Y Ménesguen

M.-C Lépy

Reference-free combined XRR-GIXRF analysis at the SOLEIL synchrotron

Keywords: X-ray reectometry, spectrometry, grazing incidence x-ray uorescence, thin lms structure

CASTOR is a new instrument, operated at the SOLEIL synchrotron facility, dedicated to the characterization of thin lms with thicknesses in the nanometer range. The instrument can combine x-ray reectivity (XRR) measurements with uorescence (XRF) acquisitions and especially total reection x-ray uorescence (TXRF) related techniques such as grazing incidence XRF (GIXRF). The instrument is now routinely installed on the hard x-ray branch of the Metrology beamline. Geometrical characterization is presented, reproducibility of measurements is studied and the reference-free GIXRF analysis is described. Some representative examples are given to illustrate the capabilities of the setup and of the combined analysis procedure.

Introduction

Multilayered thin lm stacks with layer thicknesses in the nanometer range have been commonly characterized by x-ray reectometry (XRR) on synchrotron facilities or with in-lab sources for many years [START_REF] Parratt | Surface Studies of Solids by Total Reection of X-Rays[END_REF]. Reectivity curves are acquired by varying the incident angle in the grazing incidence regime while recording the intensity of the specular reected beam. Grazing incidence x-ray uorescence (GIXRF) is a total reection x-ray uorescence analysis (TXRF) related technique [START_REF] De Boer | Glancing-incidence x-ray uorescence of layered materials[END_REF], which uses the x-ray uorescence (XRF) angle-dependent intensity at shallow incidence angles. The renewed interest for these techniques came from the elaboration of nanomaterials for applications in power electronics, electricity storage and microelectronics, which require new means of accurate characterization at the nanometer scale. The XRF signal is element-specic and contains information about the elemental composition, concentration proles and thicknesses. The combination of XRR, sensitive to the electronic density, and GIXRF, sensitive to element density, produces results of better accuracy compared to one technique alone. Some experimental facilities have already been developed to allow GIXRF-XRR combined analysis, using either synchrotron radiation [START_REF] Lubeck | A novel instrument for quantitative nanoanalytics involving complementary X-ray methodologies[END_REF][START_REF] Das | Simultaneous measurement of X-ray reectivity and grazing incidence uorescence at BL-16 beamline of Indus-2[END_REF] or laboratory x-ray sources [START_REF] Ingerle | Combination of grazing incidence x-ray uorescence with x-ray reectivity in one table-top spectrometer for improved characterization of thin layer and implants on/in silicon wafers[END_REF][START_REF] Spanier | A exible setup for angle-resolved X-ray uorescence spectrometry with laboratory sources[END_REF].

The Laboratoire National Henri Becquerel (LNHB) developed a specic instrument to perform such charaterizations at the Metrology beamline of the SOLEIL synchrotron source to propose metrological studies of thin lm deposits and support the eorts of institutes or companies in the fast-developing eld of nanolayered structures. The instrument CASTOR (Chambre d'Analyse Spectrométrique en Transmission ou en Réexion -Analysis Chamber for Transmission or Reection Spectrometry) is a seven axis goniometer based on the model developed at the Physikalisch Technische Bundesanstalt (PTB) and the Technical University of Berlin [START_REF] Lubeck | A novel instrument for quantitative nanoanalytics involving complementary X-ray methodologies[END_REF][START_REF] Lubeck | A new generation of x-ray spectrometry UHV instruments at the SR facilities BESSY II[END_REF]. This work describes the main parts of the instrument and additional equipment, the experimental conditions at the Metrology beamline of the SOLEIL synchrotron facility and the rst results obtained on selected samples.

2 Characteristics of the experimental setup 2.1 The hard x-ray branch of the SOLEIL Metrology beamline At the SOLEIL synchrotron (Saclay, France), the hard x-ray branch monochromator of the Metrology beamline is composed of a double Si(111) crystal whose Bragg angle is equipped with a rotary position encoder that requires an energy-angle calibration. The available energy range, dened by the extreme Bragg angles, is from 3 keV to 45 keV. An accurate calibration of the energy axis of the monochromator is obtained by using several pure metal foils as transmission targets and by performing energy scanning around their absorption edges. The rst derivative of the recorded absorption curves is used to determine the accurate position of the absorption edge energies, which are then set to match the ones tabulated in Deslattes [START_REF] Deslattes | X-ray transition energies: new approach to a comprehensive evaluation[END_REF]. The residual dierence between the angular position given by the encoder and the values derived from the tabulated absorption edges is presented in [START_REF] Ménesguen | High accuracy experimental determination of copper and zinc mass attenuation coecients in the 100 eV to 30 keV photon energy range[END_REF] and is kept below 0.04%.

The minimization of harmonics or stray light are managed in the same way as detailed in [START_REF] Ménesguen | High accuracy experimental determination of copper and zinc mass attenuation coecients in the 100 eV to 30 keV photon energy range[END_REF]. The contributions of the harmonics are signicant below 7 keV and in these cases, a small detuning of the second monochromator crystal is necessary to reduce their total contribution to 0.1%.

Characteristics of the CASTOR device

The CASTOR setup is a vacuum chamber equipped with a 7-axes manipulator. It is connected to the vacuum of the synchrotron beamline for each measurement campaign and must be removed afterwards, requiring a precise alignment prior to any experimental measurements. The most critical alignment is the coincidence between the vertical rotation axis and the beam path. Using a dedicated iterative procedure, the position of the CASTOR chamber perpendicular to the beam direction is dened by a motorized stage with an estimated precision of 20 to 30 micrometers. The internal manipulator of CASTOR is composed of seven axes with ve motors for the sample and two dedicated to the detection arm where several detectors are installed (see Figure 1). The samples are placed vertically on two perpendicular translation stages (namely Tz and Ts) which are mounted on a rotation stage (Rx). This three-axis platform is attached to a horizontal translation stage (Tx) which is on the upper rotation stage (Rθ). The detection arm is on a second rotation stage (Rθ2). The fourth translation (Tz2) allows selection of a detector. The Rθ and Rθ2 are the most critical axes for the experiments and are encoded for better accuracy. They can rotate freely except that collisions are prevented by a limit switch on the Rθ arm. This conguration allows the sample manipulator and the detector arm to rotate freely within the chamber. The rotation stage Rx has a range of 350°. The translation stages have 110 mm ranges except for Tx which is limited to 102 mm.

The detection arm Rθ2 is equipped with four photodiodes and an energy-dispersive detector (e.g. Silicon Drift Detector (SDD)). The photodiode holders can be equipped with commutable slits or collimators. The current conguration is composed of four 100 mm 2 photodiodes from Opto Diode [10,[START_REF] Korde | Present status of radiometric quality silicon photodiodes[END_REF] that have a typical eciency of 0.27 A/W at 6 keV photon energy. In slot no. 1, an AXUV:Al photodiode is equipped with a 300 µm vertical slit (used for alignment or measurements). The Al coating is necessary to prevent the infra-red or visible light degrading the dark current or background, improving the dynamic range of the reectivity curves. Moreover, an AXUV100G photodiode without slit, placed in slot no. 2, was absolutely calibrated using a cryogenic electrical-substitution radiometer [START_REF] Troussel | BOLUX: a cryogenic electrical-substitution radiometer as high accuracy detector in the 150-11000 eV range[END_REF].

A spectrometer such as an energy-dispersive detector is required to perform uorescence-related measurements such as GIXRF. It is installed on the chamber facing the sample (i.e. at 90°with respect to the incoming beam.). The experiments are performed with an SDD from AMPTEK (128 eV of energy resolution (FWHM) at 5.9 keV), which is mounted on a dedicated translation stage allowing it to approach as close as possible to the sample.

3 Combining x-ray reectivity (XRR) and grazing-incidence xray uorescence (GIXRF)

Alignment and measurement procedure

The alignment of the CASTOR setup is crucial prior to any sample alignment. The main feature of it being the positionning of the pivot point in the path of the beam. For that purpose, we use a 300 µm diameter pinhole placed on the sample holder to perform the scannings. The Tx translation stage (perpendicular to the beam when θ = 0 • or 180 • ) is scanned to nd the centre position of the beam within the pinhole for θ = 0 • . The same scan is performed for θ = 180 • and the average between the two Tx positions provides the correct position for which the pivot is aligned with the beam. The whole chamber is moved according to this correction. We estimate the accuracy of the procedure to be in the range 10 µm-20 µm due to the precision of the translation stage of the chamber.

The second important point in aligning the experiment concerns the samples. For each sample, we use a well-known approach. We repeatedly scan on Tx and Rθ until no change are found in the expected values with respect to the precision of the translation or rotation steps. With the translation stage Tx, we cut in half the incident beam intensity and by rotating the theta angle Rθ, we set the sample at the maximum intensity. Nevertheless, in order to verify the accuracy, we add a scanning on the photodiode rotation stage Rθ2 at a xed θ angle (rocking curve) to account for a small theta oset.

At glancing angles (i.e. the very rst points close to 0°), geometrical eects disturb the x-ray reectivity. At 0°, half of the incoming ux is directly measured by the photodiode. The limited length of the sample prevents the measurement of the real reection intensity for angles lower than ∼ 0.2 • . These geometrical eects are taken into account in our simulations.

Example of XRR measurement and simulation

As a rst example of our procedure, we recorded the reectivity of a single layer sample at the photon energies of 6 keV and 11.4 keV. The layer is a mixture of nickel, cobalt and tungsten often used as electrode in the semiconductor industry. In order to compare the results at 6 keV and 11.4 keV, the incidence angle θ is normalized to Q = 4π × E × θ with photon energy E in keV. The Figure 3 left is presenting the comparison showing a very good agreement of the two reectivities despite a small discrepancy at high angles due to a dierent noise level at the two incident energies. Moreover, the reectivity at 6 keV was performed twice from 0°to 10°with steps of 0.03°and 0.01°in order to assess for the reproductibility when changing the steps. The measurements are presented in Figure 3 (right panel) as red and black curves respectively. The two measurements are perfectly reproducible until the reectivity signal comes closer to the photodiode dark noise. These curves also show that the instrument is able to perform x-ray reectivities with a dynamical range of six decades or more despite the fact that the primary beam, delivered by a bending magnet, is unfocussed, its intensity being reduced by slits. To simulate the x-ray reectivity of a sample, we use the Parratt recursive formula [START_REF] Parratt | Surface Studies of Solids by Total Reection of X-Rays[END_REF] together with the Nevot-Croce formalism [START_REF] Névot | Caractérisation des surfaces par réexion rasante de rayons X. Application à l'étude du polissage de quelques verres silicates[END_REF] to account for the interface roughness. At glancing angles, the geometrical eects are limiting the reectivity as the sample length is limited. The geometry is also taken into account for a better representative tting at low scanning angle. Our tting procedure uses a dierential method as in [START_REF] Ingerle | JGIXA -A software package for the calculation and tting of grazing indcidence X-ray uorescence and X-ray reectivity data for the characterization of nanometer-layers and ultra-shallow-implants[END_REF] with a χ 2 cost function of XRR(θ) × θ The x-ray uorescence signal is acquired by the SDD for which the geometrical aspects are of major importance (g. 4). A tungsten collimator (thickness e and diameter c) is placed in front of the SDD and the distance to sample (d 1 ) can be adapted manually. The solid angle is calculated accordingly to the measured distances and the dimensions of the collimator and the sample as well as the incoming beam characteristics (shape and dimensions). In a rst step, we calculate the solid angle as a function of the radial distance r as every irradiated point on the sample will not contribute with the same importance to the signal of the SDD. The formula is given in Equation 1:

Ω r (r) = S(r) ¤ Sin(β) ((d 1 + e + d 2 ) 2 + (r + x) 2 + y 2 ) dxdy (1)
where d 1 is the distance between the sample and the collimator, e is the collimator thickness, d 2 is the distance between the collimator and the SDD silicon chip. The angle β is given by β = Arctan (d 1 + e + d 2 )/ (r + x) 2 + y 2 . The integral limits are expressed as a surface S (r) which is the region common to several circles: S (r) = A SDD ∧ A c1 ∧ A c2 , where A SDD is the area of the SDD active area, A c1 and A c2 are the projected area on the SDD surface of the front and back side of the pinhole respectively. Ω r (r) takes positive values for 0 ≤ r ≤ R max , R max being the maximal o-axis distance.

In order to calculate the solid angle for every incident angle, the length of the sample (l E ) and the spot size of the beam are required. The following formula (Equation 2) is applied at each individual incident angle:

Ω (θ) = 1 A s ¤ As Ω r (r) dxdy (2)
where A s is the area in common of the impinging beam spot, the size of the sample and the solid angle of the SDD delimited by the maximal radius R max .

Simulation of the emitted x-ray uorescence

To calculate the emitted uorescence of a specic element from the sample, the evaluation starts with the calculation of the x-ray standing wave eld inside the structure. The Parratt recursive algorithm gives the value of forward and backward electric elds at the centre of each layer. In order to take into account the variation of the eld and the interfaces roughnesses, our model cut the layers into thinner slabs. The detected uorescence is derived using Equation 3:

XRF i (θ) = Ω (θ) I 0 T η i τ i ω i × j W i (z j )ρ(z j )(z j -z j-1 )× E t (z j , θ) + E r (z j , θ) 2 exp(- j-1 h=1 µ i (z h )ρ(z h )(z h -z h-1 )) (3)
where T is the acquisition live time of each spectrum, I 0 is the ux of incident photons, η i is the detector full energy peak eciency at the line energy of element i, τ i is the photoelectric cross sections of element i at incoming energy and ω i the partial uorescence yields of element i (corresponding to the shell uorescence yields combined with the line ratio). In case the line is feeded from several subshells due to the Coster-Kronig cascade, this has to be taken into account as well. The parameter W i (z j ) corresponds to the weight fraction of element i at depth z j and ρ(z j ) is the density at the same depth. The emitted uorescence is reabsorbed when passing through the upper slabs and the exponential term in Equation 3 accounts for that. The solid angle Ω (θ)and the internal electric elds E t (z j , θ) and E r (z j , θ) must be calculated for each grazing angle. To meet our experimental setup, we derived the calculation of the x-ray standing wave eld for P polarization. Thus, the electric eld is replaced by H(z j , θ)/n.

As for the x-ray reectivity, the tting parameters are the layer densities, thicknesses, roughnesses and elemental weight ratio. The combination of both the XRR and GIXRF techniques that are using similar measurement procedures but dierent signals should provide better accuracy of these proles.

Example of GIXRF

We performed a GIXRF experiment on the NiCoW sample. The photon exciting energy was chosen at 11.4 keV in order to get both the K lines of Ni and Co and the L 3 lines of W. A typical spectrum is presented in Figure 5 showing the Co and Ni Kα and Kβ line contributions and the W Lα line.

The overlapping lines are tted with gaussians with standard deviations linked by a linear energy dependence to account for the spectrometrer response, and they are perfectly distinguished by the spectrum processing software COLEGRAM [START_REF] Ménesguen | COLEGRAM, a exible user-friendly software for processing of ionizing radiation spectra[END_REF]. We performed twice the experiment for two dierent distances of the SDD in order validate the calculation of the solid angle. In order to have a good accuracy close to the maximum of the GIXRF curves in a reasonable experimental time, the software controlling the experiment allows the user to choose variable angular steps. We performed the GIXRF from 0°to 2°, with 60 or 30 seconds acquisition time per spectrum at d 1 = 15 and 25 mm respectively. For all acquired spectra, we derived the peak areas of the useful lines with COLEGRAM in a batch processing mode. We simulate the XRR (Fig. 3) and GIXRF curves from our sample model and derived the best tting parameters for the densities, thicknesses and roughness, which are presented in Table 1. The GIXRF curves are presented in Figure 6 and assess for an excellent agreement of the theoretical calculations with the experimental results.

Example of application

As an application of our reference-free approach, we performed a combined XRR/GIXRF analysis on more complex samples. We used a three-layered sample and a periodic stack (three repetitions of two layers) of Al 2 O 3 /HfO 2 in dierent combinations, deposited on Si with a native oxide layer. The layers were made by atomic layer deposition (ALD). The periodic stack is simulated by individualizing each layer, which leads to a potential large increase in the number of tting parameters.

Three-layer sample

The incoming photons with an energy of 10 keV create vacancies in the K shell of aluminum, silicon and oxygen and in the L 3 shell of hafnium, avoiding any Coster-Kronig transitions. The SDD is set at d 1 = 15 mm and the incoming beam was cut at 0.1x0.5 mm. We acquired 149 spectra of 150 s from 0°t o 3°, of which we derived the hafnium Lα line area at every angular step. The spectra are processed using the COLEGRAM software [START_REF] Ménesguen | COLEGRAM, a exible user-friendly software for processing of ionizing radiation spectra[END_REF] to get the individual peak areas, using Gaussian functions. The background of all spectra were processed to take into account the step due to the presence of a peak. As Lα 2 could not be distinguished from Lα 1 , the derived peak areas account for both line contributions.

The experimental curves are presented in Figure 7. On the left, we compare the acquired XRR with the simulation and, in the centre, we compare the experimental Hf Lα line intensity with the result of the GIXRF calculation following our theoretical model. The excellent agreement between all experimental and theoretical curves gives condence in the reliability of the parameters of the structure. In Figure 7 right, we give an overview of the depth-dependent densities derived by the model. This sample was already studied in [START_REF] Hönicke | Referencefree, depth-dependent characterization of nanolayers and gradient systems with advanced grazing incidence X-ray uorescence analysis[END_REF] by the same techniques but with dierent experimental equipment and excitation energies. The three-layered sample corresponds to the S2 sample of the article. In the published analysis, the HfO 2 and Al 2 O 3 deposited materials were assumed to be stoichiometric and having bulk density, 9.68 g.cm -3 and 3.987 g.cm -3 for HfO 2 and Al 2 O 3 respectively, which is not our case since signicantly lower densities derived from our t. In order to compare our results with the ones published in [START_REF] Hönicke | Referencefree, depth-dependent characterization of nanolayers and gradient systems with advanced grazing incidence X-ray uorescence analysis[END_REF], we derive the areal mass for HfO 2 and Al 2 O 3 from the layer densities and thicknesses and we present the results of [START_REF] Hönicke | Referencefree, depth-dependent characterization of nanolayers and gradient systems with advanced grazing incidence X-ray uorescence analysis[END_REF] in Table 3 together with our results. Our result are in good agreement for Al 2 O 3 with PTB and CEA-LETI (Laboratoire d'électronique et des technologies de l'information), nevertheless a lower content maybe due to the diculty to distinguish the aluminum Kα line from the large tail of the silicon Kα peak (photon energies lower than 3 keV are not accessible on the hard x-ray branch, it would be better to excite with an energy larger than Al K absorption edge and lower than Si K absorption edge to avoid the intense Si K peak in the spectra).

Our result concerning HfO 2 are in better agreement with LETI but signicantly dierent from PTB.

Periodic stack sample

The period of the stack is composed of two layers made of HfO 2 and Al 2 O 3 , the complete stack being three repetitions of the period (see Table 4). The XRR was acquired at 6 keV and the GIXRF at 10 keV. Our model can handle combined analysis at dierent energies. The SDD is set at d 1 = 25 mm and the incoming beam was cut at 0.1x3 mm. We acquired 116 spectra of 60 s from 0°to 7°w ith a variable angular step in order to get more experimental points close to the critical angle. The spectra were adjusted the same way as previously to derive the peak area of the hafnium Lα line. The experimental XRR and GIXRF curves are respectively presented in Figure 8 left and middle; Figure 8 right presents the calculated depth-dependent densities.

As in the case of the three-layered sample, the periodic stack was studied in [START_REF] Hönicke | Referencefree, depth-dependent characterization of nanolayers and gradient systems with advanced grazing incidence X-ray uorescence analysis[END_REF], assuming bulk densities. We scaled our thicknesses and found total thicknesses of 4.0 nm for Al 2 O 3 (4.7±0.7 (PTB) and 5.7±0.5 (LETI) in [START_REF] Hönicke | Referencefree, depth-dependent characterization of nanolayers and gradient systems with advanced grazing incidence X-ray uorescence analysis[END_REF]) and 5.0 nm for HfO 2 (4.4±0.5 (PTB) and 5.1±0.3 (LETI) in [START_REF] Hönicke | Referencefree, depth-dependent characterization of nanolayers and gradient systems with advanced grazing incidence X-ray uorescence analysis[END_REF]). Once again, our excitation energy is not the best to derive the Al Kα intensities, this explains the rather weak agreement with other values. On the contrary, our deposited mass for HfO 2 is in excellent agreement with the value derived by LETI.

Conclusion

We presented a combined XRR-GIXRF analysis of several samples. At rst, we assessed our measurement procedure and calculation method on a well-known sample in dierent conditions in order to qualify our instrument and theoretical approach. Then, we applied our procedure to more complex samples in order to derive their interesting quantities. We found an excellent agreement between experimental results and tted curves of both XRR and GIXRF which supports the condence in the stack parametres derived.

This work PTB in [START_REF] Hönicke | Referencefree, depth-dependent characterization of nanolayers and gradient systems with advanced grazing incidence X-ray uorescence analysis[END_REF] LETI in [START_REF] Hönicke | Referencefree, depth-dependent characterization of nanolayers and gradient systems with advanced grazing incidence X-ray uorescence analysis[END_REF] Hf O

Figure 1 :

 1 Figure 1: A schematic representation of the 7-axis goniometer. The beam [MAK] impinges the sample from the left, the surface of the sample being vertical.

Figure 2 :

 2 Figure 2: Geometry for the XRR at shallow angles.

Figure 3 :

 3 Figure 3: X-ray reectivities of the single layer NiCoW sample. Left: comparison at 6 keV (black) and 11.4 keV (red). Right: comparison with steps of 0.01°(black) and 0.03°(red) at 6 keV, the model is in blue line.

3. 3 X

 3 -ray uorescence detection solid angle A dedicated acquisition software allows the CASTOR user to record the x-ray uorescence emitted by the sample simultaneously or sequentially with the x-ray reectivity, in order to combine both XRR and GIXRF. In both cases some geometrical aspects are of importance to model correctly the sample composition and structure in the tting procedure. The footprint of the incoming beam on the sample is primarily dened by a slit. The samples being vertical, we use a vertical slit in order to limit the horizontal divergence of the beam and allow more ux by selecting larger height. We can use several slits of 100 µm width and 1, 3 or 5 mm vertical aperture.

Figure 4 :

 4 Figure 4: A schematic representation of the detection solid angle of the SDD. The measured distances are indicated as they are used for the estimation of the solid angle.

Figure 5 :

 5 Figure 5: Experimental x-ray uorescence spectrum at 11.4 keV and θ = 0.44 • . The individuals peak functions appear in red, the complete t in blue and the background in green.

Figure 6 :

 6 Figure 6: Experimental results of the uorescence (peak area versus angle). From the left to the right, uorescence of nickel, cobalt and tungsten. First line: d 1 = 15 mm, second line: d 1 = 25 mm. The red curves are the theoretical results from the model.

Figure 7 :

 7 Figure 7: Three-layer sample: Experimental and theoretical XRR at 10 keV (left), GIXRF curve of Hf Lα line (centre), Fitted density from air to silicon substrate (right).

Figure 8 :

 8 Figure 8: Periodic stack sample : Experimental and theoretical XRR at 6 keV (left), GIXRF curve of Hf Lα line (centre), Fitted density from air to silicon substrate (right).Density (g.cm-3 ) Thickness (nm) Roughness (nm)

Table 1 :

 1 5 . The simulation (blue line) is performed with a three-layer model in order to account for density dierences in the deposition process. The model results are presented in Table1. Experimental model after the combined GIXRF-XRR analysis.

		Density (g.cm	-3 )	Thickness (nm)	Roughness (nm)
	N iCo 0.11 W 0.01	5.764		1.023	0.53
	N iCo 0.11 W 0.01	8.742		17.518	0.47
	N iCo 0.11 W 0.01	8.119		1.153	0.47
	SiO 2	2.65		1.051	0.26
	Si	2.3296			0.3

Table 2 :

 2 Experimental model after the combined GIXRF-XRR analysis.Unfortunately, our model cannot derive associated uncertainties as it is still a work in progress.

		Density (g.cm	-3 )	Thickness (nm)	Roughness (nm)
	Hf O 2	9.209		0.601	0.427
	Al 2 O 3	3.224		2.831	0.334
	Hf O 2	9.502		0.651	0.258
	SiO 2	2.65		3.577	0.261
	Si	2.3296			0.301

Table 3 :

 3 2 (ng.mm -2 ) Comparison of the determined mass depositions by the dierent analyses.

		11.7	6.8(10)	13.6(19)
	Al 2 O 3 (ng.mm -2 )	9.13	10.4(16)	11.2(8)

Table 4 :

 4 Al 2 O 3 Experimental model after the combined GIXRF-XRR analysis.

		2.052	2.118	0.353
	Hf O 2	9.332	1.818	0.300
	Al 2 O 3	2.824	1.72	0.246
	Hf O 2	9.425	1.671	0.285
	Al 2 O 3	3.482	1.849	0.247
	Hf O 2	9.559	1.676	0.140
	Si	2.3296		0.300

Acknowledgements

The authors are grateful to Pascal Mercere and Paulo Da Silva for assistance on the Metrology beamline and to the SOLEIL sta for smoothly running the facility and to Emmanuel Nolot for fruitful discussion and providing the samples. The authors are also grateful to Diane Eichert for the fruitful advices about the experimental alignment procedure.


Chatelais Nicolas
email: nicolas.chatelais@malix.univ-paris1.fr

Peyrat Mathilde
email: mathilde.peyrat@malix.univ-paris1.fr

Are Small countries leaders of the European tax competition?

Keywords: Strategic Interactions, Tax Behaviours, Spatial Econometrics, European Union, Tax Competition, Small Countries JEL Classification: E62, F21, F22, F23, H30, H32, H73, H77, R12

The aim of this paper is to develop a better understanding of the literature dealing with strategic fiscal behaviours of small EU countries using estimations of tax reaction functions of competing national governments. Deriving a simple model of tax competition in a Nash and a Stackelberg game, we use panel data and tools from spatial econometrics to examine the role of small countries in tax competition within the enlarged European Union. We find that interactions are stronger among smaller EU countries than between larges ones and rates set in small countries influence those in big countries. Finally, small countries located in the centre of the EU have more influence on tax policies choices of big countries than small countries located in the periphery of EU.

Introduction

Since the Treaty of Maastricht (1992) and the establishment of real and nominal convergence rules, several economic regulation tools such as monetary or exchange policy can no longer be used in cyclical adjustments or in response to policies in other neighbouring countries. Countries are now competing through tax and fiscal policy.

Indeed, the statutory corporate tax rate has dropped since 1995 in all European countries. This competition arises from both the recognition of different tax regimes within EU by most mobile tax bases in their location choice and the un-cooperative commitment of some countries to attract these tax bases.

Fiscal interactions between countries are derived from the existence of externalities (positive or negative) of tax and fiscal policies among countries. Consequently, choices of tax policies are not independent. In addition, countries are competing to attract mobile tax bases (capital, profits, high incomes…) localized in their neighbours.

Knowing almost perfect spatial mobility of the tax base constituted by capital and profits, each national government adopts non-cooperative behaviours when setting their corporate tax rate. This strategy aims to attract productive capital and / or profits in order to increase the production and / or income tax.

In this article, we test whether governments influence each other in their choice of tax policies and specifically in tax rate setting. We will try to determine whether small countries, defined by their geographic size, are tax competition leaders in Europe. This is based on the hypothesis of a size effect in small countries which can encourage the pursuit of a tax rate lowering strategy to attract mobile taxable bases. Indeed, the capital transfer from the big country that results is relatively high in proportion to its production or its GDP. This capital transfer offsets the initial loss of tax revenue.

The issue of strategic interactions between local governments has been often treated in the literature.

However between states, it is increasingly an important topic of research. Thus, we can make reference to works from [START_REF] Altshuler | Follow the leader? Evidence on European and US tax competition[END_REF], Devereux, Lockwood and Redoano (2002), [START_REF] Redoano | Fiscal Interactions among European Countries[END_REF] who obtain a positive Nash reaction function for European countries and those of [START_REF] Ruiz | Is there evidence of strategic corporate tax interaction among EU Countries?[END_REF], who concludes a lack of fiscal interaction in Europe. Most of tax competition models and estimates that follow result from a Nash framework.

However, we can examine if some countries do not observe fiscal and tax policies evolutions among its neighbours in order to react in a second time. Thus, Gordon (1992) and [START_REF] Altshuler | Follow the leader? Evidence on European and US tax competition[END_REF] deal with strategic interactions in a theoretical Stackelberg world. In contrast to these two articles suggesting that the United States has played a leading role towards Europe in tax competition especially with the TRA (Tax Reform Act 1986), we test empirically the role of small countries towards larger in the EU-25. In addition, we investigate which countries or which types of small countries are leaders of the tax competition. We distinguish two groups of small countries which can influence big countries. Empirically, it seems that small countries close to the centre of the European Union, notably Belgium, have the most influence on large countries at the core and therefore are the real instigators of tax competition. Thus, large countries always adopt a position in line with respect to these small countries also at the centre.

The article is organized as follows. We will first state tax competition in Europe, recalling the theoretical role played by small countries in tax competition based on a quantitative analysis (section 2). We will then present a theoretical model (section 3) that we econometrically estimate (Section 4). Finally, we will present our findings (section 5) to conclude on the important role of small countries in the European tax competition.

Tax competition within the European Union

Faced with the existence of a wide variety of taxation systems within the European Union, firms and particularly multinationals (MNF), may find beneficial to integrate this variable in their decision-making strategic processes (localization, investments...). Knowing that, some national governments can opt for an aggressive tax strategy in order to attract these productive entities, or at least their profits, and thus compel other governments to participate in the competition game due to high capital mobility. In the European Union case, it seems that small countries are at the origin of tax competition and that large countries react to this aggressive strategy.

Why small countries practise tax competition? Existence of a size effect

In the traditional literature, the notion of size is often defined by the market size. For some small countries, the tax policy has to compensate for the small size of the domestic market. If a company is located in a large country, it can take advantage of the relatively large domestic market at its disposal to exploit increasing returns to scale. In contrast, the market size being relatively restricted in a small country, companies can not thus benefit of these scale economies and prefer to locate near the large market. Therefore, governments of small countries would benefit from having a less heavy tax system in order to offset their initial market size handicap. However, on a European scale, the domestic market size is not a very relevant argument. Indeed, since tariff barriers are free or very low and transport costs are reduced, companies have free access to the European market.

The relatively small market size is not thus a convincing argument to justify a decrease in tax rate. However, another effect linked to its small size may encourage a country to practice a tax rate reduction strategy. A relatively low tax rate results in a capital transfer towards the country setting this rate. For a small country, the capital transfer that result is relatively large in proportion to its production or its GDP, which largely offsets the initial loss of tax revenues. On the contrary, larges countries have interest in less reacting to the tax competition because cutting taxrates do not attract sufficient capital relative to their production and initial losses of tax incomes are not compensated by productive transfers.

When a change of the tax rate occurs, different consequences will sustained according to the size from the country concerned. This change has an impact of width different on the capital per capita ratio depending on the size of the country. This is due to greater capital elasticity to tax rate in small countries. Thus, at the Nash equilibrium in a two country model, the small country will be encouraged to set a tax rate lower than in the big foreign country.

For example, [START_REF] Bucovetsky | Asymmetric Tax Competition[END_REF] and [START_REF] Wilson | Tax Competition with Interregional Differences in Factor Endowments[END_REF] have proposed a theoretical framework incorporating two jurisdictions having a different number of inhabitants. Originally, no capital transfer would be excepted if the two localities adopt the same tax rate. The exchange will thus take place only because the two jurisdictions have set different tax rates. As in previous models [START_REF] Tiebout | A pure theory of public expenditure[END_REF], capital taxation is used to finance a public good and leads to a capital market equilibrium involving equal after-tax returns. Consequently, tax rates set in a locality determine capital stock in the other one, and thus its revenue from taxes. Contrary to a theoretical framework with two identical jurisdictions, the capital marginal productivity elasticity to tax rate is not any more equal in both localities.

Indeed, the amount of work available is lower in the small jurisdiction, the elasticity is therefore greater. Thus, capital flight will be higher in a small country than in a large if they adopt an equal tax rate increase. This externality is even more important as the jurisdiction size is small. In conclusion, being more influenced by tax competition, the small locality may find a benefice in establishing its tax rates below those set in larges ones.

International tax competition challenges imply to refer any more to various tax bases elasticities in tax policy elaboration.

The theory [START_REF] Bucovetsky | Asymmetric Tax Competition[END_REF][START_REF] Wilson | Tax Competition with Interregional Differences in Factor Endowments[END_REF] predicts that at equilibrium, large countries of the European Union, having capital elasticity lower than smaller countries, choose higher mobile bases tax rates than smaller members.

This proposal has been empirically verified by Huizinga-Nicodème (2006), whose regressions show a significant and robust relationship between the corporate tax burden and the size of their home country measured by the logarithm of GDP.

Tax rates Reductions in small countries

In this section we will point out an overall reduction in statutory corporate tax rates since 1995 in Europe.

We will show a more pronounced decline in small countries of the European Union. Before, we justify the choice of statutory rate compared with other indicators of tax pressure in order to deal with tax competition.

Imperfect indicators of tax pressure

The theoretical or statutory tax rate is the most frequently used indicator in economics to measure the tax burden because it is relatively simple to evaluate. However, this indicator may lack of efficiency because it does not take into account the tax base or potential tax shelter. In addition, the calculation of the tax base is rather difficult to assess accurately.

To remedy this shortcoming, it was created and introduced new tools to assess the tax burden: implicit tax rates and effective tax rates or simulated tax rates. In general, implicit tax rates ("backward looking") are used to analyze the ex-post corporate tax burden and tax distribution effects. But they do not take into account the process of investment location choices or firm's marginal investments. The calculation of effective tax rates ("forward looking") may alleviate this problem.

The indicator used in this article, is the statutory corporate tax rate which is generally used in the context of international comparisons. The standard rate is the sum of the highest federal tax rate plus any more local taxes (e.g. regional). Because of its simplicity and availability, this tax rate plays an important signal effect. It is a determining factor with regard to the firm location and profit transfers among various entities of a multinational company based in several countries. The MNF will indeed try to reduce declared benefits in countries where the nominal tax rate is high and transfer them in countries where the nominal tax rate is lower.

However, the statutory tax rate observed in various countries give only a partial view of the actual tax pressure exerted on firms.

To conclude, the statutory tax rate is a good tax assessment element of a country. It is a very important element in international comparisons and which influences considerably tax competitiveness "felt" by firms. Even if the felt tax competitiveness differs from the real competitiveness, it is often the first which influences direct investment decisions and, more generally, the localization of mobile factors of production.

The nominal tax rate criterion can act as a signal effect because the taxation systems scrubland leads to difficult decision making process for firms.

We can also establish comparisons between various tax rates in Europe (table 1). Thus, when we classify countries according to various indicators of fiscal pressure (statutory, EATR, EMTR), the ranking obtained is relatively identical to statutory tax rates. A similar classification is obtained if we take into account effective tax rates calculated by [START_REF] Devereux | Corporate income tax reforms and international tax competition[END_REF].

Thus, in this article we assume that nominal tax rates are principal vectors of tax competition: their reductions are analysed as signal effects and the study of these rates gives excellent information of a government tax policy and the tax burden which it exerts on firms. To explain the statutory tax rates fall in Europe, we can split the EU in two sub-groups according to their size. The partition was done according to two criteria that are geographical area and population. Thus we get a straightforward frontier between countries:

-Large countries: Germany, France, Great Britain, Italy, Spain and Poland -Small countries: Belgium, the Netherlands, Luxembourg, Ireland, Portugal, Greece, Austria, Finland, Denmark, Sweden, Slovenia, Slovakia, Hungary, Czech Republic, Estonia, Lithuania, Latvia, Malta and Cyprus.

This partition is nuanced when we take GDP as the choice criteria, especially for Poland, but this anomaly can be explained by the past weighing on economic structures of this country.

Thus it is quite remarkable (figure 2) that small countries set corporate tax rate lower than those established in large ones. In addition, it should be noted a stronger and earlier tax rates decrease in small countries than in larges ones.

This tax strategy is characterized by an upper point's gap in 2006 and has undergone a significant change over the last decade. However, the map (1) shows the existence of strong disparities within the group of small countries but also in that of larges ones.

This tax rates decline was particularly strong in Ireland, Czech Republic, Slovakia, Portugal, Estonia, Lithuania and Cyprus. Thus, the most striking decline is attributable to Ireland (Figure 3a), whose rate has experienced a vertiginous fall from 40% to 12% in just over 10 years, showing a very clear tax strategy on behalf Irish governments.

However, this decline is relative because it has been accompanied by a tax base broadening and the removal of MNF specific arrangements. Thus, even if statutory tax rates have fallen, the implicit tax rates decline is less important.

Slovakia has also pursued a very aggressive corporate tax policy (Figure 3a), decreasing its tax rate from 40% to 19% in just over 10 years. Portugal and the Czech Republic have also changed their tax rates in the same direction, but with a less pronounced decline (Figure 3a).

We can also refer to the sharp decline in tax rates in Estonia (Figure 3b) if we take into account the taxation of profits put to reserves (0%) instead of distributed profits (23%). We can ask whether in the long term corporate tax rates could be at zero. According to some economists as Artus (2006a), Mintz (1992) and [START_REF] Weichenrieder | Why) Do we need Corporate Taxation? CESifo Working Paper Series[END_REF], this hypothesis can not be excluded.

We can already make a very important remark: tax competition is more a small countries tax strategy rather than those of new entrants (10 CEEC). Indeed, we noticed that it is not only new entrants who have lowered their tax rates, but the majority of small EU countries. Proof of this is the sharp decline in Ireland and Portugal tax rates (Figure 3a). The proposal's corollary implies that the corporate tax rate will not go up in new small entrants countries at the end of the convergence process unless this strategy is ineffective and leads to budgetary problems. We can also observe different strategic behaviours within the group of larges countries. In general, facing the noncooperative tax behaviour of smaller countries, the reaction of large countries was moderate. During the last decade, the trend has been downward, more or less, in all large countries. For example, France and, to a lesser extent, Italy and Great Britain (Figure 4) have experienced a slight decrease in tax rates, with an increase of rates in 1997 in France due to the change of government and in Italy in 2005 due to encountered budgetary difficulties. Faced with competition from smaller countries, Spain has been able to maintain its tax rate to 35% (Figure 4) thanks to the good health of its economy. Nevertheless, there should be from 2007, a decline of both the household income tax and the corporate tax rate (expected to decline by 5 percentage points between 2007 and 2011). It was in Germany that the tax rate has most strongly fallen (Figure 4), declining from 56.6% in 1995 to 38.9% in 2001. This spectacular decline can be explained by the fact that Germany had the highest corporate tax rate in Europe. Since 2001, Germany did not lower its rate, but it should not continue since a fall of this rate is planned for 2007 to 30%.

The other only large country having significantly reduced its tax rate is Poland (Figure 4). Like Germany, Poland had a rather high tax rate (40%) in 1995 (larger than France). Moreover, a very important criterion not to omit is the distance to the European centre. Indeed, Poland being on the periphery, it will tend to lower its tax rate to offset agglomeration forces and attracts firms.

Thus, we observe that small countries impose a downward pressure on large countries which are constrained to respond by reducing in turn, their tax rates on mobile bases. The theoretical tax rate evolution in all EU countries during the last decade lead to a corporate tax rate gap in favour of small countries (map 2).

Map 2: Corporate tax rates levels in 2006 3

Tax competition Model

Simple tax competition model

We use a simple tax competition model to obtain national governments reaction functions. Reaction functions developed thereafter, based on Bruckner's theoretical model (1999,2003), will treat only the competition on corporate tax rate, and not income, consumption or property tax rates, because it seems that tax competition primarily affects firms.

Thus, we will establish two types of reaction functions according to the game taken into account:

-A Nash equilibrium where all countries set their tax rates simultaneously;

-A Stackelberg equilibrium where a country or a group of countries set their tax rates first and other EU countries adopt a Follower behavior.

For simplicity, we are building a model of an economic area with two countries, generalizable with n countries, where capital (Ki) invested in the country i, i = 1.2, is perfectly mobile and labour (Li) is considered as a fixed production factor. A private consumption good Xi is produced combining production factors according to a constant returns to scale technology. This production function is given by F(Ki, Li), written f(ki), where ki is the number of capital per worker units. For simplicity, we assume the number of workers in each country i is equal to its population.

The economy is composed of two consumer goods, a private good Xi and a public good Gi.

A tax is levied on investment in each country, with t i the tax rate per unit of capital in the country i. The firm profit function in the country i is defined as follows:

i i i i i k t k r k f . . ) ( - - = Π (3.1)
With r, the common interest rate in the area The profit maximization is equivalent to equalize the marginal productivity to the interest rate plus the tax rate:

i i t r k f + = ) ( ' (3.2)
The capital net tax rate of return is equal to the marginal productivity less taxes. As capital is perfectly mobile, net returns should be equal in both countries:

r t k f t k f = - = - 2 2 1 1 ) ( ' ) ( ' (3.3)
Where r is the uniform net return. Then, we note K the capital amount available in the area, thus obtaining the following condition:

2 2 1 1 . . K L K L K + = (3.4)
Countries' capital endowments are equal and are equally distributed among workers, leading to individual endowments given by k* = K / (L1 + L2).

For simplicity, we specify an identical firm production function among countries:

2 ² . . ) ( i i i k k k f α β - = where α β , > 0 (3.5)
Thus, equations (3.3) and (3.4) determine k1, k2, and r according to tax rates t 1 and t 2 .

1 k = * ) ( . 1 2 2 1 2 1 2 k L L L L L t t + + - α (3.6) 2 k = * ) ( . 2 1 2 1 1 2 1 k L L L L L t t + + - α (3.6')
It is clear that the capital elasticity to tax rate is negative,

) .( / 2 1 2 L L L t k i i + - = ∂ ∂ α < 0.
Ceteris paribus, a tax rate increase in country i causes capital flight towards another country to equalize the net return. The rate of return derivative to the tax rate, i t r ∂ ∂ / , is also negative, indicating the net return is reduced by higher tax rates.

We define a utility function for residents, Ui, having homogeneous preferences in each country. This utility function Ui (Xi; Gi) is a trade-off between consumption of a private good Xi and a public good Gi.

Consumers' incomes (Ri) are composed of labour incomes (wages),

) ( ' . ) ( i i i i k f k k f w - =
and capital endowments returns, rk*. The country i residents' budget constraint is:

* .k r w x i i + = (3.7)
We assume the public good is fully financed by tax on capital. We thus obtain the government fiscal constraint:

i i i k t g . = (3.8)
Substituting wi in (3.7), and using (3.8), we can rewrite the utility function in the following form:

[ ] i i i i i i k t k r k f k k f U . *; . ) ( ' . ) ( + - (3.9)
Residents of country i choose t i to maximize (3.9), taking into account tax rate effects on ki and r. In addition, the tax rate is regarded as a constant.

Without loss of generality, we consider the utility function Ui as additive (linear preferences):

i i i i i i g x g x U γ + = ) ; ( (3.10)
Now the analytical framework is presented, we must consider different types of games that countries will participate.

Nash Equilibrium

Each government seeks to maximize the representative agent utility function under both public and private budget constraint. Thus, we maximize (3.10) subject to constraints (3.7) and (3.8).

i i i i i i i k t k r k f k k f U Max i . . * . ) ( ' . ) ( γ τ + + - = (3.11)
Nash equilibrium tax rates t 1 and t 2 are located at the intersection of countries 1 and 2 reaction functions. Substituting (3.6) into (3.11), the equation can be solved in order to obtain the reaction function of country 1:

* . . ) .( . 2 ² ) .( . ² 1 2 2 1 2 1 2 2 1 2 1 2 1 k t L L L L L L L L t λ γ γ +       + - + - = (3.12)
Where:

        - + +         + - + = 1 2 1 2 1 1 2 2 1 2 1 2 2 1 1 . 2 1 ) .( ² . ) .( . 2 ² )² .( L L L L L L L L L L L L γ γ α λ
In the same way, we obtain the country 2 reaction function:

* . . ) .( . 2 ² ) .( . ² 2 1 2 1 1 2 1 2 1 1 2 1 2 k t L L L L L L L L t λ γ γ +       + - + - = (3.12')
Where:

        - + +         + - + = 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 . 2 1 ) .( ² . ) .( . 2 ² )² .( L L L L L L L L L L L L γ γ α λ
Rearranging (3.12) and (3.12), we obtain

* . . / 2 1 / 1 1 2 1 1 1 1 1 k t s s t λ γ γ +       - - = (3.13) * . . / 2 1 / 1 2 1 2 2 2 2 2 k t s s t λ γ γ +       - - = (3.13') Where ) ( 2 1 2 1 L L L s + = and ) ( 2 1 1 2 L L L s + =
are each population shares in the total area population.

Second order conditions to get a maximum in the optimization program requires

) / 2 1 ( 1 1 s γ - < 0 or 2 1 1 s > γ .
Thus, equation's (13) denominator is negative. Therefore, the reaction function slope, equal to

) / 2 1 ( ) / 1 ( 1 1 1 1 s s γ γ - - , is negative when 2 1 s < 1 γ < 1
s and positive when 1 γ > 1 s . In the same way for the second country, in order to maximize its optimization program, we need to have 2 2 2 s > γ and conditions on the sign of the reaction function slope are symmetric, that is, negative when

2 2 s < 2 γ < 2 s and positive if 2 γ > 2 s .
This theoretical result indicates that country 1 will decrease (increase) its tax rate in response to an increase in country 2 when the marginal public good utility is small (large).

The intersection of the two reaction functions (3.13) and (3.13) gives the Nash equilibrium.

Equations (3.13) and (3.13') show that reaction functions both depend on country preferences and its size. Occulting the case where i γ = i s , the slope is different from zero. Thus, with econometric estimates, the empirical test to verify the strategic interactions existence will consist in checking the get parameter significance. Otherwise, this would tend to prove the absence of strategic interaction between European countries.

Suppose now both countries are of equal size, ie that L1 = L2, we obtain following reaction functions:

* . 4 1 ) 1 ( 4 . 4 1 2 1 1 1 2 1 1 1 k t t       - - +       - - = γ γ α γ γ (3.14) * . 4 1 ) 1 ( 4 . 4 1 2 1 2 2 1 2 2 2 k t t       - - +       - - = γ γ α γ γ (3.14')
Thus, if both countries have same preferences, γ 1 = γ 2 = γ , both countries will react in the same proportion to changes in tax rates in the neighbouring country.

This two countries model is also valid for an N countries economic area. We empirically test this model in the case of the European Union where we strongly suspect tax strategic behaviours. The confirmation of this theory will consist in testing the significance of the reaction function slope in each country. If this test is significant, it will clearly indicate the presence of strategic behaviours within the European Union.

Stackelberg leader extension

In contrast to the theoretical Nash framework where all countries are considered on an equal footing, the Stackelberg model will afford to highlight the presence of leader(s) element(s) in the tax competition. Thus, we derive the model in a leader / follower framework, where the leader country set its tax rate first without taking into account those in other countries. Then other follower countries will set their own taking into account the tax rate set in the first country.

For simplicity, we are always thinking in the context of a two countries model where we will adopt a two stages sequential approach (backward induction):

-Step 1: We determine the follower country's reaction function -Step 2: We determine the leader country's reaction function

Step 1: Country 1, the follower, chose its tax rate t 1 depending on the leading country tax rate t 2.

Country 1 chooses its rate t 1 to maximize its utility function U1 (X1; G1). This makes it possible to characterize the calculated previously best reaction function given by (3.13).

* . . / 2 1 / 1 1 1 1 1 1 k t s s t leader follower λ γ γ +       - - = (3.15)
Step 2: Country 2, the leader, chooses t 2 to maximize its utility function, taking into account the country 1 best reaction function (BR1) in its utility function maximizing program, we get:

* . )²] 1 ²( ) 1 ( 2 [ ] 2 ) ) )( [( 2 2 2 2 1 2 1 2 1 2 2 2 k A S A S L L L L S S AS t leader - - - - - + + - = γ α α α λ γ (3.15') With ) 2 1 ( ) 1 ( 1 1 1 1 S S A γ γ - - = > 1
The leader country sets its tax rate to maximize its inhabitant's welfare and follower countries act as in the case described previously in the Nash equilibrium.

From these two theoretical results (tax rates fixed in a Nash and Stackelberg framework), we will seek to empirically verify the earlier advanced argument where small EU countries practise tax competition.

Empirical methodology

Econometric specification

The empirical estimation underlying theory is a strategic tax competition model [START_REF] Brueckner | Testing for Strategic Interaction among Local Governments: The Case of Growth Controls[END_REF][START_REF] Brueckner | Strategic interaction among governments: An overview of empirical studies[END_REF] in a game where countries set their corporate tax rates simultaneously (Nash reaction function, [START_REF] Wildasin | Nash Equilibria in Models of Fiscal Competition[END_REF] or following the leader country's tax rate (Stackelberg reaction function). Empirically, we will estimate reaction functions in both cases for EU countries. We are in line with the few abundant literature dealing with estimation of national reaction functions (Devereux, Lockwood and Redoano, 2002[START_REF] Redoano | Fiscal Interactions among European Countries[END_REF][START_REF] Altshuler | Follow the leader? Evidence on European and US tax competition[END_REF][START_REF] Ruiz | Is there evidence of strategic corporate tax interaction among EU Countries?[END_REF].

We econometrically estimate the two following equations:

∑ ≠ + + + + + = j i i t i t i t j ij t i T d X ε η ϕ θ τ ω β δ τ , , , (4.1) ∑ ≠ - + + + + + + = j i i t i t L t i t j ij t i T d X ε η ϕ ατ θ τ ω β δ τ 1 , , , , (4.2)
Where i represent country, t, the time, τ i is the home country tax rate, τ j,t , are "neighbouring" countries tax rates, τ L,t-1 is the lagged leader tax rate, X i is an exogenous vector of domestic socio-economic control variables, ω is a weighting matrix discussed later and α, β, θ, η, ϕ are parameters to estimate, d i being potential individual effects and T , an eventually trend necessary for the estimate, and ε i, , the error term. We detail the exogenous control variables in a later part.

Thus, slopes of Nash (4.1) and Stackelberg (4.2) reaction function are respectively β and α. According to the theoretical model, these slopes can be negatives because of the desire to maintain public expenditure, but we expect that these slopes are rather positives because of the high capital elasticity to tax rates.

Econometric estimation

We chose to use panel data to carry out this empirical study. Most of studies on tax competition were realized with cross section data. The panel method has several advantages; it enables to take into account the tax competition dynamics between countries without eliminating effects related to cross section information. In addition, the panel reduces the risk of colinearity among explanatory variables, since these variables are expressed in two dimensions.

Finally, this approach makes it possible to cure the presence of time invariant omitted variables (unobservable individual correlated with some of explanatory variables).

According to the homogeneity tests procedure presented by [START_REF] Hsiao | Analysis of Panel Data[END_REF] for the following regression:

Yi,t = αi + βiXi,t + εi,t
The considered panel is then a panel with individual effects: the 26 vectors β i can be considered as identical between countries while constants α i differ among the 26 countries. Parameters α i are considered as deterministic constants specific to each 26 countries. These parameters can correspond to omitted structural variables that may explain tax rates choice in different countries. The fixed effects model provides converging estimates, even when there is a correlation between explanatory variables and unobservable characteristics.

But when estimating (4.1) and (4.2), we were confronted with several problems often present in spatial econometrics [START_REF] Anselin | Spatial Econometrics: Methods and Models Dordrecht[END_REF]):

-Endogeneity of τ i,t -The possible spatial dependence of error terms.

Variables endogeneity

The presence of strategic interactions implies that tax rates, τ i,t , of different countries are jointly determined.

Indeed, the theoretical model imposes a joint determination of tax rates in Nash equilibrium, and this simultaneity induces that the explanatory variable is endogenous what makes the use of OLS inconsistent.

There are three methods to solve this endogeneity issue:

-Estimate (4.1) with the method of Maximum Likelihood (ML)

-Using the method of Instrumental Variables or the Generalized Method of Moments [START_REF] Anselin | Spatial Econometrics: Methods and Models Dordrecht[END_REF]).

-This endogeneity problem can be avoided by assuming that interactions take place with one or more delay.

In this case, τj,t would be replaced by τj,t-n in (4.1) where the number of lags will be determined.

We chose to estimate the equation (4.1) with the Generalized Method of Moments. This method is also consistent with the second potential econometric problem that may be the errors spatial dependence.

Spatial dependence of errors

The spatial autocorrelation is the "correlation of a variable with itself (autocorrelation) attributable to the data geographic organization (spatial)" (Griffith, 1992) To check the absence of errors autocorrelation, it is important to carry out this test in order to eliminate fallacious estimation risks:

-Risk of having estimated standard deviations bias, estimators remaining however unbiased.

-Risk of common shock presence producing a correlation between tax rates of a country and those of neighbours and that this correlation wrongly leads to conclude that neighbouring rates affect the rate of a given country, whereas actually it is not the case. This is a well-known risk in spatial econometrics [START_REF] Anselin | Spatial Econometrics: Methods and Models Dordrecht[END_REF]).

The Moran statistical test is the most used in the detection of spatial autocorrelation (see Anselin & al, 1996;Anselin and Kelejian, 1997). Thus, the Moran statistics calculation allows to determine for each country if the tax rate is fixed randomly compared to the geographical proximity (in this case, the statistic is not significant), and therefore if there is a spatial autocorrelation (positive or negative) with the setting of tax rates between countries.

This confirms the existence of positive spatial autocorrelation between series.

To overcome these econometric issues, two estimation methods are possible. The first consists in regressing the equation (4.1) with the method of Maximum Likelihood taking into account the error structure described in (4.3). This method has been explored by Case et al (1993). Another approach is to use the Instrumental Variables method.

The estimation method chosen was that of Generalized Moments (GMM). [START_REF] Kelejian | A Generalized Spatial Two-Stage Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances[END_REF] have shown that with the presence of spatial dependence of errors, instrumental variables method, and therefore the Method of Moments provide a robust estimate of β. With several exogenous structural variables it seemed more conducive to use this method.

We must now take into account the neighbouring countries influence of on the domestic country by building weighting matrices.

Weighting Matrices

In reality, a country has generally more than one neighbour, it is important to order the influence of its neighbours on the domestic country by building weighting matrices. These weightings will be used to measure the impact of a neighbour on a country by assigning it a value. Thus, we will evaluate the influence for each pair of states.

Weights of the weighting matrix ω are constructed to take into account the influence of tax rates of each country considered as a "neighbour" on the home country's tax rate. In order to identify all possible influences, we opted for the construction of 3 weighting matrices:

-Geographic weight :

o Distance matrices:

Distance between countries :

∑ = j ij ij d ij d d 1 1 ω
Where ωij d are the ij elements of the weighting matrix W d and d ij is the geographical distance between country i and country j. Unlike most studies, we do not take the distance between capitals of each country, but the weighted distance calculated by Mayer and Zignago (CEPII), considering several cities in the country in order to take into account the geographic distribution of population within the country.

Distance to the centre (Belgium-Brussels) :

∑ = j ic ic d ic d d 1 1 ω
As with the previous matrix, the distance from the centre is weighted by the geographical distribution within each country.

-Economic weight :

o GDP : ∑ - - = jt jt it jt it gdp ijt GDP GDP GDP GDP ) ( 1 ) ( 1 ω
The distance weighting ω ij d , where geographically closest countries have a larger impact, seems to be the most obvious. However, with the New Geographic Economy contribution, it seemed important to build a matrix of weights from the centre, supposed to be Brussels in our study, to take into account agglomeration forces in the tax rate setting process. Indeed, more a country is close to the centre, less it will be influenced by tax choices of peripheral countries. This does not call into question our initial hypothesis that small countries are tax competition instigators, because those are mostly at the European Union periphery.

We also add a notion of "economic neighbour" to take into account the size of countries and their possible similar behaviours. Indeed, large countries having much heavier economic structures than smaller countries, it seems clear that they do not have the same capacities of reaction, either in time and magnitude of changes. Thus, we consider a GDP weighting matrix. It should be noted that this matrix may vary over time. We note that each matrix is normalized to 1, thereby encouraging comparisons and index construction such as that of Moran used to test the autocorrelation.

Data and variables

We estimate models (4.1) and (4.2) using time series data of 25 European Union countries, before the integration of Bulgaria and Romania in January 2007, plus Iceland over the period 1995 -2006. We have deliberately chosen to begin the period of study in 1995 as the majority of countries of the former Warsaw Pact and the Soviet Union have started their conversion towards a market economy only a few years after their emancipation. The explained variable, τ j,t , is represented by top statutory tax rates. These data are derived from the OECD and the European Commission database. This choice reflects the importance of this rate in the tax pressure evaluation by firms prevailing in a country. Moreover, it represents more than 40% of income from taxation [START_REF] Redoano | Fiscal Interactions among European Countries[END_REF]. We concentrate only on the corporate tax rate because it seems it is the principal tool of tax competition between states; tax competition on high incomes being more marginal, and unproven [START_REF] Devereux | Do Countries Compete over Corporate Tax Rates?[END_REF]. We use statutory tax rates rather than effective (simulated) tax rates because first ones appear to be more relevant as signal effect to the firms' attention. Indeed, Devereux, Lockwood, [START_REF] Devereux | Do Countries Compete over Corporate Tax Rates?[END_REF] have shown that the effective marginal corporate tax rate (EMTR) is not relevant in order to reflect strategic interactions. This is in line with our expectations, because this rate is more used to note the after tax return of a marginal investment.

Geographic weightings are carried out from dyadic data calculated by Mayer and Zignago (CEPII). We use weighted distances to take account of the population geographical distribution within the country. Economic weights are computed from GDP data available on the DRI database.

Finally, instrumental variables and control variables take into account for each country:

- The presence of these variables is warranted to reflect the structural constraints weight present in national economies. Indeed, a relatively large country with important public expenditure will be inclined to respond less quickly and less intensely to lower tax rates in other countries. Moreover, the presence of variables such as GDP, GDP per capita and the deficit as a % of GDP makes it possible to take into account positive and negative shocks on national economy which would affect consequently the tax rate level in the country. These data are derived from the Datastream database.

For our estimates, we have verified that all these series were stationary, whether they are dependent or explanatory variables. When that did not happen, we make them stationary to make sure of our estimates accuracy.

Results

In this part, we test countries reaction delays to other European countries tax policies. We demonstrate that there is a group of leaders in the tax competition but also succeed in characterizing this group of leading countries.

The first step is to demonstrate that small countries, taken as a whole, are tax competition leaders. Then we establish a distinction according to the geographical location of these countries within Europe. So we separate the group of small countries into two groups: "integrated" (located in the centre of Europe) and "distant" (located at the periphery). It appears that only the "integrated" group plays a significant role in corporate tax rate setting of larges countries. Lastly, the third stage is to determine whether a small country, taken individually, plays a leading role in the tax competition.

Existence of strategic behaviours in the EU

While regressing each country tax rate on tax rates of other countries in the European Union, we found the presence of strategic interactions within the European Union [START_REF] Chatelais | Comportements stratégiques fiscaux des pays en Europe[END_REF]. Table ( 4) also shows that countries observe their neighbours fiscal policies and respond the following year. Indeed, taking account of a tax reform in a neighbouring country, the home country will develop a new tax schedule by changing its rates the following year in order to stop capital flight. We note that the domestic country's reaction is more than proportional compared to the tax rates decline in countries with a common border (estimated coefficients are respectively 1.35 and 1.84 for distance to the centre and bilateral distances matrices). This can be explained by the fact that the country reacts to tax rates falls in other countries but it also anticipates its neighbour's reaction to its own lower tax rates and / or tries to attract taxable incomes. Countries react more strongly to their closest geographical neighbours' tax rate since a tax rates fall can jeopardize access to the market known as "home market bias". Indeed, a firm may locate in the neighbouring country, taking advantage of a more attractive tax system, while having access to the domestic market at a lower cost. The distance to the centre criterion also appears to be important in the decisionmaking process and in reaction. Countries, including small distant ones, will be attentive with core countries' fiscal policies and will tend to react by lowering their tax rates more sharply in order to compensate for agglomeration forces present in the centre.

However, for estimates made with the GDP weighting matrix, we found similar coefficients with one lag

(1.00) or none (1.08). Finally, we can note that estimates with two lags of delays are not significant. This suggests that countries react quickly, usually in the following year. This delay is an average, some countries will react faster, theoretically small ones, while others will take longer to respond, probably larger ones. This first empirical part of the study allows establishing the presence of strategic behaviours within countries of the European Union. In addition, it appears that countries will react more to their neighbours' tax policies after one year.

According to the theory developed in the preceding parts and initial estimates, we can assume that there are leading countries in the tax competition and that they are small countries. We thus now try to detect the influence of small

European countries on tax rates set in large countries. We carry out estimates for the group of large and small countries which enable to highlight the greatest reactivity of small countries compared to large ones. In other words, we sought to determine whether small countries implement aggressive tax strategy vis-à-vis big countries.

Small countries: leaders of the tax competition in Europe.

According to the theory developed previously, we expect that small countries of the European Union apply aggressive lowering tax rate strategy vis-à-vis larges countries in order to attract taxable incomes. The fact that countries wait to note others tax policy before making their own reforms suggest the existence of leaders in the falling tax rates process.

According to table (5), [START_REF] Chatelais | Comportements stratégiques fiscaux des pays en Europe[END_REF] find that tax interactions are stronger between small countries than in all European countries or among large countries. These interactions are weak or even non-existent within the group of larges countries (see Table 5).

The presence of strategic interactions being established, we show that small European countries are at the origin of this tax competition.

We continue with panel estimates. A group of 8 small countries will play the Stackelberg leader role. This group is defined as follows: Belgium, Denmark, Czech Republic, Ireland, Portugal, Slovakia, Lithuania and Latvia.

The average tax rate of this group was calculated using three weights: the distance from the centre, the weight of GDP and population. Results of the table (6) show estimates of the following equation conducted with the three weighting matrices (distance from the centre, distance and weight of bilateral GDP): Estimates with distance from the centre weight and bilateral distances weight produce conclusive results

τ i,t = δ + β ω ij τ j,t + θX i,t + ατ L,t-1 + ε i i≠ j ∑ (4.1)
(unlike those obtained with the GDP weight, estimated coefficients are significantly different from zero). The detailed results below were achieved with weighting by the distance from the centre. Most of large countries being in the European Union core , it appears that they are more influenced by the change in tax rates of small countries close to the centre.

However, it should be noted that there is no strategic behaviour among major countries when we add the group of small countries leaders of Stackelberg in the regression. This may indicate that the pressure that small countries impose on large ones is strong enough to that competition between major appears negligible (and is not revealed in econometric regressions).

In conclusion, we show that small countries impose a (downward) pressure on large countries' corporate tax rates. It would be interesting to make a distinction between small countries to see those having the most influence on larges ones. Indeed, we should remember that we weighted the tax rate of the eight small countries group by their distance from the centre (small countries close to the core and thus close to large ones have the most weight). We assume that small countries close to the centre and those at the periphery do not have the same impact on tax rates setting in big countries.

Small countries close to the centre are the "real" tax competition leaders

The importance of the small countries group in tax rate setting has been demonstrated. We refine this analysis by distinguishing small countries close to the centre and those at the periphery. Indeed, small countries are very numerous (20 "small" against 6 "big") in the European Union, we make a distinction within this category. Our assumption is that small countries close to the centre have the most influence on core major countries and consequently they are the real instigators of this tax competition. Big countries always adopt a position of follower with respect to these small countries integrated to the centre rather than vis-à-vis small peripheral countries.

We split our sample of small countries into two groups: "integrated ones" close to the Economic heart and "distant ones"

-"integrated": Ireland, Belgium, the Netherlands, Luxembourg, Denmark, Austria, Slovenia and Czech Republic -"distant": Sweden, Finland, Estonia, Latvia, Lithuania, Hungary, Slovakia, Greece, Portugal, Cyprus, Malta, and Iceland.

This separation was done only on geographical criteria, and the absence of Scandinavian countries in the "integrated" category is justified. The 6 big remaining countries (Germany, France, Great Britain, Spain, Italy and Poland) are the third group of countries.

The so-called "integrated" countries Table ( 7) summarizes estimates made with the group of small integrated countries. Small countries have indeed a central influence on tax rate established in large countries (for both geographical weighting matrices). Big countries always react with two years late and always strongly with a coefficient greater than 2. Small countries integrated to the centre plays a leadership role vis-à-vis big countries in tax rate setting. It should be noted in this case, there is the presence of strategic interactions among major countries. However, rates in large neighbouring countries are less important than rates set in small integrated countries. But remember that large countries such as France and Spain

have not yet done so their tax reform in 2006, which are planned for coming months. Thus small "integrated" countries are leaders in the big countries corporate tax rate setting. Now let us consider if we can demonstrate whether the same was true for small countries located at the periphery.

Small "peripheral" countries How large countries do react in dealing with changes in tax policies of small peripheral countries? Table (8) shows there is a weak influence of small distant countries or even a lack of influence if we refer to the distances matrix.

Estimated coefficients are 0.53 for weighting by the distance from the centre and 0.64 for that by the GDP. These coefficients are significantly lower than those found for the group of small countries taken as a whole or for the group of small integrated countries. We deduce from it that small peripheral European countries have very little influence on the tax rate setting in large countries.

Small countries integrated to the centre have much more influence than small distant countries on big countries tax policies and therefore play a leading role vis-à-vis larges countries. Finally, in this last part of the analysis, we demonstrate that a small individual country is playing a leading role in the tax competition. We concentrate on small core countries without neglecting some peripheral nations that have experienced tremendous reductions in their tax rate.

Existence of individual leader in the tax competition

Tests conducted earlier indicate that the group of small "integrated" countries is the tax competition ringleaders in Europe. We assume that one of these countries can be a leader: Ireland, Czech Republic, the Netherlands, Belgium or Denmark. These countries have experienced significant reductions in their tax rate, but they do not have the same location within the European Union and have different levels of development. That is why we think relevant to test whether one or several of them, taking into account their characteristics, could be leaders. reductions play an important role in tax reforms of its large neighbours. Similarly the table (10) shows that there are strategic interactions between some peripheral countries with respect to large countries. These are Slovakia, whose influence is quite weak (0.24) and Lithuania whose influence is similar to that of Ireland (0.48). In conclusion, we have highlighted the existence of a tax competition leading countries group in Europe.

We first demonstrated that this is the group of small countries. Then, refining our analysis, that only small countries at the centre of Europe (and therefore close to large countries) have a real influence on the corporate tax rate setting strategy.

Moreover, it seems that, apart from Belgium, small individual countries have very little influence on the setting of tax rates in large countries. Big countries react more to tax policies noted in small countries at the centre taken as a whole. We can ask whether in addition to a size effect, a group effect can exist. The fact that only one country lowers its tax rates has certainly less impact on large countries policies if several small countries lower tax their rates in a short period of time.

Conclusion

The tax competition in Europe has a significant impact on national economies and obliges countries to take into account their neighbours fiscal behaviours. Our theoretical argument is that small countries of the European Union are at the origin of the tax competition and impose a downward pressure on corporate tax rates. Small countries practise this non-cooperative strategy because of the existence of a size effect. Indeed, the capital transfer is relatively more important for a small country than for a large one.

We demonstrate in this study, the presence of strategic interactions between European Union countries and especially the leading role of small countries in the tax competition. Specifically, consideration by large core countries of fiscal policies of small countries integrated to the centre has been demonstrated. This is confirmed by future tax reforms in big major countries. Indeed, we note a decline in the German corporate tax rate to 38.4% in 2007 which is expected to continue in 2008 to 29.9%. The same conclusion is drawn in Spain which is expected to pass its initial tax rate (35%) to 32.5% in 2007 and then to 30% in 2008. No drop in rates has been recorded in France, but the Sarkozy government seems inclined to drop very sharply in the next two years. United Kingdom will also reduce its tax rate by two percentage points in 2008 to be decreased to 28%. Thus, we see clearly that tax rates of major

Figure 2 :

 2 Figure 2: Evolution of Statutory tax rates in small and large countries

Figures 3 (

 3 Figures 3 (a and b): Corporate tax rates evolution in small countries

Figure 4 :

 4 Figure 4: Corporate tax rates evolution in large countries

  The share of public expenditure in GDP, of over 60 years in the population

Table 1 :

 1 Country classification with respect to tax pressure level(2005)

	Ranking

24,0 26,0 28,0 30,0 32,0 34,0 36,0 38,0 40,0 1990 1992 1994 1996 1998 2000 2002 2004 2006 24,0 26,0 28,0 30,0 32,0 34,0 36,0 38,0 40,0

sources:OECD, European Comission services

Table 2 :

 2 Corporate statutory tax rates

		1995	2000	2006
	Large countries	39.5%	38.3%	34.9%
	Small countries	32.6%	31.1%	23.4%
	Indeed, according to the table (2), the tax rates gap between these two groups of countries was already on average of
	7 points in 1995, and reached more than 11 points in 2006. According to table (3), small countries are at the origin of
	the European tax competition. Indeed these ones have initiated a very significant tax-cutting process and sharply
	intensifying during the twenty-first century. For example, small countries have reduced their tax rates by more than
	28% between 1995 and 2006, while the decline in larges countries hardly established at 12%. This decline has been
	particularly strong since the beginning of 2000 with a loss of 7.7 points in corporate tax rates in small countries. It
	appears that larger countries have been forced to respond by reducing their rates also by 3.6 points.

Table 3 :

 3 Corporate tax gap

		1995-2000	2000-2006	1995-2006
	Large countries	-1.2 pts	-3.6 pts	-4.6 pts

Small countries -1.5 pts -7.7 pts -9.1 pts Map 1: Corporate statutory tax rate evolution between 1995 and 2005

Table 4 : Reaction function estimates with and without lags for all countries of UE for 1995-2006

 4

	Dependent variable : dτi (corporate tax rate)
	GMM with individuals fixed effects
			Coefficients Estimates
	Explanatory Variables:	Wcentre	Wdistance	Wgdp
	dτj	0,97**	0,95*	1,08***
		(2,15)	(1,85)	(3,92)
	dτj(1)	1,35**	1,84*	1,00**
		(2,05)	(1,85)	(2,12)
	R²	0,16 [0,11]	0,13 [0,08]	0,15 [0,14]
	Sum squared resid	889 [940]	1129 [1097]	899 [909]
	DW	1,92 [2,01]	2,02 [2,16]	2,03 [2,00]
	J-stat	7,37 [12,46]	3,55 [4,84]	7,27 [8,97]
	t-stat in bracket
	* statistic significance level at 10%, ** à 5% et *** à 1%
	Regressions statistics between [ ] concern estimations with one lag.

Table 5 : Reaction function estimates for small and large EU countries over 1995-2006

 5

	Dependent variable : dτi (corporate tax rate)
	GMM with individuals fixed effects
			Coefficients Estimates
	Explanatory Variables:	Wcentre	Wdistance	Wgdp
	dτj small countries	1,05**	1,07**	1,13***
		(2,04)	(2,07)	(4,20)
	dτj large countries	0,70*	0,78*	0,52
		(1,78)	(1,87)	(0,79)
	R²	0,12 [0,05]	0,12 [0,03]	0,13 [0,15]
	Sum squared resid	708 [290]	875 [267]	703 [224]
	DW	1,89 [2,08]	2,00 [2,04]	2,05 [1,90]
	J-stat	5,79 [0,95]	2,71 [2,97]	4,38 [4,59]
	t-stat in bracket
	* statistic significance level at 10%, ** à 5% et *** à 1%
	Regressions statistics between [ ] concern estimations with for the large countries group (Germany, France, United Kingdom,
	Italy, Spain and Poland)

Table 6 : Reaction function estimates for large EU countries over 1995-2006 with all small countries as leaders

 6

	Dependent variable : dτi (corporate tax rate)
	GMM with individuals fixed effects
			Coefficients Estimates
	Explanatory Variables:	Wcentre	Wdistance	Wgdp
	dWτ	0,48	0,06	0,87
		(1,11)	(0,14)	(1,32)
	Dleader(2)	2,18**	2,76**	2,48**
		(1,95)	(2,26)	(1,96)
	Constant	2,13*	2,39*	2,57
		(1,72)	(1,80)	(1,65)
	R²	0,07	0,09	0,03
	Sum squared resid	248	253	310
	DW	2,08	2,11	2,06
	J-stat	2,61	2,91	2,08
	t-stat in bracket
	* statistic significance level at 10%, ** à 5% et *** à 1%

Table 7 : Reaction function estimates for large EU countries over 1995-2006 with small integrated countries as leaders

 7

	Dependent variable : dτi (corporate tax rate)
	GMM with individuals fixed effects
			Coefficients Estimates
	Explanatory Variables:	Wcentre	Wdistance	Wgdp
	dWt	0,90***	1,00**	1,20**
		(2,54)	(2,65)	(2,24)
	Integrated Leader (2)	2,57*	2,17*	2,14
		(1,81)	(1,82)	(1,50)
	Constant	2,34	2,01	1,81
		(1,54)	(1,55)	(1,23)
	R²	0,15	0,11	0,09
	Sum squared resid	328	322	322
	DW	1,97	1,98	2,01
	J-stat	2,32	1,78	1,81
	t-stat in bracket
	* statistic significance level at 10%, ** à 5% et *** à 1%

function estimates for large EU countries over 1995-2006 with small peripheral countries as leaders

	Dependent variable : dτi (corporate tax rate)
	GMM with individuals fixed effects
			Coefficients Estimates
	Explanatory Variables:	Wcentre	Wdistance	Wgdp
	dWt	-0,08	0,48	0,68
		(-0,22)	(1,14)	(1,38)
	Periphery leader(2)	0,53*	0,18	0,64*
		(1,75)	(0,61)	(1,71)
	Constant	0,13	-0,05	0,87
		(0,25)	(-0,10)	(1,10)
	R²	0,26	0,18	0,14
	Sum squared resid	184	205	250
	DW	2,19	2,12	2,15
	J-stat	10,79	12,64	5,80
	t-stat in bracket
	* statistic significance level at 10%, ** à 5% et *** à 1%

Table8: Reaction

Table ( 9

 ( ) reveals that only Ireland and Belgium can be considered as individual leaders for major countries, but have different influences. While Ireland has a relatively weak impact (0.40), but quite real, Belgium tax rate

Table 9 : Reaction function estimates for large EU countries over 1995-2006 with small integrated countries as leaders

 9

	Dependent variable : dτi (corporate tax rate)
	GMM with individuals fixed effects
	Weighting by bilateral distances
				Coefficients Estimates
	Explanatory Variables:	Ireland	Denmark	Netherlands	Czech Republic	Belgium
	WT	0,87**	0,85	0,99**	1,30**	-1,64*
		(2,07)	(1,64)	(2,60)	(2,36)	(-2,34)
	leaders(2)	0,40*	0,38	0,81	-0,65	1,01**
		(1,76)	(0,56)	(1,45)	(-1,42)	(2,37)
	Constant	0,81	0,22	0,41	-0,52	-1,89**
		(1,40)	(0,23)	(0,66)	(-0,64)	(-2,37)
	R²	0,14	0,04	0,10	0,07	0,07
	Sum squared resid	287	240	277	340	231
	DW	2,02	2,09	2,02	2,25	2,09
	J-stat	6,28	2,57	4,29	7,12	8,36
	t-stat in bracket
	* statistic significance level at 10%, ** à 5% et *** à 1%

Table 10 : Reaction function estimates for large EU countries over 1995-2006 with small peripheral countries as leaders

 10

	Dependent variable : dτi (corporate tax rate)
	GMM with individuals fixed effects
	Weighting by bilateral distances
				Coefficients Estimates
	Explanatory Variables:	Slovakia	Portugal	Lithuania	Latvia
	WT	0,57*	0,65**	-0,33	-0,56*
		(1,71)	(2,03)	(-0,68)	(1,75)
	leaders	0,24*	-0,06	0,49*	0,18
		(1,70)	(-0,22)	(1,88)	(0,70)
	Constant	0,57	-0,30	-0,24	-0,05
		(0,68)	(-0,50)	(-0,65)	(-0,12)
	R²	0,05	0,12	0,34	0,14
	Sum squared resid	238	219	163	214
	DW	2,18	2,11	2,07	2,12
	J-stat	8,95	1,94	11,13	4,43
	t-stat in bracket
	* statistic significance level at 10%, ** à 5% et *** à 1%