Helmut Finner

Thorsten Dickhaus
email: dickhaus@math.hu-berlin.de

Edgeworth expansions and rates of convergence for normalized sums: Chung's 1946 method revisited

Keywords: Generalized self-normalized sum, Hermite polynomials, normal approximation, statistical algorithms, t-distribution, t-statistic

In this paper we revisit, correct and extend Chung's 1946 method for deriving higher order Edgeworth expansions with respect to t-statistics and generalized self-normalized sums. Thereby we provide a set of formulas which allows the computation of the approximation of any order and specify the first four polynomials in the Edgeworth expansion the first two of which are well known. It turns out that knowledge of the first four polynomials is necessary and sufficient for characterizing the rate of convergence of the Edgeworth expansion in terms of moments and the norming sequence appearing in generalized self-normalized sums. It will be shown that depending on moments and norming sequence the rate of convergence can be O(n -i/2 ), i = 1, . . . , 4. Finally, we study expansions and rates of convergence if the normal distribution is replaced by the t-distribution.

Introduction

In 1946, Kai-Lai Chung derived an expansion for the distribution of Student's t-statistic for an iid sequence of real valued random variables (ζ n : n ∈ N) with Eζ 1 = 0 and Varζ 1 = 1 in terms of the cumulative distribution function (cdf) Φ and derivatives of the probability density function (pdf) ϕ of the standard normal distribution N (0, 1), cf. [START_REF] Chung | The approximate distribution of Student's statistics[END_REF]. Thereby the t-statistic is defined by S n = √ nζ/s with s = n -1 n i=1 (ζ i -ζ) 2 1/2 and ζ = n i=1 ζ i /n. In modern notation, the expansion for the cdf F n (say) of S n derived by [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] can be written as

F n (x) = Φ(x) + ϕ(x) r i=1 n -i/2 q i (x) + o(n -r/2 ) uniformly in x ∈ R (1)
and is called an Edgeworth expansion. It is known to be valid if E|ζ 1 | r+2 < ∞ for some integer r ≥ 1 and the distribution of ζ 1 has a nontrivial absolutely continuous component with respect to the Lebesgue measure (cf. [START_REF] Hall | Edgeworth expansion for Student's t statistic under minimal moment conditions[END_REF]). Each q i appearing in this formula is a polynomial of order 3i -1 the coefficients of which depend on the moments α j = Eζ j 1 , j = 3, . . . , i + 2. Unfortunately, the explicit expansion given in equation (35) in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] is incorrect as noted earlier in [START_REF] Wallace | Asymptotic approximations to distributions[END_REF] and to our knowledge there seems to be no published correction. Once the well-hidden error in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] is corrected, the computation of the first polynomials q i appearing in (1) is elementary and straightforward with Chung's method. In Section 2 we extend the formulas given in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] such that in principle the q i 's are computable up to arbitrary order with an algebraic computer package.

Alternatively, the q i 's can be computed as outlined in [START_REF] Hall | The bootstrap and Edgeworth expansion[END_REF] under a broad scope of so-called 'smooth function models'. While this method applies to many other statistics, the computational effort of this method seems to be more tedious for self-normalized sums than Chung's method.

Anyhow, the first two polynomials q 1 , q 2 appearing in the expansion (1) are computed for example in [START_REF] Hall | Edgeworth expansion for Student's t statistic under minimal moment conditions[END_REF], [START_REF] Hall | The bootstrap and Edgeworth expansion[END_REF] and can be found in various textbooks. They are given by q 1 (y) = (y 2 /3 + 1/6) α 3 and q 2 (y) = (y 3 /12 -y/4) α 4 -(y 5 /18 + y 3 /9 -y/6) α 3 2 -y 3 /2. This representation shows that the rate of convergence is O(n -1/2 ) in case of α 3 = 0 and O(n -1 ) in case of α 3 = 0. Obviously, the best possible rate of convergence is O(n -1 ) and this may be the reason that usually only the first two polynomials are reported.

In [START_REF] Lehmann | Testing statistical hypotheses[END_REF], Section 11.4.2, an Edgeworth expansion for the classical t-statistic with normalization (n -1) -1 in the definition of s is given. The approximation polynomials in this case differ from the q i 's in (1). Hence, the norming sequence in the denominator of a self-normalized sum is of importance for the asymptotic behavior of these t-type statistics.

This observation provokes the question whether there exist other norming sequences for specific values of the moments α i , i ≥ 3, such that the rate of convergence can be improved. It will be shown in Section 3 that the answer is positive and that the first four polynomials q 1 , . . . , q 4 suffice to characterize the rate of convergence.

Section 4 is concerned with deriving Edgeworth-type expansions for and analyzing rates of convergence of self-normalized sums if the t-distribution is used as approximating distribution instead of the standard normal.

Chung's method revisited

Many of the derivations in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] refer to formulas and definitions taken from [START_REF] Hsu | The approximate distributions of the mean and variance of a sample of independent variables[END_REF]. Here we restrict attention to the ingredients which suffice to compute the polynomials q i in (1). We correct the main inaccuracy in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] and extend the formulas where necessary. However, since we have to omit some technical details of Chung's method here, it can only fully be understood by reading [START_REF] Chung | The approximate distribution of Student's statistics[END_REF], parts of [START_REF] Hsu | The approximate distributions of the mean and variance of a sample of independent variables[END_REF] and this paper simultaneously.

In the derivations in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] the function g defined by g(λ

) = z(1 + λ 2 z 2 ) -1/2 1 + ∞ j=1 Γ(3/2)/Γ(3/2 -j)/Γ(j + 1)(α 4 -1) j/2
(λx) j and its derivatives g (i) play a crucial role. The formulas given in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF], p. 458, fail by introducing the abbreviation z = z(1 + λ 2 z 2 ) -1/2 and ignoring that z depends on λ. The correct derivatives g (i) (0), i = 1, . . . , 4, in λ = 0 are given in Appendix A.

A further important quantity in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] y), where γ j (x, y) = O(λ j ), w pq (x, y) = ∂ p+q /(∂x p ∂y q )w(x, y) appears repeatedly in γ j (x, y) for various p, q with p + q ≤ 3r and w(x, y)

is h(ζ) = t 1 (ζ 2 -1)/(α 4 - 1) 1/2 +t 2 ζ, whereby ζ has the same distribution as ζ 1 . Let U j (t 1 , t 2 ) denote the jth cumulant of h(ζ) and define m k (t 1 , t 2 ) = -k-3 =0 i +1 U +3 (t 1 , t 2 )λ +1 /( + 3)! and Ψ k (it 1 , it 2 ) = k-3 j=1 m j (t 1 , t 2 ) j /j!. Expanding the U i 's in terms of t 1 , t 2 and replacing (it 1 ) p (it 2 ) q by (-1) p+q w pq (x, y) in Ψ k (it 1 , it 2 ) yields the representation Ψ k (it 1 , it 2 ) ≡ k-3 j=1 γ j (x,
= (2π(1 -ρ 2 ) 1/2 ) -1 exp(-(x 2 -2ρxy + y 2 )/(2(1 -ρ 2 ))). Finally, letting f pq (λ) = ∞ -∞ g(λ)
-∞ w pq (x, y) dy dx, the formal approximation in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] is given by

F n (z) = r j=0 f (j) 00 (0)/j! + r j=1 ∞ -∞ g(λ)
-∞ γ j (x, y) dy dx + o(λ r ), where f pq (λ) is approximated by its

Taylor series in λ = 0 wherever it appears in

∞ -∞ g(λ)
-∞ γ j (x, y) dy dx. In fact, the latter integral can be rewritten in terms of f pq (λ), which in turn can be expressed in terms of

I r pq (z) = ∞ -∞ x r w pq (x, z) dx. In Chung (1946) non-vanishing I r
pq 's needed for the computation of the first three polynomials q 1 , q 2 , q 3 are computed for p = 0, . . . , 3, r = p, . . . , 3. We extend Chung's table for all values of p, q by providing the full recursive regime for the computation of the I pq 's.

Lemma 2.1. The non-vanishing I pq 's are given by the following recursive relationship.

I 0 0q (z) = ϕ (q) (z), I 1 0q (z) = -ρϕ (q+1) (z), q ≥ 0, (2)
I r+1 0q (z) = -ρI r 0,q+1 (z) + rI r-1 0q (z), q ≥ 0, r ≥ 1, (3)
I r pq (z) = -rI r-1 p-1,q (z), 1 ≤ p ≤ r, q ≥ 0. ( 4
)
Proof. The equations in (2) are as in [START_REF] Chung | The approximate distribution of Student's statistics[END_REF] while ( 3) and ( 4) can be verified by making use of the definition of I r pq (z).

Remark 2.1. Equation ( 3) is interesting in itself because it is related to Hermite polynomials with positive coefficients. Let H n denote the normalized Hermite polynomials defined by

H 1 (x) = x, H 2 (x) = x 2 -1, H n+1 (x) = xH n (x) -nH n-1 (x)
and define modified Hermite polynomials by

h n (x) = -(I/ √ 2) n H n (Ix/ √ 2).
Then the coefficients of h n equal the coefficients of H n except that they are all positive. Interestingly, I r 0q = h r (ρD)(ϕ (q) ), where D denotes the differential operator. Note that (3), that is,

I r+1 0q (z) = -ρI r 0,q+1 (z) +rI r-1 0q (z) corresponds to h r+1 (x) = -xh r (x) + rh r-1 (x). Moreover, I r 0q (z) = (∂ q /∂z q )E[X r |Z = z],
where (X, Z) are bivariate normally distributed with means 0, variances 1 and covariance ρ.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

The next lemma provides the f (j) pq (0)'s for j = 0, . . . , 4, which are essential for the computation of q 1 , . . . , q 4 . Lemma 2.2. Setting I r pq ≡ I r pq (z) for r = 0, . . . , 4, we have for p, q ≥ 0 that f 00 (0) = Φ(z), f pq (0) = -I 0 p,q-1 1 N>0 (q), f

(1)

pq (0) = z(α 4 -1) 1/2 I 1 pq /2, f (2) pq (0) = (α 4 -1) z 2 I 2 pq+1 -zI 2 pq /4 -z 3 I 0 pq , f (3)
pq (0) = -3(α 4 -1) 1/2 (z 3 I 1 pq + z 4 I 1 p,q+1 )/2 + (α 4 -1) 3/2 3zI 3 pq -3z 2 I 3 p,q+1 + z 3 I 3 p,q+2 /8, f (4)
pq (0) = 9z 5 I 0 pq + 3z 6 I 0 p,q+1 + 3(α 4 -1) z 3 I 2 pq -z 4 I 2 p,q+1 -z 5 I 2 p,q+2 /2 + (α 4 -1) 2 (-15zI 4 pq + 15z 2 I 4 p,q+1 -6z 3 I 4 p,q+2 + z 4 I 4 p,q+3 )/16.
Now all ingredients for the computation of the polynomials q 3 and q 4 are collected. Computation by hand remains cumbersome. Therefore we prepared a Maple worksheet which allows the computation of the q i 's up to arbitrary order. Due to the structure of the f

(j)
pq 's, Lemma 2.2 can be extended by utilizing standard symbolic integration methods. Clearly, hardware resources will limit the number of computable q i 's. We report the result for q 3 and q 4 in Appendix B.

Remark 2.2. We also computed the q i 's with the method described in Hall (1992) up to order 6 with complete coincidence. The method proposed in [START_REF] Hall | The bootstrap and Edgeworth expansion[END_REF] involves the computation of moments of more complicated statistics and seems more time consuming.

Rates of convergence and generalized self-normalized sums

In case of α 3 = 0 we obtain q 1 (y) ≡ 0. This vanishing of q 1 for vanishing skewness of ζ 1 can be interpreted in that way that on the n -1/2 scale, the

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Edgeworth approximation of the cdf F n of the statistic S n cannot be distinguished from Φ. On the n -1 scale, however, there are always detectable differences between the two cdf's, because it is impossible to find a value for α 4 such that q 2 becomes the null polynomial. In other words, the rate of convergence of F n towards Φ can at most achieve O(n -1 ).

In the remainder of this section, we will present an approach to improve this rate of convergence for certain values of the moments α j . The idea is to utilize a generalized norming sequence a n (say) in the denominator of the t-statistic, i.e., we investigate "generalized self-normalized sums" of the type

T n = √ n ζ/ a n n i=1 (ζ i -ζ) 2 instead of S n .
Hereby, we consider norming sequences a n of the special form

a n = 1 n(1 -M j=1 C j n -j/2 ) (5)
(implying lim n→∞ na n = 1) for some integer M . We now derive a formal expansion for the generalized self-normalized sum T n . Our investigations shall lead to an expansion of the form

F Tn (t) = Φ(t) + r i=1 n -i/2 qi (t)ϕ(t) + o(n -r/2 ) (6)
if the (r + 2)-th moment of ζ 1 exists. The coefficients of the polynomials qi in (6) depend not only on the cumulants of ζ 1 , but additionally on the constants C j appearing in the formal description (5) of the norming sequence a n . Expansion (6) will prove useful in analyzing optimal rates of convergence for the self-normalized sum T n by choosing optimal norming constants C j .

Notice that

T n = S n /b n with b n = √ na n . Therefore, F Tn (t) = P(T n ≤ t) = P(S n ≤ b n t) = F n (b n t).
From (1) we get under the necessary moment

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

condition that for given r ≥ 1 it holds

F Tn (t) = Φ(b n t) + r i=1 n -i/2 q i (b n t)ϕ(b n t) + o(n -r/2 ). ( 7
)
In order to compute the qi 's, we now have to find suitable expansions of Φ(b n t) and ϕ(b n t) in terms of Φ(t) and ϕ(t). This can be done by noticing that we can express the leading terms on the right-hand side of ( 7)

(which is the expansion for F Tn ) by Φ(b n t)+ r i=1 n -i/2 q i (b n t)ϕ(b n t) = Φ(t)+ ϕ(t) h n (t) + r i=1 n -i/2 q i (b n t)g n (t)
, where the auxiliary functions h n and g n are defined by

h n (t) = Φ(b n t) Φ(t) -1 Φ(t) ϕ(t) , g n (t) = ϕ(b n t)/ϕ(t).
Hence, if we find suitable expansions for b n , h n (t) and g n (t), the work is done, because the qi 's can then simply be obtained by collecting the n -i/2 terms of the expansions of h n and g n and putting them together with the original q i 's, evaluated at the transformed argument values b n t. Indeed, the required expansions for b n and g n (t) are simple applications of the Taylor series of the square root and the exponential functions. For the expansion of h n (t), well-known asymptotic expansions for Mills' ratio are needed additionally.

We provide the expansions up to order n -2 in Appendix C.

Having expanded h n (t) and g n (t) in this manner, we finally obtain the first two qi 's as q1 (t) = α 3 t 2 /3+C 1 t/2+α 3 /6 and q2 (t

) = 3tC 1 2 /8+α 4 t 3 /12+ α 3 2 t/6 -t 3 C 1 2 /8 -α 3 2 t 3 /9 -α 3 2 t 5 /18 + α 3 C 1 t 2 /4 + tC 2 /2 -t 3 /2 -α 3 C 1 t 4 /6 - α 4 t/4
. We omit expressions for qi for i > 2 here because of lack of space.

Remark 3.1. Setting M = 2, C 1 = 0 and C 2 = 1 in (5), we get the stu- Theorem 3.1. (Rate of convergence for generalized self-normalized sums).

dentized sum Sn = √ n ζ/ (n -1) -1 n i=1 (ζ i -ζ)
Let ∆ n (x) = |F Tn (x) -Φ(x)|. (i) If α 3 = 0 or C 1 = 0, then ∆ n (x) = O(n -1/2 ). (ii) If α 3 = C 1 = 0 and (α 4 = 6 or C 2 = 3), then ∆ n (x) = O(n -1 ).
(iii) If α 3 = C 1 = 0 and α 4 = 6 and C 2 = 3 and (α 5 = 0 or C 3 = 0), then

∆ n (x) = O(n -3/2 ).
(iv) If α 3 = C 1 = 0 and α 4 = 6 and C 2 = 3 and α 5 = C 3 = 0, then

∆ n (x) = O(n -2 ).
Proof. We have to subsequently solve qi (t) ≡ 0 for C i and α i+2 for i = 1, 2, 3.

For i = 1, this is trivial. Plugging in α 3 = C 1 = 0 in q2 (t) leads to q2 (t) = (α 4 /6 -1)t 3 /2 + (C 2 -α 4 /2)t/2, hence, to the assertion in case (ii).

With α 3 = C 1 = 0 and α 4 = 6 and C 2 = 3, we obtain q3 (t) = -α 5 t 4 /20α 5 t 2 /5+tC 3 /2-α 5 /40, and part (iii) of the theorem is proven. For the proof of part (iv), we finally set α 3 = C 1 = 0, α 4 = 6, C 2 = 3 and α 5 = C 3 = 0 leading to q4 (t) = (3/2 -α 6 /45) t 5 + (α 6 /18 -11/4) t 3 + (α 6 /6 + C 4 /2 -45/4) t.

Since the coefficients in the latter polynomial are linearly independent, it is impossible to find values for (α 6 , C 4 ) such that q4 becomes the null polynomial and therefore the assertion of part (iv) follows.

Theorem 3.1 shows that the generalized self-normalized sum T n can lead to a rate of convergence of ∆ n (x) up to O(n -2 ), which is an improvement over the approximation quality of S n .

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT

latter case, T n will be denoted as t ν in the remainder. Equation ( 8) gives insight in the asymptotic magnitude of the deviation of the cdf of T n from F tν . It might be interesting to analyze if a t-approximation of T n will lead to an improved rate of convergence in comparison with a normal approximation.

Especially, we aim at answering the question if the norming constants C j can be employed to correct for higher-order moment differences of ζ 1 compared with a standard normal variate.

In order to derive the Q i 's in (8), we denote by q * i , i = 1, . . . , k, the approximation polynomials that arise for T n = t ν in (6), i.e., the qi 's for special choices M = 2, C 1 = 0 and C 2 = 1 in (5) and by setting the α j 's equal to the (central) moments of the standard normal distribution.

By subtracting the resulting expansion from the general expansion for T n , we immediately conclude that Q i (t) = qi (t) -q * i (t), i = 1, . . . , r. Carrying out these calculations explicitly, we obtain the first four q * i (t) as q * 1 (t) = q * 3 (t) ≡ 0, q * 2 (t) = -t 4 (t 2 + 1) , q * 4 (t) = -t (3 t 6 -7 t 4 + 19 t 2 + 21) /96. Consequently, the first two Q i 's are given by Q 1

(t) = q1 (t), Q 2 (t) = -α 2 3 /18t 5 - α 3 C 1 t 4 /6 -(1/4 + α 2 3 /9 -α 4 /12 + C 2 1 /8) t 3 + α 3 C 1 /4t 2 + (3C 2 1 /8 + C 2 /2 + α 2 3 /6 -α 4 /4 + 1/4)t.
Considering rates of convergence, it becomes apparent that Q 1 only vanishes for α 3 = C 1 = 0 and Q 2 only vanishes if additionally α 4 = 3 and C 2 = 1, i.e., the first four moments of ζ 1 coincide with a standard normal variate and the first two norming constants C 1 and C 2 are chosen as in the classical t-distribution case. As stated in Corollary 4.1 below, analogous requirements are also necessary for vanishing polynomials Q 3 to Q 8 . As a conclusion, utilizing the t-approximation (8) instead of the normal approximation (6) does not help to increase rates of convergence, since norming where the C k 's denote the norming constants in the denominator of T n as appearing in the definition of a n in (5).

|F Tn -F tν | = O(n -k * /2 ), where k * = min{k ∈ {1, . . . , M } : ∆α k+2 = 0 ∨ ∆C k = 0}. If no such k * exists, then |F Tn -F tν | = o(n -M/2 ).
Since each polynomial Q i or Qi , respectively, only depends on α j , j = 3, . . . , i + 2 and C j , j = 1, . . . , i, and equations ( 8), ( 9) are valid for T n = t ν , it is clear that also for M > 8 the conditions

∆α i+2 = 0 ∧ ∆C i = 0 for all i = 1, . . . , M (10)
imply Q i (t) ≡ 0 and Qi (t) ≡ 0 for all i = 1, . . . , M , i. e., conditions (10) are sufficient for vanishing polynomials up to the M -th. We conjecture that conditions (10) are also necessary conditions for any M ≥ 1 as stated in Corollary 4.1 for 1 ≤ M ≤ 8.

Remark 4.1. Maple worksheets for computing the polynomials appearing in the Edgeworth expansions investigated in this paper are available at http: //www.helmut-finner.de. One worksheet computes the polynomials q i with Chung's 1946 method up to arbitrary order. A further worksheet computes these polynomials for i ≤ 5 with the method described in [START_REF] Hall | The bootstrap and Edgeworth expansion[END_REF] based on a smooth function approach. Finally, as an additional crosscheck we provide a worksheet based on the Bartlett-Gayen method for the polynomials Qi with norming sequence (n -1) -1 .

A C C E P T E D M A N U S C R I P T

ACCEPTED MANUSCRIPT

A. Corrected derivatives g (i) (0), i = 1, . . . , 4

g (1) (0) = 1 2 z(α 4 -1) 1/2 x, g (2) (0) = -z 3 -1 4 z(α 4 -1)x 2 , g (3) (0) = -3 2 z 3 (α 4 -1) 1/2 x + 3 8 z(α 4 -1) 3/2 x 3 , g (4) (0) = 9z 5 + 3 2 z 3 (α 4 -1)x 2 -15 16 z(α 4 -1) 2 x 4 .

B. Polynomials q 3 and q 4

Theorem B.1. The polynomials q 3 and q 4 appearing in the Edgeworth expansion (1) are given by q 3 (y) = -

1

  2 and the corresponding ap-proximation polynomials are just the ones given in Section 11.4.2 of Lehmann and Romano (2005).

Corollary 4. 1 .

 1 Assume that the (M + 2)-nd moment α M +2 of ζ 1 is finite for some integer 1 ≤ M ≤ 8 and the distribution of ζ 1 has a nontrivial absolutely continuous component with respect to the Lebesgue measure. Then

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Remark 3.2. As pointed out in Remark 3.1, the studentized sum Sn can be generated from T n by setting C i = 0 for all i = 2 and C 2 = 1. According to Theorem 3.1, it can therefore only achieve a rate of convergence of O(n -1 ).

A justification for the special role of C 2 = 3 can be given by noticing that the norming a n = (n -3) -1 leads to variance standardization of T n , that is, VarT n = 1, if the ζ i are iid normally distributed as N(0, 1). The special role of α 4 = 6 in parts (ii) -(iv) of Theorem 3.1 is not clear to us.

Finally, we give an example where Theorem 3.1 (iv) applies.

Example 3.1. Let ϕ(x|σ) denote a normal density with respect to mean 0 and variance σ 2 and suppose the density u (say) of ζ 1 is a mixture of two normal densities, more precisely, let u

1 = 0, Eζ 2 1 = 1 and Eζ 4 1 = 6. Hence, noticing that Eζ 6 1 = 90 and setting C 1 = C 3 = 0 and C 2 = 3, Theorem 3.1 (iv) applies in this case.

An Edgeworth-type expansion in terms of Student's t-distribution

The investigations in Section 3 can be utilized in a very straightforward way to derive an Edgeworth-type expansion for T n of the form

in terms of Student's t-distribution with ν = n -1 degrees of freedom the cdf of which is denoted by F tν in formula (8). Note that a T n with norming sequence a n = (n -1) -1 and ζ 1 ∼ N (0, 1) is exactly t ν -distributed. In the

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

cannot be used to compensate for moment differences.

One could try to substitute ϕ in (8) by the pdf f tν of the t ν -distribution, i.e., consider an expansion of the form

Closely related expressions for F Tn for fixed n have already been investigated in 1949 by A. K. Gayen in [START_REF] Gayen | The distribution of 'Student's' t in random samples of any size drawn from non-normal universes[END_REF] based on M. S. Bartlett's paper [START_REF] Bartlett | The effect of non-normality on the t distribution[END_REF] by first deriving expressions for the corresponding pdf f Tn .

In order to derive explicit expressions for the first four Qi (t)'s in (9), we make use of the expansion

Substituting the latter expansion in (8) leads to Qi ≡ Q i , i = 1, 2, and

Unfortunately, we could only reproduce [START_REF] Gayen | The distribution of 'Student's' t in random samples of any size drawn from non-normal universes[END_REF] results up to order n -1 . Taking limits (n → ∞) in Gayen's paper also yields Qi ≡ Q i , i = 1, 2. However, the expressions of order O(n -3/2 ) associated with the factors α 3 3 and α 3 α 4 seem to be in error in [START_REF] Gayen | The distribution of 'Student's' t in random samples of any size drawn from non-normal universes[END_REF], p. 359, and also taking limits (n → ∞) in these expressions does not coincide with our result. Anyhow, we recomputed the original approximation method by Bartlett (cf. Bartlett (1935)) which underlies [START_REF] Gayen | The distribution of 'Student's' t in random samples of any size drawn from non-normal universes[END_REF] calculations and reproduced the Qi 's for i = 1, . . . , 4. Utilizing Chung's method, the derivations in Sections 3 and 4 and higher order expansions for the ratio ϕ(t)/f tν (t), we calculated the polynomials Q i and Qi up to order 8. Therewith, we can finally summarize our results regarding rates of convergence of |F Tn -F tν | as follows. To this end, denote the


Lucia Alessi

Matteo Barigozzi

Marco Capasso

Improved penalization for determining the number of factors in approximate factor models

Keywords: Number of factors, Approximate factor models, Information criterion, Model selection AMS-classification: 62M10, 91B84

 in the context of dynamic factor models. Simulations show that our method in general delivers more reliable estimates, in particular in the case of large idiosyncratic disturbances.

Introduction

Factor analysis is a very popular dimension reduction technique used in many disciplines as e.g. econometrics, statistics, signal processing, psychometrics, chemometrics. It allows to account for the "pervasive" cross-correlations present among the observed series of large datasets. Such correlations are summarized by means of few latent variables (the factors) which are common to all variables. We assume to observe an infinite sequence of nested vector stochastic processes {x nt = (x 1t . . . x nt ) , n ∈ N, t ∈ Z}, driven by a finite number r of unobserved factors:

x nt = Λ n F t + ξ nt , t ∈ Z. ( 1
)
F t is the r × 1 vector of factors, and Λ are the corresponding n × r loadings. The process

x nt is therefore represented as the sum of two components which we assume to be orthogonal: a common component Λ n F t , and an idiosyncratic component ξ nt . The latter is allowed to be mildly cross-correlated and this sense we say that ( 1) is an approximate factor model.

Typically r << n, but r is unknown, and its estimation is a crucial step in the identification of the model.

The common and idiosyncratic components are disentangled for n going to infinity, while consistency of the estimation is achieved when both n and T (the sample size) go to infinity.

From this double-asymptotic result it is clear the necessity of having a large cross section of long time series in order to estimate consistently (1). When datasets are large in both the time (T ) and the cross-section (n) dimensions, determining the number of common factors is particularly difficult as traditional information criteria as BIC or AIC, which are consistent for T diverging but for finite n, cannot be applied anymore. In this double-asymptotic framework the reference criterion for determining r is by [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF], who propose a consistent estimator as both n and T diverge. In practice, the method they propose is known to often deliver non-robust results as the number of factors can be over-or under-estimated (see e.g. the application on U.S. macroeconomic data in [START_REF] Forni | Opening the black box: Structural factor models versus structural VARs[END_REF].

The aim of this paper is to improve the penalization in the criterion by [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF].

Following [START_REF] Hallin | Determining the number of factors in the general dynamic factor model[END_REF], who propose a similar criterion in the case of the Generalized Dynamic Factor Model by [START_REF] Forni | The generalized dynamic factor model: identification and estimation[END_REF], we introduce in the penalty function a new parameter in order to tune its penalizing power. We get estimates of the number of static factors which are relatively more robust to different specifications of the criterion, namely when we have heteroskedastic and/or large idiosyncratic components. Finally, the consistency properties of our estimator are exactly the same as those of the original one by [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF].

The criterion we use is based on the key identifying assumption of this class of models. Namely, we want to find r such that all the eigenvalues of the idiosyncratic covariance matrix are bounded for n diverging. The simplest method to determine r is the "Scree-test". [START_REF] Cattel | The Scree test for the number of factors[END_REF] observed that the graph of the eigenvalues (in descending order) of an uncorrelated data set forms a straight line with an almost horizontal slope. Therefore, the point in the eigenvalue graph where the eigenvalues begin to level off with a flat and steady decrease is an estimator of the sufficient number of factors. Obviously such a criterion is often fairly subjective, because it is not uncommon to find more than one major break in the eigenvalue graph and there is no unambiguous rule to use. Following this intuition, the criterion by [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] minimizes the variance of the idiosyncratic component. Other recent criteria based on the eigenvalues of the covariance matrix are in [START_REF] Yao | Modelling multiple time series via common factors[END_REF] and [START_REF] Onatski | Determining the number of factors from empirical distribution of eigenvalues[END_REF]. Our procedure could be adapted also to these other studies, but we limit ourselves to [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF], being this the most known criterion in the field of time series analysis.

2 Determining the number of factors [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] consider an approximate factor model where they allow for serial dependence and heteroskedasticity of ξ nt , and for weak dependence between factors and idiosyncratic series (we refer to their paper for a detailed description of the assumptions). In such a model, when both n and T diverge, the factors F t and the loadings Λ n can be estimated by means of asymptotic principal components. If T > n, then, after imposing Λ n Λ n /n = I r , the estimated covariance matrix of the observables Γ T n is n × n and Λ T n are √ n times the eigenvectors associated to the r largest eigenvalues of Γ T n , and

F T t = Λ T n x nt /n. If instead T < n analogous
estimators can be obtained from a T × T estimator of the covariance matrix simply by exchanging the roles of the factors and their loadings. Theorem 1 in [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] proves consistency of these estimators and for a more general class of factors' estimators, provided that they all span the r-dimensional space of the common components.

The number of factors r is such that all eigenvalues of the idiosyncratic covariance matrix are bounded for n diverging (see assumption C in [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF]. Assume to have a given number of factors k, then the cross-sectional average variance of the idiosyncratic component is a function of k estimated factors

F (k)T t
, hence is a function of k:

V (k) = 1 nT n i=1 T t=1 x nit -λ (k)T ni F (k)T t 2 . ( 2
)
Clearly V (k) is minimized for k = n, but overparametrization can be avoided by introducing a penalty function p(n, T ). Therefore, the resulting criterion is (we consider here only the log-version of the [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] criterion being the recommended one by the authors):

r T n = argmin 0≤k≤rmax IC T n (k) = argmin 0≤k≤rmax log [V (k)] + kp(n, T ), (3)
r max being the maximum number of factors allowed. Finally, provided that p(n, T ) has the required asymptotic properties, r T n is consistent as n and T diverge (see Theorem 2 in Bai and Ng, 2002, for details).

3 Improved penalization and the choice of r

The information criterion (3) has the property, exploited also by [START_REF] Hallin | Determining the number of factors in the general dynamic factor model[END_REF] in the context of dynamic factor models, that a penalty function p(n, T ) leads to a consistent estimate of r if and only if cp(n, T ) does, where c is an arbitrary positive real number. Thus, multiplying the penalty by c has no influence on the asymptotic performance of the identification method.

However, for given finite n and T , the value of a penalty function p(n, T ) satisfying (3) can be arbitrarily small or arbitrarily large, and this indeterminacy can affect the actual result quite dramatically. [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] propose three choices for the penalty function and indicate the corresponding criteria as IC 1 , IC 2 , and IC 3 . However, only the first two are known to behave well in empirical applications. We therefore propose the two "modified" information criteria:

IC T * 1,c,n (k) = log [V (k)] + c k n + T nT log nT n + T , c ∈ R + , IC T * 2,c,n (k) = log [V (k)] + c k n + T nT log(min{ √ n, √ T }) 2 , c∈ R + .
The estimated number of factors is now also a function of c and, depending on the chosen criterion, is given by

r T c,n = argmin 0≤k≤rmax IC T * i,c,n (k) with i = 1, 2,
the consistency proof in Theorem 2 by [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] being still valid.

The degree of freedom represented by c can be exploited when implementing the criterion in practice. We follow the procedure proposed by [START_REF] Hallin | Determining the number of factors in the general dynamic factor model[END_REF]. 1 The only available information about the asymptotic behavior of r T c,n comes from considering subsamples of sizes (n j , τ j ) with j = 0, . . . , J such that n 0 = 0 < n 1 < n 2 < . . . < n J = n and τ 0 = 0 < τ 1 < τ 2 < . . . < τ J = T . For any j, we can compute r τ j c,n j , which is a non-increasing function of c.

Assume r > 0. According to the value of c, we have different behaviors of r τ j c,n j as function of j.

No penalty If c = 0 then r τ j 0,n j = r max , indeed no penalization is imposed.

Underpenalization If c > 0, but small, Theorem 2 applies but, in practice, as j increases r τ j c,n j increases to r max and would converge to r only if n and T would increase without limits.

In this case we overestimate r.

Overpenalization When c becomes large, r τ j c,n j tends to zero for any j and r is underestimated.

Due to the monotonicity of r τ j c,n j as a function of c, between the "small" underpenalizing values of c and the "large" overpenalizing values, there must exist a range of "moderate" values of c such that r τ j c,n j is a stable function of the subsample size (n j , τ j ). The stability with respect to

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

sample size can be measured by the empirical variance of r τ j c,n j as a function of j, i.e.

S c = 1 J J j=1 r τ j c,n j - 1 J J j=1 r τ j c,n j 2 .
In Figures 1(a)-1(c) and 1(b)-1(d) we show respectively the behavior of r [

FIGURE 1 HERE]

Notice that, there could be more than one interval of c satisfying these requirements, as the examples shows. In these cases, an explanation analogous to the theoretical argument by [START_REF] Hallin | Determining the number of factors in the general dynamic factor model[END_REF] in the case of dynamic factor models suggests that the relevant interval is the second stability interval, i.e. the smallest values of c for which r

ACCEPTED MANUSCRIPT

simulations that our criterion, that considers the second stability interval corresponding to c, delivers a number of factors that is always lower or equal to the correct number of factors, but it is never greater. Therefore, considering stability intervals corresponding to higher values c > c can never improve the estimate.

[FIGURE 2 HERE] 4 Simulations

In this Section, we conduct a set of simulation experiments to evaluate the performance of our proposed criterion, relative to that of the [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] criterion, in finite samples. The baseline model for all simulations is:

x it = r j=1 λ ij F tj + √ θξ it i = 1, . . . , n, t = 1, . . . , T,
with factors and factor loadings distributed as N(0, 1). We consider four Data Generating Processes (DGPs) similar to those in [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF].

DGP1 Homoskedastic idiosyncratic components ξ it ∼ N(0, 1).

DGP2 Heteroskedastic idiosyncratic components

ξ it = ⎧ ⎪ ⎨ ⎪ ⎩ ξ 1 it if t odd ξ 1 it + ξ 2 it if t even , ξ 1 it , ξ 2 it ∼ N(0, 1).
DGP3 Cross-sectional correlations among idiosyncratic components

ξ it = v it + J j =0 j=-J βv i-jt , v it ∼ N(0, 1), β = 0.2, J = max{n/20, 10}.
DGP4 Serial correlation among idiosyncratic components

ξ it = ρξ it-1 + v it , ξ it ∼ N(0, 1), v it ∼ N(0, 1), ρ = 0.5.
For each model we set r ∈ {1, 5} and θ ∈ 1 2 r, r, 3r, 5r , thus assigning to the idiosyncratic component a variance that is respectively one half, one, three or five times the variance of the common component. All parameters are set as in [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF]. We generate samples having size n = 200 and T = 200. We thus have four variance-ratio settings, and for each of them we have two values of r, one sample size and four DGPs. We set r max = 10, n 1 = 3 4 n , n j+1 = n j + 1, where [x] denotes the integer part of x, and c ∈ (0, 5] with step size of 0.01. For simplicity, we do not consider subsamples in the time dimension. For each case, we simulate 1000 samples and every time we run the original [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] criterion IC 1 , and its modified version IC * 1 .2

[TABLES 1 AND 2 HERE]

Tables 1 and2 show, for θ ∈ 1 2 r, r, 3r, 5r , the number of times in which a given number of factors is selected by IC 1 and IC * 1 . First, we consider the baseline scenario of equal variance for idiosyncratic and common components (θ = r). When there is only one factor, our criterion performs slightly worse than the original criterion. There is however one exception:

when we allow for cross-correlation of the idiosyncratic components (DGP3) our criterion does not diverge and is able to detect the correct number of factors in more than 80% of the cases. Instead, the [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] criterion applied to DGP3 always diverges giving r T c,n = r max = 10. Similar results hold for the case of a common variance which is double than the idiosyncratic variance (θ = 1 2 r). IC * 1 performs similarly to IC 1 for DGP1, DGP2 and DGP4, and performs much better for DGP3. The advantages of our criterion become clear when we raise the idiosyncratic variance to three times the common variance (θ = 3r). As already observed by [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF], a high idiosyncratic variance negatively affects the ability of ICs criteria in determining the correct number of factors. IC * 1 is able to work also under such noisy conditions. When considering DGP3, IC 1 diverges and suggest the maximum possible number of factors. When there is only one factor in the DGP, IC * 1 performs slightly worse than IC 1 for DGPs 1, 2 and 3, although this happens in less than 10% of cases. When

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

the number of factors is higher, our criterion outperforms the IC 1 providing in almost all the cases the right number of factors. See for example the case of 5 factors: out of 1000 replications, our criterion was able to retrieve the right number of factors in 999 cases for DGP1 (against 967), in 999 cases for DGP2 (against 32), and in 977 cases for DGP4 (against 418). If the idiosyncratic variance is raised to five times the common variance (θ = 5r) and we still consider 5 factors, the gain obtained by our proposed criterion is even more evident: out of 1000 replications, the right number of factors is retrieved in 998 cases for DGP1 (against 1), in 969 cases for DGP2 (against 0), and in 482 cases for DGP4 (against 0). For DGP3, again we have the same qualitative results: IC 1 diverges, while IC * 1 suggests 0 factors in 674 cases and a number between 1 and 9 factors for the remaining cases. Summing up, our refinement makes the [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] criterion a much more useful tool in practical contexts. Indeed, it enormously reduces the probability of large mistakes and always provides a reliable answer, even when, in the presence of a factor structure in the DGP, the dataset presents some features of high idiosyncratic variance or heteroskedasticity that would prevent the traditional criteria from suggesting a finite positive number of factors. Notice that this success is not a technical artifact depending on a tendency to always retrieve a positive number of factors. It is easy to check that when there is no factor structure in the simulated data, the proposed criterion suggests a number of factors equal to zero, exactly as the [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] criterion would do.

Empirical applications

We test the performance of our procedure by means of two empirical applications on macroeconomic and financial datasets. In the first case we take a dataset which has been used in many applications of factor models, (see e.g. [START_REF] Stock | Implications of dynamic factor models for VAR analysis[END_REF][START_REF] Hallin | Determining the number of factors in the general dynamic factor model[END_REF]. It [FIGURE 3 HERE] 6 Conclusions

In this paper we refine the [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] criterion, which is one of the most popular criteria available for addressing this issue. The appeal of our new method is three-fold: (i) it builds on a well known criterion, whose theoretical properties have been proved and are preserved; (ii) it improves the finite sample performance of the original criterion; (iii) it is easy to implement. In particular, our procedure is capable of giving an answer even when the original criterion does not converge. In general, we obtain more robust results with respect to the [START_REF] Bai | Determining the number of factors in approximate factor models[END_REF] criterion, especially when the variance explained by the common factors is relatively low.

This result constitutes an improvement in the analysis of datasets where comovements among variables are hidden by large idiosyncratic disturbances. For example, in financial applications we often find few common factors explaining a small percentage of the total variance. These factors are however of great importance for the structural analysis of financial markets (see e.g. [START_REF] Engle | Asset pricing with a factor-ARCH covariance structure: Empirical estimates for Treasury bills[END_REF], where the unique factor is interpreted as the "market factor"). Although some authors identify the factor decomposition by requiring the idiosyncratic components to be "small" or "negligible" (e.g. in the case of principal component analysis), such characterization is not reflecting the fundamental nature of factor models: idiosyncratic components indeed can be "large" and strongly autocorrelated, while the common component can be just white noise, i.e. serially uncorrelated. The ability of identifying the correct number of factors in large but finite and apparently heterogeneous datasets is therefore highly desirable.

  j as a function of n j and of r T c,n and of S c as functions of c, when simulating data from 2 different Data Generating Processes (DGP1 or DGP2 as defined in the next Section) with r = 5. These Figures suggest that the selection of the number of factors can be based on the inspection of the family of curves (n j , τ j ) → r τ j c,n j , indexed by c ∈ (0, c max ], trying to find values of c (i.e. curves) such that for c ±δ (δ > 0), r τ j c-δ,n j ↑ r τ j c,n j and r τ j c+δ,n j ↓ r τ j c,n j , as j → J, see Figures 1(a) and 1(c). The search can be made automatic by considering the mapping c → S c and by choosing r T c,n = r T b c,n , where c belongs to an interval of c implying S c = 0 and therefore a constant value of r T c,n as a function of c, see Figures 1(b) and 1(d).

  j is a constant function of j (the first stability interval corresponds always to the boundary solution r T c,n = r max and it is thus a non-admissible solution). The intuition behind this goes as follows: if the correct number of factors is r and the second stability interval correctly identifies it, by increasing the penalty it is generally possible to have another stability interval corresponding to a smaller number of factors r * < r, but in this case we are overpenalizing. As an empirical test of the above argument, in Figure 6 we show the estimated number of factors averaged over 100 simulations of the 4 DGPs considered in the next Section. The number of factors is r = 5 and we see that the second stability interval always delivers an estimated number r T c,n which is closer to r than the number suggested by the other intervals. In the next Section we show by A C C E P T E D M A N U S C R I P T

  comprises 132 series of monthly macroeconomic indicators of the U.S. economy from January 1960 to December 2003 for a total of 528 time observations. 3 In a second exercise we consider daily volatilities (proxied by the adjusted high-low range) of 93 assets belonging to the New A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT York S&P100, from 1 st January 2001 to 31 st December 2008 for a total of 2008 time observations. In Figures 3(a) and 3(b) we report results obtained for the macroeconomic application. IC * 1 indicates the presence of 6 factors. The original criteria IC 1 and IC 2 suggests 7 factors. In Figures 3(c) and 3(d) we report the results for the financial dataset. IC * 1 indicates the presence of 3 factors, while the original ICs always indicates 5 factors.

Figure 1 :Figure 2 :

 12 Figures and tables

  Figure 3: IC * 1 criterion for the two observed datasets; top row: macroeconomic data; bottom row: financial data; left column: r T c,n j as function of n j for different values of c; right column: r T c,n (solid line) and S c (dashed line) as functions of c.

All estimations in this paper were performed using Matlab (R2007a). The code is available at http://www.barigozzi.eu/research.html.

We only show a selection of results. Results for smaller samples with n = 50 and T = 50, for r =

3, and for the IC 2 and IC * 2 criteria, are available upon request.

Data are available at http://www.princeton.edu/˜mwatson


Keywords: Meta-Analysis, Environmental Kuznets Curve, Deforestation, Development C12, C83, Q23, O13

Highlights

• We carry out a meta-analysis of Environmental Kuznets Curve (EKC) studies for deforestation.

• We analyze the results of 71 studies, offering a total of 631 estimations.

• We investigate the incidence of choices made by authors (econometric strategy, deforestation measure, temporal coverage, geographical area, measure of economic development…) on the probability of finding an EKC. • After a phase of work corroborating the EKC, we find a turning point after the year 2001.

Mark Twain: "The report of my death was an exaggeration" New York Journal 2 June 1897

Introduction

One important question which is still at stake, even more in an economic crisis context, is whether environmental and economic objectives are compatible subjects.

This question gets high resonance in the literature devoted to the Environmental Kuznets Curve (EKC) for deforestation which is a subject of confrontation between optimistic and pessimistic views of development [START_REF] Carson | The Environmental Kuznets Curve: Seeking Empirical Regularity and Theoretical Structure[END_REF]. This literature has gained considerable expansion in economics as well as in natural sciences [START_REF] Mather | Environmental Kuznets Curves and Forest Trends[END_REF]. From the 1990s onwards, numerous studies following the idea of [START_REF] Grossman | Economic Growth and the Environment[END_REF] tested an EKC for deforestation.

Studying EKCs for deforestation is an important matter for two reasons. sustainable management of forests and valuation of forests carbon stocks [START_REF] Angelsen | Realising REDD+. National Strategy and Policy Options[END_REF][START_REF] Angelsen | Reducing Emissions from Deforestation and Forest Degradation (REDD): an options assessment report[END_REF][START_REF] Engel | Painting the forest REDD? » Prospects for mitigating climate change through reducing emissions from deforestation and degradation[END_REF]. As pointed out by [START_REF] Geist | What drives tropical deforestation? A meta-analysis of proximate and underlying causes of deforestation based on subnational case study evidence[END_REF] the role of forests in mitigating global environmental threats such as climate change and biodiversity erosion is a research imperative and has been motivating considerable efforts to the understanding of patterns and causes of the deforestation process and in fine to the derivation of policy implications [START_REF] Angelsen | Rethinking the Causes of Deforestation: Lessons from Economic Models[END_REF].

Second, EKC has drawn attention from scholars for several years (e.g. (Ma & D. I. Stern 2006). Studying EKCs for deforestation may therefore offer an interesting case in the methodology of economics from a sociological point of view. According to [START_REF] Kuhn | The structure of scientific revolutions 3 rd ed[END_REF], scientists share common beliefs and practices which circumscribe the "normal science". They are often reluctant to diverge from them since they are most of the time evaluated and published according to rules widely accepted within the normal science.

EKCs seem to belong to such a set of practices. It is indeed rather striking that several researchers seem to consider it as "econometrics as usual" i.e. a relevant hypothesis which helps understanding environmental degradation and for instance the deforestation process. Other researchers have conducted critical reviews and forcefully argued in favour of its obsolescence [START_REF] Levinson | The ups and downs of the environmental Kuznets curve[END_REF]D. I. Stern 2004). Thus there seems to be a discrepancy between, on the one side, researchers who dismiss EKCs and, on the other side, scientists who consider that EKCs are relevant. Among the latter, the EKC is still presented as one of the hypotheses explaining the forest transition process [START_REF] Barbier | The forest transition: Towards a more comprehensive theoretical framework[END_REF][START_REF] Rudel | Forest transitions: towards a global understanding of land use change[END_REF][START_REF] Mather | The Forest Transition[END_REF]).

The objective of this paper is to examine why scholars get accustomed to the EKC for deforestation or whether they have "objective" reasons to dismiss it. In contrast with existing studies or critical surveys on the EKC for deforestation, we propose to address the subject with a meta-analysis which can be considered as an attempt to identify "wheat from chaff" [START_REF] Stanley | Wheat from Chaff: Meta-Analysis as Quantitative Literature Review[END_REF] i.e. identify a set of reasons that favour or falsify empirical evidences of EKCs for deforestation.

The rest of the paper is organised as follows. Section 2 presents the debate in the literature on the EKC for deforestation. Section 3 further justifies the use of metaanalysis. Section 4 is devoted to the construction of the database. Section 5 develops on the empirical strategy and the econometric results. Section 6 discusses the results and Section 7 concludes.

Contrasted Premises and Results

The story with EKCs started with [START_REF] Grossman | Economic Growth and the Environment[END_REF] and [START_REF] Panayotou | Empirical Tests and Policy Analysis of environmental Degradation at Different Stages of Economic Development[END_REF].

According to (Beckerman 1992, p.482) : "[…] there is clear evidence that, although economic growth usually leads to environmental deterioration in the early stages of the process, in the end the best -and probably the only -way to attain a decent environment in most countries is to become rich." This optimistic premise was later mitigated by the [START_REF] Arrow | Economic Growth, Carrying Capacity, and the Environment[END_REF]) assertion according to which economic growth is not a sufficient condition for getting environmental improvements or curbing environmental degradation. It is even more doubtful to find EKCs when stock variables with strong potential feedback are expected in, for instance, ecosystems such as tropical forests.

The economic literature provides theoretical underpinnings for an EKC. Many of them model pollutants emissions -income relationships [START_REF] Andreoni | The simple analytics of the environmental Kuznets curve[END_REF][START_REF] Xepapadeas | Economywide and International Environmental Issues[END_REF]. [START_REF] Munasinghe | Is environmental degradation an inevitable consequence of economic growth: tunneling through the environmental Kuznets curve[END_REF]) develops a static model showing that appropriate policy measures (i.e. removal of subsidies on timber exports) could help alleviating the pressure on tropical forest stocks and thus "tunnelling" through the EKC for deforestation. Theoretical explanations of EKCs for deforestation were also developed with the so-called poverty-environment hypothesis as opposed to the capital or frontier model [START_REF] Geist | What drives tropical deforestation? A meta-analysis of proximate and underlying causes of deforestation based on subnational case study evidence[END_REF][START_REF] Rudel | The paths to rain forest destruction: Crossnational patterns of tropical deforestation, 1975-1990[END_REF]. The contention that economic development has a negative (positive) impact on deforestation is suggested by the former (latter). Theoretical models explaining agricultural land expansion may also be compatible with an EKC for deforestation [START_REF] Barbier | The Economics of Tropical Deforestation and Land Use: An Introduction to the Special Issue[END_REF]. It is however worth to notice that several authors are rather unsatisfied with the current theoretical literature on the EKC for deforestation and call for further developments (Roy Chowdhury & Moran 2012). Early econometric results [START_REF] Shafik | Economic Growth and Environmental Quality. Time-Series and Cross-Country Evidence[END_REF] and further ones on the existence of an EKC for deforestation were not conclusive. More than a half over more than 600 estimations published since 1992, do not corroborate the EKC (Figure 1). This corroboration. As mentioned by [START_REF] Koop | Is there an environmental Kuznets curve for deforestation[END_REF], FAO databases were mostly used.

Early FAO databases were from FAO production yearbook. Supposedly, more recent ones from Forest Resources Assessment published in most recent studies have greater reliability. Irrelevant econometric methods are also put forward. Early estimates relied on cross-section data which imply restrictive hypotheses [START_REF] Dinda | Environmental Kuznets Curve Hypothesis: A Survey[END_REF][START_REF] Koop | Is there an environmental Kuznets curve for deforestation[END_REF]. More recent studies thus implemented panel data or more generally pooled cross sectional time series data. It may be worth to examine whether improvements made in econometric devices had an impact on the existence of an EKC for deforestation.

One may thus be confused between on the one side, Stern's contention according to which "most of the EKC literature is econometrically weak" (D. I. [START_REF] Stern | The Rise and Fall of the Environmental Kuznets Curve[END_REF]Stern , p.1420) ) and empirical analyses still relying on or testing the EKC for deforestation. But one may understand the situation while remembering [START_REF] Mccloskey | The Rhetoric of Economics[END_REF]. It seems that economists need more than statistical tests to get rid of a hypothesis like the EKC. Would they give weight to the rhetoric surrounding the EKC? In this paper, we want to check whether characteristics of publications dealing with EKCs for deforestation play a role in their corroboration. We argue that conducting a meta-analysis of EKCs for deforestation equations can contribute to their critical assessment.

The Use of Meta-Analysis

A meta-analysis produces, through a set of statistical techniques, an overall summary of empirical knowledge on a specific topic. "Meta-analysis is the analysis of empirical analyses that attempts to integrate and explain the literature about some important parameter" (Stanley & Jarrell 1989, in (Stanley et al. 2008, p.163). "It can explain the excess study-to-study variation typically found in empirical economics, uncover the trace of statistical power that is associated with a false theory, and see through the distortion of publication selection and misspecification biases" (Stanley 2005, in (Stanley et al. 2008, p.163).

Before the 90's, scientists synthesised the results of the research literature through the use of narrative review. However, the limit of this method was quickly reached with the increase in the amount of information available. Indeed it seems inconceivable using this method to successfully synthesize the results of dozens of studies which, moreover, are conflicting in their results. [START_REF] Hunter | Methods of Meta-Analysis[END_REF] explain that narrative surveys developed the so-called "myth of the perfect study": a researcher may be convinced that some studies suffer from methodological flaws and exclude them from the review;

another one may consider different methodological flaws. Hence, idiosyncratic ideas may lead one to reduce available literature on a topic from 100 studies to 10 and another one from 100 to 50.

Meta-analysis allows to address the inherent limitations of the traditional narrative review by providing a more effective approach [START_REF] Stanley | Wheat from Chaff: Meta-Analysis as Quantitative Literature Review[END_REF]; it allows to set up an evaluation with several measures of "outcome"; it permits the use of moderators: some characteristics of the samples used in different studies influence the results obtained.

Moreover, meta-analysis is a good tool for decision support because it helps in the assistance in planning, in the generation of new hypotheses, and eventually, prospective meta-analysis.

Research questions are often the subject of numerous studies leading to various controversies. As mentioned earlier, the EKC for deforestation is no exception, on the contrary, as shown in different reviews of the literature on the subject. Different opinions differ on the question of the existence of an EKC for deforestation and this, especially since empirical analysis validate and invalidate this hypothesis according to econometric models, measures of deforestation, study area, temporal coverage, etc.

Since 1992, empirical work has been accumulated and we even see a significant increase in the number of studies since the 2000s (Figure 1). Finally, despite the abundance of empirical studies, it is difficult to propose clear recommendations arising directly from the relationship between economic growth and deforestation. Our approach, based on the meta-analysis, is complementary with respect to existing surveys (D. I. [START_REF] Stern | Economic growth and environmental degradation: The environmental Kuznets curve and sustainable development[END_REF]D. I. Stern 2004;[START_REF] Dinda | Environmental Kuznets Curve Hypothesis: A Survey[END_REF]). Indeed, it allows us to statistically analyse the variation in results found between existing works and in particular to analyse the influence of choices made by the authors on their results [START_REF] Stanley | Wheat from Chaff: Meta-Analysis as Quantitative Literature Review[END_REF].

There exists meta-analysis for EKC studies. These encompass however various environmental goods. [START_REF] Cavlovic | A meta-analysis of environmental Kuznets curve studies[END_REF] performed a meta-analysis of 25 studies - step is the extraction of data: it is directly applicable to studies that were selected for further analysis after the application of inclusion / exclusion criteria. The fifth step is the statistical analysis. As recommended by [START_REF] Nelson | The Use (and Abuse) of Meta-Analysis in Environmental and Natural Resource Economics: An Assessment[END_REF], we address relevant issues such as heteroskedasticity, the non-independence of regressions, the specification of the model, the non-normality of residuals and multicollinearity.

Construction of the Database

Sampling procedure and sample characteristics

This meta-analysis is based on 71 studies, offering 631 estimations. 1 In a metaanalysis, an important aspect of the work is to find studies that addressed the theme of the investigation. We conducted our research using academic databases such as "Science Direct", "Springer", "RePEc Ideas", "Mendeley", "Wiley", etc. The first step of this research was to enter keywords such as "Kuznets AND Deforestation", "Income AND Deforestation", "Development AND Deforestation", "Poverty AND Deforestation". Given that there are relevant studies which are not identified by bibliographic databases, in a second step we carried out a manual search and this by identifying journals and articles relevant to the topic. We then browsed these studies in order to identify new ones which have addressed closely the subject. A third step was necessary as we had to directly contact the authors since some studies are not accessible on the web. At the end of the research process, we got to build up an information base of nearly 102 items that we have then filtered through inclusion criteria.

At this stage, we selected eligibility criteria to be applied to the bibliographic database. In order not to bias the meta-analysis on subjective considerations, we tried to be broad in our inclusion criteria. First, we looked closely at the dependent variable used in each estimation. Once done, we decided to retain all the measures relating directly to deforestation. At the end of this inclusion work, we have reached 71 studies that were used to build the database of this article.2

Every relevant estimation was carefully examined and the information was gathered to build the database. Accordingly, we surveyed all the information given by the authors related to the EKC (coefficients associated with variables of interest), the sample (size, geographical area …), the econometric method, control variables, and all information considered necessary to carry out this meta-analysis. In terms of coefficients and their sign, they were adjusted according to the nature of the dependent variable and depending on the outcome. For example, when the dependent variable is the rate of deforestation, to obtain an EKC, the coefficient of per capita GDP should be positive and negative for the squared GDP. By cons, if the dependent variable is the stock of forest, the hypothesis of an EKC is confirmed if the coefficient of GDP is negative and that of squared GDP is positive. Also to be as broad as possible we defined our null hypothesis whenever the estimated marginal effects of GDP and GDP squared had the expected signs at the 10% level. 3 We did not try to minimise this type I error since it is detrimental to the power of the test. In brief, we define our dependent variable as EKC which equals to 1 when the existence of the EKC for deforestation is corroborated and 0 otherwise.

Moderator variables

At the end of this work, the database we had raw on hand included 631 observations and numerous differences, most of which focused on the explanatory variables used.

Therefore for the sake of synthesis, these variables were grouped into classes. Table 1 presents the description and descriptive statistics of the variables. Next we shall discuss factors that may affect the income-deforestation relationship. These are the explanatory variables used in our model.

In view of the criticisms of the EKC literature, our variables of interest are the year of publication and the econometric technique.

Regarding the time period, variables are the year of publication of the study, the initial year of data, the final year of data. Differences between studies may reflect numerous issues such an academic trend, the increasing quality and availability of deforestation data, the mix of old and recent data or econometric requirements in peerreviewed journals. For instance, [START_REF] Jordan | The Environmental Kuznets Curve: Preliminary Meta-Analysis of Published Studies[END_REF]) finds that the most recent year of publication is, the more likely to find an EKC decreases, while controlling for the nature of environmental degradation.

With respect to econometric techniques, as pointed out in critical reviews of EKCs, one may cast doubts on econometric techniques in early works. Advances in econometrics may make recent studies more credible than they were in the nineties.

Several authors argued for instance that panel data regressions are more appropriate for analysing the EKC. Indeed, deforestation is a dynamic process therefore the nature of the relationship between income and deforestation should also be dynamic. If the time dimension is ignored, a spurious income-deforestation relationship could be observed. [START_REF] Dinda | Environmental Kuznets Curve Hypothesis: A Survey[END_REF]) argues that cross-section analysis is based on restrictive econometric assumptions as to admit the same constant. What is more, using panel data also allows for controlling for substantial time-invariant unobservable variables (such as countries physical characteristics or historical factors among many others). On top of that, the EKC theory was developed in an a-spatial framework. Hence, we could expect the inclusion of spatial dimension to alter the EKC hypothesis.

In addition to the variables of interest, we propose control variables or moderators.

Concerning the measures of deforestation, we note that in early studies, the latter is not homogeneous. As part of this work 15 different measures are used. It is interesting to investigate whether these modify the income-deforestation relationship. In fact, some measures of deforestation are causes of the latter, as the expansion of agricultural land or infrastructures. [START_REF] Geist | What drives tropical deforestation? A meta-analysis of proximate and underlying causes of deforestation based on subnational case study evidence[END_REF] illustrate the differentiated effects of different causes of deforestation. We also believe relevant to include the geographical area; in fact growth path are different in developed and developing countries which can influence the occurrence of EKC and even more so that the samples of studies are based on recent data that do not capture development phases of developed countries. As for the measures of economic development, we observe some heterogeneity in the sample.

However, there is no a priori on why and in what way such differences affect the occurrence of an EKC. Regarding the publication type, according to the aim of the study, one can expect different strategies of the authors, without compromising their neutrality. In particular, as Lindhjem ( 2007) finds a significant effect of master's thesis in his meta-analysis, we can also expect an effect of this publication type. In our case, we suspect that Master students have less incentive to find an EKC, ceteris paribus. We also include the presence of control variables for inequalities, life expectancy 4 and infrastructures. We expect the inclusion of these variables to alter the incomedeforestation relationship. In fact, they are correlated with income and this may rise collinearity issues or shed light on transmission channels. Finally we add other sources of variation namely the sample size, the h-index of the publication, 5,6 the type of model estimated (quadratic or cubic) and the number of regressions presented. The lower, the better 6 This ranking is done by "Ideas Repec". At the date of last visit, journals were ranked from 1 to 3 496. So we calculated a ratio that is equal to the rank of the journal divided by the rank of the last journal listed (i.e. 3 496). However for studies that did not receive a ranking on this site, the ratio calculation was made assuming that the rank of this study will be equal to the rank of the last journal classified to which was added one.

Empirical Strategy and Results

Our response variable is a binary variable "EKC".9 Recall that we investigate the factors that influence the occurrence of an EKC. 10 We first estimate the model using robust ordinary least squares.11 Indeed, in the presence of a binary dependent variable, if the model is estimated by OLS, the errors are heteroskedastic. Therefore, it is appropriate to make a robust estimation with White-corrected standard errors.

Furthermore, given that there may be a within-study autocorrelation leading to the dependence of regressions within one article, we ran OLS with cluster-robust inference. 12 The reason is that each study estimates various regressions with different units of measures (cf. Table 2). Then, we perform a bootstrap to deal with non-normality of residuals and to get reliable standard errors. Due to the high number of variables available -some being redundant -in a metaanalysis, multicollinearity is a serious concern. Recall that multicollinearity leads to unstable coefficients and inflated standard errors. We use Variance Inflation Factors (VIFs) to detect it (cf. Table 3). Different views exist on the threshold to accept. Some argue that VIFs should not go beyond 30; others adopt a more strict value of 10. As shown in Table 3, VIF values do not exceed 10. What is more, the mean of VIFs is not considerably larger than 1, which goes in line with the most conservative rule of thumb [START_REF] Chatterjee | Regression Analysis by Example 4 th[END_REF]. We then perform the Ramsey's RESET test, on robust OLS, in order to check for omitted variables and model misspecification.13 The quality of fit is tested. The quality of the prediction in contingency tables is reported Table 5. The threshold is fixed at 36 % because in the population (546 observations), there are 36 % of "EKC" and 64 % of "no EKC". Results suggest that we reach 66.85 % of good predictions with the Probit model. To validate the robustness of our results, we use other thresholds to reject the EKC, that is to say 1% and 5% instead of 10% retained in this work. In other words, we build two new dependent variables EKC01 and EKC05. With the first one, we choose a significance level of 1% for the variables GDP per capita and its square. With the second we use a threshold of 5%. Using the 10 % threshold we have 32 % of the observations validating the EKC. It falls to 27 % for the 5 % threshold and 17 % for the 1 % one. We then estimate the Probit model. Results do not significantly change. 16

6.

Results and Discussion

Main results

Our results show that the most recent year of publication is (year), the more likely to have an EKC declines. We shall then identify the existence of non-linearity with respect to the year of publication. In the OLS regressions (robust, cluster and bootstrap cluster), we add the square of the year of publication. Both variables year and its square appear to be significant in the three models, the first one having a positive coefficient and the second having a negative one. In addition, coefficients are the same in each model (respectively 18.89 and 0.00472). The turning point is around the year 2001. 17,18 This may be attributable to several factors, including an academic fad and technology (econometric practices and better availability of data).

Empirical choices should be carefully done before deriving policies from empirical results. On this issue, our results are not clear-cut which is not in line with (D. I. Stern

2004

) contention according to which early empirical evidences of EKCs where the result of weak econometrics. 19 It is clear however from OLS regressions that the integration of a spatial dimension (econometrics10 econometrics11) decreases the likelihood of an EKC. 20 Although these results are not to be taken per se, they raise the question of the existence of spillovers between the rates of deforestation of countries. If these spillovers exist, does it change the nature of the relationship between economic development and deforestation? Existing work seem to go in that direction (e.g. Nguyen [START_REF] Nguyen | Nonlinearities and heterogeneity in environmental quality: An empirical analysis of deforestation[END_REF].

We therefore believe that future work should incorporate the spatial dimension in order to control for the existence of spatial autocorrelation [START_REF] Anselin | Spatial econometrics: methods and models[END_REF]. Regarding deforestation, there may be several sources of spatial dependence between countries.

First, national policies for forest conservation may be influenced by policies in neighboring countries resulting in a pattern of political spatial dependence. Second, there may be positive or negative ecological spillovers between countries. For instance, 17 18.89/(2*0.00472) = 2001.06 18 Our robustness check provides consistent turning points. We precisely find 1999.29 for 1 %, 2000.98 for 5 % and 2000.39 for 10 %. 19 The existence of an EKC for deforestation was seldom performed on time series data. It was however the case with other types of pollutants such like CO2 or SO2. Even with recent econometric techniques, the results on the existence of an EKC are not conclusive (see e.g. [START_REF] Bernard | The Environmental Kuznets Curve: Tipping Points, Uncertainty and Weak Identification[END_REF]). 20 It is likely that the number of spatial regressions is too small to be captured in Logit and Probit estimations. We run a regression with three dummies for econometric techniques: OLS, panel (dynamic, fixed effects and random effects) and spatial (panel and GWR). It did not change the results. OLS and panel estimations were not significant, whereas spatial estimations were negative and significant.

seeds or forest fires -if necessary to recall -do not respect political boundaries. Third, unobserved variables may be related by a spatial process. In the case of deforestation, these may be climatic or geomorphological variables. And fourth, phenomena in neighboring countries are likely to influence deforestation patterns in a given country.

For example, more stringent forest protection policies in a country may cause more deforestation in an adjacent country. As a matter of fact, there is uncertainty about the choice of the relevant spatial model (i.e. whether to estimate a spatial lag model, a spatial error model, a spatial Durbin model or a Cliff-Ord type model). All effects can occur simultaneously and there is no theoretical argument to exclude a form of spatial dependence a priori. The use of spatial econometrics is relatively recent. Theoretical [START_REF] Corrado | Where Is the Economics in Spatial Econometrics[END_REF][START_REF] Elhorst | Applied Spatial Econometrics: Raising the Bar[END_REF]) and technical literature is growing fast [START_REF] Drukker | A command for estimating spatial-autoregressive models with spatial-autoregressive disturbances and additional endogenous variables[END_REF]. Regarding the EKC for deforestation thanks to the better availability of data, their use will be facilitated and more relevant. This is an avenue of research to dig that could call into question the existence of an EKC for deforestation.

At that point, we cannot conclude on the quality of data although this concern was put forward (D. I. [START_REF] Stern | Economic growth and environmental degradation: The environmental Kuznets curve and sustainable development[END_REF]. 21 Still, we do find that the more recent data coincide with the more chance of getting an EKC. It therefore stems the following question: Is it appropriate to mix data of different quality? Our analysis cannot answer that question.

Five remarks

Sample size

The sample size (sample size) has a positive and significant effect on the existence of the EKC. This result is line with other meta-analyses [START_REF] Jeppesen | Environmental Regulations and New Plant Location Decisions: Evidence from a Meta-Analysis[END_REF]. Indeed, according to standard testing theory sample sizes increase the power of tests. In other words, larger sample sizes reduce the probability of erroneously accepting the null hypothesis.

Deforestation index

The term "deforestation" is not clear-cut. It covers a wide range of meanings. This is striking in EKC studies for deforestation. Authors use different measures (Table 1). Our results do not evidence a clear effect of the deforestation index on the probability of the occurrence of an EKC for deforestation.

Infrastructures

Deforestation is driven by numerous factors. In some studies, authors only focus on the income-deforestation nexus. In others, authors add control variables. We find that adding infrastructures variables (infrastructures) decreases the probability of obtaining an EKC. There is evidence that infrastructure expansion is a cause of deforestation [START_REF] Pfaff | What Drives Deforestation in the Brazilian Amazon?: Evidence from Satellite and Socioeconomic Data[END_REF]. However, this variable may be highly correlated to income. (D. I. [START_REF] Stern | The Rise and Fall of the Environmental Kuznets Curve[END_REF] addresses this issue and concludes that it is difficult to infer from studies that include additional control variables. We add that our results for infrastructures are not surprising. Indeed, if they are correlated with per capita income, it is normal that authors do not find a significant relationship between economic development and

deforestation. Yet, we argue that it is not enough to conclude that there is no EKC in that specific case, because such result could also be explained by the fact that this variable plays a role in channelling the effect of economic development on deforestation.

Inequality index

We find a strong positive relationship between the introduction of a control variable for inequalities (inequality) and the probability of finding an EKC. Our result does not contradict [START_REF] Koop | Deforestation, distribution and development[END_REF] and [START_REF] Heerink | Income inequality and the environment: aggregation bias in environmental Kuznets curves[END_REF]. They find that high inequality strengthens the effect of economic development on deforestation.

Publication type

If the type of publication is a Master's thesis (pub type4), then the probability of an EKC declines relative to an article, a working paper or conference paper, ceteris paribus.

This could be explained by a "publication strategy" effect. A Master's student seems not to have the same incentives to validate the existence of an EKC whereas a researcher may be more prone to evidence an EKC as a means to publish his work in an academic journal.

Publication Bias

Publication bias remains today the biggest problem facing meta-analyses. Indeed in principle, the meta-analysis must combine exhaustively all the studies that have been made in the area of interest. First there may be a bias in the reporting of results: the estimation results are more likely to be published if they are significant. This problem is referred as the "file drawer" problem by meta-analysts [START_REF] Stanley | Meta-regression analysis as the socio-economics of economics research[END_REF]. Nonsignificant results may not be submitted to academic journals (nor published as a working paper) or they may be more rejected by publishers of academic journals. Thus the literature is not a perfect representation of all work in the field. The sources of this type of bias are complex and highly dependent on the decisions of the authors as well as publishers. Secondly, studies with "good results" are often published in English. Here, the means of dissemination may also be a source of publication bias.

Also, in our analysis we do not take into account the quality of the papers. This problem is referred to as "garbage in, garbage out" [START_REF] Borenstein | Introduction to Meta-Analysis[END_REF]). However, we believe that excluding articles, on its own, would only add bias, compared to what we believe to be a "good paper". In addition, our work allows rather performing "waste management". 22 In this sense, our work examines the characteristics of the studies that are related to the occurrence of results.

Concluding Remarks

The objective of this paper was to shed light on why EKC for deforestation results differ across studies. And that, in view of raising the attention of researchers, practitioners and policy makers on the underlying causes that lead to the validation of the EKC for deforestation in the academic literature, despite critical reviews of EKCs. We have therefore constructed an original database gathering the characteristics of econometric results for EKCs for deforestation and then implemented a meta-analysis.

Our main results are the following. Recent estimates do not corroborate the EKCs for deforestation. The year 2001 appears to be a turning point. Our results also confirm that empirical choices i.e. estimators choice affect the occurrence of an EKC for deforestation with respect to the hypothesis of spatial autocorrelation in deforestation rates. The contention of (D. I. Stern 2004) of the poor quality of econometrics underlying EKCs estimates is thus given a particular flavour.

Finally, let us consider another interpretation of our results in a Popperian way. A reason why EKCs are still considered is that scholars hardly make a definitive refutation of the EKC hypothesis; they rather prefer evidencing the presence or the absence of corrobation. Indeed, in order to test the relevance of the theory underlying the EKC, scholars must consider auxiliary hypotheses. In the case of the EKC those auxiliary hypotheses are partly provided by the econometric hypotheses. When the EKC is not corroborated, scholars cannot identify whether it is the result of falsified theory or of the econometric hypotheses. This is the essence of Quine's holism [START_REF] Quine | Main Trends in Recent Philosophy: Two Dogmas of Empiricism[END_REF]: the EKC for deforestation cannot be tested in isolation. 23 The EKC story is not at its end. Building on our results, we can predict that empirical and theoretical research on EKCs is still lively until theoretical alternatives will be provided. Following Popper, we can hope that "refuted theory can be repaired to cope with the newly discovered anomalies" (Blaug 1992, p.25) Lastly, as mentioned earlier, a meta-analysis opens the way for prospective metaanalysis. In future work, we shall explore the quality of estimations. Indeed, in this article we focused on estimators. Notably, we shall focus on EKC studies using panel techniques and investigate the study-to-study variation focusing on econometric tests performed such as tests for endogeneity, multicollinearity, Hausman test, etc.

First, forest depletion

  has been given increasing attention especially with the recognition of the role of forests in the global carbon cycle. Recent developments in the literature on climate change put forward the potential role of forests in climate change mitigation (N. H. Stern 2007). The impetus was given to the so-called "Reducing emissions from deforestation and forest degradation" (REDD) device first proposed for negotiation at the Montreal UNFCCC CoP held in 2005. The Bali Road Map adopted in 2007 went beyond with the REDD+ concept by enhancing other objectives such as

Figure 1 .

 1 Figure 1. Empirical analyses of EKCs for deforestation

  bulk of mixed results seems to have encouraged several authors against the EKC. One of the most serious attack against EKCs was provided by (D. I. Stern 2004; Levinson 2002) with close titles. Critical reviews of EKC focusing on deforestation raised several empirical shortcomings which may have impacted the EKC for deforestation

  121 observations -with a focus on turning points. They cover 11 categories of environmental goods such as urban air quality or deforestation. Observations for deforestation represent only 6 % of their sample. The impact of deforestation on income turning points depends on the econometric specification of the meta-analysis.[START_REF] Jordan | The Environmental Kuznets Curve: Preliminary Meta-Analysis of Published Studies[END_REF] analyses 255 articles -373 observations -among which 8 % deal with deforestation. Results show no significant impact of deforestation on the probability of finding an EKC.The operationalization of a meta-analysis is done by following specific steps. The first step is the formulation of the subject: what are the issues and objectives of the study (central questions we would like answers to) and what are the assumptions made. The second step is the overall design of the study: this is to clarify outcomes, populations, settings, criteria for inclusion and exclusion of studies. The third step is to specify the sampling plan, the units of observation and then begin the literature search. The fourth

Table 1 .

 1 7,8 Description and descriptive statistics of variables

	Variables	Description	Mean	SD	Min	Max
	EKC	= 1 if an EKC is found, 0 otherwise	0.330	0.470
	Year	Publication year	2004.582	4.705	1992	2011
	initial data	Initial year of data	1978.182	12.908	1948	2006
	end data	Final year of data	1997.403	5.475	1984	2010
	sample size	Sample size	73.643	211.526	4	2619
	H index	H index of the publication	0.518	0.412 0.000286 1.000286
	reg	Number of regressions presented in the study	24.324	19.81	1	60
	Measures of deforestation
	deforestation1 = 1 if Agricultural land expansion rate, 0 otherwise	0.019	0.137
	deforestation2 = 1 if Arable land area, 0 otherwise	0.016	0.125
	deforestation3 = 1 if Built-up land, 0 otherwise	0.005	0.069
	deforestation4 = 1 if Crop Area, 0 otherwise	0.040	0.195
	deforestation5 = 1 if Deforestation, 0 otherwise	0.777	0.417
	deforestation6 = 1 if Forest area clearcut, 0 otherwise	0.013	0.112
	deforestation7 = 1 if Forest stock, 0 otherwise	0.071	0.258
	deforestation8	= 1 if Land use for fuel wood, charcoal, paper, and pulp, 0 otherwise	0.005	0.069
	deforestation9 = 1 if Land use for pasture, 0 otherwise	0.005	0.069
	deforestation10 = 1 if Land use to produce timber, 0 otherwise	0.005	0.069
	deforestation11 = 1 if Mangrove deforestation index, 0 otherwise	0.006	0.079
	deforestation12 = 1 if Paved roads in forest regions, 0 otherwise	0.029	0.167
	deforestation13 = 1 if Roads encroaching on forest cover, 0 otherwise	0.006	0.079
	5

Table 2 .

 2 Robust OLS equations

	VARIABLES	OLS robust	OLS cluster-robust	Boostrap OLS cluster-robust
	year	-0.0220**	-0.0220	-0.0220***
		(0.00997)	(0.0160)	(0.00812)
	initial data	0.00591**	0.00591	0.00591***
		(0.00276)	(0.00495)	(0.00218)
	end data	0.0111	0.0111	0.0111
		(0.00947)	(0.0151)	(0.00763)
	sample size	0.000474***	0.000474***	0.000474***
		(6.13e-05)	(9.72e-05)	(4.70e-05)
	H index	-0.0686	-0.0686	-0.0686
		(0.0799)	(0.170)	(0.0495)
	deforestation2	0.483	0.483*	0.483*
		(0.295)	(0.272)	(0.249)
	deforestation3	-0.468***	-0.468**	-0.468***
		(0.130)	(0.184)	(0.117)
	deforestation5	-0.0203	-0.0203	-0.0203
		(0.111)	(0.158)	(0.0909)
	deforestation6	-0.364**	-0.364*	-0.364***
		(0.149)	(0.211)	(0.123)
	deforestation7	-0.188	-0.188	-0.188
		(0.154)	(0.246)	(0.123)
	deforestation8	-0.468***	-0.468**	-0.468***
		(0.130)	(0.184)	(0.119)
	deforestation9	-0.135	-0.135	-0.135
		(0.293)	(0.184)	(0.334)
	deforestation10	-0.135	-0.135	-0.135
		(0.293)	(0.184)	(0.340)
	deforestation11	0.559***	0.559**	0.559***
		(0.175)	(0.252)	(0.146)
	deforestation12	-0.0653	-0.0653	-0.0653
		(0.162)	(0.184)	(0.146)
	deforestation13	-0.410***	-0.410*	-0.410***
		(0.143)	(0.223)	(0.127)
	deforestation14	0.477***	0.477*	0.477***
		(0.162)	(0.269)	(0.126)
	econometrics1	0.140	0.140	0.140
		(0.159)	(0.180)	(0.162)
	econometrics2	-0.684***	-0.684***	-0.684***
		(0.165)	(0.204)	(0.147)
	econometrics3	0.623***	0.623**	0.623***
		(0.137)	(0.255)	(0.109)
	econometrics6	-0.0403	-0.0403	-0.0403
		(0.0795)	(0.0984)	(0.0732)
	fe_retain	0.0940	0.0940	0.0940
		(0.0814)	(0.0868)	(0.0739)
	econometrics8	-0.435***	-0.435***	-0.435***
		(0.113)	(0.143)	(0.114)
	re_retain	-0.103	-0.103	-0.103
		(0.126)	(0.144)	(0.114)
	econometrics10	-0.406***	-0.406***	-0.406***
		(0.0920)	(0.121)	(0.0853)
	econometrics11	-0.406***	-0.406***	-0.406***
		(0.0932)	(0.121)	(0.0868)
	econometrics12	0.181	0.181	0.181
		(0.257)	(0.307)	(0.182)
	area1	0.0487	0.0487	0.0487
		(0.0949)	(0.146)	(0.0859)
	area2	-0.273**	-0.273	-0.273**
		(0.122)	(0.236)	(0.124)
	area3	-0.590***	-0.590***	-0.590***
		(0.0724)	(0.112)	(0.0670)
	area4	-0.588***	-0.588***	-0.588***
		(0.0723)	(0.112)	(0.0669)
	area5	0.103*	0.103	0.103**
		(0.0570)	(0.122)	(0.0454)
	area6	-0.00590	-0.00590	-0.00590
		(0.240)	(0.114)	(0.253)
	area8	-0.0388	-0.0388	-0.0388
		(0.0833)	(0.136)	(0.0694)
	area10	0.0532	0.0532	0.0532

Robust standard errors in parentheses; *** p<0.01, ** p<0.05, * p<0.1

Table 3 .

 3 Variance Inflation Factors

	Variables	VIF
	end data	7.28	infrastructures	2.12	econometrics11	1.33
	econometrics6	6.37	re_retain	2.12	econometrics2	1.29
	fe_retain	5.96	deforestation12	2.02	deforestation13	1.28
	year	5.88	area5	2	area3	1.26
	deforestation5	5.42	pub type3	1.7	deforestation10	1.2
	pub type4	4.53	development3	1.66	deforestation3	1.2
	deforestation7	4.45	area11	1.65	deforestation8	1.2
	deforestation2	4.04	econometrics1	1.57	deforestation9	1.2
	reg	3.75	econometrics12	1.56	development5	1.13
	area12	3.67	inequality	1.54	area6	1.11
	deforestation6	3.66	sample size	1.54	deforestation14	1.1
	pub type2	3.59	deforestation11	1.5	area2	1.09
	pub type1	3.57	area10	1.5	area4	1.09
	initial data	3.51	econometrics10	1.49	econometrics3	1.08
	H index	3.06	econometrics8	1.46	life expectancy	1.08
	development1	2.79	area1	1.39
	area8	2.26	quadratic	1.38	Mean VIF	2.44

Secondly, we estimate a Logit model and a Probit model with robust standard deviations. The choice of one distribution or the other does not induce significant differences between the results, although tests slightly favour the Probit model.

14

Table 4 presents marginal effects for the Probit model. 15 14 Tests and Logit estimations are available upon request. See Table A -2 in the Appendix. 15 Logit and Probit estimations are available upon request to authors. Logit and Probit estimations are run on 546 observations (compared with 613 for OLS) because we have few observations for some variables.

Table 4 .

 4 Probit marginal effects

	VARIABLES	Robust probit	Probit cluster-robust	Probit Bootstrap cluster-robust
	Year	-0.0269**	-0.0269	-0.0269***
		(0.0112)	(0.0176)	(0.0102)
	initial data	0.00829***	0.00829	0.00829***
		(0.00316)	(0.00531)	(0.00274)
	end data	0.0110	0.0110	0.0110
		(0.0104)	(0.0164)	(0.00879)
	sample size	0.00209**	0.00209	0.00209***
		(0.000812)	(0.00173)	(0.000711)
	H index	-0.0889	-0.0889	-0.0889
		(0.0843)	(0.180)	(0.0573)
	deforestation2	0.486**	0.486**	0.486
		(0.218)	(0.200)	(0.507)
	deforestation5	-0.0805	-0.0805	-0.0805
		(0.126)	(0.164)	(0.114)
	deforestation7	-0.216*	-0.216	-0.216**
		(0.116)	(0.181)	(0.105)
	deforestation9	-0.210	-0.210	-0.210
		(0.196)	(0.152)	(0.137)
	deforestation10	-0.210	-0.210	-0.210*
		(0.196)	(0.152)	(0.127)
	deforestation12	0.0263	0.0263	0.0263
		(0.174)	(0.233)	(0.155)
	econometrics1	0.183	0.183	0.183
		(0.232)	(0.277)	(0.181)
	econometrics6	-0.0771	-0.0771	-0.0771
		(0.110)	(0.140)	(0.108)
	fe_retain	0.105	0.105	0.105
		(0.108)	(0.122)	(0.114)
	re_retain	-0.134	-0.134	-0.134
		(0.135)	(0.151)	(0.150)
	econometrics12	0.220	0.220	0.220
		(0.279)	(0.336)	(0.201)
	area1	0.133	0.133	0.133
		(0.143)	(0.177)	(0.150)
	area2	-0.264**	-0.264	-0.264***
		(0.111)	(0.167)	(0.0793)
	area5	0.166**	0.166	0.166***
		(0.0678)	(0.124)	(0.0523)
	area6	0.0312	0.0312	0.0312
		(0.233)	(0.104)	(0.209)
	area8	0.0259	0.0259	0.0259
		(0.136)	(0.178)	(0.118)
	area10	0.156	0.156	0.156
		(0.175)	(0.207)	(0.191)
	area11	0.292**	0.292*	0.292**
		(0.124)	(0.176)	(0.118)
	development1	0.123*	0.123	0.123**
		(0.0679)	(0.133)	(0.0573)
	development5	0.418**	0.418**	0.418***
		(0.168)	(0.172)	(0.159)
	pub type1	0.0107	0.0107	0.0107
		(0.0754)	(0.134)	(0.0588)
	pub type2	-0.221	-0.221	-0.221
		(0.177)	(0.194)	(0.499)
	pub type4	-0.251***	-0.251	-0.251***
		(0.0901)	(0.153)	(0.0802)
	inequality	0.530***	0.530***	0.530***
		(0.0950)	(0.133)	(0.0605)
	infrastructures	-0.297***	-0.297***	-0.297***
		(0.0636)	(0.106)	(0.0514)
	quadratic	0.245***	0.245***	0.245***
		(0.0764)	(0.0827)	(0.0798)
	reg	0.000182	0.000182	0.000182
		(0.00202)	(0.00275)	(0.00189)
	Constant
	Observations	546	546	546
	Pseudo R²	0.1478	0.1478	0.1478

Robust standard errors in parentheses; *** p<0.01, ** p<0.05, * p<0.1

Table 5 .

 5 Contingency tables

		Probit model for EKC
	Classified	D	-D	Total
	+	156	137	293
	-	44	209	253
	Total	200	346	546
	Correctly classified : 66.85%
	Classified + if predicted Pr(D) >= .36; True D defined as EKC = 0.

Table A -

 A 2. Logit Marginal EffectsTable A -3. Probit robustness check Robust standard errors in parentheses; *** p<0.01, ** p<0.05, * p<0.1

	VARIABLES VARIABLES Year	EKC01 Robust logit -0.0936***	EKC05 -0.0630**
			(0.0361)		(0.0320)
	Year	initial data	0.0378***	-0.0276**	0.0195**
			(0.0118)	(0.0123)	(0.00865)
	initial data end data	0.0104	0.00818**	0.0111
			(0.0382)	(0.00339)	(0.0321)
	end data sample size	0.00556*	0.0113	0.00472**
			(0.00319)	(0.0110)	(0.00211)
	sample size_section H index	-0.175	0.00220**	-0.0494
			(0.301)	(0.000888)	(0.254)
	H index deforestation2	4.822***	-0.0801	0.710
			(0.736)	(0.0865)	(0.934)
	deforestation2 deforestation5	-1.314***	0.491**	-0.886**
			(0.405)	(0.212)	(0.355)
	deforestation5 deforestation7	-2.194***	-0.0897	-1.577***
			(0.698)	(0.137)	(0.486)
	deforestation7 deforestation9	-1.019	-0.209*	-0.933
			(0.830)	(0.113)	(0.811)
	deforestation9 deforestation10	-1.019	-0.200	-0.933
			(0.830)	(0.179)	(0.811)
	deforestation10 deforestation11	-1.009	-0.200	0.686
	deforestation12 deforestation12 econometrics1 econometrics1 econometrics6 econometrics6 fe_retain fe_retain re_retain re_retain econometrics12 econometrics12 area1 area1 area2 area2 area5 area5 area6 area6 area8 area8 area10 area10 area11 area11 development1 development1 development5 development5 pub type1 pub type1 pub type2 pub type2 pub type4 pub type4 inequality inequality infrastructures	(0.925) 0.249 (0.661) -0.559 (0.402) -0.125 (0.366) -0.798 (0.674) 1.743** (0.760) 0.261 (0.486) -0.894 (0.655) 0.469** (0.208) 0.168 (0.479) -0.0783 (0.490) 0.852*** (0.224) 0.922 (0.683) 0.269 (0.269) -4.576*** (0.355) -1.598*** (0.487) 1.527*** (0.432) -12.35	(0.179) 0.0257 (0.181) 0.178 (0.242) -0.0886 (0.116) 0.0869 (0.114) -0.154 (0.134) 0.188 (0.308) 0.151 (0.157) -0.248** (0.117) 0.178** (0.0720) 0.0330 (0.233) 0.0299 (0.148) 0.173 (0.188) 0.319** (0.134) 0.124* (0.0705) 0.432*** (0.151) 0.00568 (0.0766) -0.238 (0.163) -0.271*** (0.0860) 0.523*** (0.0954)	(0.898) -0.699 (0.576) 0.483 (0.613) 0.0765 (0.322) 0.278 (0.314) -0.497 (0.500) 1.101 (0.737) 0.225 (0.375) -0.916 (0.587) 0.456*** (0.175) -0.385 (0.606) -0.151 (0.411) 0.0680 (0.451) 0.565* (0.334) 0.389** (0.195) 1.356** (0.605) 0.0305 (0.215) -0.273 (0.768) -1.170*** (0.386) 0.789* (0.428) -5.542***
	infrastructures	(8.348)	-0.279***	(1.912)
		quadratic	0.771**	(0.0649)	0.828**
	quadratic	(0.392)	0.237***	(0.347)
		reg	0.0169**	(0.0769)	0.0101*
	reg		(0.00686)	0.000404	(0.00600)
		Constant	90.88**	(0.00211)	64.09**
			(36.65)		(31.70)
	Observations Observations	512	546	550
	Pseudo R² Pseudo R²	0.2908	0.1472	0.1873

Robust standard errors in parentheses; *** p<0.01, ** p<0.05, * p<0.1

The list of primary studies is available from the authors upon request (See Table A -1 in the Appendix).

For instance were excluded from the sample studies analyzing biodiversity, the ecological footprint, land vulnerability or protected areas. Also studies focusing on households were excluded.

The decision on the results was based on the significance of all coefficients of interest. For instance, in the quadratic model, as in most studies tests for joint significance were not performed, the model is considered significant if the two coefficients (i.e. per capita GDP and squared per capita GDP) are individually significant.

Note that few studies provide tests to compare models. It is therefore not possible to provide a single regression per article.

Other information was collected from studies but was excluded due to collinearity issues such as the unit for the deforestation measure, the source of data for deforestation, etc.

In the field of environmental economics, the use of a binary criterion into a meta-analysis was used, among others, by[START_REF] Jeppesen | Environmental Regulations and New Plant Location Decisions: Evidence from a Meta-Analysis[END_REF][START_REF] Schläpfer | Survey protocol and income effects in the contingent valuation of public goods: A meta-analysis[END_REF][START_REF] Jacobsen | Are There Income Effects on Global Willingness to Pay for Biodiversity Conservation[END_REF][START_REF] Kiel | The impact of Superfund sites on local property values: Are all sites the same[END_REF][START_REF] Jordan | The Environmental Kuznets Curve: Preliminary Meta-Analysis of Published Studies[END_REF]

It is important to note that, in this paper, we do not look into "effects size" which would imply using the income associated with turning points.

Several authors recommend running OLS when a binary variable is the endogenous regressor. See[START_REF] Angrist | Estimation of Limited Dependent Variable Models with Dummy Endogenous Regressors: Simple Strategies for Empirical Practice[END_REF]) for more details.

Standard errors are clustered by each study. Such approach has been used, for instance, by[START_REF] Barrio | A meta-analysis of contingent valuation forest studies[END_REF].

We find that F(3, 578) = 1.23 with Prob > F = 0.2981. Therefore there is no omitted variable / misspecification issue in the model.

When gathering information we have extracted the data source. However, this information could not be used in the meta-analysis because these variables are highly correlated with measures of deforestation and because their source very often relies on FAO data.

Appendix A -For the referees, not for publication