Keywords: Travail collaboratif, Ontologie, Cohérence sémantique, RST

OntoReST : Une ontologie basée sur RST pour le maintien de la cohérence sémantique dans l'édition collaborative Résumé : L'édition collaborative est un processus durant lequel plusieurs auteurs contribuent au contenu d'un document. La collaboration multi-synchrone est très efficace car elle permet de réduire le temps de complétude d'une tâche. Cependant elle est connue pour produire de documents incohérents. La plupart des environnements d'édition collaborative existants ne vérifient pas réelement la cohérence sémantique des documents. La vérification de la cohérence reste à la charge des auteurs. Ceci constitue un obstacle majeur pour garantir une collaboration efficace. Pour adresser ce problème, nous utilisons les technologies du web sémantique et une théorie du discours nommée Rhetorical Structure Theory (RST). Nous développons OntoReST, une ontologie basée sur RST qui aide à la détection automatique des textes incohérents. OntoReST offre aussi aux auteurs une information utile sur la structure sémantique des textes en conduisant à des documents plus cohérents.

Introduction

Collaborative writing is the process by which more than one author, in addition to sharing opinions, also contributes to the content of a document [START_REF] Lowry | Building a Taxonomy and Nomenclature of Collaborative Writing to Improve Interdisciplinary Research and Pratice[END_REF]. Collaborative writing is standard practice in technical and scientific settings; some examples include research papers, software development, proposals for funding and user manuals. When collaboration is efficiently managed, the advantages of working in a group include increased efficiency, reduced errors and the benefits of different viewpoints and expertise [START_REF] Tammaro | Collaborative Writing Is Hard to Support: A Field Study of Collaborative Writing[END_REF][START_REF] Noël | Empirical study on collaborative writing: What do co-authors do, use, and like?[END_REF]. If collaboration is poorly supported, it can lead to inconsistencies, misunderstandings, conflicts, redundant work and coordination problems. The nature of collaboration varies extensively in terms of the group writing strategies, proximity and synchronicity of group activities [START_REF] Tammaro | Collaborative Writing Is Hard to Support: A Field Study of Collaborative Writing[END_REF]. For instance, collaborative writing can be done in parallel synchronously, asynchronously or multi-synchronously [START_REF] Dourish | The Parting of the Ways: Divergence, Data Management and Collaborative Work[END_REF][START_REF] Molli | SAMS: Synchronous, Asynchronous, Multi-Synchronous Environments[END_REF][START_REF] Skaf-Molli | New Work Modes for Collaborative Writing[END_REF]. Synchronous work with a joint-writing strategy can give good results but is limited to small groups working for short periods of time. Asynchronous work with turn-taking strategies allow to work distributed in time but does not allow task parallelization. The multi-synchronous interaction mode allows people to work in parallel synchronously or asynchronously while being distributed in time and space. It is the least restrictive collaborative writing strategy. This working mode is well known in software engineering. Software engineers commonly use version control systems or distributed version control system to achieve high parallelization of tasks and reduce development time. This high level of concurrency is potentially risky and can lead to software inconsistencies. Fortunately, software engineers can compile software and run automatic tests in continuous integration strategy [START_REF] Duvall | Continuous Integration: Improving Software Quality and Reducing Risk (The Addison-Wesley Signature Series)[END_REF] to limit the risk of inconsistencies.

Unfortunately, we cannot reuse these proven efficient collaborative strategies outside the software engineering world. Currently, we do not dispose of automatic testing mechanisms for checking document consistency according to some specifications. Text documents do not have typed objects to reason about, and they cannot be "compiled" in order to verify some type safety violation.

Multi-synchronous collaboration mode greatly increases the risk of misaligned contributions by individual authors. While each section may be well constructed, they may not 'fit' logically when placed together. While this is easy to correct in short texts, the problem is much harder in large, multi-authored documents. This is what we refer to as semantic inconsistency and is the focus of this paper.

Semantic consistency is poorly supported by existing Multi-synchronous collaborative writing environments. In these environments, each author works on her own copy of the shared data. The system is correct if: [START_REF] De Bruijn | Owl dl vs. owl flight: conceptual modeling and reasoning for the semantic web[END_REF] it eventually converges to an idle state where all copies are identical (2) user intentions are preserved [START_REF] Sun | Achieving Convergence, Causality Preservation, and Intention Preservation in Real-Time Cooperative Editing Systems[END_REF]. 'Intention' means that if an operation produced an effect when generated, this effect must be observable in the same way by all users. For instance, if the author has the text AC and she wants to insert B exactly between A and C, the system will ensure that all the authors will see B between A and C regardless of other concurrent operations. Of course, intention preservations do not prevent two authors to insert the same idea twice or contradicting ideas at the same place in the document. It is clear that automatic detection of such problems is very difficult and current collaborative environments rely on authors to verify the logical connections of the content. Collaborative environments help authors by providing awareness about concurrent changes. Next, authors have to verify that each local operation is compatible with all other concurrent operations. We want to leverage this stage by providing more information about the context and impact of modifications. This requires us to define more clearly what is meant by "semantic consistency". "Semantic consistency" or "coherence" (the two words are used interchangeably in this paper) is a subjective phenomenon because different readers perceive texts differently. What we mean by coherence is the ease with which a document can be read and understood. While multiple factors such as grammar and punctuation can affect this, the logical progression of the ideas presented is perhaps the one with the most impact. The mere sequence in which the sentences and sections are laid out can significantly affect how they are understood [START_REF] Knott | A Data-Driven Methodology for Motivating a Set of Coherence Relations[END_REF][START_REF] Marcu | The Theory and Practice of Discourse Parsing and Summarization[END_REF].

(See examples in our previous papers [START_REF] Rahhal | Semcw: Semantic collaborative writing using rst[END_REF][START_REF] De-Silva | Narratives to preserve coherence in collaborative writing[END_REF]).

Research into this area revealed that linguists had developed discourse theories to guide the analysis and synthesis of text. In particular, [START_REF] Mann | Rhetorical structure theory: Toward a functional theory of text organization[END_REF] developed Rhetorical Structure Theory (RST) [START_REF] Mann | Rhetorical structure theory: Toward a functional theory of text organization[END_REF]. This theory attributes the coherence of a text to the implicit logical relationships that exist between its segments. In previous work [START_REF] Rahhal | Semcw: Semantic collaborative writing using rst[END_REF], we combine RST and merging algorithm based on the operational transformation approach [START_REF] Oster | Tombstone transformation functions for ensuring consistency in collaborative editing systems[END_REF] to address document coherence during collaborative writing.

In this paper, we use techniques from the semantic web domain to address the problem of semantic coherence during collaborative writing. More precisely, we define an ontology called OntoReST based on RST that allows to improve the quality of documents. OntoReST turns the document contents into a machine readable and structured form which allows the detection of semantic inconsistencies. It also allows document querying, reasoning and searching. Such ontologies can be used to turn text into typed objects. This allows software to be written (equivalent to compilers for programs) to check some properties about the document. Consequently, multi-synchronous collaboration interaction mode can be used in order to achieve more efficient collaboration for producing text documents.

So, first we give a brief description of RST and show how the structure of a text can be analysed using it. In section 3, we define OntoReST formally using OWL and description logic DL. Next, in section 4, we detail the required steps to manipulate this ontology during collaborative writing and describe the applied merging algorithm using an example. We also discuss related work in the field and finally, present our conclusions and directions for future work.

Rhetorical Structure Theory (RST)

There are several discourse theories developed by linguists to analyse the structure of texts. We have chosen Rhetorical Structure Theory (RST) [START_REF] Mann | Rhetorical structure theory: Toward a functional theory of text organization[END_REF] for its simplicity, precise relationship definitions and its ability to render itself to formal descriptions. RST attributes the coherence of a text to implicit logical relationships such as 'Motivation', 'Background' and 'Elaboration' that exist between sections of the text. The rest of this section gives a brief overview of how a RST analysis can be done and highlights parts of the process essential to the discussions in this paper.

Analysing a Text Using RST

The first step in a RST analysis is to divide the text into non-overlapping, functionally independent segments [START_REF] Mann | Rhetorical structure theory: Toward a functional theory of text organization[END_REF].

As an example, we use the text below to demonstrate the segmentation.

[Text 1:] [1:The problem with existing writing software is their inability to detect semantic problems in documents.] [2:OntoReST is the result of combining the techniques behind onotologies and those related to RST.] [3:When combined with existing writing tools, it can help improve the quality of documents by alerting authors to possible semantic inconsistencies.] During a bottom-up analysis, the second step is to identify logical relationships that exist between pairs of segments. For instance, in the above example, we see segment 3 to be providing motivating information to the statement in segment 2 (i.e. Motivation relationship).

Segments in a relationship can play one of two roles: a nucleus or a satellite. A nucleus is considered to be an important segment, essential to the understanding of the text. A satellite is not as critical but does provide supporting material.

More information about RST can be found in Mann and Thompson's paper (1988). Mann and Thompson defined 23 relationships, each with precise descriptions of what should go in each end of the relationship and the its expected effect on the reader. Henderson and De Silva [START_REF] Henderson | A narrative approach to collaborative writing: A business process model[END_REF], however, considered 23 to be too many for technical writing and began selecting a subset of relationships that were sufficient for analysing technical documents. In [START_REF] De-Silva | A narrative-based collaborative writing tool for constructing coherent technical documents[END_REF], a user study has shown that technical authors found a set of 9 relationships adequate for their analysis.

In the analysis, segments involved in a relationship collectively form a span. A span can in turn become part of another relationship. For instance, in our example, the span of segments 2 and 3 is identified as being in a BACKGROUND relationship with segment 1 (i.e. segment 1 provides background information that helps understand the significance of BOTH segments 2 and 3). Hence, the analysis is a recursive process and continues until all the segments are assembled into a tree of relationships as shown. This is called a RSTree. In the RSTree the arrowhead points towards the Nucleus as shown in the figure 1.

RSTrees Properties

The important point about RSTrees is that [START_REF] Mann | Rhetorical structure theory: Toward a functional theory of text organization[END_REF] [START_REF] Mann | Rhetorical structure theory: Toward a functional theory of text organization[END_REF] conjecture that producing a wellformed RSTree for a text indicates that a text is coherent. This is a useful measure in our work where we apply RST to detect incoherent texts. They define four properties that determine if a RSTree is well formed. They are:

Completedness One schema application (the root) should cover the entire text.

Connectedness Each text span/segment, apart from the span that covers the entire text, should be a minimal unit in the tree or part of another schema application.

Uniqueness Each text span/segment should have only one parent (i.e. each schema application consists of a different set of text spans/segments).

1-3 1:The problem with existing

writing software is their inability to detect semantic problems in documents.

m m m m m m m m m m m m m m m BACKGROU N D % % 2-3

H H H H H H H H H H H H H H H

2:OntoReST is the result of combining the techniques behind onotologies and those related to RST. Adjacency Only adjacent text spans/segments can be grouped together to form larger spans.

H H H H H H H H H H

We make use of these properties in OntoReST to evaluate the coherence of documents written collaboratively.

Ontologies for Semantic Collaborative Writing

An ontology describes basic concepts in a domain and defines relations among them [START_REF]Handbook on Ontologies[END_REF]. It is composed of concepts, properties, relations and restrictions on properties. We formalize the RST theory as an ontology. This allows to take advantages of the semantic Web by turning the content of document into a machine readable and structured form. It provides also a common knowledge base for the authors. Moreover, it is possible to detect automatically semantic problems and to make interesting queries on the document. For example, by selecting all the Nucleus we can produce a summary of the document.

We have defined three ontologies (see figure 2):

Document structure ontology: captures the internal structure of the document (sections, sentences, etc).

Rhetorical ontologyOntoReST : models the document in terms of its rhetorical elements (i.e. segments, spans and RST relationships). This allows the detection of semantic inconsistencies in documents.

Annotation ontology: annotates the sentences and the sections of the document. It also captures additional meta data about the document. This is helpful in classifying the documents according to their types, authors and topics.

Figure 2: Ontology Layers

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The definition of three separate ontologies allow flexible modification of the ontologies. For instance, we can modify the structure of the document, without modifying the RST information. The idea of defining three different ontologies is inspired from [START_REF] Tudor Groza | SALT -Semantically Annotated LaTeX for scientific publications[END_REF]. As our major interest is detecting semantic inconsistency, in this paper we will focus just on the rhetorical ontology OntoReST.

Rhetorical Ontology OntoReST

The rhetorical ontology captures the semantics of the text using RST. It models the segments, spans and rhetorical relations. It also uses the four properties for well-formed RSTrees to detect semantic discrepancies in the document. We do not use all 23 rhetorical relations defined by Mann and Thompson, but we only use the subset of 9 relations identified in [START_REF] Henderson | A narrative approach to collaborative writing: A business process model[END_REF]. This ontology is not tied to a specific document but it is a generic ontology applicable to several types of technical documents. A Document is composed of an ordered sequence of segments and it has a set of spans and a set of rhetorical relations. It has also the following properties: hasID: a unique identifier given by the system for the document.

hasContents: links a document to its segments, spans and rhetorical relations. The textual content of a document is the hasContent values of its all ordered segments. This property is the inverse property of containingDoc.

hasRoot: indicates the root of a document. Its inverse property is isRoot. For example, the span 1 -3 is the root of the text in the figure 1. A root is a span that has the isRoot property.

An Element can be either a Segment or a Span. It is used to avoid repetition of common properties in Segments and Spans. It has the following properties: hasID: is a unique identifier given by the system for each element.

firstPosition and lastPosition: are the position of the element in the document. They have equal value for a segment. For a span they indicate the position of the first and the last segments covered by the span in the document. We use them for the adjacency property. For example, the firstPosition of the Element 1 -3 of the figure 1 is 1 and the lastPosition is 3. The elements 1 and 3 are not adjacent because their positions are not consecutive. While the Element 2 -3 has two adjacent children.

visible: is the status of the element. Its range is boolean and has "true" as default value. It turns to "false" when the element is deleted. We add this property because we do not delete physically the element, we just mark it as invisible. This property will be set by the merging algorithm as we will see later.

hasParent: indicates the parent of an element. Each element in the document has a parent property except the root (which is the span that covers the entire text). For example, the span 2 -3 of the figure 1 is the parent of the Elements 2 and 3 and the span 1 -3 is the root of the text, it has no parent. changed : is the status of the span. It is "false" by default. It is set to "true", after deleting one of its children or the hasRstRelation property.

The changed property allows to propagate modifications to the concerned spans and relations in the RSTree. This property gives some awareness that helps the authors to localize the modified parts of the documents. This saves time and effort for the authors during the revising and the reviewing phase, and especially after integrating the modifications of other authors.

A Rhetorical Relation is the rhetorical relation that holds between two elements. It is always linked to a span. It has the following properties: hasID: is a unique identifier given by the system for each relation.

hasName: is the name of the relation such as Motivation or Elaboration. Its domain is a Relation and its range is a list of relations' names. According to the RST theory, the name of the relation specifies the order of the nucleus and the satellite i.e. nucleus is before satellite or the opposite.

hasNucleus and hasSatellite: represent the nucleus and the satellite respectively of the relation. Their values are the first and the last child of the linked span i.e. they reference the first and last child properties of the related span. We need to define the hasNucleus and hasSatellite: properties since we make separation between span and relation. This allows to avoid to reconstruct the RSTree in case of minor modifications (such as correct misspelling or rephrasing a segment without changing its meaning).

visible: is the status of the relation. Its range is boolean and has "true" as default value. It turns to "false" when the relation is deleted.

inSpan: this property is the inverse property of the hasRstRelation.

Formal Specification of OntoReST

We use both OWL (Web Ontology Language) and Description Logic to formalize OntoReST. We write the class axioms, the property axioms and the constraints in OWL DL which is the most investigated species of OWL [START_REF] De Bruijn | Owl dl vs. owl flight: conceptual modeling and reasoning for the semantic web[END_REF]. OWL DL has different syntaxes. However, the normative syntax for OWL DL is the abstract syntax. OWL can be seen as an alternate notation for Description Logic Language SHOIN (D). Table 1 presents the concepts of the OntoReST.

Table 2 presents the object and data properties. In this table, we skipped some identical properties for simplicity. Both tables represent a mapping between the OWL DL abstract syntax and the syntax of the Description Logic SHOIN (D).

OntoReST in Collaborative Writing

Using OntoReST to maintain semantic consistency during collaborative writing requires the following steps: Ontology instantiations: Each instance of concept is created locally at a user's site. So, when the author adds a sentence, the system will detect this modification as an operation and create an instance of the segment concept. The operation is then sent and integrated at all others users' sites. Eventually, when there is no modifications, the replicated instances will be the same at all sites. In the section 4.1, we define operations to instantiate this ontology.

Merging algorithms: To integrate remote modifications, we use the Tombstone Transformation Functions algorithm [START_REF] Oster | Tombstone transformation functions for ensuring consistency in collaborative editing systems[END_REF] (TTF) as detailed in the section 4.

3.

Inconsistency checker: There is a significant difference between using RST in collaborative writing and traditional applications of RST. Usually, RST is applied to a 'static' text. However, in collaborative writing, the text continually changes and its corresponding RSTree changes too. We have to check that the new RSTree respects the four properties defined in the RST theory as detailed in section 4.4.

Populating Ontology

In this section, we describe the process of ontology's instantiation during the edition. During the edition, the changes made by the authors are detected by the system as the following operations:

addSegment(position, hasID, content, sid) adds an instance of segment with the specified position and the text content. sid is the identifier of the site that generates the operation. The sid is necessary for the merging algorithm.

delSegment(position) deletes logically a segment at the given position. The visible property of the segment is set to false. There is no physical deletion of segments to ensure the convergence [START_REF] Oster | Tombstone transformation functions for ensuring consistency in collaborative editing systems[END_REF]. If the segment has a parent span, then the changed property of its parent is set to true.

addSpan(hasID, hasFirstChildID, hasSecondChildID) creates an instance of span with the required properties.

delSpan(hasID) deletes logically a span. The visible property of the span is set to false. If the span has a parent span, then the changed property of its parent is set to true.

addRelation(hasID, NucleusID, SatelliteID, SpanID, hasName) adds a rhetorical relation between the children of its linked span. For example, the hasFirstChild (hasSecondChild) of the span could be the satellite(nucleus) or the nucleus (satellite) of that relation.

delRelation(hasID) deletes logically a relation. The visible property of the relation is set to false. The changed property of the span linked to this relation is set to true.

We define only add and delete operations. Because during the merge the update operation is detected as delete followed by add by the diff algorithms.

Let us consider a scenario where an author u 1 is working on site1. She wants to write the Text 1 of the section 2 but this time with rhetorical annotations.

She modifies her local copy by generating operations. The system will detect these changes as the following sequence of operations: S =[addSegment(1, 1A, "The problem with..",1);addSegment(2, 1B, "OntoReST is..", 1); addSegment(3, 1C, "When combined with..", 1); addSpan(2-3s, 1B, 1C ); addRelation(2-3, 1C, 1B, 2-3s, "Motivation"); addSpan(1-3s, 1A, 2-3s); addRelation(1-3, 1A, 2-3s, 1-3s, "Background")]

The system will build the RSTree for AnnotatedText1 as depicted in the figure 4.

Concurrent Writing

Now consider that two authors A and B working on site2 and site3 respectively are writing the AnnotatedText1 of the figure 4. Each author has his own copy of the text.

Author A wants to delete the segment in position 2. So, he deletes this segment. In order to preserve local consistency of the RSTree, the system will propagate this deletion to its parent span and its associated relation. The span 2 -3s and the relation 2 -3 are logically deleted. And also, the parent of the span of 2 -3s will be replaced with 1 -3 s and the relation 1 -3 with 1 -3 as shown in the figure 5.

The changes performed by AuthorA will be detected by the system as the following sequence of operations P 1:

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Merging Ontological Data

In this section, we will detail through an example how we merge the above concurrent operations. AuthorA has generated P 1 and AuthorB has generated P 2 , so the copies hosted on site2 and site3 are diverging now i.e. they have different content. Both sites exchange their operations and run the integration process as depicted in figure 6. In order to converge, the system has to ensure that M erge(P 1, P 2) = M erge(P 2, P 1). Unfortunately, this property is not ensured by traditional merge algorithms.

This problem is well-know in CSCW community. The Operation Transformation (OT) framework [START_REF] Ellis | Concurrency Control in Groupware Systems[END_REF] has been developed to ensure convergence in these conditions. In [START_REF] Rahhal | Semcw: Semantic collaborative writing using rst[END_REF], we defined a set of all transformation functions that deal with concurrent operations and ensure convergence of semantically annotated documents with RST.

As shown in the figure 7, the final state is converged towards a value that has both RSTree. This value is inconsistent since it does not respect the rhetorical properties. For example, the segment at the position 1 violates the uniqueness property. Having both RSTree will help the authors to better understand the reasons of the semantic inconsistency and to locate exactly the segments of the text that are responsible of this inconsistency. In the next section, we detail our inconsistency checker.

Inconsistency Checker

In our approach we consider a document is semantically coherent if it respects the four properties of the RST (see section 2). Therefore, we formalize these properties as constraints in the OntoReST ontology. These To check the inconsistent instances of the OntoReST, we formalize the violation of the coherence properties of the RST in DL on an instance d of the document Document(d) as follows :

Connectedness Violation (ConV) For the document d, an inconsistent element that violates the connectedness property is a visible element that has no parent and that is not a root of d.

ConV ≡ Element ∃ visible.{true} ∃ containingDoc.{d} ∀ hasParent.⊥ ∀ isRoot.¬{d}
Uniqueness Violation (UniV) For the document d, an inconsistent element that violates the uniqueness property is a visible element that has a number of parents different than one. AdjV ≡ ( > firstPosition.{e} lastPosition.hasFirstChild.{s}) ( < lastPosition.{e} firstPosition.hasSecondChild.{s}).

We have implemented and verified our OntoReST in Protégé and our rhetorical constraints in Protégé Axiom Language (PAL).

Related works

In the collaborative writing domain, most work on semantic consistency are based on constraints. Some approaches allow the violation of the constraints. In case of their violation, the reparation is done either automatically like in [START_REF] Skaf-Molli | Semantic Consistency for Collaborative Systems[END_REF] or manually. Others prevent the violation of the constraints like in [START_REF] Kermarrec | The IceCube Approach to the Reconciliation of Divergent Replicas[END_REF]. If an operation violates the constraints, the operation is canceled. Both approaches in [START_REF] Skaf-Molli | Semantic Consistency for Collaborative Systems[END_REF] and [START_REF] Kermarrec | The IceCube Approach to the Reconciliation of Divergent Replicas[END_REF] bring about lost updates which is not ideal.

Constraints can be specific to an application and concern more the document structure. However, they cannot capture the co-author's understanding and logical reasoning about the text. RST provides this need by attributing relationships between its segments. These relationships create an overall effect on the reader which contributes to understanding the text better. Therefore, when authors exchange documents, they also pass on their understanding of it via the attached RST relationships. In our work, we evaluate the semantic consistency of documents using the RST properties as our constraints.

The project SALT (Semantically Annotated Latex) has some common features with our work. In [START_REF] Tudor Groza | SALT -Semantically Annotated LaTeX for scientific publications[END_REF], the authors propose a framework for authoring and annotating LaTeX documents. They develop ontology based on RST. The authors add RST-based semantic tags to their LaTeX documents while editing. SALT does not consider collaborative work. In our work, the OntoReST ontology is not just used to add semantic annotations within the document but also as a tool to evaluate the document's level of coherence during collaborative writing. By constantly maintaining and checking the four properties, we are able to detect inconsistencies and alert the authors to such areas. We anticipate that we can integrate our work easily into the SALT framework such that our RST capability can be extended into LaTeX documents too.

Conclusions and Future Work

The problem we have tackled in this paper is the lack of support in current collaborative writing tools to detect semantic problems in texts. We have combined ideas from a discourse theory called Rhetorical Structure Theory (RST) and ontologies to develop OntoReST. RST has provided us with a way to formally identify what it means for a text to be coherent (or semantically sound). RST attributes the coherence of a text to underlying relationships between its segments such as ELABORATION or SEQUENCE. These relationships create an overall effect on the reader which contributes to understanding the text better. The creators of RST also define some properties which we have formalised and used in our work to evaluate if the text is semantically sound.

OntoReST offers a novel application of semantic annotations where the focus is to achieve more coherent documents in collaborative writing. It benefits from having the advantages of ontologies such as providing a common knowledge base for authors, the ability to easily search for data through queries. We make use of a merging technique to ensure the convergence of the OntoReST, i.e. getting identical RSTrees for documents on all the authors' sites leaving no room for confusion.

This combination of different strands of research (RST, ontologies and collaborative writing) to bridge a gap in existing tools is novel and is the main contribution of our paper. As future work, we intend to evaluate its usability and identify its shortcomings.

3 :

 3 When combined with existing writing tools, it can help improve the quality of documents by alerting authors to possible semantic inconsistencies.

MFigure 1 :

 1 Figure 1: RST for Text1

Figure 3 :

 3 Figure 3: The Rhetorical Ontology OntoReST

  containingDoc: indicates the document containing the elements. A Segment is a sentence. It inherits the properties in element and has an additional property hasContent. A Span covers two adjacent segments. It is always linked to one and only one rhetorical relation. Span inherits the properties in element and has the additional following properties: hasFirstChild and hasSecondChild : are the first and the second child elements of a Span. isRoot: indicates the parent document of the root span. Its inverse property is hasRoot. hasRstRelation: indicates the rhetorical relation that exists between the children of the span. The children of a span are the nucleus and the satellite of that relation.

Figure 4 :

 4 Figure 4: RSTree of AnnotatedText1

Figure 5 :

 5 Figure 5: AnnotatedText1 of the Author A

Figure 6 :Figure 7 :

 67 Figure 6: Global merging scenario

UIE≡

  Element ∃ visible.{true} ∃ containingDoc.{d} ¬ (=1 hasParent) Completedness Violation (CompV) The document d has a number of roots different than one. CompV ≡ ¬ (= 1 hasRoot) Adjacency Violation (AdjV) For a visible span in the document d, there exists a visible element (segment or span) between the last position of its first child and the first position of its second child. C1 ≡ Span ∃ visible.{true} ∃ containingDoc.{d} C2 ≡ Element ∃ visible.{true} ∃ containingDoc.{d} Let s and e be C1(s) and C2(e):

Table 1 :

 1 The Rhetorical concepts in OWL and DL

	OWL Abstract syntax	DL syntax
	Class axioms
	SubClassOf(Document Thing)	Document
	SubClassOf(Element Thing)	Element
	SubClassOf(Span Element)	Span Element
	SubClassOf(Segment Element)	Segment Element
	EquivalentClasses(Element unionOf(Span Segment)) Element ≡ Span Segment
	DisjointClasses(Span Segment)	Span Segment ⊥
	EquivalentClasses(RheRelation)	RheRelation ≡
	unionOf(Background ... SolutionHood))	Background ... SolutionHood
	DisjointClasses(Background, Contrast)	Background Contrast ⊥, ...
	DisjointClasses(Sequence, SolutionHood)	Sequence SolutionHood ⊥
	OWL Abstract syntax	DL syntax
	Property axioms
	ObjectProperty(hasFirstChild domain(Span)	∀ hasFirstChild -.Span
	range(Element))	∀ hasFirstChild.Element
	restriction(hasFirstChild maxCardinality(1)	Span (= 1 hasFirstChild)
	minCardinality(1))
	ObjectProperty(hasNucleus domain(RheRelation)	∀ hasNucleus -.RheRelation
	range(Element))	∀ hasNucleus.Element
	restriction(hasNucleus minCardinality(1)	RheRelation (≥1 hasNucleus) (≤2 hasNucleus)
	maxCardinality(2))
	ObjectProperty(hasParent domain(Element)	∀ hasParent -.Element
	range(Span))	∀ hasParent.Span
	ObjectProperty(inSpan domain(RheRelation)	∀ inSpan -.RheRelation
	range(Span)	∀ inSpan.Span
	inverseOf(hasRstRelation))	inSpan ≡ hasRstRelation -
	DatatypeProperty(firstPosition domain(Element)	Element ∃ firstPosition.String
	range(String))
	DatatypeProperty(hasID	∀ hasID -.(Element RheRelation)
	domain(unionOf(Element RheRelation	Element RheRelation Document ∃ hasID.Integer
	Document)) range(Integer))
	DatatypeProperty(visible range(Boolean))	Element RheRelation ∃ visible.Boolean

Table 2 :

 2 The Rhetorical properties in OWL and DL

RR n°6384

http://www.inria.fr ISSN 0249-6399

Acknowledgments

We thank the Orpailleur team at LORIA, especially Jean Lieber for improving and revising the OntoReST and Yannick Toussaint for simulating discussions.


Kamal Anouar

Nicolas Aubert

Does the catering theory of dividend apply to the French listed firms?

Keywords: Dividends, Payout policy, Catering, Dividend premium, Behavioral corporate finance G02, G35

This paper tests the catering theory of dividend in the French market. It investigates how prevailing investor's demand for dividend payers proxied by the dividend premia affects the dividend policy. The dividend premia are measured at the market level and at the firm level.

We find that the market demand for dividends measured by dividend premia affects the decision to start, to continue or to omit to pay dividends and the decision to increase the dividends. However, catering theory does not seem to affect the magnitude of the dividend changes since most results are not significant.

Introduction

The decision to pay out dividend to shareholders is considered as strategic because it is related to other financial and investment decisions [START_REF] Allen | Payout policy[END_REF][START_REF] Allen | Payout policy[END_REF]. But dividend policy is still regarded as a puzzle even though it has been extensively investigated by academic literature. Several papers relaxed the hypotheses of Miller and Modigliani. Among them, Baker and Wurgler (2004a) developed the catering theory of dividend which takes into account investors' rationality limits. According to them, "managers rationally cater to investor demandthey pay dividend when investors put higher prices on payers and they do not pay when investors prefer non payers" (Baker and Wurgler, 2004a;p. 1125). They regard this managerial strategy as a consequence of investors uninformed and time varying demand for dividend paying stock and the failure of arbitrage. Managers take advantage of the uninformed demand for dividends to boost the company's share price. This strategy leads them to pay dividends when investors' demand for dividends is high and conversely. Baker and Wurgler (2004a) assume that the demand for dividends can be observed by the difference of valuation between dividend paying firms and non-dividend paying firms. A positive difference reflects the demand for dividend whereas a negative difference reflects that investors do not desire dividends. If companies' valuation is proxied by the market-tobook ratio, the probability that a company starts or continues paying dividend is higher when its market-to-book ratio is lower than paying companies' market-to-book ratio. This paper tests the catering theory of dividend in the French market. Focusing on France is appealing for several reasons. The non-financial companies distribute on average about 10% of their EBITDA between 2001 and 2015 (data from the INSEE 3 ). According to the Global dividend index published by Henderson, Europe is comfortably the second largest region in the world for dividends after northern America and the United Kingdom. Among European countries, France holds the first position by paying 25% of the dividends representing 50 billion dollars in 2013. The same study points that France is the third dividend paying country in the world after the USA and the United Kingdom since 2009 4 . In these two latter countries, previous studies emphasized that the catering theory of dividend applies because they are common law countries which is not the case for France. Besides this French singularity,

France has an intermediary model of governance [START_REF] Aste | Reforming French corporate governance: A return to the two-tier board[END_REF]. For instance, the companies have the freedom to choose to adopt a unitary or dual board of directors [START_REF] Belot | Freedom of choice between unitary and two-tier boards: an empirical analysis[END_REF]; they do not have obligations to appoint financial experts on their board [START_REF] Jeanjean | Determinants of board members' financial expertise-Empirical evidence from France[END_REF]; the board of directors can include employees [START_REF] Ginglinger | Employee ownership, board representation, and corporate financial policies[END_REF]. Belot et al (2014) identify nine European countries 5 whose legal code is inherited from the French civil law. La [START_REF] Porta | Agency problems and dividend policies around the world[END_REF] consider that one of the principal remedies to the agency problem is law. According to them, common law countries offer the best legal protection against majority shareholders whereas in civil law countries, "investors' protection is a good deal more precarious" (p. 4). They further argue that, among civil law countries, French law countries have the weakest protection. They assume that in these countries, dividend policy could be regarded as a substitute for law protection. In this substitution model, "dividends are a substitute for effective legal protection, which enables firms in unprotective legal environments to establish reputations for good treatment of investors through dividend policies" (p. 27). According to the Banque de France, another important feature of the largest French listed companies belonging to the CAC 40 index is that an important proportion of their market value is hold by foreign investors: 46, 7% at the end of 2013. This proportion reaches 42, 7% for all listed companies. More than half of the companies belonging to the CAC 40 index are actually controlled by foreign investors 6 . This exposure to foreign investment would make France particularly sensitive to the market pressure in terms of demand for dividends highlighted by the catering theory. These specific features make France an interesting case for studying dividend policy.

We use a 12 years panel data from 2001 to 2012 covering the largest French listed companies.

We investigate the behavioral imperfection documented by Baker and Wurgler (2004a) to better understand the dividend policy decision. This paper is the first attempt to test the catering theory of dividend with recent French archival data. One innovation is to use firm level measures of the catering effect. Although we validate the catering theory for the binary decision to pay dividend and its extension by [START_REF] Li | Dividend changes and catering incentives[END_REF] to the decision to increase the dividends. Most of the results regarding the magnitude of dividend changes are not significant.

The remainder of this paper is organized as follows. Section 2 presents the literature review. Section 3 describes the data and methodology. Section 4 presents the results. Section 5 discusses the results. Section 6 offers concluding remarks.

Literature review

After [START_REF] Miller | Dividend policy, growth, and the valuation of shares[END_REF], the literature mainly relaxes three hypotheses of their model: the nonexistence of tax [START_REF] Brennan | Taxes, market valuation and corporate financial policy[END_REF][START_REF] Elton | Marginal stockholder tax rates and the clientele effect[END_REF][START_REF] Poterba | New evidence that taxes affect the valuation of dividends[END_REF][START_REF] Hamon | Le marché français des actions: études empiriques 1977-1991[END_REF], the information perfection [START_REF] Bhattacharya | Imperfect information, dividend policy, and" the bird in the hand" fallacy[END_REF][START_REF] Bhattacharya | Nondissipative signaling structures and dividend policy[END_REF][START_REF] John | Dividends, dilution, and taxes: A signalling equilibrium[END_REF][START_REF] Miller | Dividend policy under asymmetric information[END_REF][START_REF] Rozeff | Growth, beta and agency costs as determinants of dividend payout ratios[END_REF][START_REF] Easterbrook | Two agency-cost explanations of dividends[END_REF][START_REF] Jensen | Agency costs of free cash flow, corporate finance, and takeovers[END_REF] and the agents' rationality [START_REF] Shefrin | Explaining investor preference for cash dividends[END_REF]. This last imperfection becomes a major source of interest with the growing influence of behavioral finance. Relaxing the investor's perfect rationality assumption in corporate finance leads to the emergence of the behavioral corporate finance approach [START_REF] Shefrin | Behavioral corporate finance[END_REF]. In their recent updated survey on behavioral corporate finance, [START_REF] Baker | Behavioral Corporate Finance: An Updated Survey[END_REF] divide the literature in two broad groups according to who is assumed to be affected by irrationality. The "market timing and catering" approach considers that rational managers face irrational investors whereas the "managerial bias" approach assumes that irrational managers face rational investors. Our paper adopts the first point of view and we refer to catering as "any actions intended to boost share prices above fundamental value" (Baker et al, 2013, p. 353). The managers who are able to perceive mispricing will intervene to remove this misevaluation. The mispricing caused by irrational investors may affect the decision of paying dividends to shareholders. According to Baker and Wurgler (2004a), the catering theory of dividend "has three basic ingredients. First, for either psychological or institutional reasons, some investors have an uninformed and perhaps timevarying demand for dividend-paying stocks. Second, arbitrage fails to prevent this demand from driving apart the prices of payers and nonpayers. Third, managers rationally cater to investor demandthey pay dividends when investors put higher prices on payers, and they do not pay when investors prefer nonpayers" (p. 1125). They test the theory by relating the decision to pay dividends to the dividend premium. The dividend premium reflects the uninformed demand for dividend paying shares. Their measure of the dividend premium is the difference between the logs of the average market-to-book ratio of dividend payers and the logs of the average market-to-book ratio of non-payers. They confirm that the dividend premium is a strong predictor of the decision to initiate dividend payout in the US between 1962 and 2000 and validate the prediction that managers respond to investors' demand. The higher the demand for dividend, the higher the propensity to pay. Baker and Wurgler (2004a) restrict the catering theory to the binary decision to pay or omit dividend. [START_REF] Li | Dividend changes and catering incentives[END_REF] extend their model to the decision to increase or to decrease the dividends and the magnitude of this change. They validate their prediction empirically. Thereafter, several empirical studies validate the theory in other countries. Baker and Wurgler (2004b) establish themselves a close empirical link between the propensity to pay dividends and catering incentives.

Catering incentives are measured by the same dividend premium proxy as in Baker and Wurgler (2004a). They identified four waves of appearing and disappearing dividends which are closely related to their dividend premium proxy. [START_REF] Ferris | God Save the Queen and Her Dividends: Corporate Payouts in the United Kingdom*[END_REF] also use aggregate data in the United Kingdom from 1988 to 2002 and validate the catering theory. They find that the dividends' disappearance in the 1990s was related to a lower dividend premium. [START_REF] Ferris | Catering effects in corporate dividend policy: The international evidence[END_REF] confirmed that managers rationally respond to the investor's demand for dividend in common law countries. In contrast, in civil law countries (including France), companies do not follow the investor's preference for dividend. [START_REF] Denis | Why do firms pay dividends? International evidence on the determinants of dividend policy[END_REF] investigated the determinants of dividend policy in six countries including France from 1994 to 2002 and do not find support for the catering theory of dividend outside of the US. [START_REF] Hoberg | Disappearing dividends, catering, and risk[END_REF] find that catering theory no longer explains payout policy in the US when risk is included in their regressions. Von Eije and Megginson (2008) also includes French companies in their sample of European companies and "find no systematic effects of a country-specific catering variable in EU companies, which suggests that catering is not an important factor influencing European payout policies" (p. 349). In France, [START_REF] Albouy | The perception of dividends by managers: Do French managers differ from their North-American peers[END_REF] use survey data to test several dividend theories and do not find support to the catering theory even though the items measuring how investors' demands are well ranked by the managers.

Data and methods

We use data on French listed companies from 2001 to 2012. All the data were collected from Thomson Datastream. All the companies belong to the CAC All Tradable (CACT) index. This index replaced the SBF 250 in 2011. Although the CACT includes more than 300 firms, the SBF 250 used to include 250 firms. Our sample covers a period ending in 2012on year after the SBF 250 is replacedand gathers data on most of companies belonging to this reference index. French listed companies are included in this index if they fulfil the Minimum Trading Velocity requirement defined by NYSE Euronext 7 . We remove financial and utility companies and those with outliers or missing values. The CACT index experienced a turnover during this period. Many firms getting in and out of the index. Since our purpose is to investigate the time varying demand for dividend, we keep only companies we can track during the whole period of time. Our sample is then a balanced dataset comprising 2652 firmyear observations (12 years  221 firms). As for the estimation method, we use random effects regressions following the result of the Hausman test. We use lagged values of all independent variables to measure how observation of investor's demand for dividend in a given year by the managers affects their decision to cater dividends the following year.

Dependent variables: the dividend payout policy

Previous research measured payout policy as a binary decision (initiate/omit, increase/decrease) or as a continuous one (the change of the dividend). We follow the same path using: (i) a dummy variable "DpsBin" taking the value of 1 if a company pays a dividend for a given year and 0 otherwise; (ii) a dummy variable "Dividend increase" taking the value of 1 if a company increases the level of the dividend yield for a given year and 0 otherwise;

(iii) two continuous variables measuring the changes of the dividend yield (the absolute value of the amount of decrease "DyDecrease" and the amount of increase "DyIncrease"). We use a log transformation of these last two variables to correct their positive skewness.

3.2.Independent variables: the dividend premia

We use firm level variables measuring the catering effect while previous papers used an annually aggregate measure of the investors' demand for dividends. This is an innovation of this paper. The dividend premium is the main measure of the investors' demand for dividend in previous papers. This Baker and Wurgler dividend premium (DP) is computed as the annual aggregate difference between the logs of the dividend payers and nonpayers average market-to-book ratios. The intuition of Baker and Wurgler (2004a) is that a positive (respectively negative) dividend premium will incentivize managers to initiate (respectively omit) dividend payout. We compute the same variable "DP3" used by Baker and Wurgler (2004a). Thus, it takes a unique value each year for all companies thus measuring a market level catering effect affecting all the companies the same way. We also create two proxies of the dividend premium at the firm level:a dummy variable "DP1" taking the value of 1 if the log of the dividend payers average market-to-book ratio for a given year is higher than the log of the average market-to-book ratio for a given company;a continuous variable "DP2" measuring the difference between the log of the dividend payers average market-to-book ratio for a given year and the log of the average market-to-book ratio for a given company. With DP1 and DP2, the interpretation of the results depends on whether the firm already pays dividends or not. We therefore can identify four different situations according to the current dividend policy: i) If the firm pays dividends and DP1 equals 1 or DP2 is positive, the rational decision is to stop paying or to decrease the dividend; ii) If the firm pays dividends and DP1 equals 0 of DP2 is negative, the rational decision is also to stop paying or to decrease the dividend; iii) If the firm does not pay dividends and DP1 equals 1 or DP2 is positive, the rational decision is to start paying or to increase the dividend; iv) If the firm does not pay dividends and DP1 equals 0 of DP2 is negative, the rational decision is to not initiate paying dividend. From this point of view, there is only one case (iii) where there is an incentive to start to pay or increase the dividend. In most instances, we report negative coefficients.

We take equal and value weighted averages of the market-to-book ratios.

Control variables

A set of control variables measure firms' characteristics: the dividend yield, the investment opportunities, the profitability, the retained earnings to the total equity, the size, the free cash flow, the systematic risk and the specific risk. This set of control variables also aims at controlling how competing explanations of the dividend policy. Continuing to pay a dividend, increasing or decreasing it and the amount of the variation do not have the same meaning according the level of the dividend yield. Because we use lagged values on all the independent variables, the dividend yield also controls for the stickiness of the dividend policy that has long been documented in the literature [START_REF] Lintner | Distribution of incomes of corporations among dividends, retained earnings, and taxes[END_REF]. Therefore, we include the dividend yield in our regressions. The dividend yield is used for instance by [START_REF] Li | Dividend changes and catering incentives[END_REF]. [START_REF] Fama | Disappearing dividends: changing firm characteristics or lower propensity to pay?[END_REF] use the total asset growth rates as a measure of the investment opportunities and the logarithm of the total assets to control for size. We expect that firms having more growth opportunities to pay out less dividend and large firms to pay more dividend. [START_REF] Fama | Disappearing dividends: changing firm characteristics or lower propensity to pay?[END_REF] find that firms that have never paid dividends have the best growth opportunities and that dividend payers are much larger than non-payers. As in [START_REF] Grullon | Dividend changes do not signal changes in future profitability[END_REF], we use the return on assets (ROA) to measure the influence of profitability on the dividend decision. We expect managers of profitable firms to be more likely to initiate or to increase dividends. DeAngelo et al. (2006) and Von Eije and Megginson (2008) use the retained earnings to the total equity as a proxy of the life-cycle theory of dividends. We use the same variable and expect a positive link between the probability of paying dividend and the retained earnings to total equity ratio. The free cash flow (FCF) variable evaluates the capacity of company to pursue the investment opportunities in the interest of shareholders and comes with agency cost [START_REF] Jensen | Agency costs of free cash flow, corporate finance, and takeovers[END_REF]. Firms with more free cash flow have more incentive to initiate or to increase dividends. Finally, we include two variables considering the systematic risk and the idiosyncratic risk following [START_REF] Hoberg | Disappearing dividends, catering, and risk[END_REF]. We expect managers to be more reluctant to pay or increase the dividends if the risks are high.

Empirical results

4.1.Descriptive analysis

As a first descriptive analysis, Figure 1 plots the equally (dashed line) and value weighted (solid line) dividend premia of our sample in the manner of Baker and Wurgler (2004a).

Recall that dividend premium proxies the investors' demand for dividend. It shows that the dividend premium does not remain stable over time suggesting a time varying demand for dividends. The differences between the values of the equally and value weighted dividend premia are important even though the values vary in the same senses until 2007-2008. From this time, the equally weighed dividend premium increases whereas the value weighted dividend premium decreases. Figure 2 plots the proportions of dividend payers between 2001 and 2012 while Figure 3 displays the proportions of companies that increased the dividend between 2001 and 2012. The patterns of these curves are very similar showing that the decisions to pay dividend and to increase vary the same way. On the contrary, the relationship between figure 1 on one hand and figures 2 and 3 on another hand are not obvious. At an aggregated level, the comparison of these figures does not suggest support of the catering theory. Indeed, the proportion of dividend payers does not seem to be driven by the dividend premium. Table 1 provides summary statistics of all the variables for the whole period investigated. During the period studied, 66.78% of the companies pay dividends and 39.14% of the companies increase the dividends. The average increase is 0.61 euros (SD=1.60) and the average decrease is 0. 53 euros (SD=1.70). The average market-to-book ratio is 2.461 euros (SD=3. 545). As for our equally weighted (Ew) measures of the dividend premium, 31% of the companies have a higher market-to-book ratio than dividend payers (DP1Ew=0:31%). DP2Ewthe variable measuring this differenceequals 0.306 on average (SD=0.716). The Baker and Wurgler aggregate annual equally weighted dividend premium equals -0.003 on average (SD=0.005) and 0.006 for the value weighted dividend premium (SD=0.009). Statistics on control variables are also reported in the table. Table 2 displays the correlation matrix between the variables included in the analysis. We do not have major collinearity issues except between the variables asset variation and return on assets. To check collinearity issue, we compute the variance inflation factors and report an average index smaller than 2.

4.2.The decision to pay, continue or omit dividends

We first examine how the prevailing investors' demand for dividend payers as well as the financial firm's characteristics affects the probability of starting or continuing to pay dividends. The results are reported in the columns labelled 1 to 6 of the Table 1. Columns 1 to 3 display the results for the 3 equally weighted dividend premia and columns 4 to 6 show the results for the 3 value weighted dividend premia. The dependent variable is coded 1 if the company pays a dividend and 0 otherwise. We run six binary logistic regressions of the decision to either start or continue paying dividend on each of our six dividend premium proxies and on our set of control variables. Five out of six coefficients associated to the dividend premia are significant indicating that the catering theory applies to the decision to pay, continue or to omit dividends. DP1 and DP2, the firm level measures of the dividend premium are always associated with negative coefficients. The coefficient of the Baker and Wurgler (2004a) dividend premium is negative when it is equally weighted and positive when it is value weighted. The coefficients associated to the size, the dividend yield and to the retained earnings are positive and significant. Firms are more likely to initiate or continue to pay dividend if they are large with a high past dividends and also when retained earnings represent a large portion of total equity. The coefficients on free cash flow are significant and positive (models 1, 3 and 6). Indeed, firms are more likely to offer dividend to shareholders if they have a high level of free cash flows in order to reduce the asymmetry information and the agency cost. The coefficients on the investment opportunities  measured by the asset growth rates  are positive and significant, suggesting that firms tend to cut dividend when they have a poor growth opportunities. The coefficients on the variables reflecting systematic and idiosyncratic risks are negative and positive respectively. The negative coefficients indicate that managers are more reluctant to start or to continue paying dividends if the systematic risks are high. Conversely, the negative coefficients show that firms are more likely to pay dividends even though the idiosyncratic risk is high reflecting certain level of overconfidence.

The coefficients associated to profitability are not significant.

4.3.The decision to increase the dividends

The second analysis studies the decision to increase or to decrease the dividends. The results are reported in the columns labelled 7 to 12 of the Table 1. Columns 7 to 9 display the results for the 3 equally weighted dividends premia and columns 10 to 12 show the results for the 3 value weighted dividend premia. Here, the binary dependent variable takes the value of 1 if a company increases its dividends and 0 otherwise. We run six binary logistic regressions of the decision to increase the dividend on each of our three dividend premium proxies and on our set of control variables. The results of the equally weighted (respectively value weighted) dividend premium proxies are reported in table 3 (models 7, 8 and 9) (respectively models 10, 11 and 12). Two out of our three dividend premium proxiesnamely DP1 and DP2are significant and negative. These results are consistent with our predictions. Only the Baker and Wurgler (2004a) annual measure (model 12) is positive and significant suggesting that managers are increasing the level of dividends if the investors demand for dividends is high.

The coefficients associated to the dividend yield are negative and statistically significant.

Firms with a weak past dividends are more likely to increase the dividends. The coefficients related to the retained earnings, risk and size are positive and significant. These results suggest that the larger the firms are, the more likely the increase is. Higher level of retained earnings and risk are associated to a higher probability to increase the dividends. All the other coefficients are not significant.

4.4.Determinants of the magnitude of dividends change

We now investigate what drives the magnitude of dividend changes. To do so, we use two continuous variables i.e. the increase and the decrease magnitude. We multiply this latter variable by minus one to get the absolute value of the decrease. Therefore, we run two separate regressions: One regression of the natural logarithm of the magnitude of the increase and the other of the natural logarithm of the magnitude of the decrease. These variables are regressed on the independent variables including our firm-level measures of the dividend premium (equally and value weighted). The results on the dividend magnitude increase are reported in table 4 (models 13 to 18). The coefficients related to our three dividend premium proxies are positive but not significant. Only the second firm level measure of the value weighted dividend premium (model 17) is significant at 5% level. The coefficients associated to the dividend yield, free cash flow and risk are positively related to the magnitude of the increase. Managers have incentives to pay more dividends if the past dividends and risk are high with an important level of free cash flows to reduce consequently the asymmetry information and the agency cost. The coefficients on the assets variation, profitability and size are negative and significant. This finding suggests that managers have no incentive to pay more dividends in the case of firms with poor profitability and even if the investments opportunities are weak to finance future investment opportunities. Finally, the coefficients on the retained earnings are not significant. The result suggests that the magnitude of the dividends increase is not driven by this variable.

The results on the magnitude of the dividend decrease are reported in table 4 (models 19 to 24). Only the coefficients related to the value weighted DP2 (positive) and DP3 (negative) in models 23 and 24. Recall that we change the sign of the magnitude by multiplying the value decrease by minus one. Consequently, the signs must be interpreted in the other direction, that is negative for DP2 and positive for DP3. These latter results are then consistent with the previous findings reported in the table 3. The other coefficients related to our proxies are not significant. The coefficients associated to the dividend yield are positive and significant suggesting that managers tend to cut the dividend more if the past dividend is high. The coefficients on the profitability and size are negative and significant. Dividends' decrease is lower for larger and low profitable companies. The coefficients on the investment opportunities, free cash flow, retained earnings and systematic risk are not significant. The results suggest that the magnitude of the dividends decrease is not driven by these variables.

Discussion

Our conclusions regarding whether the catering theory of dividend applies to France are mixed. The coefficients associated to the dividend premia suggest that the catering theory of dividend applies to France regarding the decision to start, continue or omit dividends (Baker and Wurgler, 2004a) and the decision to increase the dividend [START_REF] Li | Dividend changes and catering incentives[END_REF]. Overall, results for the value weighted dividend premia are stronger in terms of significance. Value weighted dividend premia are more accurate because they consider the relative market value of the companies in the calculations. It is very likely that the largest market capitalizations have stronger influence on the decisions to pay dividends. This is also true for the extended version of the theory of [START_REF] Li | Dividend changes and catering incentives[END_REF] concerning the magnitude of the dividend decrease.

Results are less evidential for the increase where only one model displays significant result (model 17). For the magnitude of the dividend decrease, coefficients associated to the value weighted dividend premia are significant in two models out of three.

When we closer look at the three dividend premia measures, we observe that the results obtained for our firm level variables (DP1 and DP2) are consistent to what we expected. One innovation of the paper is to use firm level measures of the dividend premia. This new way of computing dividend premia brings a more comprehensive understanding of how dividend policy is defined. We interpret the consistency of the results as a confirmation that our approach is correct. Although we believe that these new measures are consistent with the catering theory, they cannot be interpreted in the same manner as the aggregated proxy of the dividend premium, all the firms face the same dividend premium for a given year. In most papers investigating the catering theory, both the dividend policy and the dividend premium are aggregated at a yearly level for decades. Our paper takes a different approach studying the company level dividend policy. By doing so, we assume that rational managers define their dividend payout policy by observing their market valuation and comparing it with the valuation of dividends paying firms.

If we compare our results regarding the first version of the theory to previous literature, our findings are different from [START_REF] Kuo | What drives the disappearing dividends phenomenon[END_REF], [START_REF] Ferris | Catering effects in corporate dividend policy: The international evidence[END_REF] and [START_REF] Denis | Why do firms pay dividends? International evidence on the determinants of dividend policy[END_REF] who find that catering theory does not apply in civil law countries including France. Unlike our paper, these papers use aggregated dividend premium measures. We use different measures of the dividend premium with a closer focus on the French market. Using survey data, [START_REF] Albouy | The perception of dividends by managers: Do French managers differ from their North-American peers[END_REF] do not find either that the catering theory applies to France. All these papers attempt to test the first version of the catering theory: the binary decision to start or continue to pay dividend. As for the prediction of the extended catering theory developed by [START_REF] Li | Dividend changes and catering incentives[END_REF], our significant results for the value weighted dividend premia are in part consistent with it regarding the magnitude of dividends changes decision. These authors use yearly aggregated measures of the dividend premium. They find that the dividend premium affects positively the magnitude of the increases and negatively the magnitude of the decreases. In the first instance, we have the same result for model 17. Having in mind that we multiplied the magnitude of the decrease by minus one, we report a positive association between the Baker and Wurgler dividend premium and the magnitude of the decrease. The latter variable is negatively related to the firm level dividend premium. These two results differ from [START_REF] Li | Dividend changes and catering incentives[END_REF]. We then find that the decisions regarding the amount of the dividend increase and the amount of the dividend decrease are not symmetric.

We run additional analyses to check the robustness of our results. One question is whether firms with small capitalizations take the dividend premium into account when they define their dividend payout policy. The catering theory assumes that managers are very sensitive to investors' demand which may not be the case for small capitalizations. To check this effect, we run regressions on firms belonging to the CAC 40 index including the 40 largest French capitalizations. The dividend premium proxies are not associated with significant coefficients either. The clientele effect is another determinant of dividend policy according to the literature and a competing theory. For undiversified holders such as family, dividend represent an important source of income. As robustness checks, we run the regressions including the percentage of family ownership and the results do not change much. 16 out of the 24 regressions display the same results. The signs remain the same but are no longer significant for the remaining regressions. We believe it to be due to a smaller sample size because we collect data on family ownership only from 2001 to 2008.

Our results can also be explained by other factors. Baker et al (2013) distinguished two situations where agent's rationality is assumed to be limited. We adopt the "market timing and catering" approach that considers that rational managers face irrational investors. We do not investigate the dividend policy assuming managers' irrationality. Some unexpected results might be affected by management's irrationality and should be investigated by future researches. The use of firm level proxies of the dividend premium and the investigation of recent and exhaustive French data are two innovations of this paper. Using the dividend premium as a proxy of the time varying demand for dividend can be debated. The market-tobook ratio can capture other facts. For instance, it is often used as a proxy of investment opportunities but also as a strong proxy of mispricing (see, for example, [START_REF] Barberis | Mental accounting, loss aversion, and individual stock returns[END_REF][START_REF] Daniel | Overconfidence, arbitrage, and equilibrium asset pricing[END_REF]. As for the dividend policy decisions itself, we are aware that managers base their decision to pay dividend on perceived rather than actual demand for dividend. This can introduce measurement error as already pointed out by [START_REF] Li | Dividend changes and catering incentives[END_REF].

Other behavioral factors are not measured by our study whereas recent evidence shows that they can have strong influence on dividend policy. Indeed, Breuer et al (2014) investigate dividend policy in 29 countries including France and emphasize that patience, ambiguity aversion and loss aversion affect payout policy significantly. Finally, managers may look at a dividend premium with a horizon shorter than the year. When they decide their payout policy, they can be affected by valuation differences within the preceding month or even the preceding month. One can have a closer look at how this short term demand for dividends may affect dividend policy in forthcoming research.

Conclusion

This paper aims at testing the catering theory of dividend in France. This theory predicts that managers rationally cater dividend to investors when the demand for dividend is high and conversely. Baker and Wurgler (2004a) claim that the time varying demand for dividend can be proxied by the dividend premium they compute at an aggregate level in the USA. We use comparable measures of investors' demand for dividend computed at the firm level and investigate the dividend payout policy of a sample of French listed companies from 2001 to 2012. Although we validate the catering theory for the binary decision to pay dividend and its extension by [START_REF] Li | Dividend changes and catering incentives[END_REF] to the decision to increase the dividends. Most of the results regarding the magnitude of dividend changes are not significant. These findings are in line with a stream of behavioral corporate finance that regards managers as responding rationally to irrational investors' behavior.

Table 1. Descriptive statistics

Notes. Dividend increase and decrease gives the absolute value of the dividend's variation. MTBV represent the market-to-book ratio. DP1 is a dummy variable taking the value of 1 if the log of the dividend payers' average market-to-book ratio for a given year is higher than the log of the average market-to-book ratio for a given company. DP2 is a continuous variable measuring the difference between the logs of the dividend payers' average marketto-book ratio for a given year and the average market-to-book ratio for a given company. DP3 is a continuous variable measuring the difference between the logs of the dividend payers and nonpayers average market-to-book ratios (Baker and Wurgler, 2004a). P-value * significant at 5%, ** significant at 1%.

Table 2. Correlation matrix

Notes. DpsBin is a dummy variable taking the value of 1 if a company pays a dividend for a given year and 0 otherwise. Dividend increase is a dummy variable taking the value of 1 if a company increases the level of the dividend yield for a given year and 0 otherwise. Dividend variation gives the absolute variation of the dividend yield. DP1 is a dummy variable taking the value of 1 if the log of the dividend payers' average market-to-book ratio for a given year is higher than the log of the average market-to-book ratio for a given company. DP2 is a continuous variable measuring the difference between the logs of the dividend payers' average market-to-book ratio for a given year and the average market-to-book ratio for a given company. DP3 is a continuous variable measuring the difference between the logs of the dividend payers and nonpayers average market-to-book ratios (Baker and Wurgler, 2004a). P-value * significant at 5%, ** significant at 1%. Notes. DpsBin (dependent variable) is a dummy variable taking the value of 1 if a company pays a dividend for a given year and 0 otherwise. Dividend increase (dependent variable) is a dummy variable taking the value of 1 if a company increases the level of the dividend yield for a given year and 0 otherwise. DP1 is a dummy variable taking the value of 1 if the log of the dividend payers' average market-to-book ratio for a given year is higher than the log of the average market-to-book ratio for a given company. DP2 is a continuous variable measuring the difference between the logs of the dividend payers' average market-to-book ratio for a given year and the average market-to-book ratio for a given company. DP3 is a continuous variable measuring the difference between the logs of the dividend payers and nonpayers average market-to-book ratios (Baker and Wurgler, 2004a) Notes. ln(Dividend increase) or ln(Dividend decrease) are the dependent variables. Dividend increase -respectively decrease -gives the positiverespectively negative times minus one -variation of the dividend. DP1 is a dummy variable taking the value of 1 if the log of the dividend payers' average market-to-book ratio for a given year is higher than the log of the average market-to-book ratio for a given company. DP2 is a continuous variable measuring the difference between the logs of the dividend payers' average market-to-book ratio for a given year and the average market-to-book ratio for a given company. DP3 is a continuous variable measuring the difference between the logs of the dividend payers and nonpayers average market-to-book ratios (Baker and Wurgler, 2004a). Standard errors are displayed in parentheses *** p<0.01, ** p<0.05, * p<0.1.

Increase

Table 3 . The decision to pay, continue or omit dividends & The decision to increase the dividends

 3

Table 4 . How much to increase or decrease dividend? Determinants of the magnitude of dividends change

 4 . Standard errors are displayed in parentheses *** p<0.01, ** p<0.05, * p<0.1.

			The decision to pay, continue or omit dividends			The decision to increase the dividends
		Equally weighted	Value weighted	Equally weighted	Value weighted
		Model 1	Model 2	Model 3	Model 4	Model 5	Model 6	Model 7	Model 8	Model 9 Model 10 Model 11 Model 12
	DP1	-0.490**			-2.249***			-0.255*			-0. 598***
		(0.240)			(0.673)			(0.133)			(0.219)
	DP2		-0.209			-0.583***			-0.177**			-0.232***
			(0.150)			(0.0844)			(0.089)			(0.0445)
	DP3			-58.70***			46.6***			-4.74		34.07***
				(18.494)			(10. 561)			(10.70)			(5.902)
	Dividend yield	0.180*** 0.179*** 0.163*** 0.182*** 0.240*** 0.165*** -0.130*** -0.129*** -0.132*** -0.117*** -0.140*** -0.143***
		(0.041)	(0.041)	(0.041)	(0.0412)	(0.044)	(0.042)	(0.030)	(0.0299)	(0.03)	(0.029)	(0.031)	(0.030)
	Asset variation	0.004**	0.004**	0.0003	0.0054** 0.0059**	0.0002	0.00007	-0.00001	-0.00006	-0.0002	-0.0004	0.00003
		(0.002)	(0.002)	(0.0009) (0.00247)	(0.003)	(0.0008)	(0.0013)	(0.0013)	(0.0013)	(0.0014)	(0.0015)	(0.001)
	Return on assets	0.0001	0.001	0.0011	0.0006	0.0042**	0.0008	0.0015	0.0019	0.0014	0.0005	0.0019	0.0008
		(0.002)	(0.002)	(0.00163) (0.0019)	(0.0021)	(0.0017)	(0.0014)	(0.001)	(0.0014)	(0.002)	(0.0016)	(0.0015)
	Free cash flow	3.96e-07* 3.88e-07 4.7e-07** 1.91e-07	3.98e-07 3.94e-07* 2.91e-08	3.13e-08	3.18e-08	5.21e-09	8.68e-08	3.19e-08
		(2.38e-07) (2.38e-07) (2.34e-07) (2.35e-07) (2.3e-07)* (2.27e-07) (5.99e-08) (5.98e-08) (5.90e-08) (5.75e-08) (6.48e-08) (5.80e-08)
	Ret. earnings/equity 0.001**	0.002**	0.0012*	0.004**	0.0033** 0.0034** 0.0012** 0.0011** 0.0012** 0.004*** 0.0024*** 0.0029***
		(0.001)	(0.001)	(0.0006)	(0.0017)	(0.0013)	(0.0015)	(0.0005) (0.00048)	(0.000)	(0.0011)	(0.0008)	(0.0009)
	Systematic risk	-0.914*	-0.943**	-1.120**	-0.960**	-0.977*	-1.426***	0.658**	0.660**	0. 598**	0.651**	0.714**	0.381
		(0.471)	(0.472)	(0.471)	(0.484)	(0. 504)	(0.497)	(0.272)	(0.274)	(0.273)	(0.269)	(0.289)	(0.278)
	Idiosyncratic risk	0.032**	0.034**	0.033**	0.029*	0.0135	0.027*	0.033*** 0.033*** 0.034*** 0.0312*** 0.0282*** 0.026***
		(0.01)	(0.015)	(0.0159)	(0.0154)	(0.0158)	(0.016)	(0.0086)	(0.009)	(0.0086)	(0.0085)	(0.009)	(0.0081)
	Size	1.22***	1.212*** 1.332*** 1.121*** 0.986*** 1.279***	0.32***	0.319*** 0.318*** 0.225*** 0.178*** 0.315***
		(0.141)	(1.139)	(0.154)	(0.146)	(0.15)	(0.151)	(0.045)	(0.0444)	(0.0444)	(0.045)	(0.0515)	(0.0451)
	Constant	-13.77*** -13.93*** -15.69*** -10.71*** -9.22*** -14.94*** -4.71*** -4.821*** -4.876*** -3.107*** -2.394*** -4.910***
		(1.782)	(1.770)	(1.970)	(1.996)	(1.955)	(1.927)	(0.601)	(0. 598)	(0.604)	(0.676)	(0.731)	(0.612)
	Log likelihood	-653.19	-654.33	-654.36	-641.38	-617. 53	-640.94	-1320.64	-1320.48	-1313.33	-1303.81	-1281.30	-1282.27
	Wald Chi 2	124.50*** 124.16*** 121.90*** 115.79*** 138.73*** 12.72*** 102.07*** 103.06*** 100.74*** 105.16*** 101.13*** 124.06***
	Observations	2,193	2,193	2,187	2,141	2,137	2,143	2,188	2,188	2,176	2,136	2,132	2,132
	Companies	209	209	208	209	209	209	208	208	207	208	208	208

http://www.insee.fr/fr/themes/series-longues.asp?indicateur=part-dividendes-ebe.

http://www.henderson.com/getdoc.ashx?id=25805 accessed in March 2015.

Belgium, Greece, Italy, Lithuania, Luxembourg, the Netherlands, Portugal, Romania and Spain.

https://www.banque-france.fr/fileadmin/user_upload/banque_de_france/publications/BDF197-1_CAC40.pdf accessed in March 2015.

More information can be found following this link: https://indices.nyx.com/sites/indices.nyx.com/files/cac_family_rules_version_13-01_jan_2013_incl_6.9.pdf

Appendix Figure 1. Dividend premium (2001-2012)

Notes. This figure plots the dividend premium defined by Baker and Wurgler (2004a) as the difference between the logarithm of market to book of dividend paying firms and the logarithm of the market to book of nonpaying firms. These differences are computed for equally (dashed line) and value weighted (solid line) portfolios.