Ricardo And Lemke

Christian Bidard

Keywords: JEL classi…cation B12, C61, C65, D33 Dynamics, Lemke, rent, Ricardo, scarce resources

We study the economic mechanism which sustains the substitution of a marginal method for another when demand increases, in the presence of scarce resources. In those Ricardian dynamics, it is shown that the outgoing method is determined by the quantity side of the problem, the incoming method by the value side. That discrepancy explains both the possible failure of the dynamics and the possible occurrence of multiple equilibria. Conditions for existence, uniqueness and the working of the dynamics are stated. A parallel is drawn with the parametric Lemke algorithm used to solve linear complementarity problems.

1 The Ricardian dynamics [START_REF] Ricardo | On the Principles of Political Economy and Taxation[END_REF] examined the long-run dynamics of capitalism when land was the main scarce resource. With the employment of more and more workers, the production of wheat must be increased and, in a closed economy, less fertile lands are cultivated. The extension of cultivation requires a rise in the price of corn, which must be high enough to make the investment on the lower quality of land as pro…table as in any other sector of the economy. Then, the owners of the more fertile lands are in a position to demand a rent of their farmers. The level of the rent on lands of higher qualities, the …rst to be cultivated, uniformizes the production costs with those on the marginal land. An alternative to the extension of cultivation is its intensi…cation, which consists in using more expensive but more productive methods on the same land, but that possibility does not a¤ect the economic laws.

Under the retained hypotheses, which ignore technical progress, Ricardo's description of the dynamical process is deemed to be well understood, correct and general. Even if agriculture is no longer a strategic sector in many contemporary economies, an expansion of the analysis may consist in studying how the marginal factors shift over time, as a consequence of a shift in the households'demand between closely substitutable products. Our aim is more modest, or more basic, and consists in showing that the analytical grounds of the canonical Ricardian dynamics are not so clear, even in the simplest cases: these di¢ culties are illustrated in Section 3 by means of a numerical example involving a unique agricultural product and a unique land for which the dynamics fail.

We therefore return to Ricardo's original framework and study it with the help of modern tools. We consider a sequence of long-run equilibria generated by an exogeneously changing …nal demand basket. The scarce resources are called lands. Time is not taken into account explicitly because the adjustment of prices is presumed to be almost instantaneous. What makes the speci…city of the analysis with regard to comparative statics is the following observation: when a change of demand leads to a physical limit on some land, rents and prices jump to a higher level, therefore the value side of the dynamics is discontinuous; by contrast, the scarcity of wheat is met by operating one new method, be it on the same or another land, and that method is introduced at a zero activity level, so that the dynamics of activity levels are smooth. We shall see that this condition su¢ ces to de…ne the sequence of equilibria in a unique way.

Following [START_REF] Sra¤a | Production of Commodities by Means of Commodities[END_REF], it is assumed that the independent distribution variable is the given rate of pro…t: this departure from Ricardo's own hypothesis, who assumed a given real wage, is inessential and leads to analytical simpli…cations. The retained formalization is fairly general, as it allows for the joint use of several scarce resources and the joint production of several commodities by each method of production (but we consider generic cases only). The notion of long-term equilibrium for a given demand is recalled in Section 2. Section 3 explains how the methods succeed one another when demand changes and why the dynamics may fail. Section 4 states an algebraic condition for their working as well as for local and global uniqueness. Section 5 draws a close parallel between Ricardo's approach and a modern algorithm used to solve linear complementarity problems. Eventually, some references are given in Section 6.

Long-term equilibria

The data of a long-term equilibrium are the methods of production, the available areas of lands and the (nonnegative and uniform) rate of pro…t; its unknowns are the activity levels, the prices and the rents. We write down the conditions de…ning an equilibrium corresponding to an exogenously given …nal demand basket d.

Let there be n commodities, m methods of production, g grades of lands, and homogenous labor. Method i (i = 1; :::; m) is represented by a input vector a i 2 R n + of reproducible commodities, the areas i 2 R g + of land(s) used and an amount l i 2 R + of labor, while the output vector is a basket b i 2 R n + . Production takes one period. Constant returns prevail and the unit activity level of each method is arbitrary. Land is not explicitly considered as a joint output of production. Matrices A, ; B are obtained by stacking these vectors, method i being represented by the ith columns of these matrices and scalar l i . A peculiarity of the formalization here retained is that free disposal and fallowing are listed among the methods (hence, m > n + g): the input vector of free disposal is an amount of some commodity, that of fallowing an amount of land, and in both cases the output is zero.

The physical side of an equilibrium states that, for an adequately chosen semipositive vector y of activity levels (y 2 R m + ), the demand basket is obtained as the net product of the economy

(B A)y = d (1)
and that the scarcity constraints on lands are met. Let vector 2 R g + denote the available areas of lands. As fallowing is taken into account into explicitly, the scarcity constraints are written as the vector equality

y = (2)
The positive components of vector y correspond to the operated methods.

The value side of an equilibrium states that any operated method i yields the ruling rate of pro…t r (r 0) and that the non-operated methods do not pay extra pro…ts. For the operated methods, the condition is written

b T i p = (1 + r)a T i p + T i + l i ( 3
)
where p is the price vector and the rent vector (these vectors are nonnegative). labor is chosen as numéraire and the wage w 0 = 1 is paid post factum (Sra¤a's hypothesis). If fallowing is operated on some land, i.e. if that quality of land is not fully cultivated, equality (3) implies that the rent on that land is zero.

As vector y must satisfy the n + g constraints ( 1) and ( 2), the number of operated methods is at least equal to n + g, ‡ukes apart. But since a price equality with n prices and g rents holds for each operated method, it is expected that their number does not exceed n + g. On the whole, the number of operated methods is equal to n + g, ‡ukes apart. If the operated methods are known, solving the system (1)-( 2)-(3) determines the activity levels and the price-and-rent vector. Therefore, the only di¢ culty is to identify the set of operated methods.

Two comments are in order:

-It is more usual to de…ne an equilibrium by means of inequalities with complementarity relationships. Since both de…nitions are equivalent, the choice between them is a matter of convenience, and the alternative de…nition will be used in Section 5. The main advantage of the above de…nition is the constancy of the number of operated methods (including free disposal and fallowing) independently of the demand vector: the extension of cultivation is seen as a substitution of method, possibly for fallowing. With the alternative de…nition, the number of operated methods varies with demand and an extension of cultivation is seen as the introduction of a new method.

-We ignore degeneracies, and for instance only one land becomes fully cultivated at a given time. Squareness, with n + g operated methods, is a generic property, but an exception to the rule plays a major role in the dynamics: at the point of transition between consecutive equilibria, the activity level of some previously operated method drops to zero and, then, the number of operated methods falls to n + g 1.

Exit and entry rules

The dynamic approach looks at the evolution of a long-term equilibrium when demand d = d(t) changes with time and proceeds by studying the e¤ect of a change of demand on the set of operated methods. Starting from a given long-term equilibrium E 0 , a small change in d(t) triggers the dynamics. As long as lands of the same grade as those presently cultivated are available, the adjustment to demand only requires that of the positive activity levels, with no changes in the operated methods, prices and rents. The physical limit of equilibrium E 0 is reached at t = t 0 when the activity level of some operated method vanishes. (Simple as it may seem, the rule lies on the convention that the non-cultivated part of a land is deemed to be 'cultivated'by means of the fallowing method.) Flukes apart, only one activity level vanishes at t 0 . That limit de…nes an exit rule in the sense that it allows us to identify the unique method which belongs to equilibrium E 0 at t = t 0 " and will be excluded from the next equilibrium E 1 at t = t 0 + ".

At the breaking point t 0 , the price of the scarce agricultural good rises up to the level where the use of a more costly method, either on a new type of land (extensive cultivation) or on the same land (intensive cultivation), becomes pro…table. A new equilibrium is found and another quiet period with adaptations of positive activity levels opens. The identi…cation of the incoming method relies on the value side of the problem. Beyond the rises in the price of corn and in rents, the whole set of prices is also disrupted because the change in the value of corn modi…es the prices of all commodities in the production of which corn enters directly or indirectly. Let us denote the price-and-rent vector (p; ) as . Since we already know all but one of the new operated methods, and therefore all but one of the price equalities (3), the new price-and-rent vector 1 is known up to one degree of freedom. At this stage, the hypothesis of a given rate of pro…t leads to a formal simpli…cation: since the price-and-rent equations (3) form a set with as many a¢ ne equations as unknowns, the solution to all but one equations is written 1 ( ) = 0 +0 , where vector 0 is the previous price-and-rent vector, vector 0 represents the direction of the change in the price-and-rent vector, and is a still unknown scalar representing the intensity of the change.

We can go further and determine the exact value of , and therefore the incoming method and the new (potential) equilibrium. The property we use is that the pro…tability of any method at prices 0 + 0 is a monotonous function of . For the n + g 1 operated methods which are common to the previous and the new equilibrium, that pro…tability is constant and measured by the ruling rate of pro…t. For all other methods, it changes with . Consider …rst the excluded method. As it yields the normal rate of pro…t at prices that is positive (otherwise, replace 0 by 0 and by ). Consider now the methods which were not operated in the initial equilibrium, and let the value of increase progressively up from zero. At = 0, these methods incur extra costs. Since their pro…tability varies monotonously with , it is expected that, for great enough, at least one of them becomes pro…table (the existence of such a method is dealt with in Lemma 1 below). The entry rule is: when increases, the new operated method is the …rst which yields the ruling rate of pro…t. For, if continued to increase beyond that minimum level, that …rst method would pay extra pro…ts.

The exit and entry rules determine the evolution of the set of operated methods and de…ne the dynamics: the exit rule relies on the physical side of the model (some activity level vanishes), the entry rule on its value side (some method becomes pro…table). The coexistence of rules obeying di¤erent principles explains the di¢ culties met by the dynamics: it is not ensured that the incoming method meets the evolution of demand at the origin of the loss of the previous equilibrium. An illustration is given in a one-commodity model with homogenous land.

Example.

The given rate of pro…t is r = 1. The total area of land is = 100, with three available methods:

(method A) 0:4 qr. corn + 1 acre land + 1 labor ! 1 qr. corn (method B) 0:2 qr. corn + 2 acres land + 4 labor ! 1 qr. corn (method C) 0:4 qr. corn + 0:25 acre land + 2 labor ! 1 qr. corn Let the …nal demand d increase. For low levels of demand, land is partially cultivated by means of method A, the cheapest method for a zero rent. In that equilibrium E 0 , and for a wage 1 paid post factum, the price of corn is p 0 = 5. Land is fully cultivated at level d 0 = 60. For r = 1, the general solution to the system (here reduced to the unique price equation attached to method A) (1 + r)0:4p 1 + 1 + 1 = p 1 is 1 = 0 + 0 with change of method occurs at t = t 0 , either the new set of methods meets demand d(t 0 + "), and then the dynamics work locally, or the previous and the new sets of operated methods constitute two neighboring equilibria, i.e. which di¤er by one method only, both sustaining demand d(t 0 ").

The numerical example has an additional feature, as there exists an equilibrium E 2 which sustains the production of more than 60 quarters: it is made of methods B and C operating jointly at the price-and-rent vector (p 2 = 60; 2 = 4). Therefore, the sequence E 0 , E 1 , E 2 of consecutive equilibria connects low to high levels of demand. This path, however, is not economically admissible because the intermediate equilibrium assumes a decrease in the level of demand, whereas a direct jump from E 0 to E 2 at d = 60 would imply a discontinuity in activity levels.

Local and global dynamics

In this Section, we state the conditions ensuring a smooth physical transition between consecutive equilibria. Matrix (B A;

) has n + g rows and m columns. At equilibrium E 0 , let C 0 be the square matrix of dimension n + g extracted from the columns of (B A;

) and corresponding to the set M of the presently operated methods, i.e. to the positive components y 0 of y, and let be the vector obtained by stacking d and . The unique vector equality

C 0 y 0 = (4)
summarizes both conditions (1) and ( 2) and de…nes the physical side of an equilibrium. On the value side, let T 0 = (p T ; T ) be the price-and-rent vector at equilibrium E 0 and C 0 (r) be the square matrix extracted from the columns of (B (1 + r)A;

) corresponding to the positive activity levels. The price equations (3) are written more compactly as:

T 0 C 0 (r) = l T 0 (5)
When the activity level of some operated method vanishes, the reconstruction of the next equilibrium starts from equalities [START_REF] Dantzig | A complementarity algorithm for an optimal capital path with invariant proportions[END_REF], which remain satis…ed for all methods but one. Let us consider the set

D = d; 9y > 0 d << (B (1 + r)A)y; T y << (6)
(For vectors, notation x >> 0 means all its components are positive, x > 0 that its components are nonnegative and x 6 = 0, x 0 that the components are nonnegative, which is the case for activity levels, prices and rents.)

Lemma 1 The existence of an incoming method is guaranteed if the demand vector belongs to D.

Proof. Since the price-and-rent vectors 0 and 1 satisfy the same equalities (5) for all common operated methods, their di¤erence 1 0 = 0 is a solution to n + g 1 linear equalities: 0T c i (r) = 0 for i 2 M n fjg, where j is the outgoing method. Vector 0 is unique up to a factor that we choose in order that 0T c j < 0. Then the existence of a minimum positive scalar for which some inequality T 0 c k (r) < 0 is turned into the equality T 1 c k (r) = 0 is guaranteed if inequality 0T c k (r) > 0 holds some method k. If 0 has a negative component, this is the case for some free disposal method or some fallowing method. Assume that 0 is semipositive. Vector d(t 0 ) being produced by the methods i 2 M n fjg and 0 being such that 0T c i (r) = 0 for i 2 M n fjg, we have

0T (t 0 ) = 0T (d(t 0 ); ) = 0T X i2M nfjg y i (t 0 )(b i a i ; i ) 0T X i2M nfjg y i (t 0 )(b i (1 + r)a i ; i ) = X i2M nfjg y i (t 0 ) 0T c i (r) = 0.
As d(t 0 ) belongs to D, hypothesis (6) implies the existence of a semipositive vector y such that C(r)y >> (t 0 ), therefore 0T C(r)y > 0T (t 0 ) 0 and there does exist a method k such that 0T c k (r) > 0.

Up from now, it will be assumed that the demand basket belongs to or varies in D. The outgoing and the incoming methods have the following property, which results from the notion of equilibrium: at prices 0 , the outgoing method j yields the ruling rate of pro…t, whereas the incoming method k is too costly. At prices 1 , the situation is reversed. That alternative admits an algebraic expression: Lemma 2 det C 0 (r) and det C 1 (r) have opposite signs.

Proof. Let j = (1+r)p T 1 a j + T 1 j +l j p T b j = l j T 1 c j (r) > 0 be the extra costs incurred by the outgoing method j at the new equilibrium prices 1 = (p 1 ; 1 ), and k > 0 those incurred by the incoming method k at the previous prices 0 . From equalities T 1 c j (r) = l j j = T 0 c j (r) j and, similarly,

T 0 c k (r) = T 1 c k (r) k , there follows ( T 1 T 0 )( k c j (r) + j c k (r)) = 0. As T 1 c i (r) = T 0 c i (r)
= l i for the other n + g 1 common operated methods i 2 M n fjg, and therefore ( T 1 T 0 )c i (r) = 0, it turns out that the matrix with columns c i (r) and k c j (r) + j c k (r) has a zero determinant. Therefore k det C 0 (r) + j det C 1 (r) = 0 and these determinants have opposite signs.

Consider now the physical side and suppose that the dynamics work. For = (d(t 0 + ");

), a similar alternative holds: at equilibrium E 0 , equation ( 4) does not admit a solution with positive activity levels, whereas it has a solution at equilibrium E 1 . The algebraic expression of the productivity condition is similar to that of Lemma 2, matrix C i (r) being replaced by

C i = C i (0): Lemma 3
The dynamics work locally if and only if det C 0 and det C 1 have opposite signs.

Proof. (t) being the vector obtained by stacking vectors d(t) and , the physical constraints on activity levels are written as Cy = (t). For t = t 0 ", all activity levels y i (i 2 M ) are positive at equilibrium E 0 but y j (t) vanishes at t = t 0 and would become negative at t = t 0 + ". When that method j is replaced by another method k, algebraic decompositions of vector (t) lead to the formal equalities

(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 (7)
At t = t 0 both decompositions coincide, with y i (t 0 ) = y 0 i (t 0 ) > 0 (i 2 M n fjg) and y j (t 0 ) = 0, therefore y 0 k (t 0 ) = 0 (the incoming method starts at a zero level): At t = t 0 + ", y 0 i (t) remains positive by continuity (i 2 M n fjg) 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 at t = t 0 + ". Equality [START_REF] Erreygers | On the uniqueness of square cost-minimizing techniques[END_REF] shows that the n + g 1 vectors c i for i 2 M n fjg and vector y j (t)c j y 0 k (t)c k are linearly dependent, therefore y j (t) det C 0 y 0 k (t) det C 1 = 0. y j (t) and y 0 k (t) have opposite signs if and only if it is the case for the two determinants.

Lemmas 2 and 3 show that the dynamics work locally if and only if the following condition (E) holds:

(E) The sign of det C 0 (r)= det C 0 is preserved when the outgoing method is replaced by the incoming method.

To obtain a global uniqueness result, note …rst that nothing in the formal arguments prevents that d has negative components (but d = 0 is a degeneracy since it is obtained by means of less than n + g operated methods). Let us set hypothesis (H):

(H) Every method other than free disposal and fallowing requires labor.

(For existence results, it su¢ ces to assume that labor is directly or indirectly required to produce a semipositive net product.) Lemma 4 Under assumption (H), the equilibrium is unique for any d << 0.

Proof. Free disposal and fallowing sustain the production of d << 0, the associated price-and-rent vector being = 0. Conversely, the value dp of the net product is at least equal to the wages and to the rents. For d << 0, this implies that p = = 0 and that the operated methods do not use labor, therefore they are free disposal and fallowing.

If condition (E) holds everywhere, a global existence and uniqueness result is obtained:

Theorem 1 For a given rate of pro…t, let demand change in D. The dynamics work locally if and only if condition (E) holds. If condition (E) holds everywhere in D, every demand basket d in D is sustained by a unique equilibrium.

Proof. Suppose that the sign condition holds for any pair of consecutive sets. any two baskets d 0 and d 1 in the convex set D, let us link them by an oriented curve d(t) in D. An equilibrium at d 0 is progressively transferred, by means of successive transforms along the curve, to another at d 1 . If d 0 is sustained by a unique equilibrium, could another curve joining d 0 to d 1 lead to a di¤erent equilibrium at d 1 ? Along the curve, an equilibrium is transformed into a uniquely de…ned neighboring equilibrium when some activity level vanishes. The process is reversible in the sense that, if one starts from the second equilibrium and moves in the opposite direction on the same curve, its successor is the initial equilibrium (because, by construction, the equilibrium which succeeds the second equilibrium is sustained by a priceand-rent vector of the type 1 0 , and the minimality of implies that this vector is 0 ). Imagine that demand d 1 is sustained by several equilibria. By following the reverse curve from d 1 to d 0 , 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. To sum up, if condition (E) holds everywhere in D, uniqueness for some non degenerate demand basket d 0 implies global uniqueness. Lemma 4 concludes.

An alternative statement of Theorem 1 is that the multiplicity of equilibria for a given demand d is due to a failure of the dynamics somewhere on a curve leading from low levels of demand to d. It also follows from Theorem 1 that the dynamics always work when the rate of pro…t is zero. The property is due to the duality between the physical and the value sides: for a zero rate of pro…t, the pro…tability criterion and the productivity criterion coincide (golden rule).

From Ricardo to Lemke

One and a half century after Ricardo, mathematicians elaborated algorithms to …nd a solution to a class of problems called linear complementarity problems. As it turns out that the static (i.e., for a given demand vector) long-term equilibrium is of that type, we study the relationships between the Ricardian dynamics and an algorithmic approach of the Lemke type.

We now refer to the alternative de…nition of an equilibrium mentioned at the end of Section 2: an equilibrium is a nonnegative solution (p; ; y) to the system of vector inequalities with complementarity relationships

(B A)y d 0 [p] (8) y + 0 [ ] (9) ((1 + r)A B) T p + T + l T 0 [y] ( 10
)
where the variables into brackets are componentwise complementary to the associated inequalities: they are nonnegative, and zero if the corresponding component of the inequality is strict. The gap between ( 8)-( 9) and the previous formalization ( 1)-( 2) of an equilibrium is only apparent: if a commodity is in excess supply, its price is zero, the free disposal method yields the ruling rate of pro…t and its use allows us to switch from inequality [START_REF] Freni | Capitale tecnico nei modelli dinamici ricardiani[END_REF] to equality (1); the same for land. Let w be the vector obtained by stacking the left-hand sides of inequalities ( 8)-( 9)- [START_REF] Lemke | Equilibrium points of bimatrix games[END_REF], i.e. the excess supply, the non-cultivated areas and the extra costs; let z be the vector of complementary unknowns obtained by stacking the price vector p, the rent vector and the activity levels y; let q be the vector of data obtained by stacking d; and l T ; …nally, let M be the square matrix

M = 2 4 0 0 B A 0 0 (1 + r)A B T 0 3 5 A long-term equilibrium is a solution (z; w) of the system w = M z + q (11) w 0; z 0; w i = 0 or z i = 0 for any i (12)
That problem is called a linear complementarity problem and is denoted LCP(q; M ). Not all LCPs have solutions, and di¤erent techniques have been elaborated to …nd one of them, if any. Note that, if one knows the set S of the zero components of z, the complementary set S is made of zero components of w, and all other components of z and w are easily found by solving a square system of linear equations, which is assumed to be nondegenerate. The core of the problem is therefore to identify the distribution of zeroes between z and w. (Similarly, if one knows the set of non-operated methods, all characteristics of an equilibrium are easily found.) [START_REF] Lemke | Equilibrium points of bimatrix games[END_REF] conceived an algorithm to …nd a solution to the bimatrix game. Interestingly enough, the formalization of a bimatrix game as a linear complementarity problem lets appear a matrix M with two blocks of zeroes on the diagonal, as in the long-term equilibrium problem (but the vectors q di¤er). [START_REF] Lemke | Bimatrix equilibrium points and mathematical programming[END_REF] generalized the procedure, and many variants were introduced later. The problem is to identify the set S. The general principle of a Lemke algorithm is to start from an 'almost solution'S which su¤ers from some defect, then to improve it by changing one of its elements, then another, according to a mechanically determined sequence of trials, until the elimination of the defect. An important property of the procedure is that it does not return to an already examined set S. Since the number of sets is …nite, the algorithm …nds a solution or collapses at some stage (which is the necessary outcome if no solution exists). A variant of that algorithm is the parametric Lemke algorithm [START_REF] Cottle | The Linear Complementarity Problem[END_REF]): its basic idea is to determine a solution of LCP(q 1 ; M ) by transferring a known solution of LCP(q 0 ; M ) along a curve q(t) joining q 0 to q 1 : The initial defect is that the almost solution is a solution to LCP(q; M ), but for a wrong value of q. A solution at q 0 , more generally at q(t), is de…ned by the zero S components of z, the zero S components of w and the other positive components. Slight changes in q(t) are met by adjusting these positive components, with no change in the set S. The adjustment stops when some positive component of z or w, say w j (t), vanishes at some t = t 0 . Since w j (t) was positive at t 0 ", index j is in S. When q(t) continues to evolve, the next set S 0 is obtained by changing the set S and transferring index j from S to S: the constraint w j = 0 is now substituted for the previous one z j = 0. In a neighborhood of t 0 , the other positive components of (z(t 0 ); w(t 0 )) remain positive by continuity.

It turns out that, from a formal point of view, the parametric LCP algorithm is a version of the Ricardian dynamics in which vector q(t 0 ) 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. The new set S de…ned by the algorithm is a neighbor of the previous, as only one nonzero variable is changed. In the literature on LCP, the search for a new positive component or a new facet is similar to that of the incoming method as described in Section 3 and the choice is also determined by a minimum rule (Lemma 1 simpli…es the problem as it solves all di¢ culties linked to existence).

There is, however, one signi…cant di¤erence. It has been noticed that the incoming method may be less productive than the outgoing method and, then, the Ricardian dynamics stop. A similar di¢ culty is met in the LCP framework: if z k is the new variable which was zero and becomes nonzero, equality w = M z + q(t) holds at t = t 0 with z k (t 0 ) = 0, while the positive components of (w; z) satisfy complementarity conditions. At t = t 0 + ", these variables remain positive by continuity, but the algebraic decomposition w = M z + q(t 0 + ") only ensures that the value of z k (t 0 + ") is close to zero. If that component is positive, the process leads to a transfer of the solution from q = q(t 0 ") to q = q(t 0 + "). If, on the contrary, z k (t 0 + ") is negative, then z k (t 0 ") is positive and a second solution to LCP(q(t 0 "); M ) is found. By contrast with the Ricardian dynamics, the Lemke algorithm goes on by making a U-turn on the curve. The algorithm continues for lower and lower values of t ('antitone' move, in the terminology of the LCP literature). Eventually, however, another U-turn may occur (or must occur, if some condition ensuring the working of the parametric algorithm is met) and new solutions sustaining increasing levels t are obtained. A third solution sustaining q(t 0 ") is …rst reached and, …nally, a solution for level q 1 .

The point is illustrated by the discussion of the example in Section 3: for low levels of demand, method A is the cheapest (equilibrium E 0 ). At level d 0 = 60, the land is fully cultivated and the dynamics stop because the next cheapest method is less productive. On the contrary, the parametric Lemke algorithm goes on by switching to equilibrium E 1 , allowing for a temporary decrease in the net product. Method B is progressively substituted for A, and the net product drops from 60 to 40. At this stage, method C is introduced and the …nal equilibrium E 2 sustains high levels of demand. As far as one looks at the existence of long-term equilibria in a static framework, not at the dynamics, the following result illustrates the powerfulness of the algorithm: Theorem 2 For any d in D, and ‡ukes apart, the number of equilibria is odd, with one more equilibrium for which the signs of det C(r) and det C are the same.

Proof. Let us draw a curve d(t) in D, without self-intersection, from d 0 << 0 to a given vector d 1 2 D. The unique solution at d 0 is transferred along that curve, and the sequence of equilibria is mechanically de…ned by the entry and exit rules, with a reversibility property. If, for some demand vector, the same equilibrium were found twice, once for increasing values of t and next for a decreasing value, reversibility implies that the same property would hold for the successor of that equilibrium when t increases, and a contradiction is obtained by considering the …rst equilibrium which is found twice. As the number of equilibria for a given demand basket is …nite, the algorithm must reach d 1 , in spite of possible U-turns. The conclusion is that the equilibrium at d 0 is ultimately transformed into some equilibrium at d 1 .

Let us now draw a curve, without self-intersection, from d 0 << 0 to d 1 and coming back to d 2 << 0. Consider an equilibrium at d 1 . If the transfer of that equilibrium along the curve leads ultimately to either d 0 or d 2 , that equilibrium is met somewhere in the unique sequence of equilibria (such a sequence is called a path, to distinguish it from the curve d(t)) which, starting from d 0 , goes to d 2 . That path sustains an equilibrium at d 1 an odd number of times, s+1 times for increasing values of t (therefore, with det C(r) and det C of the same sign) and s times in the other direction. The alternative is that the path starting from d 1 never reaches d 0 or d 2 . Since there are …nitely many combinations of n + g operated methods, the path admits a loop. Suppose that the path itself is not a loop. Then, the …rst combination of methods which is found twice on the path has two predecessors. By the reversibility property, an equilibrium on the reverse path would have two successors, a contradiction. Therefore the path itself is a …nite loop. Everytime it reaches d 1 it de…nes a new equilibrium at that point, with as many equilibria with increasing values of t as with decreasing values. These additional equilibria do not change the balance between the two types of equilibria.

A part of the LCP literature deals with conditions ensuring the absence of antitone move: in the long-term equilibrium framework, these conditions amount to ensuring the working of the dynamics.

6 References to the literature Ricardo showed that, in normal times, the evolution of demand is met by an adjustment of activity levels with no changes in prices and rents, but prices and rents jump abruptly when a land is fully cultivated, the rise being suf-…cient to let some non-operated method yield the ruling rate of pro…t. The adjustment to demand is met by introducing a new method of cultivation on the same (intensive cultivation) or another (extensive cultivation) land. Ricardo failed to notice a third possibility, which consists in introducing a corn-saving method in industry, and was mistaken in assuming that it is always possible to get rid of rent by considering the set of marginal methods [START_REF] Bidard | The frail grounds of the Ricardian dynamics[END_REF]). [START_REF] Sra¤a | Production of Commodities by Means of Commodities[END_REF] formalization of prices and rents was later completed to take explicitly into account a demand vector and the scarcity constraints. The discovery of the possible multiplicity of equilibria for a given demand (D'Agata 1982), [START_REF] Freni | Capitale tecnico nei modelli dinamici ricardiani[END_REF] was a surprise, but the reason of these complications remained obscure because the dynamic approach had been abandoned and replaced by the search of a solution of a given system of inequalities with complementarity relationships. [START_REF] Salvadori | Land and choice of techniques within the Sra¤a framework[END_REF] transferred a result from the literature on linear complementarity [START_REF] Dantzig | A complementarity algorithm for an optimal capital path with invariant proportions[END_REF] to state existence for any demand basket in the set D de…ned by [START_REF] Erreygers | Terre, Rente et Choix de Techniques[END_REF], but the algorithm he referred to has no clear economic interpretation. Starting from geometrical considerations, [START_REF] Erreygers | Terre, Rente et Choix de Techniques[END_REF][START_REF] Erreygers | On the uniqueness of square cost-minimizing techniques[END_REF] stated the criterion (E) for local uniqueness and argued that global uniqueness might also be obtained under additional assumptions on the structure of equilibria. Theorem 2 generalizes Bidard and Erreygers's (1998) oddity result for economies without lands. The relative sign of the determinants plays a role similar to the index in the di¤erentiable version of general equilibrium theory (Mas-Colell 1985).

Conclusion

When the direct or indirect demand for a resource increases and meets a scarcity limit, the prices rise and the price system adapts itself in such a way that demand is met by means of a new method. That method may either make use of an alternative resource (as exempli…ed by the notion of extensive cultivation), or may make a more e¢ cient use of the same scarce resource (intensive cultivation), or may save the resource. Ricardo was the …rst economist to give a rather precise description of the dynamical aspects of the substitution process in the productive system, but a detailed analysis shows that the working of that process is not so clear. The ultimate reason of the di¢ culty is that prices are not a perfect scarcity index, so that the pro…tability of a method may be disconnected from its ability to meet the requirements for use. The present study has greatly simpli…ed the dynamics themselves by assuming an instantaneous adjustment and has let aside the analysis of substitutability on the demand side, a phenomenon which plays an important role in actual economies. Increasing substitutabilty either on the supply side by assuming di¤erentiability or on the demand side may favour the working of the dynamics, but the di¢ culties pointed at in a simple framework should not disappear by complexifying the problem.

, i.e. for = 0, its pro…tability at prices 0 + 0 is either above or below the normal level, depending on the sign of . Since the very notion of equilibrium requires that no method pays extra pro…ts at prices

, the property determines the sign of . Without loss of generality, we assume

= (p 1 ; 1 ); 0 = (5; 0) and 0 = (5; 1). The pro…tabilty of both methods B and C improves with , and the minimum rule designates method B as the incoming method ( = 1), hence 1 = (10; 1). But method B is less productive per acre than A, therefore the progressive substitution of B for A leads to a reduction of the net product, from 60 to 40 quarters after full substitution. The combination of methods A and C, which would solve the physical problem, is excluded because method B would pay extra pro…ts at the associated price-and-rent vector.Note that the combination of methods (A, B) de…nes an equilibrium E 1 for any demand between 40 and 60. The phenomenon is general: when a

Acknowledgements

We acknowledge Guido Erreygers and Emmanuel Hyon for discussions on Ricardo's rent theory and the Lemke algorithm, respectively.


Olfa Kaabia
email: olfa.kaabia@u-paris10.fr

Ilyes Abid
email: ilyes.abid@u-paris10.fr

Khaled Guesmi
email: khaled.guesmi@ipag.fr

Does Bayesian Shrinkage Help to Better Reflect What Happened during the Subprime Crisis?

Keywords: Contagion, subprime crisis, OECD housing markets, VAR/ BVAR models and Bayesian shrinkage. JEL Classification Codes: F47, C11, C32

We study the contagion effects of a U.S. housing shock on OECD countries over the period of the subprime crisis. Considering a large database containing national macroeconomic, financial, and trade dynamic variables for 17 OECD countries, we evaluate forecasting accuracy, and perform a structural analysis exercise using VAR models of different sizes: a standard VAR estimated by OLS and a MEDIUM and LARGE VARs estimated by a Bayesian shrinkage procedure.

Our main findings are that: First, the largest specification outperforms the smallest one in terms of forecast accuracy. Second, the MEDIUM VAR outperforms both the LARGE BVAR and the SMALL VAR in the case of structural analysis. So the MEDIUM VAR is sufficient to provide plausible impulse responses, and reproduce more realistically what happened during the subprime crisis. Third, the Bayesian shrinkage procedure is preferable to the standard OLS estimation in the case of an international contagion study.

INTRODUCTION

The subprime crisis has gripped the U.S and spread out to many countries all over the world. The current financial crisis is nothing but the result of the American mortgage crisis that leads to many adjustments in housing markets. However, the subprime crisis is not the first turmoil event occurred in the housing markets. Yet, the housing prices boom during the early 2000s has raised many questions and since that many studies focused on a possible international transmission of housing shocks across countries [START_REF] Otrok | The Global Housing Price Boom[END_REF]. So, the last subprime crisis has confirmed the fears of possible contagion effects due to liberalization of markets. In fact, since 2006, the decrease of U.S. housing prices as a result of collapsing residential investments has been followed by a wave of crises and a sharp decrease in housing prices in other economies. This last crisis, which emerged in the USA in the summer of 2006, was followed by a sharp fall in housing prices in Ireland, New Zealand, Spain and the United Kingdom. These almost parallel developments provide evidence in favor of a significant correlation across national housing markets.

There appears to be a large degree of co-movements between very different and distant countries. It seems that the U.S. housing boom and bust has spread to other parts of the world and so confirm that the United States continues to retain its place as the world's principal leading country. Nowadays, there are several possible different explanations of an international transmission of housing prices shocks. The housing prices may be driven by economic or financial fundamentals 1 , the wealth effect 2 , external news 3 , etc. In this paper, we aim at going further beyond questions of international transmission of housing prices to examine "contagion effects" which are directly associated with crisis events. Here by contagion we mean, as in the case of a pandemic, a process where a sudden change at an important date in local prices in one country affects global prices in other countries. We formulate this idea by saying that contagion may occur when a local shock affects the propagation mechanism of a large number of OECD variables. In other words, the investigation of contagion effects is to focus on the changes in the transmission mechanisms of shocks at a critical date. Our analysis is important in terms of policy implications, or even for international investors.

The scope of our paper is to contribute to the literature on international transmission. The idea is to examine the contagion effects of a U.S. housing price index shock on OECD countries, and to explore which VAR model better reflects the propagation and magnitude of the changes in the transmission processes, so as to reproduce what happened during the subprime crisis. To our knowledge, no paper in the literature has used such comparison to test and model the contagion effects.

Specifically, in our study, we address the following questions:

1. Is it necessary to deal with a large panel of data when studying contagion effects during the subprime crisis?

2. Which VAR model specification (SMALL, MEDIUM or LARGE) does reflect most faithfully what happened during the subprime crisis, in terms of forecast and structural analysis?

3. In practice, is Bayesian shrinkage a valid alternative to Ordinary Least Squares (OLS) estimation in the case of international transmission study? 1 The idea is that housing prices are likely to co-move if housing prices are driven by fundamentals, and if the cycles of fundamentals are correlated. See Goodhart and Hoffman (2008). 2 Decrease in house prices will induce households to reduce their consumption since 5% of the household income is from real estate; e.g. [START_REF] Lettau | Understanding Trend and Cycle in Asset Values: Reevaluating the Wealth Effect on Consumption[END_REF] and [START_REF] Case | Comparing wealth effects: the stock market versus the housing market[END_REF]. 3 News on housing prices in some countries may lead investors and (potential) house buyers to revise their expectations on housing prices in other countries. These revised expectations can be unrelated to changes in fundamentals akin to contagion effects in exchange rates and stock markets during the Asian crisis [START_REF] Kaminsky | On Crises, Contagion, and Confusion[END_REF].

To answer these questions we choose different sizes of VAR models to characterize such dynamics for the OECD countries in our panel. Also, we build on the results obtained by De Mol, Giannone, and Reichlin (2008) and Bańbura, Giannone and Reichlin (2010) by setting the degree of Bayesian shrinkage in relation to the cross-sectional dimension of the model, so as to compare the three models.

Our main contributions to the existing literature are: First, the novelty of dealing with a Bayesian shrinkage procedure allows us to study the impact of U.S. housing price index shock on the many OECD financial and economic variables included in our dataset in the form of impulse responses. This is particularly relevant given the most recent crisis which has been characterized by sudden shocks of large magnitude. Researchers, investors, and policy analysts are focusing on robust models allowing the reflection of negative effects on many variables in crisis period.

Second, many economic concepts need more than one variable such as the real activity. So, dealing with a Bayesian shrinkage approach, i.e. the MEDIUM/ LARGE Bayesian VARs may be, a priori, a solution to capture many concepts.

Third, another advantage of using this type of VAR models specifications is that impulse responses can be observed both for variables included only in a small VAR, and for large key variables. If housing markets are contagious, economic policy should focus on structural reforms ensuring a stable domestic market in order to limit the amplification of shocks between housing markets.

Fourth, an alternative to analyse relatively large data sets is to define a small set of factors (indicators), or group of variables, at a time, FAVAR models, for instance, see e.g. [START_REF] Christiano | The Effects of Monetary Policy Shocks: Evidence from the Flow of Funds[END_REF], [START_REF] Kim | International transmission of U.S. monetary policy shocks: Evidence from VAR's[END_REF] & [START_REF] Kaabia | Theoretical Channels of International Transmission During The Subprime Crisis To OECD Countries: A FAVAR Model Under Bayesian Framework[END_REF]. However, comparison of impulse responses across models is problematic. Our approach may be a not neglecting solution for that.

Finally, using different VARs specifications allows us to compare our results to previous findings in terms of forecast and structural analysis, in the case of international transmission process.

Dealing with a huge database, we will study the impact of the U.S. housing price index shock and analyze the contagion effects on OECD countries. Our database includes the 204 monthly following variables: real GDP, personal consumption, short-term and long-term interest rates, all share price index, effective exchange rates, housing price index, consumer price index, unemployment rate, export and import prices for each of the seventeen considered OECD countries, over the period of 1980: M1 -2006: M6 4 . It is worth emphasizing that this sample is larger and more international than related studies 5 .

The main results are: First, the largest specification outperforms the smallest one in terms of forecast accuracy. Second, the MEDIUM VAR outperforms the LARGE BVAR and the SMALL VAR in the case of structural analysis. So the MEDIUM VAR is sufficient to provide plausible impulse responses and reproduce more realistically what happened during the subprime crisis. This result is interesting in proving that a LARGE Bayesian VAR estimated over a hundred variables is not needed and produces worse forecasting, and structural analysis results than the MEDIUM VAR which has not yet been considered in the literature. Third, the Bayesian shrinkage procedure is preferable to the standard OLS estimation, in the case of an international contagion study. The rest of the paper is structured as follows: Section 2 presents the literature review. Section 3 exposes the empirical framework. Section 4 describes the data and presents the results in terms of forecast and structural analysis. Section 5 draws the appropriate conclusions.

LITERATURE REVIEW

Most of studies on contagion effects have focused on global financial markets, in general, and on asset market linkages, in particular. These studies can be broadly classified in several categories, as there is not only one definition of contagion. Surprisingly, the economists are not unanimous on a single definition of contagion 6 . From all the proposed definitions, we retain that of [START_REF] Favero | Is the international propagation of financial shocks non-linear?: Evidence from the ERM[END_REF] who focus on financial linkages and transmission process 7 . They used Vector Auto-regressions (VAR) models developed by [START_REF] Sims | Macroeconomics and Reality[END_REF] to offer an alternative to simultaneous equation models, and to detect contagion effects. Initially, Sims had emphasized the use of unrestricted VAR models as a means of modeling economic relationships. Nowadays, VAR models are standard tools in macroeconomics, and are widely used for structural analysis. They have the advantage of not imposing restrictions on the parameters, and, hence, provide a very general representation allowing the capture of complex data relationships. Other VAR-type models have been proposed as Structural VARs 8 ; however, we always notice the same criticism of the VAR approach, according to the relatively small amount of information used in VARs 9 . This issue has been especially addressed by [START_REF] Bernanke | Measuring the Effects of Monetary Policy: a Factor Augmented Vector Autoregressive (FAVAR) Approach[END_REF]. In that sense, many researches propose to add more than eight variables in a VAR model; for instance, the marginal approach proposed by Christiano, Eichenbaum and Evans (1996), or [START_REF] Kim | International transmission of U.S. monetary policy shocks: Evidence from VAR's[END_REF].They define a core set of indicators and add one variable, or a group of variables, at a time; however, comparison of impulse responses across models is problematic. Also, let mention Leeper, Sims, and Zha (1998) who increased the number of variables included by applying Bayesian priors. However, in their study, the VAR systems still contain less than 20 variables. Also, [START_REF] Stock | Implications of Dynamic Factor Models for VAR Analysis[END_REF], and [START_REF] Bernanke | Measuring the Effects of Monetary Policy: a Factor Augmented Vector Autoregressive (FAVAR) Approach[END_REF] introduced the FAVAR models. The idea is that if a small number of estimated factors effectively summarizes large amounts of information about the economy, then a natural solution to the degrees-of-freedom problem in VAR analyses  which have to be of limited dimensions  is to augment the standard VAR with estimated factors. However, many issues remain unsolved concerning the estimation approach, the number, and the nature of the factors.

So, our analysis starts from the great criticism of the sparse information set used in the VAR model, which normally does not include more than eight variables to conserve degrees of freedom. Generally, central banks use a large information set to analyze the state of the economy before making any decision. In that sense, the VAR approach may exclude important information considered pertinent in the transmission process.

EMPIRICAL FRAMEWORK

We propose to study contagion effects in the case of simulating the last subprime crisis, using different sizes of VAR models. So, we evaluate forecasting accuracy, and perform a structural exercise on the effect of a U.S. housing shock using different sizes of VARs: a SMALL one, estimated by OLS as well as a MEDIUM VAR and, a LARGE one estimated by Bayesian shrinkage.

analysis approach popularized in this literature by [START_REF] Forbes | No Contagion, Only Interdependence: Measuring Stock Market Comovements[END_REF], the VAR approach of [START_REF] Favero | Is the international propagation of financial shocks non-linear?: Evidence from the ERM[END_REF], the probability model of Eichengreen andal. (1995, 1996) and the approach of [START_REF] Bae | Analysis of the Arabidopsis nuclear proteome and its response to cold stress[END_REF]. They showed that the different definitions used to test for contagion are minor, and under certain conditions, are even equivalent. 7 The focus on relations between the transmissions of shocks through fundamental linkages has been primarily studied by [START_REF] Masson | Multiple equilibria, contagion, and the emerging market crises[END_REF], and called "Pure contagion". 8 Introduced by [START_REF] Sims | Macroeconomics and Reality[END_REF]. 9 To conserve degrees of freedom, standard VARs rarely employ more than six to eight variables.

In this section, first, we follow standard recommendations in the Bayesian literature, and build on the results of De Mol, Giannone, and Reichlin (2008), and Bańbura, Giannone and Reichlin (2010) by coping with the curse of dimensionality using Bayesian shrinkage via the imposition of priors. Second, we evaluate the forecast performance of different VAR model sizes. And finally, we deal with a structural analysis and make the impulse responses.

Setting Priors

Let a VAR model with p lags, V AR ( ) p , be as follows:

1 1 t t p t p t Y c A Y A Y u - - = + + + + ⋯ (1)
where 1, , ( ,..., )

t t n t Y y y ′ = , 1 ( , , ) n c c c ′ = … is a vector of constants; 1 , , p A A … are the autoregressive ( ) n n
× matrices and t u are independent (0, )

N ψ errors.
The Bayesian methods combine likelihood function with prior may lead, a priori, to a valid posterior density even if some parameters are not identified in the likelihood function. However, prior information becomes increasingly important as the number of parameters increases relatively to sample size. In this case, priors on the parameters 101 , , ,

p A A …
and the residual covariance matrix, ψ , should be set. In the literature, many priors suggested. For a complete review of the existing priors, the reader can refer to [START_REF] Koop | Forecasting with Medium and Large Bayesian VARs[END_REF].

We follow the standard procedure developed by [START_REF] Litterman | Forecasting With Bayesian Vector Autoregressions -Five Years of Experience[END_REF] for the VAR coefficients priors. Then, we take into account the modifications proposed by [START_REF] Kadiyala | Numerical Methods for Estimation and Inference in Bayesian VAR-Models[END_REF], and Sims and Zha (1998) for the residual covariance matrix priors.

First, according to [START_REF] Litterman | Forecasting With Bayesian Vector Autoregressions -Five Years of Experience[END_REF], a V AR ( ) p can be considered as "centered" equations around the random walk with drift as follows:

1 t t t Y c Y u - = + + (2)
The idea suggested by Litterman is to shrink all VAR coefficients towards zero except for coefficients on own lags of each dependent variable. The latter are either set to one (for variables which exhibit substantial persistence), or zero (for variables which do not). So Litterman assumes that the Minnesota prior beliefs11 are:

( ) ( ) 2 2 ² if ² if , 1 ; ² 0 otherwise otherwise ² i k k ij ij i j j i k j i k E A V A k λ δ σ λ ϑ σ  =  = =       = =           (3)
The coefficients 1 , , p A A … are assumed to be a priori independent and normally distributed. Also, the hyper parameters λ and ϑ control the overall tightness of the prior distribution around the random walk or white noise, and govern the relative importance of the prior beliefs with respect to the information contained in the data. More precisely, if 0 λ = , the posterior equals the prior and the data do not influence the estimates. If λ = ∞ , the posterior expectations coincide with the OLS estimates. In the rest of the paper, we will choose λ in relation to the size of the VAR model. As the number of variables increases, the parameters should be shrunk some more in order to avoid overfitting.

The parameter k is the lag length and the ratio 2 1 / k gives the rate at which prior variance decreases with increasing lag length. Also, 2 2 / i j σ σ reflects the different scale and variability of the data.

Second, according to [START_REF] Kadiyala | Numerical Methods for Estimation and Inference in Bayesian VAR-Models[END_REF], and Robertson and Tallman (1999), Litterman's assumption of fixed, and diagonal residual covariance matrix is somewhat unrealistic; that is why they impose a Normal Inverted Wishart prior which retains the principles of the Minnesota prior. Also, this assumption is problematic in the case of the structural analysis, where it is necessary to take into account possible correlation among the residual of different variables.

Consequently, we follow [START_REF] Kadiyala | Numerical Methods for Estimation and Inference in Bayesian VAR-Models[END_REF], and Robertson and Tallman (1999) who propose to deal with the matrix form of a V AR ( ) p model given by:

Y X U β = + (4) where 1
( ,..., ) T Y Y Y ′ = is a ( ) T n × matrix.
( )

1 ,..., T X X X ′ = is a ( ) T K × matrix, with ( ) 1 1, , , t t t p X y y - - ′ ′ ′ = … and
( )

1 K np = +
since each row contains p lags for each dependent variable, and an intercept. Also,

( ) 1 , ,..., p c A A β ′ = is the ( ) K n × matrix of coefficients and ( ) 1 ,..., T U u u ′ = is a ( ) T n × residual matrix.
The Normal Inverted Wishart prior has the form:

( ) 0 0 ( ) ( ), vec N vec β ψ β ψ ⊗ Ω ∼ and
( )

0 0 S , iW ψ α ∼ (5)
where the prior parameters 0 β , 0 Ω , 0 S and 0 α are chosen so that prior expectations, and variances of β coincide with those implied by equation (3), and the expectation of ψ is equal to the fixed residual covariance matrix ∑ of the Minnesota prior 12 .

In order to match the Minnesota moments in equation ( 3), and implement the prior of equation ( 5), it is necessary to add dummy variables as follows:

* * * Y X U β = + (6)
where

( ) * , d Y Y Y ′ ′ = ; ( ) * , d X X X ′ ′ = ; ( ) * , d U U U ′ ′ =
and the dummy observations d Y and d X are:

1 1 1 1 ( 1) 1 1 1 1 ( , , ) ( , , ) (1, , ) 0 0
; 0 0 ( , , ) 0 0 n n n np n p n d d n np n n np n diag diag diag p Y X diag δ σ δ σ σ σ λ λ σ σ η × -× × × × ×     ⊗                     = =                         … … … … … … … … … … … … … … … … … … … … … … … … … … … … (7)
So we obtain that 1 0 ( )

d d d d X X X Y β - ′ ′ = , 1 0 ( ) d d X X - ′ Ω = , 0 0 0 S ( ) ( ) d d d d Y X Y X β β ′ = - - and 0 . d T K α = -
And in that case, the posterior has the form 13 :

( )

* * 1 ( ) , ( ), ( ) vec Y N vec X X β ψ β ψ - ′ ⊗ ɶ ∼
, and ( , 2 )

d Y iW T T k ψ Σ + + - ɶ ∼ (8) with * * 1 * * ( ) X X X Y β - ′ ′ = ɶ and * * * * ( ) ( ) Y X Y X β β ′ Σ = - - ɶ ɶ ɶ
So we impose a Normal Inverted Wishart prior which retains the principles of the Minnesota prior under the condition that 1 ϑ = 14 . The posterior expectation of the coefficients coincides with the OLS estimates of the regression of * Y on * . X This expression is common in the Bayesian literature, and coincides with the posterior mean for the Minnesota prior.

After setting the priors, we will move to explain how to make the forecast analysis resulting from different VAR specifications.

Forecast Analysis

We compute point forecasts using the posterior mean of the parameters. Let ( , ) ( , ) ˆm c λ be, respectively, the posterior mean of the autoregressive coefficients, and the constant term of a given model ( , or m SMALL MEDIUM LARGE = )

ˆm j A λ ( 1, , j p ∀ = … ) and
and parameter λ . The point estimate of the one-step-ahead forecast is computed as:

( , ) ( , ) ( , ) ( , ) 1 1 1 ˆm m m m t p t p t t Y c A Y A Y λ λ λ λ -+ + = + + + ⋯ (9)
Also, the other forecasts h-steps ahead are computed recursively as follows:

( )

( , ) ( , ) ( , ) 1,
+ + + = … ( 10
)
where n is the number of variables included in the m model and h is the forecast horizon. Note that in the case of the benchmark model (random walk with drift), the prior restriction is imposed exactly, that is 0 λ = and the corresponding forecasts are denoted by (0) ˆt h t Y + , and are the same for all the specifications.

In our analysis, we compute h-step-ahead forecasts, ( , ) ˆm T h T Y λ + , using only the information up to time T. For a given forecast horizon h, in each period 0 1

, , T T H h T h

= + - - …
where H denotes the longest forecast horizon to be evaluated. 0 T and 1 T are respectively the beginning and the end of the evaluation sample.

As for the out-of-sample forecast accuracy, we compute the Mean Squared Forecast Error ( ) MSFE for the variable, , i and a horizon h as: 13 To insure the existence of the prior expectation ofψ , it is necessary to add an improper prior ( )

1 0 2 ( , ) ( , ) , , , 1 0 1 1

T h m m i h i T h i T h T T T H h

MSFE y y T T H

λ λ - + + = + - = - --+ ∑ (11)
And to compare the different specifications, we use the MSFE relative ( ) RMSFE to the benchmark as follows:

( , ) , ( , ) , (0) ,

m i h m i h i h MSFE RMSFE MSFE λ λ = (12)
We follow De Mol, Giannone, and Reichlin (2008) and Bańbura, Giannone and Reichlin (2010), and set the overall tightness, λ , to yield a desired average " Fit " for the key variables of interest during the pre-evaluation period, and then keep it fixed for the entire evaluation period. So, λ is chosen for a desired " Fit ", and is given by:

( , ) ( ) (0) 1 ( ) arg min m m i i I i msfe Fit Fit I msfe λ λ λ ∈ = -∑ (13)
In which I represents the variables included in the SMALL VAR model,

( , ) m i msfe λ
is an insample one-step-ahead mean squared forecast error evaluated using the training sample 0 1, ,

1 t T = - …
and is given for the number of lags p as follows:

( )

0 2 2 ( , ) ( , ) , 1 , 1 0 1 1
T m m i i t i t t t p msfe y y T p λ λ - + + = = - --∑ (14)
More precisely, the desired " Fit " coincides with the one obtained by OLS estimation on the VAR model that is for:

( , ) ; (0) 1 m i m V AR i I i msfe Fit I msfe λ λ =∞ = ∈ = ∑ (15)
In the next section, we will explain how to make the structural analysis resulting from the VAR models.

Structural Analysis

We follow [START_REF] Stock | Implications of Dynamic Factor Models for VAR Analysis[END_REF] and [START_REF] Bernanke | Measuring the Effects of Monetary Policy: a Factor Augmented Vector Autoregressive (FAVAR) Approach[END_REF], and divide the variables in the data into two categories: slow and fast-moving variables. This distinction is crucial because it implies that slow-moving variables do not respond contemporaneously to an initial shock. This hypothesis is equivalent to ranging the variable in an exogeneity order.

So, if we note t S as representing the slow variables, t r is the shocked variable and t Z is the fast-moving variable, we can write ( , , ).

t t t t Y S r Z =
The Structural VAR is written as follows:

0 1 1 t t p t p t Y v Y Y e - - = + + + + Ż Ż ⋯ Ż (16)
with (0, )

t e W N D ∼
and where

1 v C c - = , 1 0 C - = Ż , 1 1, ,

DATA and RESULTS

Data

Our large international dataset is drawn from Datastream, Eurostat and the Federal Reserve website, FRED -Saint Louis Fed. The data consists of monthly variables from the period 1981M1-2006M6 for 17 OECD countries, namely United States (U.S.), Canada (CAN), Finland (FIN), France (FRA), Germany (ALL), Ireland (IRL), Italy (ITA), Netherlands (NLD), Spain (ESP), Denmark (DNK), Norway (NOR), Sweden (SWE), Switzerland (SWI), United Kingdom (UK), Australia (AUS), Japan (JPN), and New-Zealand (NZL).

Our dataset includes 204 variables, with 12 variables for each of the 17 countries, encompassing a wide range of financial variables (3-Month Interest Rates, 10-Year Interest Rates, Stock Indices and Housing Price Indices, HPI), variables related to real economy (GDP, Personal Consumption, Industrial Production, Unemployment Rates), aggregate price variables (CPI), trade variables (Import and Export of Goods and Services), and Effective Exchange Rates. All the data are seasonally adjusted, and the variables are measured at constant national prices. As in the literature, interest rates are differenced, and activity variables are logarithmized. More detailed description is provided in appendix A.

In our analysis, we will consider the three following VAR specifications:

• SMALL: This is a VAR including five OECD housing price indices. The U.S. housing price index, HPI_USA, is included because we assume that it will impact the other OECD housing markets. Due to the number limitation, we choose to consider the most representative country for each of the four considered regions. So Canada will represent North America (NA); France, the European Monetary Union (EMU); the U.K, the Non European Monetary Union (NEMU); and Japan, Asian Pacific (AP). Thus, the SMALL VAR represents an international model.

• MEDIUM: This VAR includes all the seventeen housing price indices of the OECD considered countries.

• LARGE: This VAR contains all the considered 204 variables in our database.

Results

Forecast Analysis

In this section, we evaluate the forecast performance of the five considered OECD housing price indices included in the three VAR specifications: HPI_USA, HPI_CAN, HPI_FRA, HPI_UK and HPI_JPN, respectively the housing price indices of USA, Canada, France, United Kingdom and Japan, over the period going from mid-2006 until the end of 2010.

Note that for the SMALL VAR, we implement information criteria for lag selection and take the optimal lag according to the Bayesian Information Criterion (BIC). The number of lags retained is 11 p = . This choice confirms the one made by Bernanke given in Table 1 in appendix B.

[Table 1 about here] Moreover, to compare models of different sizes, we choose the Bayesian shrinkage hyper parameter in relation with the models dimensions by ensuring that the in-sample " Fit " is computed by ( , )

; (0) 1 5 m i m V AR i I i msfe Fit msfe λ λ =∞ = ∈ = ∑
so is constant and equal to 0.9572.

As the dimension increases, we set the tightness of the prior so that all models have the same in-sample " Fit " as the smallest VAR estimated by OLS. Also, we report, in Table 2, the RMSFE for the five OECD housing price indices for the three different VARs.

[Table 2 about here] So, according to this table, we notice that the MEDIUM and LARGE VARs outperform the SMALL one for the OECD countries except for the USA. For the U.S and for all lags, the SMALL VAR outperforms the other VARs. This confirms that the USA continues to retain its place as the world's principal leader and does not depend on the other OECD housing markets. It seems, clearly, that for the USA, the smaller the model, the better the forecast. For the other OECD housing markets, the MEDIUM and the LARGE VARs provide better performance than the SMALL one. Except for the USA, using a large data set helps to better forecast the OECD housing markets. This is an important result denoting that adding information, e.g. dealing with the MEDIUM or LARGE VARs, helps to improve the forecast in the case of the OECD housing markets except for the USA.

Our results are interesting and confirm the previous findings of Bernanke and al. (2005) who criticize the SMALL VAR model due to its limited number of variables. With this simple forecast exercise, we show that employing five variables produces worse forecasting results than using seventeen or 204 ones except for the U.S.

Structural Analysis

In this section, we present the results of the impulse response functions, and the variance decompositions to the U.S. housing price index shock in mid-2006 for the VARs.

We display the impulse response functions for the three models under consideration only for the five housing price indices (HPI_US, HPI_CAN, HPI_FRA, HPI_UK and HPI_JPN) included in the different VARs. We divide the variables in the data into slow and fast-moving variables: the slow-moving ones are GDP, consumption, housing price index, CPI, industrial production, export and import of goods and services, and unemployment rate. The rest of the variables e.g. stock price index, effective exchange rates, short-term and long-term interest rates are the fastmoving ones. Also, t r is the U.S. housing price index.

[Figure 1 about here]

The dotted lines indicate the posterior coverage intervals corresponding to 90% and 68% confidence levels as mentioned in the legend. At first sight, we notice that the impulse responses maintain the expected sign for all specifications, and that according to the considered information and also to the VAR specification, impulse response functions change in shape as predicted.

For the SMALL VAR, the initial shock is less persistent and become nil about a four or fivemonth horizon. The impulse responses are significant only for a two-month horizon. We can conclude that the positive U.S housing shock is very brief. We remark that the more significant the impulse responses are, the tighter the 90% and 68 % confidence intervals are, and vice versa.

For the MEDIUM VAR and except for Japan, the impulse responses have the same shape. Also, it appears that a positive American housing prices shock affects immediately and significantly the Canada, France, the U.K and Japan. The downward trend of the persistent responses for over a one-year horizon seems to be a realistic figure. It denotes that the subprime crisis effects spread over the five considered OECD countries at least for a one-year horizon which was really the case. Moreover, the tightness of the confidence bounds, compared to the ones of the SMALL VAR, is clear and denotes that the responses of the MEDIUM model are more reliable than those of the SMALL one.

For the LARGE VAR, the U.S impulse response is significant for the whole one-year horizon considered, unlike the French impulse responses which are not significant. The Canadian and Japanese impulse responses are significant only for the first month; however, the most significant impulse response is that of the U.K becoming nil in a four-month horizon.

Unfortunately, the confidence bounds are wide denoting that the impulse responses are unreliable.

The relative tight confidence bounds of the SMALL and the MEDIUM VARs indicate the precision of the response of the U.S. housing price index positive shock on the other five considered OECD housing markets, which is not the case for the LARGE VAR. In the case of the last model, the 90 and 68 % confidence bounds are very wide. So, it seems that the MEDIUM model provides better results in terms of impulse response functions. We can conclude that adding information related to the other OECD countries is a better choice and is sufficient when making an international study on OECD housing markets. At this stage, combining those conclusions with the previous ones made in the case of forecasting analysis, leads us to say that the MEDIUM VAR is preferable to the SMALL and LARGE ones.

Also, to complete our analysis, we report the impulse response functions for the seventeen OECD housing prices in the case of the MEDIUM and LARGE VARs, respectively in figures 2 and 3. In fact, the advantage of the MEDIUM and LARGE VARs, compared to the SMALL one, is that impulse responses can be observed for all the OECD countries.

[Figure 2 about here]

[Figure 3 about here] So, just as a confirmation of our previous findings, it appears, clearly, that the MEDIUM VAR is preferable to the LARGE one. This last model has very wide confidence bounds. The MEDIUM VAR provides realistic response functions reflecting what happened during the subprime crisis. The OECD did not react in the same way. It seems that the effect of the positive U.S. housing shock on OECD housing markets is significant and persistent over a one-year horizon. Moreover, the same features can be seen from the variance decomposition analysis below:

[Table 4 about here]

The results show that the size of the positive U.S. housing price index shock is clearly more pronounced for the MEDIUM model than for the SMALL or for the LARGE VARs. Besides, for the U.S. housing price index, and since the 6-month horizon, we remark that the size of the initial shock in the MEDIUM VAR is bigger than that in the LARGE VAR, and both remain bigger than that in the SMALL VAR. This suggests that the U.S. housing shock is more and more persistent, even after a 36-month horizon.

At this stage, it appears that the MEDIUM VAR is preferable to the LARGE and SMALL ones; however, we notice, that the SMALL and MEDIUM VARs produce qualitatively and quantitatively similar results. So as to complete our analysis and decide which one of the estimation methods (Bayesian Shrinkage or OLS) is more suitable in our study, we estimate the SMALL VAR by Bayesian shrinkage. Therefore, we repeat forecasting and structural analyses for the SMALL VAR using the Bayesian approach. We get exactly the same results for the SMALL whether it is estimated both by the two methods in terms of forecasting or structural exercises. So the Bayesian shrinkage is a not neglecting alternative to the OLS when dealing with more than 10 variables.

All the results confirm that the MEDIUM VAR is preferable to the SMALL and LARGE ones. Our findings are in accordance with those of Bańbura, Giannone and Reichlin (2010). Moreover, we confirm that the Bayesian shrinkage procedure is preferable to the OLS estimation while studying the contagion effects during the subprime crisis. Besides and since the SMALL VAR is criticized due to its limited number of variables, the MEDIUM VAR is a suitable model for international study.

CONCLUSION

This paper assesses the performance in terms of forecast and structural analysis of different sizes of VAR models while studying contagion effects during the subprime crisis. The great criticism addressed especially by [START_REF] Bernanke | Measuring the Effects of Monetary Policy: a Factor Augmented Vector Autoregressive (FAVAR) Approach[END_REF] is the sparse information set of the VAR model which can normally only include a few variables to conserve degrees of freedom. So, we propose to study different-sized VAR models (SMALL, MEDIUM and LARGE). In other words, we consider the following specifications: 5, 17 OECD housing price indices and 204 international macroeconomic and financial variables.

We examine both forecasting accuracy and structural analysis of the effect of a positive U.S. housing price index shock so as to study what has happened during the subprime crisis. We follow the results of De Mol, Giannone and Reichlin (2008) and Bańbura, Giannone and Reichlin (2010) by setting the degree of Bayesian shrinkage in relation to the cross-sectional dimension of the model. So, we build on standard recommendations in the Bayesian literature; as the model becomes larger, we increase the overall shrinkage so as to maintain the same in-sample " Fit " across models, and guarantee a meaningful model comparison.

Our findings show that a MEDIUM VAR estimated by Bayesian shrinkage outperforms the standard VAR and a large panel of data. So, the MEDIUM Bayesian VAR estimated over the OECD housing markets produces better forecasting and structural analysis results than the standard VAR or the LARGE one.

Consequently, the MEDIUM VAR is sufficient to provide a plausible impulse response and so reproduces what has happened during the subprime crisis. Moreover, our results prove that the Bayesian shrinkage procedure is preferable to the OLS estimation while dealing with an international study.

Appendix A: Data Transformations Details

As in [START_REF] Stock | Implications of Dynamic Factor Models for VAR Analysis[END_REF], we use the following transformation codes: 1 -no transformation (levels); 2 -first difference; 3 -second difference; 4 -logarithm; 5-first difference of logarithm and 6-second difference of logarithm. In the transformation line, ln denotes logarithm, ∆ln and ∆²ln denote the first and second difference of the logarithm, lv denotes the level of the series, and ∆lv denotes the first difference of the series. Also, following [START_REF] Bernanke | Measuring the Effects of Monetary Policy: a Factor Augmented Vector Autoregressive (FAVAR) Approach[END_REF], we divide the variables into slow moving (denoted by an asterisk: * next to the variable) and fast moving variables.

Appendix B: Tables and figures

  Mol, Giannone, and Reichlin (2008), and Bańbura, Giannone and Reichlin (2010), for more details.14 See Kadiyala and Karlsson (1997) for more details.

  the lower diagonal Cholesky matrix of the covariance of the residuals of the reduced form of a VAR be noted

.

  We follow Gordon and Leeper (1994) by generating draws from the posterior of So for each draw, ψ , we compute B ,

  and al. (2005), indicating that the series are very persistent. The estimation is based on the sample from 1982:2 to 2006:6, and the results reported are for the same overall shrinkage obtained by :

Figure 1 : 20 Figure 2 :

 1202 Figure 1: Impulse Response Functions for the five considered OECD housing price indices (HPI_USA, HPI_CAN, HPI_FRA, HPI_UK and HPI_JPN) for the Three Models (SMALL, MEDIUM and LARGE VARs)

Figure 3 :

 3 Figure 3: Impulse Response Functions for the Seventeen OECD Housing Price Indices for the LARGE VAR CAN

Table 1 :

 1 The Value of the Shrinkage Hyper Parameter λ for the Three Considered VARs

		SMALL	MEDIUM	LARGE
	λ	∞	0.1666	0.0529

Table 2 :

 2 The RMSFE for Forecast Horizons h = 1, 3, 6 and 12 for the Three VARs

	Horizons	Variables	SMALL	MEDIUM	LARGE BVAR
		USA__HPI	0.00424584	0.02014758	0.40870282
	h=1	CAN__HPI	0.23182731	0.2478295	0.09568868
		FRA __HPI	0.13655286	0.10086105	0.14867839
		UK__HPI	0.21336135	0.12402276	0.08353959
		JPN__HPI	0.07864748	0.0647852	0.3244206
		USA__HPI	0.2748464	1.32543157	0.78337481
	h=3	CAN__HPI	1.10138512	0.59779169	0.49348267
		FRA __HPI	0.88376298	0.6292799	1.20411601
		UK__HPI	1.96526665	1.79006799	0.5997567
		JPN__HPI	1.07262422	0.83355113	1.64177915
		USA__HPI	0.50348596	1.0006723	0.70535425
	h=6	CAN__HPI	0.78590664	0.71603423	1.19127467
		FRA __HPI	1.5603861	0.80525732	0.37284653
		UK__HPI	0.98245335	0.64396971	0.47369708
		JPN__HPI	1.06957526	1.05170864	1.70839684
		USA__HPI	0.35935005	0.5664477	0.83765078
	h=12	CAN__HPI	1.15101844	0.74793794	0.63268384
		FRA __HPI	1.00142644	0.95390452	0.4202525
		UK__HPI	1.34089864	0.89350743	0.56161668
		JPN__HPI	1.46271746	1.07656697	1.38529736

Table 4 :

 4 The Percentage Share of the U.S Housing Price Index Shock in the Forecast Error Variance for 1, 3, 6 and 12 Months Forecast Horizons

	Variable	Horizon	SMALL	MEDIUM	LARGE
		1	0	0	0
		3	0.00128917	3.3467212	0.96442987
		6	0.00358472	3.77802602	12.1879643
	HPI_USA	12	0.00696661	3.50771871	18.2814383
		24	0.01493674	3.2174522	19.2062481
		36	0.02173222	3.04501921	19.2249845
		1	0.76813491	4.9389675	3.1839636
		3	1.1708081	3.3715549	9.2960356
		6	3.5873512	9.9064257	8.8831806
	HPI_CAN	12	5.4869865	10.6927819	6.3593114
		24	6.6684848	12.6559531	7.905008
		36	7.408685	12.6551657	5.9241766
		1	7.7194E-06	5.45700449	0.06767973
		3	0.00244045	15.0189539	3.38306224
		6	0.00219837	15.0654537	3.42596745
	HPI_FRA	12	0.00222918	15.1476367	3.30027306
		24	0.00281384	15.1760117	3.25983811
		36	0.00268707	15.1766648	3.25125855
		1	8.76813491	1.45700449	93.1839636
		3	9.1708081	5.0189539	39.2960356
		6	10.5873512	8.0654537	28.8831806
	HPI_UK	12	9.4869865	10.1476367	20.3593114
		24	11.6684848	9.1760117	13.905008
		36	9.408685	10.1766648	10.9241766
		1	0.06767973	94.9389675	7.7194E-06
		3	3.38306224	93.3715549	0.00244045
		6	3.42596745	92.9064257	0.00219837
	HPI_JPN	12	3.30027306	92.6927819	0.00222918
		24	3.25983811	92.6559531	0.00281384
		36	3.25125855	92.6551657	0.00268707

It is considered as a broad measure of financial and economic co-movements.

For example, that of[START_REF] Stock | Implications of Dynamic Factor Models for VAR Analysis[END_REF], or even that of De Bandt and Malik (2010).

For a more complete review, the reader can refer to the study ofDungey and al. (2003, 2005). They compare the correlation

The prior on the intercept, , c is diffuse.

The reader can refer to[START_REF] Litterman | Forecasting With Bayesian Vector Autoregressions -Five Years of Experience[END_REF] for more details concerning the setting and hypothesis of each parameter.

See Kadiyala and Karlsson (1997) for more details.