Laurent Daudet

Marcela Morvidone

Bruno Torrésani

Time-Frequency and Time-Scale Vector Fields for Deforming Time-Frequency and Time-Scale Representations

Keywords: Time-Frequency Analysis, Gabor transform, Wavelets, instantaneous frequency, group delay, reassignment, group representation, Bargmann space

HAL is

INTRODUCTION

Linear time-frequency representations such as the continuous wavelet or Gabor transforms are sometimes considered as time-frequency representations with poor resolution. This comes from the fact that their square modulus may be seen as particular smoothings of Wigner functions, which are known to have sharp localization properties (at least for some particular classes of signals). The problem of "de-blurring" such linear representations has received some attention since the early attempt by Kodera, Gendrin and DeVilledary, 1 revisited more recently by Auger and Flandrin. [START_REF] Auger | Improving the readability of time-frequency and time-scale representations by the reassignment method[END_REF] More recent contributions, due to Chassande-Mottin et al, [START_REF] Chassande-Mottin | Differential reassignment[END_REF][START_REF] Chassande-Mottin | Méthodes de réallocation dans le plan temps-fréquence pour l'analyse et le traitement de signaux non-stationnaires[END_REF] and Daubechies and Maes [START_REF] Daubechies | A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models[END_REF] should also be quoted.

Reassignment methods may be seen as an attempt to restore the resolution (in time-frequency or time-scale space) which has been lost when using wavelet or Gabor transforms rather than supposedly "optimal" bilinear transforms. They aim at "sharpening" a given time-frequency representation by "deforming" it, i.e. by shifting coefficients in the time-frequency plane using an appropriate prescription for the shifts. The main idea is the following. Let R : R 2 → R 2 be a map on the time-frequency plane, called the reassignment map. To any timefrequency representation ρ ∈ L 1 (R 2 ), associate the measure on R 2 defined as follows: for any Ω ⊂ R 2 , denote by R -1 Ω the inverse image of Ω by R, i.e the set R -1 Ω = (b, ν) ∈ R 2 , R(b, ν) ∈ Ω . Then set

M (Ω) = R -1 Ω ρ(b, ν)dbdν . (1)
By construction, the total mass of ρ is preserved, which is an important property.

The main issue is the choice of the reassignment map. When ρ is the square modulus of a continuous Gabor transform (see below), a prescription has been given by Kodera, Gendrin and de Villedary, [START_REF] Kodera | Analysis of time-varying signals with small BT values[END_REF] which was reconsidered and studied later on by Auger and Flandrin. [START_REF] Auger | Improving the readability of time-frequency and time-scale representations by the reassignment method[END_REF] In such a situation, the time-frequency representation under consideration is such that its total mass equals (up to a constant) the energy of the signal, and also equals the total mass of the reassigned representation. However, a continuous Gabor transform is a complex valued function, and the reassignment does not take at all into account its argument (even though the argument is used explicitely for computing the reassignment map). Also, since only the spectrogram is reassigned (and not the CGT), the method does not provide any way of reconstructing the original signal from its reassigned time-frequency representation.

An alternative has been proposed by Daubechies and Maes, [START_REF] Daubechies | A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models[END_REF] based on the continuous wavelet transform. In that approach, the reassignment is performed on the (generally complex valued) transform itself, rather than on its square modulus. This yields the so-called synchrosqueezed wavelet transform. A main feature of that transform is that, in contrast with the other reassigned transforms, it is invertible: the signal may be reconstructed from its synchrosqueezed wavelet transform. However, let us mention that the corresponding reassignment map does not handle the argument of the wavelet transform in a natural way, in the sense that the sum in Eq. ( 1) involves complex numbers with different arguments. We shall come back to that point later on.

More recently, E. Chassande-Mottin and his collaborators [START_REF] Chassande-Mottin | Differential reassignment[END_REF][START_REF] Chassande-Mottin | Méthodes de réallocation dans le plan temps-fréquence pour l'analyse et le traitement de signaux non-stationnaires[END_REF] suggested a different interpretation of the reassignment maps, in terms of infinitesimal displacements of coefficients of the considered time-frequency representation. They proposed to use reassignment maps for building a vector field on the time-frequency plane, and use such a vector field as a starting point for further signal processing tasks. Such an approach has found several interesting applications, for example for time-frequency partition of signals. As we shall see, such an approach is also quite natural when more theoretical and abstract arguments are taken into account.

We propose in this paper a different interpretation of reassignment and reassignment maps, very much in the spirit of the differential reassignment of Refs. [START_REF] Chassande-Mottin | Differential reassignment[END_REF] and 4 . Our first remark is the fact that time-frequency representations are generally functions defined on a two-dimensional space, called the phase space, or the timefrequency space, which turns out to be an homogeneous space for some motion group: the Weyl-Heisenberg group in the Gabor case, and the affine group in the wavelet case. In both cases, the group action plays the role of translations on the time-frequency space. Therefore, it makes sense to use that particular group action for "displacing" coefficients on the phase space and for designing reassignment algorithms.

More precisely, we consider displacements on the phase space generated by vector fields, satisfying two constraints. The first constraint is a covariance constraint: covariance with respect to the underlying group action. This is an important constraint, for it ensures that the reassigned time-frequency transform will share the covariance properties of the initial transform. The second constraint is phase invariance: we impose that the argument of the time-frequency transform be unchanged by the deformation. Such a constraint originates from more "practical considerations": it has been recognized for a long time that the phase of continuous timefrequency transform may be of great importance for many tasks of signal analysis and processing. [START_REF] Portnoff | Time-frequency representations of digital signals and systems based on short time Fourier analysis[END_REF][START_REF] Delprat | Asymptotic wavelet and gabor analysis: extraction of instantaneous frequencies[END_REF][START_REF] Guillemain | Characterization of acoustic signals through continuous linear time-frequency representations[END_REF] We shall prove that such requirements are sufficient to yield the classical prescription 1 for Gabor reassignment, and a new one for the wavelet reassignment.

The paper is organized as follows. We set up our notations and describe general results in Section 2. Then we describe the case of Gabor transform and the Weyl-Heisenberg group in Section 3, and the wavelet case in Section 4. Section 5 is devoted to conclusions.

PRELIMINARIES

Throughout this paper, we shall mainly be concerned with the space of finite energy signals L 2 (R), equipped with the scalar product f, g = f (t)g(t) dt .

Among the time-frequency representations, the so-called Wigner (or Wigner-Ville) distribution (the WVD) plays a prominent role. The WVD is defined as follows. For f ∈ L 2 (R), one sets

W f (b, ν) = f b + t 2 f b - t 2 e -2iπνt dt . (2)
This defines a bounded continuous function of b and ν, which in addition belongs to all the spaces L p (R 2 ), 1 ≤ p ≤ ∞ (see Ref. [START_REF] Lieb | Integral bounds for radar ambiguity functions and wigner distributions[END_REF] for a proof). In particular, it is well known that

R 2 W f (b, ν) db dν = ||f || 2 , (3)
which suggests to interpret W f (b, ν)/||f || 2 as a probability density in the two-dimensional plane. The WVD also satisfies the following marginal properties:

R W f (b, ν) db = | f (ν)| 2 , (4)
R W f (b, ν) dν = |f (b)| 2 . ( 5
)
This in turn also suggests to introduce average quantities: namely, the instantaneous frequency

ν(b) = 1 |Z f (b)| 2 νW Z f (b, ν) dν , (6)
where Z f is the analytic signal * of f . The instantaneous frequency provides an estimate for a "local frequency" of the signal † . Similarly, the group delay may be defined as another appropriate moment of the Wigner-Ville distribution:

τ (ν) = 1 | Z f (ν)| 2 bW Z f (b, ν) db . (7)
The notion of instantaneous frequency has been criticized on the basis of physical arguments: its physical meaning can be rather questionable for complex signals. Indeed, suppose that a fonction f is the sum of two sine waves. Then its instantaneous frequency is an oscillatory function, which does carry information about the signal, but does not reflect its very nature.

In such situations one would need to introduce several such instantaneous frequencies, which would correspond to different frequency ranges. Such a program may be realized in several ways, but is most often based on other time-frequency representations. For example, given a window g ∈ L 2 (R), and letting G f (b, ν) denote the continuous Gabor transform of f ∈ L 2 (R) (details will be given in Section 3 below), one may introduce new quantities:

b(b, ν) = 1 |G f (b, ν)| 2 τ W f (τ, ξ) W g (τ -b, ξ -ν)dτ dξ , (8) ν(b
, ν) = 1 |G f (b, ν)| 2 ξW f (τ, ξ) W g (τ -b, ξ -ν)dτ dξ , (9)
which indeed provide estimates for time-frequency dependent notions of time and frequency. It is interesting to notice the similarity of the expressions defining ν(b) and τ (ν) on one hand, and b(b, ν) and ν(b, ν) on the other hand. In both case, these quantities arise as mean values of time and frequency variables, with respect to an appropriate "density", thus defined by an "integral" formula. However, the instantaneous frequency and the group delay also arise from a "differential formula", expressed as a logarithmic derivative of a Gabor transform. We shall show below that b(b, ν) and ν(b, ν) may also be defined via a logarithmic derivative, for an appropriately chosen derivative.

DEFORMATION OF CONTINUOUS GABOR TRANSFORM

Generalities

To fix the notations, let us start by describing some of the main features of the continuous Gabor transform (CGT for short). More details may be found in many textbooks, for example Ref. [START_REF] Schempp | Harmonic analysis on the Heisenberg nilpotent Lie group[END_REF] for group theoretical aspects, or Ref. [START_REF] Carmona | Practical Time-Frequency Analysis: continuous wavelet and Gabor transforms, with an implementation in S[END_REF] for signal processing applications.

Definition 1. Let g ∈ L 2 (R), g = 0. The continuous Gabor transform of any finite energy signal f ∈ L 2 (R) is the function of two variables G f (b, ν) = f, g (b,ν) , (10)
where b, ν ∈ R, and the Gaborlets g (b,ν) are shifted and modulated copies of the reference window g, defined by

g (b,ν) (t) = e 2inπν(t-b) g(t -b) (11)
Here, n is an arbitrary positive integer.

In general, one chooses n = 1, but we keep it arbitrary for the time being. Notice that our convention differs from that used in Ref. [START_REF] Chassande-Mottin | Differential reassignment[END_REF] , where the symmetrized version of the CGT ‡ is used. Both formulations may be obtained from each other modulo minor modifications. It is well known that after a suitable normalization, the Gabor transform is an isometry of L 2 (R) to the space L 2 (R 2 ) (unless otherwise stated, the measure is the Lebesgue measure

): for all f ∈ L 2 (R), G f ∈ L 2 (R 2 )
, and one has the norm equivalence (provided ||g|| = 0 !)

|G f (b, ν)| 2 dbdν = ||g|| 2 |f (t)| 2 dt . (12)
Therefore, it may be inverted by its adjoint: for all f ∈ L 2 (R), we may write

f = 1 ||g|| 2 G f (b, ν) g (b,ν) dbdν , (13)
the equality being understood in the weak L 2 -sense, i.e. in the sense of scalar products with L 2 (R) functions.

The square modulus of the CGT (called the spectrogram) enjoys interesting properties. We emphasize two of them below:

1. For f ∈ L 2 (R), |G f | 2 is a smoothed version of the Wigner transform of f : |G f (b, ν)| 2 = W f (τ, ξ) W g (τ -b, ξ -ν) dτ dξ . ( 14
)
2. The spectrogram is invariant with respect to time and frequency translations:

given f ∈ L 2 (R), let f ∈ L 2 (R) be defined by f (t) = e 2inπλt f (t -τ ), where λ, τ are fixed real numbers. Then it is easily verified that |G f (b, ν)| 2 = |G f (b -τ, ν -λ)| 2 .
However, the CGT itself is not fully covariant with respect to shifts and modulations. Indeed, if again

f (t) = e 2inπλt f (t -τ ), we have G f (b, ν) = e 2inπλ(b-τ ) G f (b -τ, ν -λ) . (15)
This is intimately related to the fact that time and frequency translations do not form a group. They in fact generate a three-dimensional group, called the Weyl-Heisenberg group.

The Weyl-Heisenberg group, and its regular representations

We first give a brief account of the harmonic analysis on the Weyl-Heisenberg group. The interested reader is invited to consult Refs. [START_REF] Schempp | Harmonic analysis on the Heisenberg nilpotent Lie group[END_REF] and 12 for more details.

The Weyl-Heisenberg group is of the form

G W H = R 2 × [0, 1[, with group law (b, ν, ϕ) • (b , ν , ϕ ) = (b + b , ν + ν , ϕ + ϕ + νb ) , (16)
and may be identified with a group of matrices

(b, ν, ϕ) ≈   1 ν ϕ 0 1 b 0 0 1   .
The unit element is (0, 0, 0), and the inverse of any (b, ν, ϕ) ∈ G W H is given by

(b, ν, ϕ) -1 = (-b, -ν, -ϕ + bν) . ( 17
) ‡ defined by G f (b, ν) = f (t)e -2iπν(t-b/2) g(t -b) dt
The Weyl-Heisenberg group is unimodular: the Haar measure µ, defined by dµ(b, ν, ϕ) = db dν dϕ (18) is both left and right invariant: for any (b

0 , ν 0 , ϕ 0 ) ∈ G W H , dµ((b 0 , ν 0 , ϕ 0 )(b, ν, ϕ)) = dµ((b, ν, ϕ)(b 0 , ν 0 , ϕ 0 )) = dµ(b, ν, ϕ) .
The complete classification of irreducible unitary representations of the Weyl-Heisenberg group is a classical result of harmonic analysis on Lie groups. [START_REF] Schempp | Harmonic analysis on the Heisenberg nilpotent Lie group[END_REF] Such representations are essentially characterized by an integral number n ∈ Z. More precisely, except for the characters, all of them act on L 2 (R), as follows:

if f ∈ L 2 (R), [π n (b, ν, ϕ)f ](t) = e 2iπn(ϕ+ν(t-b)) f (t -b) . (19)
Therefore, we have

g (b,ν) (t) = π n (b, ν, 0)g(t) . ( 20
)
Let L 2 (G W H ) denote the space of functions on G W H which are square-integrable with respect to the invariant § measure dµ(b, ν, ϕ) = db dν dϕ. A main aspect of harmonic analysis on the Weyl-Heisenberg group is the study of the so-called regular representations of

G W H on L 2 (G W H )
The left regular representation The left-regular representation

λ of G W H on L 2 (G W H ) is defined as follows: if F ∈ L 2 (G W H ), λ(g)F (h) = F (g -1 h) . (21)
In explicit form, we then have

λ(b , ν ϕ )F (b, ν, ϕ) = F (b -b , ν -ν , ϕ -ϕ + ν (b -b)) . (22)
λ is a unitary representation, and is reducible. Its reduction goes as follows. Introduce the family of spaces

H n = F ∈ L 2 (G W H ), F ((b, ν, ϕ)(0, 0, ϕ )) = F (b, ν, ϕ)e -2iπnϕ
for all (b, ν, ϕ)(0, 0, ϕ ) ∈ G W H .

(

) Note that if F ∈ H n , then one has F (b, ν, ϕ) = F (b, ν, 0)e -2iπnϕ := F n (b, ν)e -2iπnϕ . Therefore, H n L 2 (P) L 2 (R 2 ), where P = G W H /[0, 1[ R 2 . 23
The connection between the continuous Gabor transform (as well as the continuous wavelet transform) and the theory of square-integrable group representation has been observed and studied in details in Refs. [START_REF] Grossmann | Transforms associated to square integrable group representations I: General results[END_REF] and 14 .

Let now F ∈ L 2 (G W H ). Using a Fourier series decomposition with respect to the variable ϕ, we write

F (b, ν, ϕ) = n F n (b, ν)e -2inπϕ . (24)
Therefore, L 2 (G W H ) splits into a direct sum

L 2 (G W H ) = ∞ n=-∞ H n , (25)
The group acts onto H n as follows: if

F n ∈ H n and (b , ν , ϕ ) ∈ G W H , λ(b , ν , ϕ )F n (b, ν) = e 2inπν (b-b ) e 2inπϕ F n (b -b , ν -ν ) . ( 26
)
From now on, we keep the index n fixed, and we denote by λ n the map R 2 → U(L 2 (R 2 )) defined by:

[λ n (b , ν )F ](b, ν) = e 2inπν (b-b ) F (b -b , ν -ν ) . (27)
Remark: The representation λ n realizes the translations on P R 2 , viewed as an homogeneous space for the action of the Weyl-Heisenberg group. The space P, endowed with the action of G W H above, is called phase space, or Time-Frequency plane ¶ . Hence, translations of functions on the phase space differ from the usual translations, in the sense that the shift is accompanied by a phase correction.

The following property is a direct consequences of the above definitions.

Proposition 1. Let g ∈ L 2 (R), g = 0. Let f ∈ L 2 (R)
, and let G f denote its CGT, as defined in Definition 1.

Then G f ∈ H n . In addition, we have, for all h = (b , ν ,

0) ∈ G W H G πn(h)f = λ n (b , ν )G f . (28)
The infinitesimal generators of λ n are also of interest: introduce the differential operator ∇ = ( ∂b , ∂ν ) t , where for any smooth enough function F , we set

∂b F (b, ν) = ∂ b F (b, ν) (29) ∂ν F (b, ν) = ∂ ν F (b, ν) -2inπbF (b, ν) . ( 30
)
Then we have that for all

F ∈ L 2 (R 2 ), [λ n (v b , v ν )F ](b, ν) = e -v b ∂b e -vν ∂ν F (b, ν) . ( 31
)
When F is an analytic function, the latter equality is proved using infinite Taylor series expansion for F and the exponentials. The extension to L 2 (P) uses standard density arguments.

The right regular representation Similarly, one introduces the right regular representation ρ, defined by

ρ(g)F (h) = F (hg) . ( 32
)
The right regular representation leave the H n spaces invariant, and one may define

ρ n (b , ν )F n (b, ν) = e -2inπνb F (b + b , ν + ν ) (33)
Again, the infinitesimal generators of the right regular representation are of interest. As before, we introduce the modified derivative ∇ = ( ∂b , ∂ν ) t , where for any smooth enough function F , we set ∂b

F (b, ν) = ∂ b F (b, ν) -2inπνF (b, ν) (34) ∂ν F (b, ν) = ∂ ν F (b, ν) . ( 35
)
Then we have that for all

F ∈ L 2 (R 2 ), [ρ n (v b , v ν )F ](b, ν) = e v b ∂b e vν ∂ν F (b, ν) . ( 36
)
Properties The above derivatives enjoy interesting properties (in particular some covariance properties), which we summarize in the following proposition:

Proposition 2.

1. The infinitesimal generators satisfy the following commutation relations:

[ ∂b , ∂ν ] = -2inπ , [ ∂b , ∂ν ] = 2inπ ,
2. One has the following covariance properties: for all F , and for all (b, ν) ∈ R 2 ,

∇(λ n (b, ν)F ) = λ n (b, ν) ∇F , ∇(ρ n (b, ν)F ) = ρ n (b, ν) ∇F . ( 37
)
Sketch of the proof: The first assertion follows from a direct calculation. For the second one, we first remark that, as a consequence of the associativity of the group law, the left and right regular representations λ and ρ commute with each other. Therefore, they also commute with the infinitesimal generator of one another, which proves the second assertion of the proposition.

Constant Phase Deformations

Let us now return to the problem of deforming a CGT. The goal of this section is to provide a geometrical interpretation of the reassignment maps proposed and studied in Refs. [START_REF] Kodera | Analysis of time-varying signals with small BT values[END_REF] and 2 for the CGT. Let us first set the problem. We are interested in finding a vector field

v : (b, ν) → v(b, ν) = (v b (b, ν), v ν (b, ν)) t ∈ R 2
defined on the phase space, such that the argument of the CGT remains unchanged under the infinitesimal action of the Weyl-Heisenberg group. More precisely, we ask for invariance with respect to the right regular action ρ n ( v), at first order in > 0. The reasons of the choice of the right-regular representation ρ rather than λ will be given in subsection 3.5 below.

Let F be a smooth function defined on the plane. For > 0 and v ∈ R 2 , let τ v ( ) :

L 2 (R 2 ) → L 2 (R 2 ) denote the operator defined by τ v ( )F (b, ν) = ρ n (-v(b, ν)) F (b, ν) . (38)
At first order in , we have

τ v ( )F (b, ν) = F (b, ν) -v(b, ν) • ∇F (b, ν) + O( 2 ) , (39)
where the symbol "•" stands for the scalar product in C 2 .

Let now f ∈ L 2 (R), and let G f denote its CGT. Let us write

G f (b, ν) = |G f (b, ν)| e iΩ(b,ν) , (40)
where Ω(b, ν) = arg(G f (b, ν)). One immediately sees that, again at first order in ,

τ v ( )G f (b, ν) = G f (b, ν) -G f (b, ν) v • |∇G f (b, ν)| |G f (b, ν)| + iv • ∇ m Ω(b, ν) + O( 2 ) , (41)
where we have introduced the modified gradient ∇ m , defined by

∇ m Ω(b, ν) = ∂ m b Ω(b, ν) ∂ m ν Ω(b, ν) = ∂ b Ω(b, ν) -2nπν ∂ ν Ω(b, ν) (42)
Therefore, imposing that the argument of G f (b, ν) be unchanged imposes that the vector field v be of the form

v(b, ν) = σ 2π ∂ m ν Ω(b, ν) -∂ m b Ω(b, ν) = σ 2π ∂ ν Ω(b, ν) -∂ b Ω(b, ν) + 2nπν , (43)
where σ = σ(b, ν) ∈ R is a real number. For the sake of simplicity, we shall assume that the sign of σ is constant.

Gaussian windows and the Bargmann space

Up to now, we have essentially focused on the study of the argument of the transform, and imposed a "constant phase" constraint. We now focus on the study of the modulus. Let (b, ν) → v(b, ν) be a time-frequency vector field as in (43). Then we have, at first order in ,

τ v ( )G f (b, ν) = G f (b, ν) -G f (b, ν)p(b, ν) + O( 2 ) , (44)
where we have introduced the quantity

p(b, ν) = v • |∇G f (b, ν)| |G f (b, ν)| = σ 2π ∂ ν Ω(b, ν) -∂ b Ω(b, ν) + 2nπν • |∇G f (b, ν)| |G f (b, ν)| , (45)
We start with the following observation. Let us introduce the complex variable z = ν + ib, and set ∂z = ∂ν + i ∂b (46)

Then we have Proposition 3. Assume that the sign of σ(b, ν) is independent of b and ν. A sufficient condition for the sign of p(b, ν) in (45) to be constant is that the CGT G f satisfy the Cauchy-type equation

∂z G f (b, ν) = 0 (47)
Such a condition is automatically satisfied for Gaussian windows:

g(t) = Ce -πnt 2 , (48)
where C ∈ C is a constant.

The meaning of the proposition is the following: if the window g satisfies condition (47), imposing a condition of constant phase of the "displaced" transform ensures that the displacement always goes globally in the direction of increasing or decrasing values of |G f |.

Proof of the proposition: Let us assume that (47) holds, and observe that ∂b

G f G f = ∂ b |G f | |G f | , and ∂ν G f G f = ∂ ν |G f | |G f | . ∂b G f G f = ∂ m b Ω , and ∂ν G f G f = ∂ m ν Ω .
Then,

p(b, ν) = σ ∂ m ν Ω ∂ b |G f | |G f | -∂ m b Ω ∂ ν |G f | |G f | = -σ | ∂b G f | 2 |G f | 2 .
Therefore, a positive value of σ ensures that the action of τ v increases the modulus of G f .

Let us now study in more details the solutions of the Cauchy-type equation (47

). Let us express G f (b, ν) in the form G f (b, ν) = F (z, z) . ( 49
)
Then, equation ( 47) is equivalent to

∂ z F (z, z) = π 2 n(z + z)F (z, z) ,
whose solutions are of the form

F (z, z) = F(z)e -π
where F(z) is a holomorphic function. Simple manipulations finally yield the following expression, reminiscent of the Bargmann factorization 15

G f (b, ν) = G f (z) e -π 2 n(|z| 2 + z 2 2 ) , (50)
where G f (z) is a holomorphic function of z = ν + ib.

Let us now consider a window g of the form (48). Then, an explicit computation yields

G f (b, ν) = e 3π 4 nz f (t)e -2inπtz e -πnt 2 dt e -π 2 n(|z| 2 + z 2 2 ) , (51)
i.e. precisely of the form (50). This completes the proof of the proposition.

For completeness, let us notice that the Bargmann factorization obtained here assumes a form slightly different from the form obtained when using the symmetrized version of the CGT. We refer to Ref. [START_REF] Chassande-Mottin | Méthodes de réallocation dans le plan temps-fréquence pour l'analyse et le traitement de signaux non-stationnaires[END_REF] for a detailed presentation of the symmetrized case.

Application to reassignment

Let us now come back to the reassignment problem. Our goal here is to set up a rationale for obtaining a prescription for reassignment maps, from first principles. For the sake of simplicity, we assume that σ is a (positive) constant, and we set it to 1 (without loss of generality, as σ can be absorbed in the constant ). Our approach emphasizes two constraints:

1. Covariance: Let f ∈ L 2 (R) and x ∈ G W H ; let f = π n (x)f
. With the same notations as before, we know from Proposition 1 that G f = λ n (x)G f . Therefore, imposing covariance of the reassignment with respect to the action of G W H on L 2 (R) implies that one must use translations on P which commute with λ n , i.e. the representation ρ n .

2. Constant phase: this constraint ensures that at first order, the reassignment preserves phase coherence.

These two constraints lead us to the following prescription for (generalized) infinitesimal reassignment:

G f (b, ν) -→ ρ n (-v(b, ν))G f (b, ν) = e 2inπ νv b (b,ν) G f (b -v b (b, ν), ν -v ν (b, ν)) , (52)
with v = v(b, ν) as in Equation ( 43) and σ = 1. We notice in particular that if we do not take the arguments into account, or equivalently, if we limit ourselves to the reassignment of the spectrogram, the prescription

b → b -v b (b, ν) = b -2π ∂ ν Ω(b, ν) ν → ν -v ν (b, ν) = ν(1 -n ) + 2π ∂ b Ω(b, ν) (53)
coincides with the prescription given by E. Chassande-Mottin and his collaborators, [START_REF] Chassande-Mottin | Differential reassignment[END_REF][START_REF] Chassande-Mottin | Méthodes de réallocation dans le plan temps-fréquence pour l'analyse et le traitement de signaux non-stationnaires[END_REF] in the framework of the differential reassignment.

Remark: We notice that the choice = 1/n yields the reassignment maps originally proposed in Ref. [START_REF] Kodera | Analysis of time-varying signals with small BT values[END_REF] (where only the case n = 1 is considered):

     b → b = b - 1 2nπ ∂ ν Ω(b, ν) ν → ν = 1 2nπ ∂ b Ω(b, ν) (54)
and also obtained in terms of the Wigner function. At the present stage of our work, we have no explanation for this coincidence, since our point of view is that of infinitesimal reassignment.

up to the modifications needed for replacing the CGT used here with the symmetrical one used in Ref. [START_REF] Chassande-Mottin | Méthodes de réallocation dans le plan temps-fréquence pour l'analyse et le traitement de signaux non-stationnaires[END_REF] .

Remark:

We also notice that for particular choice = 1/n, we obtain directly the new locations b and ν in terms of the infinitesimal generator of the left representation λ:

       b(b, ν) = - 1 2nπ ∂ν G f (b, ν) G f (b, ν) ν(b, ν) = 1 2nπ ∂b G f (b, ν) G f (b, ν) (55)
This fact is interesting by itself, as it shows that the prescription (b, ν) → ( b, ν) is covariant with respect to the left translations λ on the time-frequency space, therefore with respect to the action of G W H on L 2 (R).

Such formulas are reminiscent of the formulas which define the instantaneous frequency and the group delay in terms of logarithmic derivatives of the analytic signal: for example

ν(t) = 1 2π ∂ t Z f (t) Z f (t) , and
τ (ν) = - 1 2π ∂ ν Ẑf (ν) Ẑf (ν) ,
(we recall that Z f is the analytic signal of f ). Again, the expressions obtained here are similar, and in both cases the derivative used is covariant with respect to the underlying group action (usual translations on the real line, and left-translations on the time-frequency space).

THE CASE OF WAVELET TRANSFORM

The case of the continuous wavelet transform (CWT) is slightly different from that of the CGT, because it is based upon a group with a different structure. However, the main features of the analysis are similar. In particular, one still has a well-defined notion of phase space P, and of left and right regular representations of the underlying group G af f on L 2 (P).

The continuous wavelet transform

Let us start by recalling the basic aspects of continuous wavelet analysis. We shall mainly limit ourselves to the space of analytic signals, i.e. the Hardy space

H 2 + (R) = f ∈ L 2 (R), f (ω) = 0 ∀ω ≤ 0 . (56)
Definition 2. Let ψ ∈ L 2 (R) be a fixed function, hereafter called the wavelet. The continuous wavelet transform (CWT) of any finite energy analytic signal f ∈ H 2 + (R) is the function of two variables

T f (b, a) = f, ψ (b,a) = 1 √ a f (t) ψ t -b a dt , (57)
where the variables b and a run over R and R + respectively, and the wavelets ψ (b,a) are shifted and scaled copies of the reference wavelet ψ, defined by

ψ (b,a) (t) = 1 √ a ψ t -b a . (58)
It may be shown (see e.g. Ref. [START_REF] Carmona | Practical Time-Frequency Analysis: continuous wavelet and Gabor transforms, with an implementation in S[END_REF] ) that if the wavelet ψ is such that

0 < c ψ = ∞ 0 | ψ(ω)| 2 dω ω < ∞ (59)
then for all the wavelet transform maps H 2 + (R) into the space L 2 (R × R + , dbda/a 2 ), and that the transform is invertible on its range: for all f ∈ H 2 + (R), one has

f = 1 c ψ T f (b, a) ψ (b,a) db da a 2 . ( 60
)
The continuous wavelet transform may be given a group-theoretical interpretation along the same lines as before.

In particular, it may be completely characterized by its covariance properties, namely covariance with respect to translations and rescalings, which generate the so-called affine group G af f .

The affine group and its representations

The affine group G af f is the group generated by elements of the form {(b, a) ∈ R×R + }, with group multiplication

(b, a)(b , a ) = (b + ab , aa ) , (61)
and may be realized as a matrix group, namely the group of matrices of the form (b, a) ≈ a b 0 1 .

The unit element is (0, 1), and the inverse of (b, a)

∈ G af f reads (b, a) -1 = - b a , 1 a . ( 62
)
Unlike the Weyl-Heisenberg group, the affine group is not unimodular. It possesses a measure µ which is left-invariant * * , defined by

dµ(b, a) = db a da a , (63)
but this measure is not right-invariant. The right-invariant measure is given by

dµ (b, a) = db da a , (64)
The affine group acts on H 2 + (R) by an irreducible unitary representation π: for all f ∈ H 2 + (R) and (b,

a) ∈ G af f , π(b, a)f (t) = 1 √ a f t -b a . ( 65
)
The representation is square-integrable, [START_REF] Grossmann | Transforms associated to square integrable group representations I: General results[END_REF][START_REF] Grossmann | Transforms associated to square integrable group representations II: Examples[END_REF] so that if ψ ∈ H 2 + (R) is such that the condition (59) is fulfilled, then the map T :

f ∈ H 2 + (R) -→ T f defined by T f (b, a) = f, π(b, a)ψ , (66)
and identified with the CWT, defines an unitary map (after an appropriate normalization) between H 2 + (R) and L 2 (G af f , dµ), as expressed by Equation (60). (73)

Then we have that for all

F ∈ L 2 (R × R + ), [ρ(v b , v a )F ](b, a) = e v b ∂b e log(va) ∂a F (b, a) . (74)
Again, the left regular and right regular representations commute with each other, and with the infinitesimal generators of one another. As before, we denote by P the phase space which is now isomorphic to the affine group itself.

Constant phase deformations and reassignment

We now address the problem of deforming a CWT. We want to provide a geometrically natural prescription for wavelet reassignment maps. We are interested in finding a vector field

v : (b, a) → v(b, a) = (v b (b, a), v a (b, a)) t ∈ R 2
defined on the phase space, such that at first order in > 0, the phase of the CWT remains unchanged under the infinitesimal (right) action of the affine group. For convenience, let us introduce the auxiliary vector

w = w(b, a) = (v b (b, a), log(v a (b, a))) . (75)
Let F be a smooth function defined on the phase space P. For > 0 and v ∈ R 2 , let τ w ( ) : L 2 (P) → L 2 (P) denote the operator defined by

τ w ( )F (b, a) = e -v b ∂b e -log(va) ∂a F (b, a) . (76)
At first order in , we have

τ w ( )F (b, a) = F (b, a) -w(b, a) • ∇F (b, a) + O( 2 ) . (77)
Let now f ∈ L 2 (R), and let T f denote its CWT. Let us write

T f (b, a) = |T f (b, a)| e iΩ(b,a) , (78)
where Ω(b, a) = arg(T f (b, a)). One immediately sees that, again at first order in ,

τ w ( )T f (b, a) = T f (b, a) -T f (b, a) w • ∇ m |T f (b, a)| |T f (b, a)| + iw • ∇ m Ω(b, a) + O( 2 ) , (79)
where, as above, we have introduced the modified gradient ∇ m , which in this case coincides with the infinitesimal generator ∇:

∇ m Ω(b, a) = ∂ m b Ω(b, a) ∂ m a Ω(b, a) = a∂ b Ω(b, a) a∂ a Ω(b, a) (80)
Therefore, imposing that the argument of T f (b, a) be unchanged under the right action of the group imposes that the vector field w be of the form

w(b, a) = σ 2π ∂ m a Ω(b, a) -∂ m b Ω(b, a) = σ 2π a∂ a Ω(b, a) -a∂ b Ω(b, a) , (81)
where σ = σ(b, a) ∈ R is a real number. Again, we obtain a form for the vector field which involves phase derivatives of the CWT.

The reassignment may be performed as before. Let us choose σ = 1 for the sake of simplicity. Given a vector field w(b, a) satisfying the condition (81), and > 0, we have the following prescription

T f (b, a) -→ T f b -a v b (b, a), a v a (b, a) , (82)
We notice that such a prescription is quite different from the prescription studied in Ref. [START_REF] Chassande-Mottin | Méthodes de réallocation dans le plan temps-fréquence pour l'analyse et le traitement de signaux non-stationnaires[END_REF] . However, it is coherent with the structure of the affine group (i.e. additive on the first variable, and multiplicative on the second one).

CONCLUSIONS

We have presented in this article a geometrical setup for interpreting the methods based upon deformations of time-frequency representations. Starting from the description of the continuous wavelet and Gabor transforms in the framework of group representations, the time-frequency space appears as an homogeneous space with respect to the group action. The action of the group defines (generalized) translations on the time-frequency space.

Deforming a time-frequency representation (in this context, a square-integrable function on the time-frequency space) may be done using a local action of the group, i.e. by group-translating the time-frequency representation with translations which depend on the considered time-frequency location. At the infinitesimal level, such local translations are completely characterized by a vector field on the time-frequency space.

Two different group actions on the time-frequency space may be considered. We have shown that covariance requirements force one to limit to the so-called "right-translation". The resulting deformations is then covariant with respect to the group action. For example, if the group under consideration is the affine group, the deformed wavelet representation of a shifted and scaled copy of a signal f equals a shifted and scaled copy of the deformed wavelet representation of the original signal f . Once the group action has been chosen, the last question is the choice of the vector field, which yields the prescription for reassignment maps in reassignment algorithms. Such prescriptions have been proposed in the literature, but little is known when it comes to interpret them geometrically. We have shown here that the classical prescription 1 for CGT reassignment is obtained by imposing that the deformation preserves the phase of the transform. This remark also provides a complementary point of view on the recently introduced "differential reassignment" methods. [START_REF] Chassande-Mottin | Méthodes de réallocation dans le plan temps-fréquence pour l'analyse et le traitement de signaux non-stationnaires[END_REF] When applied to the case of the CWT, our approach also yields a prescription for deforming a continuous wavelet transform, which differs from the standard one. Applications of the corresponding wavelet reassignment are in progress, and will be presented elsewhere.

The work presented here raises several problems. To our opinion, the main one is that of inverting a reassigned time-frequency transform. It has been shown that such an inversion is possible 5 in a particular case. It is well known that continuous time-frequency transforms are extremely redundant, and that such a redundancy provides them with a great stability with respect to perturbations. One may conjecture that there exist deformations of time-frequency representations which preserve the inversion formula, and which possess the covariance properties described above.

Another interesting question concerns the extension of the results presented here to more general situations (higher dimensions, different geometries,...). We have deliberately chosen to emphasize the group theoretical point of view, because the latter provides a good framework for generalizations. For example, the results in the Weyl-Heisenberg case should generalize to higher dimensions without much difficulties, as higher dimensional Weyl-Heisenberg groups has the same structure as the one-dimensional. Generalization of the wavelet case may be done in several directions, as there exist several candidates for affine group in higher dimensions.

  The left regular action of G af f on L 2 (G af f , dµ) is introduced as before (see (21)): if F ∈ L 2 (G af f , dµ), we write for (b , a ) ∈ G af f [λ(b , a )F ](b, a) = F b -unitary representation of G af f on L 2 (G af f , dµ).The infinitesimal generators of λ are of interest: as before, introduce the differential operator ∇ = ( ∂b , ∂a ) t , where for any smooth enough function F , we have set∂b F (b, a) = ∂ b F (b, a) (68) ∂a F (b, a) = a∂ a F (b, a) + b∂ b F (b, a) .(69)Using Taylor series expansions one may show that for all F ∈ L 2 (G af f , dµ), [λ(v b , v a )F ](b, a) = e -v b ∂b e -log(va) ∂a F (b, a) . (70) Similarly, the right-regular representation acts on L 2 (G af f , dµ ), and is defined by [ρ(b , a )F ](b, a) = F (b + ab , aa ) . (71) The corresponding infinitesimal generators are given by ∇ = ( ∂b , ∂a ) t , where for any smooth enough function F , we have set ∂b F (b, a) = a∂ b F (b, a) (72) ∂a F (b, a) = a∂ a F (b, a) .

* Given f ∈ L 2 (R), its analytic signal is the functionZ f ∈ L 2 (R) such that Z f (ν) = 2 f (ν) Θ(ν), Θ being Heaviside's step function.† Because of Heisenberg's inequality, it is extremely difficult to make sense to such a notion. However, from the intuitive point of view, the local frequency is expected to make sense for signals to which local amplitude and phase may be associated, in such a way that the amplitude varies on scales much larger than the oscillations coming from the phase.

§ in the sense dµ ((b , ν , ϕ )(b, ν, ϕ)) = dµ(b, ν, ϕ) for all (b , ν , ϕ ) ∈ GW H .

¶ In technical terms, P is a symplectic manifold (with symplectic form [(b, ν), (b , ν )] = bν -νb ).

n(z+z) 2 ,

* * in the sense that dµ((b , a )(b, a)) = dµ(b, a) for all (b , a ) ∈ G af f .

ACKNOWLEDGMENTS B.Torrésani wishes to thank E. Chassande-Mottin and I. Daubechies for stimulating discussions. L. Daudet is supported by a grant of the French Ministère de l'Education Nationale, la Recherche et la Technologie. M. Morvidone is supported by a grant of the Fondo para el Mejoramiento de la Calidad Universitaria de la República Argentina.


Thi Thanh

Xuan Tran

The Impact of Electricity Production from Renewable Sources, Nuclear Source and the Conversion of Land Use into Agricultural Land on CO2 Emissions

 suggest that the impact of economic factors such as growth and trade on the environment can be decomposed into scale, technique and composition effects. Later work of Antweiler et al.

). Thus, this study contributes to this literature and pay attention to variables which are linked to the sources of CO2 emissions. Since electricity production and the conversion of land use into agricultural land are two main single sources of carbon dioxide emissions, I examine these impact on per capita CO2 emissions. The results of estimation for a panel of 99 countries spanning the period 1971-2010 indicate that :(1) increasing the share in electricity production from nuclear and renewable sources can decrease CO2 emissions whereas (2) the conversion of land use into agriculture land raises the amount of carbon emitted.

Introduction

Following [START_REF] Antweiler | Is free trade good for the environment?[END_REF], economic literature on the impact of economic growth and liberal trade on environmental degradation has progressed in three main ways. Here, I report only two branches of literature that link directly to the problematic of this paper.

The first branch of the literature focus on the relationship between economic growth and environmental degradation. The main interest of these studies is to investigate in the Environmental Kuznets Curve (EKC) literature, which hypothesizes that environmental problems should first worse and then improve along to an increase of income.

The Kuznets curve is named for [START_REF] Kuznets | Economic growth and income inequality[END_REF] who hypothesized that income inequality first rises and then falls as economic development proceeds. This hypothesis was applied to environmental issues in the later work of [START_REF] Grossman | Environmental impacts of a north american free trade agreement[END_REF]. [START_REF] Panayotou | Empirical tests and policy analysis of environmental degradation at different stages of economic development[END_REF] called this "inverse U" shape environmental consequences of economic activities the Environmental Kuznets Curve. This empirical phenomenon of an inverted U relationship between GDP per capita and environmental indicators has led to a large and developing theoretical works on the mechanism involving the upward and the downward sloping of an EKC. Numerous theoretical works provide different possible explanations for such an inverted-U relationship between economic growth and environmental problem such as intercountries difference in abatement activities (see, for example, [START_REF] Smulders | Pollution abatement and long-term growth[END_REF], [START_REF] Andreoni | The simple analytics of the environmental kuznets curve[END_REF] [START_REF] Brock | The green solow model[END_REF]), in institutional quality [START_REF] Jones | A positive model of growth and pollution controls[END_REF], [START_REF] Jones | Endogenous policy choice: the case of pollution and growth[END_REF] pay attention on the mechanism of vote while theoretical work of [START_REF] Damania | Trade liberalization, corruption, and environmental policy formation: theory and evidence[END_REF] present an interest in the role of corruption, for example), in environmental regulations (see, for example, [START_REF] Porter | Towards a dynamic theory of strategy[END_REF], [START_REF] Copeland | International trade and green growth[END_REF]).

Compared with the findings of previous studies, [START_REF] Grossman | Environmental impacts of a north american free trade agreement[END_REF] provide a very different explanation of the inverted U relationship by decomposing the impact of economic activities on environmental quality into: the negative environmental consequences of scalar increases in economic activity (scale effect), the positive impact of a greater real income on the cleaner production technique (technique effect) and the composition of output (the mix of dirty/clean industries). Whether the impact of country's mix of industries on the environment is negative or positive depend on country's comparative advantage in pollution-intensive sectors. In an ACT model, comparative advantage is considered as a function of country's endowment of capital, labor and its level of environmental regulations.

Number of empirical works support this idea. [START_REF] Vukina | Transition to markets and the environment: Effects of the change in the composition of manufacturing output[END_REF] suggest that the at the initial stages of economic development, emissions increase as a country develops because scale effect outweighs the composition and technique effect. By contrast, at the latter stages of development,when country specializes into clean industries, also its production's technique is cleaner, the composition and technique effects are strong enough to dominate the scale effect of growth. Thus, emissions decrease along to an increase of GDP per capita.

In this branch of literature, trade openness is considered as an important factor in generating the EKC slope. Theoretical work of [START_REF] Antweiler | Is free trade good for the environment?[END_REF] implies that the impact of trade on environmental outcomes can be decomposed into trade-induced scale, trade-induced technique and trade-induced composition effect. Thus, if the EKC can be formed by combining the scale, technique and composition effect, than trade could plays an additional role in the impact of growth on environmental outcomes. Empirical evidence suggests that trade plays an important role in generating the upward sloping portion for industrializing countries by promoting exports growth of manufactured goods and downward sloping portion for industrialized countries by facilitating imports of manufactured goods [START_REF] Suri | Economic growth, trade and energy: implications for the environmental kuznets curve[END_REF]).

The second branch of the literature concentrates on the change of trade flows with liberalized trade due to inter-countries difference in level of environmental regulations. These studies ask whether dirty industries shift from rich countries with strict environmental policies (high environmental abatement costs) to poor countries with weak environmental regulations (low abatement costs), that is the so-called "industrialflights" or "pollution havens" hypothesis. By examining the change of country's mix of industries induced by liberalized trade, this branch of literature analyzes the nature of the trade-induced composition effect. If this movement of pollution-intensity industries across countries is confirmed, then trade liberalization should lead to a decrease of pollution in rich developed countries whereas poor countries will see their emissions rise with liberal trade. At the global level, "industrial-flights" phenomenon could harm the environment. Among these studies, early work of [START_REF] Antweiler | Is free trade good for the environment?[END_REF] (hereafter the ACT's model) is the one that combines these two literature in one model to estimate the impact of economic growth and trade openness on the environment. On the one hand, these authors introduce the impact of economic growth on pollution outcomes by applying EKC hypothesis. And on the other hand, they also introduce the role inter-countries differences in factor abundance and environmental regulations play in the determining of trade's pattern, thus on the compositional change of pollution induced by liberal trade. The ACT model contributes to the economic literature on the trade-growth-environment relationship in three ways:

First, ACT's model allows us to construct a bridge between theoretical works and empirical works, which is not evident in the literature. On the one hand, the model links many controversy hypotheses involving the trade-growth-environment relationship. On the other hand, using this model, [START_REF] Antweiler | Is free trade good for the environment?[END_REF] also estimate empirically the environmental outcomes of economic growth and openness to trade in 108 cities representing 43 developed and developing countries during the period 1971-1996 and found a significant relationship between trade openness and SO2 concentrations. They suggest that earlier empirical investigations failed to find a strong and convincing link between freer trade and environmental problems because they "lacked a strong theoretical underpinning". They also believe that with a well-completed theoretical framework, they are able "to look in the right directions for trade's effect". Second, as argued by [START_REF] Cole | Determining the trade-environment composition effect: the role of capital, labor and environmental regulations[END_REF], the ACT's model is "the only theoretical model to make a distinction between the competing effects of environmental regulations and physical capital endowments" on country's trade pattern. Thus, ACT's model can reveal under which economic factor, pollution intensive industries have been driven. Finally, this model also allows to estimate separately the scale, technique and composition effect of economic growth and trade on pollution outcomes.

Previous studies of [START_REF] Cole | Determining the trade-environment composition effect: the role of capital, labor and environmental regulations[END_REF] and [START_REF] Managi | Does trade openness improve environmental quality?[END_REF] have shown that the relationship between economic factors and pollution can vary by pollutant given the differences between many common pollutants, "particularly with regards to their sources" [START_REF] Cole | Determining the trade-environment composition effect: the role of capital, labor and environmental regulations[END_REF]). Thus, I suggest that factors leading to differences in carbon dioxide emissions across countries can be divided into two groups: (i) inter-X. Tran countries differences in economic factors such as income per capita, capital-labor ratio and trade openness (which are captured in a standard ACT's model, for all pollutants) and (ii) inter-countries differences in appropriate factors which are linked to source of carbon dioxide. By consequent, this work attempts to capture the impact of these main sources of carbon dioxide emissions on the amount of pollution released. Two sectors are examined: energy sector and land use land use change and forestry sector (LULUCF). 1

Among studies on the relationship between energy sector and CO2 emissions, while there have been numerous that have investigated in the relationship between energy consumption and economic growth, both in two directions, few papers including the share of total energy consumption derived from renewable sources and nuclear sources as one of potential nonpolluting factor of CO2 emissions. [START_REF] Apergis | On the causal dynamics between emissions, nuclear energy, renewable energy, and economic growth[END_REF] examine the causal relationship between CO2 emissions, nuclear energy consumption, renewable energy consumption, and economic growth for a group of 19 developed and developing countries during the period 1984-2007 using a panel error correction model. The long-run estimates indicate that there is a statistically significant negative association between nuclear energy consumption and emissions, but a statistically significant positive relationship between emissions and renewable energy consumption. Menyah & Wolde-Rufael (2010) using time series econometrics of integration and causality method to explore the causal relationship between carbon dioxide (CO2) emissions, renewable and nuclear energy consumption and real GDP for the US for the period 1960-2007. They found a unidirectional causality running from nuclear energy consumption to CO2 emissions but no causality running from renewable energy to CO2 emissions.

Several studies including renewable and non-renewable energy consumption as additional variables in an EKC estimation 2 . [START_REF] Jebli | Testing environmental kuznets curve hypothesis: The role of renewable and non-renewable energy consumption and trade in oecd countries[END_REF] find that more trade and 1 Le [START_REF] Quéré | Global carbon budget 2013[END_REF] cite burning fossil fuel for energy sector (Figure 1 shows the increasing trend of world CO2 emissions from burning fossil fuel during the period 1960-2010) and land use change due to the expansion of agriculture sector top the list of human sources of CO2 emissions.

2 The following equation is the traditional and dominant technique in the EKC literature using cross-sectional data: 3 .

E it = α i + γ t + β 1 y it + β 2 (y it ) 2 + β 3 (y it ) 3 + β 4 Z it + it , ( 1
)
where E is environmental indicators, y is per capita income, α captures site or country specific term i, γ refers to time specific term t. The vector Z capture other additional variables as population density, locational dummies, etc. While the evidence of an negative correlation between energy consumption from nuclear source and the amount of carbon released has been founded in previous studies, the role of renewable energy on CO2 reductions is still inconclusive.

After burning fossil fuel for energy sector, land use change due to the expansion of agriculture sector top the list of human sources of CO2 emissions (Le [START_REF] Quéré | Global carbon budget 2013[END_REF]). Empirical studies usually concludes that LULUCF sector is an important source of CO2 emissions4 and global net annual emissions of carbon from land use

X. Tran

and land-use change increases gradually (R. A. [START_REF] Houghton | How well do we know the flux of co2 from land-use change?[END_REF]).

Among number of activities in the LULUCF sector, the conversion of land could lead to both CO2 emissions and removals: land converted to forest land and grassland are two major sources of CO2 sequestration whereas land converted to agricultural land5 and settlements are two major sources of CO2 emissions [START_REF] Hallsworth | Mapping carbon emissions & removals for the land use, land use change & forestry sector[END_REF]). Thus, there is also a large literature on the impact of urbanization and agriculture on emissions [START_REF] Vleeshouwers | Carbon emission and sequestration by agricultural land use: a model study for europe[END_REF], [START_REF] Kalnay | Impact of urbanization and land-use change on climate[END_REF], [START_REF] Vanum | Impact of urbanization and land use changes on climate[END_REF]). Recent studies also examine the impact of the use of crop-lands for bio-fuels on greenhouses gases. [START_REF] Searchinger | Use of us croplands for biofuels increases greenhouse gases through emissions from land-use change[END_REF]).

Literature on the impact of land use change on CO2 emissions provides a chemistry/biology explanation of this phenomenon. Nowhere, however, examine this relationship under an economic view and put the impact of land use change and economic factors side by side and estimate its impact on pollution outcomes. Thus, there is a need to examine the impact of land use change on CO2 emissions.

By examining the impact on CO2 emissions of the share of electricity production from nuclear and renewable sources, also the share of agricultural land on total land, this work is novel for 2 points: (i) the first one that examines the causal relationship between the sources of electricity production and the conversion of land into agricultural land on CO2 emissions and (ii) the first one that attempts to control for variables which are linked to the sources of emissions in a standard ACT's model.

Using a large panel of 99 countries over the period 1971-2010, this study also provides a global view on how these factors can affect environmental quality around the world.

The results of estimation indicate that for one unit increase in the share of electricity production from nuclear source, CO2 emissions per capita is expected to decrease by 1 unit. And CO2 emissions per capita could increase by 2 units if the share of agriculture increases by 1 unit, holding all other variables constant. Thus, this study confirms the positive effect on the environment of fuel-switching for the production of energy; also the negative effect of the conversion of land use into agriculture land on environmental quality.

This paper is organized as follow: section 2 presents the scale, technique and composition approach in the ACT's model, also my suggestion to decompose carbon dioxide emissions from electricity production and land use conversion into scale, technique and composition effects; section 3 shows the strategy of estimation, the problem of endogeneity while examining the trade-growth-environment relationship, and data description. Section 4 reports the results of estimations. Finally section 5 concludes.

Scale, technique, composition approach

The scale-technique-composition approach is prime concern of this paper in two ways. First, the model and methodology of [START_REF] Antweiler | Is free trade good for the environment?[END_REF], which is the central focus of this study, based on the decomposition of economic factors on pollution outcomes through the scale, technique and composition effects. Second, the effects of energy sector and LULUCF sector on CO2 emissions are decomposed into scale, technique and composition effects. Thus, this decomposition shows variables linked with these main sources of CO2 emissions and that are not already controlled for by the other explanatory variables in an ACT estimation.

ACT's model

To begin with, it is useful to provide a brief summary of the ACT's model and the method of decomposition of pollution. The key purpose of the ACT model is a simple general equilibrium model when government policy and private sector behavior interact to determine the equilibrium level of pollution. According to these authors, the government determines the level of taxes on emissions τ that maximizes the sum of utility of N agents in economy. Through the level of τ , the government decides the supply of the pollution. In another hand, private sector behavior considers pollution emissions as a sacrifice to economic activities (because they should pay emissions tax τ and decides the pollution demand. Combining the two functions of pollution X. Tran demand and supply yields the reduced form of pollution emissions equation. A brief outline of the model is presented below.

Consider a small economy open economy that faces fixed world prices and produces two goods X and Y. Good X is capital intensive and therefore generates pollution during its production and good Y does not (Y is labor intensive) 6 . Consequently, the X industry jointly produces two outputs-good X and emissions Z. Let β denote the wedge between domestic and world prices induced by trade frictions (the direct and indirect costs associated with good transaction), if p x is the relative price of X (good Y is the numeraire, p y = 1) then domestic prices will differ from world prices (p w ), the function of domestic price can be written as:

p x = βp w (2)
More precisely, β > 1 if country is a dirty good X importer and β < 1 if country is a dirty good X exporter. ACT define trade liberalization as the gradual reduction in trade frictions that moves domestic price closer to world prices. Consequently, from equation ( 2), for an exporter of polluting good X, β rises with freer trade and this also raises the relative price p of the dirty good X. The composition of national output shift toward X. By contrast, β falls with freer trade for an importer of the polluting good and this also lowers the relative price p of the dirty good X. Consequently, the composition of national output shift toward clean industry Y.

Let focusing on the supply and demand side of emission Z. On the supply side, suppose that firms face a price τ ( imposed by the government) for each unit of emission they generate. The level of τ reflects the supply side of the emission Z. The government preferred a level of pollution tax that maximizes the weighted sum of each group's utilities. The composition of pollution supply is:

τ = T ype + δ 1 β + δ 2 p x + δ 3 IN C (3)
The level of emissions tax depends on the proportion of "Greens" and "Browns" 7

6 That means, for any r and w (market returns for K and L), the capital / labor ratio in X is higher than Y: Kx Lx >

Ky Ly

7 two citizen groups whom differ in their preferences over pollution in this country (variable T ype), the price level of dirty good X p x , country's trade friction β and country's real per capita income IN C.

On the demand side, emission z is defined as:

z = ex = eϕS (4)
Where e is emission intensity (the proportion of emission z for each unit of output x), ϕ is the share of X in total output. Finally, S is defined as an economy's scale 8 . In differential form it becomes

z = e + ϕ + S (5)
Where the composition of output ϕ = ϕ(k, p(β, p w , τ )), pollution intensity e = e(τ , p(β, p w , τ )) Replacing τ in the pollution demand equation on the pollution supply yields the equilibrium reduced form equation of emissions z:

z = π 1 S + π 2 k -π 3 IN C -π 4 T ype + π 5 p w + π 6 β, ( 6
)
Where all π i are positive, and none of the right hand-side variables are determined simultaneously with emissions. Emission z depends on: the scale of economic S, the composition of country's factor abundance (k/l ratio), the cleanliness of production technique (GDP per capita INC), the type of country (the proportion of "Greens" and "Browns" T), world's price of dirty good X (p w ) and finally trade friction (β).

The ACT framework provides a standard guideline on how growth and trade cause environmental problems, not for some specific ones. [START_REF] Cole | Determining the trade-environment composition effect: the role of capital, labor and environmental regulations[END_REF] have shown that the relationship between economic pollutants and pollution can vary by pollutant given the differences between many common pollutants, "particularly with regards to their sources". Thus, I focus therefore on the main sources of carbon dioxide emissions.

8 S = p 0 x x + p 0 y y

Decomposition of emissions from energy sector and LU-LUCF sector

The need to introduce the sources of pollutant

Why ignoring the sources of different pollutant can bias the results of estimation ?.

Recall the decomposition of pollution emissions at country level:

z = ex = eϕ x S (4)
Pollution emissions depends on the emissions intensity of production e, the proportion of the dirty good industry in the economy ϕ x , and finally the scale of economy S. We can rewrite equation (4) as:

z = S * x S * z x (7)
Pollution emissions z depends on the scale of economy S, the proportion of the dirty good industry in the economy x S , and finally the amount of pollution emitted z on each unit of x produced z

x . Economic factors such as GDP/km 2 (scale effect, refers to S), capital/labor ratio (composition effect, refers to z

x and GDP per capita (technique effect, refers to z x ) provide a general view on how economic growth affect on pollution outcomes.

At industry level, total pollution from manufacturing, Z, can be written as the sum of pollution from each of its component industries z i . This in turn can be written as the total value shipped from manufacturing, S, multiplied by the sum of each industry's share of total output, (ϕ = x i S ) times amount of pollution per dollar of value shipped in this industry, (e i = z i

x i ) (see, for instance, [START_REF] Levinson | Technology, international trade, and pollution from us manufacturing[END_REF])

Z = z i = S x i S * z i x i ( 8
)
Here I suggest, for example, the decomposition of CO2 emissions and SO2 emissions. 9 The main sources of CO2 emissions are burning fossil fuels, land use land use change and forestry sector, and industrial process. Total amount of CO2 emitted through these 3 sectors can be decomposed into:

z CO2 = S energy x i S energy * z i x i +S LU LU CF x j S LU LU CF * z j x j +S Indus x k S Indus * z k x k (9)
Similarly, the main sources of SO2 emissions are burning fossil fuels and industrial process.

z CO2 = S energy x i S energy * z i x i + S Indus x k S Indus * z k x k (10)
Assuming that country's government decided to implement a new policy of land management in favor of the environment (a afforestation for example), thus this environmental friendly policy can to lower CO2 emissions in this country whereas the level of SO2 emissions remains unchanged. This simple example shows that ignoring the sources of pollutants may be problematic while using the ACT's model.

Simple decomposition of emissions from electricity production and land use conversion

To simplify, I focus therefore on the pollution generated from electricity production (largest single source of CO2 emissions from energy sector) and from the conversion of land use into agricultural land (the largest single source of CO2 emissions from LULUCF sector)

Electricity production

The amount of pollution being released is the sum of pollution emitted through raw non-renewable (raw) sources, renewable sources and nuclear sources. Thus, I suggest that the amount of pollution released from electricity production can be decomposed as follows:

z electric = S

X. Tran

Total emissions from the production of electricity sector z electric depend on: (i) the scale of this sector S electric (scale effect); (ii) the amount of pollution emitted by using non-renewable, renewable and nuclear sources ( zraw sraw ; z renewable s renewable and z nuclear s nuclear , respectively) 10 and (iii) the share of non renewable, renewable and nuclear sources in the production of electricity ( sraw S electric , s renewable S electric and s nuclear S electric respectively.

The composition effect of energy sector can be shown through different sources of energy production. In fact, among several sources of energy production 11 , some generated more emissions than others. Thus, the "composition" of these different sources should have a significant impact of the compositional change of CO2 emissions.

Producing more energy from renewable sources and using fuels with lower carbon contents are ways to reduce carbon emissions.That is the so-called "fuel-switching" phenomenon.

The sources of electricity production can be divided into 3 groups: non-renewable sources (minerals, coal, oil, gas); renewable sources (including hydro-power, biomass, geothermal, solar, wind) and nuclear source. The chart below show that generating electricity from fossil fuels causes GHGs far higher than when using nuclear or renewable sources (particularly solar and wind energy, which provide electricity without giving rise to any carbon dioxide emissions).

The conversion of land use into agricultural land

The amount of CO2 emitted due to the conversion of land use is the sum of pollution emitted through the conversion of land use into agricultural land and settlements.

z LU LU CF = Land * land agriculture Land * z agriculture land agriculture +Land * land settlements Land * z settlements land settlements , ( 12
)
10 For energy sector, technique effect is shown through "energy efficiency". Energy efficiency means "using less energy to provide the same service". Following [START_REF] Patterson | What is energy efficiency?: Concepts, indicators and methodological issues[END_REF], energy efficiency is defined as the ratio of useful output of a process on energy input into a process. Thus, the higher the rate of energy efficiency, the lower emissions generated.

11 Coal, natural gas, nuclear, hydro-power, biomass, geothermal, solar, wind, petroleum, other gases.) The amount of CO2 emitted from land use conversion depend on total land use (scale effect), pollution emitted by using land converted to agricultural activities and settlements ( z agriculture land agriculture and z settlements land settlements respectively), also the share of agricultural land and settlements land in total land.

Model, Adressing Endogeneity and Data

Strategy of estimation

First, I employ a specification similar to those of [START_REF] Cole | Determining the trade-environment composition effect: the role of capital, labor and environmental regulations[END_REF] and [START_REF] Managi | Does trade openness improve environmental quality?[END_REF], where the determinants of emissions per capita at country-level using an ACT framework can be written as:

E it =α 0 + α 1 KL it + α 2 (KL it ) 2 + α 3 IN C it + α 4 (IN C it ) 2 + α 5 KL it IN C it (13) + α 6 T it + α 7 T it Rel.KL it + α 8 T it (Rel.KL it ) 2 + α 9 T it REL.IN C it + α 10 T it (Rel.IN C it ) 2 + α 11 T it Rel.KL it Rel.IN C it + P ollutant it + it ,
Where the income term (IN C) is GDP per capita, KL is the national capital-to-labor ratio, T is the trade intensity (exports plus imports on GDP), REL.KL is country k's capital-to-labor ratio measured relative to the world average, and REL.INC is country k's real income per capita measured relative to the world average.

The income term (IN C) and its quadratic (IN C

2 ) capture the scale-technique effect of economic activity on emissions. The capital-labor term (KL) and its quadratic (KL 2 ) capture the effect of factor abundance on the compositional change of emissions. The interaction term (IN C*KL) refers to the impact of an increase of income per capita on emissions at a given level of capital-labor ratio and vice-versa.

Trade intensity variable T represents the direct effect of trade openness on the compositional changes of pollutions emissions. The interaction between trade intensity and relative capital T *Rel.KL, also its quadratic term T *(Rel.KL) 2 refer to the KLE hypothesis: countries that have comparative advantage in capital-intensive sectors (which are also pollution-intensive sectors) will see its emissions raise with trade liberalization. Alternatively, countries that have comparative advantage in laborintensive sectors will see its emissions fall with trade. The interaction between trade intensity and relative income T *Rel.IN C, also its quadratic term T *(Rel.IN C) 2 refer to the EKE hypothesis: countries that have relative weak environmental regulations (also relative poor countries) will specialize in the pollution-intensive sectors whereas countries with higher environmental stringency will specialize in clean industries.

P ollutant it is a vector that captures variables which are linked to the sources of the concerned pollutant. Because the ACT's model don't account for the sources of pollutant, in equation ( 13), P ollutant it = 0. Equation 13 refers to Model A, which is a standard ACT equation. An estimation of model A can help to (i) compare the results with previous studies and (ii) verify whether an investigation for a standard ACT equation in the case of carbon dioxide is necessary.

Accounting for the share of renewable sources and nuclear source in the production of electricity, also the conversion of land use into agricultural land, the vector P ollutant it in Model A becomes:

P ollutant it = α 12 N uclear -Energy it + α 13 Renewable -Energy it + α 5 Agriculture it ( 14
)
Where N uclear -Energy it and Renewable -Energy it refer to the share of electricity production from nuclear and renewable sources of country i at time t, respectively. The variable Agriculture measures the share of agricultural land on total area. I refer to this amended form of equation ( 13) as Model B.

Adressing Endogeneity

Endogeneity is a common problem that occurs empirical literature on the tradegrowth-environment relationship. [START_REF] Cole | Determining the trade-environment composition effect: the role of capital, labor and environmental regulations[END_REF] argue that the causal direction among each pair out of these three key variables is not evident.

On the one hand, many believe that changes in per capita income will lead to higher demand for environmental quality. Both trade and growth affect per capita income, thus openness to trade and economic activities could have important effect on environmental quality.12

X. Tran

On the other hand, environmental policy can also affect economic activities and trade pattern through higher level of environmental abatement costs. That is, environmental policy could have an adverse effect on trade and economic activities.

In the case of endogeneity problem, instrumental variables (IV) are considered as an estimation that provide consistent parameter estimates. Note that an valid instrument should highly exogenous yet correlated with endogenous variable and uncorrelated with the error term.

First, standard approach to construct an instrument for trade share is proposed by [START_REF] Frankel | Does trade cause growth?[END_REF]. These authors argue that country' geography characteristics (as its distance with others, its population, etc.) should have important effects on trade and are plausibly uncorrelated with other determinants of income. Consequently, geography's component of country's trade seems to be a good instrument of openness to trade.

Second, following [START_REF] Managi | Does trade openness improve environmental quality?[END_REF], to construct an instrument for income per capita, I use a set of variables from the endogenous growth equation. Note that, trade is also endogenous in income equation. That is, country that is rich for reason other than trade may trade more13 . By consequent, geography's component of trade share is used as an instrument for real trade share variable in income equation.

The Frankel & Romer's method to construct an instrument for trade share also the [START_REF] Managi | Does trade openness improve environmental quality?[END_REF] method to construct an instrument for income per capita are reported in Appendix A and B, respectively.

Data

The study covers 99 countries from Asia (24), Europe (23), Africa (29), America (23) from 1971 to 2010. We also observe a greater correlation between capital-labor ratio and GDP per capita, that is rich countries is also capital-intensive countries.

Results

Table 4.A shows results of estimation of Model A.

Considering the non-trade variables: the results indicate a significant and positive relationship between per capita GDP and pollution in all specifications .Thus, scale dominates technique effect in the case of carbon dioxide emissions. The dominant scale for CO2 emissions has been founded in previous studies. The sign of IN C 2 is positive in the case of OLS (insignificant), GLS and Within estimation (at .1 significance level) but statistically negative in the case of IV estimations (at .05 significance level). This result seems to be similar with [START_REF] Managi | Does trade openness improve environmental quality?[END_REF], who found statistically insignificant relationship between IN C 2 and CO2 emissions in the case of OLS but statistically negative in the case of GMM estimation, also [START_REF] Cole | Determining the trade-environment composition effect: the role of capital, labor and environmental regulations[END_REF] who find no evidence of a relationship between quadratic term of income and CO2 emissions using fixed and random effects estimations.

With regard to the composition effect of economies: both two previous studies found that increases in KL also increase per capita CO2 emissions. [START_REF] Managi | Does trade openness improve environmental quality?[END_REF] also found that each additional increase in KL has a diminishing impact (the sign of (KL) 2 is negative with statistical significance). Results in table 4.A indicate that an increase in KL will increase per capita emissions with diminishing effect only while using OLS, Random and Fixed effect estimations. IV estimations yield a statistically insignificant relationship between capital-labor ratio and pollution.

The interaction term IN C * KL refers to the impact of an increase in income per capita at a given level of capital/labor ratio and vice versa. Studies of [START_REF] Cole | Determining the trade-environment composition effect: the role of capital, labor and environmental regulations[END_REF] and [START_REF] Managi | Does trade openness improve environmental quality?[END_REF] provide different results by method of estimation: significantly negative (OLS), insignificant (fixed and random effect), significantly positive (GMM). Results in column (1), ( 2), (3) show that the cross-product of K/L and S is negative with statistically significance while using OLS, Within and GLS as estimation method. However, the coefficient becomes insignificant while using IV estimation (column (4) and ( 5)).

Considering the trade variables: [START_REF] Managi | Does trade openness improve environmental quality?[END_REF]: significantly positive (OLS), insignificant (Within, GLS) and significantly negative (IV, GMM).

Table 4.B presents the results of OLS, Random effect estimation GLS and Random effect with instrumental variables estimation of the emissions of CO2 per capita using the full dataset.

Column (1), ( 2), (3) shows the results of estimations without the set of trade variables proposed by ACT while in column ( 4), ( 5) and ( 6) the set of variables that present the trade-induced composition effect. In all columns, the sign of Nuclear-Energy is negative at the .01 significance level. The result indicates that an increase in the share of nuclear energy as a source of electricity production should help to lower CO2 emissions per capita. GLS estimation and GLS with instrumental variables estimations show that for one unit increase in the share of nuclear energy on total sources of electricity production, CO2 emissions per capita is expected to decrease by 1 unit, holding all other variables constant. Results from OLS estimation (column (1) an ( 4)) seem to underestimate the impact of using nuclear energy compared to other methods of estimation.

Table 4.B also indicates that increasing the share of renewable energy in electricity production (the Renewable-energy term) could lower pollution emissions. However, the significance level of this variable changes substantially over estimation method (at the .01 significance level for OLS estimation, .05 significance level for GLS estimation and .1 significance level for GLS-IV estimation).

The effect of an increase in the share of agricultural land is positive at the .01 significance level in all columns. Results in column ( 2) and ( 3) indicate that, CO2 emissions per capita could increase by 2 units if the share of agriculture in total land increases by 1 unit, holding all other variables constant.

While adding up the set of trade-induced composition effect proposed by ACT, the coefficients of additional variables don't lose their significance, except for Renewableenergy, however the coefficient remains significant. 0.000 0.000 0.000 t statistics in parentheses, + p < 0.10, * p < 0.05, * * p < 0.01, * * * p < 0.001. In IV estimation, trade openness, per capita GDP and its square term are instrumented for using predicted openness, predicted per capita GDP, and predicted its square term, respectively.

Conclusion

While regarding the impact of the composition of an economy on pollution outcomes using a standard ACT specification, I ask whether for a given pollutant there is necessary to adding up variables that link directly to its sources. Thus, I focus on two main sources of CO2 emissions, energy sector and LULUCF sector and therefore controlling for the share of renewable and nuclear energy in total source of electricity production also the share of agricultural land on total land, respectively. Results of estimation show that (1) increasing the share of nuclear source in total source of electricity production lowers the amounts of CO2 emitted; (2) even if the variable that refer to the share of renewable source in total source of electricity production is insignificant in several specifications, but for almost estimations, the results indicate that using renewable source for the production of electricity is environmental friendly. Note that in all specifications, it appears that the magnitude of the former is greater than the latter in term of reduction of CO2 emissions; (3) and, for almost specifications, the results show that increasing the share of agricultural land in total land could cause significant detrimental impact on the environment by generating more carbon dioxide.

This study confirms the positive effect on the environment of fuel-switching for the production of energy; also the negative effect of the conversion land use change into agriculture land on environmental quality. Thus, while the debate over the impact of growth and trade on the environment is still inconclusive, there is a need to focus on the sources of important pollutants, in order to reduce the amount of pollution emitted through human activities. For instance, improving energy efficiency, switching fuel, limiting the deforestation are several solution to cut off carbon dioxide emissions.

Appendix A. Gravity Equation

The Frankel and Romer version of a gravity equation was rewritten as: ln(τ ij /GDP i ) = β 0 + β 1 lnD ij + β 2 lnN i + β 3 lnA i + β 4 lnN j + 5 lnA j + β 6 (L i + L j )

+β 7 B ij +β 8 B ij lnD ij +β 9 B ij lnN i +β 10 B ij lnA i +β 11 B ij lnN j +β 12 B ij lnA j +β 13 B ij (L i +L j )+σ ij , ( 15
)
Where N is their population, A is their area, L is a dummy for landlocked countries, B is a dummy for a common border between two countries. Variables from α 8 to α 13 represent geographical interactions between two countries whether they share a common border. This version of gravity equation can provide more information about bilateral trade on one hand, and on the other hand, to ensure that only country' geographic characteristics have been used to estimate openness to trade between two countries.

The reduced form of equation ( 15) is :

ln(τ ij /GDP i ) = a X ij + e ij , (16)
Here, X ij is the vector of variables representing geographic characteristics of country i and j.

Secondly, these authors suppose that if geography is a component of bilateral trade, than it's true for overall trade. The simple collection of data about geographic characteristics of all countries supports enough information about the level of openness to trade.

Ti = i =j e â X ij , (17)
Equation ( 17) examines the construction of international country's overall trade after obtaining the vector of coefficient α in equation ( 16). International trade's level of country i is simply knowing as an aggregate of its trade with each other countries in the world.

  more use of renewable energy help to reduce CO2 emissions in 25 OECD countries during the period 1980-2010 while adding these factors as additional variables in an quadratic form of the pollution-income equation. Similarly,[START_REF] Bilgili | The dynamic impact of renewable energy consumption on co 2 emissions: A revisited environmental kuznets curve approach[END_REF] argue that renewable energy consumption leads to CO2 reductions for a panel of 17 OECD countries during the period 1977-2010.

Figure 1 :

 1 Figure 1: Global CO2 emissions and CO2 emissions from burning fossil fuel: 1900-2011

Figure 2 :

 2 Figure 2: Greenhouse Gas Emissions from different forms of electricity generation

Table 3 .

 3 Table 1 displays summary statistics for the sample. Data of CO2 emissions per capita, the share of electricity production from nuclear source and renewable source, also the share of agricultural land on total land are from World Development Indicators -World Bank Database. Data on trade intensity (at 2005 constant price), capital, population and real GDP per capita (2005 US$) come from Penn World Table 8.1. 3 reports pairwise correlation for the variables of interest. The table indicates a strong correlation between CO2 emissions per capita and real income per capita (at 76 %), a moderate correlation between CO2 emissions per capita and trade intensity.

		Table 1: Descriptive Statistics
	Variable	Dimension	Obs Mean	Std.	Min	Max
	CO2 emissions per capita	tons	3960 4.317928 5.901309 .016115 67.4166
	GDP per capita	$ 10k	3960 1.017623 1.226213 .016093 9.796309
	Capital/Labor ratio	$ 10k/worker 3960 6.375212 7.217392 .0957449 52.85397
	Relative per capita GDP	Dimensionless 3960 .9850296 1.166349 .015525 9.544766
	Relative Capital/labor ratio Dimensionless 3960 .9263247 1.036363 .0151855 9.698598
	Exports plus imports/GDP Per cent	3960 .6819743 .4864387 .0316	4.3305
	Renewable energy	Per cent	3360 .0217085 .0511931 0	.6259649
	Nuclear energy	Per cent	3360 .0475197 .1273073 0	.7909981
	Agricultural Land	Per cent	3880 .3954565 .2140539 .01	.9055524

Table 2 :

 2 Pairwise correlation of variables of interest

		CO2 per capita GDP per capita Trade Intensity Capital-labor ratio
	CO2 per capita	1.0000
	GDP per capita	0.7607	1.0000
	Trade Intensity	0.2857	0.2922	1.0000
	Capital-labor ratio 0.7682	0.9078	0.3013	1.0000

Table 4

 4 statistics in parentheses, + p < 0.10, * p < 0.05, * * p < 0.01, * * * p < 0.001. In IV estimation, trade openness, per capita GDP and its square term are instrumented for using predicted openness, predicted per capita GDP, and predicted its square term, respectively.

	.A: The determinants of CO2 emissions per capita
		OLS	Within	GLS	Within	GLS
	Individual Effect	No	Fixed	Random	Fixed	Random
	Intrument	No	No	No	Yes	Yes
	INC	1.401 * * *	0.722 * * *	0.822 * * *	2.112 * * *	2.186 * * *
		(12.61)	(8.92)	(10.24)	(7.56)	(8.19)
	IN C 2	0.0127	0.0545 *	0.0455 *	-0.344 * *	-0.328 * *
		(0.31)	(2.55)	(2.11)	(-2.94)	(-2.94)
	KL	0.355 * * *	0.140 * * *	0.150 * * *	-0.0751 +	-0.0584
		(19.60)	(10.70)	(11.47)	(-1.70)	(-1.41)
	KL 2	-0.00689 * * * -0.00113 -0.00147 * 0.00317 + 0.00275 +
		(-5.71)	(-1.55)	(-2.00)	(1.92)	(1.76)
	KL*INC	-0.0667 * * * -0.0428 * * * -0.0433 * * * -0.0161	-0.0195
		(-5.15)	(-5.81)	(-5.84)	(-0.72)	(-0.89)
	T	0.918 * * *	0.357 * * *	0.338 * * *	1.716 * * *	1.542 * * *
		(16.72)	(8.80)	(8.32)	(7.46)	(7.49)
	T*Rel INC	-0.518 * * *	-0.320 * * *	-0.364 * * * -1.684 * * * -1.675 * * *
		(-4.58)	(-5.20)	(-5.88)	(-6.73)	(-7.05)
	T * (RelIN C) 2	0.0329	0.0267	0.0358 *	0.375 * * *	0.364 * * *
		(1.02)	(1.63)	(2.16)	(3.97)	(4.05)
	T*Rel KL	-0.819 * * *	-0.247 * * *	-0.237 * * *	0.653 * *	0.645 * *
		(-7.32)	(-3.90)	(-3.70)	(2.67)	(2.79)
	T * (RelKL) 2	0.192 * * *	0.0860 * * * 0.0938 * * *	0.0233	0.0321
		(5.10)	(4.79)	(5.17)	(0.57)	(0.83)
	T*Rel INC*Rel KL	0.125 +	0.000821	-0.00643	-0.199 +	-0.193 +
		(1.83)	(0.03)	(-0.20)	(-1.78)	(-1.80)
	cons	-1.939 * * *	-0.638 * * *	-0.723 * * * -1.256 * * * -1.274 * * *
		(-47.74)	(-17.52)	(-8.34)	(-15.10)	(-8.35)
	N	3960	3960	3960	3960	3960
	adj. R 2	0.738	0.178
	Hausman test		358.67		20.51
	Prob¿chi2		0.000		0.038

t

  Table4.A predicts a strong and significantly positive correlation between trade intensity and emissions (at .01 significance level for all specifications). The coefficients of estimation of the ERE are very similar with those of[START_REF] Managi | Does trade openness improve environmental quality?[END_REF]: the sign of T it Rel.IN C it is negative with statistical significance for all specifications at an increasing rate (the sign of T it (Rel.IN C it ) 2 is negative for all specifications. The sign of T it Rel.KL it changes with estimation method: positive in the case of Instrumental Variable estimation (both fixed and random effects), negative in the case of OLS, GLS and Within Estimation; and at increasing rate : the sign of T it (Rel.IN C it ) 2 is positive for all specifications (but insignificant in the case of IV estimations).Finally, the results for T it Rel.KL it Rel.IN C it are quite similar with those of

Table 4 .

 4 B: The role of Nuclear Energy, Renewable Energy and Land-Use Change on CO2 emissions

		Model A		Model B
		Within	2SLS	Within	2SLS	Within	2SLS
		(1)	(2)	(3)	(4)	(5)	(6)
	INC	0.722 * * *	2.186 * * *	0.637 * * *	1.963 * * *	0.811 * * *	1.485 * * *
		(8.92)	(8.19)	(8.66)	(6.09)	(10.21)	(4.50)
	INC 2	0.0545 *	-0.328 * *	0.0620 * *	-0.491 * *	0.115 * * *	-0.912 * * *
		(2.55)	(-2.94)	(3.22)	(-3.15)	(4.73)	(-3.94)
	KL	0.140 * * *	-0.0584	0.138 * * *	-0.0575	0.137 * * *	-0.0285
		(10.70)	(-1.41)	(11.43)	(-1.10)	(11.25)	(-0.88)
	KL 2	-0.00113	0.00275 + -0.000683 -0.000813 0.000799	-0.0113 * *
		(-1.55)	(1.76)	(-1.01)	(-0.47)	(1.11)	(-3.08)
	KL*INC	-0.0428 * * * -0.0195	-0.0423 * * * 0.0378	-0.0638 * * * 0.185 * *
		(-5.81)	(-0.89)	(-6.19)	(1.27)	(-8.02)	(2.84)
	T	0.357 * * *	1.542 * * *	0.301 * * *	1.582 * * *	0.232 * * *	1.477 * * *
		(8.80)	(7.49)	(7.39)	(6.32)	(5.56)	(4.31)
	T*Rel INC	-0.320 * * *	-1.675 * * * -0.232 * * *	-1.565 * * * -0.303 * * *	-0.780 * *
		(-5.20)	(-7.05)	(-4.20)	(-5.13)	(-4.83)	(-2.64)
	T * (RelIN C) 2	0.0267	0.364 * * *	0.00503	0.445 * * *	-0.0682 * *	0.830 * * *
		(1.63)	(4.05)	(0.35)	(3.58)	(-3.14)	(3.84)
	T*Rel KL	-0.247 * * *	0.645 * *	-0.311 * * *	0.523 +	-0.254 * * *	-0.0172
		(-3.90)	(2.79)	(-5.48)	(1.84)	(-4.28)	(-0.08)
	T * (RelKL) 2	0.0860 * * *	0.0321	0.0693 * * *	0.107 *	0.000792	0.516 * *
		(4.79)	(0.83)	(4.24)	(2.46)	(0.04)	(3.23)
	T*Rel INC*Rel KL 0.000821	-0.193 +	0.0311	-0.361 *	0.182 * * *	-1.228 * *
		(0.03)	(-1.80)	(1.05)	(-2.47)	(4.22)	(-3.07)
	Nuclear Energy			-1.222 * * *	-1.088 * * * -1.252 * * *	-1.492 * * *
				(-12.12)	(-6.65)	(-11.56)	(-7.99)
	Renewable Energy			-0.647 * * *	-0.531 * * * -0.647 * * *	-0.488 * * *
				(-13.49)	(-7.59)	(-13.51)	(-6.73)
	Agriculture					1.796 * * *	1.046 * *
						(9.50)	(3.19)
	cons	-0.638 * * *	-1.274 * * * 0.0195	-0.588 * * * -0.805 * * *	-0.886 * * *
		(-17.52)	(-8.35)	(0.45)	(-5.95)	(-9.35)	(-6.46)
	N	3960	3960	3280	3280	3200	3200
	adj. R 2	0.178		0.9586		0.9584
	Hausman	199.54		517.21		143.09
	Prob¿chi2

Table 5

 5 p < 0.05, * * p < 0.01, * * * p < 0.001

		: Income Equation
		lnIncome
	L.lnIncome	0.798 * * *
		(39.08)
	lnT	0.100 * * *
		(9.32)
	lnP	0.0327
		(1.44)
	lnKL	0.0329 +
		(1.76)
	lnSch	-0.165 *
		(-2.57)
	year	0.00180 * *
		(2.69)

t statistics in parentheses + p < 0.10, *

for instance, R.[START_REF] Houghton | Emissions of carbon from forestry and land-use change in tropical asia[END_REF] found that the emissions of carbon from land-use change in the 1980s accounted for approximately 75 % of the regionŠs total carbon emissions in tropical Asia

Most of these LULUCF emissions are intimately connected to agriculture, as many resulted from deforestation caused by expansion of farms into tropical forests[START_REF] Van Der Werf | Co2 emissions from forest loss[END_REF]). The conversion of land from forested to agricultural land can have a wide range effect on CO2 emissions. As introduced by the EPA (United States Environmental Protection Agency), "in countries where large areas of forest land are cleared, often for agricultural purposes or for settlements, the LULUCF (Land use, Land use change, Forestry) sector can be a net source of GHGs emissions".

see also[START_REF] Levinson | Technology, international trade, and pollution from us manufacturing[END_REF] for decomposition analysis at industry-level

see, for example,[START_REF] Jones | Endogenous policy choice: the case of pollution and growth[END_REF]

see, for example,[START_REF] Dollar | Trade, growth, and poverty*[END_REF]

Appendix B. Income Equation

We use an income equation similar to [START_REF] Managi | Does trade openness improve environmental quality?[END_REF]. Following the endogenous growth literature, country's income depend on its lagged values, trade openness, capital-labor ration, population and humain capital. Income equation can be written as:

where P is the population, Sch is index of human capital based on school attendance years, µ is the error term.

Table 5 shows results of the difference GMM estimation using instrumental variables for the income equation 18. As mentioned in the introduction, I treat trade share as endogenous in the income equation and therefore use predicted value of trade share as an instrument for real trade intensity.