dynamical systems [START_REF] Vallée | Euclidean dynamics[END_REF].

For the -Euclid algorithm, it is natural to choose, as the length of the input, the sum of the length of its components. Changing the input length yields a different probabilistic behavior, even when = 2. We show the following: in the case = 2, the mean number of iterations remains linear with respect to the (new) length but the distribution of number of iterations is now asymptotically uniform. In the case ≥ 3, our analysis exhibits a strong difference between the first phase and the following phases. In the first phase, the number of iterations has a linear mean and follows a beta law (the same as the law followed by the minimum of -1 reals i.i.d. in the unit interval). In the following phases, the number of iterations is constant on the average and follows a geometric law. These results can be expected, as Knuth wrote "In most cases, the length of the partial gcd decreases rapidly during the first few phases of the calculation, and this will make the remainder of the computation quite fast" [START_REF] Knuth | of The Art of Computer Programming[END_REF]. Our analysis shows that, in most cases, "almost all the calculation" is done during the first phase. This will make possible to compare (in the extended version of this paper) the -Euclid algorithm to another strategy proposed in [START_REF] Cooperman | Gcd of many integers[END_REF].

Plan of the paper. Sections 2, 3, 4 are devoted to the polynomial case, and Section 5 to the integer case. Section 2 describes the algorithm and states the main results, then, Sections 3 and 4 are devoted to the proofs, mainly based on analytical combinatorics: we use generating functions, built in Section 3, and used in Section 4. Section 5 explains the strong similarity between the two cases (polynomial and integer); it presents the useful generating functions (here of Dirichlet type), states the main results (only for the averagecase) and briefly describes the main steps of the proof. We also explain why distributional results may be expected.

MAIN RESULTS FOR POLYNOMIALS.

We consider the ring Fq[X] of polynomials over the finite field Fq with q elements, and the degree of a non zero polynomial x is denoted by d(x).

The plain -Euclid algorithm on Fq[X] deals with a sequence of nonzero polynomials and computes their greatest common divisor y via a sequence of -1 gcd computations between two polynomials, as we previously explained.

Each phase is a 2-Euclid Algorithm which performs a gcd computation between two polynomials, with a sequence of Euclidean divisions, as recalled as follows:

The Euclid Algorithm on (a1, a2) with d(a1) > d(a2)

a1 = m1a2 + a3 0 < d(a3) < d(a2) a2 = m2a3 + a4 0 < d(a4) < d(a3) . . . = . . . + ap-1 = mp-1ap + ap+1 0 < d(ap+1) < d(ap) ap = mpap+1 + 0
Then, gcd(a1, a2) is the last non-zero remainder ap+1. It can be chosen monic. The number of steps (here equal to p) is one of the main parameters of interest.

This paper aims at precisely understanding the random behavior of the plain algorithm. Since the algorithm is a succession of phases, it is important to describe each phase of index k (k ∈ [1.. -1]), with the following parameters:

(i) the number L k of divisions performed during the k-th phase, (ii) the degree D k of the gcd y k at the beginning of the k-th phase. We are also interested in the analysis of global parameters.

(iii) The algorithm may be interrupted as soon as D k = 0, and Π is the number of useful phases. Namely: Π = 0 if D1 = 0 and Π := max{k|D k > 0} if D1 > 0. (iv) The total number of divisions of the interrupted algorithm is defined as

L = 0 if Π = 0 and L = L1 + L2 + . . . + LΠ if Π > 0.
The possible inputs are all the sequences x formed of polynomials, and we limit ourselves to monic polynomials, without loss of generality. Then the set of inputs is Ω = U , where U is the set of monic polynomials. Here, the size of the input x is the total degree of the sequence (and not the maximum, as it is often the case), and we let

d(x) = d(x1, x2, . . . , x ) := d(x1) + d(x2) + . . . + d(x ).
The subset Ωn formed with the inputs of size n is a finite set, and it is endowed with the uniform probability. Now, the parameters of interest become random variables and we are interested in their probabilistic features.

With analytic combinatorics methodology, we prove the following two results. The first one deals with the expected values, whereas the second one describes asymptotic limit laws.

Average-case analysis.

Theorem 1 below exhibits a strong difference between the first phase and the following ones. It shows that, on average, the first phase performs a linear number of iterations which involves the entropy 2q/(q -1) of the Euclid dynamical system; then, even if the degree of the first gcd is linear with respect to the input size, the degree of the gcd is proven to be of constant order after the first phase on the average. Then, the number of divisions L k which will be performed in the following phases, together with the degrees D k of the following gcd's will be of constant order. Theorem 1. [Expectations]. When the set Ωn is endowed with the uniform distribution, the following holds:

(a) The expectation of the number of iterations L1 during the first phase is linear with respect to the size n and satisfies

En[L1] = q -1 2q n + 3q + 1 4q + O 1 n . (b) For any k ∈ [2.. -1]
, the expectation of the number of iterations L k during the k-th phase is asymptotic to a constant, and satisfies

En[L k ] = q k -1 q k -q 1 + O 1 n .
(c) The expectation of the degree of the first polynomial x1 is linear with respect to the size n and satisfies

En[D1] = n . (d) For any k ∈ [2.. -1]
, the expectation of the degree D k of the gcd y k at the beginning of the k-th phase is asymptotic to a constant, and satisfies

En[D k ] = 1 q k-1 -1 1 + O 1 n .

Limit Laws.

The following result makes more precise the results obtained in Theorem 1, and explains more deeply the difference between the first phase (k = 1) and the following phases. For k = 1, the expected degrees of the first two polynomials x1 and x2 are linear, and the number of divisions is closely related to min(d(x1), d(x2)). Then, it is natural to expect beta laws for the first phase, more precisely a beta law of parameter (1, -1), since it is the law of the minimum Y of -1 random variables i.i.d. on the unit interval, which satisfies P[Y ≥ x] = (1 -x) -1 . Such a law has a density equal to ( -1)(1 -x) -2 . For = 2, this is the uniform law. For the following phases, geometric laws are expected, since the means of the gcd degrees are of constant order.

The next result describes the asymptotic distribution of L k [Assertion (a)] and D k [Assertion (b)]. At first glance, the results do not seem to heavily depend on the index k of the phase. However, it is not true, since the two ratios p k := (q -1)/(q k -1) (in case L), r k := q 1-k (in case D), are equal to 1 for k = 1 and strictly less than 1 for k ≥ 2.

Theorem 2. [Limit laws.] When the set Ωn is endowed with the uniform distribution, the following holds: (a) The number of iterations L1 during the first phase asymptotically follows a beta law of parameter (1, -1) on the interval [0, (q -1)/(2q)] whereas, the number of iterations L k during each following phase asymptotically follows a geometric law with ratio p k := (q -1)/(q k -1). One has:

Pn [L k > n/(k + 1)] = 0, for any k. For any k, the distribution of L k satisfies when n → ∞, Pn[L k > m] = q -1 q k -1 m 1 + O m n for m = o(n),
and for m/n → c with c ∈]0,

1 k+1 q k -1 q k [ Pn[L k > m] = q -1 q k -1 m 1 - (k + 1)q k q k -1 c -1 1 + O 1 n .
(b) The degree D1 of the first polynomial x1 asymptotically follows a beta law of parameter (1, -1) on the interval [0, 1], whereas the degree D k of the gcd y k at the beginning of each following phase asymptotically follows a geometric law with ratio r k := q 1-k . One has : Pn [D k > n/k] = 0 for any k. For any k, the distribution of D k satisfies when n → ∞,

Pn[D k ≥ m] = q (1-k)m 1 + O m n for m = o(n),
and for m/n → c with c ∈]0,

1/k[ Pn[D k ≥ m] = q (1-k)m (1 -kc) -1 1 + O 1 n .

Global parameters.

The interrupted algorithm stops as soon as the gcd y k is of degree 0. Let Π(x1, . . . , x ) be the number of useful phases. Since the event [Π ≥ k] coincides with the event [D k ≥ 1] for k ∈ [1.. -1], Theorem 2 leads to the following: Theorem 3. When the set Ωn is endowed with the uniform distribution, the distribution of the number Π of useful phases satisfies Pn

[Π ≥ 0] = 1, Pn[Π ≥ ] = 0, Pn[Π ≥ k] = q 1-k 1 + O 1 n for k ∈ [1.. -1].
The following result studies the total number L of divisions performed by the interrupted version (see the proof in the appendix). With Theorems 1 and 3, the mean is easy to study. Moreover, Theorems 1 and 2 prove that the variance of L1 is quadratic whereas the variance of L k (for k ≥ 2) is of constant order. Then, the linearity of the mean with the Markov inequality entail that L has the same asymptotic distribution as L1.

Theorem 4. When the set Ωn is endowed with the uniform distribution, the total number of divisions L performed by the interrupted version of the -Euclid algorithm has an expected value En[L] equal to

q -1 2q n + 3q + 1 4q + -1 k=2 q q k + q -1 q k -1 + O 2 n .
Moreover, the number L asymptotically follows a beta distribution of parameter (1, -1) on the interval [0, (q -1)/(2q)].

GENERATING FUNCTIONS

General setting

We use the analytic combinatorics methodology, and deal with its main tool, the generating functions. We use a variable zi to mark the degree of the i-th polynomial xi, and the generating function F (z1, z2, . . . z ) of the set Ω = U , relative to the size d, is defined as

F (z1, z2, . . . z ) := x∈U z d(x 1 ) 1 z d(x 2 ) 2 . . . z d(x ) .
It is equal to the product U (z1) U (z2) . . . U (z ), where U (z) is the generating function of the set U of the monic polynomials relative to the size d, namely

U (z) = x∈U z d(x) = n≥0 q n z n = 1 1 -qz .
Most of the time, we limit ourselves to the case when all the variables zi are equal, and we write F (z) instead of

F (z, • • • , z).
For studying a parameter (or a cost C) on Ω = U d , a main tool is the bivariate generating function relative to some cost C, obtained by introducing a further variable u to mark the cost, and defined as

F (z, u) := x∈U d z d(x) u C(x) .
We are first interested in the mean value of parameter C, and we deal with the cumulative generating function

F (z) := ∂F ∂u (z, u)|u=1, and
En[C] = [z n ] F (z) [z n ]F (z) . (1)
The probability distribution of the cost C can be studied with the generating function F (z, u), via the relation

Pn[C = i] = [z n u i ]F (z, u) [z n ]F (z) .
Then, the probabilities Pn[C ≥ m] are expressed with the "cumulative bivariate generating functions" defined as

F [m] (z) := i≥m [u i ]F (z, u), and Pn[C ≥ m] = [z n ] F [m] (z) [z n ]F (z) .
(2)

Another expression for the generating function

We first obtain an alternative expression for the generating function F (z) with a product of -1 factors, each of them describing a phase of the algorithm.

Proposition 1. The generating function of the set Ω = U with the size equal to the total degree decomposes as

F (z) = U (z) = U (z ) • -1 k=1 T (z, z k )
and involves the phase-function T defined as

T (z, t) = U (z) + U (t) -1 1 -G(zt) , (3)
the generating function U (z) of monic polynomials, and the generating function G(z) of general polynomials with strictly positive degree, i.e.,

U (z) = 1 1 -qz , G(z) = (q -1)qz 1 -qz .
Proof. The Euclid algorithm first compares the degrees of x1 and x2. There are three cases:

d(x1) = d(x2), d(x1) > d(x2), d(x1) < d(x2
). In the first case, there is an extra subtraction, which can be viewed as a division with a quotient equal to 1.

In all the cases, the gcd y := gcd(x1, x2) together with the sequence of quotients (m1, m2, . . . mp) completely determines the input pair (x1, x2). More precisely, one writes (x1, x2) = (y x1, y x2) with a coprime pair ( x1, x2) and the execution of the Euclid algorithm on the pair ( x1, x2) produces the same sequence (m1, m2, . . . mp) as the pair (x1, x2). The first quotient m1 is monic (this is due to the fact that x1 and x2 are monic) and the remainder of the sequence Σ = (m2, . . . mp) is formed with general polynomials mi (no longer monic) with d(mi) ≥ 1. As previously, the total degree of Σ is d(Σ) = d(m2) + . . . + d(mp).

We now focus on the first quotient m1, and we consider the three following possible cases:

(a) If d(x1) = d(x2), then m1 = 1. (b) If d(x1) > d(x2) then d(m1) ≥ 1, d( x2) = d(Σ), d( x1) = d(m1) + d(Σ). (c) If d(x1) < d(x2) then d(m1) ≥ 1, d( x1) = d(Σ), d( x2) = d(m1) + d(Σ)
. All these remarks provide an alternative expression of the product U (z1) U (z2). Indeed, we use the two relations z

d(x 1 ) 1 z d(x 2 ) 2 = (z1z2) d(y) • z d( x 1 ) 1 z d( x 2 ) 2 , x 1 , x 2 z d( x 1 ) 1 z d( x 2 ) 2 = 1 + m 1 z d(m 1 ) 1 + z d(m 1 ) 2 Σ (z1z2) d(Σ) ,
together with the conditions previously described on the first quotient m1, the sequence Σ and the gcd y. The first factor gives rise to the generating function

1 + (U (z1) -1) + (U (z2) -1) = U (z1) + U (z2) -1
which involves the generating function U (z) of monic polynomials, whereas the second factor is expressed with the generating function G(z) of general polynomials with a strictly positive degree, under the form 1/(1 -G(z1z2)). We have then proven the following alternative form for the product

U (z1) U (z2) = U (z1z2) • T (z1, z2
). When we replace this expression into the total product U (z1) U (z2) . . . U (z ) = F (z1, z2, . . . , z ) and iterate the transformation, we obtain an alternative expression for the generating function F (z1, z2, . . . , z ) with a product of -1 factors, each of them involving the phasefunction T at points z k and t

k = z1 . . . z k F (z1, z2, . . . z ) = U (t ) • -1 k=1 T (z k+1 , t k ).
It can be useful in some studies to keep all the variables zi, but here, we let z1 = z2 = . . . = z , and we obtain an expression of the generating function F (z).

Generating functions for parameters of interest

When studying the parameter L k (number of steps in the k-th phase), we use an extra variable u which marks each step of the k-th iteration, and we deal with the generating function

T (z, t, u) = u U (z) + U (t) -1 1 -uG(zt) with t = z k , which replaces T (z, z k ) inside F (z).
When studying the parameter D k (degree of the gcd at the beginning of the k-th phase), we use an extra variable u which marks the degree of the gcd y k , and we deal with the generating function

U (t, u) = 1 1 -qut with t = z k , which replaces U (z k ) inside F (z). Finally: Proposition 2. For any k ∈ [1.. -1]
, the bivariate generating function L k (z, u) relative to the number of divisions during the k-th phase satisfies

L k (z, u) = U (z) • T (z, z k , u) T (z, z k ) .
For any k ∈ [1.. -1], the bivariate generating function D k (z, u) relative to the degree of the k-th gcd y k at the beginning of the k-th phase satisfies

D k (z, u) = U (z) • U (z k , u) U (z k ) .
With Proposition 1, the functions T (z, z k , u) and U (z k , u) admit precise expressions, and Proposition 2 leads to the expression of the bivariate generating functions L k (z, u) and

D k (z, u) L k (z, u) U (z) = u 1 -G(z k+1 ) 1 -uG(z k+1 ) , D k (z, u) U (z) = 1 -qz k 1 -quz k . (4)
Finally, taking the derivative with respect to u, we obtain the cumulative generating functions

L k (z) U (z) = 1 -qz k+1 1 -q 2 z k+1 , D k (z) U (z) = qz k 1 -qz k . ( 5
)
Extracting in (4) the coefficient of [u i ] in the bivariate generating functions and taking the sum over i ≥ m gives

L [m] k (z) U (z) = G(z k+1 ) m-1 , D [m] k (z) U (z) = (qz k ) m . ( 6
)
4. FINAL STEPS FOR THE PROOFS.

We have obtained in ( 5) and ( 6) the expressions of useful generating functions, that are always here fractional functions. It is then possible to use (1) and ( 2) to obtain an exact expression of the expectation and probability distribution of the parameters D k and L k . However, we are mainly interested in the asymptotic behaviour (as n → ∞) of these probabilistic features. Singularity analysis describes the possible transfer between the behavior of a generating function, viewed as an analytic function, near its dominant singularity (the singularity closest to 0) and the asymptotic behavior of its coefficients. More precisely, the position and the nature of the dominant singularity play a fundamental role.

Average-case analysis.

Here, the main tools are the cumulative generating functions [START_REF] Delange | Généralisation du théorème d'ikehara[END_REF]. They both admit a dominant pole in z = 1/q but the order of the pole is different according to the phase. For the first phase (k = 1), the pole z = 1/q is of order + 1 whereas for the other phases (k ≥ 2), this pole remains of order .

We now use the following result, here quite trivial since we deal with rational fractions.

Lemma 1. Consider a function f (z) = g(z)/(1-qz) j with j ≥ 2 which involves a function g(z) which is analytic in the disk |z| > 1/q and satisfies g(1/q) = 0. Then,

[z n ]f (z) = g 1 q n + j -1 j -1 q n -g 1 q n + j -2 j -2 q n-1 ,
with a remainder term in O n j-3 q n . The previous estimate also holds if g admits simple isolated poles on the punctured circle {|z| = 1/q, z = 1/q}, as soon as j ≥ 3.

We first consider the case when the phase index k is at least equal to 2. In this case, with the expressions of the cumulative generating functions D k (z) and L k (z) given in [START_REF] Delange | Généralisation du théorème d'ikehara[END_REF], the lemma applies with j = and g(z) = 1 -qz k+1 1 -q 2 z k+1 , g

1 q = q k -1 q k -q (case L), g(z) = qz k 1 -qz k g 1 q = 1 q k-1 -1 (case D).
Consider now the case when the phase index k equals 1. In this case, the integer j equals + 1, and

g(z) = qz (case D), g(z) = 1 -qz 2 1 + qz (case L).
In both cases L and D, the lemma applies. In case D, we obtain:

[z n ] D1(z) = q n n + -1 .
We remark that, in case L, the function g admits z = -1/q as a simple pole and we obtain [

z n ] L1(z) = = q -1 2q q n n + + q + 3 4 q n-1 n + -1 -1 +O n -2 q n .
In all the cases, the normalization by card(Ωn) = n+ -1 -1

q n proves Theorem 1.

4.2 A general framework for limit laws.

The "cumulative bivariate generating functions" which are useful for the study of the distributions at the k-th phase are, due to (6), of the form

U (z) • A k (z) m ,
where the function A k (z) depends on the index of the phase, and the type of parameter. One has

A k (z) = qz k (type D), (7)
A k (z) = (q -1)qz k+1 1 -qz k+1 = G(z k+1 ) (type L). (8)
In all the cases, the generating functions are expressed as a product of the function U (z) which has a pole of order at z = 1/q, with a "large power" of a function A(z). The term "large power" is used since the exponent m may depend on the size n. As the following proposition shows it, there are two main cases, according as the value of A at z = 1/q is equal to 1 or not. The first case happens for k = 1 and leads to beta distribution, whereas the second case happens for k ≥ 2 and leads to geometric distributions.

The following result is proven in the appendix.

Proposition 3. Consider the function Then, the coefficient of z n in F [m] (z) satisfies the following:

F [m] (z) = 1 (1 -z) • A(z) m ,
(a) When m/n → 0, then [z n ]F [m] (z) = n -1 ( -1)! • a m 1 + O m n .
(b) When m/n → c for some c ∈]0, a/b[ then

[z n ]F [m] (z) = n -1 ( -1)! a m 1 - b a c -1 1 + O 1 n .

End of the proof of Theorem 2.

The probabilities Pn[L k > m], Pn[D k > m] are related to the coefficient of z n in the generating functions described in [START_REF] Dixon | The number of steps in the Euclidean algorithm[END_REF], and the functions A k (z) of interest are given in [START_REF] Dolgopyat | On decay of correlations in Anosov flows[END_REF] and [START_REF] Flajolet | Analytic Combinatorics[END_REF]. Since, in case D, A k (z) is multiple of z k , and, in case L, A k (z) is multiple of z k+1 , we first deduce the equalities

Pn [D k > n/k] = 0,
Pn [L k > n/(k + 1)] = 0. After a change of variable z → z/q, the hypotheses of Proposition 3 are fulfilled for z → A k (z/q), and one has,

a k = q 1-k , b k a k = k (case D), a k = q -1 q k -1 , b k a k = (k + 1) q k q k -1 , (case L).
For k = 1, the constants a k are equal to 1, whereas they are strictly less than 1 for k ≥ 2. Finally, the normalization by card(Ωn),

card(Ωn) = n -1 ( -1)! q n 1 + O 1 n ,
proves Assertions (a) and (b) of Theorem 2.

THE INTEGER CASE.

We now briefly explain how a similar study can be performed in the case of the Euclid algorithm on integers. As usual (see for instance [START_REF] Lhote | Gaussian laws for the main parameters of the Euclid algorithms[END_REF][START_REF] Vallée | Euclidean dynamics[END_REF]), the polynomial study shows the road, and similar results are expected in the integer study, even if they are often more difficult to obtain and less precise.

The plain algorithm on integers.

In the integer case, the -plain Euclid algorithm has exactly the same structure as in the polynomial case. It is composed with -1 phases, each of them being the Euclid algorithm which performs the gcd computation between two integers. Its execution is described as in Figure of Section 2, and the degree is just replaced by the integers themselves. The main parameters of interest are the same, and D k is now the length of the gcd y k , namely its logarithm.

The possible inputs are all the sequences x formed of integers, and we limit ourselves to positive integers, without loss of generality. Then the set of inputs is Ω = N , where N is the set of positive integers. Here, the size of the input x equals the product of its components x1 • x2 • . . . • x , and the length of the input is the sum of the lengths of its components.

Here again, the subset ΩN formed with the inputs of size at most N is a finite set, and it is endowed with the uniform probability.

Dirichlet generating functions.

The generating functions are now of Dirichlet type, and the basic one is the generating function of the set N . We deal with -uples x of integers x = (x1, x2, . . . x ) and consider the generating function

F (s1, s2, . . . s ) = x∈N 1 x s 1 1 1 x s 2 2 . . . 1 x s = ζ(s1) . . . ζ(s ),
where the ζ function is the generating function of N,

ζ(s) := n≥1 1 n s .
The main case of interest s1 = s2 = . . . s = s gives rise to F (s) := F (s, . . . , s) = ζ(s) . However, we begin with the generic case when the -uple s = (s1, . . . , s ) is general.

The first step derives, as previously, a decomposition for the product ζ(s1) ζ(s2), of the form

ζ(s1) ζ(s2) = ζ(s1 + s2) • T (s1, s2),
where T (s1, s2) is the phase generating function which describes the Euclid algorithm on two integers. In the integer case, the 2-Euclid algorithm is described with the transfer operator, related to the underlying dynamical system, and introduced by Ruelle in [START_REF] Ruelle | Thermodynamic Formalism[END_REF] in a general setting. This operator deals here with the Gauss map S defined by S(x) := 1/x -[1/x], and the set G of its inverse branches, namely

G := hm(t) : t → 1 m + t ; m ≥ 1 .
It depends on a complex parameter s, and acts for s > 1 on functions defined on the unit interval as

Gs[f ](t) = h∈G |h (t)| s/2 f • h(t). (9)
The following result provides an analog of Proposition 1.

Proposition 4. The generating function of Ω = N with the size "product of inputs" can be written as

F (s) = ζ(s) = ζ( s) • -1 k=1 T (s, ks),
and involves the phase-function T defined as

T (s, t) = 1 2 (I -Gs+t) -1 • (Gs + Gt)[1](0) , ( 10
)
where Gs is the transfer operator relative to the Euclid dynamical system, defined in [START_REF] Flajolet | Continued fraction algorithms, functional operators, and structure constants[END_REF].

x2 x1 = h • g(0), x3 x2 = g(0).
Here, h := hm 1 is related to the first quotient and g = hm 2 • hm 3 • . . . • hm p is related to the sequence (m2, m3, . . . , mp).

Since the two pairs ( x1, x2) and ( x2, x3) are coprime, the denominators of each rational are expressed with derivatives,

1 x 2 1 = |(h • g) (0)| = |h (g(0))| • |g (0)|, 1 x 2 2 = |g (0)|.
Then, in the case when x1 ≥ x2, the sum

x 1 ≥ x 2 1 x s 1 1 1 x s 2 2 = 1 + h,g |h (g(0))| s 1 /2 • |g (0)| (s 1 +s 2 )/2 ,
can be expressed with the transfer operator Gs as2

1 2 (I -Gs 1 +s 2 ) -1 • Gs 1 [1](0) + 1 .
The case x2 ≥ x1 can be dealt with exchanging the roles of x1 and x2. Finally, the relations

x 1 ,x 2 1 x s 1 1 1 x s 2 2 = v 1 y s 1 +s 2 x 1 , x 2 1 x s 1 1 1 x s 2 2 entail the equality ζ(s1)ζ(s2) = ζ(s1 + s2) • T (s1, s2),
where T is defined in [START_REF] Friesen | The statistics of continued fractions for polynomials over a finite field[END_REF]. When we replace this expression into the total product ζ(s1) ζ(s2) . . . ζ(s ) = F (s1, s2, . . . , s ), and iterate the transformation, we obtain an alternative expression for the generating function F (s1, s2, . . . , s ) with a product of -1 factors, each of them involving the phasefunction T at points s k and

t k = s1 + . . . + s k , F (s1, s2, . . . s ) = ζ(t ) • -1 k=1 T (s k+1 , t k ).
It may be useful in some studies to keep all the variables si, but, here again, we let s1 = s2 = . . . = s = s, and we obtain the expression of the generating function F (s).

Dirichlet generating functions for parameters of interest

When studying the parameter L k (number of steps in the k-th phase), we use an extra variable u which marks each step of the k-th iteration, and we deal with the generating function 2T (s, t, u) = u(1 -uGs+t) -1 • (Gs + Gt) [START_REF] Baladi | Euclidean algorithms are gaussian[END_REF](0), with t = ks, which replaces 2T (s, ks) inside F (s). When studying the parameter D k (length of the gcd at the beginning of the k-th phase), we use an extra variable u which marks the length of the gcd y k , and we deal with the generating function

ζ(t, u) = n≥1 u log n n t = ζ(t -log u) with t = ks,
which replaces ζ(ks) inside F (s). Finally, we obtain an analog of Proposition 2.

Proposition 5. For any k ∈ [1.. -1], the bivariate generating function L k (s, u) relative to the number of divisions during the k-th phase satisfies

L k (s, u) = ζ(s) • T (s, ks, u) T (s, ks) .
For any k ∈ [1.. -1], the bivariate generating function D k (z, u) relative to the size of the k-th gcd y k at the beginning of the k-th phase satisfies

D k (s, u) = ζ(s) • ζ(ks -log u) ζ (ks) .
With Proposition 4, the function T (s, ks, u) admits a precise expression. Using Proposition 5, and taking the derivative with respect to u, we obtain the cumulative generating functions

D k (s) ζ(s) = ζ (ks) ζ(ks) , L k (s) ζ(s) = T (s, ks) T (s, ks) . (11)
Remark that the Dirichlet series T (s, ks) := ∂T ∂u (s, ks, u)|u=1

involves two occurrences of the quasi-inverse (I-G (k+1)s ) -1 .

Average-case analysis results.

The following result is an exact analog of Theorem 1. In particular, in Assertion (a), the entropy π 2 /(6 log 2) of the integer Euclidean system replaces its polynomial analog (2q)/(q -1) on Fq[X]. Theorem 5. [Expectations]. When the set ΩN is endowed with the uniform distribution, the following holds:

(a) The expectation of the number of iterations L1 during the first phase is linear with respect to the length log N and satisfies

EN [L1] = 6 log 2 π 2 1 log N 1 + O 1 log N . (b) For any k ∈ [2.. -1]
, the expectation of the number of iterations L k during the k-th phase is asymptotic to a constant a k which is expressed with the operator Gs at s

= k + 1, EN [L k ] = a k 1 + O 1 log N .
(c) The expectation of the length of the first integer u1 is linear with respect to the length log N and satisfies

EN [D1] = 1 log N . (d) For any k ∈ [2.. -1]
, the expectation of the length of the gcd y k at the beginning of the k-th phase is asymptotic to a constant, and satisfies

EN [D k ] = ζ (k) ζ(k) 1 + O 1 log N .
5.5 Main principles for the analysis.

We have obtained in [START_REF] Heilbronn | On the average length of a class of continued fractions[END_REF] the expressions of the cumulative generating functions. It is now possible to "extract" coefficients of these Dirichlet series, with an analog of (1).

However, for Dirichlet generating functions, it is (very often) only possible to study the sum of the coefficients for n ≤ N , and this is why we deal with the set ΩN of inputs with size at most N . Then the mean value of cost C on ΩN is obtained from the cumulative generating function as

EN [C] = n≤N [n -s ] F (s) n≤N [n -s ]F (s)
.

As previously, singularity analysis performs a transfer between the behavior of a Dirichlet generating function, viewed as an analytic function, near its dominant singularity (here, the singularity with the largest real part) and the asymptotic behavior of its coefficients. The position and the nature of the dominant singularity play a fundamental role. However, this transfer is more difficult for Dirichlet series. As previously, the basic tool is the Cauchy formula, but, here, the circles centered at 0 are replaced by vertical lines, which are not compact. For this short version, we use here the following Tauberian theorem, due to Delange [START_REF] Delange | Généralisation du théorème d'ikehara[END_REF][START_REF] Tenenbaum | Introduction à la théorie analytique des nombres[END_REF], which deals with Dirichlet series with positive coefficients, but does not provide any remainder term. This is why we only prove here, in this short version, a weak version of Theorem 5, without remainder terms. Theorem 6. Let F (s) = n≥1 ann -s be a Dirichlet series with non negative coefficients such that F (s) converges for (s) > σ > 0. Assume the following:

(i) F (s) is analytic on (s) = σ, s = σ, (ii) for some γ ≥ 0, one has

F (s) = A(s)(s -σ) -γ-1 + C(s), where A, C are analytic at σ, with A(σ) = 0. Then, as N → ∞, n≤N an = A(σ) σΓ(γ + 1) N σ log γ N [1 + (N ) ], (N ) → 0.
5.6 Sketch of proof for Theorem 5.

We apply Theorem 6 with σ = 1 to the cumulative functions defined in [START_REF] Heilbronn | On the average length of a class of continued fractions[END_REF]. We then study the series ζ(s) , ζ (s), T (s, ks), T (s, ks). All is known about the first two, and this entails a easy application of Theorem 6 in case D.

We focus now on case L, where we use precise results on the operator Gs, when it acts on a convenient Banach space B (not described here). First, the function ζs = Gs [START_REF] Baladi | Euclidean algorithms are gaussian[END_REF](0) has a pole of order 1 at s = 1, and near s = 1, one has ζs ∼ C/(s -1), where C belongs to B. Then, near s = 1, 2 δ(k,1) T (s, ks) ∼ 1/(s -1)(I -G (k+1)s ) -1 [C](0), for any k, (δ(i, j) is the Kronecker symbol). Now, the following is known: for t > 2, the operator Gt has a spectral radius strictly less than 1, and the quasiinverse (I -Gt) -1 is analytic there. On the line t = 2, the quasi-inverse (I -Gt) -1 is analytic except at t = 2 where it admits a simple pole with a residue which involves the entropy. As T (s, ks) (resp. T (s, ks)) contains one (resp. two) occurrence(s) of the quasi-inverse, the following holds: -for k ≥ 2, the series T (s, ks) and T (s, ks) have a simple pole at s = 1, -for k = 1, the series T (s, s) has a pole of order 2 at s = 1, and the series T (s, s) has a pole of order 3 at s = 1. Finally, the function L k (s) satisfies the hypotheses of Theorem 6, with σ = 1; the exponent γ equals for k ≥ 2 and + 1 for k = 1. This concludes the "proof" in case L.

And now?

We can also study the number of useful phases, and recover the classical result PN [D k = 0] ∼ 1/ζ(k). If we wish to obtain remainder terms in Theorem 5, and an analog of distributional results obtained in Theorem 2, we need the analog of Proposition 3. For this purpose, we have to deal with the Perron Formula, as for previous distributional analyses performed in integer case.

The Perron Formula provides remainder terms as soon as the Dirichlet series of interest possess a "tameness" region on the left of the vertical line s = 1, where s = 1 is their only pole and they are of polynomial growth for | s| → ∞. Classical results [START_REF] Tenenbaum | Introduction à la théorie analytique des nombres[END_REF] entail such tameness properties for the zeta function and its derivatives, and results due to Dolgopyat-Baladi-Vallée [START_REF] Dolgopyat | On decay of correlations in Anosov flows[END_REF][START_REF] Baladi | Euclidean algorithms are gaussian[END_REF] prove that they also hold for the transfer operator G2s. Then, it would be possible to obtain the analog of Theorem 2 where n, m are replaced by log N, log M and the ratios p k , r k which respectively appear in Assertions

Some experiments

The figure shows the experimental densities of L1 for = 2 (left) and = 4 (right) and clearly exhibits the uniform limit law (left) and the beta limit law (right). These experiments are obtained with 5•10 5 executions on random integer inputs (x1, . . . , x ) whose (total) length satisfies log 10 N = 10 4 .

APPENDIX 7.1 Proof of Proposition 3

The residue theorem entails

[z n ]fm(z) = -Res fm(z) z n+1 , z = 1 + 1 2iπ Cr fm(z) z n+1 dz
where Cr is the circle with radius r > 1 with r close enough to 1. Then, with hypotheses on A, the following upper bound holds when m is of the form m = cn,

1 2iπ Cr fm(z) z n+1 dz ≤ A(r) c r n 1 r(r -1)
.

The function r → (1/r)A(r) c equals a c at r = 1 and the conditions on c, a = A(1) and b = A (1) prove that it is strictly decreasing when r is close to 1. Hence, for r > 1 close enough to 1, we let ρ = A(r) c /r = θa c with θ < 1 and

1 2iπ Cr fm(z) z n+1 dz = a m O(θ n ).
Furthermore, the residue at z = 1 equals

Res

fm(z) z n+1 , z = 1 = (-1) ( -1)! d -1 dz -1 A(z) m z n+1 z=1
The successive derivatives of 1/z n+1 satisfy

d i dz i 1 z n+1 z=1 = (-1) i • n i 1 + O 1 n .
There are two cases for the derivatives of A(z) m . Case m/n → c with c > 0. Then

d j dz j A(z) m z=1 = b j • m j • a m-j 1 + O 1 m , and

Proof of Theorem 4

The variable L equals L := L1 + . . . + LΠ if Π ≥ 1 and equals 0 if Π = 0. Then

L = 0 • 1 [Π=0] + L • 1 [Π≥1] = L • 1 [Π≥1]
When Π ≥ 1, the variable L is defined as

L = Π k=1 L k = -1 k=1 L k • 1 [k≤Π] ,
so that L • 1 [Π≥1] = L and then finally L = L. We then study the variables

L k = L k • 1 [k≤Π] .
In particular, the difference L k -L k satisfies Then Theorem 3 applies and we obtain for k ≥ 2,

L k -L k = L k • 1 [k>Π] ,
En[ L k ] = q -1 q k -1 + q q k + O 1 n ,
and for k = 1,

En[ L1] = En[L1] + O 1 n .
We conclude with the linearity of the mean.

We now prove the beta limit law. We split the random variable L into two random variables:

-the main random variable L1, which admits a beta limit law (Theorem 2), -the remainder random variable R = L -L1 which is "negligible" w.r.t L1; more precisely, Theorem 1 entails the estimate En[|R|] = o(En[L1]).

The next proposition shows that, in this situation, the sum L = L1 + R asymptotically follows the same beta law as L1. This provides an extension of the result obtained in [START_REF] Lhote | Gaussian laws for the main parameters of the Euclid algorithms[END_REF] for gaussian laws. The random variable X + Y asymptotically follows the same law as X.

Proof. Consider a sequence δn which will be made precise later, and define the two events E and F as

E = [X + Y < c • γn], F = [|Y | ≤ δn].
The hypotheses on Y and the Markov inequality lead to

Pn[E ∩ F c ] ≤ Pn[F c ] = O rnγn δn . ( 12
)
On the other side, the following inclusions hold,

[X ≤ c • γn -δn] ∩ F ⊂ E ∩ F ⊂ [X ≤ c • γn + δn]. ( 13
)
The rightmost inclusion in [START_REF] Knopfmacher | The exact length of the euclidean algorithm in Fq[X][END_REF] and the Lipschitz condition on f entail the upper bound

Pn[E ∩ F ] ≤ f c + δn γn + O( n) = f (c) + O n + δn γn . (14)
The leftmost inclusion in [START_REF] Knopfmacher | The exact length of the euclidean algorithm in Fq[X][END_REF], together with [START_REF] Hensley | The number of steps in the Euclidean algorithm[END_REF] and the Lipschitz condition entail the lower bound

Pn[E ∩ F ] ≥ f (c) + O n + rnγn δn + δn γn . (15)
With relations ( 12), ( 14) and [START_REF] Lhote | Gaussian laws for the main parameters of the Euclid algorithms[END_REF], we obtain

Pn[E] = f (c) + O n + rnγn δn + δn γn .

Then the optimal choice δn = γn √ rn concludes the proof.

To derive the proof of Theorem 4, the previous proposition was applied with n = Θ 1 n , rn = Θ 1 n , γn = n, a = q -1 2q

and

f (c) = 1 -1 - c a -1
.

  where A(z) is analytic on the disk |z| ≤ ρ with ρ > 1 and satisfies a := A(1) = 0, b := A (1) > 0 and for |z| close enough to 1, |A(z)| ≤ A(|z|).

Proof.

  The Euclid algorithm first compares the two integers x1 and x2. There are three cases: x1 = x2, x1 > x2, x1 < x2. In all the cases, the gcd y := gcd(x1, x2) together with the sequence of quotients (m1, m2, . . . mp) completely determines the input pair (x1, x2). More precisely, one writes (x1, x2) = (y x1, y x2) with a coprime pair ( x1, x2) and the execution of the Euclid algorithm on the pair ( x1, x2) produces the same sequence (m1, m2, . . . mp) as the pair (x1, x2), with now remainders xi which satisfy xi = y xi. The execution of the Euclid algorithm on the pair ( x1, x2) with x1 > x2 builds continued fraction expansions

  (a) and Assertions (b) of Theorem 2 are replaced by p k := λ ((k + 1)/2) (case L), r k := exp(1-k) (case D).Here, λ(s) is the dominant eigenvalue of the operator G2s (when acting on the functional space B), which plays a central role in many analyses of Euclidean type. It satisfies λ(s) = 1, and, for k = 3, we recover the constant λ(2) which plays a central role in Euclidean dynamics, notably in the analysis of the Gauss lattice reduction algorithm[START_REF] Daudé | An average-case analysis of the gaussian algorithm for lattice reduction[END_REF][START_REF] Flajolet | Continued fraction algorithms, functional operators, and structure constants[END_REF].

1 ( - 1

 11 , inserting these relations into the Leibnitz formula for the derivative provides the estimate Res fm(z) z n+1 , z = 1 = -a m n -This concludes the proof of Assertion (b).Case m = (n) • n with (n) → 0. Thend j dz j A(z) m z=1 = a m n j b a j O( (n) j )and, inserting these relations into the Leibnitz formula for the derivative provides the estimateRes fm(z) z n+1 , z = 1 = -a m n -1 ( -1)! [1 + O( (n))] 1 + O 1 n .

  and the relations [k> Π] = [D k = 0] ⊂ [L k = 1], entail the equality L k • 1 [k>Π] = 1 [k>Π], and, finallyEn[L k ] -En[ L k ] = En[1 [k>Π] ] = Pn[k > Π] = 1 -Pn[k ≤ Π].

Proposition 6 .

 6 Consider two random variables X and Y defined on Ω. Assume that there exist (i) a function f : [0, a] → [0, 1], strictly increasing, Lipschitz, with f (0) = 0 and f (a) = 1, (ii) three sequences γn → ∞, n → 0, rn → 0, such that, one has, for all c ∈]0, a[, for all n,Pn[X < c • γn] = f (c) + n En[|Y |] = rn • γn.Then the following holds, for all c ∈]0, a[, for all n,:Pn[X + Y < c • γn] = f (c) + O ( n + √ rn) .

The factor (1/2) is here to take into account the fact that any rational of ]0, 1] admits two continued fraction expansions: the proper one and the improper one.

Acknowledgements. Thanks to the two ANR Projects: -ANR BOOLE (ANR 2009 BLAN 0011) -ANR MAGNUM (ANR 2010 BLAN 0204)


Mohamed El

Hedi Arouri
email: mohamed.arouri@edhec.edu

Adel Ben Youssef
email: adel.ben-youssef@unice.fr

Hatem M'henni
email: hatem.mhenni@topnet.tn

Christophe Rault
email: christophe.rault@univ-orleans.fr

Empirical Analysis of the EKC Hypothesis for SO2 Emissions in Selected Middle East and North African Countries

Keywords: JEL Classification: C23, O11, Q25, Q28 Environmental Kuznets Curve, Sulfur Dioxide emissions, Economic Growth, panel data, MENA countries

Studying the impact of economic growth on the environment in the context of developing countries has become of increasing economic importance in recent years. Alarming international reports show that pollutants emissions are growing at their highest level ever, particularly in the South countries. This paper implements recent bootstrap panel unit root tests and cointegration techniques to investigate the relationship between Sulfur dioxide emissions and real GDP for 12 MENA countries over the period 1981-2005. Our investigations lead to the result that no evidence is found for the EKC hypothesis for 10 country of the region. However, the EKC hypothesis is valid for the case of Egypt and Tunisia; the two most industrialised and diversified economies in our sample. At the same time, our findings show that EKC is not valid for the region when considered as a whole.

INTRODUCTION

Environmental Sustainability Index (ESI) report of 2005 1 suggests that most Middle East and North African (MENA) countries perform badly in environmental quality management. The same conclusion is found by the new version of the ESI called the Environmental Performance Index (EPI) 2 in 2010.

According to this index, the MENA region can be divided into two distinct groups. The first one is composed of Algeria, Egypt, Iran, Jordan, Lebanon, Morocco, Syria and Tunisia. These countries perform relatively well in terms of environmental burden of disease and indoor air pollution. They also have roughly average results on most other indicators, but poor air pollution performance. Their scores on urban particulates and industrial carbon dioxide performance scores fall far below other clusters. The second group is composed of Bahrain, Libya, Kuwait, Oman, Qatar, Saudi Arabia, United Arab Emirates, Sudan and Yemen. This cluster is comprised of mainly fossil fuel producing and processing nations and too lowincome countries. They perform well on the environmental burden of disease but poorly on outdoor air pollution. Their scores are among the lowest in some of the water indicators, but most notably, they have the worst greenhouse gas per capita performance of all the clusters. Therefore, the question of sustainability of growth in MENA Countries is becoming central. On the one hand, environmental constraints may lead to lower the growth for the region in a context of a demographic boom associated with a high rate of unemployment. On the other hand, new opportunities and benefits from technological transfer may lead to better trend of growth and sustainability. One of the most important questions that arise in this context is to understand the nature of the relationship between economic growth and environmental quality in MENA countries.

According to the Environmental Kuznets Curve (EKC) hypothesis, as income increases, environmental degradation (emissions) increases as well until some threshold level of income is reached after which the environmental degradation begin to decline. An extensive literature has shown that the EKC hypothesis is validated for different local 1 Between 1999 and 2005, the Yale and Columbia team published four Environmental Sustainability Index reports (http://sedac.ciesin.columbia.edu/es/esi/) aimed at gauging countries" overall progress towards "environmental sustainability". 2 The 2010 EPI ranks 163 countries on 25 performance indicators tracked across ten well-established policy categories covering both environmental public health and ecosystem vitality. These indicators provide a gauge at a national government scale of how close countries are to established environmental policy goals. A pilot exercise was conducted in 2006 and a complete report was published in 2008.

pollutants in Organisation for Economic Cooperation and Development (OECD) countries.

Thanks to their income per capita levels, most of OECD countries have shifted from the first branch of the curve to the second one in pollutants especially for local pollutants like Sulphur dioxide, PMP10, NOx and River pollutants. An extensive literature has discussed the theoretical foundation of such findings. The picture seems quite different for developing countries, especially Middle East and North African (MENA) countries, where findings are often inconclusive (see Arouri et al. (2012) and references therein).

Our article investigates the MENA countries as a region as well as at a country level by taking advantage of recent advances in the econometrics of non-stationary panel data econometric techniques and seemingly unrelated regression (SUR) methods. We choose to use sulphur dioxide (SO 2 ) emissions as the environmental quality proxy in order to test the existence of the Kuznets curve in the studied MENA countries. At least, three main arguments justify this choice. First, emissions of SO 2 are among the most important forms of energyrelated pollution in the region. They originate primarily from stationary sources in the industrial and power-generating sector, and SO 2 emissions are known for their adverse effects on human health and the natural environment (EPA 2007). Second, although global sulphur emissions have increased over much of the last century, levels have begun to decline in recent decades. By 2000, global emission was approximately 25 percent lower than the peak level observed in the 1980s. Nevertheless, this decline was not homogenous across world regions.

Whereas emissions are still rising in many developing countries, industrialized countries experienced an especially strong decline with emission reductions of more than 50 percent from 1970 to 2000. Obviously, it is important to understand if this decline is also observed in MENA Countries. Finally, SO 2 is produced by burning fossil fuels and is primarily emitted from stationary sources in the industrial generating sector (Olivier et al. 2005). Specifically, lignite and hard coal have high Sulphur contents so that their combustion is responsible for a large part of global SO 2 emissions. Yet, since the beginning of the 1970"s, more end-of-pipe technologies, such as flue-gas desulfurization have been adopted to filter Sulphur dioxide.

The existence of an EKC can reveals the extent to which MENA countries are adopting these new technologies.

Thus, the aims of our article are threefold. First, we check for the existence of EKC in the 12 countries belonging to MENA countries in matter of SO 2 . Second, we assess the theoretical values of the turning points from which the economic growth improves the environmental quality in these countries. Third, we investigate the nature of the causality relationship between economic growth and emissions of SO 2 .

The rest of paper is organised as follows. Section 2 surveys the theoretical foundation of the EKC and offers a synthesis of empirical studies on this topic. Section 3 presents the data, introduces the econometric methodology and discusses the empirical results. Section 4 recommends the appropriate policies and concludes.

LITTERATURE REVIEW

Theoretical explanations of the nature of the SO2-economic growth relationship

Generally the impacts of economic development on environment are disaggregated into three macro determinants: scale effect, technique effect, and composition effect [START_REF] Grossman | Economic Growth and the Environment[END_REF]Copeland and Taylor 2004;Brock and Taylor 2006). The scale effect refers to the fact that increases in output require more inputs, and, as a by-product, imply more emissions.

Economic growth therefore exhibits a scale effect that has a negative impact on the environment (Arrow, 1995). The technique effect refers to the invention of new technologies which are environmental friendly and to the application of these new technologies in production which in turn lead to the reduction of the pollution of the environment (Andreoni and Levinson, 2001). The impact of the technique effect is theoretically positive (de Bruyn 1997, Han andChatterjie, 1997). The composition effect stems from changes in production of an economy caused by specialization (from agriculture or/and basic industries to high-tech services). All else equal, if the sectors with high emission intensities grow faster than sectors with low emission intensities, than composition changes will result in an upward pressure upon emission (Dasgupta, Mody, Roy, and Wheeler, 1995). The expected impact of the composition effect is positive deriving from the Rostow evolution postulate. Due to the different nature of these individual effects, the overall impact of growth on the environment is ambiguous (Grossman and Krueger (1991[START_REF] Panayotou | Demystifying the Environmental Kuznets Curve: Turning a Black Box into a Policy Tool[END_REF]), and Cole (2004)).

Taking into account the nature of the Sulfur Dioxide as specific pollutant, several explanatory factors were proposed in order to explain the nature of the relationship between economic growth and SO 2 emissions: (i) the decomposition of the economy structure, (ii) adoption of new technologies and innovation, (iii) demographic factors like the structure of population or population density, (iv) environmental regulation, institutions and control system, and, (v) energy consumption structure.

According to De Bruyen (1997) change in industrial structure is the main factor affecting trends in SO 2 emissions. [START_REF] Stern | The Rise and fall of the Environmental Kuznets Curve[END_REF]Stern ( , 2005) ) asserts that changes in technology can lead with time to reductions in pollution-lowering of EKC-in both developing and developed countries. Case studies, particularly in China, show that pollution-reducing innovation and standards may be adopted with relatively short time lags in some developing countries. " [START_REF] Stern | The Rise and fall of the Environmental Kuznets Curve[END_REF] proposes that at middle-income levels, rapid growth can overwhelm these clean-up efforts, which have more effect in slower-growing higher income countries". Several articles show that population density is negatively correlated with sulfur dioxide emissions [START_REF] Selden | Economic growth and environmental quality: is there a «Kuznets» curve for air pollution emissions?[END_REF], Cole and Neumayer, 2004and Farzin and Bond, 2006). The main explanations are lower transportation requirements and higher environmental preferences in populated areas. Population compositional change has also considerable environmental policy implication. Recently, Menz and Kühling (2011) show that "societies with a low population and young and high proportion of senior citizens emit more Sulfur Dioxide". They verified these facts for 25 OECD countries from 1970 to 2000. In the same study, Menz and Kühling argue that three factors actually determine national SO 2 emissions: total national energy consumption, importance of fossil fuels with high Sulfur contents in the process of energy generation (energy mix), and usage of en-of-pipe technologies.

Empirical validation of the EKC hypothesis for sulfur dioxyde emissions

Sulfur emissions show the most typical environmental Kuznets curve among the air pollutants. A wide range of publications shows that SO2 EKC is empirically validated for most OECD countries. Since the findings of [START_REF] Selden | Economic growth and environmental quality: is there a «Kuznets» curve for air pollution emissions?[END_REF] and [START_REF] Grossman | Economic Growth and the Environment[END_REF] a plethoric literature has examined the EKC hypothesis for SO 2 at both regional and country levels. [START_REF] Stern | Progress on the Environmental Kuznets Curve?[END_REF] claims that, in most OECD countries, the evidence for the inverted-U relationship is found only for a subset of environmental measures, e.g. air pollutants such SO 2 or suspended particulates. These findings are supported by those by Cole et al. (1997), [START_REF] Selden | Economic growth and environmental quality: is there a «Kuznets» curve for air pollution emissions?[END_REF], Stern andCommon (2001), Halkos (2003) and [START_REF] Markandya | Empirical Analysis of National Income and SO2 Emissions in Selected European Countries[END_REF]. Finally, Wang (2010) confirm the existence of Long-run Sulfurincome relationship for 19 OECD countries during the period 1870-2001.

Understanding past emission patterns in OECD countries has numerous insights for future emission projection especially in developing countries. One of the most important case studies is China. China has experienced rapid economic growth during last two decades and one can ask if this rapid growth has led to the validation of EKC for SO 2 . Recent works find that even in developing countries like China, SO 2 emissions and GDP per capita are following an EKC. For instance, Gao et al. (2011) found that EKC is valid for SO 2 emissions for 29 Chinese provinces during the period between 2000[START_REF] Ahmed | Costs of environmental degradation: An analysis in the Middle East and North Africa region[END_REF]. Mou et al. (2011) ) establish for the same period that the relationship between economic growth and SO 2 emissions in the biggest city in China (Chongqing) is following an EKC. In contrast, Vincent (1997) shows that there is no confirmation of SO 2 EKC for Malaysia.

As for MENA countries, using cointegration analysis [START_REF] Chebbi | Trade openness and CO2 emissions in Tunisia[END_REF] reveal a positive linkage between trade openness and per capita SO 2 emissions and a negative linkage between economic growth and per capita SO 2 emissions in the long-run. [START_REF] Fodha | Economic growth and environmental degradation in Tunisia: An empirical analysis of the environmental Kuznets curve[END_REF] provide support for a long-run N-shape relationship between per capita SO 2 emissions and per capita GDP. The authors confirm the EKC hypothesis. Akbostanci et al. (2009) study the relationship between SO 2 emissions, energy consumption and economic growth in Turkey at two levels: the national level and for the 58 Turkish provinces. The authors found a monotonic and increasing relationship at the national level. However, they found an N shaped curve at provinces level. Their findings do not support the EKC.

Regarding the turning point values, the results show a large dispersion across different studies. According to Lieb (2003) the reported turning points for SO 2 range from USD 2 900 to USD 98 200 (in PPP USD 1985). The calculations are very sensitive to the estimation methods and the econometrics used. Recent studies show that turning points for most OECD countries range from US$ 5 000 to US$ 10 000. Stern and Common (2001) show that the turning point in OCCD countries is US$ 9 000. [START_REF] Markandya | Empirical Analysis of National Income and SO2 Emissions in Selected European Countries[END_REF] found a turning point at US$ 11 900 in 1990 PPP dollars. The turning points in non-OECD countries are extremely high and show unilateral increase for most of them.

Our work extends the finding of this literature by examining the situation in MENA countries at two levels: the whole region and the country level. [START_REF] Stern | The Rise and fall of the Environmental Kuznets Curve[END_REF] asserts that a large portion of EKC literature is statistically weak and when these statistical problems are taken into account and appropriate techniques are used, EKC cannot exist. We challenge this view in our paper and we show that using recent and appropriate econometrics leads to the existence of EKC for SO 2 in MENA Countries 3 .

3 Another concern is related to the environmental indicators" measurements. The "measures of the environmental degradation fall in two general categories: emission of the pollutants and environmental concentrations of pollutants" (Kaufman et al., 1998, p210). These two measurements illustrate different aspects of the environmental degradation situation and neither of them can offer a comprehensive description. "Emission directly measures the amount of pollutants generated by economic activities during a period without regarding to the size of the area into which the pollutants are emitted". It is actually a flow measurement for the polluting

DATA AND EMPIRICAL RESULTS

In what follows, after introducing the series of our data set we start by testing for unit roots in our variables. If the variables are non-stationary in our country panel, we test for the existence of long-run cointegration relationships and investigate their magnitude. Finally, we estimate panel error correction models (ECM) in order to examine the interactions between short-and long-run dynamics of our environmental variables.

Data set and panel unit root tests

We investigate the relationship between SO 2 emissions and GDP in MENA region using recent panel econometric methods. As several MENA countries have signed Kyoto protocol, there are still concerns regarding the environmental problems. To conduct our empirical analysis, we need the following variables for all studied MENA countries:

-the SO 2 emissions (S); -the per capita real GDP (Y).

We collect data form World Bank Development Indicators (WDI). Our data are annual and cover the period 1981-2005 for the following MENA countries: Algeria, Bahrain, Egypt, Jordan, Kuwait, Lebanon, Morocco, Oman, Qatar, Saudi Arabia, Tunisia and UAE. The variables S and Y are measured in metric tons per capita and constant 2005 international dollar, respectively.

At first, we empirically investigate the following model based on variables in natural logarithms:

it it i it i i it Y c Y b a S      2 (1)
The coefficients b and c represent the long-run elasticity estimates of SO 2 emissions with respect to real GDP and squared real GDP, respectively. According to the discussion above, under the EKC hypothesis an increase in income is associated with an increase in SO2 emissions(b>0) and there is an inverted U-shape pattern at which point an increase in income leads to lower SO 2 emissions (c<0). capacity of economic activities. "The concentration measures the quality of pollutants per unit area without regarding to the activity that emitted them", it is more like a stock measurement describing the final result of the encounter between emission, abatement efforts and the self-purification capacities of nature. As concentration is a more direct environmental quality indicator and has more direct impact on productivity and public health, [START_REF] Selden | Economic growth and environmental quality: is there a «Kuznets» curve for air pollution emissions?[END_REF] believe it should be easier to obtain an inverted-U curve for concentration than for emission indicators.

The first step of the analysis is to look at the data properties. Two classes of tests allow investigating the presence of a unit root: The first generation panel unit-root tests (including Hadri, 2000;[START_REF] Im | Testing for unit roots in heterogeneous panels[END_REF], were developed on the assumption of the crosssectional independence among panel units (except for common time effects), and may be at odds with economic theory and empirical results. On the other hand, second generation tests (for instance, [START_REF] Choi | Combination Unit Root Tests for Cross-Sectionally Correlated Panels[END_REF][START_REF] Moon | Testing for a Unit Root in Panels with Dynamic Factors[END_REF] relax the assumption of cross-sectional independence, allowing for a variety of dependence across the different units. To test for the presence of such cross-sectional dependence in our data, we have implemented the simple test of [START_REF] Pesaran | General Diagnostic Tests for Cross Section Dependence in Panels[END_REF] and have computed the CD statistic. This test is based on the average of pair-wise correlation coefficients of the OLS residuals obtained from standard augmented Dickey-Fuller regressions for each individual. Its null hypothesis is cross-sectional independence and is asymptotically distributed as a two-tailed standard normal distribution.

Results available upon request indicate that the null hypothesis is always rejected regardless of the number of lags included in the augmented DF auxiliary regression (up to five lags) at the five percent level of significance. This confirms that the MENA countries are, as expected, cross-sectionally correlated, which can indeed reflect here the presence of similar regulations in various fields such as environmental policies, trade, customs, tourism, legislation, business administration, and increasing financial and economic corporation.

To determine the degree of integration of our series of interest (S, Y, and Y 2 ) in our panel of 12 MENA countries, we employ the bootstrap tests of [START_REF] Smith | More Powerful Panel Unit Root Tests with an Application to the Mean Reversion in Real Exchange Rates[END_REF], which use a sieve-sampling scheme to account for both the time series and cross-sectional dependencies of the data through bootstrap blocks. The specific tests that we consider are denoted t , LM , max , and min . t is the bootstrap version of the well known panel unit root test of [START_REF] Im | Testing for unit roots in heterogeneous panels[END_REF],

1 1 N i i LM N LM   
 is a mean of the individual Lagrange Multiplier (LM i ) test statistics, originally introduced by Solo (1984), max is the test of [START_REF] Leybourne | Testing for unit roots using forward and reverse Dickey-Fuller regressions[END_REF], and

min = 1 1 N i i N min  
 is a (more powerful) variant of the individual Lagrange Multiplier (LM i ), with min min( , )

i fi ri

LM LM 

, where fi ri LM and LM are based on forward and backward regressions (see [START_REF] Smith | More Powerful Panel Unit Root Tests with an Application to the Mean Reversion in Real Exchange Rates[END_REF] for further details). We use bootstrap blocks of m=20. 4 All four tests are constructed with a unit root under the null hypothesis and heterogeneous 4 The results are not very sensitive to the size of the bootstrap blocks.

autoregressive roots under the alternative, which indicates that a rejection should be taken as evidence in favour of stationarity for at least one country.

The results, shown in Table 1 suggest that for all the series (taken in logarithms) the unit root null cannot be rejected at the five percent level of significance in our country panel for the four tests. 5 We therefore conclude that the variables are non-stationary in our country panel. 6 Null hypothesis: unit root (heterogeneous roots under the alternative).

Panel long-run relationship

Given that all the series under investigation are integrated of order one, we now proceed with the two following steps. First, we perform 2 nd generation panel data cointegration tests (that allow for cross-sectional dependence among countries) to test for the existence of cointegration between S and Y, Y 2 contained in X. Second, if a cointegrating relationship 5 The order of the sieve is permitted to increase with the number of time series observations at the rate T 1/3 while the lag length of the individual unit root test regressions are determined using the [START_REF] Campbell | Pitfalls and Opportunities: What Macroeconomists should know about Unit Roots[END_REF] procedure. 6 The lag order in the individual ADF type regressions is selected for each series using the AIC model selection criterion. Another crucial issue is the selection of the order of the deterministic component. In particular, since the cross-sectional dimension is rather large here, it may seem restrictive not to allow at least some of the units to be trending, suggesting that the model should be fitted with both a constant and trend. However, since the trending turned out not to be very pronounced, we have considered that a constant is enough in our analysis. Actually, the results of the bootstrap tests of [START_REF] Smith | More Powerful Panel Unit Root Tests with an Application to the Mean Reversion in Real Exchange Rates[END_REF] are not very sensitive to the inclusion of a trend in addition to a constant in the estimated equation (see Statistic b in Tables 1). We have of course also checked using the bootstrap tests of [START_REF] Smith | More Powerful Panel Unit Root Tests with an Application to the Mean Reversion in Real Exchange Rates[END_REF] that the first difference of the series are stationary, hence confirming that the series expressed in level are integrated of order one.

exists for all countries, we estimate for each country the cross-section augmented cointegrating regression

 S it   i   i X it   1 S t   2 X t  u it , i  1 ,...,N; t  1 ,...,T (2)
by the CCE estimation procedure proposed by [START_REF] Pesaran | Estimation and inference in large heterogenous panels with multifactor error structure[END_REF] that allows for cross-section dependencies that potentially arise from multiple unobserved common factors. The cointegrating regression is augmented with the cross-section averages of the dependent variable and the observed regressors as proxies for the unobserved factors. Accordingly, t S and t X denote respectively the cross-section averages of S and X i in year t. Note that the coefficients of the cross-sectional means (CSMs) do not need to have any economic meaning as their inclusion simply aims to improve the estimates of the coefficients of interest.

Therefore, this procedure enables us to estimate the individual coefficients γ i in a panel framework. 7In addition, we also compute the CCE-MG estimators of [START_REF] Pesaran | Estimation and inference in large heterogenous panels with multifactor error structure[END_REF]. For instance, for the γ parameter and its standard error for N cross-sectional units, they are easily obtained as follows:

N N i CCE i MG CCE      1   , and N SE CCE i N i MG CCE ) ( ) ( 1        
, where

CCE i  ˆ and ) ( CCE i   
denote respectively the estimated individual country time-series coefficients and their standard deviations.

We now use the bootstrap panel cointegration test proposed by [START_REF] Westerlund | A Panel Bootstrap Cointegration Test[END_REF]. This test relies on the popular Lagrange multiplier test of [START_REF] Mccoskey | A Residual-Based Test of the Null of Cointegration in Panel Data[END_REF], and makes it possible to accommodate correlation both within and between the individual cross-sectional units. In addition, this bootstrap test is based on the sieve-sampling scheme, and has the advantage of significantly reducing the distortions of the asymptotic test.

Another appealing advantage is that the joint null hypothesis is that all countries in the panel are cointegrated. Therefore, in case of non-rejection of the null, we can assume that there is cointegration between S and its potential determinants contained in X.

The asymptotic test results (Table 2) indicate the absence of cointegration. However, this is computed on the assumption of cross-sectional independence, not the case in our panel.

Consequently, we also used bootstrap critical values. In this case we conclude that there is a long-run relationship between sulfur dioxide emissions and potential determinants, implying that over the longer run they move together. [START_REF] Westerlund | A Panel Bootstrap Cointegration Test[END_REF].

Estimation of the long-run parameters

Given the evidence of panel cointegration, the long-run pollution income relations can be further estimated by several methods for panel cointegration estimation. We estimate the above equation to assess the magnitude of the individual γ i coefficient in the cointegrating relationship with the CCE estimation procedure developed by [START_REF] Pesaran | Estimation and inference in large heterogenous panels with multifactor error structure[END_REF], which addresses cross-sectional dependency.

 S it   i   1i Y it   2 i Y it 2  u it , (3) with 1,..., , 1,..., i N t T  
, and the respective estimation results are reported in Tables 3. The results show that there are an inverse U-shaped relationships between per capita pollution and per capita GDP for all studied MENA countries, expect Kuwait, UAE and Qatar. For Tunisia, the elasticity is 1.430-0.870Y with the threshold income of 1.644 (in logarithms). EKC hypothesis seems to hold in this case. We reach the same conclusion in the case of Egypt. However, for Algeria the elasticity of SO 2 emissions per capita with respect to real GDP per capita in the long-run is 1.105-0.04Y with the threshold income of 27.625 (in logarithms) which is very high (when transformed in dollars) compared to its level of real GDP in that period. In contrast, for Saudi Arabia, the elasticity of SO2 emissions with respect to real GDP is 1.037-0.704Y, implying a threshold income of only 1.473 (in logarithms), which is very low compared to the Saudi real GDP.

Therefore, we have to point out that for most countries where we found an EKC, we are confronted to the problem of the position of the threshold compared to the level of real GDP reached by each country during the period. Our calculations lead us to conclude that none of the studied cases verified this particular EKC hypothesis except Egypt and Tunisia.

Egypt noticed a remarkable improvement in sulfur dioxide concentrations during the first years of the 2000, whereas daily average concentrations were ranged between 20-40 μg/m3 which is lower than the limit stated in the Executive Regulation of Environment Law 4 /1994 (150 μg/m3). This improvement is due to the efficient use of fuel in power stations and industrial sector, reducing diesel fuel usage in these sectors and expands in natural gas usage.

The actions related to rationalisation of energy use in Tunisia were mainly focused on stepping up the actions of mandatory and periodic energy audits and signing performance contracts in the industry, transport and services sectors. Since the end of the nineties, pilot projects in the field of energy conservation were implemented in the housing and services sectors, and encouraging the use of energy saving equipments, appliances and materials.

Besides, several programmes were pursued in relation to cogeneration in the industry sector, energy efficiency in street lighting networks, and rationalisation of energy use in the administration and public facilities. Also, as part of implementing the State policy in the field of energy substitution and directing consumption towards less costly energy, effort was invested in pursuing the programme of promoting the use of natural gas as a fuel in the transport sector and fostering the use of natural gas powered air conditioning in the services sector.

Finally, the results from the common correlated effects mean group (CCE-MG) method are reported in Table 4. The last table shows that the elasticity of SO 2 emissions per capita with respect to real GDP per capita in the long-run is 0.250-0.054Y with the threshold income of 4.630 (in logarithms), which is not supportive of the EKC hypothesis in the MENA region. This result was expected given the number of countries producing oil and gas in our sample8 .

Estimation of a panel error-correction model

In the previous sub-section, we have estimated the long-run relationships between S and Y, Y 2 for our panel of 12 MENA countries, using the common correlated effects mean group (CCE-MG) estimates (see Tables 4).

Having established the long-run structure of the underlying data and given that there exists a long-run relationship for all countries in our four panel sets, we turn to the estimation of the complete panel error-correction model (PECM) described by equations ( 5):

  ) 5 ( , 0 1 1 1 b X S X S S it p j it it i j it j p j j it j it                      
We use the Pooled Mean Group (PMG) approach of [START_REF] Pesaran | Pooled mean group estimation of dynamic heterogeneous panels[END_REF], with long-run parameters obtained with CCE techniques, in order to obtain the estimates of the loading factors λ i (weights or error correction parameters, or speed of adjustment to the equilibrium values), as well as of the short-run parameters β j and θ j for each country of our panel. Consequently, the loading factors and short-run coefficients are allowed to differ across countries.9

The lag length structure p is chosen using the Schwarz (SC) and Hannan-Quinn (HQ) selection criteria, and by carrying out a standard likelihood ratio testing-down type procedure to examine the lag significance from a long-lag structure (started with p=4) to a more parsimonious one. Afterwards, in order to improve the statistical specification of the model, we implemented systematically Wald tests of exclusion of lagged variables from the short-run dynamic (they are not reported here) to eliminate insignificant short-run estimates at the 5% level. We tested the residuals from each PECM model for the absence of heteroscedasticity, autocorrelation, and we can report that they are not subject to misspecification. The results of the PECM estimations based on (5) are reported in Tables 5, only for significant short-run estimates at the 5% level. countries have built recently a capacity to manage environmental problems and, especially, air pollution, the non-decline of SO2 emissions as GDP increases may be explained by corruption (Leitao, 2010). In his study, Leitao found an inverted U-shaped curve between Sulfur dioxide and economic growth. However, the author suggests that different incomepollution paths across countries are found due to corruption. Most of the considered countries perform badly in matter of corruption. While laws in matter of Air pollution exist, the enforcement of laws and control are ineffective due to corruption.

Second, contrary to CO 2 emissions that are more linked to consumer behaviour and are non-source pollution, SO 2 emissions are more closed to producers" behaviour and are local pollutant. The non-decline of SO 2 in MENA countries can be explained also by little or absence of change in matter of adoption of new technologies (end-of-pipe technologies).

While, in OECD countries there is a fall in SO 2 emissions due to massive adoption of new technologies in matter of desulfurization, this is not the case in most of MENA countries.

Changes in the technological behaviour may lead in the near future to a change in the relation between economic growth and SO 2 emissions.

Moreover, the rapid growth of energy demand and especially for electricity generation may explain poor performances in matter of SO 2 reduction in MENA countries. It is well known that electricity generation plants emit high levels of Sulfur dioxide. MENA countries are facing or will face shortage in this domain. As Krane (2010) states "for the six states of Golf Cooperation Council (GCC)…are unable to meet their own fast rising demand for domestic energy, mainly natural gas feedstock for electricity generation." These countries are going to face shortage in the near future due to their fast demand in matter of energy10 . Energy conservation options are envisaged. Some GCC countries are investing in nuclear power to generate energy and other MENA countries are investing in renewable energy generation (Ghaddar, 2010).

Third, most of the considered countries are based on primary sector (Rentier States11 ) and their move toward a service economy is low. They have not yet reached a positive regime where the effect of growth on SO 2 emissions is environmental improving. The economic composition of these economies is changing slowly. Some non-oil countries like Egypt and Tunisia are changing their structural economic composition and are performing better than the other countries in the sample.

At the country-level, our results show that EKC is not verified for the studied countries except Egypt and Tunisia. Our results confirm those by [START_REF] Fodha | Economic growth and environmental degradation in Tunisia: An empirical analysis of the environmental Kuznets curve[END_REF] using a different methodology. The authors show an evidence for Sulfur Dioxide EKC for Tunisia. Their main explanation relies on the enforcement of laws and the effectiveness of the control of plants, which are responsible of SO 2 emissions. The Economic structure of Tunisia dominated by services may also explain this result. In the case of Egypt, the result is explained by technological change and adoption of new technologies. In fact, Egypt has shifted to cleaner technology in matter of electricity generation. Generation plants are more using gas and less burning oil. This shift and more effective regulation lead to an improvement of the situation. Our results show that for these two countries the values of the turning points are very close to those found for OECD countries.

In the case of Gulf Cooperation Council countries (GCC), the shift towards more energy efficiency could improve their performance (Doukas et al., 2006). These countries are exploring new policies, but this reorientation has not yet resulted in the development of consistent strategies and policies (Reiche, 2010). At the same time, one must mention that several initiatives of renewable energy were taken in Algeria, the kingdom of Saudi Arabia and other MENA countries like the pioneering project of Masdar Sustainable City12 . These initiatives are expected to improve the situation in the next years. Actual efforts and policies changes are not captured by actual statistics and, the EKC is not verified at the country level in most cases. However, we think that all these initiatives are improving the situation.

REFERENCES.

Akbostancı E. , Turut-Asık S. and Tunç G. (2009)

Table 1a -

 1a Panel unit root tests of Smith et al. (2004) for S and Y, Y 2 (1981-2005) *

			Sulfur Dioxide Emissions (S)
	Test	Statistic	Bootstrap	Statistic	Bootstrap
		(a)	P-value*	(b)	P-value*
	t	-1.309	0.738	-2.021	0.685
	LM	3.197	0.287	4.456	0.617
	max	-0.537	0.952	-1.610	0.750
	min	1.650	0.518	3.264	0.670
			Per Capita Real GDP (Y)		Square of Per Capita Real GDP (Y 2 )
	Test	Statistic	Bootstrap	Statistic	Bootstrap	Statistic	Bootstrap	Statistic	Bootstrap
		(a)	P-value*	(b)	P-value*	(a)	P-value*	(b)	P-value*
	t	-1.521	0.492	-2.446 3	0.152	-2.393	0.187	-2.157	0.198
	LM	3.891	0.123	5.841	0.133	4.692	0.264	3.504	0.384
	max	0.216	0.865	-0.685	0.974	0.327	0.846	-0.687	0.784
	min	2.177	0.224	1.954	0.993	2.161	0.237	1.972	0.814

Notes: (a) Model includes a constant. (b) Model includes both a constant and a time trend. * Test based on Smith et al. (2004). Rejection of the null hypothesis indicates stationarity at least in one country. All tests are based on 2,000 bootstrap replications to compute the p-values.

Table 2 -

 2 Panel cointegration between sulphur dioxide emissions and potential

	determinants (1981-2005)
		LM-stat Asymptotic	Bootstrap
			p-value	p-value #
	Model with a constant term	11.272	0.000	0.859
	Notes: bootstrap based on 2000 replications.
	a -null hypothesis: cointegration of sulphur dioxide emissions and potential determinant
	series.
	# Test based on

Table 3 -

 3 Individual country CCE estimates for 12 MENA countries for the sulfur dioxide emissions and potential determinants

	Country				Y			Y 2	Constant
		γ 1				t-Stat	γ 2	t-Stat	α i	t-Stat
	Algeria	1.105		2.342	-0.020	-2.513	2.754	2.115
	Egypt	2.066		2.949	-1.08	2.569	5.551	2.781
	Jordan	1.833		2.679	-0.841	-3.689	4.849	2.898
	Lebanon	4.513		4.073	-2.203	-4.253	-4.136	-2.417
	Morroco	1.102		3.826	-0.343	-2.598	-0.205	-2.532
	Tunisia	1.430		4.218	-0.435	-1.959	5.539	8.508
	Bahrain	0.829		3.890	-0.203	-2.335	-3.672	-1.912
	Kuwait	-5.165	-4.787	2.179	3.405	-14.200	-5.190
	UAE	-2.310	-2.826	0.778	2.129	11.713	5.117
	Oman	4.657		3.571	-1.490	-2.282	5.295	3.616
	Qatar	-2.964	-2.715	1.813	2.772	-10.205	-5.818
	Saudi	1.037		2.465	-0.352	-3.770	-4.734	-4.574
	Note the coefficients of the variables	E	t	and	X	t

1 of equation (

2

) have not been reported in the table.

Table 4 -

 4 Results for common correlated effects mean group (CCE-MG) estimations,

	12 MENA countries (1981-2005) for SO 2 emissions (1) X= (Y, Y 2 )
	Constant	-3.42
		(-2.76)
	Y	0.250
		(5.28)
	Y 2	-0.027
		(-4.37)
	Note: t-statistics are in parentheses.

Table 5 -

 5 Panel Error-Correction estimations for S it , X= (Y, Y 2 ),

		D S it-1	DY t	DY it-1	D Y 2 it	D Y 2 t-1	Loading
								factor λ i
		0.40	0.41	-			-0.015	-0.10	-0.52
		(2.08)	(3.14)				(-2.76)	(2.82)	(-4.95)
	Algeria
		-	0.55	-			-0.21	-0.04	-0.86
	Egypt		(2.23)				(-4.48)	(-2.72)	(-3.22)
		-	0.57	-			-0.18	-0.05	-0.76
	Jordan		(2.76)				(-4.36)	(-3.37)	(-4.13)
		-	0.58	-			-0.17	-0.03	-0.81
	Lebanon		(1.98)				(-4.76)	(-3.18)	(-3.38)
			0.43	-			0.13	-0.09	-0.78
	Morroco	-	(2.21)				(3.77)	(-2.85)	(-5 .72)
		-	0.15	-			-0.05	-	-0.27
	Tunisia		(2.13)				(-2.39)	(-2.75)
		0.28	0.10	0.07	-0.03	-	-0.14
	Bahrain	(2.48)	(2.10)	(2.91)	(-2.53)	(-2.62)
		-	0.05	-			-0.21	-0.01	-0.65
	Kuwait		(2.63)				(-4.48)	(-2.28)	(-3.41)
		-	0.31	-			-0.38	-	-0.05
	UAE		(4.42)				(-5.25)	(-2.72)
		-	0.21	-			-0.21	-	-0.58
	Oman		(3.21)				(-4.48)	(-3.27)
		-	0.25	-			-0.22	-0.07	-0.45
	Qatar		(2.74)				(-3.79)	(5.14)	(-5.21)
		-	0.22	-			-0.08	-0.07	-0.40
	Saudi		(2.16)				(-2.82)	(-2.94)	(-3.52)
		intercept	Y	Y 2
	CCE-MG	-3.19	0.37	-0.21
		(-6.15)	(4.19)	(-4.48)
	Notes: The estimations are obtained from the Pooled Mean Group approach with long-run parameters estimated with
	CCE techniques. The coefficients of the variables	E	t	and	X	t 1	of equation (2b) have not been reported in the
	table. t-statistics are in brackets. S-Sulfur Dioxide Emissions; Y -Per Capita Real GDP; Y 2 -Square of Per Capita
	Real GDP.

  : "The relationship between income and environment in Turkey: Is there an environmental Kuznets curve?". Energy Policy 37, 861-867. Arouri M., Ben Youssef A., M"Henni H. and C. Rault (2012), "Energy Consumption, Economic Growth and CO2 Emissions in Middle East and North African Countries", Energy Policy, Forthcoming. Arrow K., Bolin B, Costanza R, Dasgupta P, Folke C, Holling C, Jansson B, Levin S, Maler K, C.Perrings and D.Pimentel (1995),"Economic Growth, Carrying Capacity, and the Environment, Ecological Economics 15(2), 91-95. Bai, J., Ng, S. (2004). "A PANIC Attack on Unit Roots and Cointegration", Econometrica, 72 (4), 127-1177.

Note that in order to estimate the long-run coefficients we have also implemented the Pooled Mean Group (PMG) estimators (see[START_REF] Pesaran | Estimating long-run relationships from dynamic heterogeneous panels[END_REF],[START_REF] Pesaran | Pooled mean group estimation of dynamic heterogeneous panels[END_REF]), which allowed us to identify significant differences in country behaviour. However, we only report the results of the Common Correlated Effects (CCE) estimators developed by[START_REF] Pesaran | Estimation and inference in large heterogenous panels with multifactor error structure[END_REF], since they allow taking unobservable factors into account, which would not be the case of the PMG estimators.

The burning of fossil fuels is the most significant source of air pollutants such as SO 2 , CO, certain nitrous oxides such as NO and NO 2 (known collectively as NO x ), SPM, volatile organic compounds (VOCs) and some heavy metals. It is also the major anthropogenic source of carbon dioxide (CO 2 ), one of the important greenhouse gases.

Note that before considering equation (3), we first used a Wald statistic to test for common parameters across countries (i.e λ i = λ, and γ i =γ, for i=1,...,N) with the CCE techniques of[START_REF] Pesaran | Estimation and inference in large heterogenous panels with multifactor error structure[END_REF], that allow common factors in the cross-equation covariances to be removed. We found that only the null hypothesis γ i =γ, for i=1,…,N was not rejected by data, whereas the speeds of adjustment λ i vary considerably across countries (results are available upon request).

Qatar is an exception among these countries.

The term Rentier States connotes a country that derives most of its national income from the external sale of natural resources.

MENA countries are estimated to have a potential to generate 630,000,000 megawatts of solar power and also 75000 megawatts of wind power potential(Ghaddar, 2009).

Research (ERF) for the project « INCOME LEVEL AND ENVIRONMENTAL QUALITY IN THE MENA COUNTRIES: DISCUSSING THE ENVIRONMENTAL KUZNETS CURVE HYPOTHESIS».

Results from Table 5, allow checking for two sources of causation: (1) the lagged difference terms (short-run causality) and/or (2) the error correction terms (long-run causality). The causality from GDP to SO 2 emissions depend on the level of economic growth. As for the long-run dynamics, the loading factor, which measures the speed of adjustment back to the long-run equilibrium value, is significantly negative in all cases confirming that all the variables of our model move together over the long run. Thus, the long-run equilibrium deviation has a significant impact on the growth of SO 2 emissions.

CONCLUSION AND POLICY IMPLICATIONS

Our article had three aims. First, we investigated the existence of EKC for the MENA region taken as a whole in the matter of Sulfur dioxide. Second, we tested for the existence of EKC for each country. Finally, we explored the nature of the causality relationship between economic growth and emissions of SO 2 . Compared to previous works, we took advantage of implementing more robust and recent bootstrap unit root tests and panel cointegration techniques to investigate the relationship between economic growth and SO 2 emissions in MENA countries.

Departing from the hypothesis that the 12 countries are homogenous and looking at the regional-level, our results reject the EKC hypothesis for the MENA region. This finding can be explained by at least three complementary arguments. First, although most of MENA