Franc ¸ois Charoy Jörn Franke
email: joern.franke@sap.com

Cédric Ulmer

F Charoy
email: charoy@loria.fr

Handling Conflicts in Autonomous Coordination of Distributed Collaborative Activities

Coordination between different organizations in dynamic situations, such as a disaster response, is challenging. Organizations are autonomous and coordinate the situation from their point of view. There is no central authority to coordinate all operations. To coordinate their actions, organizations need to exchange information on what they are doing. However, they cannot share everything with everybody due to privacy, regulatory or strategic reasons. Currently, only e-mail, telephone or fax are used to exchange information. This makes detecting and handling of conflicting views on the situation very difficult. We propose an approach for inter-organizational process management for these kinds of dynamic scenarios. It allows different organizations to share selected activities by replicating them in the different workspaces of the organizations. State changes of shared activities are propagated optimistically. We explain detecting and handling of two different types of conflicts that can occur in this setting. We provide an implementation and interviews to validate the concepts.

I. INTRODUCTION

According to Gartner and McKinsey the management of activities in flexible distributed processes becomes more and more important for many organizations [START_REF] Olding | Expand your bpm horizons by exploring unstructured processes[END_REF]. Even more important, we can say that not only the processes require flexibility, but also the coordination of activities by several autonomous organizations with shifting goals. In this paper, we consider processes that take place in the "real world" involving humans belonging to different organizations. We use the disaster response management domain as a critical example where coordination is important and where goals can change during an event. These processes cannot be fully described in a structured manner due to their dynamic nature. From an inter-organizational perspective we also argue that it is not possible to create a global shared process that is coordinated by one entity. Each organization coordinates its own activities based on its experience and governance rules. However, in order to coordinate with each other, these organizations need to share information about what they are doing and how they rely on activities of other organizations. Today, it is mostly done using communication tools that are not the best tools for coordination (e.g. email, phone or fax). It is difficult to establish an adequate situation overview and to model coordination using these tools. We explain in this paper how an activity management system can be designed to address this problem. We assume that there is no central coordination, but a network of organizations that need to synchronize their actions. The network structure is not known in advance. Organizations need to exchange information about what they plan to do, are doing or have done. They may also ask others to do something. This will require to share information about activities and to allow concurrent changes of this information. This may cause conflicts and we will show how they can be detected and handled. Our work is based on an approach presented in [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]. This paper is an extension taking into account the inter-organizational dimension, where contrary to a centralized solution, not everything can be shared amongst the different organizations (e.g. the police cannot share information related to crime investigations with the fire fighters). This has, for example, regulatory, privacy or strategic reasons. The main contribution of this paper is how conflicts caused by this can be detected and handled in this distributed setting.

In the next section, we describe a realistic use case in the field of disaster response management where distributed coordination is needed. It has been developed together with end users [START_REF] Döweling | Soknos -an interactive visual emergency management framework[END_REF]. We explain how the coordination, i.e. activities and their dependencies, is modeled (based on the framework proposed in [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]) in section three. We describe then how this model is leveraged on the inter-organizational level to coordinate activities of different organizations in section four. Particularly, the focus is resolving conflicts when sharing activities and integrating them in the processes of different organizations. We describe the architecture in section five and the implementation in section six. In section seven, we discuss end user feedback. Finally, we describe related work and give an outlook on future research.

II. USE CASE

To illustrate our work, we present here a use case that demonstrates the need for flexibility and coordination among autonomous organizations. It has been derived from a realistic disaster response use case developed in the SoKNOS project [START_REF] Döweling | Soknos -an interactive visual emergency management framework[END_REF] together with end users, such as fire fighter and police commanders. The organizations need to work together, but none of them is hierarchical superior to the other. They form an organizational response network. In the simplified version of the use case three organizations respond to a flood: the police, the fire fighters and the military. The military is responsible for protecting a chemistry plant from the flood. They fill sandbags, transport sandbags and build a dam to fulfill this objective. The fire fighters are building a dam to protect a residential area from the flood. They rely on the military to provide sandbags to them. The police has to evacuate the residential area in case of a flood. They have to determine people, warn the people, order shelter or transport people. Some of these actions depend on the outcomes or on the status of others. Police activities depend on the success of the dam construction. The fire fighters rely on the delivery of sandbags by the military. It is beneficial for each organization to have knowledge about what is actually happening and what may concern the other. But of course, each organization needs also to keep some of their actions private (i.e. everything cannot be shared) for privacy reasons or due to internal policies. The organizational network can be extended at any time. For example, different regions or states may provide additional command centers for supporting the coordination among different disaster sites. In this case, actions may also have an impact on each other (building dams at two different places for instance). During our research within the SoKNOS project and interactions with end users (e.g. workshops) we found out that current means, such as email, telephone or fax, cause some problems for coordination. For example, it is almost impossible to get an accurate overview of the status of all ongoing actions. Some organizations may think that something is happening while it has failed (an order to close an airport has been given and is assumed to be completed while it is not). It can be very difficult to detect these conflicting views using the traditional means. This leads to confusion about the current situation.

We argue that a process based approach to manage this coordination can address these challenges. In such an approach, we consider that the organizations are autonomous and coordinate the situation from their point of view. There is no central entity defining how to do the coordination in detail for all organizations.

III. A FRAMEWORK FOR COORDINATION OF ACTIVITIES

We describe in this section how activities and temporal dependencies are modeled by users to describe explicitly coordination (based on the framework explained in [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]). The cited work describes also how this model is verified and executed and we will reuse this functionality in the following sections.

Definition 1 An activity type at d = (S, st, se, f, G) represents the management lifecycle of an activity with S is a finite set of activity states, st ∈ S describes the start state of an activity type, se ∈ S describes the end state of an activity type (i.e. a state where no further transition is possible), st = se a start state is not an end state and f : S → S is a transition function defining the possible transitions from one state to another for one activity type. The specification of the activity type can be extended by governance rules G = {g 1 , .., g n }. They describe who can transit from one state to another, e.g. g x ⊆ f is the transition function of the role "x". Fig. 1 illustrates an example for such an activity type. The white circle describes the start state and the black circle describes an end state. Other states are "Plan", "Execute", "Idle", "Fail", "Cancel" and "Finish". We do not allow strongly connected components (i.e. cycles) in the activity type, because this causes confusion (cf. [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]). For example, it would be possible to go from state "Execute" to "Fail" and vice versa. This is difficult to display and understood by the user. Particularly, when the status is shared with other users.

Definition 2 An activity is defined as a i = (uid, name, cat, cs, P ) where uid is a unique identifier of the activity, name describes the activity, cat is the activity type of the activity, cs is the current state of the activity. On creation it must be the start state st of its activity type. P is the set of participants assigned by the creator of an activity to a governance role in cat.G. An activity can change its state in parallel to other activities without affecting them. However, a dependency can be established between activities, if it is perceived by the user as important. Any further data can be attached to the activity. Definition 3 A temporal dependency is defined as d i = (a s , s s , a d , s d , type) with a s is the source activity, s s is the state of the source activity, a d is the destination activity, s d is the state of the destination activity and type is the type of temporal dependency. We use Allen's proposed time interval relationships for describing different types of temporal dependencies [START_REF] Allen | Maintaining knowledge about temporal intervals[END_REF] (see Fig. 2 illustrating seven of them and omitting six inverse dependencies). The dependency changes its state ("Violate" or "Neutral") depending on the order of the state changes of the associated activities (cf. [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]). An example for a temporal dependency is the dependency "overlaps" between the states "Execute" of activity "A" and "B". This means that activity "A" has to enter state "Execute" and then later activity "B" can enter state "Execute". Activity "A" has to leave state "Execute" before activity "B" does. If this does not happen in this order then the dependency is violated. This can happen, because the situation requires this or people of other organizations are not aware of this dependency. In this case the user is made aware of this and can take appropriate actions, such as communicating with the stakeholders of the activities or by creating new ones.

We will provide in the subsequent sections examples demonstrating how this model is used on the interorganizational level to coordinate activities. We explain the nature of conflicts that may occur in this distributed setting, how they can be detected and managed.

IV. COORDINATION AND CONFLICT ON THE INTER-ORGANIZATIONAL LEVEL

As we explained in the use case section, organizations need to share information about their actions to coordinate. In this section, we will describe how they can coordinate by sharing activities. We consider that each organization maintains an activity workspace (AW) containing all its own activities and dependencies. A person from one organization can decide to share an activity with a person of another organization. The selection of organizations is based on an existing social network (e.g. the fire fighter commander knows the police commander) and is out of the scope of this paper. The person of the other organization can then decide to insert this activity in its AW. This preserves autonomy of both organizations. The shared activity is then replicated in the AWs of both organizations and can be managed like any other activity of the workspace. New dependencies can be created from and to this activity. The status of this activity can also be changed in both AWs and changes are propagated optimistically. Thus, both organizations will have a shared view on the current operations. Optimistic replication means that concurrent state changes can occur that may conflict or have different outcomes regarding dependencies violation. In the remainder of this section, we detail the general principles of the approach and our proposal to deal with conflicts.

A. Sharing of Activities

In our approach, participants model activities and dependencies on an activity workspace (AW). They can share some activities with other participants of another AW. They can establish dependencies between shared activities and their own activities. In Fig. 3 we describe an example for sharing of activities. In the first step (T1), the fire fighter commander shares the activity "Build Dam" with the police commander. In a second step (T2), the police commander has integrated the shared activity "Build Dam" in his AW and created a dependency from the shared activity to his activity "Warn People".

B. Updating States of Shared Activities

We describe in this section how state changes of shared activities are propagated to all AWs, where the activity is replicated. We propose to do this optimistically, i.e. we propagate the state change and detect as well as handle conflicts afterwards. This allows coordinating in an instant like with traditional means. A pessimistic approach would mean to lock the activity for a period of time in which no state changes can be entered by the user. This would limit unnecessarily the possible interaction with the system (cf. also [START_REF] Greenberg | Real time groupware as a distributed system: Concurrency control and its effect on the interface[END_REF]). A pessimistic approach can lead to inaction, because people have to wait until they can provide input or receive input to do action. This is contrary to what happens in disaster response management. Thus, a pessimistic approach is not possible in our use case. In Fig. 3, we illustrate in step three (T3) that the fire fighter commander changed the activity "Build Dam" to state "Execute". The AW of the fire fighter commander sends the update to the AW of the police commander, because the activity "Build Dam" has been shared with the police commander before.

We presented in [START_REF] Franke | Coordination and situational awareness system for inter-organizational disaster response[END_REF] a protocol for optimistically propagating state changes. The underlying assumption is that messages arrive eventually. The outcome of the protocol is that state changes are applied in any AW where the shared activity is replicated. Applying a state change in a model means detecting if dependencies are violated by it (cf. for a detailed explanation [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]). Since the protocol only provides optimistic replication, we need to detect and handle conflicts afterwards. We describe two important types of conflicts that we have identified in the next subsections.

In the next two subsections we describe detecting and handling of two different types of conflicts that can occur after optimistic propagation by this protocol.

C. Detecting and Handling Conflicts of one Shared Activity

The first type of conflict occurs when the state of one shared activity is changed into two different states concur- rently. For example, when considering the activity type in Fig. 1, a conflict can occur if one person sets an activity based on this activity type to state "Cancel" and the other one to state "Fail" concurrently. This type of conflict is illustrated in Fig. 4. The activity "Build Dam" is changed by the commander in the command center to state "Cancel" and by the commander in the field to state "Fail" in the third step (T3). If the protocol above is used, the conflict can be detected based on the activity type and the history of state changes of the activity. In the activity type, it is impossible to transit from state "Execute" to "Fail" and at the same time from "Execute" to "Cancel". Thus, there is a conflict. Definition 4 Conflicting state change history: Let σ y = (s 1 → s 2 , .., s n-1 → s n ) be the execution history with the state changes s 1 → s 2 , .., s n-1 → s n of activity y based on the activity type at. A conflict occurs when: ∃((s i → s j ) ∧ (s i → s k ) ∧ (s j = s k )), i = 1..n -1. This definition means that there is conflict in the history of state changes if there are two or more state changes originating from the same state (s i ) of the same activity.

As mentioned, we assume that all AWs sharing the activity have eventually the same elements in their execution history. We cannot go twice through the same state without causing conflicts in the history with the activity type. This is only possible when the activity type has cycles, which we excluded by definition.

However, if this conflict is handled manually by the user then it cannot be guaranteed that it will be resolved eventually. Thus, we propose an automated approach. Our approach is inspired from [START_REF] Lamport | Time, clocks, and the ordering of events in a distributed system[END_REF], but we adapted it to our requirements, where we do not have a central authority. It is based on the governance roles defined by the creator of an activity (see previous section) to resolve automatically the conflict. For example, the commander in the command center has shared the activity "Build Dam" with the commander in the field and both perform conflicting state changes. The commander in the command center changes the state to "Cancel" and the commander in the field to "Fail". Since the commander in the command center has the accountability for the activity, he is higher in the role hierarchy than the commander in the field who is responsible. both AWs is "Cancel" for the shared activity "Build Dam". We now present an algorithm to handle conflicting state changes of the same activity automatically. We assume that the governance roles of the activity type form a hierarchy (e.g. accountable > responsible > informed). If the roles are on the same level of the hierarchy then the conflict can be highlighted to the users and they can resolve the conflict manually. Every time a conflicting state change of the same activity is detected, each AW performs algorithm 1. This algorithm ensures eventually that there is one agreed activity state, when there was one conflict.

Using algorithm 1, the AWs select the state set by the highest governance role. This means each AW reaches eventually the same state for the activity, because they receive eventually a history of all state changes. In more complex activity types there is the possibility of several conflicts. For example, let's assume the activity type of the activity "Build Dam" is extended by adding two further states "Complete Failure" and "Partial Failure" after the state "Fail". This means there can be a conflict, when the activity is changed from "Execute" to "Finish" by the fire fighter commander in the field and from "Execute" to "Fail" by the fire fighter commander in the command center. Then, the military commander changes it from "Fail" into "Partial Failure" and the fire fighter commander in the command center changes into "Complete Failure". There are now two conflicts. The algorithm can be extended to resolve several conflicts by applying it to all conflicts and by removing state changes causing the conflicts from the history. We omit the details here due to space restrictions. It should be noted that the algorithm is not about handling "wrong" states. Although the states are conflicting, each party (fire fighter commander in the command center, fire fighter commander in the field or military commander) may have justified reasons for its own view. The main goal of the algorithm is to converge to a common view based on strategic direction and defined governance roles. This does not require additional communication, because we assume that eventually the history of state changes of an activity contains the same items in all AWs.

D. Detecting and Handling Conflicts of Shared Activities with Dependencies

The second type of conflict can occur when two shared activities, connected via one dependency, change their state and this may lead to the case that the same dependency in different AWs is in different states (e.g. in one "Neutral" and the other "Violated"). This conflict is different from the previous one and the situation causing it is illustrated in the upper part of Fig. 5. The military commander has shared the activity "Transport Sandbags" with the fire fighter commander in the command center. The fire fighter commander has created a dependency "overlaps" to his own activity "Build Dam". The own activity "Build Dam" has been shared with the fire fighter commander in the field. In the bottom part of the figure, we illustrate the problem as a sequence diagram. We assume that the military commander changes the activity "Transport Sandbags" to the state "Execute" and the fire fighter commander in the field changes the activity "Build Dam" to state "Execute". The fire fighter commander in the command center cannot determine the order of state changes properly, because there might be delay when transmitting the state changes. This means the state change of the military commander is received after the state change of the fire fighter commander in the field, although the military commander changed it before the fire fighter commander. This can also lead to a different temporal order of state changes in different AWs, because each AW can receive state changes in different orders. This means they have a conflicting view on the situation. This implies that we need to ensure global causality eventually, so that all participants have the same view on the situation.

Definition 5 Eventual global causality: a x : s i < a y : s i+1 → C j (a x : s i ) < C j (a y : s i+1 ) ∀AW j = 1, ..n sharing activity x and/or y. This definition means that when state change a x : s i happens before state change a y : s i+1 then this needs to be equally observed in all AWs where the shared activities are replicated.

Lamport [START_REF] Lamport | Time, clocks, and the ordering of events in a distributed system[END_REF] introduced the notion of virtual time in distributed systems. It is similar to the idea in Definition 5. Virtual time progresses in terms of events, i.e. time stands still when there is no event. An event in our approach is a state change. This notion allows defining of global "happen before" relationships between state changes in each AW. Using this notion, we can also detect in the example which state change happened before the other one (cf. also our activity framework for the non-distributed setting [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]). Ensuring Definition 5 means we need to find a function C for each AW, so that it is able to order the events in the same order like the other AWs. Vector clocks [START_REF] Mattern | Virtual time and global clocks in distributed systems[END_REF] address this by using a vector containing the clock (event counter) of each AW n : V = (c 1 , .., c n ). Every time an AW i propagates a state change s to the other AWs, it increases only its counter of the vector clock (i.e. the i-th item of the vector): ??? Order ??? Time Shared Fig. 5. Example for a situation that can cause a conflicting view on causality an order by comparing the vectors of different state changes using the following clock function C: s i (with clock vector V x ) is partially ordered before s j (with clock vector V y ), if : V x [k] ≤ V y [k]∀k (otherwise they are simultaneous). It is always possible to create this partial order (cf. [START_REF] Mattern | Virtual time and global clocks in distributed systems[END_REF] for proofs of these concepts). The ordered state changes can be inputted into the state machine representing the dependency (cf. [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]) to detect if a dependency is violated or not and since it is the same order it will always be the same result in all AWs.

V [i] = V [i] + 1.
Although the vector clock approach seems to be suitable for our purposes, it has one drawback, because not everything is shared with everybody. This may lead to a situation where it is not possible to establish causality, because some of the AWs do not know about each other. For example, let us assume that in the situation illustrated in Fig. 5 the military commander in his AW changes the activity "Transport Sandbags" into state "Execute" and propagates the state change (with Vector clock V M ilitary = ((1, "M ilitary"), (0, "F ireF ighterCommandCenter"))) to the AW of the fire fighter commander in the command center (illustrated in the upper part of the figure). The fire fighter commander in the field changes the activity "Build Dam" into state "Execute" and propagates the state change to the AW of the fire fighter commander in the command center (with vector clock V F ireF ighterF ield = ((1, "F ireF ighterF ield"), (0, "F ireF ighterCommandCenter")). The fire fighter commander in the command center is never able to establish causality in this case, because the AW of the military and the AW of fire fighter in the field do not know their vector clocks. This is undesired, because it would make the definition of temporal dependencies in this special case useless.

We solve this problem by introducing the following rule in our protocol:

When a vector clock with a state change is received then the AW i increases its own clock c i and sends the updated clock vector to all AWs it has activities shared with.

The effect of this rule can be illustrated via the previous example. Let us assume that the AW of the fire fighter commander in the command center receives the state change from the military. It then updates its vector clock and sent it (V F ireF ighterCommandCenter = ((0, "F ireF ighterF ield"), (1, "F ireF ighterCommandCenter"))) to the AW of the fire fighter commander in the field as well as the AW of the military. When now the fire fighter commander in the field changes the activity "Build Dam" to "Execute" it propagates the state change together with the updated vector clock (V F ireF ighterF ield = ((1, "F ireF ighterF ield"), (1, "F ireF ighterCommandCenter"))) to the AW of the fire fighter commander in the command center. The AW of the fire fighter commander in the command center is now able to establish causality according to Definition 5.

Approaches in distributed systems (e.g. [START_REF] Bartlang | Dhtflex: A flexible approach to enable efficient atomic data management tailored for structured peer-topeer overlays[END_REF]) expect that everything is shared among everybody and thus the problem does not occur there. This also means that our approach has an advantage in our scenario over the other approaches.

V. ARCHITECTURE

We implemented our concepts mentioned before to do student experiments with the prototype. They are implemented as an extension to Google Wave that enables instant collaboration with optimistic replication of shared documents ("Waves"). Shared documents can be distributed between different servers of different organizations (illustrated on Fig. 6 as different Wave servers). A "Wave" can have participants from different servers. The reason for choosing a collaboration platform over a simpler platform was to show how our approach works in the context of different tools needed for disaster response management (e.g. text exchange, maps, images or videos). Furthermore, it provides the infrastructure for implementing sharing of activities. Google Wave can be extended in two different ways: "Gadget" and "Robot". A "Gadget" can be inserted into a "Wave" and is a graphical user interface to provide additional collaboration functionality (e.g. collaborating on images or collaborative modeling). It is rendered within the Google Wave Web Client in a web browser. A "Robot" can be added as an automated participant to a "Wave" and can react on events in "Waves" and modify them. It can also create further "Waves". Google Wave is in process to be fully open-sourced platform [START_REF] Google | Wave in a box[END_REF].

We illustrate the architecture of our extension in Fig. 6 in context of the Wave Federation Architecture [START_REF] Google | Wave in a box[END_REF]. Activities and dependencies can be modeled in a special "Wave" called "AW-Wave" containing a "Gadget" providing the necessary functionality. The "Gadget" is called "AW-Gadget" and stores its data (e.g. the model) in the "AW- Wave". The "AW-Wave" can be compared to an AW. A robot is a distributed application on the Google App Engine or any other server. It is responsible for propagating the state changes of activities to different "AW-Waves". Activities themselves are linked to special "Waves" called "Activity-Waves". People can collaborate in this activity, e.g. they can insert pictures, write text or work collaboratively on a map of the situation. An activity can be shared by inviting a participant to an "Activity-Wave". This means Google Wave provides already a mean for sharing and replicating activities as required by our approach. The robot makes the shared activity available in the "AW-Waves" of the participant who has been invited to the "Activity-Wave". The activity is then shown in the "AW-Gadgets" of the "AW-Waves" of the participant and the participant can replicate it into his/her models. They can also create dependencies to their own activities. State changes can be initiated via the "AW-Waves", where the activity is replicated. The "AW-Gadget" stores for this a state change as well as the vector clock in the "AW-Wave". The robot copies then the state change to all "AW-Waves" where the activity is replicated. According to the rule mentioned in the previous section, the robot copies the vector clocks to all "AW-Waves" with which activities have been shared. The "AW-Gadgets" can create the global order of state changes based on the vector clocks. They also highlight violation of dependencies to the user and they display to the user when there have been conflicting state changes according to Definition 4.

VI. IMPLEMENTATION

We illustrate a screenshot of our extension in Fig. 7. We support a graphical modeling notation, but as an alternative the activities and dependencies can be represented in a table. The later view seems to be more easy to use when many activities need to be created. The figure illustrates the activity workspace of the fire fighter commander in our use case. It uses a graphical modeling notation. The fire fighter commander sets the activity "Build Dam" conflicting to the field commander to state "Fail" and "Cancel" respectively.

VII. DISCUSSION

We started validation of our approach by presenting it to four experienced disaster managers. The general approach of activity management has been commented positively by the disaster managers (cf. [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF], [START_REF] Franke | A model for temporal coordination of disaster response activities[END_REF]). The problems of traditional means for coordination have been recognized by a fire fighter commander (Southern California): "[..] the pile of messages in the inbox, [which contains] the reality as a situation [..] being able to put them in context and update them and coordinate them to create a common picture is the difficulty".

For example, a fire fighter commander (Washington, DC) highlights that it allows measuring the progress of the situation and managing shifting goals (end states):

"there is a couple things [about your approach], [..] it is a good way to measure progress, and the second thing is that you recognize that the end state [goal] will change because of the dynamic situation of the incident you are involved in [..] the end state may need to be modified or you might have intermediate type of objectives".

He further says that sharing of activities (missions) is important: "[You are able to] define specific objectives that need to be accomplished in order to meet that end state [..] and then [these] objectives [are] transmitted as mission statements to the ground [..] those folks at the ground level [define] the mission, [plan] the mission , [develop] the tasks and tactics [and have to] make time-critical decisions in order to meet that mission".

Another fire fighter commander (Southern California) confirms the previous statements: "[The approach] addresses the problem of coordination and sharing goals and objectives from one organizations to other and it is that communication [to update] information, [such as] progress as far as plan, changes made, it is that communication link that inherently seems to be the crux of all problems. [If this] sharing does not occurs [then] a lot of information stays within each independent organization [..] Without that knowledge we duplicate services, we actually implement plans that interfere with the others, goal and objectives".

Our approach can help them to have a more meaningful and accurate situation awareness on the current state of their own and others' activities. The fire fighter commander (South France) highlights that our approach can make a difference in situation where coordination is the issue: many organizations are involved, it is a geographically distributed situation, there needs to be time to plan and there needs to be time for communication between different actors.

VIII. RELATED WORK

In [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF] several process management systems, addressing a disaster response scenario, are compared. They do not take into account an inter-organizational setting. Interorganizational coordination of activities has mostly been addressed in the area of business process management systems. Aalst and Weske [START_REF] Van Der Aalst | The p2p approach to interorganizational workflows[END_REF] propose to split a previously defined public process in several parts and let every involved party execute its part in a distributed fashion. Grefen et al. define the public process as a contract between organizations [START_REF] Grefen | Cross-flow: Crossorganizational workflow management in dynamic virtual enterprises[END_REF]. Schulz et al. [START_REF] Schulz | Facilitating cross-organisational workflows with a workflow view approach[END_REF] describe a view-based approach which is similar to the previous approach. Fdhila et al. [START_REF] Fdhila | A flexible approach for automatic process decentralization using dependency tables[END_REF] propose to execute a public process as a choreography. The approaches in [START_REF] Grundy | Serendipity: integrated environment support for process modelling, enactment and work coordination[END_REF], [START_REF] Rinderle | Evolution of process choreographies in dychor[END_REF] allow defining a public process and each involved party can deviate from it. All these systems require to some extent defining a global public process or choreography before the process execution. Based on our end user interactions, we believe this is not an acceptable assumption for autonomous organizations that may even work together for the first time. It is also not always obvious when and if they have to work together. The processes in our use case are not only defined top-down, but also bottomup. They can be defined vertically and horizontally in a network of organizations. We do not require definition of a public process in advance. Furthermore, these approaches consider mostly sequential processes, which is not a realistic assumption for disaster response processes [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]. They are also limited with respect to detection of concurrent conflicts. Our approach can be compared with the unified activity management approach in [START_REF] Harrison | Roles and relationships for unified activity management[END_REF]. However, they do not consider an inter-organizational distributed setting.

We find many approaches addressing detecting and handling of conflicts in distributed collaborative work (cf. [START_REF] Greenberg | Real time groupware as a distributed system: Concurrency control and its effect on the interface[END_REF]). For example, collaborative image [START_REF] Sun | Consistency maintenance in real-time collaborative graphics editing systems[END_REF] or text editing [START_REF] Imine | Formal design and verification of operational transformation algorithms for copies convergence[END_REF] with optimistic change propagation. These approaches deal with conflicts in unstructured documents (e.g. text or images) and not in structured models like our concept.

Other approaches deal with the consensus problems in distributed systems (e.g. [START_REF] Lamport | The part-time parliament[END_REF], [START_REF] Bartlang | Dhtflex: A flexible approach to enable efficient atomic data management tailored for structured peer-topeer overlays[END_REF]). The consensus problem is different from our concept, because it deals with faulty processes (i.e. processes that send different values to different processes). The problem there is to find a consensus between processes what are correct values. In our approach, such a consensus is not possible, because each party has a different view on the real world (e.g. the command center has a more distant strategic view and the people in the field a more operational view) and consensus protocols cannot deal with this problem. Furthermore, the actors may have different goals and processes that interact with each other, but not in the sense of a distributed algorithm. In our scenario, the notion of faulty processes does not exist. This is why we proposed another approach based on governance roles to resolve conflicting views.

IX. CONCLUSION

We described in this paper how the coordination of autonomous organizations can be supported by an activity management system. The basic idea is to allow the organizations to share selected activities with each other by replicating them in the different workspaces of the organizations. We presented two different conflicts that can occur when optimistically propagating state changes of shared activities. We explained how they can be detected and also provided mechanisms for handling them. Pessimistic propagation would lead to inaction, requires a constant reliable communication and the dynamics of the situation cannot be captured in time. Using optimistic propagation allows capturing the situation (activities and their state) in time and it enables detection of conflicting views, which is seen as beneficial by disaster managers. Thus, even traditional means only allow optimistic propagation of information. The proposed solution can already overcome some problems with traditional means for ad-hoc coordination (e.g. email or phone) or even more formal BPM approaches (see related work), because they have only limited capabilities for detecting and handling concurrent conflicting views on dynamic processes which are only partially known by each organization. In the future, we want to explore how to establish the social network between actors sharing activities based on pre-defined plans. This also includes the definition of governance roles. The approach has been commented positively by disaster managers [START_REF] Franke | A model for temporal coordination of disaster response activities[END_REF], [START_REF] Franke | Collaborative coordination of activities with temporal dependencies[END_REF]. We have already started first evaluations in form of experiments with students. We expect that our approach is applicable to other dynamic collaborative scenarios, such as organization of large events (e.g. Olympic Games) or development projects. Our concepts presented here can also be adapted to extend other constraint-based process management systems to the inter-organizational level (e.g. [START_REF] Van Der Aalst | Decserflow: Towards a truly declarative service flow language[END_REF]).

Fig. 1 .

 1 Fig. 1. Example for an activity type with governance roles

Fig. 2 .

 2 Fig. 2. Types of supported temporal dependencies

  It attaches its vector clock V to the state change. An AW receiving a state change can now put them in

Fig. 7 .

 7 Fig. 7. Screenshot of our prototype

  This means the final state in

	input : Execution history σ of one activity with
	conflicting state changes
	conflictingstatechangeslist ←
	GetConflictingStateChanges(σ);
	chosenStateChange ← conflictingstatechangeslist[0];
	for i ← 0 to conflictingstatechangeslist.size -1 do
	if chosenStateChange.role <
	conflictingstatechanges[i].role then
	chosenStateChange ←
	conflictingstatechangeslist[i];
	end
	end
	SetCurrentState(chosenStateChange)
	Algorithm 1: Handle conflicting state changes of one
	activity

X. ACKNOWLEDGEMENTS

The research was partially funded by the French Ministry of Research within the RESCUE-IT project [23]. We thank the domain experts for providing us detailed insights.


Corrado Andini
email: andini@uma.pt

Mincer Equation

Pedro Telhado Pereira

Santiago Budría

Günther Lang

Joop Hartog

A Dynamic Mincer Equation with an Application to Portuguese Data

Keywords: Return to Schooling, Wage Level, Panel Data Mincer Equation, Return to Schooling, Wage Level, Panel Data. JEL codes: I21, J31, C23

This paper argues in favor of a dynamic specification of the Mincer equation, where past observed earnings play the role of additional explanatory variable for current observed earnings. A dynamic approach offers an explanation why the return to schooling in terms of observed earnings is not independent of labor-market experience, as suggested by some recent empirical evidence for the United States.

Introduction

A seminal work by Jacob [START_REF] Mincer | Schooling, Experience and Earnings[END_REF] has been the starting point of a large body of literature dealing with the estimation of a wage equation where the logarithm of hourly earnings is explained by schooling years, labor-market experience, and experience squared. Within this framework, the coefficient of schooling years is usually interpreted as being the return to an additional year of schooling in terms of observed earnings.

An excellent synthesis of the research papers adopting the Mincer equation as underlying framework has been provided by David Card (1999). The reviewed works generally focus on the estimation of the average impact of schooling on earnings, by means of both ordinary least squares and instrumental-variable techniques.

Today, 'the state of the art' described by Card looks outdated. This is partly because the last decade was characterized by a special interest in adopting the Mincer equation for identifying the effect of schooling not only on the mean but also on the shape of the conditional wage distribution, by using the quantile-regression techniques due to [START_REF] Koenker | Regression Quantiles[END_REF]. Starting from a seminal work by [START_REF] Buchinsky | Changes in the U.S. Wage Structure 1963-1987: Application of Quantile Regression[END_REF], the last few years saw the publication of numerous estimates of the schooling-coefficient along the conditional wage distribution, with the frequent finding that education has a positive impact on within-groups wage inequality, as suggested by [START_REF] Martins | Does Education Reduce Wage Inequality? Quantile Regression Evidence from 16 Countries[END_REF] among others. Additional results using instrumental-variable-quantile-regression techniques have been provided by [START_REF] Arias | Individual Heterogeneity in the Returns to Schooling: Instrumental Variables Quantile Regression Using Twins Data[END_REF], [START_REF] Lee | Endogeneity in Quantile Regresssion Models: A Control Function Approach[END_REF] and Andini (2006a).

In spite of its wide acceptance within the profession, the spread of the framework developed by Mincer over the last forty years has not been uncontroversial. Some authors criticized the Mincerian framework by arguing that the equation is not able to provide a good fit of empirical data; some stressed that the average effect of schooling on earnings is likely to be non-linear in schooling; some suggested that education levels In summary, despite some critical voices, the history of human-capital regressions seems characterized by a generalized attempt of consistently estimating the coefficient of schooling (both on average and over the conditional wage distribution), under an implicit acceptance of the theoretical interpretation of the schooling-coefficient itself.

Nevertheless, the important issue of the theoretical interpretation of the schoolingcoefficient has been recently rediscovered and discussed by [START_REF] Heckman | Earnings Functions, Rates of Return, and Treatment Effects: the Mincer Equation and Beyond[END_REF], who empirically tested several implications of the classical Mincerian framework, using Census data for the United States. Among other implications of the Mincerian approach, the authors tested and often rejected the implication that the return to schooling in terms of observed earnings is independent of labor-market experience.

On the lines of [START_REF] Heckman | Earnings Functions, Rates of Return, and Treatment Effects: the Mincer Equation and Beyond[END_REF], our paper will provide additional theoretical and empirical arguments against the usual interpretation of the coefficient of schooling in the standard Mincer equation. Indeed, we will argue that the return to schooling in terms of observed earnings is, in general, dependent of labor-market experience. As we will see, the latter result can be easily derived from a dynamic specification of the Mincer equation where past observed earnings play a role in explaining current observed earnings.

The reminder of this paper is as follows. Section 2 describes the standard theory behind 3 uses the dynamic Mincer equation to show that, in general, the return to schooling in terms of observed earnings is not independent of labor-market experience. Section 5 presents estimation results for the dynamic Mincer equation, which are consistent with the theoretical arguments proposed in section 3 and section 4. Section 6 concludes the manuscript.

Static Mincer equation

This section presents the theoretical foundations of the standard [START_REF] Mincer | Schooling, Experience and Earnings[END_REF] equation as recently reported by [START_REF] Heckman | Fifty Years of Mincer Earnings Regressions[END_REF]. Therefore, we make no claim of originality at this stage and mainly aim at helping the reader with notations and terminology adopted in the next sections 1 .

+ + = 1 t 0 j j j 0 t ) k r 1 ln( E ln E ln .
Under the assumptions that:

• schooling is the number of years s spent in full-time investment2

( 1 k ... k 1 s 0 = = =
),

• the return to schooling in terms of potential earnings is constant over time

( = = = 1 s 0 r ... r
) ,

• the return to post-schooling investment in terms of potential earnings is constant

over time ( = = = 1 t s r ... r ) ,
we can write expression (3) in the following manner:

(4)

= + + + + = 1 t s j j 0 t ) k 1 ln( ) 1 ln( s E ln E ln ,
which yields to:

(5 In order to build up a link between potential earnings and labor-market experience z, Mincer assumes that the post-schooling investment linearly decreases over time, that is:

) = + + 1 t s j j 0 t k s E ln E ln .
(6) = + T z 1 k z s where 0 s t z =
, T is the last year of the working life and ) 1 , 0 ( .

Therefore, using ( 6), we can re-arrange expression (5) and get:

(7) 2 0 t z T 2 z T 2 s E ln E ln + + + .
In addition, following Mincer, we are interested in potential earnings net of postschooling investment costs, which are given by:

(8) 2 0 t z T 2 z T T 2 s E ln T z 1 E ln + + + +
or alternatively by: (9) Finally, assuming that observed earnings are equal to net potential earnings at any time t (a key-assumption, as we will see in the next section):

(10

) = T z 1 E ln w ln t t
, and, using expression (9), we obtain the standard Mincer equation:

(11) 2 t z z s w ln + + + .
We will label expression (11) as static Mincer equation in order to distinguish the latter from the dynamic equation obtained in the next section.

Dynamic Mincer equation

Let us start stressing again that the standard 15), this section provides several definitions of returns to schooling, which will be useful for the empirical application in the next section.

Static return to schooling in terms of net potential earnings

To begin, we find of interest stressing that the total return to schooling in the static model ( 11) is given by the following expression:

(16) ! ! = ! ! + s w ln s w ln z s t
and is constant over the working life, meaning independent of labor-market experience z. Further, because of assumption (10), the return to schooling in terms of observed earnings and the one in terms of net potential earnings coincide 5 .

We will label as static return to schooling in terms of net potential earnings and will show, in section 4.3, that our interpretation of in terms of net potential rather than observed earnings is the most appropriate.

Returns to schooling in terms of observed earnings

The dynamic model ( 15) allows obtaining the evolution of the schooling return over the entire working life. For instance, at time s, expression (15) can be written as follows:

(17) and the total return to schooling is given by:

(22) 2 2 s ) 1 ( ) 1 ( s w ln ) 2 ( + + ! ! = + .
Therefore, at time z s + , the return to schooling in terms of observed earnings is given by the following expression:

[ ] Z 3 2 z s ) 1 ( .... ) 1 ( ) 1 ( ) 1 ( 1 s w ln ) z ( + + + + + ! ! = + ,
and is, in general, dependent of labor-market experience z.

Clearly, at the end of the working life, the total return in terms of observed earnings is as follows:

(24) [ ] T 3 2 T s ) 1 ( .... ) 1 ( ) 1 ( ) 1 ( 1 s w ln ) T ( + + + + + ! ! = + .

Dynamic return to schooling in terms of net potential earnings

The return in expression ( 23) is, in general, lower than the return in expression ( 16), although the first converges to the latter as labor-market experience z increases. Indeed, for a value of ) 1 , 0 ( , the following expression holds:

(25) = " " # ) 1 ( 1 1 ) z ( lim ) ( z .
Therefore, the dynamic model ( 15) is able to provide a measure of comparable 6 with expression ( 16). We will label ) (" as dynamic return to schooling in terms of net potential earnings. 12 forever. To the extent of T being a finite number, the return to schooling in terms of observed earnings ) z ( can never be equal to , but in the very special case of 1 = .

Final remarks

It is easy to prove that the following inequalities hold:

(26) < < < ) T ( ) z ( ) 0 (
for every z and T such that

" < < < T z 0 and 0 > , if ) 1 , 0 ( .
In addition, one can verify that:

(27) < = = = 0 ) T ( ) z ( ) 0 (
for every z and T such that

" < < < T z 0 and 0 > , if 0 = .
Finally, it is easy to show that:

(28) = = = ) T ( ) z ( ) 0 (
for every z and T such that

" < < < T z 0 and 0 > , if 1 = .

Empirical application

Based on the static Mincer equation ( 11) and its dynamic version (15), we compare estimation results from the following two empirical models:

F o r P e e r R e v i e w 13 (29) it 5 2 it 4 it 3 i 2 1 it z z s w ln µ + µ + µ + µ + µ = with ( ) 0 z , z , s Exp 2 it it it it 5 = µ and ( ) 0 z , z , s Quant 2 it it it it 5 = µ & for each & (30) it 6 2 it 5 it 4 i 3 1 it 2 1 it z z s w ln w ln ( + ( + ( + ( + ( + ( = with ( ) 0 w ln , z , z , s Exp 1 it 2 it it it it 6 = ( and
( ) 0 w ln , z , z , s Quant 1 it 2 it it it it 6 = ( &
for each & using both ordinary least squares and quantile-regression techniques.

Therefore, following the most recent practices and using two different approaches, we look at the impact of schooling not only on the mean but also on the shape of the conditional wage distribution.

Notice that & is an indicator of the regression quantile. Further, notice that a number of potentially-relevant additional explanatory variables are disregarded because the aim of the application consists of comparing results from two simple and alternative models: a static model and a dynamic model.

We use data for Portuguese male workers extracted from the European Community Household Panel (ECHP), in the period of 1994-2001. Our unbalanced panel is described in Table 1. To avoid distortions due to outliers, following a common procedure, we exclude individuals whose hourly earnings are very high (10 times the average) or very low (0.10 times the average).

Based on sections 3 and 4, we will refer to the estimate of:

• 3

( as the return to schooling ) 0 ( ;

• 2 ( as the bargaining power of the employer 1 ;

• 2 1 ( as the bargaining power of the employee ; Specifically, Martins and Pereira find a coefficient of schooling years, estimated using ordinary least squares, that is four percentage-points higher than our estimated coefficient (12.6% vs. 8.2%, see Table 2, column 4). Further, we observe a similar gap regarding the impact of schooling on within-groups wage inequality, computed as difference 9 between the coefficient at the ninth decile of the conditional wage distribution and the coefficient at the first decile (8.9% vs. 4.9%).

Nevertheless, despite the reported gaps, our results for 1994-2001 are not really in contrast with those of Martins and Pereira for 1995 because there is evidence of a decreasing trend in both average returns and within-groups-wage-inequality measures in Portugal, after a peak in 1995 10 .

Conclusions

Being conceived as a long-run equilibrium model, the standard Mincer framework disregards short-run earnings dynamics and assumes that current net potential earnings are equal to current observed earnings at any point in time. This framework, however, has some strong empirical implications which may or may not be consistent with data.

Particularly, as argued by [START_REF] Heckman | Earnings Functions, Rates of Return, and Treatment Effects: the Mincer Equation and Beyond[END_REF], one of the empirical implications of the classical Mincerian model, the independence of the return to schooling in terms of observed earnings from labor-market experience, seems often rejected by empirical tests 9 Notice that, using our terminology, this difference measures the impact of schooling on within-groups net potential wage inequality.

10 See [START_REF] Pereira | Education and Earnings in Portugal, presented at the "1ª Conferência Sobre o Desenvolvimento Económico Português no Espaço Europeu[END_REF] and [START_REF] Budría | Educational Qualifications and Wage Inequality: Evidence for Europe[END_REF]. This finding is very interesting and deserves, in our view, further investigation.

Our paper is primarily intended to offer an explanation why the return to schooling in terms of observed earnings may be dependent of labor-market experience. With an eye on real-life labor markets, we start our analysis from the hypothesis that observed earnings do not instantaneously adjust to net potential earnings, and argue in favor of a dynamic specification of the Mincer equation, where past observed earnings contribute to explain current observed earnings together with current net potential earnings. Within our dynamic framework, the return to schooling in terms of observed earnings turns out to be dependent of labor-market experience. We also provide empirical evidence in favor of a dynamic approach, using longitudinal data for Portuguese male workers over the period of 1994-2001. This evidence is roughly consistent with our earlier findings using data on male workers from the US Longitudinal Survey of Youth (see Andini, 2006b).

F

  years in the wage equation. As a matter of example,[START_REF] Murphy | Empirical age-earnings profiles[END_REF] maintained that the standard Mincer equation provides a very poor approximation of the true empirical relationship between earnings and experience, while Trostel (2005) argued that the average impact of an additional year of schooling on earnings varies with the number of completed schooling-years.

the

  Mincer equation. Section 3 develops the theoretical foundations of a new Mincer equation that we label dynamic, in contrast to the standard static framework. Section 4

3

  Notice that the symbol of equality ) (= in expression (4) becomes a symbol of rough equality ) ( in expression (5).

  Expression (25) helps to show that our interpretation of in terms of net potential rather than observed earnings is the most appropriate because nobody can live and work

  Figure 1Plotted estimation results based on quantile regression

Page 4 of 24 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

	(3)
											F o r
																	P
	Jacob Mincer argues that potential earnings today depend on investments in human e
	capital made yesterday. Denoting potential earnings at time t as t E , Mincer assumes e r
	that an individual invests in human capital a share t k of his/her potential earnings with a return of t r in each period t. Therefore we have: R e v
	(1)	E	t	1 +	=	E	t	1 (	+	r t	k	t	)	i e
																	w
	which, after repeated substitution, becomes:
	(2)	E	t	=	j t	=	0 1	1 (	+	r	j	k	j	)	E	0
	or alternatively
	1 Although not original, we believe that section 2 is crucial for the paper.
																	4

=

  Expression (13) basically implies that the first observed wage at time s depends on both net potential earnings at time s and the minimum wage w at time 1 s , which is set by law and therefore independent of schooling years.

	Submitted Manuscript
	the possibility of a margin-formation between observed earnings and net potential 4. Returns to schooling
	earnings should not be ruled out a-priori. Empirically, this implies that observed Based on model (11) and on model (
	earnings may not coincide with net potential earnings, although the first generally
	depend on the latter.
	Second, observed earnings are sticky as already documented in the literature 4 . This Notice that the coefficient can be interpreted as the bargaining power of the
	means that observed earnings do not instantaneously adjust to changing environments employee. Indeed, if the employee has full bargaining power ( 1 = ), observed earnings
	and exhibit path-dependence. That is, current observed earnings partly depend on past are equal to net potential earnings as in the expression (10) of the standard Mincer
	F o r observed earnings, for several reasons such as multi-period labor contracts both in F o r framework. However, if the employer has full bargaining power ( 0 = ), observed wages are completely independent of net potential earnings and equal to the legal minimum at any time s t . Reality is likely to range between these two extreme F o r
	P P scenarios, and expression (12) allows capturing this fact. Section 5 will provide P
	e e r e e r e empirical evidence supporting (12). If we use expression (9) to replace net potential earnings in equation (12), then (12) e r
	R e v i e current net potential earnings. Hence, at any time s t , observed earnings are given by R e v becomes: (14) ( ) 1 t 2 t w ln ) 1 ( z z s w ln + + + + R e v the following expression: i e i e
	s section simply argues that observed earnings do not instantaneously adjust to net t . This w (12) 1 t t t w ln ) 1 ( T z 1 E ln w ln + = w or alternatively w
	potential earnings because of two reasons. (15) 1 t t s w ln ) 1 ( w ln + + +	z	+	z	2
	First, employee's skills are not the only determinant of observed earnings. Schooling with [ ] 1 , 0 and the following initial-condition:
	and post-schooling investments provide individuals with net potential earnings, which is equal to (11) if the employee has full bargaining power, i.e. setting 1 = . We
	meaning skills required to earn a given amount of money. However, observed earnings (13) 1 s 1 s w ln w ln = . will label expression (15) as dynamic Mincer equation.
		7 8 9

Mincer equation assumes, in expression (10)

, that observed earnings are equal to net potential earnings at any time are the result of both employee's skills and employer's willingness to pay. Since reallife labor markets are characterized by asymmetric information and wage-bargaining, unionized and non-unionized industries. However, despite the existing evidence (both at macroeconomic and microeconomic level) on the autoregressive nature of observed earnings, this stylized fact has not received enough attention in Mincerian studies so far.

Based on the above arguments, this section maintains that assumption (10) can be modified such that current observed earnings depend on both past observed earnings and current net potential earnings. For computational simplicity, we assume that current observed earnings are a Cobb-Douglas function of both past observed earnings and 4 See Taylor (1999) for a good survey.

Page 8 of 24

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Page 10 of 24 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

																			Submitted Manuscript
	and the return to schooling is given by:
	(18)	(	0	)	=	!	! s w ln	s		.
	Analogously, at time	1 s + , expression (15) can be written as follows:
	(19)	ln	w	s	1 +			F r o s w ln ) 1 ( +	+	s	+	1	+	2 1
	P and the total return to schooling is given by:
																		e
	(20)	) 1 (	=	!	s w ! ln	s	1 +	+			1 (	e r ) .
	At time	R v e s + , expression (15) is as follows: 2
	(21)	ln	w	s	+	2						+	1 (	)	ln	w	s	1 +	+	s	+	2	+	i e 2 2
																				w
		ln	w	s					+	1 (	)	ln	w	s	1	+	s	+	0	+	0	2
	5 See expression (8).
																				10

Page 14 of 24 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  Quadros de Pessoal data-set, while we use longitudinal data for the period of 1994-2001, from the ECHP.

	Submitted Manuscript
	1995, extracted from the so-called
	F o r F o r F o r
	P P P
	e e e
	e r e r e r
	R e v R e v R e v
	i e i e i e
	w w w
	16

Page 17 of 24 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

  Census data on white and black male workers in the United States.

	Submitted Manuscript
	performed using
	F o r
	P
	e
	e r
	R e v
	i e
	w
	17

Table 1 Sample statistics based on ECHP data for Portuguese male workers: 1994-2001

 1

	o r
	P
	e
	e r
	R e v
	i e
	w

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 2 Estimation results based on ordinary least squares and quantile regression

 2

	Column	(1)	(2)	(3)	(4)	(5)
	Theoretical parameter	(	0	)	1	("	)
	Empirical parameter	(	3	(	2	2 1 (	μ	2	2 1 3 ˆ( (
	&	=	0	1 .	0.007	0.853	0.147	0.053	0.046
							(0.001)	(0.010)	(0.002)
	&	=	0	2 .	0.002	0.935	0.065	0.062	0.034
							(0.000)	(0.007)	(0.001)
	&	=	0	3 .	0.003	0.954	0.046	0.068	0.055
							(0.000)	(0.003)	(0.001)
	&	=	0	4 .	0.003	0.953	0.047	0.074	0.063
							(0.000)	(0.003)	(0.001)
	&	=	0	5 .	0.004	0.943	0.057	0.079	0.069
							(0.000)	(0.004)	(0.001)
	&	=	0	6 .	0.006	0.916	0.084	0.087	0.075
							(0.000)	(0.004)	(0.001)
	&	=	. 0	7	0.010	0.890	0.110	0.091	0.088
							(0.000)	(0.004)	(0.001)
	&	=	0	8 .	0.014	0.855	0.145	0.097	0.099
							(0.001)	(0.008)	(0.001)
	&	=	0	9 .	0.022	0.798	0.202	0.102	0.108
							(0.002)	(0.011)	(0.001)
	OLS	0.012	0.843	0.157	0.082	0.079
							(0.001)	(0.004)	(0.001)
	Standard errors in parentheses. Estimates of 2 ( , 3 ( and 2 µ are all significant at 1% level.

It is assumed that schooling starts at the beginning of life.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

& & & & &

& & & & & &

Acknowledgements I gratefully acknowledge financial support from the European Commission (EDWIN Project, HPSE-CT-

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

µ as the static return to schooling in terms of net potential earnings ; The empirical validation of model ( 15) obviously requires that:

• 2 ( ranges between 0 and 1, both on average and over the conditional wage distribution.

Particularly, if one agrees that the bargaining power of the employer is generally higher than the bargaining strength of the employee, then we may reasonably expect that:

• 2 ( ranges between 0.5 and 1, both on average and over the conditional wage distribution.

Further, from expressions ( 16) and (18), we may expect that:

• 3 ( is lower than 2 μ , both on average and over the conditional wage distribution;

• 3

( is roughly equal to the product between 2 μ and 2 1 ( , both on average and over the conditional wage distribution;

• 3

( is positively correlated with 2 1 ( over the conditional wage distribution.

Finally, from expressions ( 16) and ( 25), one can reasonably expect that:

is roughly equal to 2 μ , both on average and over the conditional wage distribution.

Estimation results are reported in Table 2 and Figure 1. Surprisingly, the empirical analysis does not reject any of our six theoretical predictions. The results based on ordinary least squares are very satisfactory. Regarding the quantile-regression results, the only case where we obtain less satisfactory results is related to the second decile of the conditional wage distribution.

Compared to the standard model ( 29), the main advantage of model ( 30) consists of allowing for the estimation of the return to schooling in terms of observed earnings at several stage of the working life, by replacing estimation results for 2 ( and 3 ( into expression (23) 7 .

A final note is about our specific results for the standard Mincer equation ( 29) in comparison with previous estimates for Portugal.

Despite the existence of several studies using Portuguese data and quantile-regression techniques 8 , to the best of our knowledge, the only two recent journal articles adopting the basic Mincerian specification (29) as underlying empirical framework are due to Pereira and Martins (2002a) and [START_REF] Martins | Does Education Reduce Wage Inequality? Quantile Regression Evidence from 16 Countries[END_REF]. In both the two studies, the authors focus on male workers as we do, but they use cross-sectional data for the year of 7 See, for instance, expression (22).

8 Examples are provided by [START_REF] Hartog | Changing Returns to Education in Portugal during the 1980s and Early 1990s: OLS and Quantile Regression Estimators[END_REF], [START_REF] Machado | Earnings Functions in Portugal 1982-1994: Evidence from Quantile Regressions[END_REF][START_REF] Machado | Earnings Functions in Portugal 1982-1994: Evidence from Quantile Regressions[END_REF]Andini (2006a)