Elise Garrot

Sébastien George

Patrick Prévôt

Design of an Assistance Tool to Support the Tutor in the Setting-up of Learning Situations

Keywords: Computer-based environments for human learning, ontology, expert system, instrumentation of tutor's activity, tutoring model, learners' characteristics, modelling

Currently, tutors roles in distance learning are not clearly defined and few tools support them in their functions. Some tools help tutors to monitor learners and to interact with them, but no tool assist them in the setting-up of learning sessions. The activities are created by the instructional designer who envisages standard scenarios without knowing learners. Thus, the aim of our research project is to create a system to help tutors to prepare learning situations, by adapting them to learners' characteristics. The first phase of our work consists in creating a general model of on-line tutoring. Then, this model is integrated in the system in the form of a frame-based ontology containing learners' characteristics and parameters of learning situations. An inference engine then creates links between these characteristics by reasoning on the ontology. Finally, some rules deduce relations between ontology elements in order to give the tutor advice.

INTRODUCTION

With the Internet generalization, information and communication technologies (ICT) play a prominent part in distance learning. It involved a new definition of teacher roles, even if up to now these are rather badly defined. Our work contributes to the identification of tutor's role, in particular during collaborative learning situations. It consists on the one hand, in modelling the place and the role of the tutor within this learning situation and, on the other hand, in designing a system aiming at helping the tutor to take into account learners' characteristics during the settingup of learning sessions. In this article, we first explain the research issues which led us to supply an assistance tool to the tutor. The next section proposes a model of online tutoring. The sections that follow present the design of the system, of which we show an example of implementation in the form of rules which advice the tutor. The final section presents an overview of our work in progress.

RESEARCH ISSUES

According to a socio-constructivist approach [START_REF] Doise | The Social Development of the Intellect[END_REF], interactions between learners play a dynamic role to individual learning. In this way, collaborative learning activities are more and more used in distance education. This kind of activities, like project-based learning, has already proved to be useful [START_REF] George | Project-Based Learning as a Basis for a CSCL Environment: An Example in Educational Robotics[END_REF]. Indeed, the setting-up of collaborative activities is not easy, due to the fact that it must be adapted to learners' needs and characteristics [START_REF] Inaba | Learners' Roles and Predictable Educational Benefits in Collaborative Learning -An Ontological Approach to Support Design and Analysis of CSCL[END_REF]. In most distance courses, tutor has mainly a psychological and methodological role to support learners. Our approach of tutor's roles is based on the view of [START_REF] George | Complexité des situations pédagogiques e-learning dans un contexte multi-culturel, collaboratif et synchrone[END_REF] and [START_REF] Denis | Roles and Competencies of the e-Tutor (Learn Nett project)[END_REF] for whom the tutor is seen like a pedagogue who has the possibility to build and to adapt activities to learners' needs, as long as the instructional designer envisaged a variety of activities and possible situations of learning. In this case, tutor's role is not limited to monitor and support learners: it also consists in preparing specific learning situations from existing generic situations. The instrumentation of tutor's activity in distance learning environments is still little developed. Research was rather centred on the characterization and the standardization of learning activities in order to assist authors in the designing of scenarios, and tutors to support them to monitor learner's activities and interact to solve difficulties [START_REF] Després | Synchronous Tutoring in Distance Learning[END_REF]. Noting this deficit, we thus wished to offer to tutors a tool to assist and advise them in the setting-up of learning sessions, by creating links between learners' characteristics and activities parameters. Figure 1 highlights the role of the assistance system for the tutor to help him to set-up learning sessions. Learning situations consist of a set of activities carried-out by a learners group engaged in the same objective. Project, case study, problems resolution… are examples of learning situations.

MODELLING OF THE ON-LINE TUTORING

The modelling work was realized with the software MOT1 (Modeling using Object Types), an object-oriented modelling tool, developed by the Tele-University of Quebec [START_REF] Paquette | Generic and Specific Process-Oriented Models using Knowledge Types for Learning and Advising[END_REF]. The objective of this modelling is to identify actors and resources of collaborative learning situations, as well as their links with the tutor. This tutoring model centred on the tutor declines in fourteen diagrams MOT, developing several levels of details, an example of which we present in this part. This work led to the identification of:

• Actors of the course: tutor, learner, workgroup, instructional designer, conception and realization team, and computer designer. • Educational variables which represent roles and degrees of freedom given to tutors.

• Resources which the tutor has at his/her disposal during the course.

• Learners' characteristics which could influence their learning [START_REF] Brusilovsky | Adaptive hypermedia[END_REF]): learner's knowledge and behaviour, his/her experience, identity profile (curriculum-vitae, cultural origins, interests and habits), needs and objectives, and cognitive capacities (figure 2). The system must be able to adapt to learning situations which vary according to the type of online courses and tutoring model applied. That's why we chose to separate knowledge and reasoning, the system being based on ontology.

Ontology has the advantage to make explicit what is regarded as implicit in the field (Kasai et al., 2004), to use a vocabulary comprehensible by all actors, to re-use and make evolve this vocabulary.

Concerning the implementation, we chose the software Protege20002 , a tool for modeling and knowledge acquisition developed by the University of Stanford, in the United States. The plug-in JessTab3 , integrated into Protege2000, makes it possible to introduce the knowledge stored by Protege2000 into a data base, in order to be inferred by some rules written in the inference engine Jess4 , an expert system independent of Protege2000.

Ontological model

The developed ontology resumes the concepts brought out during the first work of tutoring modelling. Furthermore, it associates to each concept properties and relations. For example, in order to associate to learners their characteristics, we created two different concepts: Actor and Actors' characteristics (Figure 3). Both concepts are connected by properties of the class Actor which are instances of the class Actors'characteristics.

Figure 3: Properties associating to learners their characteristics in Protege2000

To give a structure and describe learning situations, we referred to the granularity levels of the standard IMS-LD (IMS Learning Design) 5 . This model, based on the EML standard [START_REF] Koper | Modeling Units of study from a pedagogical perspective -The pedagogical meta-model behind EML[END_REF], describes a formal way to represent the structure of a Unit of Learning and the concept of a pedagogical method specifying roles and activities that learners and support persons can play using learning objects.

In the ontology, activities are decomposed in the following way:

Course -> Learning situation -> Activity ->Resources and communication tool -> Media

The course corresponds to the unit of learning described in IMS-LD. Learners carry out a set of activities within a same learning situation. In a case study-based learning, an activity can be the problem formalization, searching for causes or solutions. Some resources (web resources, books, paper documents …) and means of communication (chat, forum, e-mail…) are offered to actors before and during learning sessions. They represent the learning environment. A resource is composed of media (picture, video, sound…).

IMPLEMENTATION OF RULES TO ADVICE THE TUTOR

The implementation of the system consists in writing different types of inference rules, having no ambition to be totally educationally valid at this time of the project; they only show the feasibility of the system. The types of rules are the following ones:

• Rules which create links between learners' characteristics.

• Rules which deduct advices to the tutor concerning the type of pedagogy to be applied for each learner, according to some of his/her characteristics. • Rules which create links between activities' parameters and learners' characteristics by advising the tutor an activity to attribute to a learner according to some variables. For example, let us consider a learner which has just validated an activity. We have developed a rule which advises the tutor an activity to attribute to the learner and the modality associated (individual or collective), according to the need of autonomy declared by him/her. (if (and (eq ?pre_required_activities ?validated_activities) (eq (slot-get ?activity modality) individual-or-collaborative)) then (foreach ?need (slot-get ?learner has-as-needs-and-objectives) (if (eq (slot-get ?need autonomy) TRUE) then (printout t "Propose to learner "?name" to do the activity "?activity_name "individually."crlf))

CONCLUSION AND FUTURES DIRECTIONS

We conceived a tutoring model which details the functioning of online tutoring: tutor's roles and degrees of freedom, learners' characteristics, resources and means of communication available. We implemented this model in the form of an ontology which integrates all the concepts, by specifying their properties and relations. Finally the development of the system containing rules, which apply on the classes instances of the ontology, showed the feasibility of our assistance system to set-up learning situations. Moreover, the diversity of rules shows the flexibility of the system and the many prospects offered. Our research will be directed towards two axes: the evolution of the tutoring model and the complete implementation of the system. We are in relation with several tutors of various formations which are interested in working in collaboration on this subject. It will be as many possible grounds for our future experiments. Concerning the tutor's assistance system, we want to give tutors the possibility to modify and create themselves learning rules, thus ensuring a feed-back on uses and a good appropriation of the tool. We have also in prospect to associate an interactions analysis agent to automatically feed the system with data concerning interactions between learners. This information will guide the tutor in the setting-up of sessions to come.

Figure 1 :

 1 Figure 1: Role of the tutor's assistance tool to set-up learning situations

Figure 2

 2 Figure 2 is an example of a MOT diagram. It describes learners' characteristics the tutor can identify before a course. To each characteristic are associated resources which can help to obtain them. A questionary and an interview with the tutor are examples of resources.This representation makes it possible to highlight links between procedures (symbolized by circles), concepts (symbolized by squares) and actors (symbolized by hexagonal shapes). These links can be composition links (labeled C) and input/output links (labeled I/P).

Figure

  Figure 2: Learners' characteristics identification

http://www.licef.teluq.uquebec.ca/anglais/real/mot.htm

http://protege.stanford.edu/

http://www.ida.liu.se/~her/JessTab/

http://herzberg.ca.sandia.gov/jess/


Garrot Élise
email: elise.garrot@insa-lyon.fr

George Sébastien
email: sebastien.george@insa-lyon.fr

Prévôt Patrick
email: patrick.prevot@insa-lyon.fr

A PLATFORM TO SUPPORT A VIRTUAL COMMUNITY OF TUTORS IN EXPERIENCE CAPITALIZING

Keywords: Virtual Communities of Practice, On-line Tutor, Knowledge Management System, Communication and Information Technologies

Nowadays, Information and Communication Technologies change the way students learn. Many researches are centered on the developing of new forms of pedagogy and software for student learning. But ICT also involve changes in teachers roles, which are not yet well defined in particular for on-line tutors. In this article, we develop new ways for tutors to define themselves their "professional identity", their functions and "good practices". By providing a platform to develop communities of practice of tutors, we aim at giving them the possibility to share knowledge, experiences and to refer to them in their day-to-day practice. We present a prototype of tutors interface to record and retrieve experiences stored as knowledge in a database.

INTRODUCTION

Nowadays, virtual communities concern several areas. We can talk about communities of practice in companies to develop innovation or communities of interests with the development of forums, blogs and wikis on Internet. Everybody is concerned with this new way of social communication.

In education, Information and Communication Technologies (ICT) involve new learning techniques, which are used by companies and universities with on-line courses and training. But it involves a new definition and new roles for the teacher in distance learning, and particularly for the tutor. It is now the time to study the way to use ICT to support tutors in the tasks that one assigns to them.

In the first section of this article, we study the changes involved by ICT in education and the problems we want to solve. In the second section, we present communities of practice (CoPs) as a means to help tutors. Then, we present the design of a knowledge management system which can be used by this community of tutors to support them in their work and to capitalize on their experiences. We also show a prototype of tutor interface to store and retrieve useful experiences. Finally, we conclude with future directions and all the possibilities implied by the definition of tutor roles and by the platform we develop.

RESEARCH ISSUES

The more and more important roles played by Internet and especially by ICT have changed the way we learn. Distance learning moved from a mail to an on-line mode. We emphasize benefits of this new form of education for students, like course individualization and flexibility in terms of space and time.

Computer mediated education involves new roles for teachers. In a traditional classroom, the teacher is the designer of his/her course and often transmits course contents to students. There is a direct relation between them and students. Teachers can see if the class is in trouble. Sometimes, if the size of the group allows it, learners can ask questions in real time and everyone can ear the answer. There is a delineated time for learning. But, at a distance, the teacher's jobs are divided into several distinct roles. Time and space marks are lost [START_REF] Garrot | A system to support tutors in adapting distance learning situations to students[END_REF]. Now there are two people: the instructional designer and the tutor. Contents are created by the instructional designer who is an expert on teaching area. Contents are then delivered to students often without any relation with the instructional designer. The tutor role is divided into several functions which vary according to distance courses and are generally classified into four parts: pedagogical, organisational, relational and technical. Tutoring usually consists in interacting with learners to assist them in building knowledge and competencies and to assess them. Now, research tends to develop more and more educational software and on-line courses to diversify learning activities for students, like project based learning or case studies. It is also centred on the characterization and the standardization of learning activities in order to assist instructional designers in the designing of scenarios.

But according to Casey and Greller (2005), the development and technical community is realizing that "there is much more to teaching than delivering the 'right' content and organising the 'right' student activities". For the course to be effective, Mc Pherson and Nunes (2004) insist on the importance of the role both of the tutor and the instructional designer: "the focus is frequently placed on design and developing information and communication technology (ICT) based environments and insufficient attention is given to the delivery process. These efforts have little chance of succeeding without a tutoring team that has appropriate online tutoring skills necessary to explore and maximize the designed environments."

Currently, tutors have scripted courses that they try to adapt to their students without help. Tutors do their work as best they can because:

• The instrumentation of tutor activity in distance learning environments is still little developed [START_REF] Dufresne | Vers un modèle générique d'assistance aux acteurs du téléapprentissage[END_REF].

• They have no means to monitor the change of their activities. It is necessary to make them aware that traditional courses cannot become "on-line learning" by modification or adaptation. It requires a fundamental restructuring. Concerning the first point, an important problem is that there are no traces of their work (decisions taken, events produced, scenario configuration or personalization, quality of the knowledge building process). Therefore there is no possibility of sharing practices and reusing them in other contexts.

As a result, tutors do not really know what they have to do and what the best way to do it is. Tutoring is not yet considered as a separate profession and tutors' work is not well defined. In literature, distance tutors are named differently, according to the functions that one assigns to them: moderator, facilitator, online tutor, online moderator, e-moderator, coach, distance education tutor, e-tutor… A tutor is usually a single person who does not know how to situate him/herself between students and instructional designers.

So, not only do ICT change the way students learn, but also the way teachers work. We study how ICT can contribute to the recognition and definition of a specific role of tutor. Today it is advisable in education to ask the question of how ICT, having modified the teacher role, can now contribute to define it and help "tutors".

So, our research purpose is to bring a solution to solve the problems for tutors defined in this section, which are: • Lack of professional identity.

• Lack of help in their day-to-day practice.

• Lack of practice sharing. Casey and Greller (2005) assert that "good teaching […] is reflective and reflexive about the experience of teaching and incorporates lessons learnt from the experience of teaching into teaching practice". Tutors must examine their own work. They also have to confront their ideas in order to determine "good practices", and so have the possibility to refer to these. According to a socio-constructivist approach of learning, collaboration between tutors can help each one to build new competencies and reinforce existing ones. It refers to the notion of "proximal zone of development" defined in [START_REF] Vygotsky | Thought and language[END_REF].

TUTORS IN A VIRTUAL COMMUNITY OF PRACTICE

That is why we suggest that tutors develop their competencies and define their work and good practices in the context of a community of practice. We now describe how this approach can contribute in solving all the problems previously defined.

COMMUNITIES OF PRACTICE

According to [START_REF] Wenger | Cultivating communities of practice[END_REF], communities of practice (CoPs) are "groups of people who share a concern, a set of problems, or a passion about a topic, and who deepen their knowledge and expertise in this area by interacting on an ongoing basis." They operate as "social learning systems" [START_REF] Snyder | Communities of practice in government: leveraging knowledge for performance[END_REF] where practitioners connect to solve problems, share ideas, set standards, build tools, and develop relationships with peers. Organizations and researchers use a variety of terms which refer to CoPs: knowledge communities, knowledge networks, learning networks… In order to situate a CoP in reference to all these terms, [START_REF] Snyder | Communities of practice in government: leveraging knowledge for performance[END_REF] give a definition: "A community of practice is a particular type of network that features peer-to-peer collaborative activities to build member skills".

Other important aspects of CoPs are pointed out by Scarbough and Swan (1999): "Socially, CoP are the fabrics of knowing as members of CoP acquire communal identity around a shared passion, relationships, roles and ways of intermingling common knowledge, practices and approaches". This definition highlights the social aspect of CoPs. Members of a CoP, by interacting around a same subject, define common practices, and create identity which is common to all the members.

A virtual community of practice is a specific kind of CoP, resulting from computer mediation. According to [START_REF] Koh | Knowledge sharing in virtual communities: an e-business perspective[END_REF], a virtual community is "a group of people with common interests or goals, interacting for knowledge (or information) sharing predominantly in cyberspace." Relationships in virtual CoPs are now software mediated (e.g. forum, blog or wiki) generally classified by topic of interest. [START_REF] Fernback | Virtual communities: Abort, retry, failure?[END_REF] pointed out the need for a virtual CoP to have "a specified boundary or place (e.g., a conference or chat line) that is symbolically delineated by topic of interest."

In the context of our study, we take up these main characteristics of virtual CoPs: • Members share an interest, roles, a concern, a set of problems or a passion.

• They aim at building members skills and deepening their knowledge and expertise.

• Members acquire a common identity.

• They need a specified boundary or place.

• Practitioners define common knowledge, practices and approaches.

By putting these characteristics together, we notice that creating a CoP for tutors could be a solution to the problems highlighted in the previous section. CoPs give the possibility for tutors from different backgrounds to define and acquire a professional identity, to share and develop their knowledge, competencies, expertise and to define "good practices" to apply in their work. But we now have to define the organization for a specific CoP of tutor, and the tools to support their interactions.

TUTORS IN COMMUNITIES

There are tutors in many countries in the world and they all have their own practices and their own background. Tutors usually have another job and do tutoring as an "extra", along with their job. Some of them are teachers in university, researchers, and company employees… So they have different competencies and expertise to share with others and to acquire by participating in a CoP.

According to Mc Pherson and Nunes ( 2004), for successful online learning, tutors must have subject matter expertise, technical skills, and also be educationalists with pedagogical, information and communication skills that are necessary to manage and facilitate online learning. But, due to their various backgrounds and the lack of training, only a few tutors have all these requirements. That is why it is important to favor the development of CoPs in which tutors will improve their expertise and competencies. Since "communities are emergent" [START_REF] Brown | Organizational learning and communities of practice[END_REF], we can not "create" CoPs but we could provide the infrastructure in which they will emerge.

We base our work on interviews of tutors, in order to know their needs and opinion. Regarding, on the one hand, the tutor role and, on the other hand, characteristics of CoPs, we can distinguish three types of CoP of tutors (Fig. 1).

Tutors who monitor the same type of activities:

As we said previously, there are more and more various activities offered to learners in distance learning: project-based learning, case study… These activities can be divided into several distinct phases: for example individual or collaborative phases, distance or face-to-face phases... So, tutor functions vary according to these different types of phases. Communities can be created for tutors who supervise the same type of activity, in order to improve their practice and to share their experiences.

Tutors who have the same function in a course:

According to the on-line courses, the training, the type of activities or the university, different functions are assigned to tutors. We sum up these possible functions, detailed in [START_REF] Garrot | A system to support tutors in adapting distance learning situations to students[END_REF]. The tutor can be:

• An assessor: the tutor assesses the learning group's productions and activities, and the competencies and knowledge acquired by a learner.

• An intellectual catalyst: to incite learners to have individual reflection, to express themselves publicly and to develop their ideas [START_REF] Denis | Roles and Competencies of the e-Tutor (Learn Nett project[END_REF].

• A moderator of interactions: the tutor develops and regulates interactions [START_REF] Dillenbourg | What do you mean by 'collaborative learning'?[END_REF] and sets up the group dynamics.

• A psychological and emotional support: a human mediator to motivate, encourage, stimulate learners [START_REF] Lentell | The Importance of the Tutor in Open and Distance Learning[END_REF] and create a friendly learning environment.

• A facilitator of learning: for a given learning situation, tutor can provide documents or advise adapted resources to guide learners and give them feedback on the assessment.

• A technical support: the tutor helps learners to get acquainted with tools and provides technical support in case of trouble.

• A pedagogical architect: tutor has scope to adapt the learning situations created by the instructional designer to learners' needs and characteristics.

Tutors who use the same subject matter:

In order to develop their knowledge and competencies in their subject matter, it is important to provide tutors with topics. Furthermore, the tutor's role can vary according to the subject matter concerned, e.g. computer science, humanities or corporate knowledge management.

Activity based community

Function based community

Subject matter based community

Tutor Tutor Tutor Tutor Tutor Tutor

INTERNET

Figure1. CoPs of tutors as a global community

So, we can see that CoPs of tutors can be defined by a university or a company, they should be defined by common interests and common practices: "CoPs are important as they weave the organization around competencies without reverting to functional structures" [START_REF] Pan | Bridging Communities of Practice with Information Technology in Pursuit of Global Knowledge Sharing[END_REF]. A tutor can only participate in a specific community or in two or three types of CoP. The advantage of defining several communities around shared practices is to create more knowledge and develop more interactions than in a global community [START_REF] Pan | Bridging Communities of Practice with Information Technology in Pursuit of Global Knowledge Sharing[END_REF]. Furthermore, within an organization perceived as a set of communities, not simply of individuals, separate community perspectives can be amplified by interchanges among communities [START_REF] Brown | Organizational learning and communities of practice[END_REF].

A KNOWLEDGE MANAGEMENT SYSTEM

According to a socio-technical approach, [START_REF] Koh | Knowledge sharing in virtual communities: an e-business perspective[END_REF] distinguish two important points in order to develop CoPs: usability and sociability. Concerning sociability, [START_REF] William | Four smart ways to run online communities[END_REF] suggested three virtual community managing strategies: member development, community asset management and community relationship management. More specifically, in order to run a virtual community intelligently, a clear vision, opinion leaders, off-line activities, rules/roles, and useful contents based on expertise are required. In addition to these common social characteristics of successful communities, [START_REF] Whitaker | The factors enabling and inhibiting the development of agricultural Internet virtual communities: An Australian case study[END_REF] highlight the importance of technology factors such as consistent and compatible software, which are very important in the context of a virtual community.

In [START_REF] Pan | Bridging Communities of Practice with Information Technology in Pursuit of Global Knowledge Sharing[END_REF], the authors look at the use of information technology to link communities in order to create global knowledge sharing. They draw conclusions of the implementation of a Knowledge Management System (KMS) in a world company. They assure that it is important to create common knowledge CoP as opposed to a single global community or multi-region forum arrangement. They emphasize the need to provide multiple channels/forums for diverse knowledge sharing needs and preferences.

Von [START_REF] Krogh | Developing a knowledge-based theory of the firm[END_REF] distinguishes three stages of KM: capturing knowledge, sharing and transferring knowledge, generating new knowledge. To overcome these stages, we implement a system to facilitate the capturing of tutors knowledge (their experiences, useful documents and links, references…). The knowledge sharing and generating will be the result of the activity of CoPs. Since "the structural process that is associated with community is communication" [START_REF] Fernback | Virtual communities: Abort, retry, failure?[END_REF], we have to provide to CoPs with the suited software to develop interactions.

So, in this section, we present our platform functionalities and tools to support CoPs interactions and knowledge sharing. We also define the organization of CoPs: leaders, rules and roles.

Functionalities of the system and network architecture

In order to choose the tools to support the CoPs of tutors, we first define functionalities that tutors need: • To interact with other tutors about experiences and to determine "good practices" (the level of interaction defines the level of participation in the community).

• To refer to an expert tutor in a field of competencies (technical, subject matter…).

• To relate and store an experience.

• To record documents, useful links and other resources.

• To retrieve contextual experiences stored as knowledge.

To support these functionalities, we choose to develop a platform with a database and two communication tools: email and forum. As illustrated in Fig. 2, there are different information and communication flows:

• From a tutor to a community (by forum).

• From a tutor to an expert tutor (by email).

• From the database to a tutor interface.

• From a tutor interface to the database.

Each tutor can access an interface (we show a prototype of this interface in the next section) to connect to the database, in order to store or search experiences. When referring to an experience, tutors can contact a tutor or access a forum. In order to support each community, a forum is associated to each type of community. For example, we propose: • a forum for tutors who monitor collaborative activities,

• a forum about the function of interaction moderator, • or a forum about sciences courses.

Threads of forum can be stored in the database. To summarize, the database contains various types of data:

• Information about tutors (users of the platform).

• Information about activities defined by parameters.

• Experiences (text files), entered by a tutor directly in the base or thread of forum.

• Good practices (text files) which are threads of forum or a document written by the community.

• Documents stored by a tutor (useful resources).

Tutor interface for knowledge searching and retrieval

With our system, we want tutors to record experiences. An experience could be what they set up in their course and whether it works or not. For example, a tutor can adapt difficulties to a specific learner and improve his/her learning. It must be an interesting testimony for the other tutors. But, to be effective, this action must be performed short, because tutors do not have enough time for this. That is why we have to develop a simple interface which is well defined and easy to use.

We choose to create a matrix in which tutors can store experiences according to a function or a specific phase of the course (Fig. 3). The first line of the matrix corresponds to the several phases of a course, which are parameterized activities.

When accessing the matrix for the first time, the tutor determines parameters (illustrated in a cell) of each activity in the course:

• Individual or collective activity.

• Distance or face-to-face activity. • Synchronous or asynchronous activity. Fig. 3 illustrates a case study which has four activities. The tutor can click on different cells to store or search an experience. For example:

• is a cell in the intersection of a column and a line. The tutor accesses experiences which correspond to the function of motivational support and the second phase of the case study (cause searching).

• is a cell in first column. The tutor accesses all experiences linked to the motivational support function.

• is a cell in first line. The tutor accesses all experiences related to the second phase. The number in each cell indicates the number of experiences which correspond to a specific function and a type of activity. Darker is the color of the box, more there are stored experiences. So, with this matrix,

Figure 2 .

 2 Figure 2. Mediated interactions through the platform supporting CoP of tutors

CONCLUSION AND FUTURE DIRECTION

We studied a new way to use ICT in education in order to help tutors to learn and develop competencies and expertise in the context of communities of practice. As members of CoPs, tutors can share their experiences and define good practices. All this knowledge will be easily accessible through our platform. Our interface enables tutors to rapidly retrieve the knowledge they need in the context of the activity they are monitoring.

Presently, we are developing this platform with the above described functionalities. At the same time, we are in contact with several tutors from different backgrounds who provide advice on the platform design. First, we will offer only one or two forums, in order to test the efficiency of the interface and the database. This experiment will give us first results to make the platform evolve.

tutors can store and access the experiences which concern them very quickly. We will also develop an interface to give them the possibility to make query on other knowledge and resources.

Organization of the global community supported by the platform

To develop and enable the community to survive, we must have leaders for each type of community previously defined. According to the emergence concept, we give tutors the possibility to determine themselves as leaders. A leader is:

• An expert (about a tutor function, a subject area or a type of activity).

• A moderator of the forum associated to the community.

• A referral agent for tutors who are in trouble (interactions by mail).

The leader has to choose useful forum threads to store in the database. A same tutor can become the leader of several different communities.


Jean-Bernard Chatelain
email: jean-bernard.chatelain@univ-paris1.fr

Kirsten Ralf
email: kirsten.ralf@esce.fr.

Robert Kollman

Sumudu Kankanamge

Gregory Levieuge

Efrem Castelnuovo

Georges Overton

Can we Identify the Fed' s Preferences?

Keywords: JEL classi…cation numbers: C61, C62, E31, E52, E58 Ramsey optimal policy, zero-credibility policy, Identi…cation, Central bank preferences, New-Keynesian Phillips curve, Working capital channel

published or not. The documents may come     Can we Identify the Fed' s Preferences?

Introduction

Can we test if the Fed follows Ramsey optimal policy under the limited credibility of quasi-commitment with a non-zero probability to commit [START_REF] Debortoli | Monetary regime switches and central bank preferences[END_REF], [START_REF] Schaumburg | An investigation of the gains from commitment in monetary policy[END_REF], Roberts (1987)) versus the Fed follows a policy with a complete lack of credibility with a zero probability to commit [START_REF] Oudiz | International Policy Coordination in Dynamic Macroeconomic Models[END_REF])? Are the Fed's preferences facing the same identi…cation problem as Taylor rule parameters for both types of policies? [START_REF] Cochrane | Determinacy and Identi…cation with Taylor Rules[END_REF] found that the simple Taylor rule parameters are not identi…ed in new-Keynesian models including forward-looking in ‡ation, but only the auto-correlation parameters of non-observable shocks.

Beginning with [START_REF] Simon | Dynamic Programming under Uncertainty with a Quadratic Criterion Function[END_REF] certainty equivalence property of the linear quadratic regulator, the quadratic loss function describing policy-maker's preferences is used for modelling stabilization policy [START_REF] Duarte | A feasible and objective concept of optimal monetary policy: The quadratic loss function in the postwar period[END_REF]). It took more than two decades for the estimation of Fed's preferences to begin with [START_REF] Salemi | Revealed preference of the Federal Reserve: using inversecontrol theory to interpret the policy equation of a vector autoregression[END_REF], using inverse control [START_REF] Salemi | It's what they do, not what they say[END_REF]). [START_REF] Salemi | Revealed preference of the Federal Reserve: using inversecontrol theory to interpret the policy equation of a vector autoregression[END_REF] took into account two structural breaks for three monetary policy regime during the period 1947-1992 using monthly data. [START_REF] Salemi | Revealed preference of the Federal Reserve: using inversecontrol theory to interpret the policy equation of a vector autoregression[END_REF] includes a careful investigation of identi…cation issues when the monetary policy transmission mechanism is a backward-looking vector auto-regressive (VAR) model. These 2 Ramsey versus Zero-credibility Monetary Policy

Monetary Policy Transmission Mechanism Including or not the Cost of Working Capital

The reference new-Keynesian Phillips curve is the monetary policy transmission mechanism:

t = E t [ t+1 ] + x t + u t where > 0, 0 < < 1

where x t represents the welfare-relevant output gap, i.e. the deviation between (log) output and its e¢ cient level. t denotes the rate of in ‡ation between periods t 1 and t. u t denotes a cost-push shock. denotes the discount factor. E t denotes the expectation operator. The cost push shock u t includes an exogenous auto-regressive component: u t = u t 1 + " u;t where 0 < < 1 and " u;t i.i.d. normal N 0; 2 u [START_REF] Adolfson | Optimal Monetary Policy in an Operational Medium-Sized DSGE Model[END_REF] where denotes the auto-correlation parameter and " t is identically and independently distributed (i.i.d.) according to a normal distribution with constant variance 2 u . The reduced-form parameter (denoted ) of the slope of the new-Keynesian Phillips curve relates in ‡ation to marginal cost or to the output gap. It depends on four structural parameters: the representative household discount factor , the household's elasticity of substitution between each di¤erentiated goods ", the measure of decreasing returns to scale of labor in the production functions of the …rms , and the proportion of …rms who do not reset their price each period [START_REF] Gali | Monetary Policy, In ‡ation, and the Business Cycle[END_REF], chapter 3):

( ; "; ; ) = (1 ) (1 ) (1 ) (1
+ ") :

In the new-Keynesian model, labor is assumed to be the only input in the production function. The cost of capital channel of monetary policy is ruled out by assumption. Christiano, Trabandt and Walentin (2010) assume that labor cost is …nanced by working capital. The marginal cost in the new-Keynesian Phillips curve is then a linear combination of labor and capital cost (Christiano, Trabandt and Walentin (2010), equation [START_REF] Havranek | Habit formation in consumption: A meta-analysis[END_REF]:

(1 + ) x t + (1 ) + i t (3)
We estimate two polar cases. When material inputs are not used in production ( = 1) and when no labor cost is …nanced by working capital ( = 0), the cost of labor is the only cost taken into account and the output gap is the policy instrument. When material inputs are all used in production ( = 0) and when all labor cost is …nanced by working capital ( = 1), the cost of capital is the only cost taken into account with a slope of the new-Keynesian Phillips curve denoted i . Then, the Federal funds rate is the policy instrument with an interest smoothing parameter i for the Fed's preferences.

Monetary Policy Regimes and Central Bank Behavior

In a monetary policy regime indexed by j, a policy maker has a period loss function 1 2 ( 2 t + x;j x 2 t ) for the labor cost model or the period loss function includes an interest smoothing term 1 2 ( 2 t + i;j i 2 t ) for the cost of working capital model. Fed's preferences are represented by the relative cost x;j > 0 or i;j > 0 of changing its policy instrument. A policy maker may re-optimize on each future period with exogenous probability 1 q strictly below one ("stochastic replanning" [START_REF] Roberds | Models of policy under stochastic replanning[END_REF], "quasi commitment" [START_REF] Schaumburg | An investigation of the gains from commitment in monetary policy[END_REF][START_REF] Kara | Monetary policy under imperfect commitment: Reconciling theory with evidence[END_REF] or "loose commitment" (Debortoli and Nunes, 2014)).

Following [START_REF] Schaumburg | An investigation of the gains from commitment in monetary policy[END_REF], we assume that the mandate to minimize the loss function is delegated to a sequence of policy makers with a commitment of random duration. The degree of credibility is modelled as if it is a change of policy-maker with a given probability of reneging commitment and re-optimizing optimal plans. The length of their tenure or "regime" depends on a sequence of exogenous i.i.d. Bernoulli signals f t g t 0 with E t [ t ] t 0 = 1 q, with 0 < q < 1. If t = 1; a new policy maker takes o¢ ce at the beginning of time t. Otherwise, the incumbent stays on. A higher probability q can be interpreted as a higher credibility. As seen below, this leads to use a "credibility adjusted" discount factor q in the policy maker's optimal behavior.

Secondly, in this new monetary policy regime indexed by k, there may be a switch of the transmission mechanism parameter (the slope of the Phillips curve) k including in particular, a switch of the representative household's elasticity of substitution between each di¤erentiated goods " k . Thirdly, the new policy maker may have new preferences

x;k . If the policy maker is maximizing welfare, its preferences are given by the new values of structural parameters of the transmission mechanism x;k = k " k (Gali (2015)). More generally, and stepping beyond the welfare-theoretic justi…cation for (1), one can interpret

x as the weight attached by the central bank to deviations of output from its e¢ cient level (relative to price stability) in its own loss function, which does not necessarily have to coincide with the household's. Because structural parameters may change for a new regime k, long run equilibrium values may also change in the estimation.

Under regime j, policy plans solve the following problem (omitting subscript j for the central bank preferences x , transmission mechanism parameter , the auto-correlation of the cost-push shock and its variance of its disturbances " t ):

V jk (u 0 ) = E 0 t=+1 X t=0 ( q) t 1 2 2 t + x x 2 t + (1 q) V jk (u t ) (4)
s.t. t = x t + qE t t+1 + (1 q) E t k t+1 + u t (Lagrange multiplier t+1 ) u t = u t 1 + " t ,8t 2 N, u 0 given.
The utility the central bank obtains is next period objectives change is denoted V jk . Since when objectives change, the central bank loses its commitment, this value function depends on the policies of the alternative regime. In ‡ation expectations are an average between two terms. The …rst term, with weight q is the in ‡ation that would prevail under the current regime upon which there is commitment. The second term with weight 1 q is the in ‡ation that would be implemented under the alternative regime, which is taken as given by the current central bank. The key change is that the narrow range of values for the discount factor 2 [0:99; 1] for quarterly data (4% discount rate at most) is much wider for the "credibility weighted discount factor" of the policy maker:

q 2 ]0; 1].
In what follows, the notation will correspond to q, in order to simplify notations and because the theoretical model does not allow to identify separately and q for the estimation.

VAR Representation of Ramsey optimal policy

Di¤erentiating the Lagrangian with respect to the policy instrument (output gap x t ) and to the policy target (in ‡ation t ) yields the …rst order conditions (see appendix 2):

@L @ t = 0 : t + t+1 t = 0 @L @xt = 0 : x x t t+1 = 0 ) x t = x t 1 x t x t = x t+1 = x ( t t )
that must hold for t = 1; 2; ::: The central bank's Euler equation ( @L @ t = 0) links recursively the future or current value of central bank's policy instrument x t to its current or past value x t 1 , because of the central bank's relative cost of changing her policy instrument is strictly positive x > 0. This non-stationary Euler equation adds an unstable eigenvalue in the central bank's Hamiltonian system including three laws of motion of one forward variable (in ‡ation t ) and of two predetermined variables (u t ; x t ) or (u t ; t ).

The natural boundary condition 0 = 0 minimizes the loss function with respect to in ‡ation at the initial date:

0 = 0 ) x 1 = x 0 = 0 and 0 = x x 0 6 = 0
It predetermines the policy instrument which allows to anchor the forward-looking policy target (in ‡ation). The in ‡ation Euler equation corresponding to period 0 is not an e¤ective constraint for the central bank choosing its optimal plan in period 0. The former commitment to the value of the policy instrument of the previous period x 1 is not an e¤ective constraint. The policy instrument is predetermined at the value zero x 1 = 0 at the period preceding the commitment. Combining the two …rst order conditions to eliminate the Lagrange multipliers yields the optimal initial anchor of forward in ‡ation 0 on the predetermined policy instrument x 0 .

Ljungqvist and Sargent (2012, chapter 19) seek the stationary equilibrium process using the augmented discounted linear quadratic regulator solution of the Hamiltonian system [START_REF] Anderson | Mechanics of Forming and Estimating Dynamic Linear Economies[END_REF]) as an intermediate step (Chatelain and Ralf (2017) algorithm). The policy instrument is exactly correlated with private sectors variables:

x t = F t + F u u t : (5)
We estimate three observationally equivalent estimations of Ramsey optimal policy structural VAR of the two observable variables (in ‡ation and the policy instrument), using feasible generalized non-linear least squares for a system of equations (see appendix 2).

The …rst estimation estimates the two stable eigenvalues: the exogenous autocorrelation of cost-push shock , the reduced form eigenvalue and the reduced from policy rule parameter with respect to in ‡ation F :

t+1 x t+1 = (1 ) 1
F ( 1) F + t x t + t+1 1 F " t+1 (6)
The auto-correlation coe¢ cient is into the matrix of the VAR of in ‡ation and the policy instrument. It does not appear into the disturbances, which are now white noise, a necessary condition for estimating a VAR. We assume an i.i.d. measurement error t+1 in order to avoid the stochastic singularity of the perfect collinearity of the in ‡ation equation predicted by the Ramsey optimal policy theory (appendix 2). A high R 2 for the in ‡ation equation would signal that the variance of this measurement error is su¢ ciently small to remain close to the theoretical model.

t+1 x t+1 = 1 F (1 ) 1 F 1 F 1 F 1 F + 1 F 1 F ! t x t + t+1 1 F " t+1 (7)
The second estimation estimates the two stable eigenvalues: the exogenous , the reduced form eigenvalue and the ratio of two structural parameters x = 1 " where " is the household's elasticity of substitution between each di¤erentiated goods, if the central bank maximizes the representative household welfare.

t+1 x t+1 = (1 ) (1 ) x x + t+1 x t+1 + t+1 1 F " t+1 (8)
The third estimation substitutes the reduced form stable eigenvalue by its nonlinear function of the three structural parameters ; x ; in the above VAR of the second estimation.

= 1 F = 1 2 1 + 1 + 2 x s 1 4 1 + 1 2 x 2 1 q = (9)
This requires an identi…cation restriction that we set on the discount factor . We estimate separately three structural parameters: the auto-correlation of the cost-push shock , the preference of the Fed x , and the slope of the new-Keynesian Phillips curve . Only three structural parameters can be identi…ed: , , F or , , x or , ( q) ; ( q) for a given value of the discount factor q (see appendix 2):

( ) = 1 q F ) x ( ) = 1 1 F ( q) : (10)
If initial values of in ‡ation and of the policy instrument (in deviation from their equilibrium values) were perfectly measured at the date of commitment, the ratio x would be over-identi…ed by the optimal initial anchor of forward in ‡ation on the predetermined policy instrument equation:

x = 0 x 0 : (11)
The semi-reduced form cost-push shock rule parameter F u requires an identi…cation restriction, for example, setting a value for (see appendix 2):

F u ( ) = 1 1 F < 0: (12)
The standard error u of cost-push shock is computed using the standard error of residuals ";x of the output gap rule equation in the VAR [START_REF] Adjemian | Evaluation de la politique monétaire dans un modèle DSGE pour la zone euro[END_REF]. It requires an identi…cation restriction, because it depends on F u :

u ( ) = ";x F u ( ) = 1 F ";x : (13)
The standard error of the measurement of the in ‡ation equation (which is theoretically predicted to be zero) and its covariance with the cost push shock x = F u xu are also available.

One identifying equation is missing in order to identify the remaining four structural parameters ( x ; ; ; u ) and the negative feedback rule parameter F u . We set an iden-ti…cation restriction on the credibility weighted discount factor to a given values in the estimations.

For each regime, we estimate the slope of the Phillips curve j , the elasticity of substitution between each di¤erentiated goods ( 1 " j = x;j j ), the Fed's preferences x;j , the auto-correlation of the cost-push shock j and its variance 2 u;j . We calibrate the two parameters q. The measure of decreasing returns to scale of labor in the production functions of the …rms and the proportion of …rms who do not reset their price each period face an under-identi…cation problem detailed in the next section.

In…nite Horizon Zero-Credibility Policy

With quasi-commitment, the probability of not reneging commitment could be in…nitely small (near-zero credibility), but it remains strictly positive: for example, q = 10 7 > 0 (Roberts (1987)) with q 2 ]0; 1]. Before Roberts (1987) stochastic replanning article, [START_REF] Oudiz | International Policy Coordination in Dynamic Macroeconomic Models[END_REF] invented an in…nite horizon zero-credibility policy, which holds when the policy maker re-optimizes with certainty for all future periods. This forever complete lack of credibility corresponds to q = 0. This zero-credibility policy is mentioned as "discretion" by [START_REF] Gali | Monetary Policy, In ‡ation, and the Business Cycle[END_REF] with a very minor change with respect to [START_REF] Oudiz | International Policy Coordination in Dynamic Macroeconomic Models[END_REF] detailed in the appendix. Testing Roberts (1987) against [START_REF] Oudiz | International Policy Coordination in Dynamic Macroeconomic Models[END_REF] amounts to test the assumption of a non-zero probability of not replanning and reneging commitment q 2 ]0; 1] against a zero probability of not reneging commitment q = 0 for ever, so that [START_REF] Oudiz | International Policy Coordination in Dynamic Macroeconomic Models[END_REF] equilibrium is unlikely.

The central bank minimizes its loss function subject to the new-Keynesian Phillips curve and such that private sector and the central bank policy instrument reacts only to the contemporary predetermined variable u t at all periods t with a perfect correlation. These zero-credibility rules are determined by time-invariant rule parameters N ZC and F u;ZC (indexed by ZC) which are optimally chosen for all periods:

t = N ZC u t and x t = F u;ZC u t ) x t = F ;ZC t with F ;ZC = F u;ZC N ZC (14)
The subscript TC is for zero-credibility policy. The central bank zero-credibility rule responds only to in ‡ation with a perfect correlation. Substituting the private sector's in ‡ation rule and the policy rule in the loss function:

max f ;xtg 1 2 E 0 +1 X t=0 t 2 t + x x 2 t = max fFu;ZC;NZCg 1 2 N 2 ZC + x F 2 u;ZC u 2 0 1 2
The central bank …rst order condition is:

0 = N u;ZC @N u;ZC @F u;ZC + x F u;ZC F ;ZC = F u;ZC N ZC = 1 x @N u;ZC @F u;ZC
Substituting the private sector's in ‡ation rule and the policy rule in the in ‡ation law of motion leads to the following relation between N ZC on date t, N ZC;t+1 and F u;ZC :

t = E t [ t+1 ] + x t + u t ) N ZC u t = N ZC;t+1 u t + F u;ZC u t + u t N ZC = N ZC;t+1 + F u;ZC + 1
The central bank foresees that N ZC;t+1 = N ZC in its optimization (Oudiz and Sachs' (1985), appendix):

N ZC = F u;ZC + 1 1 = F ;ZC N ZC + 1 1 ) @N u;ZC @F u;ZC = 1
The endogenous rule parameters are increasing function of the central bank cost of changing the policy instrument x . They are bounded by limit values of x 2 ]0; +1[:

0 < t;ZC u t = N ZC ( x ) + = x (1 )
x (1

) 2 + 2 < N = 1 1 1 < x t;ZC u t = F u;ZC ( x ) + = x (1 ) 2 + 2 < 0 1 < x t;ZC t;ZC = F ;ZC ( x ) + = F u;ZC N ZC = x (1 ) < x < 0
The policy instrument (the output gap) is exactly negatively correlated (F ;ZC < 0) with the policy target, in ‡ation.

Speci…cation tests of Ramsey versus Zero-Credibility Policy 3.1 Sign restrictions of reduced form rule parameters

Oudiz and Sachs (1985) equilibrium implies a bifurcation (a sharp qualitative change) of the economy dynamic system with respect to the later paper of Roberts (1987) stochastic replanning equilibrium. Proposition 1. Opposite sign restrictions. The in ‡ation policy rule parameter F is positive and bounded using Roberts (1987) stochastic replanning equilibrium. The in ‡ation policy rule F ;ZC is negative and unbounded using [START_REF] Oudiz | International Policy Coordination in Dynamic Macroeconomic Models[END_REF] zerocredibility equilibrium. There is a saddle-node bifurcation on the in ‡ation eigenvalue 0 < < 1 < ZC when shifting from negative-feedback of Ramsey optimal policy under quasi-commitment equilibrium (eigenvalue ) to the positive-feedback mechanism of in…nite horizon zero-credibility equilibrium (eigenvalue ZC ).

Proof: The reduced form in ‡ation rule parameters F ;ZC of zero-credibility policy (respectively F of Ramsey optimal policy) is an a¢ ne negative function of the in ‡ation eigenvalue ZC (respectively ). The in ‡ation rule parameters satis…es the following inequalities:

F ;ZC = 1 1 x < 0 < 1 < F = 1 < 1 (15)
The in ‡ation eigenvalues satis…es the following inequalities:

0 < = 1 2 1 + 1 + 2 x s 1 4 1 + 1 + 2 x 2 1 < 1 ( 16
)
1 1 < ZC = 1 F ;ZC = 1 + 1 1 1 2 x ( 17
)
The policy instrument (or the Lagrange multiplier on forward-looking in ‡ation) is optimally predetermined for Ramsey optimal policy whereas it is forward-looking with in…nite horizon zero-credibility policy. This implies an additional stable eigenvalue for Ramsey optimal policy with respect to zero-credibility policy, according to Blanchard and Kahn (1980) determinacy condition. QED.

Figures 1 and2: In ‡ation rule parameter and in ‡ation eigenvalues functions of x for Ramsey optimal policy (solid line) and zero-credibility policy (dash line) for = 0:995, = 0:340; = 0:99.

Cost of changing policy instrument Eigenvalue

The estimated parameters = 0:995, = 0:340 for a given = 0:99 of the Ramsey optimal policy model are found during Volcker-Greenspan's Fed starting 1979q3-2006q2 in the next section. Figure 1 plots the positive-feedback in ‡ation rule parameter F ;ZC < 0 of zero-credibility policy (dashed line) and respectively the negative-feedback in ‡ation rule parameter F > 0 of Ramsey optimal policy (continuous line) as non-linear increasing (respectively decreasing) function of the relative cost of changing the policy instrument

x . The estimated parameters correspond to the point: x = 4:552 and F = 0:447. Figure 2 plots the related eigenvalue ZC of zero-credibility policy (and respectively the eigenvalue of Ramsey optimal policy) as non-linear decreasing (respectively increasing) function of the relative cost of changing the policy instrument x . The estimated parameters correspond to the point x = 4:552 and = 0:856.

For an in…nite cost of changing the policy instrument ( x ! +1, the Fed has maximal inertia): the negative-feedback rule parameter F decreases to its lowest value (zero). The stable in ‡ation eigenvalue tends to one for Ramsey optimal policy. For zero-credibility policy, the positive-feedback rule parameter F increases up to zero (with negative values). The unstable in ‡ation eigenvalue ZC decreases towards to 1= > 1.

For a near-zero cost of changing the policy instrument ( x ! 0, the Fed is an in ‡ation nutter), the negative-feedback rule parameter F increases to its largest bounded positive value. The stable in ‡ation eigenvalue tends to zero for Ramsey optimal policy. For zero-credibility policy, the positive-feedback rule parameter decreases to minus in…nity (with negative values). The unstable in ‡ation eigenvalue ZC tends to plus plus in…nity.

Speci…cation tests of zero-credibility policy

Time-consistent policy is described by a permanent anchor of in ‡ation on the output gap and by two AR(1) processes of in ‡ation and of the output gap. Variables, such as in ‡ation ( t + ) are not computed as deviations of equilibrium (already denoted t ). Estimates of equilibrium values ( ; x ) are then sample mean values found in the estimates of intercepts. The reduced form zero-credibility policy policy rule to be tested (which corresponds to a permanent anchor of in ‡ation on the output gap) allows to estimate the reduced form zero-credibility policy rule parameter F ;ZC :

x t + x = F ;ZC ( t + ) + (x F ;ZC ) + " x ;t with (18)
" x ;t = 0 for all dates, R 2 = 1 and F ;ZC < 0 We assume i.i.d. measurement errors t for the in ‡ation equation of Ramsey optimal policy. By the same token, for the in…nite horizon zero-credibility policy, we also assume an i.i.d. measurement error " x ;t , for the policy rule equation, in order to avoid the stochastic singularity of the perfect negative correlation predicted by the theory. A high R 2 for the in ‡ation equation would signal this measurement error to be su¢ ciently small and close to the theoretical model. For example, one can perform a one-sided test of a composite null hypothesis of a simple correlation very close to minus one: H 0;ZC;SS : r x < 0:99. A …rst speci…cation test of the negative sign restriction of zero-credibility policy is more informative.

Negative sign restriction (NSR): H 0;ZC;N S : r x < 0

A positive sign of the zero-credibility rule suggests Ramsey optimal policy as an alternative, with an opposite sign on the policy rule parameter F . The simple correlation between the output gap and in ‡ation provides another estimate of the zero-credibility policy rule parameter F ;ZC = r x ";x = "; < 0. It is equal to the one found using the ratio of standard errors of residuals of the AR(1) estimations for in ‡ation and for the output gap : F ;ZC = ";x = "; only if the following condition is satis…ed:

r x = 1 (stochastic singularity).
The rejection of stochastic singularity is expected to be stronger in the zero-credibility policy rule with a lower R 2 (because it does not include a lagged dependent variable) than the one of the Ramsey in ‡ation equation (which includes lagged in ‡ation). Hence, a second speci…cation test of zero-credibility policy versus Ramsey optimal policy is a Durbin (1970) test of the auto-correlation of residuals of these measurement errors of the zerocredibility positive-feedback policy rule. The measurement errors of the zero-credibility policy rule should not be auto-correlated according to the theory of zero-credibility policy: White noise disturbances: H 0;ZC;Durbin : ";x = 0 for " x ;t = ";x " x ;t + t with t i.i.d.

(20) If measurement errors are correlated, this alternative hypothesis suggests that at least one lagged policy instrument is missing in the regression of the policy rule. This is exactly a reduced form of the Ramsey optimal policy rule which also depends on in ‡ation. Finally, we can perform another test. In the zero-credibility feedback rule, the policy instrument is a linear function of the policy target. The two AR(1) process for in ‡ation and the output gap have common auto-correlation parameters:

t + = ( t 1 + ) + (1 ) + N ZC " u;t with " u;t i.i.d. ( 21
) x t + x = (x t 1 + x ) + (1 ) x + F ;ZC N ZC " u;t with " u;t i.i.d. ( 22
)
We can test the hypothesis of a common auto-correlation parameter for the policy target (in ‡ation) and for the policy instrument (output gap):

Common auto-correlation (CA):

H 0;ZC;CA : = x ( 23
)
Proposition 2: For zero-credibility policy, only the auto-correlation coe¢ cient of the cost-push shock can be identi…ed. But the four other structural parameters are not identi…ed: the Fed's preferences parameter x , the slope of the new-Keynesian Phillips curve , the discount factor and the variance of the cost-push shock 2 ";u . If the Fed's preferences is a welfare loss function, where Fed's preferences parameter is endogenous ( x = " ), the representative household's elasticity of substitution between each di¤erentiated goods " is not identi…ed.

Proof. If common auto-correlation hypothesis is not rejected, the AR(1) estimates of the policy target and of the policy instrument identify the auto-correlation parameter of the non-observable cost-push shock: . The ratio of the standard errors of residuals of each AR(1) estimations of in ‡ation and output gap provides another estimate of the zero-credibility reduced form rule parameter F ;ZC , (if r x = 1), if the hypothesis of a negative sign is not rejected:

F ;ZC = ";x = "; (24)
The variance 2 "; of perturbations of the in ‡ation AR(1) process is:

2

"; = N 2 ZC 2 ";u ) N 2 ZC = 2 "; 2 ";u (25)
The cross equations covariance "; x between the residuals of both AR(1) process of in ‡ation and of the output gap does not allow to identify either the private sector reduced form parameter N ZC anchoring in ‡ation on the cost-push shock or the variance of the cost-push shock 2 ";u . The simple correlation between the two residuals is predicted to be exactly negatively correlated (r "; x = 1):

"; x = ";x "; 2 "; 2 ";u 2 ";u = ";x "; < 0: (26)
It is not possible to identify at least one of these four remaining structural parameters separately, because the identi…ed parameter F ;ZC does not depend only on one of these four structural parameters:

F ;ZC = 1 1 x < 0: (27)
Three identifying equations are missing in the case of zero-credibility policy. QED. Finally, the tests of reduced form parameters of bivariate VAR(1) of zero-credibility policy versus Ramsey optimal policy are not feasible. The exact multicollinearity (exact correlation) between regressors (current output gap and in ‡ation) imply a bivariate VAR(1) with in…nite coe¢ cients with denominator including the term 1 r 2

x equal to zero:

x t+1 t+1 = +1 1 1 +1 x t t + N ZC F ;ZC N ZC " t (28)
The zero-credibility policy equilibrium predicts that out-of-equilibrium behavior corresponds to a non-stationary bivariate VAR including one unstable eigenvalue ZC and one stable eigenvalue , which cannot be estimated. By contrast, the stationary structural VAR(1) of output gap and in ‡ation with Ramsey optimal policy allows to identify a larger number of structural parameters.

x t+1 t+1 = a b c d x t t + F u;R 0 " t
Two additional reduced form parameters (b; c) are available, because the stable subspace of the VAR process is of dimension two with Ramsey optimal policy instead of dimension one with zero-credibility policy.

Tests with Output Gap as Policy Instrument

Speci…cation tests

The annualized quarter-on-quarter rate of in ‡ation and the congressional budget o¢ ce (CBO) measure of the output gap are taken from Mavroeidis' (2010) online appendix (detailed information at the end of this paper's appendix). The pre-Volcker sample covers the period 1960q1 to 1979q2 and the Volcker-Greenspan sample runs until 2006q2. The period of Paul Volcker's tenure is 1979q3 to 1987q2. The period of Alan Greenspan's tenure is 1987q3 to 2006q1.

According to Debortoli and Nunes (2015), a structural break corresponds to an new initial anchor of forward in ‡ation on the output gap for Ramsey optimal policy with …nite horizon. Clarida, [START_REF] Clarida | Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory[END_REF] and [START_REF] Mavroeidis | Monetary Policy Rules and Macroeconomic Stability: some new Evidence[END_REF] The pre-tests of the null hypothesis of a quasi perfect negative correlation H 0 : r x < 0:99 between observed in ‡ation and observed output gap are rejected. The test uses Fisher's Z transformation using the procedure corr with the software SAS. The threshold of the composite null hypothesis 0:99 is far away from the 95% single tail con…dence interval, where the lowest 95% con…dence limit reported in table 3 is at most equal to 0:53 for the period beginning from 1982q1 (…gure 4). The opposite null hypothesis H 0 : r (x t ; t ) = 0 is not rejected before 1979q3. zero-credibility policy predicts a perfect correlation for the anchor of in ‡ation expectations with the output gap. If the zerocredibility policy equilibrium occurred before 1979q3, we do not reject the null hypothesis H 0 : r (E t 1 ( t ) ; t ) = 0 that the rational expectations of in ‡ation are orthogonal to observed in ‡ation.

The pre-tests of the null hypothesis of the auto-correlation of residuals H 0 : ";x = 0 are strongly rejected, with a point estimate at least equal to 0:89 (…gure 5). These tests gives a hint of model misspeci…cation. They suggest an omitted lagged policy instrument in the policy rule. When it is included in Ramsey optimal policy rule, the R 2 increases from 16% (table 3, last line) to 93% (table 6, last line) beginning in 1982q1. Table 4 investigates the auto-correlation and unit roots of in ‡ation and output gap. The output gap and in ‡ation are highly auto-correlated (respectively 0.93 and 0.86), except when in ‡ation excludes the 1981 disin ‡ation for the period after 1981q4. For the period 1982q1 to 2006q2, the in ‡ation auto-correlation coe¢ cient falls in the 95% con…dence interval 0:6 0:14 and it is statistically di¤erent from the output gap autocorrelation coe¢ cient in the 95% con…dence interval 0:95 0:06 (…gures 4 and 5). As the zero-credibility policy equilibrium predicts that the auto-correlation of the output gap and of in ‡ation should be the same, this is an additional test against zero-credibility policy, which holds for the period 1982q1 to 2006q2.

There is a negative auto-correlation of residuals " for in ‡ation and a (statistically signi…cant at the 5% level) positive auto-correlation of residuals for the output gap. The column DF reports the p-value of the Dickey-Fuller test of unit root with one lag without trend. The column PP reports the p-value of the Phillips-Perron test of unit root, which takes into account auto-correlation, with one lag without trend. The null hypothesis of a unit root is rejected for in ‡ation after 1979q2 and after 1981q4.

Tests of Ramsey Optimal Policy

Table 5 presents estimates of structural parameters, Table 5 report estimates of three structural VAR estimations for ( ; R ; F ;R ), ; R ; x and ( ; ( ) ; x ( )) with two given values for the discount factor = 1 or = 0:99 for the Volcker-Greenspan period. With these three estimations, delta method is not necessary to compute the standard errors of parameters in each case. Maximum likelihood did not converge for pre-Volcker period (unconstrained VAR corresponds to complex conjugate eigenvalues, which are excluded because of exogenous real auto-correlation of cost-push shock). Post 1982q1 estimations converged to unlikely estimates.

Table 5: Ramsey optimal policy structural parameters The cost-push shock faces is extremely persistent, close to a unit root, with estimate close to one. The ratio x is statistically signi…cant. If the Fed's preferences are identical to (welfare) household's preferences), then x = 1 " and b " = 0:07. Table 6 compares the reduced form parameters of the VAR of Ramsey optimal policy with parameters of an unconstrained VAR. The in ‡ation equation of the VAR are the same up to the third decimal of all statistics. For the output gap equation, Ramsey optimal policy slightly over-estimates the persistence of the output gap: its VAR autocorrelation parameter shifts from 0:93 to 1. For the output gap rule, the auto-correlation of residuals fell from 0:92 with zero-credibility policy to 0:29 with Ramsey optimal policy, but it remains statistically signi…cant. As well, the in ‡ation equation has a statistically signi…cant auto-correlation ( 0:25).

Dates F x = 1 " ( ) x ( ) F u;R ( ) u ( ) 79q3
The reduced form Ramsey optimal policy rule of the structural VAR is observationally equivalent to the LQR …rst step representation including the non-observable cost-push shock, when taking into account the other equations of the Hamiltonian system: Ramsey : x t = 0:995x t 1 0:064 t 1 2:861" u;t or x t = 0:447 t 2:861u t for = 0:99 (29) The policy instrument responds to two variables for Ramsey stable subspace of dimension two. The policy instrument responds to one variable in the zero-credibility policy stable subspace of dimension one. The reduced form policy rule for zero-credibility policy is:

Time-consistent: x t = 0:22 t (30)
5 Tests with Federal Funds Rate Smoothing and Working Capital Channel

Speci…cation tests

Assuming the polar case where all labor cost is …nanced by working capital instead than no labor cost at all, the federal funds rate is used here as a policy instrument. Its variance enters into the period loss function 1:00 14:51

Standard deviations are in parentheses below the mean. The means of in ‡ation are lower after Volcker than before Volcker. Excluding the period 1979q3 to 1981q4, in particular the sharp disin ‡ation which occurred during 1981 (…gure 3), the standard error of in ‡ation decreases by half from 2.03 to 1.08. The di¤erence of means between the policy interest rate and in ‡ation increased after Volcker.

Table 3B: Pre-test of zero-credibility policy rule dates obs The tests of the null hypothesis of a quasi perfect negative correlation H 0 : r i < 0:99 between in ‡ation and federal funds rate have been replaced by the usual tests of the null hypothesis: H 0 : r i = 0, because r i > 0. The negative sign of the zero-credibility policy rule is rejected, with large t statistics. The Durbin and Breusch-Godfrey tests strongly reject the lack of serial correlation (for one or two lags) with p-value below 10 4 . Tests of the null hypothesis of the …rst order auto-correlation of residuals H 0 : ";i = 0 are rejected, with a point estimate at least equal to 0:60 and at most 0:80. The in…nite horizon zero-credibility positive-feedback mechanism does not …t the data when the cost of capital is taken into account into the monetary transmission mechanism. Finally, the Taylor principle (an in ‡ation coe¢ cient F larger than one) is not satis…ed before Volcker and satis…ed after Volcker.

r i t r i = 0 : p R 2 i F ;D c ";i < 79q3
Table 4B investigates the auto-correlation of in ‡ation and federal funds rate. In ‡ation and Federal funds rate are highly auto-correlated (respectively 0.93 and 0.86), except for the period 1982q1 to 2006q2, the in ‡ation auto-correlation coe¢ cient falls in the 95% con…dence interval 0:6 0:14 and it is statistically di¤erent from the federal funds rate auto-correlation coe¢ cient in the 95% con…dence interval 0:95 0:02. In…nite horizon zero-credibility policy predict that the auto-correlation should be the same, which is not the case after 1982q1.

Table 4B. Auto-correlation of in ‡ation and federal funds rate dates obs. var. There is a negative auto-correlation of residuals " for in ‡ation and a positive autocorrelation of residuals for federal funds rate. The column DF reports the p-value of the Dickey-Fuller test of unit root with one lag without trend. The column PP reports the p-value of the Phillips-Perron test of unit root, which takes into account auto-correlation, with one lag without trend. The null hypothesis of a unit root is rejected at the 5% threshold for in ‡ation after 1979q2 and after 1981q4. It is rejected for federal funds rate at the 10% level after 1981q4 only for the Phillips-Perron test.

Tests of Ramsey optimal policy

Table 5B reports estimates using four structural VAR estimations for ( ; C ; F ;C ), for ; C ; i i and for ( ; ( ) ; i ( )) with the identi…cation restrictions for the credibility adjusted discount factor. We check that these three estimations are observationally equivalent. In particular, each estimate satisfy the theoretical relations with the others implied by Ramsey optimal policy. When the federal funds rate is the policy instrument, the estimations only converged to plausible values for the pre-Volcker period, before 1979q3. The implicit rule parameter on the cost-push shock is computed using

F u ( ) = 1 1
F and the variance of the cost push shock is u ( ) = ";i Fu( ) with the root mean square error of the residuals of the policy instrument equation equal to ";i = 0:876.

Table 5B: Ramsey optimal policy structural parameters Dates 1: 676 0:522

F i i ( ) i ( ) F u;C ( ) u ( ) < 79q3
Estimates of structural parameters are plausible values. The Fed's reduced form rule parameter F is signi…cantly di¤erent from zero at the 5%. To reach the statistical signi…cance at the 5% level of both i ( ) and i ( ) and not only one of the two parameters is obtained for identi…cation restrictions of the credibility adjusted discount factor 2 [0:92; 0:94]. This corresponds to a Fed's probability to commit in this range: q 2 [0:93; 0:95] for an identi…cation restriction of the discount factor 0:99. For > 0:94, only the Fed's preference i ( ) is statistically signi…cant. For < 0:92, only the slope of the new-Keynesian Phillips curve for working capital i ( ) is statistically signi…cant. When the identi…cation restriction on increases, both estimates i ( ) and i ( ) increase, but not at the same pace.

Table 6B shows that the reduced form parameters of the structural VAR(1) (rows S) are identical to the unconstrained VAR(1) estimates (rows U) up to the third decimal. 0:869 0:01 0:947

(1) For the in ‡ation equation, there is Granger causality from lagged federal funds rate to in ‡ation. The exogenous cost-push shock auto-correlation is close to the autocorrelation of in ‡ation in the VAR. The residuals are not auto-correlated controlling for endogenous lagged in ‡ation and federal funds according to Durbin's test. The autocorrelation estimate is 0:12, It not statistically di¤erent from zero (Durbin's test: t = 0:77, p = 0:44).

(2) For the federal funds rate rule equation, the auto-correlation 0:976 is relatively close to a unit root. The endogenous stable eigenvalue corresponds to the auto-correlation of the federal funds rate.

One can compare our modest results with two recent papers using implicit functions of the parameters within a reduced form VAR including three US time-series (in ‡ation, output gap and Federal funds rate). Givens (2012) …nds out that Volcker-Greenspan period corresponds to zero-credibility policy whereas [START_REF] Matthes | Figuring Out the Fed-Beliefs about Policymakers and Gains from Transparency[END_REF] …nds it corresponds to Ramsey optimal policy. The ratio of the estimated weight on output in the loss function in commitment versus discretion in Givens ( 2012 For the transmission mechanism, Givens (2012) estimate of the intertemporal elasticity of substitution is 0:0089 (estimated standard error 0:0035, 95% con…dence interval [0:0019; 0:0159]) for Ramsey optimal policy and 0:0002 for zero-credibility policy, with an estimated standard error 0:0001 so that zero belong to the 95% con…dence interval. [START_REF] Matthes | Figuring Out the Fed-Beliefs about Policymakers and Gains from Transparency[END_REF] estimate is 70 times larger 1=1:61 = 0:62 for both policies. The estimated standard error of this parameter is not disclosed. Givens (2012) estimate of the slope of the new-Keynesian Phillips curve is b = 0:0045, with estimated standard error 0:0016 and 95% con…dence interval [0:0019; 0:0159]), for Ramsey optimal policy and 0:0047 for discretion with estimated standard error 0:009 so that zero belongs to the 95% con…dence interval ("Estimates of are small but within the range typical of the literature." p.1043). In the discretion regime, both transmission mechanism parameters for the Federal funds rate not statistically different from zero. This result supports the hypothesis of monetary policy ine¤ectiveness during 1982-2006. [START_REF] Matthes | Figuring Out the Fed-Beliefs about Policymakers and Gains from Transparency[END_REF] estimate is 75 times larger: b = 0:7=(1 + 0:99) = 0:35: "The estimate found here, while being at the other end, is not unheard of ". The estimated standard error of this parameter is not disclosed.

Givens (2012) assumes one lag of in ‡ation, two lags of output and one lag for Federal funds rate. [START_REF] Matthes | Figuring Out the Fed-Beliefs about Policymakers and Gains from Transparency[END_REF] assumes one lag of in ‡ation and two exogenous auto-regressive forcing variables. The more lags of observable exogenous variables are added in the speci…cation of the new-Keynesian Phillips curve, the more likely the weak identi…cation of some parameters occurs [START_REF] Mavroeidis | Identi…cation issues in forward-looking models estimated by GMM, with an application to the Phillips curve[END_REF]

Conclusion

Using closed form solutions of Ramsey optimal policy and zero-credibility policy, we take exactly into account the identi…cation restrictions related to the dimension of the stable subspace of each policy. Ramsey optimal policy with quasi-commitment has a comparative advantage with respect to in…nite horizon zero-credibility policy for modelling persistence with fewer parameters. Further work may extend this approach to other transmission mechanisms than the new-Keynesian Phillips curve with labor cost channel or with working capital channel.

In what follows, refers to q to simplify notations. The solution of the Hamiltonian system are based on the demonstrations of the augmented discounted linear quadratic regulator in [START_REF] Anderson | Mechanics of Forming and Estimating Dynamic Linear Economies[END_REF], following the steps in Chatelain and Ralf (2017c): The characteristic polynomial of this upper square matrix is:

2 1 + 1 + 2 x + 1 = 0
The Hamiltonian matrix has two stable roots and ( is denoted in Gali (2015)) and one unstable root 1 . The determinant of the matrix is

1 = 1 . Then < q 1 <
1 . The trace of the matrix is

x t = x t+1 = x ( t t )
x t = F t + F u u t + F z z t = x ( t t ) =

x (P t + P u u t + P z z t t ) ) F =

x (P 1), F u =

x P u and F z =

x P z Proposition 3: From LQR to [START_REF] Gali | Monetary Policy, In ‡ation, and the Business Cycle[END_REF] vector basis (replace policy target by policy instrument).

One has:

1 F u = 1 A 1 = 1 1 1 1 1 = (1 ) 1 = (
u t+1 x t+1 = N 1 (A + BF) N u t x t + N 1 " t 0 u t t = N 1 u t x t t = 1 F x t A t
One has:

N 1 (A + BF) N = 0 (1 ) F A
Which is [START_REF] Gali | Monetary Policy, In ‡ation, and the Business Cycle[END_REF] representation of the solution:

x t = x t 1 + (1 ) F A u t 1 = x t 1 + x 1 u t 1

QED

Because the auto-correlation of the policy instrument x t and the auto-correlation of the cost-push shock are competing to explain the persistence of the policy instrument x t , this partial adjustment model with serially correlated shocks has a problem of identi…cation and multiple equilibria [START_REF] Griliches | Distributed lags: a survey[END_REF], [START_REF] Blinder | More on the speed of adjustment in inventory models[END_REF] Because the auto-correlation of the policy instrument x t and the auto-correlation of the cost-push shock are competing to explain the persistence of the policy instrument x t , this partial adjustment model with serially correlated shocks has a problem of identi…cation and multiple equilibria [START_REF] Griliches | Distributed lags: a survey[END_REF], [START_REF] Blinder | More on the speed of adjustment in inventory models[END_REF], [START_REF] Mcmanus | Multiple optima and asymptotic approximations in the partial adjustment model[END_REF], [START_REF] Fève | Monetary Policy Dynamics in the Euro Area[END_REF]). This VAR(1) can be written as:

x t = R x t 1 + t and t = t 1 + " ;t where t =

x R

(1 R ) u t . It is an AR(2) model of the policy instrument rule:

x t = R x t 1 + (x t 1 R x t 2 ) + " ;t x t = b 1 x t 1 + b 2 x t 2 + " ;t with b 1 = R + and b 2 = R :

The structural parameter and the semi-structural parameter R are functions of reduced form parameters b 1 and b 2 solutions of:

X 2 b 1 X b 2 = 0
which are given by:

  ): b D = b C = 0:0987=0:1351 = 0:7 and 10 times larger in Matthes (2015) b D = b C = 0:49=0:07 = 7.

x 1 A 1 :

 11 characteristic polynomial is equal to zero. The policy rule parameter with respect to in ‡ation depends on x :Hamiltonian system function of the stable eigenvalue (eliminating x Proposition Solution of Ricatti and Sylvester equation: Rule parameters P u and P z of the response of the Lagrange multiplier on in ‡ation to exogenous variables:

Table 2 :

 2 Summary statistics of in ‡ation and output gap

	consider the beginning

Table 4 :

 4 Auto-correlation of in ‡ation and output gap dates obs. var.

			r	R 2	c	"	"	DF	P P
	< 79q3 78	t	0:86 0:74 0:88 (0:06)	0:62

Table 6 :

 6 The new-Keynesian Phillips curve parameter is relatively large. The Fed's preference parameter x is relatively large (although not unheard of in previous estimations). for the period 1979-2006. In ‡ation and output gap structural (S) versus unconstrained (U) VAR dates obs. var. S/U

				t 1	x t 1	c	"	R 2
	79q3 108	t	S	0:85	0:009 0:428

  2 t + i i 2 t as an interest smoothing model. Figure 2 represents the time series of in ‡ation and federal funds rate. Table 2 presents summary statistics before and after the 1979q3 and 1982q1 structural breaks. Table 2B: Summary statistics of in ‡ation and federal funds rate

		dates obs. mean min max	after: obs. mean min max
	t	< 79q3 78	4:39 (2:71)	0:59 11:79	79q3 108 3:18 (2:03)	0:64 10:93
	i t < 79q3 78	5:47 (2:42)	1:68 12:09	79q3 108 6:56 (3:76)	1:00 17:78
	t	< 82q1 88	4:86 (2:90)	0:59 11:79	82q1 98	2:64 (1:08)	0:64 5:61
	i t < 82q1 88	6:84 (3:74)	1:68 17:78	79q3 98	5:76 (2:84)

Table 6B

 6B

	: In ‡ation and federal funds rate structural (S) versus uncon-
	strained (U) VAR
	dates obs. var. S/U	t 1	i t 1	c	"	R 2	R 2
	< 79q3 77	t	S	0:521 0:488	0:471 (0:340)	0:12 (0:16)	0:804 0:06 0:550 (0:092)
	< 79q3 77	t	U	0:52 (0:09)	0:49 (0:10)	(0:343) 0:471	(0:16) 0:12	0:804 0:06 0:550
	< 79q3 77	i t	S	0:025 0:976 0:316 (0:226)	0:55 (0:12)	0:869 0:01 0:947 (0:047)
	< 79q3 77	i t	U	0:025 (0:065)	0:976 (0:07)	0:317 (0:252)	0:55 (0:12)

  , Mavroeidis et al. (2014), Dees, Pesaran, Smith, Smith (2009), Canova and Sala (2009), Komunjer and Ng (2012) and Iskrev (2012)).

  , McManus et al. (1994), Fève, Matheron Poilly (2007)). Hence, we compute the representation of Ramsey optimal policy as a bivariate VAR of two observable variables (in ‡ation and the policy instrument).Proposition 4: From LQR to a bivariate VAR with observable variables:

	8 > > > > < > > > > :	8 > > > > <	t+1 u t+1 t x t x t+1 = = t+1	(1 0 F AF 0 1 i t = F t + AF u t ) A 1 t u t = M 1 (A + BF) M	t u t = M + t x t	0 " t u t t + M 1 0 " t
	,	> > > > :	t u t		=	1 1 A u t = 1 0 1 AF A t + 1 t x t AF x t = M 1	t x t
	Then:
	1		0		(1	) A	1	0		1
	F AF	0				F AF
	=	(	1) F		(1	+	) 1 F	=	x	(1	) (1 +	) x
	QED.
	8 NOT FOR PUBLICATION
	8.1 Appendix 3: Identi…cation issue for reduced form including
	a non-observable AR(1) shock.

The new-Keynesian Phillips curve can be written as a function of the Lagrange multiplier where > 0, 0 < < 1 and 0 < q < 1 (Debortoli and Nunes (2014, appendix A). We keep Gali (2015) chapter 5 t+1 notation of the Lagrange multiplier with one step ahead subscript: it corresponds to Debortoli and Nunes (2014) notation t . Our notation for the stable eigenvalue corresponds to Debortoli and Nunes (2014) notations " y = 1= ".

Demonstration: It uses the method of undetermined coe¢ cients of Anderson, Hansen, McGrattan and Sargent's (1996), section 5. The solution is the one that stabilizes the state-costate vector for any initialization of in ‡ation 0 and of the exogenous variables u 0 in a stable subspace of dimension two within a space of dimension three ( t ; t ; u t ) of the Hamiltonian system. We seek a characterization of the Lagrange multiplier t of the form:

To deduce the control law associated with vector (P ; P u ; P z ), we substitute it into the Hamiltonian system:

We write the last two equations in this system separately:

It follows that:

The …rst equation is such that:

The method of undetermined coe¢ cients implies for the …rst term:

For the second term:

(1

)

For the third term:

(1 )

1 QED Proposition 2: Optimal policy rule parameters formulas:

Demonstration:

The …rst order condition relates Lagrange multiplier to the policy instrument: 2 . If 6 = 0 and 6 = R , two sets of values for R and are observationally equivalent. The …rst solution is such that R > and the second solution is such that R < . The larger , the larger the identi…cation issue, because it increases the gap between a more inertial monetary policy with lower correlation of monetary policy shocks and a less inertial monetary policy, that we cannot distinguish. The ADLQR representation and Gali (2015) representation of the stationary solution of the VAR(1) of optimal policy are not useful to identify parameters, because they include the cost-push shock u t which is not observable.

The reduced form estimated variance provides another equation with a theoretical positive sign restriction Substituting the private sector's in ‡ation rule [START_REF] Canova | Back to square one: Identi…cation issues in DSGE models[END_REF] and policy rule [START_REF] Castelnuovo | Model uncertainty, optimal monetary policy and the preferences of the Fed[END_REF] in the in ‡ation law of motion (1) and comparing it with the forcing variable law of motion (2) leads to the following relation between N ZC on date t, N ZC;t+1 and F u;ZC :

A myopic central bank does not notice that N ZC;t+1 = N ZC [START_REF] Gali | Monetary Policy, In ‡ation, and the Business Cycle[END_REF]) in its optimization:

This …rst order condition of the central bank optimization is substituted into the new-Keynesian Phillips curve equation, where, only at this stage, players of the game discover that it is assumed N ZC;t+1 = N ZC;t = N ZC . Gali's (2015) solutions are:

In zero-credibility equilibrium [START_REF] Oudiz | International Policy Coordination in Dynamic Macroeconomic Models[END_REF]), the central bank does foresees that N ZC;t+1 = N ZC in its optimization, with the following solutions, that we consider for the remaining part of the paper:

In Oudiz and Sachs'(1985) general solution, this is the condition after substitutions of the private sector's rule (matrix N ZC ) and the policy maker's rule (matrix F u;ZC ) for both dates t and t + 1 into the law of motion of the private sector dynamics:

with general notations and equalities with Gali's (2015) transmission mechanism:

In [START_REF] Oudiz | International Policy Coordination in Dynamic Macroeconomic Models[END_REF], N ZC;t+1 = N ZC;t at all dates, whereas Gali (2015) assumes myopia (or N ZC;t+1 = 0) for the policy maker. This assumption changes the initial jump of in ‡ation, impulse response functions of in ‡ation and the output gap and welfare. It does not change the identi…cation problem of discretion raised in this paper, because the stable subspace of discretion have the same dimension (one) using the reference [START_REF] Oudiz | International Policy Coordination in Dynamic Macroeconomic Models[END_REF] discretion equilibrium or [START_REF] Gali | Monetary Policy, In ‡ation, and the Business Cycle[END_REF] and [START_REF] Clarida | The Science of Monetary Policy: a new Keynesian Perspective[END_REF] myopia assumption.

Appendix 5: Code for 3 estimations of Ramsey optimal policy

The …rst estimation estimates the two stable eigenvalues: the exogenous , the reduced form and the reduced from policy rule parameter with respect to in ‡ation F . The second estimation estimates the two stable eigenvalues: the exogenous , the reduced form and the ratio of two structural parameters x (the Fed's preference divided by the slope of the new-Keynesian Phillips curve). This estimate of x is an estimate of The third estimation requires an identi…cation restriction that we set on the discount factor ( = 1 or = 0:99). We estimate separately three structural parameters: the auto-correlation of the cost-push shock , the preference of the Fed x , and the slope of the new-Keynesian Phillips curve .