R Tieulent
email: tieulent@in2p3.fr

A Béteille

J.-Y Grossiord

V Kakoyan

S Kox

M Méziane

J.-F Muraz

P Pillot

Results from the test bench of the Geometry Monitoring System of the ALICE Muon Spectrometer

We present the results obtained with the test bench of the Geometry Monitoring System (GMS) for the ALICE Muon Spectrometer. It consists in a mock up, reproducing at full scale, three half planes of the chambers 6, 7 and 8 of the spectrometer. We show that the GMS is able to measure transverse displacements with an accuracy of 1.5 μm. We show also that the resolution deteriorates by a factor 3 to 4 when thermal gradients are generated.

Results from the test bench of the Geometry

Monitoring System of the ALICE Muon Spectrometer 1 Introduction ALICE is the only experiment dedicated to the study of nucleus-nucleus collisions at the LHC. Its aim is the physics of strongly interacting matter at extreme energy densities, where is expected the formation of a new phase of matter, the quark-gluon plasma (QGP). One of the most promising probes of the QGP is the production of heavy quarkonium states (J/Ψ, Ψ , Υ, Υ , Υ ) which will be detected via their muonic decays in a forward muon spectrometer [START_REF]ALICE dimuon forward spectrometer[END_REF][START_REF]ALICE dimuon forward spectrometer : addendum to the TDR[END_REF]. Its tracking system consists of ten planes of cathod pad chambers and has to measure the invariant mass of the dimuon system with a resolution, ∆M µ + µ -/M µ + µ -, of about 1%. In particular, in order to separate the different particles of the upsilon family (M µ + µ -∼ 10 GeV/c 2 ), it is necessary to have a mass resolution of about 100 MeV/c 2 . This requirement determines the spatial resolution of the muon tracking chambers, which is about 100 microns. This implies that the positions of all tracking chambers have to be monitored with a resolution better than 40 µm [START_REF] Tieulent | ALICE internal note[END_REF]. The Geometry Monitoring System (GMS) is the device dedicated to this office.

The GMS is made of an array of about 460 optical sensors which are installed on platforms located at each corner of the tracking chambers. Two different types of optical devices were adopted: the BCAM and the so-called Proximity. In both devices the image of an object is projected on a CCD sensor through a lens. Then the analysis of the captured image provides a displacement measurement. The most relevant difference between the two systems is represented by the luminous object used. For the BCAM, which is a longrange system, the object is a pair of point-like LEDs, while it is a coded mask for the Proximity which is used for small distances. Figure 1 shows an example of images taken with a BCAM (left) and a Proximity (right). A complete description of the system can be found in [START_REF] Tieulent | ALICE internal note[END_REF]. Figure 2 shows a schematic view of the system. The red solid lines represent the optical lines. The GMS is composed of three parts. The first one is the Longitudinal Monitoring System (LMS). Its role is to monitor the relative displacements between the chamber planes. It is composed of BCAM lines linking two neighboring stations and Proximity lines linking the two chambers of each station. The second part is called the Transverse Monitoring System (TMS). It monitors the flatness of the chamber supports. It is only composed of BCAM optical lines. And finally, the last part of the system is the External Monitoring System (EMS). It is composed of eight BCAM lines linking chamber 9 to the cavern walls in order to monitor the absolute displacements of the entire spectrometer.

A simulation of the GMS was developed in order to define the number, the type and the position of the optical devices and to evaluate the performance of the system for the determination of the chamber displacement. In this simulation, various effects as installation accuracy, intrinsic accuracy of the optical devices or thermal gradient effects were included. A complete description of this simulation can be found in [START_REF] Pillot | ALICE internal note[END_REF].

Description of the test bench

In order to test the GMS geometry reconstruction program, a test bench of the GMS was built. This bench is a full-scale, aluminium-made mock-up of the three half-planes of chambers 6,7 and 8 of the muon tracking system. It was designed, realized and installed at the LPSC of Grenoble. A picture of the bench is shown in figure 3. As the first goal of this bench was to test that the reconstruction program was able to retrieve displacements, we decided to make one of the chambers mobile (chamber 7, see figure 3). Therefore, this chamber was installed on four translation stages, two in the Z direction and two in the X direction, allowing us to have translations in these two directions. The chamber was fixed to the stages using two vertical axes. It is therefore also possible to rotate it around the vertical direction Y. Figure 4 shows a schematic drawing of the displacement system (top) and an example of possible displacement (bottom). Figure 5 shows a detail picture of chamber 7 fixations. The motors used on the stages are stepping motors. Each step corresponds to a displacement of 5 µm. Studies of the accuracy and the repeatability of these motors were done using a comparator with a precision of 1 µm. This study showed that the standard deviation of the difference between the actual and the requested displacement of the stage controlled with the stepping motor is of the order of 10 µm [START_REF] Meziane | Summer student's report[END_REF].

The three chambers were equipped with the same number of optical devices as it will be in the ALICE final design, i.e.:

• one BCAM line at each corner for the link between chambers 6 and 7,

• one Proximity line at each corner for the link between chambers 7 and 8,

• six BCAM lines in the plane of each chamber for the monitoring of the deformations of the structure.

Survey of the structure

As it is shown in references [START_REF] Tieulent | ALICE internal note[END_REF] and [START_REF] Pillot | ALICE internal note[END_REF], an error in the installation of the optical devices induces a loss of resolution in the reconstruction program. The requested accuracy on the optical device position is at the millimeter level.

In order to characterize the test bench geometry, a survey of the complete structure was carried out by the CERN survey group [START_REF] Mergelkuhl | [END_REF]. Each platform supporting the optical devices is equipped with three reference points (named L, M and H, as shown in figure 6). The survey was done in two steps. First, the positions of all reference points were measured and compared to the theoretical positions in 3D space. The left plot of figure 7 shows the differences in the Z direction between the theoretical 3D positions and the measured ones. Their dispersions are of the order of 4 mm with some maximum values of the order of 1 cm. Each platform position in the Z direction was then corrected using the three adjustment screws equipping the platforms (see figure 6). Then, the reference point positions were measured again. The right plot of figure 7 shows the new difference distribution after that adjustment. The dispersion was reduced to about 0.6 mm with a maximum deviation within 2 mm. Table 1 gives the final results from the survey with respect to the 3D theoretical positions and orientations. We can see that chamber 6 is too high by 8 mm and chamber 7 is rotated Chamber

dX (m) Y + dY (m) Z + dZ (m) dθ X (rad) dθ Y (rad) dθ Z (rad) 6
3.6 10 -3 0.79 + 8.4 10 -3 0.00 + 2.2 10 -5 2.0 10 -5 1.5 10 -5 -3.9 10 -5 7 2.2 10 -3 1.28 + 2.8 10 -3 2.59 -6.2 10 -5 -1.5 10 -5 -7.9 10 -5 -2.0 10 -3 8 6.8 10 -6 1.28 + 1.8 10 -4 3.07 -1.1 10 -4 -8.2 10 -5 5.1 10 -5 -7.1 10 -4

Table 1: Offsets of the measured positions (dX, dY and dZ) and orientations (dθ X , dθ Y and dθ Z ) of the chambers with respect to the theoretical ones (nominal values of X, θ X , θ Y and θ Z are equal to zero, therefore they are not written here).

around the Z direction by 2 mrad (∼0.1 degree). This misalignment of the chambers is small enough so that each object stays within the acceptance of the optical devices. Therefore no change on the structure was done. The chamber positions were entered in the reconstruction program to improve the reconstruction accuracy.

Intrinsic resolution of the optical devices

In order to optimize the resolution of the reconstruction program, it is necessary to know the intrinsic resolution of each optical device at the time the measurement were done. This is mainly the resolution in the LED spot finding centroid on the CCD, which includes the thermal fluctuation effects. This quantity is used as weight for each measurement during the χ 2 minimization. This resolution is measured over a long time period run. Images of each optical device are taken many times without any displacement of the chambers.

Figure 8 shows the resolution for a typical BCAM. It is of the order of 0.1 µm on the CCD. It corresponds to an angular resolution of about 1.5 µrad (the latter has to be multiplied by the distance between the source and the CCD to obtain the position resolution of the LED source. The distance between chambers 7 and 8 is about 2 m therefore the resolution on the position of the source LED is about 3 µm). The Proximity resolution is better and is typically of the order of 1 µm for the position of the coded mask.

Tests of the reconstruction program

The reconstruction program was developed with simulation purposes in order to define the GMS setup, in terms of optical device number, types and positions, and to estimate its resolution. A detail report on this study can be found in reference [START_REF] Pillot | ALICE internal note[END_REF]. Figure 9 shows a scheme of the reconstruction program procedure. The goal is to retrieve the displacements of the chambers between the initial time T 0 and the current time T i . T 0 corresponds to the time of the calibration run, where straight muon tracks are collected in order to determine the initial position of the tracking chambers. Using the sets of images obtained at T 0 and T i , we can determine for each image the displacement of the spot positions on the CCD (∆ Image M es ). The reconstruction program is searching, using a χ 2 minimization procedure, to determine chamber displacements (∆ Chamber) that are able to reproduce the same displacements of the images. In order to check the performance of the reconstruction program to retrieve displacements, we used the translation tables to induced known displacements. Figure 10 shows the results of a test where chamber 7 was translated in the X direction. The left plot shows the displacement measured by the reconstruction program as a function of the induced one. We moved chamber 7 by steps of 100 µm over a range of 2 mm. The measured slope is 0.9999. The right plot shows the residues between the induced displacements and the measured ones. We have here a gaussian shape with a mean compatible with zero and a sigma of about 1.5 µm which is the resolution of the reconstruction convoluted with the Figure 11 shows, with a similar presentation, the result of a test where chamber 7 was translated along the Z direction. We moved it first by steps of 5 µm and then by steps of 50 µm over a range of 250 µm. The sigma of that distribution is about 7.4 µm. The measured slope is 0.986. Contrary to the previous result, a systematic difference of about 1% exists between the measured and induced displacements. This effect has an impact on the mean value of the residues, which is about 2 µm. We were able to retrieve this systematic effect by simulating a systematic shift in the position and orientation of the optical devices. Actually, we discovered that the plates of the platform are not perfectly flat. This induces a mispositioning of the optical elements. In order to minimize this problem, we asked the manufacturer for a better flatness quality for the plates of the final ALICE setup.

As expected, the GMS resolution in the Z direction is worse than in the transverse direction one. This was already shown by the simulation and comes from the fact that the intrinsic resolution of the optical device is better in the direction transverse to the optical axis than in the optical axis direction. We had then reduced this effect by tilting at different angles the four optical lines linking two chambers in order to mix the two directions.

The last test was made by rotating chamber 7 around the vertical axis. Plots of figure 12 show the results of this measurement as a function of the induced angle. The measured slope is 0.9912 with an error of 0.0001. It is clear, as in the previous measurement, that we have a systematic error of 1% between the induced and the retrieved rotations. Again, we were able to reproduce this effect by introducing in the simulation a systematic mispositioning of the optical devices. In order to improve the results of this test bench we will need to survey the platforms themselves in order to know the position of the optical devices more precisely. As seen in the right panel of figure 12, the rotation angle is reconstructed with a resolution of 4 µrad.

Figure 13 shows the results of a three days long run without any induced displacement of the chambers. The three plots from top to bottom represent, versus the time, the reconstructed translation and rotation of chamber 7 and the temperature of the room during this period. We can see that a correlation exists between the room temperature and the chamber translation (and/or) rotation. The temperature variations induced a small deformation of the two steel rails (λ steel = 1.2 10 -5 • C -1 ) on which the three chambers are fixed, and also a deformation of the aluminum chamber structure (λ alu = 2.4 10 -5 • C -1 ).

The deviations are of the order of few tens of µm (or µrad) over the three day period. Into the ALICE cavern, where the temperature variation is of the order of 1 degree, we can expect the same order of magnitude of movement in a stable running condition.

From the plots on figure 13 we can extract the reconstruction resolution for the different quantities. In particular, the resolution for the translation along the Y direction (T Y ) and for the rotation around the Z axis (θ Z ) (the two most important parameters for the muon spectrometer geometry because they have a direct impact on the invariant mass resolution) are respectively 1.1 µm and 0.3 µrad for chamber 7. The resolution for chamber 6 is worst (σT Y 7µm) due to the fact that its position is monitored only by BCAMs which have a worst intrinsic resolution than Proximity. These resolutions are the best we can expect as the running condition in which there were calculated are optimal (no temperature gradient in the room, calm air condition, no vibration).

T X T Y T Z θ X θ Y θ Z
6 Effects of thermal gradients Chamber 6 was equipped with a heating pad in order to mimic the power being dissipated by the front end electronics of the tracking chambers. The pad has three separated zones (figure 14). Each zone is equipped with a potentiometer, in order to adjust the power separately. The power dissipated by each zone was tuned according to the temperature map of the station 3 in natural convection which was obtained by simulation [START_REF] Salasca | Progress on cooling of stations 3, 4 and 5[END_REF]. Figures 15 and 16 show the reconstruction resolutions in the Y position for chambers 6 and 7 respectively. The left plots show this resolution without the heater and the right ones with the heater turned on at its nominal value on chamber 6. We can see that the effect of the thermal gradient is to deteriorate the reconstruction resolution by a factor 4 for chamber 6. On the contrary, it has no effect on the position of chamber 7. In the reconstruction program, chamber 8 is supposed to be fixed. Therefore, the resolution on the position of chamber 7 is driven by the Proximity resolution which links chambers 7 and 8. This explains why the reconstruction resolution for chamber 7 is not affected by the thermal gradients. The resolution obtained on chamber 6 is about 23 µm. As it has been shown in the reference [START_REF] Tieulent | ALICE internal note[END_REF], the resolution should be improved with the utilization of cooling fans. In stations 1, 2 and 3, cool air will be injected at the bottom for stations 1 and 2, and at the bottom and top for the station 3. Thermal study of this cooling has been done for station 3 [START_REF] Salasca | Progress on cooling of stations 3, 4 and 5[END_REF]. We can expect that the resolution obtained here (23 µm) can be reduced. Further tests will be done using two fans blowing air at room temperature.

Conclusion

In this note, the first results given by the Geometry Monitoring System test bench are presented. The primary goal of this mock up was to test the reconstruction program. It shows that the reconstruction program is able to retrieve with an accuracy of 1.5 µm (in the transverse direction) the displacement induced by the translation stages relatively to chamber 8. Tests with thermal gradient on chamber 6 were done. They show that the resolution on that chamber, relatively to chamber 7, deteriorates from 7 µm (without any thermal gradient) to 23 µm when thermal gradients are generated using an heating pad. This resolution is still better than the requirements which are of the order of 40 µm [START_REF] Pillot | ALICE internal note[END_REF]. Further tests will be done using the two fans that are supposed to reduce those gradient effects.

Figure 1 :

 1 Figure 1: Images taken with a BCAM (left) and a Proximity (right).

Figure 2 :

 2 Figure 2: General view of the GMS setup. The red solid lines on this figure represent the optical lines (but are not physical links) between the tracking chambers.

Figure 3 :

 3 Figure 3: Picture of the test bench.

Figure 4 :Figure 5 :

 45 Figure 4: Scheme of the translation system.Rotation axis

Figure 6 :

 6 Figure 6: Picture of a platform showing the three reference points.

Figure 7 :

 7 Figure 7: Difference along the Z direction between the theoretical reference point positions and the measured ones before (left) and after adjustment (right).

Figure 8 :

 8 Figure 8: Intrinsic resolution of a typical BCAM for the four different measurements. The units of the horizontal scale are micro-meters.

Figure 9 :

 9 Figure 9: Scheme of the reconstruction program procedure. resolution on the real displacement measured with the comparators.

Figure 10 :

 10 Figure 10: Left: displacements measured with the reconstruction program along X versus induced ones for chamber 7, right: residues. Measured disp. (µm) Induced displacement (µm) Disp. (Meas -Induced) (µm) Slope = 0.9999 ± 0.0001

Figure 11 :Figure 12 :

 1112 Figure 11: Left: measured displacements along Z versus induced ones for chamber 7, right: residues. Measured disp. (µm) Induced displacement (µm) Disp. (Meas -Induced) (µm) Slope = 0.986 ± 0.004

Figure 13 :

 13 Figure 13: Reconstructed translations (top) and rotations (middle) of chamber 7 versus time, and temperature (bottom) during the same period. T Y and θ Z are the most important parameters for the spectrometer as they have a direct impact on the invariant mass resolution.

Figure 14 :

 14 Figure 14: Scheme of the heating pad with its three separated zones of heating. The temperature indicated are the ones at contact.

Figure 15 :

 15 Figure 15: Reconstruction's resolution (in µm) of the Y position of chamber 6, left: without any heating, right: with the heater turned on at its nominal power (Note the change in scale).

Figure 16 :

 16 Figure 16: Reconstruction's resolution (in µm) of the Y position of chamber 7, left: without any heating, right: with the heater turned on at its nominal power(Note the change in scale).

Acknowledgements

We would like to thank G. Damieux-Verdeau, J.-C. Malacour and S. Roudier from the LPSC workshop, and A. Benoit, G. Gelin and F. Mounier from the IPNL workshop for the construction of the test bench. Also, we would like to thank J.-C. Ianigro for its work in the design of the mechanical supports of the optical devices, and D. Mergelkuhl and V. Prasad from the CERN survey group for their work. We kindly thank P. Stassi and A. Castera from instrumentation groups of the LPSC and IPNL respectively for their help in the definition of the project. The work of V. Kakoyan was partially supported by Calouste


J Amar

J F Carpantier

C Lecourt

GCC Sovereign Wealth Funds: Why do they Take Control?

Keywords: JEL classification: F310, F31, G15 Sovereign Wealth Funds, Cross-border Majority Acquisitions, Ordered Logit model, GCC

In this paper we examine the investment strategy of sovereign wealth funds (SWFs) of the Gulf Cooperation Council (GCC) countries. GCC SWFs are considered as relatively opaque investors and strongly politicized, raising some concerns for perceived political and security risks. We investigate what are the drivers of majority crossborder equity acquisitions made by these institutional investors over the period 2006-2015. Using both Logit and ordered Logit models, we test if the usual determinants of SWFs investments still stand when we look at influential (> 10%) or majority (> 50%) acquisitions made by GCC SWFs. We find that GCC SWFs do not consider financial characteristics of the targeted firms when they acquire large cross-border stakes but rather the characteristics of the country (countries in the European union and/or countries with a high level of shareholders protection), suggesting that their motives may go beyond pure profit maximization. We also find that transparent funds are more likely to take influential or majority stakes and that they do so predominantly in non-strategic sectors. Overall, our results indicate that even if GCC SWFs do not seek only for financial returns, acquiring majority stakes is not a lever for GCC governments to get strategic interests in the target countries.

Introduction

Defined by the IMF (2004) as "government-owned investment funds set up for a variety a macroeconomic purposes" such as stabilisation, saving for future generations and investments in socio-economic projects, sovereign wealth funds (SWFs) have sharply grown over the last decade, with resources estimated to be USD 7.3 trillion in June 2017, thanks to high oil prices, financial globalisation and sustained global large imbalances.1 SWFs have recently attracted considerable public attention. While the size and rapid growth of SWFs suggest that they have become major players in the world, buying large stakes in companies and giving government's exposure to sectors they may otherwise be unable to achieve, their objectives and behavior are not well understood. In particular, the opaqueness surrounding their structure and activities is a major concern in host countries, as it is unclear whether SWFs behave like governments or institutional investors: "the prospect of significant investments by SWFs potentially giving foreign countries control over important parts of an investee country's economy has emerged as a political issue" [START_REF] Greene | Sovereign wealth funds: a measured assessment[END_REF]. This is particularly the case of SWFs originating from the Gulf Cooperation Council (GCC) countries 2 , by which the amount accumulated has dramatically increased since 10 years due to the increasing prices of commodities such as oil and natural gas. SWFs of GCC countries manage around 40% of SWFs global assets. The SWFs of these countries are broadening their investment portfolios and focusing on achieving higher returns. Consequently, they have invested all over the world during the last decade with the bulk of them focused on developed countries and in particular Europe. It has become common news to hear that one of these GCC funds is in the process of buying, planning to buy or investing in a major institution in Western countries.

A revealing example is the full acquisition by Qatar Investment Authority (QIA) in 2011 of the popular football club Paris St. Germain. The same SWF has played the part of deal-maker with the Glencore acquisition of Xstrata in 2013 and with Glencore again by buying stakes in Russian oil company Rosneft in December 2016. In June 2016, the Public Investment Fund (PIF) of Saudi Arabia has announced to have taken a USD 3.5 billion stake in the taxi company Uber, in order to diversify the economy of the country by investing in sectors less dependent on oil. These examples illustrate well the fact that the motives of GCC funds can be other than pure profit maximisation of the financial investment and reveal their capacity to take the control or to be able to influence companies involving the strategic national interests.

While there is an extensive literature that investigates the determinants of SWFs investment decisions (see [START_REF] Amar | Are sovereign wealth funds decisions based on country factors? Working paper[END_REF]; [START_REF] Ciarlone | Escaping financial crisis? Macro-evidence from sovereign wealth fund's investment behaviour[END_REF]; [START_REF] Knill | Bilateral political relations and sovereign wealth fund investment[END_REF]; [START_REF] Kotter | Friends or foes? Target selection decisions of sovereign wealth funds and their consequences[END_REF]; [START_REF] Megginson | Determinants of sovereign wealth fund crossborder investments[END_REF] among others) only few papers address the question of the determinants of cross-border majority purchases. [START_REF] Karolyi | State capitalism's global reach: Evidence from foreign acquisitions by state-owned companies[END_REF] analyze cross-border majority acquisitions of government-led acquirers, [START_REF] Heaney | Sovereign wealth fund investment decisions: Temasek holdings[END_REF] analyze the determinants of Temasek Holding's level of investment and [START_REF] Murtinu | Sovereign Wealth Funds' Internationalization Strategies: The Use of Investment Vehicles[END_REF] show that SWFs are more likely to use investment vehicles when they take crossborder majority stakes. This is, however, a key question as it is clear that SWF activism, i.e. the acquisition of large or majority acquisitions, attracts more hostility and generates more severe political opposition by host-country governments as was for example illustrated in 2006 by the failed attempt by Dubai World Ports to acquire P&O. Cross-border majority purchases also bring regulators to require a higher level of transparency as evidenced recently by the EC Regulation dated 11 July 2017 that forces SWFs to provide more information for majority acquisitions in European companies. This question matters in the ongoing opportunity-threat debate, as it indicates what role SWFs want to take in their targets, and whether it differs across sectors, firms or countries.

The aim of this paper is to fill this gap in the literature by identifying the determinants of (minority and majority) cross-border acquisitions of SWFs originating from Gulf countries.

Though SWFs are generally seen as heterogeneous investors with respect to their source and size of assets, organizational structure, governance, risk factor and their objectives, GCC SWFs present some key characteristics that make them a distinct group among SWFs.

First, they are all funded by commodity revenues (mostly oil) meaning that their proceeds are extremely dependent on oil prices. Second, they are considered as relatively opaque investors and strongly politicized. Third, they come from autocratic countries. Finally, they are located in a same region, with common language and religion.

In particular, we study what determines the GCC SWFs' decisions to take control or large stakes in foreign firms. More specifically, we shed light on the real intention of SWFs when they decide to acquire a majority stake: Do GCC SWFs take cross-border majority stakes based on the financial health of the target firms? Based on the sectors, potentially strategic ones? Based on country specific characteristics (political or macroeconomic ones)?

Using an original large-scale database including both data on announced cross-border stakes done by GCC SWFs between 2006 and 2015, macroeconomic data on target countries as well as financial data on listed target firms, we use Logit models as well as ordered Logit models to explain the motivation of GCC SWFs to take cross-border large (> 10%) or majority (> 50%) acquisitions.

Anticipating on our results, we find that financial characteristics of the target firms do not matter in the decision-making process of majority acquisition meaning that the decision of cross-border acquisition is not oriented towards risk-return and profit maximization objectives. Taking a majority stake is a specific decision going beyond investment purpose, where data tell that financial dimension play only a minor role, if any. In the same way, we find that transparent GCC SWFs are more likely to acquire large stakes and that they do so predominantly in: i) countries of the European Union; ii) countries with a high level of shareholders protection; iii) and in non-strategic sectors. These results altogether suggest that even if the motivations behind GCC SWFs majority acquisitions are other than pure profit maximisation, taking large stakes is not a mean for governments to acquire strategic interests abroad.

The paper is organized as follows. The next section introduces the hypotheses for analyzing GCC SWFs cross-border investment decisions. Section 3 provides details on our data.

Section 4 presents the methodology and Section 5 reports our empirical findings.

Review of literature and hypotheses

There is an extensive literature investigating to what extent SWFs investment decisions differ from those of other institutional investors. [START_REF] Bernstein | The investment strategies of sovereign wealth funds[END_REF] explain that the presence of politicians inside the board of SWFs could lead to the search of strategic objectives and finally to financial and political destabilization. [START_REF] Dyck | Sovereign wealth fund portfolios[END_REF] show that a part of SWFs portfolio is oriented toward the development of their domestic countries, indicating that investment decisions of SWFs are distorted by political considerations. [START_REF] Chhaochharia | Sovereign wealth funds: their investment strategies and performance[END_REF] find that SWFs show strong biases compared to other investors. More specifically, they find that SWFs are more likely to invest in countries that share a common culture and that they display industry biases, investing predominantly in oil company stocks. [START_REF] Knill | Bilateral political relations and sovereign wealth fund investment[END_REF] find that SWFs are more likely to invest in countries with which they have weaker political relations, implying that SWFs may invest, at least partly, for non-financial motives.

The findings that SWF investment decisions are dissimilar to those of traditional institutional investors can be explained by several factors. First, SWFs are sovereign-owned institution, which may be managed either by the ministry of finance or by a board composed of government officials. Unlike other funds, the politics or the structure of the fund owned/controlled directly by the government may influence asset allocation decisions. Second, in terms of social welfare, governments have broader goals than wealth maximisation of the firm, such as the development of the national economy or the maximisation of the employment level. Third, according to the natural resources curse theory (see among others [START_REF] Sachs | Natural resource abundance and economic growth[END_REF][START_REF] Sachs | The curse of natural resources[END_REF][START_REF] Sala-I-Martin | Addressing the natural resource curse: an illustration from Nigeria[END_REF] or [START_REF] Smith | Oil wealth and regime survival in the developing world, 1960-1999[END_REF]), countries with weak institutions generally have natural resource wealth that leads to resource dependency and rentierism. Even if the declared objective of these SWFs is to ensure that the proceeds from natural resources rents will be channelled through a transparent, accountable and professionally managed fund, they may be a mean for these autocratic countries to embezzle natural resources revenues in order to invest abroad [START_REF] Carpantier | Emergence of sovereign wealth funds[END_REF]).

While there is an extensive literature that investigates the determinants of SWFs investment decisions, only few papers address the question of the determinants of cross-border majority purchases. [START_REF] Karolyi | State capitalism's global reach: Evidence from foreign acquisitions by state-owned companies[END_REF] analyze cross-border majority acquisitions of government-led acquirers but do not focus on SWFs. [START_REF] Heaney | Sovereign wealth fund investment decisions: Temasek holdings[END_REF] analyse the determinants of the level of SWFs investment, but their analysis focus on the Singaporean fund, Temasek Holdings. [START_REF] Murtinu | Sovereign Wealth Funds' Internationalization Strategies: The Use of Investment Vehicles[END_REF] show that taking cross-border majority stakes is one driver of the use of investment vehicles by SWFs.

Large or majority acquisitions raise very specific questions for the following reasons.

First, a large or a majority acquisition, potentially signals an activist stance and willingness to engage in effective corporate governance activities. Second, it can be part of a general commercial or industrial development strategy of the home country of the SWF. It allows the development of joint ventures and eases the strengthening of expertises or industrial complementarities relevant from the home country perspective (typically the downstream integration of energy value-chain). Third it signals that the investment might go beyond passive portfolio management and mean-variance optimization framework, which is quite obvious knowing that SWFs are government-owned entities.

Given these features, we now analyse the determinants of SWF majority acquisitions through 4 key hypotheses.

H1 -GCC SWFs take cross-border majority stakes without considering the financial characteristics of the target.

As documented by [START_REF] Bernstein | The investment strategies of sovereign wealth funds[END_REF], the presence of politicians inside the board of the SWFs (which is the case for all GCC SWFs) leads to the search of strategic objectives not necessarily related to financial characteristics. An abundant literature [START_REF] Chhaochharia | Sovereign wealth funds: their investment strategies and performance[END_REF], [START_REF] Ciarlone | Escaping financial crisis? Macro-evidence from sovereign wealth fund's investment behaviour[END_REF], [START_REF] Dyck | Sovereign wealth fund portfolios[END_REF]Morse (2011), Johan et al. (2013), [START_REF] Knill | Bilateral political relations and sovereign wealth fund investment[END_REF] among others) has shown that policy factors drive SWFs investment decisions.

At the same time, part of the literature shows that SWFs investment decisions are also financial [START_REF] Fernandes | The impact of sovereign wealth funds on corporate value and performance[END_REF], [START_REF] Kotter | Friends or foes? Target selection decisions of sovereign wealth funds and their consequences[END_REF], [START_REF] Megginson | Determinants of sovereign wealth fund crossborder investments[END_REF]). [START_REF] Fernandes | The impact of sovereign wealth funds on corporate value and performance[END_REF] shows that SWFs tend to act as prudent investors, taking stakes in large firms with proven profitability. He also concludes that, given their long term investment horizon, SWFs have no preference for liquid stocks. [START_REF] Kotter | Friends or foes? Target selection decisions of sovereign wealth funds and their consequences[END_REF] find that SWFs tend to invest in large firms located in financially developed countries, exhibiting financial difficulties. They then deduce that SWFs are similar to institutional investors in their preference for target firm characteristics. [START_REF] Megginson | Determinants of sovereign wealth fund crossborder investments[END_REF] analyze the determinants of SWFs investment from the country perspective. They test if SWFs are purely commercial investors facilitating cross-border corporate investments or if their investment strategies are biased by political objectives. Their results suggest that SWFs make investment decisions principally for commercial purposes.

To the extent that control decisions potentially go beyond pure portfolio risk-return management, we expect the financial factors to have a low weight, if any, in the decision process.

In other words, we test whether GCC SWFs take cross-border (full or partial) acquisitions based on financial variables, meaning that the decision of cross-border acquisition is, or not, oriented towards risk-return and profit maximisation objectives.

H2 -GCC SWFs take the degree of financial and economic development of the target country into account before taking large or majority stakes.

Several papers study the impact of target country economic, financial and institutional development on the SWFs investment strategies. Some authors [START_REF] Megginson | Determinants of sovereign wealth fund crossborder investments[END_REF] and [START_REF] Amar | Are sovereign wealth funds decisions based on country factors? Working paper[END_REF] among others) find that country factors do not explain SWFs investment decisions. On the contrary, [START_REF] Knill | Bilateral political relations and sovereign wealth fund investment[END_REF] find that SWFs are more likely to invest, and for larger amounts, in countries that have a level of economic development close to theirs. [START_REF] Ciarlone | Escaping financial crisis? Macro-evidence from sovereign wealth fund's investment behaviour[END_REF] find that SWFs tend to invest in countries that have a higher degree of economic development which is consistent with [START_REF] Karolyi | State capitalism's global reach: Evidence from foreign acquisitions by state-owned companies[END_REF] who show that government-controlled investments flow from emerging to developed economies. Furthermore, the financial openness of the target country is one of the drivers of SWFs investments. [START_REF] Amar | Are sovereign wealth funds decisions based on country factors? Working paper[END_REF] and [START_REF] Ciarlone | Escaping financial crisis? Macro-evidence from sovereign wealth fund's investment behaviour[END_REF] show indeed that SWFs are more likely to invest in countries that are financially opened. Finally, some authors find that the level of investors protection significantly explain SWFs investment decisions. [START_REF] Knill | Bilateral political relations and sovereign wealth fund investment[END_REF] show that SWFs are more likely to invest in a country with a high level of investors protection, but when they do so, they tend to invest smaller amounts, which is consistent with [START_REF] Ciarlone | Escaping financial crisis? Macro-evidence from sovereign wealth fund's investment behaviour[END_REF] but not with [START_REF] Megginson | Determinants of sovereign wealth fund crossborder investments[END_REF] who find that a higher level of investors protection leads to larger amounts.

If the literature tells us that economic development, financial openness and the level of investors protection matter in SWFs investment decisions, the way these factors impact the decision is not unanimously accepted. These different results in the literature may be explained by the great heterogeneity of this group of investors. As we use here a homogenous group of SWFs (i.e. the GCC SWFs), we would like to test whether GCC SWFs take crossborder majority acquisitions by considering the economic and financial development of the target country.

H3 -GCC SWFs are more likely to take majority equity stakes in countries where there are bilateral trade agreements.

Majority acquisitions are expected to be part of a broader partnership between countries.

We consider the impact of bilateral trade, financial or political agreements between SWF's and host countries on the acquisition decision. In line with [START_REF] Hoeckman | The political economy of the world trading system: from GATT to WTO[END_REF] and [START_REF] Murtinu | Sovereign Wealth Funds' Internationalization Strategies: The Use of Investment Vehicles[END_REF], we refer to bilateral trade agreements as reflection of political relations and decisions among countries. As explained by [START_REF] Murtinu | Sovereign Wealth Funds' Internationalization Strategies: The Use of Investment Vehicles[END_REF], international trade agreements first allow the government to signal a credible lasting commitment to liberal economic policies, limited intervention in the domestic economy and peaceful relations. Second, trade agreements are useful to reinforce political power.3 GCC countries have free trade agreements with some countries such as Singapore (GSFTA), or some European countries (The EFTA is a free trade agreement between GCC countries and Iceland, Lichtenstein, Norway and Switzerland). The existence of bilateral trade agreements between SWF's and target countries should facilitate the entrance of SWFs in foreign countries by reducing the risk perception related to SWFs investments and therefore by mitigating the potential fear against their acquisitions. Some papers find that SWFs tend to invest in countries that are identified as trade partners [START_REF] Megginson | Determinants of sovereign wealth fund crossborder investments[END_REF]). In the same way, [START_REF] Murtinu | Sovereign Wealth Funds' Internationalization Strategies: The Use of Investment Vehicles[END_REF] find that bilateral trade agreements reduce the use of corporate investment vehicles for SWFs wishing to take cross-border majority acquisitions.

Related to this literature, we expect GCC SWFs to be more likely to take majority acquisitions overseas in countries bound by bilateral trade agreements.

H4 -GCC SWFs target cross-border companies operating in strategic industries.

The hostility of the host countries' public opinion and governments towards SWFs can be explained by the fact that SWFs might seek stakes in strategic sectors as defense, finance, telecommunication, energy or transportation [START_REF] Dyck | Sovereign wealth fund portfolios[END_REF]). This is the reason why many governments want to hinder foreign SWFs investments when the target is a strategic infrastructure or a sensitive firm operating in a strategic sector [START_REF] Karolyi | State capitalism's global reach: Evidence from foreign acquisitions by state-owned companies[END_REF], [START_REF] Knill | Bilateral political relations and sovereign wealth fund investment[END_REF]). [START_REF] Fernandes | The impact of sovereign wealth funds on corporate value and performance[END_REF] gives some examples of regulatory/enforcements efforts in order to hinder SWF investments: "The German government has announced that it would introduce controls on investments by SWFs, especially if they seek stakes in strategic sectors. French President Nicolas Sarkozy has announced that he would use his country's state-owned bank (Caisse des Dépôts et Consignations) to help protect French companies against potentiel takeover threats posed by SWFs".

As explained by [START_REF] Murtinu | Sovereign Wealth Funds' Internationalization Strategies: The Use of Investment Vehicles[END_REF], the risk of political and financial destabilization for the host country is higher when the target investment is a strategic infrastructure.

Two deal-level characteristics can explain the fear of the host country: i) the sector of the foreign target company; ii) the degree of control on this company. The larger the SWF's control on target firms operating in strategic industries, the more hostile host countries will be.

According to the literature on SWFs [START_REF] Bernstein | The investment strategies of sovereign wealth funds[END_REF], [START_REF] Karolyi | State capitalism's global reach: Evidence from foreign acquisitions by state-owned companies[END_REF], Murtinu and Scalera (2016)), three factors contribute to increase the probability of cross-border majority acquisitions in strategic industries: i) the undemocratic and authoritarian nature of the countries where SWFs originate from; ii) the high level of foreign currency reserves; iii) the involvement of politicians in the board of SWFs. As GCC combines these three factors, we expect that GCC SWFs target cross-border companies operating in strategic industries.

Furthermore, according to the portfolio allocation theory, SWFs may be used to diversify the industrial base of their home country by targeting foreign industries in which their country is under-represented. The political motives of cross-border SWFs acquisitions in strategic industries can therefore be explained by the search of a greater industrial diversification or a better access to lower-cost resource inputs. Regarding resource-rich countries like GCC, these countries are dependent on revenues from sales of energy, which makes these economies extremely vulnerable to changes in oil prices. It is fundamental for these countries to diversify their revenues by targeting foreign industries in which the country is under-represented [START_REF] Sturm | The Gulf Cooperation Council Countries. Economic structures, recent developments and role in the global economy[END_REF]). Therefore, the search of a greater industrial diversification may be a result of reducing dependency on oil revenues but also the search of strategic industrial gains in order to control access to technologies with the aim of maximizing long-term returns [START_REF] Seznec | The Gulf sovereign wealth funds: myths and reality[END_REF]). We then complement the sectoral focus by testing whether cross-border acquisitions are under-represented in energy sector.

3 Data and descriptive analysis

The SWF sample

According to the IMF (2004), "Sovereign wealth funds are government-owned investment funds set up for a variety a macroeconomic purposes". Considering this definition, we conducted a comprehensive search of all existing GCC SWFs and ultimately get 15 entities. We find 7 SWFs in the United Arab Emirates, 3 SWFs in Saudi Arabia, 2 SWFs in Oman, 1 in Bahrain, 1 in Kuwait and 1 in Qatar. Names, inception dates, estimated sizes are reported in Table 1. We then conducted a search of all wholly-owned subsidiaries of these funds using the online database Thomson Reuters Eikon and the funds' websites, which are also reported in Table 1, together with the Linaburg-Maduell transparency index (the higher the index, the more transparent the SWF) and the announced main objective(s) of the funds.

Investment data

We use Thomson Reuters Security Corporation's (SDC) Platinum Mergers and Acquisitions database to collect data on announced cross-border acquisitions done by GCC SWFs directly The Linaburg-Maduell transparency index was developed at the SWF Institute by C. Linaburg and M. Maduell. It rates SWFs according to their level of transparency from 1 to 10. The higher the index is, the more transparent the fund is. For more details on the index construction, see: http://www.swfinstitute.org/statistics-research/linaburg-maduell-transparency-index/.

or by their wholly-owned subsidiaries. We doubled checked this list and completed the missing acquisitions by using the online database Factiva. We collected a number of data items, including information about the targeted firms (name, country), information about the SWFs (name, subsidiary, country), the date of the transaction, the pre-and post-acquisition share of the SWF in the targeted firm and the deal value, if disclosed. This search yields a sample of 124 cross-border acquisitions from GCC SWFs in 28 target countries over the period 2006-2015.

Our dependent variable is a discrete variable characterized by specific ownership share aimed at capturing the degree of acquisition in a target firm. We identify three levels of control, with thresholds at 10% and 50% of the target value. The first threshold follows the definition of Foreign Direct Investments (FDIs) according to which "a FDI (...) reflects the objective of a resident in one economy obtaining a lasting interest in an enterprise resident in another economy. (...) A direct investment relationship is established when the direct investor has acquired 10 percent or more of the ordinary shares or voting power of an enterprise abroad" (?). The second threshold is set at the majority stake. So, the first category of deals -called DEALS -10% -include transactions where the SWFs take shares of the target such that their total holding is less than 10% (minor stakes). The second category -called DEALS + 10% -collects the transactions leading to stakes larger than 10% but inferior to 50% (influential/large stakes).4 Following ?, we identify a third level of investment -called DEALS + 50% -which includes transactions where the SWFs take shares such that the holdings are at least 50% of the target (control/majority stakes).

Table 2 presents the number of cross-border deals by GCC SWFs over the period 2006-2015. We first note that 11 SWFs (out of the 15 SWFs identified for GCC countries) were found to effectively take cross-border acquisitions over 2006-2015, mainly those of Qatar and UAE, which together account for more than 90% of the deals. We also note that 40% of the deals are small deals (< 10%), 32% are minority deals (larger than 10% but inferior to 50%) and finally 27% of the deals are majority transactions (larger than 50%).

Figure 1 shows the evolution of GCC SWFs foreign investments over the period 2006-2015. This figure reveals that GCC SWFs have a tendency to acquire large stakes as the average post-acquisition share is larger than 19% every year. During the financial crisis (2008)(2009), SWFs made more investments but have reduced the size of their stakes. This is linked with the large number of investments made in financial institutions, such as Qatar Investment Authority investing in Barclays Bank or Abu Dhabi Investment Authority investing in Citigroup. After the crisis, the number of acquisitions decreased in number but not in value. After 2013, the average post-acquisition share was higher than 30%. for more than 10% of the deals (even 20% for the energy sector). We also note that 50% of the deals in the luxury sector are majority deals, while they represent only 14% and 16%

respectively for finance and industry sectors.

Explanatory variables

We employ a number of variables that should potentially explain the decision to acquire minority/majority stakes for GCC SWFS. These variables relate to macroeconomic and institutional target country factors, firm-level characteristics as well as the type of investment sectors. Details on variables construction and source are presented in Table 5.

Target firm/sector specific variables: In order to test whether target firm-level variables are determinants of minority/majority acquisitions for GCC SWFs (H1), the variables collected for each of the targeted firms included in the sample can be allocated to the broad classifications, performance, risk and liquidity. Returns on assets (ROA) is used in order to capture target firm performance, LIQU IDIT Y is the liquidity ratio of the target firm, DEBT is the long term debt level of the firm and DEBT /ASSET S is the ratio of long term debt on assets.

These four financial variables are based on the three years preceding the investment in order to take into account the information available at the time of the investment.

In addition to variables related to the financial performance of the firm, we also use the target firms' sectors as explanatory variables (LUXURY, FINANCE, ENERGY, INDUSTRY, METAL)and test whether GCC SWFs have an incentive to target crossborder companies operating in strategic industries as explained in H4.

Target country specific variables:

In order to test whether the economic development of the target country matters in the GCC SWF-led acquisition activity (see H2), we include several country-level variables that have been shown in the literature to be related to international investment choices of SWFs [START_REF] Fernandes | The impact of sovereign wealth funds on corporate value and performance[END_REF]). Like [START_REF] Karolyi | State capitalism's global reach: Evidence from foreign acquisitions by state-owned companies[END_REF], we use the anti-self-dealing index (ANTISELF) constructed by [START_REF] Djankov | The law and economics of self-dealing[END_REF]

SWF specific variables

Finally, we include variables measuring characteristics of each GCC SWF, including its size (SIZE ) measured as the value of the assets under management, a transparency index (TRANSPARENT ) which is a dummy variable equals 1 if the Linaburg-Maduell

Transparency index of the fund is higher than 5 and 0 otherwise and a dummy called SUBSIDIARY indicating if GCC SWFs are more prone to take majority acquisition through a wholly-owned subsidiary as in [START_REF] Murtinu | Sovereign Wealth Funds' Internationalization Strategies: The Use of Investment Vehicles[END_REF] .

Additional controls

We finally include the WTI oil price level (OILPRICES ) because commodity trade resources may be the main driver of GCC SWFs strategies and a dummy variable that identifies the financial crisis (CRISIS ), equals 1 if the transaction occurs in 2008 or 2009, and zero otherwise. In order to test whether GCC SWFs are more likely to take majority stakes in countries where there are bilateral trade agreements (H3), two proxies of bilateral trade agreement are considered: FTAFORCE which is a dummy variable equals 1 if there is a free trade agreement in force between the SWF's country and the targeted firm's country, and 0 otherwise and FTANEGO which is a dummy variable equals 1 if there is a free trade agreement under negotiation between the SWF's country and the targeted firm's country, and 0 otherwise.

Empirical Part

We first provide some preliminary insights into the data by estimating two separate Logit models based on the 10%, and 50% thresholds respectively. We then estimate a multinomial ordered Logit model designed to better fit the investment decision-making process of SWFs.

We finally discuss our results and compare them to the main outcomes of the literature on SWFs' investment decisions.

Preliminary findings with binary Logit models

We model the probability of taking a majority (resp. large) stake in the target firm as a function of the explanatory variables described above (target firm variables, target country variables, SWF specific variables as well as additional control variables), by relying on the Logit specification. The model is defined as follows:

P rob(Y ij = 1|X) = Λ(Xβ) = e Xβ 1 + e Xβ (1)
where Y ij is a dummy variable equal to 1 if the fund i takes a stake of 10% or more (resp.

50% or more) in a cross-border firm j and to 0 otherwise, where β is the coefficient vector, X is the vector of explanatory variables and Λ(.) is the standard logistic distribution function.5

Relying on a distribution function, like the logistic, ensures that we get values between zero and one, as we expect from a probability. 6 We thus test our hypotheses respectively on the determinants of GCC SWFs large (+10%) and majority (+50%) cross-border acquisitions and report the estimation results in Table 6 for full models (columns (1) and ( 3)) and then, by applying a general-to-specific backward-selection approach, for parsimonious models (columns ( 2) and ( 4)).

First, the results of the Logit estimations, based on the parsimonious models, show that the large or majority acquisition decisions are not necessarily determined by the same set of explanatory variables. We thus note that the variables dlogREER, POLITY, LUXURY, SIZESWF, FTANEGO, ROA and CRISIS are significant in the model with threshold at 50% but not in the model with threshold at 10%. We find the reverse for the variables TRANSPARENT, EU DUMMY, OILPRICES, FINANCE and ENERGY. This difference of results in the two binary Logit models confirms the relevance of considering different thresholds in our analysis. 3) and ( 4)), the endogenous variable (Y ij ) is a dummy variable equals to zero if the posttransaction share of the SWF i in the target j is smaller than 10% (resp. 50%), and equals to one if its is larger or equal. Columns (1) and (3) report the estimates for the full models, while columns (2) and ( 4) report the parsimonious model resulting from a backward-selection approach with significance level for removal of the model set at 10%. Concerning firm-level factors, the results support H1 as we find no evidence that GCC SWFs would consider the financial characteristics of the target firm. Except for the variable ROA, all the variables on the financial characteristics of the target firm are not significant.

INFLUENTIAL STAKE (+10%) MAJORITY STAKE (+50%) (1) (2) (3) (4)
Consistently with [START_REF] Kotter | Friends or foes? Target selection decisions of sovereign wealth funds and their consequences[END_REF] and [START_REF] Bernstein | The investment strategies of sovereign wealth funds[END_REF] who find that SWFs invest in distressed firms, we find that GCC SWFs prefer to take cross-border majority acquisitions in firms with low profitability (ROA), suggesting that GCC SWFs are passive shareholders with a long-run investment horizon.

Regarding H2 which stresses that GCC SWFs take cross-border acquisitions by considering the economic and financial development of the target country, the significance of REER, POLITY, ANTISELF and EU DUMMY clearly reveals that country factors are essential in the GCC SWFs acquisition decision making-process. REER is positively related to majority acquisitions, meaning that these funds are more likely to take majority stakes in countries where the real effective exchange rate is high, i.e. where there is a loss of price competitiveness. In the same way, we find that GCC SWFs are more prone to take large stakes or the control of a firm in countries where there is a high quality of investors protection, unlike

? who find that cross-border majority acquisitions of government-led acquirers are weakly related to anti-self dealing index differences. In addition, the variable EU DUMMY is found significant in the sense that GCC SWFs target countries of the European Union when they take influential (10% or more) stakes. This result is consistent with [START_REF] Hay | The increasing interest of Middle Eastern sovereign wealth funds for Europe: presenting original data[END_REF] who find that Europe is the privileged destination for Middle Eastern SWFs.7

Unlike [START_REF] Megginson | Determinants of sovereign wealth fund crossborder investments[END_REF] and [START_REF] Knill | Bilateral political relations and sovereign wealth fund investment[END_REF], we do not find a significantly positive role for bilateral trade or political agreement in facilitating influential or majority acquisitions in target countries. H3 is by consequence not supported by the data as the variable FTAFORCE is never significant and FTANEGO is found negatively significant for decisions to take majority stakes and not significant in the large stake decision setup.

In order to analyse the political motivation of GCC SWFs, we have tested whether these funds seek majority stakes in strategic sectors as explained in the H4. We find that the dummy LUXURY is significantly positive in the second model, meaning that GCC SWFs are more prone to take majority stakes in the luxury sector. On the other hand, the variables FINANCE and ENERGY are negatively related to influential acquisitions made by GCC SWFs. Concerning the financial sector, our result can be explained, first by regulatory/enforcements efforts made by developed countries (especially US and the European countries) in order to hinder SWFs majority acquisitions in this sector, and second by the large size of firms operating in this sector (a large amount may correspond to a minority stake). In the same way, we find that cross-border influential acquisition (more than 10%) of Gulf SWFs are under-represented in the energy sector, indicating that resource-rich countries, that are extremely dependent on revenues from oil and gas, try to diversify their revenues by targeting foreign industries in which the country is under-represented.

Concerning the SWF-level variables, we find that the probability for GCC SWFs of taking an influential (10% or more) stake in a cross-border industry is positively related with the transparency of the fund. Similarly, the probability of taking majority (50% or more) stake is negatively related to the size of the SWF. A small and/or transparent SWF are therefore less likely to raise hostility and political pressure from the host country's government increasing therefore the probability of influential/control stakes. The result is in line with [START_REF] Murtinu | Sovereign Wealth Funds' Internationalization Strategies: The Use of Investment Vehicles[END_REF] who find that opaque SWFs are more likely to invest cross-border through an investment vehicle than transparent SWFs in order to show a passive investment approach and reduce the political pressure in the host country.

Turning to our control variables, we unsurprisingly see that oil trade resources are a driver of GCC SWFs large acquisitions. [START_REF] Hay | The increasing interest of Middle Eastern sovereign wealth funds for Europe: presenting original data[END_REF] also find that the number of acquisitions has followed the same orientation than crude oil prices. Interestingly, our results show that GCC SWFs have limited their cross-border majority acquisitions (50% or more) during the financial crisis.

Ordered Logit Analysis

The model

Although informative, the separate (binary) Logit estimates are not fully satisfactory. First, the results for our two specifications (thresholds at 10% and 50%) are not always in line and their consistency cannot be easily tested in such setting. Second, the SWF investment decision is a 1-step decision in a 3-choice setting and not a set of two independent binary decisions as implied by the Logit approach. We thus need a multinomial discrete (three) choice model able to capture an efficient/simultaneous estimate of the parameters since SWFs have to decide either to stay minority shareholders (stake of less than 10%) in the foreign industry or to take more than 10% but less than 50% in order to have a significant influence on the management of the target or to take the control of the firm (more than 50%).

The ordered Logit model meets these requirements and addresses well for the specificities of the SWFs investment decision making-process. 8 Number of studies such as [START_REF] Ederington | Classification Models and Bond Ratings[END_REF] or Poon (2007) illustrate the relevance of considering ordered Logit models in explaining and predicting similar ordered categorical variables such as bond or credit rating (see also [START_REF] Wooldridge | Econometrics of cross section and panel data[END_REF] and [START_REF] Long | Regression models for categorical dependent variables using stata[END_REF]).

In our set-up, we define Y ij , the degree of acquisition of the fund i in a target j as follows:

9             
Y = 0 if the post-transaction share is less than 10% Y = 1 if the post-transaction share is between 10% and 50% Y = 2 if the post-transaction share is more than 50%

(2)

Similarly to the (binary) Logit model, the ordered Logit model can be derived from an

8 Ordered Logit models rely on the parallel regression assumption. A more general model, so-called generalized ordered Logit model [START_REF] Long | Regression models for categorical dependent variables using stata[END_REF]) relaxes this assumption. We performed a Wald test developed by [START_REF] Brant | Assessing proportionality in the proportional odds model for ordinal logistic regression[END_REF] in order to discriminate between the ordered Logit model and the generalized ordered Logit model. Results confirm that the ordered Logit model is not rejected and best fit our data. Results are available upon request. 9 We will no longer explicitly refer to ij subscripts in view to simplify notations underlying latent variable Y * determined as follows:

     Y * = Xβ + e, e | X ∼ Λ(0, π 2 3 ) Y = k if α k < Y * ≤ α k+1 k = 0, 1.2 (3)
where β is the coefficient vector, X is the vector of explanatory variables, Λ(.) is the standard logistic distribution function, α 0 = -∞, α 3 = +∞, and α 1 and α 2 are unknown parameters.

As Y * crosses unknown thresholds α 1 or α 2 , we move up the ordering of alternatives. For example, with a very low Y * , i.e. smaller than α 1 , we get a minority stake. For a very high Y * , i.e. larger than α 2 , we get a majority stake.

We can then easily compute the conditional choice probabilities:

                 P (Y = 0 | X) = P (Y * ≤ α 1 | X) = 1 1 + exp(Xβ -α 1 ) P (Y = 1 | X) = P (α 1 < Y * ≤ α 2 | X) = 1 1 + exp(Xβ -α 2 ) - 1 1 + exp(Xβ -α 1 ) P (Y = 2 | X) = P (Y * > α 2 | X) = 1 - 1 1 + exp(Xβ -α 2 ) (4)
The cut-points α 1 , α 2 and the parameters β are estimated by maximum likelihood. The interpretation of the βs are of limited interest as they relate to Y * while our focus is on the categorical variable Y . We will thus report in the results section the partial effects of X k on the probabilities, which are computed as follows for the continuous explanatory variables:

                 ∂P (Y = 0 | X)/∂X k = - exp(Xβ -α 1 ) (1 + exp(Xβ -α 1 )) 2 β k ∂P (Y = 1 | X)/∂X k = exp(Xβ -α 2 ) (1 + exp(Xβ -α 2 )) 2 - exp(Xβ -α 1 ) (1 + exp(Xβ -α 1 )) 2 β k ∂P (Y = 2 | X)/∂X k = exp(Xβ -α 2 ) (1 + exp(Xβ -α 2 )) 2 β k (5)
The partial effects on P (Y = 0 | X) and P (Y = 2 | X) are unambiguously determined by the sign of β k , while the sign is not conclusive for the effect on the intermediate category.

Since partial effects are conditional on specific values for X, we follow common practice by setting the variables at their average values. As these partial effects are of little relevance when X k is a dummy variable, we will in these cases rather compute the partial effects for indicator variables as follows:

∂P (Y = k | X)/∆X k = P (Y = k | X -k , X k = 1) -P (Y = k | X -k , X k = 0) (6)
where X -k , the explanatory variables other than X k , are set at their average values.

Results

We report the results of the general and parsimonious ordered Logit model estimations in Table 7. We also report the marginal effects based on the parsimonious model in Table 8.

The estimates of the ordered Logit model confirm the results of the bivariate Logit models on several aspects. First, regarding financial characteristics of the targeted firm, once again we find that variables representing the financial health of the targeted firm are not significant.

It means that the financial characteristics of the target firm do not matter in the control decision making-process for GCC SWFs.

Secondly, we find confirmation that GCC SWFs prefer to take the control of a target firm in nations where there are better investment protections. A 0.10 increase in the ANTISELF variable (which stands within the range 0-1), decreases the probability to take a minority stake (less than 10%) by 4.68 percentage points and increases the probability to take a majority acquisition (more than 50%) by 3.59 percentage points, holding all other variables at their means. In the same way, we find that the probability to take a majority (resp. minority) stake increases (resp. decreases) by 21.5 percentage points (resp. 30.5) when the target country is in the European Union.

Thirdly, in line with the results of the bivariate Logit analysis, and [START_REF] Karolyi | State capitalism's global reach: Evidence from foreign acquisitions by state-owned companies[END_REF] who find that bilateral trade is positively associated with SWF investment, suggesting that variables explaining the investment decision can differ according to the degree of investment acquisition. Our result also indicates that GCC SWFs differ from other institutional investors in their investment strategies as [START_REF] Roque | The determinants of international equity investment: Do they differ between institutional and noninstitutional investors[END_REF] show that bilateral trade significantly contributes to increase institutional investors' international equity investments.

Regarding H4 which posits that GCC SWFs target cross-border companies operating in strategic industries, the estimates of both bivariate Logit models and the ordered Logit model give more contrasted results. In Table 7, the variables FINANCE and ENERGY are negatively significant, which means that the probability of having influential or majority stakes decreases when the target firm belongs to one of these sectors in contrast with minority stakes (less than 10%). In line with the bivariate Logit model, the variable LUXURY is significant in the ordered Logit model, meaning that GCC SWFs do target particularly luxury industry in their takeover decision. These results suggest that cross-border acquisitions taken by GCC SWFs are not a way for these countries to acquire strategic interests abroad.

On the other hand, it may be a consequence of the regulations aiming at preventing SWFs to take significant stakes in strategic sectors.10

Not surprisingly, the probability of having influential or majority takeovers increases with the transparency of the fund. When a fund is transparent, the likelihood to take a minority stake decreases by 26.1 percentage points, while the likelihood to take a majority stake (more than 50%) increases by 24.3 percentage points. SWFs that are managed in a non-transparent way have a tendency to increase the host country fears. Consequently, transparent SWFs have less trouble taking influential stakes in cross-border companies.

At last, results of the ordered Logit model unsurprisingly confirm that oil trade resources are a driver of GCC SWFs majority acquisitions. GCC SWFs are, indeed, financed by the proceeds from petroleum and gas.

Conclusion

While there is an extensive literature investigating the SWFs investment decision makingprocess, only few papers address the question of the determinants of SWFs cross-border equity majority acquisitions. This is, however, a key question as it is clear that SWF ac- Unlike results found in the literature investigating the determinants of SWFs cross-border investments, our results indicate that target firm-level characteristics do not matter in the decision of GCC SWFs cross-border majority acquisition. It means that the financial characteristics of the target firm are not criteria considered in the control decision making-process for GCC SWFs. A major factor determining the pattern of cross-border majority acquisition for this type of investors is the level of shareholder protection in the target country. More precisely, we find that GCC SWFs prefer investing in countries presenting a high level of shareholder protection, preferably in the European Union. In addition, we find that GCC SWFs do not target strategic sectors (except the Luxury sector if we see this sector as strategic) when taking influential or majority stakes, suggesting that cross-border equity majority-border acquisitions are not a way for these countries to acquire strategic interests abroad. The transparency of the fund is also a significant factor determining cross-border majority stake which can be explained by the fact that a transparent SWF likely reduces the hostility and political pressure from the host country's government increasing therefore the probability of influential stake. This result suggests that SWFs wishing to be involved in foreign firms management should improve their degree of transparency. Finally, contrary to the literature on the determinants of cross-border SWFs investments, we do not find some evidence that SWFs prefer to take large or majority acquisitions in countries where there are bilateral agreements. This result indicates that SWFs differ from other institutional investors in their investment strategies.

Overall, our results shed new light on SWFs investment strategy, indicating that even if their objectives may go beyond pure profit maximization, acquiring majority stakes is not a lever for governments to get strategic interests in the targeted country. These results may be of interest for the regulators seeking the optimal regulatory response to the activism of SWFs.

Figure 1 :

 1 Figure 1: Evolution of GCC SWFs Foreign Investments This Figure presents the number of deals and the average post-acquisition stake of cross-border investments led by GCC SWFs over the period 2006-2015. The graph excludes investments of Qatar Investment Authority in Xstrata which consists in 39 acquisitions of less than 1% of the target firm.

  measuring the ex-ante and ex-post effectiveness of regulation and enforcement against violators. The Chinn-Ito financial openness index of the target country (logKAOPEN ) initially introduced by Chinn and Ito (2006) measures the target country's degree of capital account openness.POLITY captures the level of authority of the targeted country as defined by the Polity IV database. Like in[START_REF] Hay | The increasing interest of Middle Eastern sovereign wealth funds for Europe: presenting original data[END_REF], a regional dummy (EU DUMMY ) for Europe is also included. This dummy variable equals 1 if the target country is in Europe and 0 otherwise. As macroeconomic performance indicators, the Gross Domestic Product per capita (GDP ) of the target country is included in order to capture the relative size of the target country economy. At last, we consider the real effective exchange rate variations of the target country in logarithmic form (dlogREER) as indicator of gain/loss of competitiveness.

  tivism generates more hostility by host-country governments. Understanding the patterns and motivations for SWF cross-border majority acquisition is therefore an important and understudied research topic. Using a unique database of 124 cross-border equity acquisitions from GCC SWFs in 28 target countries over the period 2006-2015, we aim to fill this gap in the literature. Based on ordered Logit models, we test if the usual determinants of SWFs investments stand in the case of majority acquisitions made by a distinct group among SWFs, namely by GCC SWFs.

Table 1 :

 1 GCC SWFs characteristicsThis table describes GCC SWFs' main characteristics. Column (3) gives the date of the creation of each fund. Column (4) gives the list of each fund's wholly-owned subsidiaries, based on the authors' researches. Column (5) gives the assets under management of each fund in USD billion.Column (6) gives the Linaburg-Maduell transparency index of each fund. a Column (7) gives the objective(s) of the funds, based on the SWFs websites.

	Objective(s)			Reserve investment fund			Reserve investment fund	Reserve investment fund	Saving; Reserve investment	fund; Domestic economic sup-	port	Reserve investment fund	Domestic economic support	Domestic economic support	Reserve investment fund	Reserve investment fund	Reserve investment fund	Reserve investment fund			Domestic Economic Support		Reserve investment fund			Reserve investment fund	Domestic economic support
	Transpa-	rency index		8			6	4		4		5	nc	4	4	9	5	nc			10		3			5	3
	AUM			11.1			592	6		9		256	25	160	632.3	66.3	773	110			66.3		15			183	1.2
	Wholly-owned	Subsidiaries	Atbahrain B.S.C; Bahrain Real Es-	tate Investment; Gulf Aviation	Academy	Reserve fund for future genera-	tions; General reserve fund; Saint	Martins Property Co				Qatar Airways; Qatar holding llc;	Qatar Sport Investments		Sanabil Investment		CEPSA ; Nova Chemicals Corp.;	Aabar investment	Harina Company Limited ; Lux-	inva SA; Tawreed Investments	Al Hilal Bank PJSC	Mubadala Capital and Real Es-	tate; Mubadale Capital LLC;	Mubadala Commercial Finance;	Mubadala Petroleum; Mubadala	technology	Dubai Holding ; Dubai World; Is-	tithmar world; Dubai Ports World;	Dubai world Africa service; Dubai	Airport Free Zone Authority; Emi-	rates airlines; Emirates Group;	Emirates national oil co LTD; Na-	tional Bonds Corporation	Al-ghail power
	Creation			2006			1953	2006		1980		2005	1980	2008	1952	1984	1976	2007			2002		2007			2006	2005
	Name			Bahrain Mumtalakat Holding			Kuwait Investment Authority	Oman Investment Fund		State General Reserve Fund		Qatar Investment Authority	Kingdom Holding Company	Public Investment Fund	SAMA Foreign Holding	Abu Dhabi International	Petroleum Investment Company	Abu Dhabi Investment Authority	Abu Dhabi Investment Council		Abu Dhabi Mubadale Develop-	ment Company	Emirates Investment Authority			Investment Corporation of Dubai	RAK Investment Authority
	Country			Bahrain			Kuwait	Oman				Qatar	Saudi Arabia			UAE
																														a

Table 2 :

 2 Overview of SWFs' transactionsThis table presents the number of deals and majority deals by GCC SWFs over the period 2006-2015. DEALS represents the total number of deals involving GCC SWFs. DEALS -10% represents the number of deals in which the post-acquisition stakes owned by the SWFs is lower than 10%. DEALS+10% represents the number of deals in which the post-acquisition stakes owned by the SWFs is higher or equal to 10% and lower than 50%. DEALS + 50% represents the number of deals in which the post-acquisition stakes owned by the SWFs is higher or equal to 50%.

	SWF country	DEALS	DEALS-10%	DEALS+10%	DEALS+50%
	Abu Dhabi International Petroleum Investment Company	19	6	7	6
	Abu Dhabi Investment Authority (UAE)	11	8	1	2
	Abu Dhabi Mubadala Development Company (UAE)	11	2	5	4
	Investment corporation of Dubai (UAE)	20	6	6	8
	Kuwait Investment Authority	6	3	2	1
	Oman Investment Fund	2	1	0	1
	Public Investment Fund (SA)	1	0	1	0
	Qatar Investment Authority	50	21	18	11
	RAK Investment authority (UAE)	1	1	0	0
	SAMA Foreign Holding (SA)	1	1	0	0
	State General Reserve Fund (Oman)	2	1	0	1
	Total	124	50	40	34
		100%	40%	32%	27%

Table 3

 3 gives the geographic distribution of GCC SWFs cross-border acquisitions. More than half of the transactions done by GCC are located in Europe and they are mainly majority acquisitions (64% of the deals are post-transaction shares of 10% or more and 32%

are shares of 50% or more). The other target regions are North America (almost 13% of total deals) with mainly minority investments (63% of deals less than 10% of shares) and East and Southeast Asia (10% of total deals) with larger stakes (62% of deals of more than

Table 4

 4

gives the sectorial repartition of GCC SWFs acquisitions and majority acquisitions over 2006-2015. Energy, finance, industry and luxury are key sectors and account each

Table 3 :

 3 Geographic repartition of GCC SWF cross-border investmentsThis table presents the number of deals and majority deals by target region of cross-border investments led by GCC SWFs over the period 2006-2015. DEALS represents the total number of deals involving GCC SWFs. DEALS -10% represents the number of deals in which the post-acquisition stakes owned by the SWFs is lower than 10%. DEALS + 10% represents the number of deals in which the post-acquisition stakes owned by the SWFs is higher or equal to 10% and lower than 50%. DEALS + 50% represents the number of deals in which the post-acquisition stakes owned by the SWFs is higher or equal to 50%.

	Region target	DEALS	DEALS-10%	DEALS+10%	DEALS+50%
	Africa	3	3	0	0
		100%	100%	0%	0%
	Central Asia	6	3	1	2
		100%	50%	17%	33%
	East and Southeast Asia	13	3	8	2
		100%	23%	62%	15%
	North America	16	10	3	3
		100%	63%	19%	19%
	Oceanic Bassin	8	4	2	2
		100%	50%	25%	25%
	South America	1	0	1	0
		100%	0%	100%	0%
	Europe	77	27	25	25
		100%	35%	32%	32%
	Total	124	50	40	34
		100%	40%	32%	27%

Table 4 :

 4

	Sector target	DEALS	DEALS-10%	DEALS+10%	DEALS+50%
	Energy	30	14	9	7
		100%	47%	30%	23%
	Finance	14	8	4	2
		100%	57%	29%	14%
	Industry	19	9	7	3
		100%	47%	37%	16%
	Luxury	18	5	4	9
		100%	28%	22%	50%
	Real Estate	9	3	3	3
		100%	33%	33%	33%
	Transportation	7	5	1	1
		100%	71%	14%	14%
	Other	27	6	12	9
		100%	22%	44%	33%
	Total	124	50	40	34
		100%	40%	32%	27%

Sectoral repartition of GCC SWF cross-border investments

This table presents the number of cross-border deals and majority deals by target sectors led by GCC SWFs over the period 2006-2015. DEALS represents the total number of deals involving GCC SWFs. DEALS -10% represents the number of deals in which the post-acquisition stakes owned by the SWFs is lower than 10%. DEALS + 10% represents the number of deals in which the post-acquisition stakes owned by the SWFs is higher or equal to 10% and lower than 50%. DEALS + 50% represents the number of deals in which the post-acquisition stakes owned by the SWFs is higher or equal to 50%.

Table 5 :

 5 Description of the variables

	Variables	Description

dlogREER Consumer price index-based real effective exchange rate of the targeted country. Mean of dlog over t -1, t -2 and t -3. Positive variation is an appreciation Bruegel Database logKAOPEN Logarithm of the normalized KAOPEN index of the targeted country in t -1. Initially introduced by Chinn and Ito (2006), this index measures a country's degree of capital account openness

Table 6 :

 6 Logit models -influential stakes and majority stakes This table reports the results of separate estimations of Logit models with robust errors. In models (1) and (2) (resp. (

Table 7

 7

provides evidence against H3 according to which GCC SWFs are more likely to take majority stakes in countries where there are bilateral trade agreements, as both proxies for bilateral trade agreements are non-significant. This result is inconsistent with Johan et al. (

2013

),

[START_REF] Knill | Bilateral political relations and sovereign wealth fund investment[END_REF]

Table 7 :

 7 Ordered Logit modelsThis table reports the results of the estimations of the ordered Logit model with robust errors, where the endogenous variable Y ij takes the value 1 if the post-transaction share of the SWF i in the target j is smaller than 10%, the value 2 if the share lies between 10% and 50% and the value 3 if it is larger or equal to 50%. Column (1) reports the estimates for the full model, while column (2) reports the parsimonious model resulting from a backward-selection approach with significance level for removal of the model set at 10%.

		ologit full	ologit pars.
		(1)	(2)
	TRANSPARENT	1.298**	1.200**
		(0.625)	(0.488)
	SUBSIDIARY	0.283
		(0.494)
	EUDUMMY	0.975	1.288***
		(0.733)	(0.408)
	ANTISELF	2.546**	1.971***
		(1.009)	(0.707)
	OILPRICE	2.102*	1.903*
		(1.160)	(1.027)
	ROA	-0.029
		(0.032)
	DEBTASSETS	-1.221
		(1.072)
	LIQUIDITY	0.040
		(0.072)
	DEBT	0.001
		(0.008)
	ASSETS	0.001
		(0.003)
	CRISIS	-0.869*
		(0.505)
	FTAFORCE	0.433
		(0.711)
	FTANEGO	-0.291
		(0.632)
	dlogREER	5.280*
		(2.997)
	GDP	-0.013
		(0.017)
	logKAOPEN	0.590
		(0.718)
	LUXURY	1.173*	0.987*
		(0.652)	(0.538)
	FINANCE	-1.350*	-1.422**
		(0.799)	(0.684)
	ENERGY	-1.109	-1.114**
		(0.770)	(0.518)
	INDUSTRY	-0.826
		(0.826)
	SIZESWF	-0.002
		(0.001)
	POLITY	0.036
		(0.047)
	α 1	9.757*	9.922**
		(5.429)	(4.585)
	α 2	11.543**	11.532**
		(5.518)	(4.634)
	Log-likelihood	-113.210	-121.388
	AIC	274.420	260.777
	BIC	342.107	286.159
	* p<0.10, ** p<0.05, *** p<0.01
	Robust standard-errors are between parentheses.

Table 8 :

 8 Marginal effects at means of the ordered Logit parsimonious modelThis table reports for continuous variables the respective marginal effects of ANTISELF and OILPRICE on the probabilities to take minority, influential and majority stakes, all other variables being held constant at their means, and for dichotomous variables the respective marginal effects for an investment of being made in Europe, in the luxury/finance/energy sector and made via a rather transparent SWF, on the probabilities to take minority, influential and majority stakes, all other variables being held constant at their means. The estimates are derived from the benchmark (parsimonious ordered Logit) model reported in column (2) of Table7.

		Predict Y=0	Predict Y=1	Predict Y=2
		Minority Stake	Influential Stake	Majority stake
	Continuous explanatory variables -∂P (Y = k | X)/∂X k
	ANTISELF	-0.468 ***	0.110	0.359 ***
	OLIPRICE	-0.452 *	0.106	0.346 *
	Binary explanatory variables -∂P (Y = k | X)/∆X k
	EUDummy	-0.305 ***	0.090	0.215 ***
	LUXURY	-0.209 **	0.000	0.208
	FINANCE	0.340 **	-0.153	-0.188 ***
	ENERGY	0.270 **	-0.097	-0.173 ***
	TRANSPARENT	-0.261 ***	0.018	0.243 ***

* p<0.10, ** p<0.05, *** p<0.01. Based on robust standard-errors.

According to the Sovereign wealth Fund Institute, the assets managed by these funds were estimated to be USD 3,2 trillion in September

2007, which means that the size of these funds has more than doubled since the beginning of the financial crisis (source: www.swfinstitute.org).2 GCC member states are Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and the United Arab Emirates. The United Arab Emirates are a federation of seven emirates, including Abu Dhabi, Dubai and Ras al-Khaimah, which all have their own SWF.

An example is the trade agreement between the US and 11 Asia-Pacific countries (the Trans-Pacific Partnership), where the exclusion of China by the US reveals the political nature of this agreement.

10% is also the relevant threshold used in the European legislation for defining "qualifying holdings" (Article 92 of Directive 2001/34/EC) and in the above-mentioned Commission Delegated Regulation (EU) 2017/1946 of 11 July 2017.

Similar results are obtained by relying on probit rather than Logit models. Results are available on request.

The Logit model can also be derived from underlying behavioural assumptions, which lead to a latent variable representation of the model (see[START_REF] Wooldridge | Econometrics of cross section and panel data[END_REF] and the next sub-section where this approach is presented in a more general 3-discrete choice framework).

GCC SWFs have several partnerships with European companies. For example, the SWF of Abu Dhabi has partnerships with Airbus and Total from France, Siemens from Germany or Rolls-Royce from UK.

Such regulations are in place in many developed countries such as the United States or the European Union countries.


Pierre Ablin

Jean-François Cardoso

Alexandre Gramfort

Beyond Pham's algorithm for joint diagonalization

), and propose a new quasi-Newton method for its optimization. Through numerical experiments on simulated and real datasets, we show that the proposed method outperforms Pham's algorithm. An open source Python package is released.

Introduction

The task of joint diagonalization arises in several formulations of the blind source separation problem. In [START_REF] Cardoso | Blind beamforming for non-gaussian signals[END_REF], independent component analysis is performed by joint-diagonalization of a set of cumulant matrices. In [START_REF] Belouchrani | A blind source separation technique using second-order statistics[END_REF], the separation of nonstationnary signals is carried by joint-diagonalization of a set of autocorrelation matrices. Finally, in [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF], joint-diagonalization of a set of covariance matrices separates Gaussian sources that have non-stationnary power.

Consider a set of n symmetric square matrices C (C 1 , • • • , C n ) of size p × p. Its approximate joint diagonalization consists in finding a matrix B ∈ R p×p such that the matrix set BCB (BC 1 B , . . . , BC n B ) contains matrices that are as diagonal as possible, as measured by some joint-diagonality criterion. This paper considers the joint diagonalization of positive matrices, defined as the minimization of the (non-convex) criterion

L(B) = 1 2n n i=1 log det diag(BC i B ) -log det(BC i B ) , (1)
This criterion was introduced by Pham in [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF] who derived it as the negative log-likelihood of a source separation model for Gaussian stationary sources.

Pham [START_REF] Pham | Joint approximate diagonalization of positive definite hermitian matrices[END_REF] proposes in its seminal work a block coordinate descent approach for the minimization of L. Each iteration of this method guarantees a decrease of L. Further, when there exists a matrix B * such that B * CB * contains only diagonal matrices (that is, if the set is exactly jointly diagonalizable), then in the vicinity of B * , the algorithm converges quadratically. Since then, only a few other papers have focused on minimizing this criterion. In [START_REF] Joho | Newton method for joint approximate diagonalization of positive definite hermitian matrices[END_REF], the Newton method is studied. However, this method is not practical as it requires solving a p 2 × p 2 linear system at each iteration. In [START_REF] Todros | Fast approximate joint diagonalization of positive definite hermitian matrices[END_REF], it is proposed to minimize an approximation of the cost function. The authors do so by alternating minimization over columns. This method minimizes an approximation of L, and scales in O(p 4 ).

In this paper, we propose a quasi-Newton method for the minimization of L.

We use sparse approximations of the Hessian which are cheap to compute and match the true Hessian when the set is jointly diagonalized, granting quadratic convergence. The algorithm as a cost per iteration of O(n × p 2 ), which is the natural scaling of the problem since it is the size of the dataset. Through experiments, we show that the proposed method outperforms Pham's algorithm, on both synthetic and real data.

Notation:

The identity matrix of size p is denoted I p . The Frobenius scalar product between two p × p matrices is noted as

M |M a,b M ab M ab . The corresponding norm is ||M || F = M |M . Given a p × p × p × p tensor H, the weighted scalar product is M |H|M a,b,c,d H abcd M ab M cd . The Kronecker symbol δ ab is 1 is a = b, 0 otherwise.

Study of the cost function

The cost function L(B) is defined on the group of invertible matrices, with log det(BC i B ) acting as a barrier. Its minimization is performed by iterative algorithms. To exploit the multiplicative structure of the group of invertible matrices, we perform relative updates on B [START_REF] Cardoso | Equivariant adaptive source separation[END_REF]. The neighborhood of an iterate B (t) is parameterized by a small p × p matrix E as B (t+1) = (I p + E)B (t) . The second-order (in E) Taylor expansion of the loss function

L((I p + E)B) = L(B) + G|E + 1 2 E|H|E + o(||E|| 2 ) . (2)
where the p × p matrix G(B) is the relative gradient and the p × p × p × p tensor H(B) is the relative Hessian. Simple algebra yields:

G ab = 1 n n i=1 D i ab D i aa -δ ab , and
H abcd = δ ac 1 n n i=1 D i bd D i aa -2 D i ab D i ad D i aa 2 + δ ad δ bc
(3) where we define D i = BC i B for i = 1, . . . , n. Eq. [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF] shows that H aaaa = 0, consistent with the scale-indeterminacy of the criterion: L(ΛB) = L(B) for any diagonal matrix Λ. The criterion is flat in the direction of the scale matrices.

Complexity:

The cost of computing a gradient is O(p 2 × n). It is the natural complexity of an iterative algorithm as it is the same cost as computing the set BCB . Computing the Hessian is O(p 3 × n), and computing H -1 G is O(p 6 ). This is prohibitively costly when p is large compared to a gradient evaluation. As a consequence, Newton's method is extremely costly to setup.

Relative Quasi-Newton method

In this section, we first introduce an approximation of the Hessian and then derive a quasi-Newton algorithm to minimize (1).

Hessian approximation

The accelerated algorithm presented in this article is based on the massive sparsification of the Hessian tensor when matrices D i are all diagonal. Indeed, it that case, it becomes:

Habcd = δ ac δ bd Γ ab + δ ad δ bc -2δ abcd Γ ab = 1 n n i=1 D i bb D i aa (4)
This Hessian approximation has three key properties. First, it is cheap to compute, at cost O(p 2 × n), just like a gradient. Then, it is sparse and structured. It only has 2p 2 non-zero coefficients, and can be seen as a block-diagonal operator, with blocks of size 2. Indeed, for a p × p matrix M , we have:

[ HM ] ab [ HM ] ba = Γ ab 1 -2δ ab 1 -2δ ab Γ ba M ab M ba . (5)
The following lemma establishes the positivity of the approximate Hessian:

Lemma 1
The approximation H is positive with p zero eigenvalues. If the matrices C i are independently sampled from continuous densities, with probability one, the other p 2 -p eigenvalues are strictly positive.

Proof: Using eq. ( 4), one has HE ii = 0, where E ii is the matrix with 1 for its (i, i) coefficient, and 0 elsewhere. Thus H has p zero eigenvalues, and the associated eigenvectors are the E ii for i = 1 • • • p. The p 2 -p other eigenvalues are the eigenvalues of the 2 × 2 blocks introduced in eq. ( 5). The diagonal coefficients of the blocks are the Γ ab > 0. The determinant of a block is given by Γ ab Γ ba -1. Cauchy-Schwartz inequality yields Γ ab Γ ba ≥ 1, with equality if and only if D i aa ∝ D i bb . This happens with probability 0, concluding the proof. Finally, the approximation is straightforwardly inverted by inverting each 2 × 2 blocks. The Moore-Penrose pseudoinverse of H, H+ , satisfies:

[ H+ G] ab = Γ ba G ab -G ba Γ ab Γ ba -1 , ∀a = b , (6)
and [ H+ G] aa = 0. The cost of inversion is thus O(p 2 ).

Algorithm

The proposed quasi-Newton method uses -H+ G as search direction. Following from the positivity of H, this is a descent direction. Unfortunately, there is no Algorithm 1: Quasi-Newton method for joint-diagonalization Input : Set of matrices C, number of iterations T . Initialize B = I p . for t = 1, • • • , T do Compute the gradient G using eq.( 3) Compute the Hessian approximation H using eq.4

Compute the search direction -H+ G using eq. ( 6) Do a backtracking line search in that direction to find a step size α decreasing L Set B ← (I p -α H+ G)B end Output: B guarantee that the iteration B ← (I p -H+ G)B decreases the cost function. Therefore, we resort to a line-search to find a step α > 0 ensuring L((I pα H+ G)B) < L(B) (condition ( * )). This is done using backtracking, starting from α = 1 and iterating α ← α 2 until the condition ( * ) is met. The full algorithm is summarized in Algorithm. 1.

Quadratic convergence: Like Pham's algorithm, the proposed algorithm enjoys quadratic convergence when the matrix set is jointly diagonalizable. Indeed, assume that there exists a matrix B * such that B * CB * contains only diagonal matrices. Then, by construction, H(B * ) = H(B * ). It follows that the method converges quadratically in the vicinity of B * .

Experiments

Setting

For the experiments, three data sets are used -coming either from synthetic or real data -and we compare our approach to Pham's algorithm. The code to reproduce the experiments is available online 1 . We set n = 100 and p = 40.

Initialization: For a dataset C, the algorithms start from a whitener (whitening matrix): writing P DP = 1 n n i=1 C i with P orthogonal and D diagonal, the initial matrix is taken as B (0) = D -1 2 P . Metrics: To compare the speed of convergence of the algorithms, we monitor the diagonalization error L(B), and the gradient norm ||G(B)||. The first quantity goes to 0 if the dataset is perfectly diagonalizable, while the second should always converge to 0 since the algorithm should reach a local minimum.

Synthetic data: We proceed as in [START_REF] Ziehe | A fast algorithm for joint diagonalization with non-orthogonal transformations and its application to blind source separation[END_REF] for generating synthetic datasets. We generate n diagonal matrices of size p × p, (D 1 , • • • D n ) for which each diagonal coefficient is drawn from an uniform distribution in [0, 1]. Then, we generate a random 'mixing' matrix A ∈ R p×p with independent normally distributed entries. Finally, in order to add noise, we generate

n matrices R 1 , • • • , R n ∈ R p×p of normally distributed entries. The dataset is then C = (C 1 , • • • , C n ) with: C i = AD i A + σ 2 (R i )(R i ) , (7)
where σ controls the noise level. In practice we take σ = 0 (experiment (a), perfecty jointly-diagonalizable set) or σ = 0.1 (experiment (b)).

Magnetoencephalography (MEG) data:

We use the MNE sample dataset [START_REF] Gramfort | MNE software for processing MEG and EEG data[END_REF]. From n matrices containing p signals of T samples, X 1 , • • • , X n ∈ R p×T , corresponding to time segments of MEG signals, we jointly diagonalize the set of covariance matrices C i = 1 T X i X i (experiment (c)). This practical task is in the spirit of [START_REF] Pham | Blind separation of instantaneous mixtures of nonstationary sources[END_REF]. Results are displayed in Figure 1 4

.2 Discussion

In experiment (a), where the dataset is exactly jointly diagonalizable, we observe the expected quadratic rate of convergence for both the proposed algorithm and Pham's method. We also observe that breaking the model (experiments (b) and (c)) makes convergence fall back to a linear rate. As expected, the cost function does not go to 0 in those cases. We observe that the proposed quasi-Newton algorithm outperforms Pham's method by about an order of magnitude on each experiment.

1Fig. 1 :

 1 Fig. 1: Comparison of the algorithms on three datasets. (a): Jointly diagonalizable synthetic dataset. (b): Same dataset with added noise, rendering perfect joint diagonalization impossible.(c): Real data, covariance matrices from MEG signals. Note the log-log scale.

* This work was supported by the Center for Data Science, funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02, and the European Research Council (ERC SLAB-YStG-676943).