Jean Berstel

Clelia De Felice

Dominique Perrin

Christophe Reutenauer

Giuseppina Rindone

Recent results on syntactic groups of prefix codes

Introduction 1 2 Automata, monoids and groups 2 3 Holonomy groups 5 4 The degree of syntactic groups 10 5 Codes with empty kernel 15

Introduction

Finite permutation groups can be considered as building blocks of finite transformation monoids. Indeed, by the Krohn-Rhodes Theorem, any finite transformation monoid divides a semidirect product of permutation groups and three element transformation monoids.

Given a transformation monoid M on a finite set, defined by a finite set of generators, it is in general non-trivial to obtain a description of the groups contained in M . For example, there is no simple algorithm to check that all these groups are trivial (see [START_REF] Stern | Complexity of some problems from the theory of automata[END_REF]).

This paper, which is of expository nature, contains a new presentation of the techniques that allow to compute the groups contained in a transformation monoid. We use this technique to present a survey on some recent results that allow to describe the groups contained in transformation monoids. These transformation monoids arise as the transition monoid M of the minimal automaton recognizing the submonoid generated by a prefix code X. The maximal groups contained in M are called the syntactic groups of X. We treat two cases. In the first case, we discuss the problem of building bifix codes with the minimal possible number of elements having a given syntactic group. In the second one, we show that, for a particular class of prefix codes, all proper syntactic groups are cyclic.

The first case uses Sturmian words, which are a basic notion of combinatorics on words (see [START_REF] Lothaire | Algebraic Combinatorics on Words[END_REF]). Using Sturmian words one may prove that any transitive permutation group G of degree d and rank k is a syntactic group of a bifix code with (k -1)d+1 elements [START_REF] Berstel | Bifix codes and Sturmian words[END_REF]. We describe this construction here (Theorem 4.2). This result is an improvement of several previous ones due to Schützenberger [16] and to part of the authors of this paper [START_REF] Perrin | On syntactic groups[END_REF].

The second case uses prefix codes such that no element of them is an internal factor of another one, called prefix codes with empty kernel. We describe here the result proved in [START_REF] Berstel | On the groups of codes with empty kernel[END_REF]: if X is a prefix code with empty kernel, all proper syntactic groups of X are cyclic (Theorem 5.1).

The paper is organized as follows.

In Section 2, we give some basic definitions concerning automata, monoids and groups. A more detailed presentation can be found in [START_REF] Berstel | Codes and Automata[END_REF].

In Section 3, we define the holonomy groups of a transformation monoid. This notion is due to Eilenberg [START_REF] Eilenberg | Automata, Languages, and Machines[END_REF]. We show how it is related to the classical notion of group in a monoid, as presented for example in [START_REF] Lallement | Semigroups and combinatorial applications[END_REF] or [START_REF] Berstel | Codes and Automata[END_REF]. We recall the notion of Schützenberger representation and show how it may be used to compute generators of the holonomy groups. We also give an alternative method to compute such a set of generators based upon the notion of fundamental group of a graph.

In Section 4, we present a result from [START_REF] Berstel | Bifix codes and Sturmian words[END_REF] according to which any transitive permutation group of degree d which can be generated by k elements is a syntactic group of a bifix code with (k -1)d + 1 elements (Theorem 4.2). The proof uses Sturmian words. We recall the necessary definitions of Sturmian and episturmian words. We illustrate the construction on several examples.

In Section 5, we define the kernel of a set of words. We present the main result of [START_REF] Berstel | On the groups of codes with empty kernel[END_REF] which states that the proper syntactic groups of a prefix code with empty kernel are regular and cyclic (Theorem 5.1). We give several examples of this situation including ones using the well-known Černý automata.

Automata, monoids and groups

Let A be a finite set. We denote by A * the free monoid on the set A. A deterministic automaton A = (Q, i, T ) on the alphabet A is given by a set Q of states, an initial state i ∈ Q, a set T ⊂ Q of terminal states and a partial map from

Q × A into Q denoted (q, a) → q • a. An edge from p to q labeled a is a triple (p, a, q) ∈ Q × A × Q such that p • a = q. A path is a sequence c = (f 1 , f 2 , . . . , f n ) of consecutive edges f j = (q j , a j , q j+1 ). The word a 1 a 2 • • • a n is the label of the path.
For a state p ∈ Q and a word w ∈ A * , we denote p • w = q if there is a path labeled w from p to the state q and p • w = ∅ otherwise.

The set recognized by the automaton is the set of words w ∈ A * such that i • w ∈ T .

When the initial state and the terminal state need not be specified, we also denote an automaton A = (Q, E) with Q its set of states and E its set of edges.

Example 2.1 The automaton on the left of Figure 2.1 recognizes the words without any occurrence of bb. We use an incoming arrow to denote the initial state and an outgoing arrow to denote the terminal states. Thus, for the automata of Figure 2.1, state 1 is initial and terminal and state 2 is terminal. An automaton A is trim if for any q ∈ Q, there is a path from i to q and a path from q to some t ∈ T .

An automaton is called simple if it is trim and if it has a unique terminal state which coincides with the initial state.

For a set X ⊂ A * , we denote by A(X) the minimal automaton of X. The states of A(X) are the nonempty sets u

-1 X = {v ∈ A * | uv ∈ X} for u ∈ A * .
The initial state is the set X and the terminal states are the sets u -1 X for u ∈ X. Its edges are the triples (p, a, q) such that p = u -1 X and q = (ua) -1 X for some u ∈ A * . The minimal automaton A(X) is trim and recognizes the set X. A prefix code is a set of words which does not contain a proper prefix of one of its elements. The dual notion of a suffix code is defined symmetrically. A bifix code is a set of words which is simultaneously a prefix code and a suffix code.

Let X ⊂ A * be a prefix code. There is a simple automaton which recognizes the submonoid X * generated by X, namely the minimal automaton of X * . Conversely, any set is recognized by a simple automaton is a submonoid generated by a prefix code.

Let A = (Q, i, T ) be an automaton. For w ∈ A * , we denote ϕ A (w) the partial map from Q to Q defined by pϕ A (w) = q if p • w = q. The transition monoid of A is the monoid of partial maps from Q to Q of the form ϕ A (w) for w ∈ A * . It is denoted M (A). The degree of a permutation group G on a set R is the cardinality of R. The permutation group G is transitive if for any r, s ∈ R there is some g ∈ G such that rg = s.

A group automaton is a simple automaton A = (Q, 1, 1) on the alphabet A such that the maps ϕ A (a), for all letters a ∈ A, are permutations of Q. The transition monoid of a group automaton is a permutation group on Q. Its degree is Card(Q). Since the automaton is trim, the group is transitive.

The following result gives equivalent definition of submonoids recognized by group automata (see [START_REF] Berstel | Bifix codes and Sturmian words[END_REF]).

Proposition 2.6

The following conditions are equivalent for a submonoid M of A * .

(i) M is recognized by a group automaton with d states.

(ii) M = ϕ -1 (H), where H is a subgroup of index d of a group G and ϕ is a surjective morphism from A * onto G. (iii) M = H ∩ A * ,
where H is a subgroup of index d of the free group on A.

Let Z be the minimal generating set of a submonoid satisfying one of the conditions of Proposition 2.6. It is a bifix code called a group code. The integer d is its degree. The transition monoid of the minimal automaton of Z * is the group of Z, denoted G(Z). Note that, since a group automaton is minimal, the group automaton recognizing Z * is unique (up to the names of the states). Note also that the degree of Z is equal to the degree of the permutation group G(Z) and also to the index of the subgroup generated by Z. Eilenberg uses the term holonomy group in a sligthly different sense. The definition of the holonomy group in [START_REF] Eilenberg | Automata, Languages, and Machines[END_REF], is the following (see also [START_REF] Michael | Algebraic Automata Theory[END_REF]). Let J be the set of subsets of Q which are of the form Qm for m ∈ M or have cardinality at most one. Let I ⊂ Q and let J (I) be the set of J ∈ J such that J ⊂ I, J = I and which are maximal with respect to this property. Any element of Stab(I) defines a permutation of the set J (I). Indeed, for J ∈ J (I) and m ∈ Stab(I), the set Jm is in J (I) (it is maximal because m defines a permutation of the elements of J contained in I). The holonomy group, as defined in [START_REF] Eilenberg | Automata, Languages, and Machines[END_REF], is the permutation group on J (I) obtained in this way. Thus our definition differs in defining the action of the elements of Stab(I) on I itself instead of the set J (I).

Groups in monoids.. Let M be a monoid. A group in M is a subsemigroup of M which is isomorphic to a group. Note that the neutral element of a group contained in M needs not be equal to the neutral element of M .

A group in M is maximal if it not included in another group in M . For any idempotent e in M , there is a unique maximal group contained in M and containing e. It is denoted by G(e). The following property is well-known (see [START_REF] Berstel | Bifix codes and Sturmian words[END_REF]). Let G be a group in a transformation monoid on Q as above. The canonical representation of G is the group of permutations which is formed of the restrictions of the maps in G to their common image. We denote by G e the canonical representation of the group G(e). By Proposition 3.1, this representation is faithful.

For any set I ⊂ Q, there exists idempotents e of M such that I ⊂ Qe. Indeed, the neutral element of M is such an idempotent. We say that the idempotent e covers exactly the set I if I ⊂ Qe and Qe is minimal for this property. Note that any idempotent e covers exactly the set Qe.

The following statement shows that for any I ⊂ Q, the group Group(I) is obtained as the restriction to I of a group in M . Proposition 3.2 Let M be a transformation monoid on a finite set Q. For any I ⊂ Q, let e be an idempotent which covers exactly I. Then Group(I) is the restriction to I of the group G(e) ∩ Stab(I).

Proof. Let H = G(e) ∩ Stab(I). The restriction to I of an element of H is in Group(I). Conversely, let m ∈ Stab(I) and let g = eme. Since M is finite, there is an integer n such that h = g n is idempotent. Since I ⊂ Qe, we have e ∈ Stab(I) and thus g ∈ Stab(I), which implies that h ∈ Stab(I) and I ⊂ Qh. On the other hand, since h ∈ M e, we have Qh ⊂ Qe. By the minimality of Qe, we obtain Qh = Qe. This implies h = e. Thus the submonoid generated by g is a cyclic group containing e, which implies that g belongs to G(e) and thus to H. Since m and g have the same restriction to I, this shows that any element of Group(I) is obtained as the restriction to I of an element of H.

The following result shows in particular that the holonomy groups of a monoid M relative to the image of an idempotent are isomorphic with maximal groups contained in M . Holonomy groups of automata. The holonomy groups of an automaton are the holonomy groups of its transition monoid.

Let A = (Q, E) be an automaton and let I ⊂ Q. Let w ∈ A * be such that ϕ A (w) ∈ Stab(I). The restriction of ϕ A (w) to I is a permutation which belongs to Group(I). It is called the permutation of I defined by w. To compute a set of generators of a holonomy group, one may use two different methods. The first one is to compute the appropriate Schützenberger representation.

The Schützenberger representation. We recall here a description of the Schützenberger representation of a transformation monoid. For a more detailed description, see [START_REF] Lallement | Semigroups and combinatorial applications[END_REF] or [START_REF] Berstel | Codes and Automata[END_REF] (where the more general case of a monoid of unambiguous relations is treated).

Let M be a transformation monoid on a finite set Q. Let e be an idempotent of M and let I = Qe. Let I be the set of subsets J of Q such that Im = J and Jn = I for some m, n ∈ M . For each J ∈ I let m J , m ′ J ∈ M be such that

Im J = J, Jm ′ J = I with the normalization conditions m I = m ′ I = e and em J m ′ J = e (3.1)
for all J ∈ I. It is always possible to choose the m J , m ′ J in such a way that condition (3.1) is satisfied since em J m ′ J belongs to the group in M containing e.

Note that for any J ∈ I, the restriction of m ′ J m J to J is the identity on J. Indeed, let j ∈ J. Since J = Im J there is an i ∈ I such that j = im J . Then, using (3.1), we have

jm ′ J m J = im J m ′ J m J = iem J m ′ J m J = iem J = im J = j.
For J, K ∈ I and m ∈ M such that Jm = K, we define an element (J, m, K) of Group(I) as the restriction to I of m J mm ′ K . It is straightforward to verify that if Jm = K and Kn = L, then (J, mn, L) = (J, m, K)(K, n, L). Indeed, for any i ∈ I, we have

i(J, m, K)(K, n, L) = im J mm ′ K m K nm ′ L = im J mnm ′ L = i(J, mn, L)
since im J m ∈ K and m ′ K m K is the identity on K. Thus, we have the following result (see Proposition 9.2.1 in [START_REF] Berstel | Codes and Automata[END_REF]). Proposition 3.5 Let S be a set of generators of M . The permutations (J, m, K) for m ∈ S and J, K ∈ I with Jm = K form a set of generators of the group Group(I).

The Schützenberger representation of M relative to the idempotent e is the map which assigns to m ∈ M the I × I matrix with elements in Group(I) ∪ 0 defined by

µ(m) J,K = (J, m, K) if Jm = K 0 otherwise
When M is the monoid of transitions of an automaton A, the Schützenberger representation is defined by a transducer as in the following example.

Example 3.6 Let A be the automaton of Example 3.4. Set ϕ = ϕ A and M = ϕ(A * ). Let e be the idempotent ϕ(a 3 ). The set I is composed of

I = {1, 2, 3}, J = {1, 4, 5} and K = {1, 2, 5}. We choose m J = ϕ(b), m ′ J = ϕ(a 2 ba 2 ), m K = ϕ(ba), m ′ K = ϕ(aba 2
). The Schützenberger representation of M relative to e is the transducer of Figure 3

.3. 1, 2, 5 1, 4, 5 1, 2, 3 a | (123) b | (1) a | (1)
a | (23) An alternative method. An alternative method can be used to compute a set of generators of the holonomy groups of an automaton. Let A = (Q, E) be an automaton on a finite set Q of states. Let M be the transition monoid of A. Let e be an idempotent of M and let I = Qe. Let I be, as above, the set of subsets J of Q such that Im = J and Jn = I for some m, n ∈ M . Let G be the graph with I as set of vertices and the triples (J, a, K) ∈ I × A × I such that J • a = K as set of edges. We consider (J, a, K) as an edge from the vertex J to the vertex K with label a.

We also consider in the graph G inverses of the edges. If f = (J, a, K), then f -1 is an edge from K to J labeled a -1 . A generalized path in G is a sequence of consecutive elements from the set of edges and their inverses. Its label is the element of the free group on A obtained by concatenating the labels of its elements. The iterated simplification of consecutive edges which are mutually inverse generates an equivalence on the set of generalized paths. The set of classes of generalized paths from I to I forms a group denoted by H. The group H is called the fundamental group at the point I (see [START_REF] Lyndon | Combinatorial Group Theory[END_REF] for example). It is well-known that H is a free group and that a basis of H can be obtained as follows. Let T ⊂ E be a spanning tree with root I of the graph G. Since G is strongly connected, there is for every J in I a unique path p J in T from I to J.

Then the set X = {p J f p -1 K | f = (J, a, K) ∈ E \ T } is a basis of H called the Schreier basis relative to T (see [START_REF] Lyndon | Combinatorial Group Theory[END_REF]).

Let S ⊂ E be a set of edges such that S -1 is a spanning tree with root I of the reversal of the graph G. In this way, for each vertex J of G, there is a unique path q J from J to I using edges in S.

The following result shows that each choice of the pair T, S gives a set of generators of the holonomy group Group(I).

Proposition 3.7 The set Y = {p J f q K | f = (J, a, K) ∈ E \ T } is a basis of H.
The restriction to I of the labels of the elements of Y generates Group(I).

Proof. We first observe that for any f = (J, a, K) ∈ E, we have p J f q K ∈ Y . It is true by definition if f / ∈ T . Otherwise, we use an induction on the length of q K . Since T is a tree, we cannot have K = I and thus q K is not empty. Let g = (K, b, L) be the first edge of q K . Since the set of paths in S contains the suffixes of its elements, we have q K = gq L . Similarly, since T is a tree, p K = p J f . We have p K gq L ∈ Y by induction hypothesis. Since p J f q K = p K gq L , we have proved the claim.

Since Y ⊂ H = X , it is enough to prove that X ⊂ Y . We prove that, for f = (J, a, K) ∈ E \T , we have

p J f p -1 K ∈ Y . It is true if q K is empty since then p J f is in X ∩ Y .
Otherwise, let g = (K, b, L) ∈ E be the first edge of q K . Since the set of paths in S contains the suffixes of its elements, we have

q K = gq L . Then p J f q K = (p J f p -1 K )(p K gq L ) and thus p J f p -1 K = p J f q K (p K gq L ) -1 .
We have p J f q K ∈ Y by definition and p K gq L ∈ Y by the preliminary remark. Thus p J f p -1 K ∈ Y . Finally, the labels of the elements of Y form the submonoid Stab(I) and thus their restrictions to I form the group Group(I).

Note that, in the definition of Y , every element appears only once. Indeed, assume that p J f q K = p J ′ f ′ q K ′ for two edges f = (J, a, K) and f ′ = (J ′ , a ′ , K ′ ) in E \ T . Suppose that p J is a prefix of p J ′ . If it is a proper prefix, then f is an edge of the path p J ′ and thus is in T , a contradiction. Thus p J = p J ′ and f = f ′ .

The degree of syntactic groups

Let X be a prefix code and let A be the minimal automaton of X * . A syntactic group of X is a holonomy group of A relative to the image of an idempotent of M (A). Thus, by Proposition 3.3, a syntactic group of X has the form Group(I) for a set I = Qe, which is the image of an idempotent e in the monoid M (A).

Recall some terminology concerning permutation groups. Let G be a permutation group G on a set S. The order of G is its cardinality and its degree is the cardinality of the set S. The group G is said to be transitive if for any r, s ∈ S there is an element g of G such that rg = s.

Let us call minimal rank of a group G the minimal cardinality of a generating set for G. We will discuss the following conjecture [START_REF] Perrin | Sur les groupes dans les monoïdes finis[END_REF].

Conjecture 1 (Rank Conjecture) Let X be a finite bifix code and let G be a transitive permutation group of degree d and minimal rank k. If G is a syntactic group of X, then Card(X) ≥ (k -1)d + 1.

The inequality is related to Schreier's Formula. Indeed, if X is a basis of a subgroup of index d in a free group on k generators, then by Schreier's Formula, Card(X) = (k -1)d + 1.

The degree of nonspecial groups. Let X be a prefix code and let

A = A(X * ). A syntactic group G of X is called special if ϕ -1
A (G) is a cyclic submonoid. In particular a special syntactic group is cyclic.

The following result is from [START_REF] Perrin | On syntactic groups[END_REF]. This shows that the conjecture is true for groups of minimal rank 2. The theorem is clearly not true for special syntactic groups since Z/nZ is a syntactic group of X = a n for any n ≥ 1. This theorem was proved before by Schützenberger [START_REF] Schützenberger | A property of finitely generated submonoids of free monoids[END_REF] with a weaker bound (Card(X) ≥ d) but with a more general hypothesis (an arbitrary set X of words instead of a prefix code).

The general idea is that some parameters in the transition monoid of the minimal automaton of X * (such as the degrees of the holonomy groups) can be bounded in terms of Card(X) only, instead of the sum of the lengths of the words of X.

The proof of Schützenberger uses the Critical Factorization Theorem (see [START_REF] Lothaire | Combinatorics on Words[END_REF]).

The following result (from [START_REF] Berstel | Bifix codes and Sturmian words[END_REF]) shows that the bound in the Rank Conjecture can be reached for any transitive permutation group. Theorem 4.2 Any transitive permutation group of degree d which can be generated by k elements is a syntactic group of a bifix code with (k-1)d+1 elements.

We first give two examples illustrating the result. We will see afterwards the general construction. The holonomy group relative to {1, 2, 3} is the symmetric group S 3 generated by (123) defined by a and (23) defined by baa (see Example 3.4). Such a construction can be used to realize any group generated by a d-cycle α and another permutation β using

X = a d ∪ {a i ba d-(i+1)β | 0 ≤ i ≤ d -1}.
Example 4.4 Let X be the bifix code with 5 elements represented below.

The action on the sets of states with four elements is shown in Figure 4.2 below.

The word ba defines the permutation (18)(24) and the word aba the permutation (14)(28). Thus the regular representation of (Z/2Z) 2 is a syntactic group of this code. The proof of Theorem 4.2 relies on the notion of Sturmian and episturmian word that we describe now.

Episturmian words. Given a set F of words over an alphabet A, the right (resp. left) order of a word u in F is the number of letters a such that ua ∈ F (resp. au ∈ F ). A word u is right-special (resp. left-special ) if its right order (resp. left order) is at least 2. A right-special (resp. left-special) word is strict if its right (resp. left) order is equal to Card(A). In the case of a 2-letter alphabet, all special words are strict. For an infinite word x, we denote by F (x) the set of factors of x. By definition, an infinite word x is episturmian if F (x) is closed under reversal and if F (x) contains, for each n ≥ 1, at most one word of length n which is right-special.

Since F (x) is closed under reversal, the reversal of a right-special factor of length n is left-special, and it is the only left-special factor of length n of x. A suffix of a right-special factor is again right-special. Symmetrically, a prefix of a left-special factor is again left-special.

As a particular case, a strict episturmian word is an episturmian word x with the two following properties: x has exactly one right-special factor of each length and moreover each right-special factor u of x is strict, that is satisfies the inclusion uA ⊂ F (x) (see [START_REF] Droubay | Episturmian words and some constructions of de Luca and Rauzy[END_REF]).

A Sturmian set is the set of factors of a strict episturmian word.

It is easy to see that for a strict episturmian word x on an alphabet A with k letters, the set F (x) ∩ A n has (k -1)n + 1 elements for each n. Thus, for a binary alphabet, the strict episturmian words are just the Sturmian words, since a Sturmian word has one right-special factor for each length and its set of factors is closed under reversal. An episturmian word s is called standard if all its left-special factors are prefixes of s. For any episturmian word s, there is a standard one t such that F (s) = F (t). This is a rephrasing of Theorem 5 in [START_REF] Droubay | Episturmian words and some constructions of de Luca and Rauzy[END_REF]. Let G be a transitive permutation group of degree d on a set Q which can be generated by k elements. Let A be a k-letter alphabet and let ϕ : A * → G be a surjective morphism from A * onto G. Let 1 be an element of Q. The map q → qϕ(a) defines an automaton A = (Q, 1, 1). It is trim since G is transitive and thus it is a group automaton. Furthermore, G is the transition monoid of A. Let Z be the group code generating the submonoid recognized by A. Then G(Z) = G by definition. Thus Theorem 4.2 follows from the following result from [START_REF] Berstel | Bifix codes and Sturmian words[END_REF]. Theorem 4.7 Let A be an alphabet with k elements. Let Z ⊂ A * be a group code of degree d. Let F be a Sturmian set. The set X = Z ∩ F has (k -1)d + 1 elements and G(Z) is a syntactic group of X.

We illustrate the construction on three examples. In each case, we start from a group code Z and we compute the minimal automaton of X * , where X = Z ∩F . We exhibit in each case a set I ⊂ Q such that Group(I) is equivalent to G(Z). We have already seen (Example 4.4) that the holonomy group relative to {1, 2, 4, 8} is the regular representation of (Z/2Z) 2 .

Example 4.10 Let A = {a, b, c} and let F ⊂ A * be the set of factors of the Tribonacci word (see Example 4.6). Let ϕ : A * → (Z/2Z) 3 be the morphism defined by ϕ(a) = (1, 0, 0), ϕ(b) = (0, 1, 0) and ϕ(c) = (0, 0, 1). Let Z be the group code generating the submonoid ϕ -1 (0, 0, 0) and let X = Z ∩ F . The code X has 17 elements and the minimal automaton of X * is given in Table 4.1. The word abacaba is an element of F of rank 8. Its image is the set I = {1, 2, 7, 9, 19, 21, 30, 32}. The action on the 8-element sets is shown on Figure 4 defined respectively by the three words on the left. Thus this group is the regular representation of (Z/2Z) 3 .

Codes with empty kernel

A word s is an internal factor of w if w = rst with r, t nonempty. The kernel of a set of words X is the set of words in X which are internal factors of the words of X. We will state in this section results on syntactic groups of prefix codes with empty kernel. There are two important particular cases of prefix codes with empty kernel.

The first case is that of semaphore codes. A semaphore code is a set of the form X = A * S \ A * SA + for some nonempty set S ⊂ A + . Equivalently, a semaphore code is a maximal prefix code with empty kernel (see [START_REF] Berstel | Codes and Automata[END_REF]).

The second particular case is that of infix codes. An infix code is a set of words which does not contain any proper factor of its elements. It is a bifix code with an empty kernel.

Let X be a prefix code and let F be the set of internal factors of X. Let A = A(X * ) be the minimal automaton of X * . A syntactic group G of X is proper if there is a word w / ∈ F such that ϕ A (w) ∈ G. Note that if G is proper, then for any g ∈ G there is a word w / ∈ F such that ϕ A (w) = g.

Recall that a permutation group G is regular if no element of G distinct from the identity has a fixpoint.

The following result is from [START_REF] Berstel | On the groups of codes with empty kernel[END_REF] where it is stated in the more general case of codes instead of prefix codes. Theorem 5.1 Let X ⊂ A + be a prefix code with empty kernel. Any proper syntactic group of X is a finite regular cyclic group. M of A contains groups which are cyclic of order 1, 2 or 3. For example, ab contains the cycle (134) while a contains the cycle [START_REF] Lothaire | Algebraic Combinatorics on Words[END_REF]. Let us note an interesting feature of this example. For any word w in (a ∪ babb * aba) * , we have ϕ A (w) ∈ Stab({1, 2}) and w defines a cyclic group of order 2 (see Figure 5.2). But the set of these words is a submonoid which is not cyclic and not even finitely generated. This example shows another interesting feature. Consider the group G containing ϕ(ba). The neutral element of G is e = ϕ(baba) and its set of fixpoints is {1, 2, 3, 6}. The group G is of degree 4. It is composed of the permutation (16)(23) and the identity. It is thus of order 2 (note that this gives an example of a syntactic group which is not transitive). Actually, M does not contain any cyclic group of order 4. From groups to monoids. Let G be a transitive permutation group on R = {1, 2, . . . , n} and let ϕ : A * → G be a surjective morphism. Let H be the subgroup of G which fixes 1. Let Z be the bifix code generating the submonoid ϕ -1 (H). Let X be the set of elements of Z which have no proper factor in Z. Let M be the transition monoid of the automaton A(X * ).

The following result appears in [START_REF] Berstel | On the groups of codes with empty kernel[END_REF].

Proposition 5.4 The set X is a finite infix code. The syntactic groups of X, except possibly the group reduced to the neutral element of M , are cyclic and regular of degree at most n.

Note that the group reduced to the neutral element of M is of degree equal to the number of states of A(X * ) which may be larger than n. ware Semigroupes [START_REF] Froidure | Algorithms for computing finite semigroups[END_REF]). Moreover

• ab defines the cycle (1 6 11 4 9).

• a 2 ba 2 b 2 defines the permutation (1 12)(5 11) of degree 4 but there is no cyclic group of order 4 in M (A).

• a and b define cycles of degree 3.

• a 4 ba 5 defines the identity on {1, 2} and a 2 bab 2 a 2 ba 4 defines the transposition (1 2).

Černý automata. Let A be a complete deterministic automaton on a set Q of states. A word w is called synchronizing for A if the set Q • w of states reached after reading w has only one element. The automaton A is called synchronized if there exists a synchronizing word for A. Recall that Černý's conjecture [START_REF] Černý | Poznamka k homogenym s konecnymi automati[END_REF] expresses that a complete deterministic synchronized automaton with n states has a synchronizing word of length at most (n -1) 2 . The Černý automaton of order n, denoted C n is the following automaton on the alphabet A = {a, b}. Its set of states is the set {0, 1, . . . , n -1} and the transitions are defined by

i • a = i + 1 mod n, i • b = 1 if i = 0 i otherwise
The automata C n for n = 4, 5, 6 are represented in Figure 5.5.

For each n ≥ 1, the automaton C n is synchronized. Indeed, it is easy to verify that the word (ba n-1 ) n-2 b is a synchronizing word of length (n -1) 2 . It is known that there is no synchronizing word of length less than (n -1) 2 [START_REF] Černý | Poznamka k homogenym s konecnymi automati[END_REF]. We use Theorem 5.1 to prove the following result. The proof uses semaphore codes (see the beginning of Section 5). Proof. Let X n be the prefix code generating the stabilizer of 0 in C n , which is the submonoid formed by the words x ∈ A * such that 0 • x = 0. We have

X n = a(b * a) n-1 ∪ b(b * a) n-1
It is easy to verify that X n is a maximal prefix code which has empty kernel. Thus it is a semaphore code. The automaton A = C n , with 0 as initial and terminal state, is the minimal automaton of X * n . Let G be a syntactic group of X n . Since ϕ A (a n ) is the neutral element of M (A), the set ϕ -1 A (G) contains words which have an arbitrary high number of occurrences of a, and thus which are not factors of X. Thus G is proper and we conclude that it is cyclic and regular by Theorem 5.1. By Proposition 3.2, the holonomy groups of A are the restriction to I of the group G(e)∩Stab(I) for some idempotent e in M (A) with image J containing I. Since Group(J) is a syntactic group of X, it is cyclic and regular. Thus the same holds for the holonomy groups of A.

Note that the semaphore code X n above is the product of the two semaphore codes A and (b * a) n-1 (the product of two semaphore codes is a semaphore code by Proposition 3.5.12 of [START_REF] Berstel | Codes and Automata[END_REF]).

We do not know if there is a relation between the property of Černý automata expressed by Proposition 5.8 and the fact that they are an extremal exemple for the shortest length of synchronizing words. A connexion between Černý's conjecture and holonomy groups has already been established. Indeed, it is known that the conjecure holds for aperiodic automata, i.e. automata with trivial holonomy groups [18].

We give below examples showing the following features of the automata C n for n = 4, 5, 6: All holonomy groups are cyclic of order and degree k for all k with 1 ≤ k ≤ n.

We do not know if this property holds for all automata C n .

Example 5.9 The automaton C 4 has cyclic holonomy groups of order and degree 1, 2, 3, 4. Indeed, the table below indicates the permutations defined by each word on the first row. It does not seem easy to describe in general the shape of the words defining a cycle of length k for 1 < k < n -1.

Figure 2 . 1 :

 21 Figure 2.1: Two automata

Example 2 . 3 Figure 2 . 2 :

 2322 Figure 2.2: Two simple automata

Example 2 . 4

 24 The transition monoid of the automaton on the left of Figure2.2 has six elements, images of 1, a, b, ab, ba and bb respectively.

Example 2 . 5

 25 The transition monoid of the automaton on the right of Figure 2.2 is the regular representation of the cyclic group Z/2Z.

Example 2 . 7 3 . 2 . 8 Figure 2 . 3 :

 2732823 Figure 2.3: Two group automata

Proposition 3 . 1

 31 Let G be a group in a transformation monoid M . All elements of G have the same image I. The restriction of the elements of G to I is a faithful representation of G as a permutation group on I.

Proposition 3 . 3

 33 Let M be a transformation monoid on a finite set Q. Let e be an idempotent in M and set I = Qe. Then Group(I) = G e . Proof. By Proposition 3.2, and since G(e) ⊂ Stab(I), Group(I) is the restriction to I of the group G(e), which is by definition G e .

Example 3 . 4 Figure 3 . 1 :Figure 3 . 2 :

 343132 Figure 3.1: An automaton with holonomy group S 3 .

Figure 3 . 3 :

 33 Figure 3.3: The Schützenberger representation

Example 3 . 8

 38 Consider again the automaton of Example 3.4 and the idempotent e = ϕ(a 3 ) as in Example 3.6. Set also I = {1, 2, 3}. The group H is generated by the loop (I, a, I) and the cycle (I, baa, I). Thus Group(I) is generated by α(a) = (123) and α(baa) = (23) where α(w) denotes for w ∈ A * the restriction to I of ϕ(w).Example 3.9 Let A be the automaton represented in Table3.1. Set ϕ = ϕ A and let e = ϕ(a 5 ). The action of A on the sets with 5 elements is represented in Figure3.4. Set I = {1, 2, 3, 4, 5} and J = {1, 2, 6, 7, 8}. Set f = (I, a, I), g = (I, b, J), h = (J, b, J) and k = (J, a, I). Let us choose the tree T reduced to the edge g and the set S reduced to the edge k.

Figure 3 . 4 :

 34 Figure 3.4: The action on 5 element subsets

Figure 3 . 5 :

 35 Figure 3.5: The Schützenberger representation

Theorem 4 . 1

 41 Let G be a permutation group of degree d. If G is a nonspecial syntactic group of a prefix code X, then Card(X) ≥ d + 1.

Example 4 . 3

 43 Let X = {aaa, aaba, ab, baa}. The minimal automaton of X * is the automaton of Figure3.1.

Figure 4 . 2 :

 42 Figure 4.1: An automaton

Example 4 . 5 Example 4 . 6

 4546 Set A = {a, b}. The Fibonacci morphism is the substitution f : A * → A * defined by f (a) = ab and f (b) = a. The Fibonacci word x = abaababaabaababaababaabaababaabaab • • • is the fixpoint f ω (a) of f . It is a Sturmian word (see [12] Example 2.1.1). We call Fibonacci set the set of factors of the Fibonacci word. Consider the following generalization of the Fibonacci word to the ternary alphabet A = {a, b, c}. Consider the morphism f : A * → A * defined by f (a) = ab, f (b) = ac and f (c) = a. The fixpoint f ω (a) = abacabaabacababacabaabacabacabaabacab • • • is the Tribonacci word . It is a strict standard episturmian word (see [9]).

Example 4 . 8 Figure 4 . 3 :

 4843 Figure 4.3: A code with syntactic group S 3

Figure 4 . 4 :

 44 Figure 4.4: A code with syntactic group (Z/2Z) 2

Figure 4 . 5 :

 45 Figure 4.5: The action on 8-elements subsets It shows that the holonomy group relative to I is generated by the three

Example 5 . 2 Figure 5 . 1 :

 5251 Figure 5.1: The automaton A(X * )

Figure 5 . 2 :

 52 Figure 5.2: The action of A * on the 2-element subsets reachable from {1, 2}

Figure 5 . 3 :

 53 Figure 5.3: An infix code with a group of degree 4 and order 2.

Example 5 . 5 Figure 5 . 4 :Example 5 . 6 Example 5 . 7

 55545657 Figure 5.4: The infinite group code Z and the finite code X

Figure 5 . 5 : 6 Proposition 5 . 8

 55658 Figure 5.5: The automata C 4 , C 5 , C 6

a ab a 2 b a 3 bab a 3 bab a 2 b a 3

 33 (0123) (123)[START_REF] Lyndon | Combinatorial Group Theory[END_REF] [START_REF] Berstel | On the groups of codes with empty kernel[END_REF] Example 5.10 The automaton C 5 has cyclic holonomy groups of order and degree 1, 2, 3, 4, 5. Indeed, the table below indicates the permutations defined by each word on the first row. a ab a 3 b a 2 b a 4 b (01234) (1234) (142) (13) (1) Example 5.11 The automaton C 6 has cyclic holonomy groups of order and degree 1, 2, 3, 4, 5, 6. Indeed, the table below indicates the permutations defined by each word on the first row. a ba 5 b a 5 b (012345) (12345) (1532) (135)(13) (1)It is easy to see that, in general, (i) a defines the n-cycle (01• • • n -1) (ii) ab defines the n -1 cycle (12 • • • n -1).(iii) a n-1 b defines the cycle (1).

Table 4 .

 4 1: The transitions of the automaton A(X * )

	permutations on the right below:
	caba	(1, 19)(2, 21)(7, 30)(9, 32)
	bacaba	(1, 30)(2, 32)(7, 19)(9, 21)
	abacaba (1, 32)(2, 30)(7, 21)(9, 19)

Table 5 .

 5 1: The transitions of A(X * )

Preprint submitted to ElsevierJuly 31, 2011


Michel Lubrano
email: michel.lubrano@univ-amu.fr

Abdoul Aziz

Junior Ndoye

Bayesian Unconditional Quantile Regression: An analysis of recent expansions in wage structure and earnings inequality in the U.S. 1992-2009

Keywords: Oaxaca-Blinder decomposition, Bayesian inference, quantile regression, unconditional quantile, influence function JEL classification: C1, C11, C13, C14, C21

We develop a reliable Bayesian inference for the RIF-regression model of Firpo, Fortin and Lemieux (Econometrica, 2009) in which we first estimate the log wage distribution by a mixture of normal densities. This approach is pursued so as to provide better estimates in the upper tail of the wage distribution as well as valid confidence intervals for the Oaxaca-Blinder decomposition. We apply our method to a Mincer equation for analysing the recent changes in the U.S. wage structure and in earnings inequality. Our analysis uses data from the CPS Outgoing Rotation Group (ORG) from 1992 to 2009. We find first that the largest part (around 77% on average) of the recent changes in the U.S. wage inequality is explained by the wage structure effect and second that the earnings inequality is rising more at the top end of the wage distribution, even in the most recent years. The decline in the unionisation rate has a small impact on total wage inequality while differences in returns to education and gender discrimination are the dominant factors accounting for these recent changes.

Introduction

Introduced by [START_REF] Koenker | Regression quantiles[END_REF], quantile regression models aim at modeling the effect of the explanatory variables on the conditional distribution of the outcome variable. They have been increasingly used in empirical labour market studies, to describe parsimoniously the entire wage conditional distribution (see e.g. [START_REF] Buchinsky | Changes in the U.S. wage structure 1963-1987: An application of quantile regression[END_REF][START_REF] Chamberlain | Quantile regression censoring and the structure of wages[END_REF], [START_REF] Machado | Earning functions in Portugal 1982-1994: Evidence from quantile regressions[END_REF]. Several competing methods of estimation in both classical and Bayesian frameworks have been recently developped (see for instance [START_REF] Yu | Bayesian quantile regression[END_REF]Moyeed 2001, Kozumi and[START_REF] Kozumi | Gibbs sampling methods for Bayesian quantile regression[END_REF] for the Bayesian side). Since any quantile can be used in any part of the outcome distribution, the quantile regression models are more flexible and more robust to outliers than the classical mean regression models.

While the conditional quantile regression models can be useful, they are very restrictive. First a change in the distribution of covariates may change the interpretation of the coefficient estimates. This point is illustrated for instance in [START_REF] Powell | Unconditional quantile regression for exogenous or endogenous treatment variables[END_REF]. To overcome this restriction, [START_REF] Firpo | Unconditional quantile regressions[END_REF] have proposed a new regression method which evaluates the impact of changes in the distribution of the explanatory variables on the quantiles of the unconditional distribution of the outcome variable. Second, the property that, in the popular Oaxaca-Blinder decomposition method of a simple linear regression, differences in unconditional means is equal to differences between conditional means, is no longer valid for conditional quantile regressions. As explained in e.g. [START_REF] Bazen | Econometric Methods for Labour Economics[END_REF], with conditional quantile regressions, the difference in unconditional quantiles is not equal to difference in conditional quantiles. This question has received several answers in the literature, see e.g. [START_REF] Juhn | Wage inequality and the rise in returns to skill[END_REF], [START_REF] Dinardo | Labor market institutions and the distribution of wages, 1973-1992: A semiparametric approach[END_REF], or [START_REF] Machado | Counterfactual decompositions of changes in wage distributions using quantile regressions[END_REF], but none of these methods can be used to decompose general distributional measures in the same way that the means can be decomposed using the conventional Oaxaca-Blinder method. However, the method of [START_REF] Melly | Decomposition of differences in distribution using quantile regression[END_REF] and the RIF-regression method of [START_REF] Firpo | Unconditional quantile regressions[END_REF] can perform a detailed decomposition very much in the spirit of the traditional Oaxaca decomposition for the mean [START_REF] Firpo | Decomposition methods in economics[END_REF].

In this paper, we develop a reliable Bayesian inference method for the RIFregression model of [START_REF] Firpo | Unconditional quantile regressions[END_REF] in which we estimate parametrically the log wage distribution by a mixture of normal densities. The mixture of normal densities is pursued so as to produce a better fit in the tails of the wage distribution. Our first empirical results show that the presence of a heavy right-hand tail in the wage distribution makes unreliable the usual density kernel estimate used in the RIF-OLS method of [START_REF] Firpo | Unconditional quantile regressions[END_REF] as it leads to an overestimation of the RIF-regression coefficients for the corresponding quantile. Our parametric approach provides better estimates of the RIF-regression coefficients in the upper tail.

Standard errors obtained by classical RIF-OLS are slightly smaller than our Bayesian standard errors due to the fact our Bayesian approach takes a better account of parameter uncertainty of the density estimation and is pursued so as to propose valid confidence intervals for the Oaxaca-Blinder decomposition.

We illustrate our approach, analysing the recent trends in U.S. wage structure and earnings inequality. The recent rise in earnings dispersion in U.S. is more of a shock as it started quite a long time ago. The literature dealing with the causes of this wage dispersion has literally exploded over the past decades. Several competing explanations have been offered. [START_REF] Bound | Changes in the structure of wages in the 1980's: An evaluation of alternative explanations[END_REF] attribute the changes to the skill-biased technological progress which increases the rate of growth of the relative demand for highly educated and "more-skilled" workers (see also Mincer 1993, Katz and[START_REF] Katz | Changes in the wage structure and earnings inequality[END_REF]. [START_REF] Murphy | The structure of wages[END_REF] stress the impact of the globalisation which increases the rate of unskilled immigration workers and led to a decrease in the growth of the relative supply of skills (see also [START_REF] Katz | Changes in relative wages, 1963-1987: Supply and demand factors[END_REF]. [START_REF] Dinardo | Labor market institutions and the distribution of wages, 1973-1992: A semiparametric approach[END_REF] focus on changes in labour market institutions, in wage setting norms including the decline in unionisation, on the erosion of the real and relative value of the minimum wage.

Atkinson ( 2008) is inclined to be careful about these now traditional explanations and suggests to take seriously the new models of earnings formation. In his book, he reviews other alternative explanations such as [START_REF] Rosen | The economics of superstars[END_REF]'s model of superstars and hierarchical models. He provides a complete descriptive analysis for the changing distribution of earnings in different OECD countries. He argues that "while the race between technology and education is appealing, a constantly rising demand for educated workers does not lead to a constantly rising wage premium but to a stable wage differential, the size of which depends on the speed of a country's response to shortages of qualified workers". Our paper does not refute these explanations but aims simply at measuring the alternative role of some factors such as union, education, experience and gender for explaining the recent changes in the U.S. wage structure and in earnings inequality.

Most previous studies on changes in the structure of wages in the U.S. have used wage data from the March Current Population Survey (CPS) (see e.g. Buchinsky 1994 and others). We use instead the hourly wage data from the Outgoing Rotation Group (ORG) supplements of the CPS as [START_REF] Lemieux | Increasing residual wage inequality: Composition effects, noisy data, or rising demand for skill?[END_REF] and [START_REF] Firpo | Decomposing wage distributions using recentered influence function regressions[END_REF]. The ORG CPS files provide a better data set for measuring changes in hourly wage distribution than the March CPS as they give a better representation of the dispersion of wages for each and every hour worked in the labour market, regardless of who is supplying this hour [START_REF] Bernstein | Has wage inequality stopped growing?[END_REF]. Since January 1992, the CPS has changed the coding scheme of its education attainment question from completed years to degree acquired. [START_REF] Kominski | Measuring educational attainment in the current population survey[END_REF] show that the new educational attainment item provides more relevant and useful data for current and future analysis. Over our period of analysis (1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009), unconditional quantile regressions show that earnings inequality is more rising in the top end of the wage distribution while an Oaxaca-Blinder decomposition shows that for a large part (around 77% on average), changes in the U.S. wage structure is explained by the wage structure effect (differences in yields of initial characteristics). If the decline in unionisation has an impact on the recent changes in wage structure, this effect is mainly operating for low wages. For higher wages the increase in returns to education and gender discrimination are the dominant factors.

The paper is organised as follows. In section 2, we review the conditional quantile regression models when using a likelihood function that is based on the asymmetric Laplace distribution [START_REF] Yu | Bayesian quantile regression[END_REF], and we show the limitations of MCMC methods and Oaxaca-Blinder decomposition procedure used for conditional quantile regression. In section 3, we present a reliable Bayesian inference for the RIF-regression of [START_REF] Firpo | Unconditional quantile regressions[END_REF] in which we estimate the log wage distribution by a mixture of three normal densities. In section 4, we provide an Oaxaca-Blinder decomposition procedure using our RIF-regression method, and we show how to obtain reliable standard errors for each component of the decomposition using the draws of the RIF-regression coefficients together with a procedure of Rao-Blackwellisation. Section 5 illustrates the approach by using the CPS-ORG sample from 1992 through 2009. Section 6 concludes.

Conditional quantile regression models

Consider the linear regression model

y i = x ′ i β τ + ǫ i , (1)
where (y i , x i ), i = 1, 2, • • • , n are independent observations, y i being the response variable and

x ′ i = (1, x i1 , • • • , x ik
) being the (k + 1) known covariates. In the next paragraph, τ will mean the τ th quantile.

β ′ τ = (β 0τ , • • • , β kτ ) represents the (k + 1)
unknown regression parameters, and ǫ i , i = 1, • • • , n are the error terms which are supposed independent and identically distributed. The unbiased estimation of β in a usual regression model requires that E(ǫ i |x i ) = 0, whatever the distribution of ǫ i .

A quantile regression model considers the same linear regression as (1), but this time, the zero expectation assumption for ǫ is replaced by the assumption that the τ th quantile of ǫ is equal to zero. If f (.) is the density of ǫ, this means that

0 -∞ f τ (ǫ i )dǫ i = τ.
(2)

The quantile regression estimator for β τ , βτ first proposed in [START_REF] Koenker | Regression quantiles[END_REF] does not consider a specific distribution for ǫ (so that f (.) is left unspecified).

It is simply given as the solution of the following minimisation problem min

β 1 N n i=1 ρ τ (y i -x ′ i β τ ), (3)
where ρ τ (.) is the check function or loss function defined as

ρ τ (u) = u × (τ -1I(u < 0)), (4)
where 1I(.) is the indicator function. As this loss function is not differentiable (as a quadratic loss function would be), one has to use linear programming techniques to solve this problem.

2.1 Using the asymmetric Laplace distribution [START_REF] Yu | Bayesian quantile regression[END_REF] have proposed to specify the distribution of ǫ using the Asymmetric Laplace Distribution (ALD):

f (ǫ i |τ ) = τ (1 -τ ) σ exp - 1 σ ρ τ (ǫ i ) . (5)
This density automatically fulfill the quantile restriction condition (2). For a symmetric Laplace process, the maximum likelihood estimator of the mean parameter is equal to the sample median. This property is generalised here for all quantiles so that the maximum likelihood estimator based on the complete likelihood

L(y i |β τ , σ τ , τ ) = σ -n τ τ n (1 -τ ) n exp - 1 σ τ i ρ τ (y i -x ′ i β τ ) (6)
provides exactly the same value as that provided by the estimator proposed of [START_REF] Koenker | Regression quantiles[END_REF] for β τ . With however the same difficulties as the loss function ρ τ (u) (4) is not differentiable at zero. A Bayesian approach does not lead to the same difficulties; the likelihood function (times the prior) has to be integrated and differentiability plays no role in integration.

Bayesian Inference for conditional quantile regression

To make inferences on the parameter of interest β τ and σ τ , given τ and the observations on (X, Y ), one has to specify a prior density for β. The posterior distribution of β τ , π(β|y) is proportional to

π(β, σ|y) ∝ L(y|β)π(β, σ),
where L(y|β) is the likelihood function given in (6) and π(β, σ) is the prior distribution of β and σ. [START_REF] Yu | Bayesian quantile regression[END_REF] show that for any type of prior, including an improper prior, the posterior moments exist. They choose an improper prior and no conjugate prior is available when the model is presented in this form. The posterior density has to be integrated out by a MCMC method. [START_REF] Yu | Bayesian quantile regression[END_REF] make use of the simple random walk Metropolis with a Gaussian proposal. The method is available as package bayesQR in R. As noted in [START_REF] Kozumi | Gibbs sampling methods for Bayesian quantile regression[END_REF], the random walk Metropolis maybe difficult to tune, because a different tuning parameter has to be chosen for every value of τ so as to get an acceptation rate of around 25%. [START_REF] Kozumi | Gibbs sampling methods for Bayesian quantile regression[END_REF] propose a location-scale mixture representation of the asymmetric Laplace distribution that allows to find analytical expressions for the conditional posterior densities of the model. With these tools, they can propose first a conditional natural conjugate prior and second a Gibbs sampler. The merit of the Gibbs sampler is to avoid the specification of a candidate density and of a tuning parameter. The normal-inverted-gamma prior combines nicely with the conditional likelihood in the Gibbs sampler. We can note however that it seems difficult to elicit an informative prior, because we should specify different hyper-parameters for each quantile. The Gibbs sampler has an important drawback compared to a direct Metropolis approach which is its extreme slowness due to the fact that one has to draw random numbers in an inverted generalised Gaussian for each observation and this is a slow operation which has to be done for each observation separately.

Oaxaca-Blinder decomposition and quantiles

The popular Oaxaca-Blinder decomposition [START_REF] Oaxaca | Male-female wage differentials in urban labor markets[END_REF][START_REF] Blinder | Wage discrimination: reduced form and structural estimates[END_REF]) makes use of the property that, in a linear regression, the difference in unconditional means is equal to the difference between conditional means. If

y i = X i β + e i , then E(y i ) = E(X i )β.
Applying this simple result to a Mincer wage equation where y is the log wage, we can explain the mean wage gap between for instance males and females as

E[y mi -y f i ] = E[X mi β m + e mi ] -E[X f i β f + e f i ] = E[X mi ]β m -E[X f i ]β f = [E(X mi ) -E(X f i )]β m + E(X f i )[β m -β f ].
This decomposition is estimated by replacing the expected value of the covariates by their sample mean and the β by their regression estimates. In a classical framework, this will be the OLS estimator, in a Bayesian framework the posterior expectation is used as a first approximation. This equation means that mean wage differences are explained first by the difference in average characteristics multiplied by the male coefficient (composition effect) and secondly by the difference in yield of female average characteristics expressed by βmβf (structure effect). This results is not directly transposable to quantile regression as in a quantile regression E(e i ) = 0. We would like to explain the difference between two unconditional quantiles as a function of the conditional quantiles. As recalled in [START_REF] Firpo | Decomposition methods in economics[END_REF], the difference in unconditional quantiles is not equal to the difference of conditional quantiles. This question has received several answers in the literature (see e.g. [START_REF] Juhn | Wage inequality and the rise in returns to skill[END_REF][START_REF] Dinardo | Labor market institutions and the distribution of wages, 1973-1992: A semiparametric approach[END_REF][START_REF] Machado | Counterfactual decompositions of changes in wage distributions using quantile regressions[END_REF][START_REF] Melly | Decomposition of differences in distribution using quantile regression[END_REF]), but none of these methods can be used to decompose general distributional measures in the same way as means can be decomposed when using the conventional Oaxaca-Blinder method. [START_REF] Juhn | Wage inequality and the rise in returns to skill[END_REF] have proposed a "plug-in" procedure of Oaxaca decomposition which allows for the distribution of the error term to depend on the covariates. But in the presence of heteroscedasticity, this method produces misleading results. [START_REF] Dinardo | Labor market institutions and the distribution of wages, 1973-1992: A semiparametric approach[END_REF] have proposed a reweighing procedure using a kernel density estimation. However, if there are too many variables, it becomes impossible to estimate counterfactual distributions non-parametrically. [START_REF] Machado | Counterfactual decompositions of changes in wage distributions using quantile regressions[END_REF] have proposed a simulation method to compute the wage structure sub-components of the detailed decomposition using a Monte Carlo approach. These components are computed by sequentially switching the coefficients of the quantile regressions for each covariate from their estimated valued. But, this method does not provide a consistent effect since the effect of the reweighed covariate of interest gets confounded by other covariates correlated with that same covariate. [START_REF] Firpo | Decomposition methods in economics[END_REF] show that the method based on the estimation of RIFregressions proposed in [START_REF] Firpo | Unconditional quantile regressions[END_REF] is more consistent for estimating the detailed components of both the wage structure and the composition effects. This is the method that we shall discuss in the next section and use as a basis for a Bayesian implementation.

Unconditional quantile regression

The Influence Function (IF ), first introduced by [START_REF] Hampel | The influence curve and its role in robust estimation[END_REF], describes the influence of an infinitesimal change in the distribution of a sample on a real-valued functional distribution or statistics ν(F ), where F is a cumulative distribution function. The IF of the functional ν is defined as

IF (y, ν, F ) = lim ǫ→0 ν(F ǫ,∆y ) -ν(F ) ǫ = ∂ν(F ǫ,∆y ) ∂ǫ | ǫ=0 (7)
where F ǫ,∆y = (1ǫ)F + ǫ∆ y is a mixture model with a perturbation distribution ∆ y which puts a mass 1 at any point y. The expectation of IF is equal to 0. Firpo et al. ( 2009) make use of ( 7) by considering the distributional statistics ν(.) as being the quantile function (ν(F ) = q τ ) in order to find how a marginal quantile of y can be modified by a small change in the distribution of the covariates. They make use of the Recentered Influence Function (RIF ), defined as the original statistics plus the IF so that the expectation of the RIF is equal to the original statistics.

Considering the τ th quantile q τ defined implicitly as τ = qτ -∞ dF (y), [START_REF] Firpo | Unconditional quantile regressions[END_REF] show that the IF for the quantile of the distribution of y is given by

IF (y, q τ (y), F ) = τ -1I(y ≤ q τ ) f (q τ ) ,
where f (q τ ) is the value of the density function of y evaluated at q τ . The corresponding RIF is simply defined by

RIF (y, q τ , F ) = q τ + τ -1I(y ≤ q τ ) f (q τ ) , (8)
with the immediate property that E (RIF (y, q τ )) = RIF (y, q τ )f (y)dy = q τ .

The illuminating idea of [START_REF] Firpo | Unconditional quantile regressions[END_REF] is to regress the RIF on covariates, so the change in the marginal quantile q τ is going to be explained by a change in the distribution of the covariates by means of a simple linear regression:

E[RIF (y, q τ |X)] = Xβ. (9)
They propose different estimation methods: a standard OLS regression (RIF-OLS), a logit regression (RIF-Logit) and a nonparametric logit regression. The estimates of the coefficients of the unconditional quantile regressions, βτ obtained by a simple Ordinary Least Square (OLS) regression (RIF-OLS) are as follows:

βτ = (X ′ X) -1 X ′ RIF (y; q τ ). ( 10
)
The practical problem to solve is that the RIF depends on the marginal density of y. [START_REF] Firpo | Unconditional quantile regressions[END_REF] propose using a non-parametric estimator for the density and the sample quantile for q τ so that an estimate of the RIF for each observation is given by

RIF (y i ; q τ ) = qτ + τ -1I(y ≤ qτ ) f (q τ ) .
Standard deviations of the coefficients are given by the standard errors of the regression.

However, the RIF-regression models of [START_REF] Firpo | Unconditional quantile regressions[END_REF] present some limitations.

-First, if the wage distribution is characterised by a heavy right-hand tail, the kernel density estimation may under-smooth the tail density estimates, leading to unreliable inference for the upper quantile regression coefficients. To overcome this problem, we propose a semi-parametric approach to estimate the distribution of log-wages using a mixture of normal densities.

-Second the classic RIF-OLS estimation does not take into account the uncertainty introduced by the use of a point estimate for f (q τ ). A Bayesian approach should help to remove this difficulty.

Bayesian inference for the RIF-regression model

We model the distribution of the observed log-wages by a mixture of K normal densities f (y|θ) indexed by θ = (θ k ) k=1,...,K , where θ k = (µ k , σ 2 k , p k ), and (µ k , σ 2 k ) are the component specific mean and variance. If each component is sampled with probability p k , then the density function f (y|θ) is written as:

f (y|θ) = K k=1 p k f (y|θ k ), (11)
where

f (y|θ k ) = 1 σ k √ 2π exp - (y -µ k ) 2 2σ 2 k .
Bayesian inference for mixture of normal densities relies on a rewriting of the likelihood function using a data augmentation representation which leads to a Gibbs sampler. Details of the approach can be found in e.g. [START_REF] Robert | Monte Carlo Statistical Methods[END_REF] or [START_REF] Frühwirth-Schnatter | Finite Mixture and Markov Switching Models[END_REF].

The RIF for a quantile regression can then be reformulated as follows

RIF (y i ; q τ ) = y(θ, τ ) = qτ + τ -1I(y ≤ qτ ) f (q τ |θ) ,
where qτ remains the natural estimator of the τ th quantile while θ is an unknown parameter. The quantile regression model becomes

y(θ, τ ) = Xβ(θ, τ ) + ǫ. ( 12
)
where ǫ is now normal with zero mean and variance σ 2 . This is a conditional linear regression, conditional on the value of θ. In fact, this problem can be treated sequentially. We first estimate the marginal density of y by means of the given mixture of normal densities in (11). Given this estimation, we run the linear regression (12). More precisely, we first derive the posterior density of θ by a Gibbs sampler, store the posterior draws of θ and then treat model ( 12) given the former Gibbs output.

Marginal moments of β are obtained by averaging over the draws of θ.

The conditional posterior density of β in ( 12) is Student with

ϕ(β|θ, τ, y, X) = f t (β|β * (θ), s * (θ), M * , n), (13)
where, if we suppose a non-informative prior for β and σ

2 M * = X ′ X, β * (θ) = M -1 * X ′ y(θ, τ ), s * (θ) = y(θ, τ ) ′ (I N -X(X ′ X) -1 X ′ )y(θ, τ ). ( 14
)
Marginal moments are obtained by integrating out θ. This integration can be approximated easily when we have posterior draws of θ, noted θ j :

E(β|y, τ ) = β * (θ, τ )ϕ(θ|y)dθ ≃ 1 m m j=1 β * (θ j ), (15)
Var(β|y, τ

) = M -1 * n -2 s * (θ, τ )ϕ(θ|y)dθ ≃ 1 m(n -2) M -1 * m j=1
s * (θ j ). ( 16)

Let us give a brief sketch of the procedure for estimating θ. The Bayesian approach for estimating mixture relies on a data augmentation representation. For each observation y i of (y 1 , . . . , y n ) from ( 11), we associate a missing variable z i that indicates its component of origin. The conditional likelihood function of the sample is

L(µ k , σ 2 k |y, z) ∝ σ -n k k exp - 1 2σ 2 k s 2 k (z) + n k (µ k -ȳk (z)) 2 , ( 17
)
where the sufficient statistics are

ȳk (z) = 1 n k i∈Z k log y i , s 2 k (z) = 1 n k i∈Z k (log y i -ȳ) 2 , n k = 1I(z i = j).
We can specify conjugate prior densities for all the parameters with a conditional normal prior for µ k , an inverted gamma2 prior for σ 2 k and a Dirichlet prior on p k . Combining these prior densities with the conditional likelihood function in (17), we obtain a conditional Student posterior density for µ k , and an inverse gamma conditional posterior density for

σ 2 ϕ(µ k |x, z) ∝ f t (µ k |µ * k , s * k , n * k , ν * k ) (18) ϕ(σ 2 k |x, z) ∝ f iγ (σ 2 k |ν * k , s * k ), (19)
where

ν * k = ν 0 + n k , s * k = s 0 + s 2 k (z) + n 0 n k n 0 + n k (µ 0 -xk (z)) 2 , µ * k = n o µ 0 + n k ȳk n * k , n * k = n 0 + n k ,
and where µ 0 , n 0 , s 0 and ν 0 are the hyperparameters of the prior densities for the mixture. We propose the following MCMC algorithm which combines inference for θ and β τ in a sequential process.

1. Set p (0) , µ (0) , σ 2 (0) , the number of draws m and select τ .

2. Compute the τ th quantile q(τ ) of the log wages and M = (X ′ X) -1

3. Begin loop on j = 1, . . . , m

(a) Begin loop on k = 1, . . . , K i. Generate z (j) i from P z (j) i = j|p (j-1) k , µ (j-1) k , σ 2(j-1) k , y i ∝ p (j-1) k f y i |µ (j-1) k , σ 2(j-1) k for each observation i ii. Compute n (j) k = n i=1 I z (j) i =j , s (j) k = n i=1 I z (j) i =j y i iii. Generate p (j) k from D γ 1 + n (j) 1 , . . . , γ k + n (j) k , iv. Generate µ (j) k from ϕ(µ (j) k |z (j) , y) v. Generate σ 2(j) k from ϕ(σ 2(j) k |y, z (j) ) (b) End loop on k (c) Compute y(τ ) (j) = q(τ ) + τ -1(y≤q(τ )) k p (j) k f (q(τ )|µ (j) k ,σ 2(j) k ) (d) Store β (j) * = M Xy(τ ) (j) (e) Store s (j) * = y(τ ) (j) ′ y(τ ) (j) -Xy(τ ) (j) ′ M Xy(τ ) (j)
4. End loop on j 5. Compute the mean of β *

Compute the mean of s

* × M n-2
As a by-product of this algorithm, we obtain draws from an approximation to the posterior density of θ, ϕ(θ) that will be useful for the derivation of Oacaca-Blinder decomposition.

Oaxaca-Blinder decomposition and RIF-OLS

The Oaxaca-Blinder method is very useful for decomposing differences in mean wages between two periods into a wage structure effect and a composition effect. For the unconditional quantile regression, the Oaxaca-Blinder decomposition procedure based on the RIF-regression model provides a detailed decomposition of the differences in mean wages between two periods [START_REF] Firpo | Decomposition methods in economics[END_REF]). If we label by A and B the two different groups, the RIF-regressions for each group g, (g = A, B) are given by

y g (θ, τ ) = X g β g (θ, τ ) + ǫ g , g = A, B. (20)
The differences in mean quantile wages between the two groups are then given by

E(y B (θ, τ )|X B ) -E(y A (θ, τ )|X A ) ∆ O (θ,τ ) = XB (β B (θ, τ ) -β A (θ, τ )) ∆ β (θ,τ ) + ( XB -XA )β A (θ, τ ) ∆ X (θ,τ ) . ( 21
)
as E(ǫ g |X) = 0 in the RIF regression. The first right hand component, ∆ β (θ, τ ) is interpreted as the difference in yields of given individual characteristics corresponding to the second period (the wage structure effect). The second right hand term, ∆ X (θ, τ ) is the component associated with differences in the characteristics themselves (the composition effect) as they have evolved between the two periods. The three quantities in (21) are conditional on θ which has now to be integrated out. Formally,

∆ O (θ, τ )ϕ(θ)dθ = XB [β B (θ, τ ) -β A (θ, τ )]ϕ(θ)dθ + ( XB -XA ) β A (θ, τ )ϕ(θ)dθ. ( 22
)
We want to compute the posterior marginal expectation and posterior marginal variance of the two components of the Oaxaca-Blinder decomposition. With ( 22), we still do not have an estimator. We can produce an estimator if we replace β g (θ, τ ) by β * g (θ, τ ) in ( 22), which means replacing the parameter by its posterior conditional expectation. Following this way, the marginal expectation of the composition and wage structure effects can be evaluated in a straightforward way:

E[∆ β (τ )] = XB 1 m m j=1 (β B * j -β A * j ) (23) E[∆ X (τ )] = ( XB -XA ) 1 m m j=1 β A * j , (24)
where β A * j ; β B * j = β A * (τ, θ (j) ); β B * (τ, θ (j) ) are the draws of the RIF-regression coefficients obtained from the Gibbs output θ = θ (j) m j=1 . The expectation of the total effect is just the sum of the two components expectations.

Remark:

We could also proceed in another way. Conditionally on a draw of θ, say θ j , we can compute the hyperparameters in ( 14) and then using (13), we can get m posterior draws of β g . As consequently m draws for ∆ O (τ ), ∆ β (τ ) and ∆ X (τ ). Once we have these m draws, we can compute the mean and variance of the Oaxaca decomposition. See [START_REF] Radchenko | A Bayesian approach to decomposing wage differentials[END_REF] for a similar implementation in the framework of the usual linear regression. Note that the first method should produce more precise results as it corresponds to a Rao-Blackwellisation.

Standard errors for the Oaxaca-Blinder decomposition

Most empirical studies which use the Oaxaca-Blinder decomposition procedure do not indicate how standard errors are obtained. As E(y(θ, τ )|X) = X′ β * (θ, τ ), a well defined approximate variance estimator for the conditional mean of the RIF is given by:

V(E[y(θ, τ )|X]) = V[ X′ β * (θ, τ )] = X′ V[β * (θ, τ )] X,
as X is supposed to be constant (see Jann 2008 for a classical approach and an alternative derivation when X is supposed to be random). As this is a conditional expectation, we have to integrate out θ to obtain the marginal variance of β as given in ( 16).

Following the lines given in [START_REF] Oaxaca | Calculation of approximate variances for wage decomposition differentials[END_REF], the conditional variances of ∆ β (θ, τ ) and ∆ X (θ, τ ) are easily obtained and when θ is integrated out, we get the following estimates which are transformations of ( 16):

4 Application: Trends in U.S. wage structure and earnings inequality 1992-2009

Over the past two decades, the U.S. experienced a sharp rise in wage inequality accompanied by large increase in wage differentials by skill groups. A large and growing empirical literature attempts at explaining the changes in the U.S. wage structure by using a variety of data sets. As stressed by [START_REF] Firpo | Decomposing wage distributions using recentered influence function regressions[END_REF], these various explanations can all be summarised in terms of the respective contributions of various sets of factors such as education, experience, unions and gender. This paper illustrates the approach developed above by measuring the contributions of some factors and explaining the recent changes in the U.S. wage structure and earnings inequality.

The data

This paper uses the Current Population Surveys (CPS),1 Outgoing Rotation Groups (ORG).2 We take the monthly earnings files for January 1992 through May 2009.

We decide to focus our attention on three years (1992,2001,2009) to cover the main features of the recent period and their evolution. We use the weekly wage divided by the number of hours worked in order to get an homogeneous definition of hourly wages. 3 We deflate these wages by the annual average CPI which is respectively 140.2, 177.1 and 214.5 for these three years. Since January 1992, the CPS has changed the coding scheme of its education attainment question from completed years to degree actually acquired. The new coding scheme details 16 categories for education 4 which include the highest level of school completed or the highest degree received. Our education variable will indicate the official number of years needed to reach the acquired education level. It will represent the efficient number of years of schooling.

Most of the studies concerning wage dispersions in the U.S. cover the period 1973-1989 in order to provide a comparison basis between the different papers. We found marked differences between our sample period 1992-2009 and the previous period 1973-1989. For instance, Melly (2005) indicates that mean and median real wages declined between 1973 and 1989. For the new period, between 1992 and 2009, we have a constant rise of real wage together with a sharp increase in inequality at the end of the period. See Table 1 for detailed figures. This evolution is also depicted in the estimated wage densities. In Figure 1, we display a non-parametric estimate of the wage density. We notice that the distribution of real wages is characterised by a heavy right tail in 2009.

The model

The formulation we adopt is a standard Mincer equation:

ln(y i ) = β 0τ + β 1τ Educ i + β 2τ Exp i + β 3τ Exp 2 i + β 4τ Union i + β 5τ F em i + ǫ iτ , (27)
where (y i , i = 1, • • • , n) is the hourly real wage for workers. We have introduced education (number of years), experience and its square, the union status and gender. Potential experience is calculated as the age minus the assigned years of education minus 6, rounded down to the nearest integer value, min(age-education-6, age-18). Education is the official number of years needed to reach the acquired education level.

As a point of comparison, we have first estimated this equation using the procedure of [START_REF] Firpo | Unconditional quantile regressions[END_REF] and we reported the results in Table 2. As already explained, this estimator requires the use of a non-parametric estimation of the data density. With Figure 1, we see that it is quite difficult to obtain a smooth estimate for the right tail with a unique window size. Figure 2 indicates that a non-parametric density estimate of the log wages is also problematic. This lack of smoothness may disturb the classical RIF-OLS. The adjusted mixture of three normal densities (see also Figure 2) provides of course a much smoother picture. We now turn to Bayesian inference results, using a non informative prior for β and σ. Posterior means and standard deviations are reproduced in Table 3, using 10000 draws for each year and the same quantiles τ = 0.10, 0.50 and 0.90.

The comparison of Tables 2 and3 motivates the following comments. First, the posterior means are very comparable to the classical estimates in the body of the log wage distribution (10 th and 50 th quantiles). However, there is a difference between the coefficient estimates of the covariates in the right tail (90 th quantile) for the years 2001, 2009 (but not for 1992) that we can explain by the difference in smoothness between the two different methods for estimating the log wage densities. The presence of a fat right tail in the distribution of 2009 might be the main explanation. In fact, the kernel density estimation may undersmooth the tail of the distribution when it is characterised by a heavy tail. This imply that the classical RIF-regression coefficients are overestimated in the upper tails (90 th quantile) of the wage distribution in 2009. This might have an impact on the results of the Oaxaca-Blinder decomposition using classical RIF-regression method of [START_REF] Firpo | Unconditional quantile regressions[END_REF] (see differences in Tables 4 and6).

Second, the posterior standard deviation are most of the time larger than their classical counterpart. In the Bayesian approach, we take into account the uncertainty contained in the first step estimation of the log-wage density. This might have consequences on the significance of wage inequality decomposition. Nevertheless, all the coefficients are well estimated with rather small standard errors.

Economic interpretation

Let us now detail the economic interpretation of Table 3. Between 1992 and 2009, the return to education has increased in all parts of the distribution. But the yield rose sharply for the median wages (4.6% to 7%)5 and for higher wages (4.8% to 9%). This provides an explanation for the rise in wage inequality (at constant education composition). The return to experience is much lower than that of education, even if it has risen over the period for all the categories. It is much higher for the first decile and for the median than for the last decile. This should reduce wage inequalities.

The evolution of the yield of being member of a union is paradoxical. In 1992, this was very profitable for median wages to be a union member with a wage differential of 49%. The yield of being unionised decreased while climbing up the wage ladder.

It becomes negligible (3% on average) for high wages. When we now look at the end of the period, the yield of being unionised has decreased for low wages, a fact already noticed in the literature, but has increased slightly for median and high wages. The last covariate concerns gender. Being a woman has always meant having a lower wage. This is especially true here for median and high wages, but not so for low wages. This gender discrimination has risen over the period for all the categories, but this only in terms of intercept earnings. As a final comment, the constant term for the lowest quantile is traditionally interpreting as measuring the effect of the minimum wage. The minimum wage was raised slightly before 1992 and 2009, but not around 2001. The constant term for 2001 is lower than for 1992, showing the readjustment of the labour market. The rise of the constant term in 2009 reflect nicely the next rise of the minimum wage.

Oaxaca-Blinder decomposition

The results of the Oaxaca-Blinder decomposition are given in Table 4 for the Bayesian approach and in Table 6 for the classical estimates as a point of comparison. These estimates are very comparable for the composition effect because β 1992 is roughly the same with classical and Bayesian RIF. The most important change is concentrated on the wage structure effect because β 2009 is much different with the two estimation methods, but these difference are concentrated in the tails (10 th and 90 th percentiles).

From now on, we shall report only the Bayesian results.

Total effects are all significant. We note that there was a large increase of 16% for the first percentile, that the increase is very moderate for median wages (7%) and comparatively huge for the last percentile (47%) over a period of 18 years.

Composition effects represent around 30% of the total effect for the median group, but only around 10% for the lowest and highest groups. Composition effects cannot explain the large increases at both end of the earning distribution. Nevertheless, we can notice that education represents the major part of the composition effect especially for the highest quantile. The other composition effects plays a weaker role, while being still significant. The decline in the unionisation rate is significant for all quantiles but cannot be regarded as a main explanation for wage inequality, contrary to what was a convincing explanation in a previous period (see [START_REF] Dinardo | Labor market institutions and the distribution of wages, 1973-1992: A semiparametric approach[END_REF] because the rates of decline are rather small.

Most of the explanation about the evolution of wages inequality relies on structure effects. We must first notice that the total wage structure effect is not significant for the median quantile, so we shall concentrate on results concerning the two extreme of the distribution. The constant term is only significant for the lowest decile, depicting the influence of the minimum wage, completed by a strong influence of experience, a weaker influence of education. Unionisation rate is not significant. The large wage increase in the highest quantile is due to a much higher reward of education (40%), compensated by an a slight increase in female discrimination (5%). The other factors are either not significant or have a very small coefficient.

Let us now consider the Oaxaca decomposition computed over the more recent period 2001-2009 of 9 years in order to see if there was an acceleration in the trends of wage inequality. The results are reported in Table 5. -0.00073 (0.000065) -0.00059 (0.000056) -0.00016 (0.000076) Union -0.00294 (0.00013) -0.00425 (0.00012) -0.00021 (0.00016) Female 0.000059 (0.000006) 0.00012 (0.000005) 0.00012 (0.000007) Italics correspond to coefficients for which 0 is contained in an HPD interval.

For the lowest quantile, the total increase is in line with the total period and significant. The wage structure effect is again not significant. For median wages, there is no significant total change. On the contrary for the highest quantile, the increase is strongly significant and denote a large acceleration in wage increase corresponding to a large increase in the yield of higher education compensated partly by an increase in gender discrimination.

Conclusion and summary

In this paper, we have proposed a reliable Bayesian inference for the RIF-regression of [START_REF] Firpo | Unconditional quantile regressions[END_REF] in which we have first estimated the log wage distribution using a mixture of normal densities and then provided marginal posterior densities for the quantile regression parameters. As a by-product, we were able to provide an Oaxaca-Blinder decomposition together with its standard deviations. Our first empirical results show that in the presence of a heavy right-hand tail in the wage distribution, the kernel estimation leads to unwanted variability in the RIF-OLS method of [START_REF] Firpo | Unconditional quantile regressions[END_REF] for the highest quantiles. Our parametric approach, using a mixture of normal densities on log wages provides a smoother fit for this upper tail and provides better estimates for the highest quantile regression coefficients. Bayesian standard errors are more realistic as they take into account the uncertainty of the first stage density estimation.

We have illustrated our method on a Mincer equation for the U.S. covering the period 1992-2009 in order to analyse the most recent changes in the wage structure and the earnings inequality. Most of the evolution of the period are concentrated on the extreme quantiles. The median wages do not experience very significant changes. The lowest wages have increased due to the yield of experience while the highest wages have experienced an enormous acceleration in the yield of education. The composition effects are rather low.

Writing the RIF as a linear conditional expectation provides a simple solution both for the quantile regression and the Oaxaca decomposition. However, it is only a local approximation. Bayesian exploration of this question should be continued using a non-linear framework, at the cost of making an Oaxaca-like decomposition more difficult.

Figure 2 :

 2 Figure 2: Fitting a mixture of three normal densities on real log-wages

Table 1 :

 1

		Hourly real wage dispersion
	for the U.S. recent period
		1992 2001 2009
	q 0.10	6.90	7.45	8.00
	q 0.25	9.36 10.30 10.00
	Mean	18.00 20.05 24.49
	Median 14.52 15.53 15.62
	q 0.75	22.02 23.98 26.49
	q 0.90	31.37 36.01 46.12
	Gini	0.352 0.369 0.455
	N	62107 63409 47837
	Figure 1: Real wage density estimates

Table 3

 3

	: Bayesian unconditional Quantile regression coefficients
	with a mixture of normal densities on log wages
		10th percentile	50th percentile	90th percentile
	Cst
	1992	1.592 (0.0124)	2.074 (0.0139)	2.917 (0.0178)
	2001	1.511 (0.0167)	1.952 (0.0145)	2.911 (0.0196)
	2009	1.659 (0.0154)	1.803 (0.0189)	2.900 (0.0295)
	Educ
	1992	0.021 (0.0010)	0.046 (0.0011)	0.048 (0.0014)
	2001	0.033 (0.0014)	0.063 (0.0012)	0.063 (0.0016)
	2009	0.024 (0.0013)	0.071 (0.0016)	0.089 (0.0025)
	Exp
	1992	0.008 (0.00053)	0.010 (0.00060)	0.007 (0.00077)
	2001	0.011 (0.00073)	0.011 (0.00063)	0.006 (0.00085)
	2009	0.013 (0.00067)	0.018 (0.00082)	0.009 (0.00129)
	Exp 2 * 100 1992	-0.0090 (0.00077) -0.0097 (0.00087) -0.0052 (0.00111)
	2001	-0.0118 (0.00104) -0.0095 (0.00090) -0.0026 (0.00122)
	2009	-0.0159 (0.00100) -0.0220 (0.00123) -0.0106 (0.00191)
	Union
	1992	0.170 (0.0070)	0.399 (0.0079)	0.026 (0.0102)
	2001	0.225 (0.0104)	0.326 (0.0089)	0.016 (0.0121)
	2009	0.165 (0.0098)	0.431 (0.0120)	0.036 (0.0188)
	Female
	1992	-0.024 (0.0055)	-0.124 (0.0061)	-0.119 (0.0079)
	2001	-0.069 (0.0074)	-0.143 (0.0064)	-0.146 (0.0086)
	2009	-0.071 (0.0067)	-0.175 (0.0083)	-0.216 (0.0129)

Table 4 :

 4 Oaxaca Blinder decomposition using Bayesian RIF-regression with a mixture of normal densities on log wages, CPS ORG 1992 -

	2009

Table 5 :

 5 Oaxaca Blinder decomposition using Bayesian RIF-regression with a mixture of normal densities on log wages, CPS ORG 2001 -

	2009

Table 6 :

 6 Oaxaca Blinder decomposition using Classical RIF-regression with kernel density estimation on log wages,CPS ORG 1992[START_REF] Firpo | Unconditional quantile regressions[END_REF]

		10th percentile	50th percentile	90th percentile
	Total effect	0.1484 (0.00394)	0.0695 (0.00491)	0.3857 (0.00825)
	Wage structure
	Total	0.136 (0.0356)	0.048 (0.0444)	0.348 (0.0749)
	Cst	0.132 (0.0180)	-0.175 (0.0224)	-0.153 (0.0378)
	Educ	-0.002 (0.0149)	0.186 (0.0186)	0.545 (0.0315)
	Exp	0.081 (0.0228)	0.175 (0.0285)	0.102 (0.0480)
	Exp 2	-0.052 (0.0134)	-0.119 (0.0168)	-0.081 (0.0285)
	Union	-0.004 (0.0015)	-0.002 (0.0019)	0.002 (0.0032)
	Female	-0.019 (0.0041)	-0.017 (0.0051)	-0.067 (0.0087)
	Composition
	Total	0.0121 (0.0010)	0.0211 (0.00118)	0.0372 (0.00145)
	Educ	0.0141 (0.00068)	0.0315 (0.00078)	0.0316 (0.00096)
	Exp	0.0094 (0.00062)	0.0120 (0.00072)	0.0081 (0.00088)
	Exp 2	-0.0036 (0.00031)	-0.0040 (0.00035)	-0.0020 (0.00043)
	Union	-0.0080 (0.00033)	-0.0193 (0.00038)	-0.0012 (0.00047)
	Female	0.00017 (0.000037) 0.00088 (0.000043) 0.00080 (0.000053)

The CPS is the monthly household survey conducted by the Bureau of Labor Statistics to measure labor force participation and employment. 50-60,000 households per month are queried. This is not really a panel survey since households are not followed if they move. They include the March CPS file and the Outgoing Rotation Group (ORG) files.

The ORG files correspond to the set of every household that enters the CPS interviewed each month for 4 consecutive months, and then ignored for 8 months.

The ORG files are often used because they include a direct observation of the hourly wage, which thus has not to be computed as the ratio between the weekly wage and the number of worked hours. However, many individuals did not answer to that question, so we prefer to compute a ratio in order to keep the maximum number of observations. And anyway, apart from a few aberrant values, our ratio series gave similar figures as the one given by the hourly series.

The 16 categories include: no diploma; high school graduate; some college but no degree; associate degree in college (occupational or vocational program); associate degree in college (academic program); bachelor's degree, master's degree, professional school degree; and doctorate degree

As underlined inBazen (2011, Table 1.1, p. 21), in a log linear regression, coefficients can be interpreted as percentages only for small values. For higher values, one has to use the formula exp(β i ) -1.