Shiling Wang
email: shilingwang0621@gmail.com

Francis Colas
email: francis.colas@inria.fr

Ming Liu
email: mingliu@cityu.edu.hk

Francesco Mondada
email: francesco.mondada@epfl.ch

Stéphane Magnenat
email: stephane@magnenat.net

Francesco Mondada Mobots

EPFL Lsro

Stéphane Magnenat Mobots

Localization of inexpensive robots with low-bandwidth sensors

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

scalable. Our solution is based on the classical Markov [START_REF] Fox | Markov localization for mobile robots in dynamic environments[END_REF] and Monte Carlo [START_REF] Dellaert | Monte Carlo localization for mobile robots[END_REF] Localization frameworks that can be seen as respectively a dense and a sampling-based implementation of a recursive Bayesian filter. While these approaches are commonly used in robots with extensive sensing capabilities such as laser scanners, their implementation on extremely low bandwidth sensors is novel, and raises specific questions, such as which distance the robot must travel for the localization to converge.

In this paper, we deploy these algorithms on the Thymio II differential-wheeled mobile robot. The robot reads the intensity of the ground using two infrared sensors (Fig. 1, middle, circled red) and estimates the speed of its wheels by measuring the back electromotive force, which is less precise than encoder-based methods. For evaluation purposes, a Vicon tracking system (http://www.vicon.com/) provides the ground-truth pose (Fig. 1, right). Our first contribution is a predictive model of the necessary distance to travel to localize the robot. Our second contribution is a detailed analysis of the performances of the two implementations in comparison with ground-truth data. Our last contribution is an experimental validation of online, real-time localization (Fig. 1, left). The source code and all experimental data are available online 1 and the system can be seen running in a video 2 . Fig. 1 The block scheme of the online system (left), the Thymio II robot with the placement of its ground sensors (middle), and markers for tracking its ground-truth pose by a Vicon system (right).

Bayesian Localization

Related work

The main challenges of solving the localization problem on affordable mobile robots are the constraints on the environment and the limited information content that inexpensive sensors can typically provide. The work of Kurazume and Nagata [START_REF] Kurazume | Cooperative positioning with multiple robots[END_REF] first raised the idea of performing inexpensive localization through the cooperative positioning of multiple robots. Prorok et al. [START_REF] Prorok | Low-cost collaborative localization for large-scale multi-robot systems[END_REF] is a modern work representative of this approach. These authors have used an infrared-based range and bearing hardware along with a distributed Monte Carlo Localization approach, allowing a group of robots to localize down to a precision of 20 cm. However, this methods would require fixed robots acting as beacons to provide absolute positioning. Moreover, both radio and infrared-based range and bearing systems require complex electronics. Finally, when the cost of all robots is added, the system is far from cheap.

A cheaper approach is to use walls around an experimental arena to localize. For example, Zug et al. [START_REF] Zug | Design and implementation of a small size robot localization system[END_REF] have developed an algorithm using an array of triangulationbased infrared distance sensors. A Kalman filter algorithm is applied to localize the robot within a 2 by 1 m box. No experimental result is provided, but a simulation shows the estimated error to be within 2 cm. Dias and Ventura [START_REF] Dias | Absolute localization for low capability robots in structured environments using barcode landmarks[END_REF] used two horizontal lines from a VGA camera to read barcodes on the walls of an arena and localize an e-puck robot. Their system employs an Extended Kalman Filter (EKF) algorithm and reaches a precision of 1 cm and 5 • . However, these systems require no obstacles between the robot and the walls, and thus are not scalable to a large number of robots.

This problem can be alleviated by detecting a known pattern visible on the ceiling and fusing this information with odometry. This approach was proven effective for localization using high-quality cameras [START_REF] Dellaert | Monte Carlo localization for mobile robots[END_REF]. Focusing on a low cost, Gutmann et al. [START_REF] Gutmann | Challenges of designing a low-cost indoor localization system using active beacons[END_REF] have developed a system using only 3 or 4 light detectors, able to localize a mobile robot in a room with an accuracy of 10 cm. However, this method requires a controlled ceiling arrangement and empty space over the robots.

Another approach is to exploit the ground for localization. Park and Hashimoto [START_REF] Park | An approach for mobile robot navigation under randomly distributed passive rfid environment[END_REF] proposed to localize a mobile robot over a ground equipped with randomly distributed passive RFID tags. The average localization error of this method is lower than 10 cm. However, this approach requires the ground to be equipped with tags which can become expensive and tedious to deploy when the area grows.

The system proposed in this paper builds on the idea of using information from the ground for absolute localization, with performance comparable with related work. To the best of our knowledge, existing solutions for localization of multi-robot systems either rely on an expensive adaptation of the environment using markers or beacons, or necessitate the embedding of expensive sensing capabilities on every robot. In contrast, our system is affordable since it only requires a low number of inexpensive infrared sensors onboard the robot (which are typically used as cliff detectors), and a simple grayscale pattern on the ground that can be achieved using a printed poster.

Model

The generic Bayesian filter used to estimate the pose of a robot in its environment uses the following variables:

• X 1:t 2-D pose at times 1..t, consisting of x, y coordinates and an angle θ .

• Z 1:t observations at times 1..t, consisting of the output of the sensors measuring the grayscale intensity of the ground. • U 1:t odometry at times 1..t, consisting of the left and right wheel speeds.

It is classically formulated as a recursive Bayesian filter with the following joint probability distribution:

p(X 1:t , Z 1:t ,U 1:t ) = p(Z t |X t )p(X t |X t-1 ,U t )p(U t )p(X 1:t-1 , Z 1:t-1 ,U 1:t-1 ). (1)
This filter is based on a few assumptions. First it assumes that the current observation is independent on the past observations, the past states and odometry commands conditionally to the current state. It also features the Markov assumption: the next state is independent on former states, commands and observations conditionally to the previous state and the current command. Finally, it assumes that the actions are independent from the past. All these assumptions are standard and can be found in Kalman filtering and other classical models.

This joint probability distribution allows to formulate the problem as the estimation of the pose X t at time t given the observations Z 1:t and the commands U 1:t :

p(X t |Z 1:t ,U 1:t ) ∝ p(Z t |X t ) ∑ X t-1 p(X t |X t-1 ,U t )p(X t-1 |Z 1:t-1 ,U 1:t-1 ).
(

) 2
This inference involves two distributions to be specified: the observation model p(Z t |X t ) and the motion model p(X t |X t-1 ,U t ), whose parametrizations depend on the robot. In addition, we define a self-confidence metrics and outline the implementation.

Observation model.

Our observation model p(Z t |X t ) = ∏ N sensors i=0,1 p(Z i t |X t )
assumes all sensor noises to be independent, and the ground color to be in the range of [0, 1] (0 being black and 1 being white). We take p(Z i t |X t ) ∼ N (v, σ obs ), for robot pose X t , sensor i, and a corresponding ground intensity v according to the map. The parameter σ obs is selected based on the knowledge of the sensor. Thymio II has two sensors (Fig. 1, center). For binary (black and white) patterns, we chose σ obs = 0.5. For grayscale images, based on measurement on a Thymio II, we set σ obs to 0.15.

Motion model. Based on the model of Eliazar et al. [START_REF] Eliazar | Learning probabilistic motion models for mobile robots[END_REF], we assume that the motion has a Gaussian error model, hence p(X t | X t-1 ,U t ) ∼ N (µ t , Σ t ). The mean µ t is built by accumulating the estimated displacements by dead-reckoning between times t -1 and t. Therefore, if ∆ x t , ∆ y t , ∆ θ t are the displacement between t -1 and t, expressed in the robot local frame at t -1, µ t is:

µ t =   x t y t θ t   with x t y t = x t-1 y t-1 + R(θ t-1 ) ∆ x t ∆ y t θ t = θ t-1 + ∆ θ t (3)
where R(θ ) is the 2-D rotation matrix of angle θ . The 3 × 3 diagonal covariance matrix Σ t is a function of the distance traveled, the amount of rotation, and two parameters α xy , α θ :

Σ t =   σ 2 xy,t 0 0 0 σ 2 xy,t 0 0 0 (α θ |∆ θ t |) 2   (4) with σ xy,t = α xy ∆ x 2 t + ∆ y 2 t .
To cope with the possibility of the robot being kidnapped and therefore its pose becoming unknown, a uniform distribution with a weight p uniform is added to X t . The parameters α xy , α θ and p uniform are estimated using maximum likelihood (Sect. 5).

Implementations. We compare two variants of this filter. In Markov Localization, the distributions are discretized using regular grids [START_REF] Fox | Markov localization for mobile robots in dynamic environments[END_REF]. For our experiments, the x, y cell resolution is 1 cm and the angular resolution varies from 20 • (18 discretization steps for 360 • ) to 5 • (72 discretization steps for 360 • ). The estimated pose is the coordinates of the cell of maximum probability. In Monte Carlo Localization [START_REF] Dellaert | Monte Carlo localization for mobile robots[END_REF], the distributions are represented using samples in a particle filter. In order to extract a single pose estimate out of the many particles, we find a maximum density area in which we average the particles. It is similar to a 1-point RANSAC scheme [START_REF] Fischler | Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography[END_REF]. We implemented both algorithms in Python with some Cython (http://www. cython.org) procedures used for time-critical inner loops. The algorithms run on an embedded computer or laptop (Fig. 1, left). Thymio II is programmed through the ASEBA framework [START_REF] Magnenat | ASEBA: A Modular Architecture for Event-Based Control of Complex Robots[END_REF], which connects to Python using D-Bus.

Self confidence. We define a self-confidence term that corresponds to the ratio of the probability mass of p(X t ) that is within a distance d xy and an angle difference d θ to the estimated pose. In our experiments, we use d xy = 3 cm and d θ = 10 • .

Theoretical analysis of convergence

One can estimate the time required for the robot to localize itself in a given space, by comparing the information needed and the information gained while traveling.

Information need. For the Markov Localization approach, there is a given number of discrete cells. The amount of information needed to unambiguously specify one among them all is H loc = log 2 (N cells ), with N cells the number of cells.

Information gain. We can estimate the information gain at each time step. Let us consider the case of a binary map and a binary sensor. This sensor ideally yields 1 bit of information per measurement. In practice, there is a loss in information due to the sensor noise, characterized by the p correct probability of the sensor to be correct:

H noise = H b (1 -p correct ), (5)
where H b is the binary entropy function:

H b (p) = -p log 2 (p) -(1 -p) log 2 (1 -p).
We also need to take into account that the sensor measurements are not completely independent. For example, when the robot is not moving, it always observes the same place and thus cannot really gain additional information besides being sure of the intensity of the current pixel. In a discretized world, we thus need to estimate the probability of having changed cell in order to observe something new, which depends on the distance traveled and the size of the cells. This problem is equivalent to the Buffon-Laplace needle problem of finding the probability for a needle thrown randomly on a grid to actually intersect the grid3 [START_REF] Laplace | Théorie analytique des probabilités[END_REF]. In our case, the probability of changing cell is given by:

p diff = 4dh -d 2 πh 2 , (6)
with d the distance traveled and h the size of the cells. We can then compute the conditional entropy for two successive ideal binary measurements O t-1 and O t separated by d based on the conditional probability. There are two cases: either the robot has not moved enough to change cell (with probability 1p diff ) and the new observation is the same as the old, or the robot has changed cell (with probability p diff ) and the new observation has the same probability to be the same or the opposite of the old. This can be summarized by the following conditional probability distribution (b/w = black/white):

p(O t = {b/w} | O t-1 = {b/w}) = (1 -p diff ) + p diff /2 p diff /2 p diff /2 (1 -p diff ) + p diff /2 (7)
After rearranging the terms of the conditional entropy, the loss of information due to redundancy in the traveled distance is:

H loss,d = 1 -H b (p diff /2). (8)
There is also redundancy between several sensors placed on the same robot. The probability that they see the same cell based on the distance between them is exactly the same as the probability of a sensor to see the same cell after a displacement of the same distance. The information loss due to the redundancy from the sensor placement is noted H sensors and follows the same formula as H loss,d , but with d being the distance between the two sensors in Eq. 6.

Finally, we can approximate the information that our robot gathers at each time step by assuming that the trajectory of the robot does not loop (no redundancy between distant time steps) and that the trajectories of the sensors are independent. The information gathered in a time step is then: [START_REF] Park | An approach for mobile robot navigation under randomly distributed passive rfid environment[END_REF] This formula also ignores the uncertainty in the robot motion. With these assumptions, it is an upper bound on the average information gain.

H(O 1 t , O 2 t | O 1 1:t-1 , O 2 1:t-1 ) = H(O 1 t , O 2 t | O 1 t-1 , O 2 t-1 ) = H(O 1 t | O 1 t-1 ) + H(O 2 t | O 2 t-1 ) -H(O 2 t | O 1 t ) = 2 • (1 -H noise -H loss,d ) -H sensors
Outlook. Faster localization can be achieved by moving at a greater speed to reduce redundancy in the successive measurements, with proportional increase in sampling frequency. If designing a new robot, better sensors would reduce cross-over noise. Setting the sensors apart would also reduce the redundancy between their information but, for our specific grid size, they are sufficiently separated in Thymio II.

Empirical analysis of performance

To evaluate the performance of our localization algorithms, we remotely controlled the robot and recorded datasets covering all possible robot motions:

• trajectory 1 and 2: The robot alternates forward and backward movements with rotations on spot, at a speed of 3-5 cm/s. • linear trajectory: The robot simply goes straight ahead along the x-axis of the map, at a speed of 3-5 cm/s. • trajectory with kidnapping: The robot alternates phases of forward movement, at a speed of 15 cm/s, and turning on spot. To test the algorithm's ability of recovering from catastrophic localization failures, we perform "robot kidnapping" by relocating the robot to another part of the map every minute.

The robot moves on a 150×150 cm ground pattern containing 50 × 50 cells of 3×3 cm, each randomly black or white (Fig. 1, right). The robot is connected to ROS to synchronize its sensor values and odometry information with ground-truth data from Vicon, sampled at a period of 0.3 s. We chose this period so that with basic trajectories, at maximum speed the robot travels approximately half the length of one cell between every sample. Parameter estimation. We estimated the noise parameters of the motion model using maximum likelihood, considering the error between the ground truth and the odometry data in the local frame between two time steps. Using trajectory 1 and trajectory 2, we found the values for α xy and α θ to be in the order of 0.1. Similarly, using trajectory with kidnapping, we also found the value for p uniform to be in the order of 0.1.

Basic localization. Fig. 2 shows the error in position and orientation for the first two trajectories. In these plots, distance traveled represents the cumulative distance traveled by the center point between the two ground sensors of the robot.

For the Markov Localization approach, all discretization resolutions allow the robot to localize down to a precision of 3 cm and 5 • . However, in trajectory 1, the resolution of 18 discretization steps is not enough to keep tracking the orientation at a distance traveled of 50 cm and 80 cm. These both correspond to the robot rotating on spot. Finer discretizations do not exhibit this problem, they are more robust and have similar precision. Therefore, we see that an angular discretization of 36 (10 • resolution) is sufficient to provide accurate tracking. In trajectory 2, we see that an angular discretization of 54 allows for a better angular precision than 36, but 72 does not improve over 54. All discretizations provide equal position precision.

For the Monte Carlo Localization approach, we see that on trajectory 1, the robot localizes already with 50k particles, but in twice the distance it takes with 100k particles. Increasing the number of particles beyond this value only marginally decreases localization time. While 50k particles are sufficient to localize on this run on average, in some trials, the robot loses orientation when it turns on spot. On trajectory 2, using 50k particles is not enough to localize the robot. Increasing this number to 100k leads to a good localization, except after the robot has traveled Fig. 2 The error between the estimation by the localization algorithm and the ground truth on trajectory 1 and trajectory 2. For Monte Carlo Localization, the solid lines show the average over 10 trials, while the light dots show the individual trials.

130 cm. This corresponds to a long moment during which the robot rotates on spot, leading to less information acquisition, and therefore degraded performances.

Overall, both approaches have similar accuracy. When angular precision is critical, the Monte Carlo Localization approach might achieve better performance, as the Markov Localization approach is limited in precision by its angular discretization. Fig. 3 The error between the estimation by the localization algorithm and the ground truth on 10 segments from the first two trajectories. The solid lines show the median while the light dots show the individual trials.

Markov Localization for different number of discretization angles

Monte Carlo Localization for different number of particles

Distance to converge. Fig. 3 shows the error in position for 5 different starting points in each of the first two trajectories, in which the robot moves at a speed of 3-5 cm/s. We see that, with the Markov Localization approach, the correct pose is found after about 20 cm. There are also outliers: this happens when the robot is turning on spot, in which case there is not enough information to localize the robot. With 400k particles, the Monte Carlo Localization approach also converges after about 20 cm. Decreasing the number of particles quickly increases the distance needed for convergence, reaching 60 cm for 100k particles. Using only 50k particles, some trajectory segments fail to converge within 80 cm length.

It is interesting to compare these distances with theoretical estimates (see Sect. 4). One difficulty is that the theoretical model specifies the probability p correct of the sensor to be correct. In our observation model, instead, we specify σ obs the noise of the sensor. Therefore, we propose to compute p correct from σ obs by considering a sensor positioned over a black ground, and by assuming that all values below a threshold of 0.5 (0 being black and 1 being white) are correctly read:

p correct = p(Z < 0.5|X = 0) = 0.5 -∞ N (0, σ obs ) (10)
In these runs, we have assumed σ obs = 0.5, leading to p correct = 0.84 (Eq. 10) and H noise = 0.63 (Eq. 5). Our robot moves at 3 cm/s with a time step of 0.3 s on a grid of 3 cm×3 cm cells in which the color is known to be similar; this yields H loss,d = 0.33 (Eq. 8). Moreover, the sensors are 2.2 cm apart, which yields H sensors = 0.041. As such the robot gathers on average at most 0.036 bit per time step, or 0.040 bit/cm (Eq. 9). With a 150 cm×150 cm environment discretized with cells of 1 cm and 5 • angle, the amount of information needed for the localization is H loc = 20.6. This means that, on average, the robot should not be localized before traveling around 520 cm 4 . This distance is far larger than the observed one of about 20 cm. The reason is that using σ obs = 0.5 for the binary case was an arbitrary choice and therefore the value of p correct does not correspond to the reality.

Computing the value of p correct to match the observed convergence distance by inverting the computation, we find 0.97, corresponding to σ obs = 0.26. If we use σ obs = 0.15 as measured on grayscale images, we have p correct = 0.99957. This leads to a minimum theoretical localization distance of about 14.3 cm. This value is slightly lower than our experimental results, which makes sense as it is a lower bound. Moreover, this low bound would be attained with a perfect filter observing a perfect pattern. Our filter is not perfect, because we have run it with an overestimated σ obs of 0.5. Nevertheless, the filter works well, showing that it degrades gracefully with respect to imprecision in its parameters. This is important in practice, as users might not be able to provide extremely precise parameters.

Effect of map size. Fig. 4 shows the effect on the distance of convergence when the map size is reduced. The robot runs linearly on one quarter of the map, while the Markov Localization is performed on the whole map, half of it, and a quarter of it. We see that reducing the map size does reduce the distance traveled necessary to converge, in accordance to the theory, because less information have to be acquired to reduce the uncertainty of the pose. Fig. 5 The error between the estimation by the localization algorithm and the ground truth, and the self confidence of the algorithm, on the run with kidnapping. For Monte Carlo Localization, the solid lines show the median over 10 trials, while the light dots show the individual trials. The gray areas show the time during the robot is being held in the air at the occasion of kidnapping.

Robot kidnapping. Fig. 5 shows the error in position, orientation and the self confidence for the run with kidnapping. In this run, the robot is kidnapped twice, after having traveled 550 cm and 1000 cm. It takes the robot approximately 100 cm to re-localize, and does so successfully with both Markov and Monte Carlo Localization approaches. This difference of distance with previous runs is mostly due to the speed of the robot, which is about 5 times faster. With the Markov Localization approach, all discretization resolutions are approximately equivalent in position performance, but 18 leads to a lower orientation precision, as well as to a lower self confidence, due to the large discretization step. Other resolutions have a confidence of about 0.5 when the robot is localized, and this value drops below 0.1 after kidnapping, clearly showing that the algorithm is able to assess its status of being lost.

With the Monte Carlo Localization approach, the robot localizes most of the time with 100k particles, and always with 200k particles or more. With 50k particles, the robot eventually localizes, but this might take more than 2 meters of traveled distance. We see that the self confidence increases with more particles, and, similarly to the Markov Localization approach, drops after kidnapping. This confirms the effectiveness of the self confidence measure. 1 The execution duration of one step for the two algorithms.

Computational cost. Table 1 shows the execution duration of one step for the two algorithms with different parameters. These data were measured on a Lenovo laptop T450s with an Intel Core i7 5600U processor running at 2.6 GHz, and are averages over the two first trajectories. We see that with the Markov Localization approach, the duration scales linearly with the number of discretization angles. With the Monte Carlo Localization approach, the scaling is linear but amortized (50k particles is not twice faster as 100k). This is due to the selection of pose estimate, which uses a RANSAC approach and is therefore independent of the number of particles. For similar localization accuracy, the Monte Carlo Localization approach is slower than the Markov Localization approach, and therefore we suggest to use the former with an angular discretization of 36 in practical applications. However, the Monte Carlo Localization approach might be preferred if a high angular precision is required, but at least 100k particles are necessary for proper localization. Fig. 6 The image, estimated trajectory and self confidence for different 59 × 42 cm grayscale patterns printed on A2 papers with a resolution of 1 pixel/cm. The robot is remotely controlled to draw a 8-shape figure, with similar motor commands for each image. Markov Localization with 36 discretization angles used. The color of the lines vary from red (confidence 0) to green (confidence 1). Trajectories are plotted starting from a confidence level of 0.2.

Real-time localization

on A2 paper sheets with a resolution of 1 pixel/cm. We used Markov Localization with an angular discretization of 36 and a spatial resolution of 1 cm; the localization is performed every 0.4 s. The localization algorithm runs on a Lenovo laptop X1 3rd generation with an Intel Core i7-5500U processor running at 2.4 GHz, at a CPU load of approximately 80 %. The algorithm localizes the robot globally on all images. The images with more contrast lead to a more robust and faster localization, while the ones with lower contrast lead to more imprecise trajectories. In Fig. 6, we see that the parts of trajectories in red are less precise than the ones in green. This shows that the self confidence measure is effective in assessing the quality of the localization. Embedded use. The runtime cost is dominated by the motion model, as it involves convoluting a small window for every cell of the probability space. With the current parameters, this leads to approximately 250 floating-point multiplications per cell. For comparison, VideoCore R IV, the built-in GPU in the Raspberry Pi 3 (costing e 33), has a peak processing power of 28 GFLOPS. Assuming that, due to limitations in bandwidth and dependencies between data, it can only be used at 10 % of its peak floating point throughput, it could run a GPU version of our algorithm at 10 Hz on a map of 1.8 by 1.8 meters with a resolution of 1 cm. Therefore, our approach can truely be used for distributed collective experiments with affordable robots.

Scalability. The computational speed of the system and the necessary distance to converge scale linearly with the size of the localized area. Also, a prolonged use of the robot might wear off the ground, slightly shifting the grayscale intensities and jeopardizing the localization. Nevertheless, modern printing options allow to use strong supports that only degrade slowly with time. Therefore, our system can be helpful for conducting collective experiments in a laboratory environment.

Optimality. An interesting question is what is an optimal pattern. However, this question has two sides. On the one hand, a pattern could be optimal in term of quantity of information, allowing the robot to localize as fast as possible. In that case, a white-noise pattern would be ideal. On the other hand, if the robot has to interact with humans, or only be monitored by humans, such a pattern would be highly disturbing. Our experimental results with paintings and drawings show that, even with a pattern having a lot of regularities and symmetries, our system is still able to localize. Therefore, we believe it is usable with natural patterns such as photographies, drawings and grayscale schematics.

Conclusion

In this paper, we have implemented and empirically evaluated Markov-and Monte Carlo-based approaches for localizing mobile robots on a known ground visual pattern. Each robot requires only inexpensive infrared sensors and approximate odometry information. We have shown that both approaches allow successful localization without knowing the initial pose of the robot, and that their performances and computational requirements are of a similar order of magnitude. Real-time localization was successful with a large variety of A2 grayscale images, using the Markov Localization approach with 36 discretization steps for angle. Should larger patterns be desired, the code could be further optimized by implementing it in the GPU, allowing it to run on inexpensive boards such as the Raspberry Pi.

In addition, we have outlined, and empirically validated, a method to estimate the localization performance in function of the sensor configuration. This method provides a guide for taking decisions about the placement of sensors in a future robot design: localization performance can be improved by placing the sensors far apart on a line perpendicular to the direction of movement of the robot; moreover, more sensors allow for collecting more information, if they are separated by the size of the smallest visual structure in the map.

These contributions to the state of the art enable absolute positioning of inexpensive mobile robots costing in the e 100 range. In the context of distributed collective robotics, they provide a solid base on which to build navigation or behavioral algorithms.

Fig. 4

 4 Fig.4The error between the estimation by the localization algorithm and the ground truth, using Markov Localization, for different map sizes on linear trajectory. The solid lines show the median over 3 different trajectory parts, while the light dots show the individual parts.

Fig. 6

 6 Fig.6shows the performance of the online, real-time localization of a Wireless Thymio II (Fig.1, left). We tested different grayscale images of 59 × 42 cm, printed

Table

	duration [s] 0.64	1.43	2.15	2.94	1.97	2.97	6.08	12.22
	algorithm		Markov Localization			Monte Carlo Localization
	parameter	18	36	54	72	50k	100k	200k	400k
			discretization angles			particles

The needle is the segment joining the start and end points of the robot movement and the grid is the borders of the cells.

see https://github.com/epfl-mobots/thymio-ground-localisation/ blob/master/theory/d_conv_from_pcorrect.py

Acknowledgements The authors thank Emmanuel Eckard and Mordechai Ben-Ari for insightful comments on the manuscript and Ramiz Morina for his drawings. This research was supported by the Swiss National Center of Competence in Research "Robotics".


Julien Pinter
email: julien.pinter@gmail.com

Christian Bordes

Massimo Giuliodori

Gunther Capelle--Blancard

Rodolphe Dos

Santos Ferreira

Michael Mcmahon

Evren Örs

Jézabel Couppey--Soubeyran

Lucas Rossini

Yannick Lucotte

Marc Pourroy

Aurélien Leroy

Amine Tazi

Kenneth De Beckker

The Eurozone deposit rates' puzzle: choosing the right benchmark 1

Keywords: deposit rates, euribor, cointegration, panel estimates, banks JEL classification: E43, E50, G10, G21

The paper proposes an alternative benchmark to the EURIBOR to analyze the post--crisis puzzling behavior of deposit rates in the Eurozone. Using bank--level CDS data for 6 major euro--countries, we build a simple country--level index for banks' cost of unsecured funding. The use of this index instead of the traditionally used EURIBOR restores the cointegration relationship between deposit rates and their reckoned opportunity cost. It also suggests that deposits have actually not been significantly over--remunerated in most euro area countries since the financial crisis, in contrast with what is often argued. Our index appears as a good alternative to the EURIBOR, which we show has become irrelevant for many countries.

Introduction

In this study, we focus on the main instrument of bank funding in the Eurozone, namely deposits from non--banks (European Central Bank 4 , 2012a) 5 . A puzzling feature about banks deposits is that they seem to have been remunerated at relatively high rates compared to bank's wholesale funding costs since the 2008 financial crisis. Deposit rates, which were closely following the Euro Interbank Offered Rate (EURIBOR henceforth) before the crisis, have diverged substantially from it since then (Figure 1). Banks thus seem to have been willing or constrained to pay more for deposits than what they pay for funding on the wholesale market. This is particularly true for periphery countries (Figure 7 and8, Appendix 1). In an economy such as the Eurozone where banks are the main vector of monetary policy transmission and where high bank funding costs have been a key factor preventing lending rates from adjusting downwards [START_REF] Illes | Why did bank lending rates diverge from policy rates after the financial crisis?[END_REF], the issue appears of prime importance.

Figure 1: Deposits rates and the EURIBOR in the Eurozone

Source: ECB

Few papers mention explanations for this post--crisis phenomenon. ECB (2012a) and ECB (2009) mention the persistent risks on unsecured funding, pushing banks to attract deposits at higher prices. ECB (2012a) also insists on the regulation incentives to increase stable source of funding. Daracq [START_REF] Daracq Paries | The Retail Bank Interest Rate Pass--Through: The Case of the Euro Area During the Financial and Sovereign Debt Crisis[END_REF] mention banks' reluctance to pass low rates on their depositors.

This paper takes another approach and argues that this "puzzle" is mainly due to the fact that the EURIBOR has become an inappropriate proxy for banks wholesale funding costs, and thus has become irrelevant as a benchmark. The reason for the use of EURIBOR as a benchmark is that banks traditionally price deposits relative to the cost of unsecured funding 4 ECB henceforth.

5 At the end of 2011, non--bank deposits were on average about 35% of banks liabilities, with a median value reaching 43.5% (ECB, 2012a). Hence, deposits weigh heavily on bank funding costs in the Eurozone. [START_REF] Cadamagnani | A Bank within a Bank: How a Commercial Bank's Treasury Function Affects the Interest Rates Set for Loans and Deposits[END_REF][START_REF] Sander | Convergence in euro--zone retail banking? What interest rate pass--through tells us about monetary policy transmission, competition and integration[END_REF]. This cost of unsecured funding was tightly linked to EURIBOR for the majority of banks before the crisis, but it then substantially diverged from it as bank funding conditions became severely heterogeneous [START_REF] Ecb | Heterogeneity in Euro Area financial conditions and policy implications[END_REF]. In effect, this means that the cost of unsecured funding for Spanish banks for instance have substantially diverged from the cost of unsecured funding for German banks after the crisis.

Using the EURIBOR for these two countries will lead one to misleadingly assume similar or connected costs of funds, thereby leading to distorted conclusions.

Using the standard theory on the structure of interest rates, we build an index for banks' cost of unsecured funding at the country--level, using a risk--free rate and a weighted average of banks' risks premium (as measured by their CDS). This index is very close to the EURIBOR before the 2008 financial crisis in each country we analyze (Figure 2), but then substantially diverges from it in most countries, especially for periphery--countries. We obtain two results from the use of our index. First, the use of our index restores the cointegration relationship between deposit rates and the cost of unsecured funding. Panel cointegration tests indeed fail in the post crisis period with the EURIBOR, but lead the expected result with our index. Running similar tests at the country--level shows that the relationship with the EURIBOR has become especially irrelevant for periphery countries, unlike the relationship with our index. Our index thus appears as a more relevant benchmark than the EURIBOR to understand deposit pricing in the Eurozone. Second, the use of our index shows that deposits have not been significantly over--remunerated since the 2008 financial crisis, unlike what is commonly argued. Building counterfactuals rates from standard panel models, we show that 72% of the seemingly "over--remuneration" of deposits is simply due to the use of the EURIBOR as a benchmark. The rest of the paper is organized as follows: section 2 presents the conventional theoretical framework for deposit pricing, reviews the shortcomings of the EURIBOR and explains how we build our index. Section 3 presents the data and the empirical method. Section 4 presents the empirical results, and section 5 the graphical analysis. Section 6 concludes.

Theoretical considerations

2.1 The conventional framework for deposit pricing It is common in academic studies since the work of [START_REF] Cottarelli | Financial structure, bank lending rates, and the transmission mechanism of monetary policy[END_REF] at least to link deposit or lending rates to money market rates, the later being considered as a good proxy for the marginal cost of funding. In the conventionally used model, the bank interest rate (BR) is set by banks depending on a marginal cost price (MC): BR = θ 0 + θMC The coefficient θ 0 is a markup over the banks' marginal costs due to transaction costs, while the size of the parameter θ depends on the demand elasticity for the banking product 6 . In the case of deposits, MC can be seen as the opportunity cost [START_REF] Sander | Convergence in euro--zone retail banking? What interest rate pass--through tells us about monetary policy transmission, competition and integration[END_REF]. If banks were operating effectively in a perfectly competitive world, moves in MC would be entirely reflected in BR: the coefficient θ would be equal to one. Such an approach is widely used in the literature, for example in De Bond (2002), [START_REF] Sander | Convergence in euro--zone retail banking? What interest rate pass--through tells us about monetary policy transmission, competition and integration[END_REF][START_REF] De Graeve | Competition, transmission and bank pricing policies: Evidence from Belgian loan and deposit markets[END_REF], ECB (2009), [START_REF] Rocha | Interest rate pass--through in Portugal Interactions, asymmetries and heterogeneities[END_REF], Van Leuvensteijn et al. (2013[START_REF] Daracq Paries | The Retail Bank Interest Rate Pass--Through: The Case of the Euro Area During the Financial and Sovereign Debt Crisis[END_REF], [START_REF] Leroy | Heterogeneous Monetary Transmission Process in the Eurozone: Does Banking Competition Matter[END_REF] among others. We do not call into question this approach, but the way it is empirically implemented.

For deposit rates, the marginal cost price considered usually corresponds to the price of unsecured funding. This approach is justified by the fact that banks traditionally price their retail products in relation to the marginal cost of unsecured funding [START_REF] Cadamagnani | A Bank within a Bank: How a Commercial Bank's Treasury Function Affects the Interest Rates Set for Loans and Deposits[END_REF]. This cost is usually approximated by a market interest rate for unsecured funding, the EURIBOR in the case of the Eurozone. As it appears in De Bond (2002) e.g., the underlying idea behind this widely spread custom is that market interest rates are seen as the most appropriate marginal cost prices due to their "accurate reflection of the marginal costs faced by banks". Consistently, the vast majority of empirical studies analyzing the pricing of banks' retail products in the Eurozone use EURIBOR rates as benchmarks7 . We are arguing that this custom is not appropriate anymore.

6

In the case of deposits, switching costs, information asymmetries and market power will for example generally imply θ<1. Hutcheson (1995) provides solid theoretical foundations on this point.

The irrelevance of the EURIBOR in the post--crisis world

The EURIBOR is computed based on a survey in which some representative banks (by their activity on the euro money market) say how much they would charge a prime bank for an unsecured loan on the money market [START_REF] Taboga | What is a prime bank? A Euribor--OIS spread perspective[END_REF]. Prime banks are supposed to be the less fragile banks, what makes the EURIBOR sometimes called the "best rate for the best banks" 8 . To the extent that loans with no guarantee are considered, the EURIBOR is widely used as a proxy for the cost of unsecured funding in economic studies 9 .

The problems of using the EURIBOR as a benchmark for the cost of unsecured funding have been revealed by the 2008 financial crisis. A dramatic change that occurred is that funding conditions have become extremely heterogeneous (ECB, 2012b). As a consequence, the EURIBOR has become by essence non--representative of the funding conditions for the majority of the banks, especially for the banks located in the periphery of the Eurozone, which have been perceived as much more risky. Figure 3 illustrates this fact by showing the average CDS of the 10 banks with respectively the lowest and highest CDS in our sample. As funding conditions became extremely heterogeneous the rate at which non--prime banks could borrow became heterogeneously disconnected from the rates at which prime banks can borrow. Under these conditions, econometric or graphical analysis using the EURIBOR will automatically fail to capture any meaningful long--term relationship with deposit rates for non--prime banks 10 .

8

See [START_REF] Taboga | What is a prime bank? A Euribor--OIS spread perspective[END_REF] for a discussion on the EURIBOR. 9

It is worth mentioning that the EURIBOR can be an appropriate benchmark for banks others than "prime banks" also when the risk premium the later would pay when borrowing unsecured (compared with prime banks) is negligible or roughly constant over time. This can explain why the use of the EURIBOR as a proxy for the cost of unsecured funding in pre--crisis economic studies used to be not problematic.

10

In addition to the heterogeneity in funding conditions, two other observations may contribute to make the EURIBOR less relevant: the number of "prime banks" has severely declined [START_REF] Taboga | What is a prime bank? A Euribor--OIS spread perspective[END_REF] and the liquidity on the unsecured interbank market has severely decreased since the financial crisis [START_REF] Babihuga | Bank Funding Costs for International Banks[END_REF]Financial Times, 2010). The former argument implies that the EURIBOR has become of direct relevance for only few banks in the post--crisis world. The later implies that the EURIBOR may have become an irrelevant proxy even for prime banks' cost of unsecured funding. Since the

A new proxy for banks' unsecured funding costs

We build a new index for banks' cost of unsecured funding at the country--level 11 . Similar to [START_REF] Illes | Why did bank lending rates diverge from policy rates after the financial crisis?[END_REF], we use the standard structure of interest rates to proxy the interest rate on unsecured debt for bank n at time t:

Interest rate n,t = risk--free rate n,t + risk premium n,t + liquidity premium n,t The risk--free rate is proxied by the Overnight Index Swap rate (OIS rate henceforth) for a given maturity, which is supposed to reflect only the policy interest rate expectations 12 . The risk premium is proxied by the Credit Default Swap (CDS henceforth) of the bank n in percentage points for comparable maturities 13 . Thus our index for the real cost of unsecured funding for bank n at time t is simply:

𝐶𝑜𝑠𝑡 𝑜𝑓 𝑢𝑛𝑠𝑒𝑐𝑢𝑟𝑒𝑑 𝑓𝑢𝑛𝑑𝑖𝑛𝑔 !,! = 𝑂𝐼𝑆 ! + 𝐶𝐷𝑆 !,! With 𝑂𝐼𝑆 ! the OIS rate at time t for the maturity considered and 𝐶𝐷𝑆 !,! the CDS of the bank for comparable maturities in percentage points. Using a weighted average of this indicator for all the banks in the country i we obtain an index at the country level for the cost of unsecured funding:

𝐼𝑛𝑑𝑒𝑥 !,! = 𝐷𝑒𝑝𝑜𝑠𝑖𝑡 𝑠ℎ𝑎𝑟𝑒 !,! * 𝐶𝑜𝑠𝑡 𝑜𝑓 𝑢𝑛𝑠𝑒𝑐𝑢𝑟𝑒𝑑 𝑓𝑢𝑛𝑑𝑖𝑛𝑔 !,! ! !!!
Where 𝐷𝑒𝑝𝑜𝑠𝑖𝑡 𝑠ℎ𝑎𝑟𝑒 !,! is the customer deposits of the bank n in proportion of the total deposits of the banks in country i (weighting criteria) 14 .

In contrast with the EURIBOR, our index thus takes into account the heterogeneity in funding conditions between prime and non--prime banks. A simple way to see this is that we will now have two different costs of funds for German and Spanish banks (Figure 9, Appendix 1).

EURIBOR is based on declarations rather than on actual rates, it may indeed have become disconnected from the rates at which banks can effectively borrow on the unsecured market, for example by issuing debt.

11 Building a proxy is necessary insofar as there are no monthly data available for bank--specific unsecured debt with a given maturity.

12 See Taboga (2014) for a discussion.

13

We do not explicitly take into account liquidity premia for two reasons: first there is a lack of appropriate data, second CDS already contain a liquidity premium that might thus make up for this omission.

Empirical approach

3.1 Methodology We consider the standard framework in which deposit rates are linked to the cost of unsecured funding, discussed in subsection 2.115 . We consider the usual specification with EURIBOR (1) and the specification with our index (2).

𝐷𝑒𝑝𝑜𝑠𝑖𝑡 𝑟𝑎𝑡𝑒 !,! = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ! + 𝛽 𝐸𝑈𝑅𝐼𝐵𝑂𝑅 !,! + 𝜀 !,! (1) 𝐷𝑒𝑝𝑜𝑠𝑖𝑡 𝑟𝑎𝑡𝑒 !,! = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ! + 𝛽 𝐼𝑛𝑑𝑒𝑥 !,! + 𝜀 !,! (
2) The deposit pricing behavior of banks should imply a linear relationship between deposit rates and the cost of unsecured funding in the long run. No other variable is needed if this long--term relationship holds in theory 16 . In econometrics terms, considering the variables as integrated of order 1, this means that the two variables should be cointegrated. In what follows we perform cointegration tests with, in turn, EURIBOR and our index used as a proxy, in a panel set--up at first and then for each country of our sample.

Data

Our dataset consists of monthly data from 2003:1 to 2015:1.

We use country--level deposit rates from ECB. In order to precisely capture the pricing behavior of banks, we use deposit rates with an agreed maturity a) with a maturity lower than 1 year b) with a maturity between 1 and 2 years and c) with a maturity over 2 years. Deposits from households and non--profit institutions are considered in that they represent the most important share of deposits and the one appearing as highly remunerated (Figure 1).

We obtain banks' CDS from Markit and restrict our sample to retail banks. We select the countries based on data availability for bank CDS17 : Austria, Germany, France, Italy, Spain, Netherlands. One--year CDS contracts on senior unsecured debt are considered in order to closely match deposits' maturity. Banks' total customer deposits data used for weighting are from Bankscope and linearly interpolated. OIS rates for a maturity of 3 months and 1 year are considered.

Standard panel unit--root tests confirm that all our data are integrated of order 1 (Appendix 3), thus making a cointegration analysis appropriate.

Cointegration tests: EURIBOR versus our index 4.1 Panel analysis

We first make cointegration tests in a panel framework, where we naturally expect more power from the tests given the potential correlation of shocks hitting eurozone economies. Following the connected literature [START_REF] Rocha | Interest rate pass--through in Portugal Interactions, asymmetries and heterogeneities[END_REF][START_REF] Bernhofer | New evidence of heterogeneous bank interest rate pass--through in the euro area[END_REF]Leroy and Lucotte, 2014 ;[START_REF] Illes | Why did bank lending rates diverge from policy rates after the financial crisis?[END_REF] and given our small N dimension, we primarily use the Westerlund tests [START_REF] Westerlund | Testing for error correction in panel data[END_REF] for cointegration (see Appendix 3). We focus on the post crisis period, where the puzzle appears (results for the pre--crisis period are given in Appendix 4 and indicate cointegration with both the EURIBOR and our index). In order not to take into account the huge volatility following Lehman brother's bankruptcy at the end of 2008, we start our analysis in 2009:1. Table 1 reports the results, where in the first column we stick to the basic approach of [START_REF] Westerlund | Testing for error correction in panel data[END_REF] and in the second column we account for cross--sectionnal dependence using the bootstrap approach of [START_REF] Westerlund | Testing for error correction in panel data[END_REF] with 800 replications (maximum allowed in our software). Note: The Ga and Gt test statistics test H0: a i = 0 for all i versus H1: a i < 0 for at least one i, where a i is the error correction coefficient (see [START_REF] Westerlund | Testing for error correction in panel data[END_REF] and Appendix 3). These statistics start from a weighted average of the individually estimated a i 's and their t--ratio's respectively. Rejection of H0 should be taken as evidence of cointegration of at least one of the cross--sectional units. The Pa and Pt test statistics pool information over all the cross--sectional units to test H0: a i = 0 for all i vs H1: a i < 0 for all i. We use an automatic selection of lags from the AIC criteria. We impose a constant in the cointegration relationship as suggested by the theory. Following the literature, EURIBOR 3months is used for the cointegration relationship with deposits with a maturity lower 1 year (Deposit rates < 1 year), EURIBOR 1 year otherwise. Similarly, OIS 3 months is used in our index for deposits with a maturity lower than 1 year, OIS 1 year otherwise.

Documents de travail du Centre d'Economie de la Sorbonne -2016.53

All in all, the results indicate that there is no cointegration relationship between deposit rates and the EURIBOR in the post--crisis world, unlike what could be found in the pre--crisis period (see Appendix 4).

Only for deposits with a maturity from 1 to 2 years the Gt statistic suggests the presence of cointegration for at least one panel, but the result barely holds when we consider the robust version of the Westerlund test. In contrast, the cointegration relationship can be inferred with our index, and this for all the deposit maturities considered here.

In Appendix 4, we show that the conclusions are roughly similar when we consider an alternative (residual--based) cointegration test, namely the Pedroni test, although the test is less adapted to our small N case.

Country--level analysis

To better understand what drives these findings, we run cointegration tests at the country-level. We apply the Engle and Granger method. That is we estimate equations ( 1) and ( 2) directly and test for stationarity of the estimated residuals by estimating the following model in each country:

∆𝜀 ! = α𝜀 !!! + 𝑢 ! (3) And testing for H0: α = 0 (unit root) vs H1: α ≠ 0 (no unit root) using the interpolated Dickey--Fuller critical values as a reference 18 . Note that we include lags of the independent variable, with a selection based on the AIC criteria 19 . Results are given for the 5% threshold, we indicate in Table 3 directly whether H0 is rejected or not since the software we use (STATA) doesn't report the p--values. As we can see on Table 2, the data do not give support to any cointegration relationship between EURIBOR and the deposit rates for France, Italy and Spain. In contrast, the cointegration relationship is always captured by our index. Interestingly, the relationship fails with the EURIBOR in countries in which banks have become perceived as riskier.

Documents de travail du Centre d'Economie de la Sorbonne -2016.53

Graphical analysis: EURIBOR versus our Index 5.1 Basic observation

When using our index the picture regarding deposit remuneration changes dramatically, especially for periphery--countries (Figure 5).

Deposits do not appear anymore as significantly over--remunerated since the beginning of the financial crisis 20 (Figure 4 and5). Only after 2013 some persistent "over--remuneration" appears, although often limited. This trend is present in all the six countries we analyzed (Figure 9, Appendix 1) with the exception of the Netherlands. This indicates that other factors that were not present before 2013 are at work since then. It could be argued that the fact we used 1 year CDS premia rather than the ones of a shorter maturity could partly explain the lower gap with our proxy. We take this into account hereafter by basing our comparisons on predicted rates from two different models: the results clearly indicate that this choice doesn't have any key impact on our conclusions. Note also that the picture is barely different when we use CDS 6 months (available for fewer data) instead of 1 year, with sometimes a proxy showing even higher rates than when we use CDS 1 year (what could be explained by the fact the short term risk is sometimes perceived as higher than the long term one). In our sample the difference between the index we could build with data for CDS 6 months and the one we use here is only 0.096 basis points on average, thus really minor.

Observation based on predicted rates

In order to quantify to which extent the misuse of the EURIBOR contributes to make deposit rates appear over--remunerated, we look at the level of deposit rates which is predicted by a simple model using the EURIBOR as a key variable, and then compare this prediction with the one from a similar model using our index as a key variable. The model used is the standard Panel Mean--Group model [START_REF] Pesaran | Estimating long--run relationships from dynamic heterogeneous panels[END_REF]. We estimate the parameters of the models for the pre--crisis period, and then use them to predict deposit rates (details of the estimates are given in Appendix 5).

As we can see on Figure 6, using the EURIBOR leads us to predict significantly lower deposit rates than when we use our index (Figure 10 and 11 in Appendix 1 shows the picture for each country). A simple computation shows that 72% of the seemingly over--remuneration of deposits is actually due to the use of the EURIBOR as a benchmark. Source: Authors' estimates, ECB

Conclusion

This study shows that the EURIBOR has become an inappropriate benchmark to analyze the pricing of deposit rates in the post--crisis period in the Eurozone. We argued that the reason for this is that the EURIBOR doesn't faithfully reflect the cost of unsecured funding for Eurozone banks anymore, given the substantial heterogeneity in bank funding conditions. In contrast, the index we have built appears more relevant. It captures the post--crisis heterogeneity in funding conditions and restores the cointegration relationship between deposit rates and the cost of unsecured funding.

Our index shows that banks' deposits have actually not been significantly "over-remunerated" since the 2008 financial crisis. We find that 72% of the seemingly over-remuneration of deposits is actually explained by the use of the wrong benchmark, namely the EURIBOR. This observation both attenuates the veracity of the statements made in previous works (ECB, 2009;ECB, 2012a;[START_REF] Daracq Paries | The Retail Bank Interest Rate Pass--Through: The Case of the Euro Area During the Financial and Sovereign Debt Crisis[END_REF] and lays the foundations for future research: the behavior of deposit rates can't be analyzed with the EURIBOR anymore.

Further research remains necessary to understand which new factors have been driving deposit rates in the recent years. Our analysis indeed reveals that new factors have been influencing the pricing of deposit rates in Eurozone countries since 2013. For such subsequent research our paper shows the necessity of using alternative proxies to the EURIBOR. We provide a simple and coherent one. The Westerlund test is part of the so--called "regression based" tests. The underlying idea is to test for the absence of cointegration by determining whether there exists an error correction mechanism for individual panel members or for the panel as a whole. Consider the following Error Correction Model (ECM henceforth), where all variables in levels are assumed to be I(1):

∆𝑦 !,! = 𝑐 ! + 𝑎 ! 𝑦 !,! -𝛽 ! 𝑥 !,! + 𝛼 !,! ∆𝑦 !,!!! ! !!! + 𝛾 !,! ∆𝑥 !,!!! ! !!! + 𝑢 !,! (4)
𝑎 ! provides an estimate of the speed of error--correction towards the long run equilibrium for that series i. The 𝛼 !,! and 𝛾 !,! coefficients take into account the short term adjustments between the variables. From this equation, [START_REF] Westerlund | Testing for error correction in panel data[END_REF] computes four statistics. The Ga and Gt test statistics test H0: 𝑎 ! = 0 for all i versus H1: 𝑎 ! < 0 for at least one i. These statistics are built from a weighted average of the individualy estimated 𝑎 ! and their t--ratio's respectively. Rejection of H0 should therefore be taken as evidence of cointegration of at least one of the cross--sectional units. The Pa and Pt test statistics pool information over all the cross--sectional units to test H0: 𝑎 ! = 0 for all i vs H1: 𝑎 ! < 0 for all i. Rejection of H0 should therefore be taken as evidence of cointegration for the panel as a whole.

The main advantages of these tests are that they are very flexible (one can decide on the number of lags J and K to include) and that they allow for an almost completely heterogeneous specification of both the long--and short--run parts of the ECM. This is a major advantage as compared with the alternative Pedroni test (see here below). Westerlund also provides a robust version of these tests which takes into account potential cross--sectional dependence. In this case, robust critical values can be obtained through bootstrapping (see [START_REF] Westerlund | Testing for error correction in panel data[END_REF]).

In our estimates, similar to [START_REF] Sander | Convergence in euro--zone retail banking? What interest rate pass--through tells us about monetary policy transmission, competition and integration[END_REF] we allow for a maximum of 4 lags for the differenced variables and then based the selection of lags on the AIC criteria. We naturally include a constant as our model imposes for the ECM. Robust critical values were obtained considering 800 replications for the bootstrapping.

--Pedroni test [START_REF] Pedroni | Critical values for cointegration tests in heterogeneous panels with multiple regressors[END_REF][START_REF] Pedroni | Panel cointegration: Assymptotic and finite full sample properties of pooled time series tests with an application to the PPP hypothesis[END_REF] introduced seven test statistics that are computed from the basic regressions of one of the pre--supposed cointegrated variable on the other, similar to the Engle and Granger methodology in essence but in a panel framework here.

The test has the advantage to allow the coefficients to vary across individuals, but the disadvantage of making the assumption that the short term relationship relation between the variable is similar to the long term one (same coefficients). It is best adapted to the cases in which both the N and T dimensions are high, therefore the tests is likely to suffer from size Documents de travail du Centre d'Economie de la Sorbonne -2016.53 distortions in our small N model (the reason for which we didn't consider it as our baseline test).

Two kinds of statistics are introduced: group--means statistics (that average the results of individual country test statistics) and panel statistics (that pool the statistics along the within--dimension). Within each category, non--parametric (phi and t) and parametric (ADF and v) test statistics are presented. In the cases in which T is lower than 100, [START_REF] Pedroni | Panel cointegration: Assymptotic and finite full sample properties of pooled time series tests with an application to the PPP hypothesis[END_REF] reports that the ADF statistics have the best power, with v and phi performing worse. For this reason, we focus only on the ADF and t statistics in our study.

The test accounts for cross--sectional dependence by time--demeaning the variable. We naturally apply this method in our model, and we select the number of lags in the residual analysis based on the AIC criteria, still departing from a maximum of 4 lags. Under the null hypothesis of "no cointegration", the test statistics are distributed N (0,1).

  Households Deposits up to one year Documents de travail du Centre d'Economie de la Sorbonne -2016.53

Figure 2 :

 2 Figure 2: Authors' index for the cost of unsecured funding versus EURIBOR

  Documents de travail du Centre d'Economie de la Sorbonne -2016.53

Figure 3 :

 3 Figure 3: Top 10 versus Bottom 10 Eurozone banks CDS

  1-Year CDS Higer 10 Banks 1-Year CDS Documents de travail du Centre d'Economie de la Sorbonne -2016.53

18

  We didn't include any constant or trend in (3) thus we don't need to use theMcKinnon critical values. 19 With a maximum of 4 lags and a minimum of 2 lags in order to avoid any biais affecting the test (setting a minimum doesn't impact our results). Documents de travail du Centre d'Economie de laSorbonne -2016.53

Figure 4 :

 4 Figure 4: EURIBOR, Authors' index and Deposit rates in the Eurozone

  to one year Documents de travail du Centre d'Economie de la Sorbonne -2016.53

Figure 5 :

 5 Figure 5: EURIBOR, Authors' index and Deposit rates in the Eurozone: Core versus Periphery countries

  Documents de travail du Centre d'Economie de la Sorbonne -2016.53

Figure 6 :

 6 Figure 6: Predicted deposit rates with the EURIBOR and with authors' index versus actual deposit rates in the Eurozone:

  to one year Predicted Deposit rates with EURIBOR model Predicted Deposit rates with authors' index model Documents de travail du Centre d'Economie de la Sorbonne -2016.53

Figure 9 :

 9 Figure 9: EURIBOR, Authors' index and Deposit rates in the Eurozone, by countries

Figure 11 :

 11 Figure 11: Share of deposits over--remuneration due to the use of the wrong proxy (deposits with an agreed maturity lower that one year)

Table 1 :

 1 Westerlunds

					tests for cointegration (2009:1 -2015:1)
					Westerlunds tests of cointegration
				Basic		Robust	to	cross--sectional
							dependence
		Variables Ga	Gt	Pa	Pt	Ga	Gt	Pa	Pt
			statistic	statistic	statistic	statistic	statistic	statistic	statistic	statistic
	With	Deposit	0.729	0.248	0.675	0.420	0.641	0.309	0.721	0.542
	EURIBOR	< 1 year
		Deposit	0.076	0.001	0.441	0.125	0.135	0.048	0.596	0.416
		between
		1 and 2
		years
		Deposit	0.267	0.153	0.214	0.046	0.315	0.478	0.451	0.461
		> 2 years
			0.000	0.000	0.000	0.000	0.001	0.006	0.003	0.003
		Deposit	0.002	0.011	0.000	0.000	0.034	0.037	0.054	0.031
		> 2 years

Table 2 :

 2 Test for cointegration in subsamples(2009:1 -2015:1)

	Do the estimated residuals of equation (1) or (2) contain a unit root
	(test at the 5% threshold)?

Table A3 :

 A3 Pedroni tests for cointegration, pre--crisis period (2003m1 -2008m9)

		Pedroni test statistics
		Panel		Group--means
	t	t	ADF	t	ADF
	With EURIBOR Deposit < 1 year	--1.833	2.1	1.11	1.749

See for example[START_REF] Sander | Convergence in euro--zone retail banking? What interest rate pass--through tells us about monetary policy transmission, competition and integration[END_REF][START_REF] De Graeve | Competition, transmission and bank pricing policies: Evidence from Belgian loan and deposit markets[END_REF], ECB(2009), ECB (2012a),[START_REF] Rocha | Interest rate pass--through in Portugal Interactions, asymmetries and heterogeneities[END_REF], Banerjee et al. (2013), Van Leuvensteijn et al. (2013), Daracq Paries et al. (2014), Leroy and Lucotte (2015) among others. Documents de travail du Centre d'Economie de la Sorbonne -2016.53

We also considered weighting CDS by banks' total assets: the indicator obtained by doing so was very similar, so that the results presented here were roughly unchanged. Documents de travail du Centre d'Economie de laSorbonne -2016.53

Note that we do not question this aspect of the literature in this paper. The pricing of deposits using the cost of unsecured funding as a benchmark is often presented as a traditional feature of banks' pricing behavior (seeCadamagnani et al. 2015 eg) and widely spread in the empirical literature. We focus on improving this strand of the literature.

Competition in the banking sector is neglected as we have a short period (7 years, see subsection 3.2) in which this factor is likely not to play a significant role and is possibly taken into account through the constant term.

In order to get a representative proxy a country is selected if we have data for at least four major banks (see Appendix 2). Documents de travail du Centre d'Economie de laSorbonne -2016.53

Source: Authors' estimates, ECB Documents de travail du Centre d'Economie de laSorbonne -2016.53

Note: The t--stat must be compared to the distribution of N(0,1): if | t--stat | > 1,96 then the null "no cointegration" is rejected at the 5% threshold, if | t--stat | > 2,57 then the null is rejected at the 1% threshold. Documents de travail du Centre d'Economie de laSorbonne -2016.53

Documents de travail du Centre d'Economie de la Sorbonne -2016.53

This work was achieved through the Laboratory of Excellence on Financial Regulation (Labex ReFi) supported by PRES heSam under the reference ANR--10--LABX--0095. It benefitted from a

French government support managed by the National Research Agency (ANR) within the project Investissements d'Avenir Paris Nouveaux Mondes (invesments for the future Paris--New Worlds) under the reference ANR--11--IDEX--0006--02.

Appendix

2 0 0 3 m 1 2 0 0 6 m 1 2 0 0 9 m 1 2 0 1 2 m 1 2 0 1 5 m 1 2 0 0 3 m 1 2 0 0 6 m 1 2 0 0 9 m 1 2 0 1 2 m 1 2 0 1 5 m 1 2 0 0 3 m 1 2 0 0 6 m 1 2 0 0 9 m 1 2 0 1 2 m 1 2 0 1 5 m 1

Austria

France Germany 2 0 0 3 m 1 2 0 0 6 m 1 2 0 0 9 m 1 2 0 1 2 m 1 2 0 1 5 m 1 2 0 0 3 m 1 2 0 0 6 m 1 2 0 0 9 m 1 2 0 1 2 m 1 2 0 1 5 m 1 2 0 0 3 m 1 2 0 0 6 m 1 2 0 0 9 m 1 2 0 1 2 m 1 2 0 1 5 m 1 Appendix 3: Details on the empirical approach

Italy

Results of the unit root tests

To study the order of integration of our variables, we use the standard panel unit--root tests used in the literature we are in line with [START_REF] Rocha | Interest rate pass--through in Portugal Interactions, asymmetries and heterogeneities[END_REF]Van Leuvensteijn et al., 2013;Leroy and Lucotte, 2014), namely the Im--Pesaran--Shin test and the Hadri test. These tests are suitable for our small N dimension and accounts for potential heterogeneity, while allowing us to take advantage of our panel dimension. We use the whole sample period when possible (2003:1 -2015:1). We do not include a time trend in the tests since it is clear there is no such trend in the data, and select the number of lags with the AIC criteria (with a maximum of 5). The Im--Pesaran--Shin test has as a nul hypothesis (H0) that all the series have a unit root (versus H1: at least one serie has a unit root). Table A1 shows that for all the series we deal with, H0 cannot be rejected at the conventional thresholds level of 5%. The Hadri test has as a nul hypothesis that all the series are stationary (versus H1: at least one serie has a unit root). As we can see on Table A1, H0 is clearly rejected for all the series we deal with. Doing the tests for the first difference leads us to reject the unit root hypothesis (or for the Hadri test not to reject the stationarity hypothesis) so that we infer the variables are I (1). For the EURIBOR, the standard Augmented Dickey--Fuller test is used, and also shows the series are I(1). Note: The Ga and Gt test statistics test H0: a i = 0 for all i versus H1: a i < 0 for at least one i, where a i is the error correction coefficient (see [START_REF] Westerlund | Testing for error correction in panel data[END_REF] and Appendix 3). These statistics start from a weighted average of the individually estimated a i 's and their t--ratio's respectively. Rejection of H0 should be taken as evidence of cointegration of at least one of the cross--sectional units. The Pa and Pt test statistics pool information over all the cross--sectional units to test H0: a i = 0 for all i vs H1: a i < 0 for all i. We use an automatic selection of lags from the AIC criteria. We impose a constant in the cointegration relationship as suggested by the theory. Following the literature, EURIBOR 3 months is used for the cointegration relationship with deposits with a maturity lower 1 year (Deposit rates < 1 year), EURIBOR 1 year otherwise. Similarly, OIS 3 months is used in our index for deposits with a maturity lower than 1 year, OIS 1 year otherwise.

Documents de travail du Centre d'Economie de la Sorbonne -2016.53

Appendix 5: Pool--mean Group estimates used for rates predictions We use the standard Panel Mean--Group model of [START_REF] Pesaran | Estimating long--run relationships from dynamic heterogeneous panels[END_REF] to estimate the relationship between deposit rates and the EURIBOR (respectively, our index) in the pre-crisis world. The use of an Error--Correction Model is justified by the fact we find cointegration between deposit rates and the EURIBOR (respectively our index) on that period (Appendix 4). The Pooled Mean--Group model is favored over the Mean--Group model following the results from the standard Hausman test. The model estimated is thus the following:

with 𝑦 being deposit rates with a maturity lower than one year, 𝑥 our index in the model (A) and the EURIBOR 3 months in the model (B). 𝑎 is the error--correction coefficient, it provides an estimate of the speed of error--correction towards the long run equilibrium. The 𝛼 !,! and 𝛾 !,! coefficients take into account the short term adjustments: 𝛼 !,! are the J coefficients for the lags of the dependent variable (the first difference of deposit rates, D.dep_rate) and 𝛾 !,! the K coefficients for the lags of the first difference of the EURIBOR (D.EUR3m) or our index (D.index). We set J = K = 3 for both models for the sake of simplicity. The results of the Pooled Mean--Group estimates are given in Table A5. Predictions are then built for each country using the coefficients obtained here.