex [START_REF] Zhou | Small Sample Tests of Portfolio Efficiency[END_REF]. Moreover, [START_REF] Shanken | Multivariate Tests of the Zero-Beta CAPM[END_REF] shows that Gibbons ' (1982) test tends to over-reject MV efficiency in finite samples. 2 Levy and Roll (2010) (henceforth, LR) offer a novel likelihood-ratio test for MV efficiency. This test is based on implicitly estimating the zero-beta rate by determining the minimal changes to sample parameters that make a market proxy efficient. 3On the other hand, the first geometric test of [START_REF] Basak | A Direct Test for the Mean-Variance Efficiency of a Portfolio[END_REF] (henceforth, BJS) is based on the "horizontal distance" between the portfolio whose MV efficiency is in question and its same-return counterpart on the MV efficient frontier. 4 Unfortunately, some portfolios lack such a counterpart [START_REF] Gerard | International Portfolio Diversification: Currency, Industry and Country Effects Revisited[END_REF], which in turn limits the applicability of the BJS test. By contrast, the "vertical test" proposed in this paper circumvents this limitation. Indeed, the vertical inefficiency measure proposed by [START_REF] Kandel | Portfolio Inefficiency and the Cross-Section of Expected Returns[END_REF], [START_REF] Wang | Efficiency Loss and Constraints on Portfolio Holdings[END_REF][START_REF] Li | Diversification Benefits of Emerging Markets Subject to Portfolio Constraints[END_REF], namely the difference between the portfolio's expected return and the expected return of its same-variance counterpart on the MV efficient frontier, is well defined for any portfolio.

Our contribution is twofold. First, we define the vertical test statistic for MV efficiency, establish its asymptotic distribution, and compare its size and power performances to those of the LR and BJS tests through Monte Carlo simulations. While no clear hierarchy emerges for small samples, the vertical test outperforms its competitors for large samples as it exhibits equivalent power with a smaller size. Secondly, we re-examine the market portfolio MV efficiency using the three tests under review (LR, BJS and the vertical tests). Irrespectively of the number of stocks in the universe, we find that the market portfolio is never MV efficient according to both the BJS and the vertical tests. For the LR test, the conclusion depends on the value given to the coefficient α, which determines the relative weight assigned to sample mean changes against standard deviation changes. In other words, the LR test reaches no clear-cut and definitive conclusion regarding the market portfolio's efficiency. Although still frail, the evidence points to the inefficiency of the market portfolio, supporting the [START_REF] Roll | A Critique of the Asset Pricing Theory's Tests. Part I: On Past and Potential Testability of the Theory[END_REF] critique of the CAPM.

The paper is organized as follows. Section 2 presents the vertical test and its asymptotic properties. Section 3 assesses the size and power of the vertical test and its two competitors.

Section 4 tests the Black CAPM on the U.S. equity market. Section 5 concludes.  are respectively given by:

The Vertical Test of Mean-Variance Efficiency

   T t t R T 1 1  (1) ' ' 1 ˆ1        t T t t R R T (2)    T t t r T 1 1  (3)     T t t r T 1 2 2 ) ( 1 ˆ  ( 4
)
where t R and t r are the date-t returns on the N primitive assets and on portfolio P, respectively.

As illustrated by Figure 1, the "horizontal distance" underlying the BJS test measures of portfolio P inefficiency is the difference between the variance of P and the variance of its same-expected-return counterpart on the efficient frontier. Our vertical test is conceived by transposing the BJS (2002) methodology to the vertical inefficiency measure introduced by [START_REF] Kandel | Portfolio Inefficiency and the Cross-Section of Expected Returns[END_REF], [START_REF] Wang | Efficiency Loss and Constraints on Portfolio Holdings[END_REF][START_REF] Li | Diversification Benefits of Emerging Markets Subject to Portfolio Constraints[END_REF]. Hence, the vertical test statistic 5 is the distance between the expected return of portfolio P and the expected return of its same-variance MV efficient counterpart. The estimated distance, denoted by  ˆ, is the solution to the following program:

                           p i for t s i p i i ,..., 1 , 0 , 1 ' . . ' min ˆ1 2           (5)
5 Another possibility would be to take the minimal Euclidian distance between portfolio P and the efficient frontier. This approach would certainly be more elegant, but would also be much more tedious as it would mix up first and second order parameters.

The following proposition states that, under the null that portfolio P is MV efficient, estimator  ˆ asymptotically follows a normal distribution: Proposition 1  ˆ asymptotically follows a normal distribution:

) , 0 ( ) ( 2    N T   as   T . ( 6
) with                    V V    2
, where  represents the asymptotic covariance matrix of the distinct elements of  ˆ,  ˆ,  ˆ, and ˆ, and

        V
 is given by (A2) in Appendix A.

Proof: See Appendix A.

As for the BJS test, this asymptotic result does not require normality assumptions on the asset returns. Moreover, as demonstrated in Appendix A, this result holds both with and without short-selling restrictions.

Power and Size Performances

In this section, we assess the size and power of the vertical test and compare its performances to those of the BJS and LR tests. To this end, we simulate series of returns drawn from the investment universe imagined by [START_REF] Das | Portfolio Optimization with Mental Accounts[END_REF], including three assets with jointly normal returns having the following parameters: (8) [START_REF] Das | Portfolio Optimization with Mental Accounts[END_REF] interpret the first asset as a bond, the second as a low-risk stock, and the third as a highly speculative stock. For the sake of comparability,6 we focus here on the case where short-selling is allowed.

           25 . 0 10 . 0 05 . 0             
We simulated 1,000 series of returns of lengths 60, 120, 180, and 240, respectively. In each case, two groups of portfolios were composed. The portfolios in the first group were generated on the efficient frontier in order to estimate the risk of type I error (false rejection of the true hypothesis that portfolios are mean-variance efficient). The portfolios in the second group were generated below the efficient frontier to estimate the risk of type II error (failure to reject the false hypothesis).

We follow the assessment of statistical tests suggested by [START_REF] Wasserman | All of Statistics: A Concise Course in Statistical Inference[END_REF]. This procedure is based on power maximization (i.e., minimization of the risk of type II error) for a given small size (i.e., risk of type I error). Figure 2 features all tested portfolios on a grid in the MV plane. To each of them, we successively apply the BJS, LR, and vertical tests.

                           p i for t s i p i i ,..., 1 , 0 , 1 ' . . ' min ˆ1 2           (9)
Under the null that portfolio P is MV efficient,  ˆ asymptotically follows a normal distribution:

) , 0 ( ˆ2   N T  as   T .
The [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF]

                , , , min arg * *, ) , ( d N N            
where distance d is defined by:

     2 1 2 1 ˆ1 ) 1 ( ˆ1 , , ,                          N i i i i N i i i i N N d             ( 10
)
and  is a coefficient determining the relative weight assigned to deviations in means relative to the deviations in standard deviations. 7For simplicity, [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF]

                                  N N C * 0 0 0 * 0 0 * * 0 0 0 * 0 0 * * 2 1 2 1                 (11)
Where C ˆ is the sample correlation matrix. In that way, only the variances have to be estimated.

Under the hypothesis that the N original assets follow a jointly normal distribution, the likelihood ratio is given by:

                                   ' * * * * log 1     trace N T ( 12
)
This test statistic asymptotically follows a chi-square distribution with N 2 degrees of freedom.

The choice of the trade-off parameter  in Equation ( 10) is instrumental to the implementation of the LR test. Indeed, a low (resp. high) value of would create a bias towards standard deviations (resp. means). In extreme cases ( 0   and 1  

), the asymptotic distribution of the LR test statistic degenerates into a chi-square with N degrees of freedom. In our performance assessments, we follow [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF] and set the value of α to 0.75.

False Rejection of Efficient Portfolios

We first assess the type I error. The four simulated efficient portfolios have expected returns of 10%, 15%, 20% and 25%, respectively. The rejection frequencies of the null of portfolio efficiency at the 5% probability level are displayed in Table 1. 8 The results show that the size is uniformly the lowest for the vertical test, followed by the LR test. Nevertheless, the vertical test, and to a lesser extent the LR test, exhibit rejection frequencies that lie below the theoretical threshold of 5%. 8 The results for the 1% and 10% probability levels are given in Table B1 in Appendix B.

Rejection of Inefficient Portfolios

We now apply the three MV efficiency tests under review to thirteen portfolios simulated as inefficient in order to assess the probability of falsely concluding that the portfolio was efficient. The results are given in Table 2 for 5% probability.9 For sample sizes below 180, the power is the lowest for the vertical test, and the highest for the BJS test. However, for larger samples, the vertical test outperforms both the BJS and the LR tests since its size is the lowest for an equivalent power. On the whole, Tables 1 and2 indicate that the vertical test rejects the null of MV efficiency less frequently than the two other tests.

The differences in power and size between the vertical test and the BJS test might look surprising since both are similar in spirit, namely they are both built from a geometric onedimensional measure of inefficiency in the MV plane. This counterintuitive result stems from the fact that the standard deviation of the vertical measure of inefficiency is higher than the standard deviation of the horizontal measure used in the BJS test. Indeed, the standard deviations of both tests depend on the absolute values of the weighting loads of the testedportfolio efficient counterpart. However, the efficient "vertical counterparts" are mostly located on the top of the efficient frontier while the efficient "horizontal counterparts" are mostly located at the bottom of the efficient frontier. Since absolute weighting loads are typically higher on the top of the efficient frontier (riskier portfolios are less diversified), the vertical distance is subject to higher standard deviations than the horizontal BJS test.

Consequently, the t-statistic generally takes lower values for the vertical test than for the BJS test, and hence the former rejects MV efficiency less frequently than the latter. This feature is particularly relevant when short-selling restrictions are imposed (see [START_REF] Best | On the Sensitivity of Mean-Variance Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results[END_REF][START_REF] Green | When Will Mean-Variance Efficient Portfolios Be Well Diversified[END_REF][START_REF] Britten-Jones | The Sampling Error in Estimates of Mean-Variance Efficient Portfolio Weights[END_REF].

Robustness Checks on the Slope of the Efficient Frontier

Both the horizontal and vertical measures of portfolio inefficiency are restricted to a single dimension in the MV plane. They are, therefore, sensitive to the slope of the efficient frontier.

For this reason, we check the robustness of our previous findings by substantially modifying the slope of the efficient frontier. This is achieved by running simulations under two alternative scenarios for the expected return on the speculative stock (15% and 35% respectively instead of 25%) while keeping all other parameters in Equation ( 8) unchanged.

As Figure 3 shows, the first case (15%) produces a flatter efficient frontier, whereas the second (35%) leads to a steeper MV efficient frontier. The minimum-variance portfolios of the three efficient frontiers still remain very close to each other. As previously, we apply the three efficiency tests to a grid of efficient and non-efficient simulated portfolios. The results are reported in Tables C1 to C4 in Appendix C. They can be summarized as follows. For the flat efficient frontier, the BJS test produces the highest size distortions, while the vertical test exhibits the lowest. Given that the BJS test outperforms the other two tests in terms of power irrespective of the sample size, a reasonable procedure for practical use is to combine the BJS and the vertical tests when the MV efficient frontier is flat. In the case of a steep efficient frontier, the results are similar to those obtained in the benchmark case. The vertical test exhibits the lowest size distortions, and its power strongly increases in comparison to the benchmark case, especially for small samples. On the whole, our results

show that the vertical test is preferable when the efficient frontier is steep and samples are large.

Is the Market Portfolio Efficient?

In this section, we apply the BJS, the LR, and the vertical tests of MV efficiency to the capitalization-weighted market portfolio made up of the 100 largest U.S. stocks 10 ). 12 In each case, we select the largest stocks of the sample. For the LR test we follow the original paper when assessing MV efficiency and use a value of α equal to 0.75. As a robustness check, we also test the MV efficiency for a value α (0.98), which gives a similar importance to deviations from variance and mean. 13Lastly, we apply the three tests to equally-weighted portfolios as robustness checks.

Figure 4 shows the efficient frontiers (without short-selling restrictions) made of 10, 50 and 100 assets, respectively, and the corresponding market portfolios. Noticeably, the MV characteristics of the market portfolio are stable with respect to the number of assets, but the efficient frontier becomes steeper when N increases. In particular, this feature shows that all configurations explored in Section 3 are realistic.

Table 3 summarizes the outcomes of the three tests. Two findings stand out. Firstly, for all sample sizes, both the BJS and the vertical tests reject the null of market portfolio efficiency.

Regardless of the number of stocks in the universe, the market portfolio is never MV efficient.

Similar results are found for equally-weighted portfolios (see Table 4).

Secondly, for all values of N, the LR test does not reject market portfolio efficiency for α = 0.75, confirming the findings of [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF]. 14 However, for α = 0.98 the LR test rejects market portfolio efficiency. This indicates that the LR test is sensitive to the value taken by parameter α. In fact, for α higher than 0.902 MV efficiency is always rejected by the LR test. 14 Even though our sample period is longer than in [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF]. On the whole, our findings support [START_REF] Roll | A Critique of the Asset Pricing Theory's Tests. Part I: On Past and Potential Testability of the Theory[END_REF] over [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF]. Indeed, while the conclusion of the LR test depends on the trade-off coefficient , the two other tests unequivocally conclude that the market portfolio is never MV efficient. The validity of the zero-beta CAPM, relying on the efficiency of the market portfolio, is thus strongly called into question. In a nutshell, the fundamental contributions of both [START_REF] Roll | A Critique of the Asset Pricing Theory's Tests. Part I: On Past and Potential Testability of the Theory[END_REF] and [START_REF] Ross | The Capital Asset Pricing Model (CAPM), Short-Sale Restrictions and Related Issues[END_REF] remain highly relevant for portfolio management.

Conclusion

This paper develops a new test of portfolio MV efficiency based on the realistic assumption that all assets are risky. The test is based upon the vertical distance of a portfolio from the efficient frontier. While the evidence is mixed for small samples, our test outperforms the previous MV efficiency tests proposed by [START_REF] Basak | A Direct Test for the Mean-Variance Efficiency of a Portfolio[END_REF] and [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF] for large samples since it produces lower size distortions for comparable power. The empirical analysis shows that the LR test is sensitive to the value taken by the nuisance parameter determining the relative weight assigned to sample-mean changes against standard-deviation changes. Furthermore, both the vertical and horizontal tests are based on intuitive measures in the MV plane and are, therefore, easy to visualize, which makes them more appealing than the LR test.

The ideally balanced distance in the MV plane remains, however, the orthogonal distance.

Even though a test based on this distance is feasible in theory, deriving its closed-form asymptotics could prove challenging. We leave this for further work. Meanwhile, the best alternative for practitioners to test portfolio efficiency is probably the dual approach combining the vertical and horizontal tests. In the final decision, the weight to be allocated to each test should then take into account the curvature of the efficient frontier.

The existing MV efficiency tests could be improved in several ways. The LR test could be generalized by relaxing the short-selling restriction. For all tests, implementing the jackknifetype estimator of the covariance matrix developed by [START_REF] Basak | Jackknife Estimator for Tracking Errors Variance of Optimal Portfolios[END_REF] could offer a promising extension since this estimator produces a more accurate covariance matrix than the sample one.

Our empirical application to the U.S. equity market highlights that the market portfolio is not MV efficient, invalidating the zero-beta CAPM. Consequently, regarding the [START_REF] Roll | A Critique of the Asset Pricing Theory's Tests. Part I: On Past and Potential Testability of the Theory[END_REF] versus [START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF]

Y R r r R R vec R U  
and its sample mean,U , is:

  1 1 1 1 ' : ' : : ' T T t t t t t t t U U R Y r w T T      
Let (.) g denote the function that maps vector U to vector V :

V vec w vec Y w Y g U g                                                  ² ˆˆ) ( 2 ˆ) ' ( ) (          
By applying the delta method, when T tends to the infinite, we have: and0  being the covariance matrix of t U , and from BJS (2002BJS ( , p. 1208)):

    ( ) ( ) (0, ) T V V T g U g N       (A1) where ' 0 D D   (A2)            j i U g D
                                             1 : 2 0 : 1 0 0 0 0 0 2 2 2 1 2  pv p p pv pv p pv p p U x K I K I x g D T  (A3) Where ( 1) 2 p p pv   ;     1 ) 1 ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) 1 ( : 0 0 : : 0                  i p i i p i i p i p i i i p i I K   ; i  ˆ stands
for the i th element of  ˆ, and Z I stands for the identity matrix of rank Z.

The asymptotic distribution of vector V is given by (A1). Let us now move to the vertical distance,  ˆ, which is a derivable function of vector V . Consequently, the delta method establishes that the asymptotic variance 2

 of  ˆ is                       V V   ˆ, where derivative V   ˆ
needs to be computed. With this aim, we express that  ˆminimizes the following Lagrangian function:

            ' ) ' ( ) 1 ' ( ' ˆ2 2 1         l (A4)
By differentiation, we have:

V V l V                      ) ' 2 ' ' ' ( ˆ2 1     0 : 1 : 0 : ' 1 : 0 : ²) : : ² : 2 : : 2 : 2 : ² ( : 0 1 2 1 2 1 2 1 1 1 2 pv p p p                    (A5)
From the first order condition applied to (A4), we obtain:

) 2 ( ' 0 2 1 1                 v l p And consequently:     1 : 0 : ²) : : ² : 2 : : 2 : 2 : ² ( : 0 0 : 1 : 0 : ' ˆ2 1 2 1 2 1 1 1 2 1         p p p pv V               (A6)
Combining the results in (A1), (A4) and (A6), we obtain the asymptotic variance 2  of the vertical distance  ˆ:

                       V V    2 (A7)
When there are no short-selling restrictions, the efficient frontier is modified because the sole constraint applied to  is that its components add up to one. Let *  denote the vertical distance in this case. The modified Lagrangian function is:

) ' ( ) 1 ' ( ' * 2 2 1                  l (A8)
By differentiating both sides of (A7), we get:

    1 : 0 : ²) : : ² : 2 : : 2 : 2 : ² ( : 0 0 : 1 : 0 : ' * 2 1 2 1 2 1 1 1 2 1         p p p pv V l              (A9) Lastly, substituting V l   * in (A8) by V l   * from (A5)
gives the asymptotic variance * 2  of the vertical distance *  when there are no short-selling restrictions. Its expression stands as:

                       V V * * * 2    (A10)

Figure 1 .

 1 Figure 1. Horizontal and vertical distances between portfolio P and the efficient frontier

Figure

  Figure 2. Efficient frontier and tested portfolios

Figure 3 .

 3 Figure 3. The three efficient frontiers under consideration

Figure 4 .

 4 Figure 4. Efficient frontiers and market portfolios for the 10, 50 and 100 largest U.S. stocks, respectively. January 1988 -December 2010

Table 1 . Rejection frequencies (in percent) at the 5% probability level for the efficient portfolios

 1 Note: BJS:[START_REF] Basak | A Direct Test for the Mean-Variance Efficiency of a Portfolio[END_REF] test; Vertical: vertical test; LR: Levy and Roll (2010) test. T is the sample size.

			T	B JS	Ver tic al	LR
			60	7.6	0.6	3.7
		10%	120 180	5.5 5.1	0.4 0.4	1.8 1.4
			240	4.1	0.2	1.3
	Expected return	15% 20%	60 120 180 240 60 120 180	6.1 6.4 5.1 4.6 8.6 5.8 5.4	0.6 0.4 0.0 0.0 0.6 0.4 0.3	2.9 1.9 1.3 1.5 3.1 1.7 1.5
			240	4.6	0.2	1.6
			60	6.4	0.6	2.8
		25%	120 180	6.3 5.6	0.4 0.0	1.7 1.5
			240	4.9	0.0	0.0

Table 2 . Rejection frequencies (in percent) at the 5% probability level for the inefficient portfolios

 2 [START_REF] Basak | A Direct Test for the Mean-Variance Efficiency of a Portfolio[END_REF] test; Vertical: vertical test; LR: Levy and Roll (2010) test. T is the sample size.

											Variance
					5%			10%			15%			20%			25%
			T	BJS Vertical	LR	BJS Vertical	LR	BJS	Vertical	LR	BJS	Vertical	LR	BJS	Vertical	LR
			60	89.8	49.1	66.6	94.4	62.0	76.4	96.8	69.0	76.7	96.8	70.3	79.9	98.2	72.3	80.6
		10%	120 99.2 180 100.0	85.4 96.7	93.9 99.1	100.0 100.0	93.4 98.9	96.4 99.6	100.0 100.0	94.7 99.5	96.2 99.7	100.0 100.0	96.6 99.3	95.9 99.3	99.7 100.0	96.4 99.9	96.1 99.4
			240 100.0	99.5	99.9	100.0	99.7	100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0	99.9
	Expected return	15% 20%	60 120 180 240 60 120 180				71.5 92.1 98.8 99.8	24.8 51.7 75.3 88.9	35.1 64.5 86.5 93.8	86.5 98.5 99.6 100.0 35.6 56.3 73.6	38.0 72.6 92.7 97.9 5.2 12.9 25.8	55.4 87.2 97.4 99.5 5.9 12.2 24.8	89.4 99.1 100.0 100.0 64.5 84.2 95.7	49.9 83.1 96.2 99.2 19.2 41.7 67.1	66.7 92.5 98.9 99.9 27.2 53.0 75.6	93.8 99.7 100.0 100.0 75.7 93.8 99.5	55.4 86.8 97.6 99.7 28.5 56.3 81.4	72.3 94.9 99.5 99.9 44.6 71.6 90.5
			240							83.8	38.1	36.6	99.0	82.0	89.9	99.8	93.0	97.0
			60													31.9	3.3	5.6
		25%	120 180													44.6 58.2	9.2 14.7	11.5 19.0
			240													72.0	24.0	28.6
	Note: BJS:

Table 3 . MV efficiency tests for the capitalization-weighted market portfolio

 3

		Annualized
	Nb. of stocks	Expected Return	Volatility (in %)	BJS test	Vertical test	LR test ( = 0.75)	LR test ( = 0.98)
		(in %)
	10	14.84	15.49	-3.11(0.00)	1.28 (0.10)	6.09 (1.00)	161.27 (0.00)
	20	15.55	16.36	-4.58 (0.00)	2.14 (0.02)	15.54 (1.00)	579.43 (0.00)
	30	14.92	15.63	-4.67 (0.00)	2.32 (0.01)	18.87 (1.00)	773.40 (0.00)
	40	15.21	15.64	-5.25 (0.00)	2.94 (0.00)	28.49 (1.00)	1597.15 (0.00)
	50	15.05	15.48	-5.54 (0.00)	3.25 (0.00)	37.61 (1.00)	2562.73 (0.00)
	60	15.20	15.54	-5.90 (0.00)	3.78 (0.00)	48.73 (1.00)	3357.71 (0.00)
	70	15.27	15.40	-6.56 (0.00)	4.46 (0.00)	65.54 (1.00)	3106.69 (0.00)
	80	15.33	15.31	-6.53 (0.00)	4.58 (0.00)	76.76 (1.00)	3491.16 (0.00)
	90	15.23	15.22	-6.83 (0.00)	4.74 (0.00)	89.71 (1.00)	3542.50 (0.00)
	100	15.25	15.22	-7.17 (0.00)	5.05 (0.00)	102.27 (1.00)	4045.07 (0.00)

Coefficient  denotes the MV trade-off in the LR test statistic. p-values are given in parentheses.

Table 4 . MV efficiency tests for the equally-weighted market portfolio

 4 Coefficient  denotes the MV trade-off in the LR test statistic. p-values are given in parentheses.

		Annualized
	Nb. of stocks	Expected Returns	Volatility (in %)	BJS test	Vertical test	LR test ( = 0.75)	LR test ( = 0.98)
		(in %)
	10	14.29	14.95	-3.22 (0.00)	1.33 (0.09)	6.78 (1.00)	197.70 (0.00)
	20	15.34	16.79	-4.56 (0.00)	2.18 (0.01)	15.75 (1.00)	706.71 (0.00)
	30	14.32	15.50	-4.54 (0.00)	2.39 (0.01)	19.37 (1.00)	979.52 (0.00)
	40	15.17	15.72	-4.99 (0.00)	2.90 (0.00)	28.48 (1.00) 1771.03 (0.00)
	50	14.79	15.47	-5.27 (0.00)	3.23 (0.00)	36.90 (1.00) 2681.93 (0.00)
	60	15.22	15.76	-5.65 (0.00)	3.75 (0.00)	47.80 (1.00) 3381.66 (0.00)
	70	15.39	15.46	-6.14 (0.00)	4.36 (0.00)	64.71 (1.00) 3453.09 (0.00)
	80	15.53	15.28	-6.00 (0.00)	4.45 (0.00)	75.95 (1.00) 3938.86 (0.00)
	90	15.21	15.13	-6.29 (0.00)	4.60 (0.00)	89.03 (1.00) 4137.95 (0.00)
	100	15.30	15.17	-6.68 (0.00)	4.92 (0.00)	102.12 (1.00)	4535.09 (0.00)

  controversy, our findings indicate that[START_REF] Roll | A Critique of the Asset Pricing Theory's Tests. Part I: On Past and Potential Testability of the Theory[END_REF] scepticism on the validity of the CAPM seems to survive the recent rehabilitation attempts made by Levy

	and Roll (2010).
	t	' t	t	t	t	t	2	t	t	t r	t w

Table B2 . Rejection frequencies (in percent) at the 1% probability level for the inefficient portfolios

 B2 [START_REF] Basak | A Direct Test for the Mean-Variance Efficiency of a Portfolio[END_REF] test; Vertical: vertical test; LR:[START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF] test. T denotes the sample size.

											Variance
					5%			10%			15%			20%			25%
			T	BJS	Vertical	LR	BJS	Vertical	LR	BJS	Vertical	LR	BJS	Vertical	LR	BJS	Vertical	LR
			60	78.5	14.7	55.3	86.8	24.4	65.8	92.0	32.4	70.4	92.7	35.7	73.2	95.3	38.8	75.4
		10%	120 96.8 180 99.7	44.1 75.2	87.6 97.7	99.5 99.9	65.4 89.2	92.6 99.0	99.6 100.0	71.1 93.0	92.8 99.0	99.7 99.9	75.0 93.2	93.2 98.5	99.5 100.0	76.5 95.6	91.4 98.5
			240 99.9	91.1	99.7	100.0	97.9	99.5	100.0	98.5	100.0 100.0	99.0	100.0 100.0	98.7	99.8
	Expected return	15% 20%	60 120 180 240 60 120 180				49.8 76.1 91.1 96.8	3.2 13.7 31.9 47.5	23.8 51.6 75.2 89.0	68.1 92.2 98.0 99.7 17.1 32.6 46.3	9.0 30.7 59.2 79.6 0.7 0.7 3.1	42.4 77.3 95.1 98.5 3.1 5.0 12.1	78.3 96.2 99.6 100.0 41.9 64.6 83.7	18.0 42.9 75.2 90.5 2.5 8.3 19.1	55.8 86.5 97.1 99.3 16.3 39.3 63.6	84.8 98.4 99.8 100.0 56.4 81.5 95.0	17.7 51.2 79.2 96.0 4.9 18.6 40.9	61.2 89.3 98.7 99.9 31.0 56.4 83.4
			240							59.6	6.5	19.4	94.3	38.2	80.0	98.6	60.5	94.9
			60													13.9	0.1	3.6
		25%	120 180													20.4 31.7	0.8 0.9	4.9 9.0
			240													44.0	2.0	14.2
	Note: BJS:

Table B3 . Rejection frequencies (in percent) at the 10% probability level for the inefficient portfolios

 B3

			Company		Annualized	Annualized		Market
					mean return	volatility (in %)	capitalization
					(in %)					in billion USD
			5%	10%		Variance 15%			20%	as of December 31, 25%
	Expected return	10% 15% 20%	T 60 120 99.9 BJS 94.7 180 100.0 240 100.0 APACHE EMERSON ELECTRIC Vertical LR 70.6 71.8 96.8 BJS 95.1 95.6 100.0 99.4 99.6 100.0 100.0 Vertical 79.5 98.7 99.9 99.9 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 LR BJS Vertical LR BJS Vertical LR BJS Vertical LR 79.7 98.8 84.5 80.7 98.6 86.8 83.6 98.9 87.9 2010 83.1 98.1 100.0 98.7 97.9 100.0 99.2 96.9 99.9 98.9 97.6 99.8 100.0 100.0 99.8 100.0 99.7 99.6 100.0 100.0 99.9 21.4 35.3 43.5 10.9 22.1 43.0 60 81.3 45.0 42.9 91.5 60.8 63.0 93.2 69.9 71.3 96.2 76.0 TARGET 17.2 28.1 42.6 76.4 120 96.5 74.8 72.4 99.6 89.8 90.8 99.8 94.0 94.9 99.8 96.2 97.2 180 99.6 90.4 90.1 100.0 97.7 98.2 100.0 99.6 99.4 100.0 99.5 99.6 240 99.9 97.0 96.4 100.0 99.3 99.6 100.0 99.9 99.9 100.0 100.0 100.0 60 50.3 15.7 8.8 75.4 37.7 34.0 83.3 51.8 51.0 120 71.3 35.1 19.4 91.0 66.0 61.8 96.7 77.5 78.8 180 84.3 48.7 33.5 98.1 84.8 82.8 99.9 93.9 93.5 HONEYWELL INTL. 13.1 30.2 41.5 ELI LILLY 9.3 27.1 40.4 MEDTRONIC 17.7 26.0 39.8 UNITEDHEALTH GP. 30.6 35.1 39.7
			240 60 DOW CHEMICAL		92.0 8.6	63.9	46.2	99.8 35.4	95.5	94.0	99.8 43.7 39.6	98.4 14.0	98.4 7.8
		25%	120 180 COLGATE-PALM. 240 TEXAS INSTS.		14.5 19.0			23.2 41.8			59.8 72.3 38.8 82.9 38.2	22.8 34.6 49.2	15.4 25.0 37.1
			ANADARKO PETROLEUM		16.6			34.7			37.7
			BANK OF NEW YORK MELLON	Note: see Table B1. 13.6		30.9			37.5
			HALLIBURTON		15.0			37.5			37.1
			WALGREEN		16.9			26.3			35.9
			DEERE		15.8			29.5			35.1
			LOWE'S COMPANIES		22.7			35.7			34.6
			DEVON ENERGY		25.5			39.3			33.9
			NIKE 'B'		24.8			33.6			33.2
			SOUTHERN		8.8			17.5			32.1
			PNC FINL.SVS.GP.		8.8			29.1			31.9
			DANAHER		23.1			28.5			30.8
			CORNING		19.7			52.0			30.2
			NEWMONT MINING		10.9			38.9			29.9
			BAXTER INTL.		10.3			24.8			29.5
			FEDEX		14.6			31.0			29.3
			CARNIVAL		17.6			34.6			28.0
			CELGENE		37.1			68.4			27.8
			EXELON		8.6			22.9			27.5
			GENERAL DYNAMICS		13.8			26.1			26.8
			AFLAC		20.1			32.1			26.6
			ILLINOIS TOOL WORKS		14.2			24.5			26.5
			JOHNSON CONTROLS		16.4			29.7			25.9
			HESS		13.6			28.9			25.8
			KIMBERLY-CLARK		9.1			20.2			25.7
			TRAVELERS COS.		9.8			25.9			25.6
			FRANKLIN RESOURCES		22.6			34.2			25.4
			DOMINION RES.		5.9			17.3			25.2
			BAKER HUGHES		12.2			35.7			24.7
			CSX		13.0			26.8			24.2
			DUKE ENERGY		6.1			20.4			23.6
			STATE STREET		17.6			32.8			23.3
			NORFOLK SOUTHERN		11.9			26.8			22.8
			AUTOMATIC DATA PROC.		12.2			21.5			22.8
			GENERAL MILLS		10.1			18.3			22.6
			THERMO FISHER		17.3			30.9			22.0
			SCIENTIFIC
			CUMMINS		20.4			39.0			21.8
			NEXTERA ENERGY		6.9			18.5			21.6
			STRYKER		23.5			32.6			21.3

In reaction to these criticisms, several authors[START_REF] Shanken | Multivariate Tests of the Zero-Beta CAPM[END_REF][START_REF] Shanken | Testing Portfolio Efficiency when the Zero-Beta Rate is Unknown: A Note[END_REF][START_REF] Zhou | Small Sample Tests of Portfolio Efficiency[END_REF][START_REF] Velu | Testing Multi-Beta Asset Pricing Models[END_REF][START_REF] Beaulieu | Finite Sample Identification-Robust Inference for Unobservable Zero-Beta Rates and Portfolio Efficiency with Non-Gaussian Distributions[END_REF] provide lower and upper bounds to the test p-values.

Small variations in expected returns and volatilities may indeed lead to significant changes in the MV efficient frontier[START_REF] Best | On the Sensitivity of Mean-Variance Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results[END_REF][START_REF] Britten-Jones | The Sampling Error in Estimates of Mean-Variance Efficient Portfolio Weights[END_REF].

The null hypothesis is that the "horizontal distance" is zero. BJS derive the asymptotic distribution of this distance. Interestingly, the BJS test can be implemented with and without restrictions on short-selling. Besides, the BJS test can also be used to compare efficient frontiers[START_REF] Ehling | Geographic versus Industry Diversification: Constraints Matter[END_REF][START_REF] Drut | Sovereign Bonds and Socially Responsible Investment[END_REF].

LR solely apply their test to cases where short-selling is allowed. Actually, the performances of their test when short-selling is restricted have not been investigated so far.

See Equation (2) in[START_REF] Levy | The Market Portfolio May Be Mean/Variance Efficient After All[END_REF].

The results corresponding to the 1% and

10% probability levels are given in TablesB2 and B3in Appendix B, respectively.

We selected the 100 largest stocks of the S&P 500 index.

The data are extracted from the Datastream database. Descriptive statistics are given in Appendix D.

In reality, individual investors rarely hold portfolios containing 100 assets[START_REF] Barber | Trading Is Hazardous to Your Wealth: The Common Stock Investment Performance of Individual Investors[END_REF][START_REF] Polkovnichenko | Household Portfolio Diversification: A Case for Rank-Dependent Preferences[END_REF][START_REF] Goetzmann | Equity Portfolio Diversification[END_REF]. The diversification benefits tend to be exhausted once an equity portfolio contains several tens of stocks[START_REF] Evans | Diversification and the Reduction of Dispersion: An Empirical Analysis[END_REF][START_REF] Elton | Risk Reduction and Portfolio Size: An Analytical Solution[END_REF][START_REF] Statman | How Many Stocks Make a Diversified Portfolio?[END_REF].

This value is actually very close to the 0.98-value considered in LR as more realistic than the 0.75 used to test the MV efficiency.

Appendix A: Proof of Proposition 1

We first derive the asymptotic distribution of the vertical distance,  ˆ, defined in Equation (5) in the case where short-selling is forbidden. At the end of this Appendix, we extend the results to the case where short-selling is allowed Let x be a k-dimensional vector, and denote )' ,..., , (

. Consider a symmetric matrix B of order k, and ] : ...

where i B is the th i column of B . Let ) (B vec be the stacked vector of the columns of B:

Next, let V be the vector formed by stacking the sample mean of t R , the elements of ) cov( t R , the sample mean of t r , and the sample variance of t r :

Vector V thus summarizes the first and second moments of the sample returns. Similarly to BJS (2002) Note: see Table B1. Note: see Table B1. Note: see Table B1.