B Chomette

J-J Sinou

Crack detection based on optimal control

Techniques for optimal control to increase system security attract a great deal of interest from industry. The presence of transversal cracks can considerably modify the dynamics of a system. In the case of closed-loop systems, i.e. controlled systems, these faults can cause the destabilization and variation of control performance. Consequently, such variations can be used to detect transversal cracks. Therefore the present study proposes an investigation into the possibility of detecting structure modification based on estimated control performance by using the Rational Fraction Polynomial (RFP) algorithm. This method is applied numerically to a multi-cracked controlled truss. Both the optimal control of the cracked structure and the possibility of detecting the presence of cracks by monitoring the evolution of control performance are studied. The efficiency of the proposed method is demonstrated through numerical simulations corresponding to different crack orientations and locations.

Introduction

Cracks and thus structural modifications cause robustness problems, particularly reduced performance, in structures controlled by modal control. One way to detect cracks is to use modal testing in which changes in modal parameters such as variations in frequencies and mode shapes are used to detect damage. A review based on the detection of structural damage through changes in frequencies was discussed by [START_REF] Salawu | Detection of structural damage through changes in frequency: a review[END_REF]. Moreover, [START_REF] Dilena | Identification of crack location in vibrating beams from changes in node positions[END_REF] proposed damage identification based on changes in the nodes of mode shapes. They demonstrated that appropriate use of resonances and antiresonances can be used to avoid the non-uniqueness of damage location for symmetrical beams. One of the limitations of the indexes based on the change in the modal properties of changes in cracked structures is the use of only resonant frequencies and mode shapes for the vibrational description of mechanical structures. This has led to extending damage detection based on the use of the Frequency Response Function. For example, Faverjon andSinou (2008, 2009) proposed a damage assessment technique based on the constitutive relation error updating method for the non-destructive detection and sizing of single and multiple open cracks in beams. They demonstrated that it is possible to detect the number of cracks in the beam and estimate both crack positions and sizes with satisfactory precision, despite the presence of noise levels and the use of a limited number of degrees of freedom for the identification procedure. Comprehensive surveys can be found in the papers of [START_REF] Doebling | Damage Identification and Health Monitoring of Structural and Mechanical Systems from Changes in Their Vibration Characteristics: A Literature Review[END_REF], [START_REF] Sohn | A Review of Structural Health Monitoring Literature: 1996-2001[END_REF] and [START_REF] Sinou | A review of damage detection and health monitoring of mechanical systems from changes in the measurement of linear and non-linear vibrations[END_REF] for structural health monitoring and damage detection techniques in the civil and mechanical engineering communities. However little effort has been dedicated to discussing the capability of a controlled multi-crack truss structure to detect cracks by using modal parameters and controlling performance. Vibration control, by using optimally located actuators and sensors [START_REF] Hac | Sensor and actuator location in motion control of flexibles structures[END_REF], is one way of controlling structures and limiting vibration damage [START_REF] Chomette | Damage reduction of on-board structures using piezoelectric components and active modal control -application to a printed circuit board[END_REF]. Several active control strategies such as active collocated control [START_REF] Preumont | Vibration Control of Active Structures: An introduction[END_REF] or passive strategies [START_REF] Badel | Piezoelectric energy harvesting using a synchronized switch technique[END_REF]) can be used. Active modal control is a solution for targeting the control energy only on the vibration modes of interest with a minimum number of components [START_REF] Gaudiller | Active control of flexible structures using a minimum number of components[END_REF]. In the case of modal control based on a Luenberger observer [START_REF] Luenberger | An introduction to observers[END_REF], structural modifications lead to changes in control performance [START_REF] Chomette | Semi-adaptive modal control of on-board electronic boards using identification method[END_REF], and such changes can be used to detect cracks. Moreover, identification algorithms can be used to obtain structure modal parameters [START_REF] Mark | Global curve fitting frequency response measurements using the rational fraction polynomial method[END_REF] and control variations in performance. Therefore we propose to demonstrate the efficiency of an optimal control and identification procedure for a complex structure with multiple cracks. This study focuses not only on crack depth and position but also crack orientation, which can drastically modify the dynamic behavior of damaged mechanical structures. We will thus show that the evolution of control parameters in relation to other parameters (such as crack orientation) can be early indicators of the presence of a crack in a mechanical structure. The paper is organized as follows. Section 2 summarizes the concept underlying crack modeling and the model of the complete truss structure. The third part deals with the control and identification procedure. Then, control performance variation will be applied as a sensitive indicator of structural integrity. The efficiency of the methodology proposed will be investigated in the last part of the paper through different numerical simulations of a multi-cracked controlled truss with different cracks configurations.

Equation of motion of the system

In this section, the modeling of a transverse crack is proposed first, after which the equation of motion for the complete system with multi-cracks is presented.

Crack model

The modeling of a transverse crack that was described extensively in [START_REF] Sinou | Damage Assessment Based on the Frequencies' Ratio Surfaces Intersection Method for the Identification of the Crack Depth, Location and Orientation[END_REF] is briefly presented here. Mayes and [START_REF] Davies | The vibrational behaviour of a multi-shaft, multi-bearing system in the presence of a propagating transverse crack[END_REF] demonstrated that a transverse crack in a shaft can be represented by reducing the second moment of area ∆I of the element at the location of the crack ∆I

I 0 = R l (1 -ν 2 ) F (µ) 1 + R l (1 -ν 2 ) F (µ) (1)
for a circular beam. I 0 , R , l, and ν are the second moment of the area, the shaft radius, the length of the section and the Poisson ratio, respectively. µ is the non-dimensional crack depth and is defined by µ = h R . F (µ) is the fractional change in the second moment of the area measured at the crack face by considering a series of experimental tests for different transverse cracks. It should be noted that there are two values of F (µ) for a given transverse crack, one for each of the orthogonal directions in the plane of the crack. The cross section of the shaft at the location of the crack has asymmetric area moments of inertia about the neutral axis of bending due to the presence of the crack. Then, for each orientation χ of the crack (see Figure 1(a)), the crack model is derived by assuming that the portion of the chordal crack lying below the neutral axis will be opened under the influence of self weight bending. As previously proposed by [START_REF] Lees | The vibration signature of chordal cracks in asymmetric rotors[END_REF], the compliance functions given in Equation 1 are evaluated in the two orthogonal directions by assuming equivalence with the chordal crack of equal effective area. This procedure is sufficient to reflect the main physics of the open and closed portions for a particular crack depth and orientation by assuming beam self-weight, as previously mentioned by [START_REF] Lees | The vibration signature of chordal cracks in asymmetric rotors[END_REF]. So, the cross-section of the beam at the location of the crack has asymmetric area moments inertia I η and I ξ about the centroidal axes η and ξ that are defined as

I η = I η
(2)

I ξ = I ξ -A X2 (3)
The area moments of inertia I η and I ξ contain the reduced moment of area ∆I for each of the orthogonal directions η and ξ, as indicated in Equation 1. The complete expressions are given in [START_REF] Sinou | The influence of cracks in rotatingshafts[END_REF][START_REF] Sinou | Damage Assessment Based on the Frequencies' Ratio Surfaces Intersection Method for the Identification of the Crack Depth, Location and Orientation[END_REF]. X defines the new centroid due to the presence of the crack (i.e. X = 2 3A R 3 (2µ -µ 2 )

3 2 ) and A is the sector area of crack (i.e.

A = R 2 (1 -µ) 2µ -µ 2 + cos -1 (1 -µ) ).
Using the standard finite element formulation, the stiffness contribution K η,ξ

crack of an open crack about the centroidal axes η and ξ for a two nodes Timoshenko beam element of length l and Young's modulus E (corresponding to the flexible degree-of-freedom [u 1 ; v 1 ; θ 1 ; ψ 1 ; u 2 ; v 2 ; θ 2 ; ψ 2 ], see Figure 1(b)) can be defined at the crack location

K η,ξ crack = E l 3             12I η 0 0 6lI η -12I η 0 0 6lI η 12I ξ -6lI ξ 0 0 -12I ξ -6lI ξ 0 4l 2 I ξ 0 0 6lI ξ 2l 2 I ξ 0 4l 2 I η -6lI η 0 0 2l 2 I η 12lI η 0 0 -6lI η 12lI ξ 6lI ξ 0 4l 2 I ξ 0 4l 2 I η             (4)
Finally, the effects of the crack on the axial load terms can be written by superimposing the axial stiffness terms over the bending stiffness.

Model of the complete truss structure

After assembling the different beam elements and the stiffness contribution of the multi-cracks, the equations of motion of the truss structure are given by

Mẍ + C ẋ + (K -K multi-cracks ) x = F e (5)
where ẍ, ẋ and x are the acceleration, velocity and displacement of the degrees-of-freedom of the cracked system. M, C and K are the mass, damping and stiffness matrices respectively. F e defines an external force vector (given by F e = Fe iωt where F defines the force amplitude). K multi-cracks is the final stiffness matrix due to the presence of multi-cracks. In this paper, an active modal control based on a Linear Quadratic Gaussian algorithm is applied to the multi-cracked structure in order to demonstrate the multi-cracks detection possibility in the case of controlled structures. Indeed, modal control is based on a model of the structure and its performances depends on the model's precision. To aid understanding, it should be borne in mind that a slight difference between the model and the structure can lead to a decrease in performance or instability in the system controlled. In this short section of the study, this decrease in robustness is investigated to detect cracks caused by a slight variation of modal parameters.

The linear control need to use a linear model which can be described in the state form by

ẋ = Ax + Bu + Gw y = Cx (6)
where A is the dynamical system matrix, B and C are the actuator and sensor matrices respectively and G is the disturbance matrix. x and y are the state and the output vectors respectively. u is the control and w the disturbance. In the case of one actuator and one disturbance point, u and w are scalar quantities. The state matrices can be written for n modes

A = 0 n,n diag (ω i ) -diag (ω i ) -diag (2ξ i ω i ) (7) B = 0 n,1 Π a (8) G = 0 1,n Π f (9) C = Π c ω -1 0 1,n (10)
x = ωq q (11)

where q is the modal displacement vector, ω i and ξ i are the natural frequencies and modal Rayleigh damping respectively. Π a , Π c and Π f are the actuator, sensor and disturbance vectors respectively in the modal basis, defined as

Π a = φ t K a (12) Π c = φ t K c (13) Π f = φ t K f (14)
where K a , K c and K f are the actuator, sensor and disturbance vectors of degrees of freedom, respectively, and φ is the mode shapes matrix. The control u is solution of the Linear Quadratique Regulator problem and is calculated by using the state vector x and the control gain matrix K u = -Kx.

(15)

As modal state vector x cannot be measured, a Luenberger observer [START_REF] Luenberger | An introduction to observers[END_REF]) is used to reconstruct the state vector x. The estimated state vector can be written

ẋ = A m x + B m u + L (y -ŷ) ŷ = C m x ( 16
)
where (A m , B m , C m ) is the state model of the Luenberger observer used to reconstruct the state vector. L is the observer gain calculated by using the Linear Quadratic Gaussien algorithm.

Frequency Response Function calculation

The transfer function of the uncontrolled system can be written for one sensor as

H BO (jω) = y w = C s (jωId -A s ) -1 G s (17)
where (A s , B s , C s ) is the state model that represents the structure in the numerical simulations. The transfer function of the system controlled can be written for one sensor as

H BF (jw) = y w = C s jωId-A s -B s K jωId-A m -B m K-LC m -1 LC s -1 G s (18)
The two state models (A s , B s , C s ) and (A m , B m , C m ) are assumed to be equal in the case of an uncracked structure. This configuration will be called the nominal configuration. In the case of a cracked structure, modifications between these two models can lead to modifications in the dynamics of the closed loop system. In this paper, these modifications are estimated by using the RFP algorithm in order to detect the crack. Thus the system (A s , B s , C s ) which represents the structure can be calculated for different crack orientation from χ = 0˚to χ = 180˚. For example, the numerical transfer function in open and closed loop can be written for χ = 180˚H

BO (jω) = y w = C 180˚( jωId -A 180˚) -1 G 180˚( 19) H BF (jω) = y w = C 180˚ jωId-A 180˚-B 180˚K jωId-A m -B m K-LC m -1 LC 180˚ -1
G 180( 20)

where (A 180˚, B 180˚, C 180˚) is the system which represents the cracked truss with a transversal crack for χ = 180˚. The mode shapes variation is taken into account in the modal matrices B 180˚, C 180˚a nd G 180˚b ased on actuator, sensor and disturbance vectors. Indeed actuator, sensor and disturbance vectors are calculated using the cracked mode shapes matrix φ 180Π

a 180˚= φ 180˚t K a (21) Π c 180˚= φ 180˚t K c (22) Π f 180˚= φ 180˚t K f (23)

Optimal placement

In the case of complex structures, like truss structures, the actuators and sensors must be located optimally in order to control the system. The location of the optimal control component is calculated by using controllability and observability gramians theory. The controllability or observability problem can be linked to the crack detection problem. The best point to detect a crack is a point where all the modes affected by the crack are observable. Consequently, it corresponds to the best point for controlling the structure. The optimal node is obtained from the [START_REF] Hac | Sensor and actuator location in motion control of flexibles structures[END_REF] criterion

P I = 2m i=1 λ i 2m 2m i=1 λ i ( 24
)
where m is the mode number used in the computation and λ i are the eigenvalues of the Gramian matrix W c defined as

W c = ∞ O
e Aτ BB t e A t τ dτ (25)

Identification procedure

The aim of the identification procedure is to estimate the variation of the modal parameters. The variation of the modal parameters (i.e. frequency, damping and modal amplitude constant variation), and the control performance variation are estimated by using the Rational Fraction Polynomial (RFP) algorithm [START_REF] Mark | Global curve fitting frequency response measurements using the rational fraction polynomial method[END_REF] applied on the complex Frequency Response Function (FRF) obtained by using the truss model. The estimated FRF can be written as

H (ω) = n k=1 a k ω 2 k -ω 2 + 2ξ k ω k ω (26)
where n in the mode number, ω k and ξ k are respectively the frequencies and modal damping, a k is the modal amplitude constant of the FRF.

4 Numerical application

The multi-cracked truss studied

The procedure explained in the previous section is applied numerically to the model of the cracked truss presented in figure 2. The truss is composed of 170 elements and 123 nodes and is clamped on nodes 122 and 123. The disturbance is applied on node 111. The Hac and Liu criterion calculated on the truss to optimally locate the controller is shown in figure 3(a) for nodes 1 to 120. The optimal location for the control is located on node 114. This location does not depend on the disturbance point. It means that nodes 111 to 121 (zoom in figure 3(b)) are the most sensitive to the behavior of the full truss. This node is chosen to apply the collocated control and to get observation of the vibration. The displacement used by the Luenberger observer to reconstruct the state vector is measured on this node. The control calculated by the LQ controller is then applied on this node. The truss model includes 20 modes and the control is performed by focusing on the first six modes.

In the following study, we focus our attention on modes 1 to 6 of the truss. Identification of the modal parameters (frequency, damping and amplitude) is performed with only one collocated actuator-sensor. Two cases are studied in order to demonstrate the efficiency method in different configurations. First, one transversal crack is located on element 52 and two straight cracks are located on elements 15 and 39 respectively (with depth µ = 1). Secondly, one transversal crack is located on element 12 and two straight cracks are located on elements 46 and 29 respectively (with depth µ = 1). The transversal crack orientation can change from χ = 0˚to χ = 180˚. When χ = 0˚, the crack is fully open and when χ = 180˚, the crack is totally closed.

As previously explained, it is assumed that the static deflection is much greater that the dynamic response of the cracked system. So, for each orientation of the crack, the crack model is derived by assuming that the portion of the chordal crack lying below the neutral axis will be opened under the influence of self weight bending (Lees and Fiswell, 2001;[START_REF] Sinou | The influence of cracks in rotatingshafts[END_REF][START_REF] Sinou | Damage Assessment Based on the Frequencies' Ratio Surfaces Intersection Method for the Identification of the Crack Depth, Location and Orientation[END_REF].

The identified modal parameters and the mode shapes for the uncracked structure are shown in table 1 andfigure 4. For the interested reader, it may be noted that variation of mode shapes are not used in the following sections to detect the presence of cracks.

Frequency

Multi-cracked uncontrolled truss

Firstly, we consider the evolution of the cracked orientation from χ = 0˚to χ = 180˚for the transversal crack and the two straight cracks. The effects on the modal parameters are illustrated in figures 5 and 6 for the two cases. The estimation is carried out in the frequency range 0-2500 Hz with 10 modes. The crack orientation from χ = 0˚to χ = 180˚causes a frequency variation (-8.8% for the first mode, -3.1% for the second mode, -6.7% for the third mode, -1.9% for the fourth mode, -4.5% for the fifth mode and -3.0% for the sixth mode in the case 1; -8.8% for the first mode, -3.1% for the second mode, -6.4% for the third mode, -2.6% for the fourth mode, -4.5% for the fifth mode and -3.1% for the sixth mode in the case 2). Consequently, one of the classical results previously reported by certain researchers is Moreover, the orientation of the crack causes a damping variation (-0.4% and -0.5% for the first mode, -0.3% and -1.5% for the second mode, -6.7% and -7.3% for the third mode, 5.1% and 5.1% for the fourth mode, 0.5% and 0.5% for the fifth mode and 0.3% and 0.3% for the sixth mode in the cases 1 and 2 respectively). Consequently, damping increases or decreases with crack closure depends on the mode. Finally, the crack orientation causes a variation of modal amplitude (-1.2% and -1.2% for the first mode, -0.3% and -0.3% for the second mode, -0.6% and -0.6% for the third mode, -0.2% and -0.1% for the fourth mode, -0.7% and -0.6% for the fifth mode and -4.1% and -2.1% for the sixth mode in the cases 1 and 2 respectively). Consequently, modal amplitude decreases with crack closure. These first results indicated that a crack can be detected by estimating modal parameters.

Multi-cracked controlled truss

Due to the effects of cracks that have been observed in the previous section, impacts of the variations on a multi-cracked controlled truss are undertaken. Indeed, in the case of controlled structure, the damping and amplitude variation can modify the controller efficiency. That is why these variations have to be identified. First of all, estimated modal parameters of the nominal uncracked controlled truss are detailed in table 2. Then, the incidence of crack orientation on the controlled structure is shown in figure 7(a) to figure 7(f) in the first case and in figure 8(a) to figure 8(f) in the second case for each mode. The references, i.e. the uncontrolled and controlled FRF of the uncracked truss corresponding to the nominal configuration, are shown in solid and dotted black lines respectively. The LQ gain K and the LQG gain L are obtained by using the model (A m , B m , C m ) of the uncracked truss and are not modified when the crack orientation χ is changed. Consequently, for each crack orientation, there are some differences between the real structure represented by the system (A χ , B χ , C χ ) and the observer model (A m , B m , C m ). These differences cause variations in control performance and underline variation of crack orientations in the case where estimating the variation modal parameters is difficult. For example, we consider the evolution of the cracked orientation from χ = 0˚to χ = 180f or the transversal crack and the two straight cracks on the controlled truss. The effects on the modal parameters are illustrated in figures 9 and 10 for the two cases. The crack orientation from χ = 0˚to χ = 180˚causes a frequency variation (-11,2% for the first mode, -4,0% for the second mode, -7,4% for the third mode, -2,0% for the fourth mode, -5,3% for the fifth mode and -6,9% for the sixth mode in the case 1; -11,1% for the first mode, -4,1% for the second mode, -7,1% for the third mode, -2,6% for the fourth mode, -5,2% for the fifth mode and -3,8% for the sixth mode in the case 2). Consequently, controlled system frequencies Frequency the first mode, -0,6% and 1,0% for the second mode, -807,9% and -620,9% for the third mode, -141,8% and -154,9% for the fourth mode, -91,8% and -112,9% for the fifth mode and -48,4% and -217,5% for the sixth mode in the case 1 and 2 respectively). Finally, the crack orientation causes a variation of modal amplitude (-0,47% and -0,5% for the first mode, -0,12% and -0,1% for the second mode, -12,4% and -11,2% for the third mode, -2,9% and -2,8% for the fourth mode, -3,8% and -4,0% for the fifth mode and -3,6% and -4,0% for the sixth mode in the case 1 and 2 respectively). For the reader comprehension, in the case of controlled truss, the variations due to the crack orientation in the frequency response function result from structural truss modification and from changes in the controller dynamics. Indeed the variation of frequencies and mode shapes induce some differences between the modal matrices A m and B m of the uncracked truss used in the Luenberger observer, and the modal matrices A s and B s of the cracked truss. These differences modify the controller dynamics and cause a large variation of the active damping induced by the control. Also a large variation of magnitude on the controlled frequency response function can be observed, particularly for the third and fifth modes. In the case of uncontrolled truss, the variations due to the crack orientation in the frequency response function result only of structural truss modifications. These results demonstrate that crack orientation causes large changes in control performance depends on the mode, that can be positive or negative. These variations could be used as an indicator of crack orientation variation by undertaking links between the crack orientation and control reduction variation in future works.

Conclusion

This study proposes an investigation into the efficiency of an optimal control and identification procedure for a complex structure with multiple cracks. This method is applied numerically to a multi-cracked controlled truss by considering not only evolutions of crack depth and position but also crack orientation, which can drastically modify the dynamic behavior of damaged mechanical structures. Both the optimal control of the cracked structure and the possibility of detecting the presence of cracks by monitoring the evolution of control performance are studied. It is shown that the evolution of control parameters and modal parameters (particu-larly modal damping) of the controlled system in relation to other parameters (such as crack orientation) can be indicators of the presence of a crack in a mechanical structure. Especially, it is observed that the reduction in decibels obtained with the control of the multi-cracked system can be a way to predict the appearance of crack. Finally, this approach could be the first step of a health monitoring strategy in order to reduce vibration in a cracked structure. Future works should be linked between the crack orientation and control reduction variation.

  Figure 1: Description of the crack element

Figure 2 :

 2 Figure2: The control loop using the Luenberger observer applied to the multi-cracked truss structure and the FRF measurement points in two cracks cases (-: control loop, -.-: FRF measurements between disturbance and sensor points)

Figure 3 :

 3 Figure 3: Optimal placement for the control of the truss

Figure 4 :

 4 Figure 4: Mode shapes of the uncracked truss

Figure 5 :

 5 Figure 5: Estimated modal parameters (frequency, damping and amplitude) for χ = 0˚to χ = 180˚in the case 1 and uncracked modal parameters (ref.)

Figure 6 :

 6 Figure 6: Estimated modal parameters (frequency, damping and amplitude) for χ = 0˚to χ = 180˚in the case 2 and uncracked modal parameters (ref.)

Figure 7 :

 7 Figure 7: Influence of a transversal crack on a multi-cracked controlled truss for χ = 0˚to χ = 180˚in the case 1 and uncracked reference in black line (-.-: uncontrolled truss, -: controlled truss)

Figure 8 :

 8 Figure 8: Influence of a transversal crack on a multi-cracked controlled truss for χ = 0˚to χ = 180˚in the case 2 and uncracked reference in black line (-.-: uncontrolled truss, -: controlled truss)

Figure 9 :

 9 Figure9: Estimated modal parameters (frequency, damping and amplitude) for the controlled truss for χ = 0˚to χ = 180˚in the case 1 and uncracked modal parameters(ref.)

Figure 10 :

 10 Figure 10: Estimated modal parameters (frequency, damping and amplitude) for the controlled truss for χ = 0˚to χ = 180˚in the case 2 and uncracked modal parameters (ref.)

Table 1 :

 1 Modal parameters of the uncracked system

	[Hz]	Modal damping	Modal amplitude [dB]
	141,8	9,97×10 -4	-55,78
	433,0	9,93×10 -4	-64,67
	857,0	1,57×10 -3	-79,09
	1238,1	2,28×10 -3	-81,48
	1414,1	2.53×10 -3	-78,74
	1499,8	3,20×10 -3	-97,99

Table 2 :

 2 Modal parameters of the uncracked controlled truss increase with crack closure. Due to the control performances modification, the orientation of the crack causes a very large damping variation on the controlled truss (9,7% and 9,9% for

	[Hz]	Modal damping	Modal amplitude [dB]
	141,1	0,136	-76,28
	431,8	0,029	-78,55
	850,8	0,011	-86,18
	1235,2	0,003	-83,09
	1407,5	0,012	-84,35
	1496,5	0,029	-99,60

Acknowledgments

The authors gratefully acknowledge the financial support of the French National Research Agency through the Young Researcher program ANR-07-JCJC-0059-01-CSD2.


Marilyne Huchet

Chantal Le Mouël

Mariana Vijil
email: mariana.vijil@rennes.inra.fr

Marilyne Huchet-Bourdon

The relationship between trade openness and economic growth: some new insights on the openness measurement issue

Keywords: growth, openness, diversification, quality, dynamic panel estimation JEL: F43, O11, O40

-Introduction

In spite of the wave of liberalizations undertaken during the last 30 years, the debate on the links and causality between openness, growth and income distribution is still open [START_REF] Rodriguez | Trade Policy and Economic Growth: A Skeptic's Guide to the Cross-National Evidence[END_REF]. Empirical evidence tends to show that in the long run more outwardoriented countries register higher economic growth performances (e. g., among others, [START_REF] Sachs | Natural Resource Abundance and Economic Growth[END_REF][START_REF] Edwards | Openness, productivity and growth: what do we really know?[END_REF][START_REF] Frankel | Does Trade Cause Growth?[END_REF][START_REF] Dollar | Trade, Growth, and Poverty[END_REF][START_REF] Lee | Once Again, Is Openness Good for Growth?[END_REF]. More recently, using broader databases and cross-section or panel-data estimations, [START_REF] Chang | Openness can be good for growth: the role of policy complementarities[END_REF] and [START_REF] Freund | Trade, regulations, and income[END_REF] also show that trade openness has a positive effect on income and that this positive relationship is enhanced by complementary policies.

According to some authors however (e.g., [START_REF] Rodriguez | Trade Policy and Economic Growth: A Skeptic's Guide to the Cross-National Evidence[END_REF] the debate on the causality between trade openness and growth is still far from being closed since most of the work supporting this empirical evidence suffer from, at least, two serious shortcomings that make their results to be questioned. The first shortcoming lies in the way trade openness is measured. The second one results from retained estimation methods.

Reviewing the existing literature on openness and growth shows that there is not a clear definition of trade openness. For many authors trade openness implicitly refers to trade policy orientation and what they are interested in is to assess the impact of trade policy or trade liberalization on economic growth. For other authors however, trade openness is a more complex notion covering not only the trade policy orientation of countries but also a set of other domestic policies (such as macroeconomic policies or policies related to law and institutions for instance) which altogether make the country more or less outward oriented. In such a case, what the authors are interested in is to measure the impact of global policy orientation on economic growth. Finally, one may adopt an even more global view of trade openness covering not only the policy dimension but also all other non-policy factors that clearly have an impact on trade and on the outward orientation of countries. Factors such as geography and infrastructures, for instance, do affect trade and the outward orientation of countries, whatever their policy orientation.

Many different measures of trade openness have been proposed and used in empirical analyses of the relationship between openness and growth. They more or less relate to the three alternative definitions of openness mentioned above. In line with the trade policy orientation definition, some authors have retained measures based on trade restrictions/distortions, such as average tariff rates1 , average coverage of quantitative barriers, frequency of non-tariff barriers or collected tariff ratios (see, e.g., [START_REF] Pritchett | Measuring Outward Orientation in LDCs: Can It Be Done?[END_REF][START_REF] Harrison | Openness and growth: A time-series, cross-country analysis for developing countries[END_REF][START_REF] Edwards | Openness, productivity and growth: what do we really know?[END_REF][START_REF] Yanikkaya | Trade openness and economic growth: A cross-country empirical investigation[END_REF]. Obviously these indicators are very imperfect partial measures of the overall restrictions/distortions induced by trade policies. [START_REF] Frankel | Does Trade Cause Growth?[END_REF][START_REF] Irwin | Does Trade Raise Income? Evidence from the Twentieth Century[END_REF][START_REF] Frankel | An Estimate of the Effect of Common Currencies on Trade and Income[END_REF][START_REF] Dollar | Trade, Growth, and Poverty[END_REF]. Their main advantage is that the data required to compute them are available for nearly all countries and over a rather long period. Their main weakness is that they are outcome-based measures and as such are the result of very complex interactions between numerous factors so that it is never clear finally what such measures empirically capture exactly. Another limitation of these trade dependency ratios lies in their endogeneity in growth regressions, which requires specific estimation techniques (such as instrumental variables techniques as in Frankel and[START_REF] Frankel | Does Trade Cause Growth?[END_REF][START_REF] Irwin | Does Trade Raise Income? Evidence from the Twentieth Century[END_REF][START_REF] Irwin | Does Trade Raise Income? Evidence from the Twentieth Century[END_REF] identification through heteroskedasticity techniques as in [START_REF] Lee | Once Again, Is Openness Good for Growth?[END_REF]. This last limitation may in fact be extended to all trade openness measures and constitutes the second shortcomings in existing empirical evidence on openness and growth that has been pointed out by [START_REF] Rodriguez | Trade Policy and Economic Growth: A Skeptic's Guide to the Cross-National Evidence[END_REF]. As argued by [START_REF] Lee | Once Again, Is Openness Good for Growth?[END_REF], all measures of openness are generally closely linked to the growth rate. Hence, this is likely that all measures of openness are jointly endogenous with economic growth, which may cause biases in estimation resulting from simultaneous or reverse causation. Various methods have been used to remedy this problem and there is still a debate among scientists about which method is the most appropriate (see, e.g., [START_REF] Dollar | Trade, Growth, and Poverty[END_REF]Kray, 2004, and[START_REF] Lee | Once Again, Is Openness Good for Growth?[END_REF].

In this paper, we adopt the most global definition of trade openness and our aim is to contribute to the on-going debate on the growth effect of trade openness. Relative to the existing literature, our contribution is the following. Firstly, we argue that trade openness is a multidimensional concept that cannot be summarized to a single measure such as the most commonly used trade share (calculated as the sum of imports and exports over GDP).

Secondly, recent developments in growth theory and in international economics have provided new insights on the relationship between trade and growth, which call for additional measures of trade openness. Hence, we propose a more elaborated way of measuring openness taking into account two additional dimensions of countries' integration in world trade: quality and diversification.

Endogenous growth theory has provided results on the positive growth effect of trade through innovation incentives, technology diffusion and knowledge dissemination (see, e.g., [START_REF] Young | Learning by doing and the dynamic effects of international trade[END_REF][START_REF] Grossman | Innovation and growth in the World economy[END_REF]. Inspired from these theoretical developments, Hausmann et al. (2007) proposed an analytical framework linking the type of goods (as defined in terms of productivity level) a country specializes in to its rate of economic growth. In order to test empirically for this relationship, they then defined an index aimed at capturing the productivity level (or the quality) of the basket of goods exported by each country. Their growth regression results showed that countries exporting goods with higher productivity levels (or higher quality goods) have higher growth performances. These results suggest that what countries export matter as regards the growth effect of trade. Hence our measurement of trade openness should consider this quality dimension as a complement to the volume (or the dependency) dimension.

Monopolistic competition trade models with heterogenous firm and endogenous productivity provide theoretical results supporting the positive growth effect of trade through both increased variety of products and improved productivity due to the exit of less efficient firms (e.g., [START_REF] Melitz | The impact of trade on intra-industry reallocations and aggregate industry productivity[END_REF]. Based on this literature, [START_REF] Feenstra | Export variety and country productivity: Estimating the monopolistic competition model with endogenous productivity[END_REF] developed a model allowing to link, for each country, relative export variety to average productivity and then to GDP growth.

Using [START_REF] Feenstra | New product varieties and the measurement of international prices[END_REF]'s index of export variety, they tested this relationship on the basis of exports to the US for a panel of 48 countries over the period 1980-2000. Their empirical results indicated that there is a positive and significant relationship between export variety and average productivity. Once again these results suggest that the structure of countries' exports matter regarding the growth effect of trade. Hence, our measurement of trade openness should also consider this variety/diversification dimension.

Our empirical application draws on the [START_REF] Frankel | An Estimate of the Effect of Common Currencies on Trade and Income[END_REF]'s model, which has been extended to take into account our set of three indicators of trade openness: trade dependency ratio, export quality index and export variety index. Estimations are performed on annual data over the period 1995-2009 for an unbalanced panel of 157 countries. We use a generalized method of moments (GMM) estimation approach developed for dynamic panel data models in order to deal with potential endogeneity biases due to omitted variables, simultaneity and measurement error.

Our results confirm that countries exporting higher quality products and countries exporting more diversified products grow more rapidly. The impact of the trade dependency ratio is found to be non-linear: it is lower (even negative) for countries which exports are slightly diversified or concentrated on low quality products.

The remainder of the article is organized as follows. In the next section, we present the specification of performed growth regressions and the retained econometric methodology.

Section 3 reports and discusses empirical results, while section 4 concludes.

-Specification of growth regressions and econometric methodology

Inspired from the [START_REF] Frankel | An Estimate of the Effect of Common Currencies on Trade and Income[END_REF]'s approach we retain the following specification:

t i t i t i t i t i t i t i t i t i Z GDP M X inst school GDP I n pop GDP pop GDP , , 7 , 6 , 5 , 4 , 3 , 2 1 , 1 0 , ln ln ln ln                                                 (1)
where the dependent variable is the logarithm of GDP per capita of country i for the year t, with GDP corresponding to gross domestic product and pop to the total population. Explanatory variables that derive from neoclassical growth theory appear on the first line of equation ( 1): n is the annual growth rate of the population of country i, (I/GDP) and school correspond respectively to the investment ratio and the level of education to account for human capital investment. The variable inst is an institutional variable which is a proxy for the quality of institutions. Variables of particular interest are reported on the second line of equation ( 1).

Aggregates exports and imports are denominated X and M. Dividing the sum of them by GDP, obtain the usually called openness ratio that, in the following, we will refer as the trade dependency or the trade ratio. Z denotes other variables that may influence GDP per capita like, typically here, our two additional trade openness measures: the export quality and the export variety/diversification indices.

The export quality index is computed according to the [START_REF] Haussmann | What you export matters[END_REF]'s approach. In order to test for the robustness of the estimated relationship between export diversification/variety and growth, we consider four alternative export diversification/variety indicators: the usual Herfindhal and Theil concentration indices, the number of active export lines and the export variety index based on the extensive margin as used in [START_REF] Feenstra | Export variety and country productivity: Estimating the monopolistic competition model with endogenous productivity[END_REF]. Extensive details on the definition and computation of these indices can be found in [START_REF] Vijil | Insertion des pays en développement dans le commerce mondial : diagnostic et typologie[END_REF].

Finally, to account for potential non linearity in the relationship between the trade ratio and the growth rate, Z may also include interactive variables between this trade ratio and the export quality or the diversification/variety indices.

More specifically, three alternative specifications are considered:

-a basic specification:

t i t i t i t i t i t i t i t i GDP M X inst school GDP I n pop GDP pop GDP , , 5 , 4 , 3 , 2 , 1 1 , 0 , ln ln ln ln                                               (2)
-an extended specification with the export quality index (quality):

t i t i t i t i t i t i t i t i t i t i t i quality GDP M X quality GDP M X inst school GDP I n pop GDP pop GDP , , , 7 , 6 , 5 , 4 , 3 , 2 , 1 1 , 0 , ) ln( * ln ) ln( ln ln ln ln                                                          (3)
-an extended specification with the alternative export diversification/variety indices (diversification):

t i t i t i t i t i t i t i t i t i t i t i ation diversific GDP M X ation diversific GDP M X inst school GDP I n pop GDP pop GDP , , , 7 , 6 , 5 , 4 , 3 , 2 , 1 1 , 0 , * ln ln ln ln ln                                                          (4)
Our empirical application is based on 157 pooled cross-country data over the period 1995-2009. As most explanatory variables are likely to be jointly endogenous with economic growth, we use the GMM estimators developed for dynamic panel data models [START_REF] Arellano | Some tests of specification for panel data: Montecarlo evidence and an application to employment equations[END_REF][START_REF] Arellano | Another look at the instrumental-variable estimation of error-components models[END_REF]. Within the GMM approach, one may choose the difference estimator, which considers the regression equation in differences, or an estimator combining the regression equation in differences and the regression equation in levels into one system.

Following [START_REF] Arellano | Another look at the instrumental-variable estimation of error-components models[END_REF], [START_REF] Blundell | Initial conditions and moment restrictionsin dynamic panel data models[END_REF] and [START_REF] Chang | Openness can be good for growth: the role of policy complementarities[END_REF], we also retain the system approach. Indeed, this estimator is more suitable when the explanatory variables are highly persistent over time, like in the case of growth models [START_REF] Bond | GMM estimation of empirical growth models[END_REF], or when the heterogeneity is relatively important.

Departing from this general model:

t i t i t i t i t i X y y , , 1 , , '            (5) t i i t i , ,            0 , ,       t i i t i i     for i =,…,N and t =2,…,T.
Where y is the dependant variable, X is the vector of explanatory variables, i  and t  denote respectively unobserved country-and time-effect and t i,

 is the idiosyncratic disturbance term, system-GMM estimator implies running a GMM procedure on the following system of equations,

) ( ) ( ' ) ( 1 , , 1 , , 2 , 1 , 1 , ,             t i t i t i t i t i t i t i t i X X y y y y     (6) t i t i t i t i t i X y y , , 1 , , '            (7)
Nevertheless, instead of doing a "first-difference transformation" as is usually case to remove the unobserved individual effect in equation ( 6), we perform a "forward orthogonal deviation" transformation. That is, for a variable w the transformation will be:

             t s is it it it t i w T w c w 1 1 ,
where the sum is taken over all available future observations it T ,

and it c is ) 1 (  it it T T
. This way of dealing with heterogeneity allows us to preserve sample size in our unbalanced panel while still been able to use past values of explanatory variables as instruments [START_REF] Arellano | Another look at the instrumental-variable estimation of error-components models[END_REF].

On the assumption of absence of serial correlation in the idiosyncratic disturbance term t i,

   0 , ,   s i t i   for i = 1, ….
, N and t s  , and that the initial conditions are predetermined

  0 , 1 ,   s i i y 
for i =,…,N and t =2,…,T.

the differenced equation ( 6) can be instrumented by lagged levels of explanatory variables [START_REF] Arellano | Some tests of specification for panel data: Montecarlo evidence and an application to employment equations[END_REF], using the following moment conditions:

    0 * 1 , , 2 ,      t i t i t i y   (8)     0 * 1 , , 2 ,      t i t i t i X   (9)
For t=3,…,T.

Furthermore, under the additional assumption that for each country i, the correlation between the explanatory variables and the country-specific effect is constant over all periods [START_REF] Arellano | Another look at the instrumental-variable estimation of error-components models[END_REF][START_REF] Blundell | Initial conditions and moment restrictionsin dynamic panel data models[END_REF], the regression in levels (7) can use the additional moments conditions:

      0 * , 2 , 1 ,       t i i t i t i y y         0 * , 2 , 1 ,       t i i t i t i X X  
This assumption that past changes in explanatory variables are uncorrelated with the fixed effect and the current error in level are only valid under the hypothesis described above about the initial conditions of the data generating process. For example, this means that in an empirical growth model, deviations of 1 , i y from long-run steady-state values, controlling for the covariates 1 , i X , must not depend on fixed-effects. [START_REF] Bond | GMM estimation of empirical growth models[END_REF] argue that this assumption may be valid in Solow growth frameworks, thus allowing us to use the System-GMM estimator in our model. Again, there is also a need of absence of serial correlation in the idiosyncratic disturbance term t i,  .

Furthermore, under the assumption that among our explanatory variables, both the population growth rate and the time-specific effect are likely to be strictly exogenous, the contemporaneous observations of these exogenous variables are used as additional instruments (Arellano and Bover,1995;[START_REF] Blundell | Initial conditions and moment restrictionsin dynamic panel data models[END_REF].

Finally, in order to test for the appropriateness of our retained instruments in the growth regressions, we consider two specification tests. The first one is the Hansen test of overidentification for which the null hypothesis is that the chosen instruments are valid. Because this test can be seriously weakened with instrument proliferation, we decide to collapse the instrument matrix in order to reduce their number [START_REF] Roodman | A Note on the Theme of Too Many Instruments[END_REF]. The second one examines whether the idiosyncratic disturbance term

t i,
 is serially correlated. The test is performed on the first-differenced error term (that is, the residual of equation ( 6) in differences) and the null hypothesis is that the latter is second-order uncorrelated. In both cases, failure to reject the null hypothesis gives support to the retained specification.

-Data and results

Data

We use annual data over the period 1995-2009 for an unbalanced panel of 157 countries. Most required data are extracted from the World Bank WDI (World Development Indicators) database. This is the case for the following variables. The dependent variable is computed using the GDP per capita based on purchasing power parity (expressed in constant 2005 US dollars). The annual population growth rate (n) is calculated based on total population figures (midyear estimates). The investment ratio (I/GDP) is proxied through the gross fixed capital formation in percentage of GDP, while the gross enrolment ratio at secondary level is used as a proxy for secondary education level (school). Finally, the trade ratio (X+M/GDP) is computed using GDP as well as values of exports and imports of goods and services in current USD dollars.

Remaining explanatory variables rely on data issued from various other databases. For the institutional variable inst, which is supposed to measure the quality of institutions, we use the rule of law estimate provided by the World Bank Governance Indicators database. This estimate captures perceptions of the extent to which agents have confidence in and abide by the rules of society, and in particular the quality of contract enforcement, property rights, the police, and the courts, as well as the likelihood of crime and violence.

. The export quality index and the alternative export diversification/variety indices are computed based on export values in current US dollars extracted from both the WDI database and the CEPII international trade database BACI (at a HS6 disaggregated level). Further details on the definition and computation of these indicators can be found in [START_REF] Vijil | Insertion des pays en développement dans le commerce mondial : diagnostic et typologie[END_REF]. Source: Authors' calculations

Empirical results

Regression results are reported in Table 2. The results of the basic regression with the trade ratio as the single trade openness measure (corresponding to equation specification (2)) are presented in the first column. The results of the regression extended to the export quality index as an additional measure of trade openness (corresponding to equation specification (3)) are reported in the second column. The last four columns of Table 2 present the results of the regression extended to the diversification/variety index (corresponding to equation specification (4)), each column reporting the results obtained with each of the four alternative retained indices.

Results of the basic regression (2) are consistent with the previous empirical literature: all estimated coefficient have the expected sign. One may notice however that few of these coefficients are significant: this is the case for the lagged GDP per capita, the investment ratio and the trade ratio variables only. As in [START_REF] Frankel | An Estimate of the Effect of Common Currencies on Trade and Income[END_REF], the population growth rate is found to have a negative but non-significant growth effect. However, contrary to [START_REF] Frankel | An Estimate of the Effect of Common Currencies on Trade and Income[END_REF] and [START_REF] Chang | Openness can be good for growth: the role of policy complementarities[END_REF] our results suggest that secondary education level would not significantly affect growth. As in previous studies the trade ratio is found to have a positive impact on growth, with a coefficient significant at the 5% level. Finally, Hansen and AR(2)

specification tests indicate that in both cases the null hypothesis cannot be rejected, which gives support to our retained specification.

These basic results still hold for alternative specifications (3) and ( 4) except for the coefficient of the trade ratio. Indeed, corresponding results in Table 2 show that an interesting pattern of non linearity in the growth effect of the trade ratio emerges when additional measures of trade openness are considered.

More specifically, we see in specification (3) that the coefficient of the trade ratio becomes significantly negative while the coefficient of the cross variable between both trade openness indicators is significantly positive, suggesting a non linearity in the growth effect of the trade ratio: for countries which exports are concentrated on low quality products (i.e., export quality index lower than exp(0.793/0.072) = 60 711 current US dollars, increasing the share of trade in GDP has a negative impact on growth; at reverse, this impact is positive for countries registering an export quality index which is higher than this threshold; in that case, the higher the export quality index, the greater the positive growth effect of the trade ratio. This last result suggests that the beneficial growth impact of increasing trade share is higher when countries' exports are concentrated on higher quality products. One may interestingly notice that in our sample, countries exhibiting the lowest export quality index belong in majority to the Low Income Countries category as defined by the World Bank2 , and that most of them exhibit an export quality index which is lower than the 60 711 current US dollars calculated threshold (the list of these countries may be found in [START_REF] Vijil | Insertion des pays en développement dans le commerce mondial : diagnostic et typologie[END_REF].

Results are similar when considering the diversification/variety dimension of trade openness.

Last column of Table 2 shows that the coefficient of the export variety index is positive and significant, suggesting, as stated by [START_REF] Feenstra | Export variety and country productivity: Estimating the monopolistic competition model with endogenous productivity[END_REF], that countries exporting more diversified products grow more rapidly. As in the previous case, the coefficient of the trade ratio becomes negative and significant while the coefficient of the interaction effect is positive and significant, suggesting that the growth effect of the trade ratio is non-linear. More specifically, increasing the trade ratio induces higher growth rate only for countries which export variety index is greater than 0.79 (i.e., 0.114/0.144). In that case, there is some pattern of complementarities between trade volume and trade (export) variety in the sense that both contribute jointly to increasing growth. Inversely, for countries with an export variety index below the 0.79 threshold, increasing the trade ratio is likely to be detrimental in terms of growth rate. One may relate this last result to the discussions on the so-called "natural resource curse" issue (and its "Dutch disease" part). As explained in [START_REF] Lederman | In search of the missing resource curse[END_REF], a number of studies have shown empirical evidence of a negative relationship between natural resource abundance and economic growth performances of countries. Various channels through which this negative relationship would operate have been proposed. Among them some directly relate to the idea of a negative growth impact of concentration in export. Finally, one may notice that in our sample, nearly all countries belonging to the Low Income Countries category exhibit an export variety index lower than the 0.79 calculated threshold [START_REF] Vijil | Insertion des pays en développement dans le commerce mondial : diagnostic et typologie[END_REF].

Similar results are obtained when the export variety index is replaced by the number of active export lines or by the Theil concentration index. In this last case however, the interpretation of coefficients is reversed since the higher the index, the higher the concentration, thus the lower the diversification. Hence, the negative sign of the coefficient of the Theil index indicates a negative effect of export concentration, hence a positive effect of export diversification on growth. In the same way, the negative coefficient of the cross variable between the trade ratio and the Theil index indicates a joint positive effect of trade volume and trade product diversification on growth. 4), major oil exporter countries have been removed from the sample. These countries exhibit very high quality export baskets and low export diversification/variety indices (in fact very high concentration -Herfindhal and Theil-indices and very low number of active lines and export variety index), which biases the estimation results (for more detailed discussion, see [START_REF] Vijil | Insertion des pays en développement dans le commerce mondial : diagnostic et typologie[END_REF]. Estimation method: two-step GMM system (Arellando and Bover, 1995;[START_REF] Blundell | Initial conditions and moment restrictionsin dynamic panel data models[END_REF] with [START_REF] Windmeijer | A finite sample correction for the variance of linear efficient two-step GMM estimators[END_REF] correction.

Instruments used:

Difference equation is instrumented by 3 rd and further lags of the level of predermined and weakly exogenous explanatory variables, contemporaneous population growth rate and time dummies. Instruments are collapsed.

Level equation is instrumented by 3 rd and further lags of first-differentiated predermined and weakly exogenous explanatory variables, contemporaneous population growth rate and time dummies. Instruments are collapsed. Source: Authors' calculations.

-Conclusion

This paper investigates the relationship between trade openness and growth. Starting from the idea that trade openness cannot be fully characterized through trade volumes only we propose to account for two additional dimensions: trade quality and trade diversification. Then, standard growth regressions are performed where, among the explanatory variables, the commonly used trade ratio is complemented by the [START_REF] Haussmann | What you export matters[END_REF]'s export quality index or the Feenstra (1994)'s export variety index. Our empirical application is based on annual data over the period 1995-2009 for an unbalanced panel of 157 countries. As most explanatory variables are likely to be jointly endogenous with economic growth, we use the system GMM estimator developed for dynamic panel data models.

Our empirical results show clearly that countries exporting higher quality products or more diversified products grow more rapidly. Furthermore, our results indicate that an interesting pattern of non linearity in the growth effect of the trade ratio emerges when export quality or export diversification measures of trade openness are considered. We show that increasing trade volumes as a percentage of GDP may be detrimental to growth when countries' exports are highly concentrated on low quality products or on a few products only. At reverse, providing that exports are concentrated on high quality products or sufficiently diversified, increasing trade share is always beneficial to growth.

Such results clearly show that the relationship between trade openness and growth is not straightforward and that the growth effect of trade is very difficult to capture empirically. Our paper calls for a more careful definition as well as measurement of trade openness.

To this regard, there is not so much good reason to limit our analysis to export quality and diversification indices. Indeed theoretical literature shows that as regards the growth effect of trade quality and diversification, the import dimension also matters. Hence, this calls for further investigations taking into account both import and export quality and diversification dimensions.

  Furthermore, data required to compute such indicators are most often available for only a limited set of countries and years. Corresponding to the global policy orientation definition, various "qualitative" indices allowing for classifying countries according to their trade and global policy regime have been proposed (see, e.g., the 1987 World Development Report outward orientation index, the Sachs and Warner, 1995, openness index or the Wacziarg and Welch, 2003, Sachs, Warner, Wacziarg and Welch openness index). Such indices unfortunately provide only a very rough classification of countries (from rather closed to rather open). Also many of the data required to construct these indices are available only for a few countries and at one point in time. Finally, measures based on trade volumes, which have been very commonly used in empirical analyses, rather relate to the most global definition of trade openness. Trade dependency ratios are the most popular of these measures (see, e.g.,

Table 1

 1 summarizes some basic descriptive statistics for all variables used in growth regressions.

Table 1 . Descriptive statistics for used variables

 1

	Variables	Obs.	Mean	Std. Dev.	Min.	Max.
	GDP per capita (PPP constant US$)	2488	10161.83	11937.56	150.81	84043.17
	Annual population growth rate (%)	2547	0.01	0.01	-0.04	0.13
	Investment ratio (%)	2344	22.12	8.39	-23.76	113.58
	Secondary schooling (%)	1868	72.64	32.02	5.17	161.78
	Rule of law	1980	-0.10	1.00	-2.69	1.96
	Trade ratio	2451	0.87	0.47	0.15	4.38
	Export quality index (cur.US dollars)	2928	74337.27	31608.83	9532.60	187607.2
	Herfindahl index	2928	0.17	0.20	0.01	1
	Theil index	2928	4.84	1.68	1.56	8.52
	Number active export lines	2928	1989.80	1666.77	1	5015
	Export variety index	2928	0.60	0.29	0.01	1

Table 2 . Growth regressions results

 2

	Explanatory variables	(2)	(3)		(4)
	Constant	-0.159	-0.623	-0.194	-0.264	-0.498	-0.087
		(0.097)	(0.335)*	(0.128)	(0.200)	(0.167)***	(0.215)
	GDP per capita (log, lagged value)	0.990	0.970	0.976	0.979	0.954	0.960
		(0.009)***	(0.018)***	(0.010)***	(0.017)***	(0.019)***	(0.015)***
	Annual population growth rate	-0.505	-0.320	-0.434	0.052	-0.199	-0.138
		(0.349)	(0.405)	(0.554)	(0.481)	(0.551)	(0.533)
	Investment ratio (log)	0.055	0.053	0.064	0.070	0.069	0.058
		(0.016)***	(0.024)**	(0.022)***	(0.026)***	(0.020)***	(0.033)*
	Secondary schooling (log)	0.016	0.049	0.044	0.059	0.062	0.061
		(0.016)	(0.024)**	(0.022)**	(0.023)**	(0.029)**	(0.027)**
	Rule of law	-0.008	-0.004	0.001	-0.011	0.003	0.008
		(0.007)	(0.015)	(0.014)	(0.016)	(0.018)	(0.016)
	Trade ratio (log)	0.043	-0.793	-0.017	0.104	-0.381	-0.114
		(0.017)**	(0.398)**	(0.031)	(0.054)*	(0.201)*	(0.078)
	Export quality index (log)		0.044
			(0.037)
	Trade ratio (log) X Export quality		0.072
	index (log)		(0.035)**
	Herfindhal index			-0.063
				(0.125)
	Trade ratio (log) X Herfindhal index			-0.058
				(0.126)
	Theil index				-0.012
					(0.010)
	Trade ratio (log) X Theil index				-0.028
					(0.011)**
	Number of active export lines					0.051
						(0.019)***
	Trade ratio (log) X Number of active					0.049
	export lines					(0.026)*
	Export variety index						0.077
							(0.028)***
	Trade ratio (log) X Export variety						0.144
	index						(0.084)*
	Observations	1228	801	801	801	801	801
	Number of countries 1	157	113	113	113	113	113
	Hansen test p-value	0.299	0.235	0.194	0.397	0.386	0.296
	AR(2) test p-value	0.138	0.459	0.419	0.382	0.372	0.393
	Robust standard errors in parentheses
	* significant at 10% level ; ** significant at 5% level ; *** significant at 1% level
	1 As regard specifications (3) and (

And/or other characteristics of the tariff distribution: tariff dispersion, frequency of tariff picks, etc.

The World Bank classifies countries according to their yearly Gross National Income (GNI) per capita, computed using the World Bank Atlas method. For 2008, the groups are: low income (LIC) with a $975 GNI per capita or less; lower middle income (LMIC) with a $976 to $3 855 GNI per capita; upper middle income (UMIC) with a $3 856 to $11 905 GNI per capita and high income (HIC) with a $11 906 or more GNI per capita.

Similar results are obtained with the Herfindhal index. However, in this case the coefficient of the trade ratio is non-significant. We thus do not comment this equation's results.


Jeremy Clotault

Emmanuel E Geoffriau

Cécile Dubois-Laurent

Sébastien Huet

Vanessa Soufflet-Freslon

Mathilde Briard

Didier Peltier

Carotenoid gene nucleotide diversity reflects carrot history and selection

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LCYE, CHXE and ZEP).

Compared to other species, a quite high single nucleotide (SNP) frequency was found for these genes (1/22 bp on average; 1/11 to 1/38 bp range). The haplotype diversity ranged from 0.523 to 0.851, with 9 to 15 haplotypes per gene. However, this high diversity was mainly due to silent or synonymous sites. The nucleotide diversity was shown to be structured by cultivar geographical origin, reflecting the species history, but also stressing out the consequence of this result when studying carotenoid genetics using association approaches. A second important factor was shown to be the gene position in the carotenoid pathway. When looking for signatures of selection, CRTISO, LCB1 and LCYE genes, located in a central position in the pathway, displayed a pattern consistent with a diversifying selection. However, the impact of selection varies depending on the root color group. Upstream genes, such as PDS, displayed a negative selection pattern, and may have been subjected to high constraints due to their overall importance for the subsequent pathway. Besides better understanding the functioning of this important pathway, our results provide valuable information for the identification of critical genes for carotenoid genetic studies, and about carrot evolutionary genetics and root color history.