Carlos Alberto

Cavichioli Gonzaga

Marc Jungers
email: marc.jungers@univ-lorraine.fr

Carlos A C Gonzaga

Jamal Daafouz
email: jamal.daafouz]@ensem.inpl-nancy.fr

Stability analysis of discrete-time Lur'e systems ⋆

Keywords: Lyapunov function, Lur'e systems, bounded sector nonlinearity, absolute stability, saturation, basin of attraction estimate

come

Introduction

The class of systems formed by a feedback interconnection of a linear system and a cone bounded nonlinearity has been investigated, either in continuousor discrete-time domains. It was initially proposed for continuous-time systems, and so-called the Lur'e problem, [START_REF] Khalil | Nonlinear Systems -Third Edition[END_REF][START_REF] Lur | On the theory of stability of control systems[END_REF][START_REF] Popov | Absolute stability of nonlinear systems of automatic control[END_REF]. Two different types of Lyapunov functions were mostly considered for stability analysis: the classical quadratic one and the Lur'e-type, which has a quadratic term and a term of the nonlinearity integral. By considering this function, the nonlinearity appears directly in the Lyapunov time-derivative. It is well known that this class of functions is more general and leads to less conservative sufficient LMI conditions than the quadratic function. For discrete-time systems, likewise, great efforts have been made to reduce the conservatism of the quadratic function. Hence, the original form of the Lur'e-type function, with the nonlinearity ⋆ A preliminary version [START_REF] Gonzaga | Stabilization of discrete-time nonlinear systems subject to input saturations: a new Lyapunov function class[END_REF] of this paper was presented at IFAC World Congress 2011 at Milano. Corresponding author M. Jungers. Tel. +33 (0)3 83 59 57 04. Fax + 33 (0) 3 83 59 56 44. integral, was considered as candidate function for stability analysis issues [START_REF] Haddad | Parameter-dependent Lyapunov functions and the discrete-time Popov criterion for robust analysis[END_REF][START_REF] Kapila | A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach[END_REF][START_REF] Gomes Da | Stability and stabilization of linear discrete-time subject to control saturation[END_REF] (see also references therein). In these references, it is assumed that the nonlinearity derivative (or discrete variation) is bounded. According to this assumption, the resulting integral term in the Lyapunov difference is upper bounded.

The discrete-time version of the Lur'e problem was also studied in a general framework developed in [START_REF] Iwasaki | Generalized quadratc Lyapunov functions for nonlinear/uncertain systems analysis[END_REF], for feedback interconnections of linear systems and nonlinearity or uncertainty. It is proposed a class of Lyapunov functions which is quadratic in the system state and in the feedback element. The stability conditions are derived for an auxiliary system, defined recursively from the original one. However, it is known that the Lur'etype function in the continuous-time framework is not quadratic in the nonlinearity.

Among different cone bounded nonlinearities, the actuator saturation is one of the most studied nonlinearities in both time-domains. In general, only local stability can be guaranteed and the exact determination of the basin of attraction is a hard task. The main challenge, then, is to improve the estimate of this set. The level sets of the Lyapunov function are usually considered and for which the quadratic function gives ellipsoidal sets. Moreover, two frameworks based on quadratic Lyapunov functions draw attention for modelling the saturation as LMI conditions. In [START_REF] Hu | Analysis and design for discrete-time linear systems subject to actuator saturation[END_REF], the saturation function of a given linear state feedback is located inside the convex hull of hyperplanes defined by the control gains and a set of auxiliary gains. The condition for an ellipsoid to be a contractive set in the single-input case is necessary and sufficient. The second approach, in [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF], describes the saturation by the dead-zone nonlinearity which satisfies a general (local) bounded sector condition. As exposed in [START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF], the computational burden of this approach is less dependent on the system dimension than the first one, because the number of LMI to be checked is smaller.

A more general class of systems is the one combining two types of nonlinearities such as the output dependent cone bounded one and the input saturation. Despite it represents many practical control problems, this problem has not been widely studied in the literature. For continuous-time domain, in [START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF], the stability analysis and closed-loop stabilization are covered by LMI optimization problems based on the quadratic Lyapunov function. The equivalent class of systems in the discretetime is considered in [START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF]. In this case, only the control synthesis problem is solved also using LMI optimization based on the quadratic function.

The main contribution of this paper is to propose a new class of Lyapunov functions including the cone bounded nonlinearity. This function is able to relax the assumption about the nonlinearity bounded variation. Based on the proposed Lyapunov function, we tackle two different stability analysis problems. Sufficient conditions to ensure the global stability of discrete-time Lur'e systems are formulated by LMI conditions. Also, it is shown that this LMI condition is related to the discrete-time strictly positive real lemma applied for an auxiliary system transfer matrix. The second class of systems is the same studied in [START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF], with a saturating control law included. Sufficient BMI (or LMI via imposing a weighting matrix) conditions are formulated to cover the local stability analysis problem. An optimization problem under these LMI constraints is proposed to maximize the size of an estimate -given by the level set of the proposed Lyapunov function -of the basin of attraction. This estimate, as illustrated in the numerical examples, might present disconnected and nonconvex sets. Also, it is highlighted that the new Lyapunov function is able to lead to a less conservative estimate than an ellipsoidal domain.

The paper is organized as follows: in Section 2, we present the first class of discrete-time systems studied and our motivation describing some additional assumptions considered in the frameworks based on the Lur'etype function. In Section 3, a new type of Lyapunov function dependent on the cone bounded nonlinearity is presented. In Section 4, sufficient LMI conditions for the global stability analysis problem is proposed and followed by an academic example to illustrate the result. In addition, we show that the global stability and the existence of the proposed Lyapunov function is related to the strictly positive realness property of an auxiliary system transfer matrix. In Section 5, it is presented the closed-loop formulation of discrete-time Lur'e systems and a given feedback control law subject to saturation. Then, a LMI optimization problem is proposed to solve the local stability analysis problem. Several examples are given, as well, to illustrate our framework and some special features. Concluding remarks are presented in Section 6.

Notations. For any vector x ∈ R n , x ≥ 0 states that, ∀ ℓ = 1, . . . , n, its components x (ℓ) are nonnegative. Also, for two vectors x, y of R n , x ≥ y states that x (ℓ)y (ℓ) ≥ 0. x is related to the Euclidian norm of vector x. The transfer function associated with a given state-

space realization G(z) ≃   A B C D   is defined by G(z) = C (zI -A) -1 B + D. A square matrix A ∈ R n×n is called Schur if all
its eigenvalues are located inside the open unit circle. A (ℓ) denotes the ℓ-th row of matrix A. For two symmetric matrices, A and B, A > B means that A -B is positive definite. A ′ denotes the transpose of matrix A. I m (0 m×n ) is the m-order identity matrix (m × n-order null matrix). The operator diag(x) describes a diagonal matrix obtained from vector x. Also, diag(A; B) is a block diagonal matrix of matrices A and B. ⋆ means the symmetric blocks in matrices. For a symmetric positivedefinite matrix M ∈ R n×n , the ellipsoidal set E(M, γ) associated with M is given by {x ∈ R n ; x ′ M x ≤ γ} and the shortcut E(M ) = E(M, 1) will be used.

Discrete linear systems under cone bounded nonlinearity

Consider the following discrete-time system including a nonlinearity ϕ(•)

x k+1 = Ax k + F ϕ(y k ), ∀ k ∈ N (1) y k = Cx k , (2)
where x k ∈ R n , y k ∈ R p are respectively the state and output vector of the system (1)-( 2) at the instant k ∈ N.

The matrices A, C and F have appropriate dimensions.

The system nonlinearity will satisfy the following assumption:

Assumption 1 The nonlinearity ϕ(•) : R p → R p verifies a cone bounded sector condition and is assumed to be decentralized [START_REF] Khalil | Nonlinear Systems -Third Edition[END_REF].

This assumption means that:

ϕ(0) = 0 and ϕ(•) ∈ [0 p , Ω],
where Ω ∈ R p×p is a positive diagonal matrix.

Thus, we have that ϕ (ℓ) (y) [ϕ(y) -Ωy] (ℓ) ≤ 0, ∀y ∈ R p , ∀ℓ = 1, • • • , p, and the following inequality

SC(ϕ(•), y, Λ) = ϕ ′ (y)Λ[ϕ(y) -Ωy] ≤ 0 (3)
is verified, for any diagonal and positive matrix Λ = diag{λ q } q=1;••• ;p ∈ R p×p . Note that Ω is given by the designer. It is simple to show, from Assumption 1, that the relation

(3) is equivalent to [Ωy] ′ (ℓ) [ϕ(y)-Ωy] (ℓ) ≤ 0, ∀ℓ = 1, • • • , p; ∀y ∈ R p , which implies, with Λ diagonal positive definite, that 0 ≤ ϕ ′ (y)Λϕ(y) ≤ ϕ ′ (y)ΛΩy ≤ y ′ Ω ′ ΛΩy, ∀y ∈ R p . (4)
We briefly consider the scalar version of this class of systems in order to expose the motivation of our results.

Motivation

In the scalar case, the discrete-time Lur'e system is described by

x k+1 = ax k + f ϕ(y k ), ∀k ∈ N, (5)
y k = cx k , (6)
where a, f and c are scalars. The continuous nonlinearity ϕ(•) : R → R is assumed to be time-invariant and satisfies Assumption 1. By having ϕ(0) = 0, the origin is an equilibrium point, and ϕ(y

)[ϕ(y) -ωy] ≤ 0, ∀ y ∈ R, with ω > 0.
The Lyapunov stability of this class of systems was studied by considering the widespread Lur'e-type Lyapunov function, [START_REF] Khalil | Nonlinear Systems -Third Edition[END_REF], which takes the cone bounded nonlinearity into account. In the scalar case, this function is defined as v(x, ϕ(•)) = πx 2 + η y 0 ϕ(s)ds, for some π > 0 and η ≥ 0. The function is inspired by the continuoustime framework and can be considered as a candidate function due to the time invariance of nonlinearity ϕ(•), which guarantees the positiveness of the integral. To our knowledge, the function v was first used in the discretetime domain, in [START_REF] Pearson | On the assymptotic stability of a class of saturating sampled-data systems[END_REF], for stability of input-saturating sampled-data systems. Because of the time invariance of ϕ(•), the following integral term appears in the Lyapunov difference I = y k+1 y k ϕ(s)ds. An extensive literature (see [START_REF] Pearson | On the assymptotic stability of a class of saturating sampled-data systems[END_REF][START_REF] Gp Szegö | On the absolute stability of a certain class of sampled-data control systems[END_REF][START_REF] Jury | On the absolute stability of a certain class of nonlinear sampled-data systems[END_REF][START_REF] Jury | On the absolute stability of nonlinear sampled-data systems[END_REF][START_REF] Sharma | On the absolute stability of multivariable discrete-time nonlinear systems[END_REF][START_REF] Kapila | A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach[END_REF][START_REF] Gomes Da | Stability and stabilization of linear discrete-time subject to control saturation[END_REF]) is forced to make additional assumptions about the slope of the nonlinearity to define an upper bound of I and to conclude on the stability. The spirit of these assumptions is to upper and/or lower bound the slope of the nonlinearity to allow a bound of the integral I via a trapezoidal rule. Assumptions on the slope prohibit treating odd (ϕ(-y) = -ϕ(y), ∀y ≥ 0) nonlinearities such as ϕ(y

) = 0, if y ∈ [0, 1] or y ∈ [3, +∞) and ϕ(y) = (|y| -2) 2 + 1 if y ∈ [1, 3],
which is two half-circles with center at y = 2 and y = -2.

The choice of the Lyapunov function v can be justified in the continuous-time original Lur'e problem because it allows the nonlinearity to be directly considered in the Lyapunov derivative. However, by using this type of function in discrete-time, conservative assumptions on the slope are necessary to conclude the stability. In the following section, a candidate Lyapunov function will be presented, not quadratic in ϕ(•) for the discrete-time domain, which is able to conclude the stability under assumption 1, without requiring any assumption on the slope of the nonlinearity.

A new class of Lyapunov functions

We present a new candidate Lyapunov Function to be associated with the system (1)-( 2). This candidate function is composed of a quadratic term with respect to the state and a cross term between the state and the nonlinearity:

V : R n × R p -→ R, (x; ϕ(Cx)) -→ x ′ P x + 2ϕ ′ (Cx)∆ΩCx, (7)
where P ∈ R n×n is a symmetric positive definite matrix and ∆ ∈ R p×p is a diagonal positive semidefinite matrix. One should point out that the quadratic Lyapunov function is recovered with ∆ = 0 p and, thus, it may be considered as a particular case of function [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF]. From inequalities (4), we have

V (x) ≤ V (x; ϕ(Cx)) ≤ V (x), (8)
with V (x) = x ′ P x and V (x) = x ′ (P + 2C ′ Ω ′ ∆ΩC)x.

The function V (x; ϕ(Cx)) can be considered as a candidate, because it verifies the following properties:

• V (x; ϕ(Cx)) ≥ 0, ∀x ∈ R n , due to the left inequality in (8); • V (x; ϕ(Cx)) = 0, if and only if x = 0, because of inequality [START_REF] Gonzaga | Stabilization of discrete-time nonlinear systems subject to input saturations: a new Lyapunov function class[END_REF] and

P > 0. • V (x; ϕ(Cx)) is radially unbounded, i.e., V (x; ϕ(Cx)) → ∞ when x → ∞.
The Lyapunov difference, which will be treated in the sequel, is defined by

δ k V = V (x k+1 ; ϕ(Cx k+1 )) - V (x k ; ϕ(Cx k )).
The level set associated with V (x; ϕ(Cx)) and γ > 0 is given by L V (γ) = {x ∈ R n ; V (x; ϕ(Cx)) ≤ γ} which is, clearly, related to the two ellipsoids associated with the upper and lower-bounds quadratic functions V (x) and V (x)

E(P + 2C ′ Ω ′ ∆ΩC, γ) ⊆ L V (γ) ⊆ E(P, γ). (9)
In Fig. 1, it is possible to see an example of the candidate function, for n = 2, which is bounded by two paraboloids. Only the part of the surfaces (x 2 ≥ x 1 ) is drawn. The level sets associated with a given γ and the functions V , V and V are depicted in Fig. 2. It is possible to see the Inclusion [START_REF] Haddad | Parameter-dependent Lyapunov functions and the discrete-time Popov criterion for robust analysis[END_REF]. One of special features of the Lyapunov function ( 7) is to provide disconnected and nonconvex level sets. The shape and number of the sets will directly depend on the nature of ϕ(•). In the follow- ing sections the candidate Lyapunov function [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF] will be considered as a main tool to study several stability analysis problems.

Global stability analysis and frequency domain interpretation

Based on the function V (x; ϕ(Cx)), a framework is provided for studying the problem of global stability of systems (1)-( 2). In addition, we present a frequency domain condition related to the existence of our candidate function.

Global stability

We address the problem of global stability analysis, which is defined as Problem 1 (Global Stability Analysis) Under Assumption 1, determine a matrix P = P ′ > 0 n and a diagonal matrix ∆ ≥ 0 p which allow to guarantee the global stability of system (1)-( 2) for any nonlinearity ϕ(•) ∈ [0 p ; Ω].

The following statement gives a sufficient condition to solve the Problem 1.

Theorem 2 For the class of systems defined by ( 1)-( 2), if there exists a matrix G ∈ R n×n , a symmetric positive definite matrix P ∈ R n×n , a positive semidefinite diagonal matrix ∆ ∈ R p×p and positive diagonal matrices

T, W ∈ R p×p , such that the LMI        P -G ′ -G G ′ A G ′ F 0 n×p ⋆ -P Π 1 A ′ Π 2 ⋆ ⋆ -2T F ′ Π 2 ⋆ ⋆ ⋆ -2W        < 0, (10)
is verified, with

Π 1 = C ′ Ω [T -∆] ; Π 2 = C ′ Ω [W + ∆] , (11)
then the function V (x; ϕ(Cx)) is a Lyapunov function and the origin of system (1)-( 2) is globally asymptotically stable.

Proof 3 By having Inequality [START_REF] Hu | Analysis and design for discrete-time linear systems subject to actuator saturation[END_REF] verified, it implies P -G ′ -G < 0 and P > 0. Hence, G is of full rank and we have -G ′ P -1 G ≤ P -G ′ -G (see [START_REF] Daafouz | Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF]). The change of basis diag(G -1 ; I n+2p ) and a Schur complement ( [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]) lead to the following inequality

    A ′ F ′ 0     P     A ′ F ′ 0     ′ +     -P C ′ Ω [T -∆] A ′ C ′ Ω [W + ∆] ⋆ -2T F ′ C ′ Ω [W + ∆] ⋆ ⋆ -2W     < 0.
(12) Due to the structure of the nonlinearity ϕ(•), the equation

(x ′ k ϕ ′ (y k ) ϕ ′ (y k+1 )) ′ = 0 is equivalent to x k = 0.
Thus by multiplying Inequality [START_REF] Jury | On the absolute stability of a certain class of nonlinear sampled-data systems[END_REF] on the right by (x ′ k ϕ ′ (y k ) ϕ ′ (y k+1 )) ′ and on the left by its transpose, we get the following inequality

δ k V -2SC(ϕ(•), y k+1 , W )-2SC(ϕ(•), y k , T ) < 0, ∀x k = 0.
(13) Since nonlinearity ϕ(•) verifies a global sector condition, Inequality (13) defines an upper bound for the Lyapunov difference, implying δ k V < 0, for any x k = 0.

Remark 4

The stability of matrices A and A + F ΩC is necessary for the feasibility of inequality [START_REF] Hu | Analysis and design for discrete-time linear systems subject to actuator saturation[END_REF]. These necessary conditions are obtained by considering the bounds of the sector condition (3), which are ϕ(y k ) = 0 and ϕ(y k ) = Ωy k . . For a given initial condition x 0 = (100; -50) ′ , the trajectory associated with system (1)-( 2) is asymptotically stable, as depicted in Fig. 3.

! " #! #" "! $" ! $" "! %" #!! & '()#* ++& '()$*
,-./+' Fig. 3. Time response of system (1)-( 2) for a given initial condition -state components: x k,(1) with '+' and x k,(2) with '×'.

Frequential Interpretation

It is possible to state a frequency domain condition for Inequality [START_REF] Hu | Analysis and design for discrete-time linear systems subject to actuator saturation[END_REF]. This condition will be derived by means of the strict positive realness of a square transfer matrix. By definition, [16, Lemma 5], a discrete-time square transfer matrix G(z) is strictly positive real if it is asymptotically stable and G(e jθ ) + G ′ (e -jθ ) is positive definite for all θ ∈ [0, 2π]. Let us define the following auxiliary system matrices

A = A; B = F 0 n×p ; C = (∆ -T )ΩC -(∆ + W )ΩCA ; D = T -(∆ + W )ΩCF 0 p W . (14)
Thus, the following theorem establishes a connection between the frequential approach and the global stability condition of Theorem 2.

Theorem 5 Suppose that there exists a positive semidefinite diagonal matrix ∆ ∈ R p×p , positive diagonal matrices T, W ∈ R p×p such that, for A, B, C, D given by ( 14), the 2p × 2p-transfer matrix G(z) = C (zI -A) -1 B + D is strictly positive real. Then, the function ( 7) is a Lyapunov function that proves global asymptotic stability of system (1)-( 2), for any nonlinearity ϕ(•) verifying the sector condition (3).

Proof 6

The proof is based on the strictly positive real lemma, [START_REF] Kottenstette | Relationships between positive real, passive dissipative, & positive systems[END_REF]Lemma 7] by considering a given transfer matrix G(z) and from which an associated asymptotically

stable realization G(z) ≃   A B C D   is given. Then, G(z) is
strictly positive real iff there exists a matrix P = P ′ > 0 solution of the following inequality

A ′ PA -P A ′ PB -C ′ ⋆ -(D + D ′ ) + B ′ PB < 0. (15)
Hence, if there exists matrices ∆, T, W such that G(z) is strictly positive real, then matrix P verifies Inequality [START_REF] Khalil | Nonlinear Systems -Third Edition[END_REF].

By replacing the auxiliary system matrices as defined in [START_REF] Kapila | A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach[END_REF] in Inequality [START_REF] Khalil | Nonlinear Systems -Third Edition[END_REF], one gets Inequality [START_REF] Jury | On the absolute stability of a certain class of nonlinear sampled-data systems[END_REF], with P = P. If we multiply Inequality (15) on the right by [x ′ ϕ ′ ϕ ′ + ] ′ and its transpose on the left, considering the shortcuts x k = x; ϕ = ϕ(y k ); ϕ + = ϕ(y k+1 ), one has

x ′ (A ′ P A -P )x -2(ϕ + ) ′ W ϕ + + ϕ ′ (F ′ P F -2T )ϕ + 2x ′ (A ′ P F ϕ + A ′ C ′ Ω ′ (∆ + W )ϕ + -C ′ Ω ′ (∆ -T )ϕ) + 2(ϕ + ) ′ (∆ + W )ΩCF ϕ ≤ 0,
which is Inequality [START_REF] Jury | On the absolute stability of nonlinear sampled-data systems[END_REF]. Thus, function ( 7) is a Lyapunov function that proves asymptotic stability of system (1)-( 2) for any nonlinearity ϕ(•) verifying the sector condition (3).

Discrete time Lur'e systems subject to input saturation

Consider, now, a more general class of discrete-time systems composed of a linear part, a cone bounded nonlinearity and saturating inputs:

x k+1 = Ax k + F ϕ(y k ) + Bsat(u k ); ∀ k ∈ N (16) y k = Cx k , (17)
where x k ∈ R n , u k ∈ R m and y k ∈ R p are, respectively, the state, control input and output vector of the system ( 16)-( 17) at instant k. Matrices A, B, F and C are real matrices of appropriate dimension. The saturation is defined as follows: sat(u k

) (ℓ) = sign((u k ) (ℓ) ) min ρ (ℓ) , (u k ) (ℓ) , ∀ ℓ = 1, ..., m.
The scalar ρ (ℓ) > 0 means the symmetric saturation limit of the ℓ-component of the control input.

The vector ρ ∈ R m is assumed to be given. In the sequel, the control law provided by [START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF] will be considered:

u k = Kx k + Γϕ(y k ), (18)
where m×n-matrix K is the state feedback gain and m× p-matrix Γ is the nonlinear feedback gain associated with ϕ(•). If Γ = 0 m×p , the nonlinearity ϕ(•) value must be available either by model estimation or measuring. The saturation will be described as a dead-zone nonlinearity

Ψ(u k ) = u k -sat(u k ). (19
) By replacing u k given by ( 18) and ( 19) in ( 16), the closedloop model is described, with A cl = A + BK and F cl = F + BΓ as

x k+1 = A cl x k + F cl ϕ(y k ) -BΨ(u k ). ( 20
)
Remark 7 Similar to the global stability analysis, the control law [START_REF] Pearson | On the assymptotic stability of a class of saturating sampled-data systems[END_REF] gains should be set such that matrices A cl and A cl + F cl ΩC (A cl + F ΩC in case of a linear state feedback) are both Schur.

Let us define the following set necessary to describe the dead-zone belonging to a generalized sector condition. For a given matrix

H ∈ R m×(n+p) , the set S(H, ρ) is defined by S H, ρ = {θ ∈ R n+p ; -ρ ≤ Hθ ≤ ρ}. Lemma 8 Consider m × (n + p)-matrices, K = [K Γ] and J = [J 1 J 2 ]. If the vector x k = [x ′ k ϕ ′ (y k )]
′ is an element of S( K -J, ρ), then the nonlinearity Ψ(u k ), with u k is defined in Equation ( 18), satisfies the following sector condition, with U ∈ R m×m diagonal and positive definite

SC u k = Ψ ′ (u k )U [Ψ(u k ) -J 1 x k -J 2 ϕ(y k )] ≤ 0. (21)
Proof 9 This proof is straightforward from [23, Lemma 1](see also [START_REF] Castelan | Control design for a class of nonlinear continuous-time systems[END_REF]).

As the main result of this section, the new class of Lyapunov function will be employed to study the problem of local stability for system ( 16)- [START_REF] Lur | On the theory of stability of control systems[END_REF]. Let us define this problem Problem 2 (Local Stability Analysis) For a given set of gains K ∈ R m×n and Γ ∈ R m×p of the control law [START_REF] Pearson | On the assymptotic stability of a class of saturating sampled-data systems[END_REF], determine a region, in the state space, as large as possible included in the basin of attraction B 0 of the system ( 16)- [START_REF] Lur | On the theory of stability of control systems[END_REF], under Assumption 1.

In general, the task of analytically determining the basin of attraction B 0 is difficult. Thus, we will use the level set L V (1) of the proposed Lyapunov function.

Theorem 10 For given matrices K ∈ R m×n , Γ ∈ R m×p , consider as variables, matrices G ∈ R n×n , J 1 ∈ R m×n , J 2 ∈ R m×p , a symmetric positive definite matrix P ∈ R n×n , a positive semidefinite diagonal matrix ∆ ∈ R p×p and positive definite diagonal matrices

Q, T, W ∈ R p×p , U ∈ R m×m . The inequalities          P -G ′ -G G ′ A cl G ′ F cl -G ′ B 0 n×p ⋆ -P Π 1 J ′ ⋆ -2W          < 0, ( 22
)     P Π 3 (K -J 1 ) ′ (ℓ) ⋆ 2Q (Γ -J 2 ) ′ (ℓ) ⋆ ⋆ ρ 2 (ℓ)     > 0, ∀ℓ = 1, • • • , m, (23)
where Π 1 , Π 2 are defined in [START_REF] Iwasaki | Generalized quadratc Lyapunov functions for nonlinear/uncertain systems analysis[END_REF] and

Π 3 = C ′ Ω (∆ -Q),
are respectively bilinear (BMI) and linear (LMI) matrix inequalities and allow to obtain an estimate of the basin of attraction B 0 given by the level set L V (1), induced by the Lyapunov function [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF].

Proof 11 Like in the proof of Theorem 2, Inequality [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF] implies P -G ′ -G < 0 and P > 0, such that G is of full rank, and so -G ′ P -1 G ≤ P -G ′ -G. By applying the change of basis diag(G -1 ; I n+2p+m ) with a Schur complement, one gets the inequality

       -P C ′ Ω [T -∆] J ′ 1 U A ′ cl C ′ Ω [W + ∆] ⋆ -2T J ′ ⋆ -2W        +        A ′ cl F ′ cl -B ′ 0        P        A ′ cl F ′ cl -B ′ 0        ′ < 0. (24)
In the following, by multiplying Inequality (24) on the right by

(x ′ k ϕ ′ (y k ) Ψ ′ (u k ) ϕ ′ (y k+1
)) ′ = 0 and by its transpose on the left, it leads to inequality

δ k V -2SC u k -2SC(ϕ(•), y k+1 , W )-2SC(ϕ(•), y k , T ) < 0.
(25) Also, by applying a Schur complement on Inequality [START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF] with respect to the last block, we obtain Inequality

P Π 3 ⋆ 2Q - 1 ρ 2 (ℓ) (K -J 1 ) ′ (ℓ) (Γ -J 2 ) ′ (ℓ) (K -J 1 ) ′ (ℓ) (Γ -J 2 ) ′ (ℓ) ′ > 0. ( 26
)
By multiplying Inequality (26) on the right by x k = ( x ′ k ϕ ′ (y k ) ) ′ and on the left by its transpose, one has the following inequality

V (x k ; ϕ(y k )) + 2SC(ϕ(•), y k , Q) ≥ 1 ρ 2 (ℓ) ( K -J) (ℓ) x k 2
. Due to the fact that nonlinearity ϕ(•) verifies the global bounded sector condition, and by noting K and J as defined in Lemma 8, we have

V (x k ; ϕ(y k )) ≥ 1 ρ 2 (ℓ) ( K -J) (ℓ) x k 2 , which induces the inclusion L V (1) ⊂ S K -J, ρ . (27
) Thus, the general sector condition [START_REF] Gp Szegö | On the absolute stability of a certain class of sampled-data control systems[END_REF], related to the dead-zone, is verified inside the level set L V (1). Due to inclusion (27), the local sector condition SC u k ≤ 0 for the dead-zone is verified inside L V (1). This implies, with the aid of the Inequality (25), that δ k V < 0, ∀ x ∈ L V (1) (x = 0). Hence, the asymptotic stability is proved inside the set L V (1).

Based on the BMI and LMI in Theorem 10, it is possible to build an optimization problem maximizing the size of L V (1) or of the ellipsoid E(P + 2C ′ Ω ′ ∆ΩC). The choice has been made to define the size of an ellipsoid as its minor axis. Its maximization consists in minimizing the greatest eigenvalue of P + 2C ′ Ω ′ ∆ΩC. This can be interpreted as maximizing the size of a ball included in the ellipsoid. Other choices are also possible, such the trace minimization, the maximization along certain directions or the volume maximization under modified constraints (see for more details and a discussion between these criteria [22, Section 2.2.5.1] and [START_REF] Durieu | Trace versus determinant in ellipsoidal approximation with application to state estimation[END_REF]). By introducing the variable R ∈ R p×p , and a scalar µ, the optimization problem min µ, G, P, J1, J2, Q, R, T, U, W, ∆ µ subject to BMI [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]; LMI [START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturations[END_REF] and LMI

µI n -P -C ′ Ω [R + ∆] ⋆ 2R > 0 (28)
maximizes the radius of the ball E(µI n ) verifying

E(µI n ) ⊂ L V (1), (29)
because, by multiplying the Inequality (28) on the right by xk = ( x ′ 0 ϕ ′ (Cx 0 ) ) ′ and on the left by its transpose, one gets the following inequality µx ′ 0 x 0 + 2SC(ϕ(•), Cx 0 , R) ≥ V (x 0 ; ϕ(Cx 0 )). As ϕ(•) verifies the bounded sector condition, the inclusion (29) is then satisfied.

Remark 12

The Inequality ( 22) is a BMI. This induces numerical difficulties. By imposing U = I m , it becomes a LMI which is more convenient to cope numerically with, even if it leads to a suboptimal solution. An other way to circumvent the bilinear nature would be to consider U = αI m , where α is a free scalar and using a line-search procedure (or to consider an iterative algorithm similar to [START_REF] Gomes Da | Stability and stabilization of linear discrete-time subject to control saturation[END_REF] or to [START_REF] Gonzaga | Stabilization of discrete-time nonlinear systems subject to input saturations: a new Lyapunov function class[END_REF]). U = I m will be considered in the sequel of the paper for a sake of clarity.

In the following, numerical examples are shown to highlight some special features related to Theorem 10.

Example 2: Consider system ( 16)-( 17) controlled by a linear state feedback with n = 2; p = m = 1; ρ = 1.5, Ω = 0.9, where ϕ(y) = Ω The LMI set of [START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF] provides K = -0.3324 -1.0006 such that matrix A cl +F ΩC has eigenvalues in {-0.1047; 0.6588} and we wish to estimate the basin of attraction. This estimate is performed by applying Theorem 10 and is compared to the one obtained via the quadratic Lyapunov function in the framework of [START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF]. By applying Theorem 10, one gets µ = 0.2810, with ∆ = W = R = 0.0381; T = 0.2424; Q = 0.2323; J 2 = 0.5188; P = G = 0.0418 0.0173 0.0173 0.2305 ; J 1 = -0.0804 -0.6335 . By applying the method proposed in [START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF], the estimate of B 0 is given by the ellipsoid E( P ), where P = 0.0865 -0.0470 -0.0470 0.4310 .

With respect to the basin of attraction B 0 estimate, it is possible to see, in Fig. 4, that our estimate, given by the set L V (1), presents nonconvex and disconnected sets (solid lines). The bounding paraboloids (dot-dashed lines) and the largest sphere E(µI n ) (dashed line) are also depicted, in addition of the ellipsoid E( P ) (dotdashed lines). For this example, we point out that the ellipsoid E( P ) is included in our estimate L V [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. By comparing the areas of both estimates, A E( P ) = 16.7742 and A L V (1) = 28.3666, we can stress here that the improvement is around 65%. Two initial conditions, x 1 0 = (4; -1.45) ′ and x 2 0 = (-2.55; 1.88) ′ , are shown in Fig. 4 (dot-square-star and dot-diamond-cross). Both are settled in the disconnected sets of L V (1). The trajectories are depicted as dot-star and dot-diamond respectively and it should be pointed out that every point is placed inside L V (1). It is exposed in Fig. 5 the trajectories with respect to the discrete time, x 1 0 at top and x 2 0 at bottom. One can notice that the system is asymptotically stable.

Example 3: Consider, now, the system ( 16)- [START_REF] Lur | On the theory of stability of control systems[END_REF] with n = p = 2, m = 1, ρ = 5, Ω (1) = 1.5, Ω (2) = 1 where ϕ (ℓ) (y) = Ω (ℓ) . Bounding ellipsoids (red dot-dashed lines), the largest sphere E(µIn) (dashed lines) and the ellipsoid estimate E( P ) provided in [START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF] (black dot-dashed lines).

y (ℓ) (1+cos(30y (ℓ) )) 2 ∀ ℓ = 1, 2. The
! " # $ % & ' ( ) * "! (% (# ! # + # , -./01, ! " # $ % & ' ( ) * "! (# ! # % + " , 1
-./01, The control gains, associated with the optimal solution of the framework proposed in [START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF], are given by K = Applying the result of [START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF] ficient conditions, the question about the gap between the basin of attraction and its estimate L V (1) may arise. That is, is there a better estimate (for instance the ellipsoid set E(P )) of the basin of attraction? Hence, we have analyzed the trajectories related to initial conditions belonging to this set, x 0 ∈ E(P ), including the set L V (1).

The initial conditions associated with an unstable trajectory are depicted as '+' markers on Fig. 6. There is no marker inside L V (1), as expected, but it can be seen that numerous points x 0 ∈ E(P )/L V (1) are associated with an unstable trajectory of the system ( 16)-( 17). This implies that E(P ) cannot be an estimate of the basin of attraction (at least for this nonlinearity ϕ(•)). In addition, the points located between the disconnected sets L V (1) draw attention and point out that the set L V (1) is here well suitable to estimate B 0 . Thus, we have exposed an interesting feature of this new Lyapunov function because the level set L V (1) is a less conservative estimate of the basin of attraction B 0 for the class of systems defined by ( 16)-( 17) than an ellipsoid.

Remark 13 To illustrate Remark 12, we consider U = αI m and the line-search procedure on Example 2. The comparison in Table 1 underline that our choice α = 1 -in this case -is not so conservative.

Conclusion

In this paper, it is presented a new Lyapunov function which was considered to address several problems of stability of discrete-time Lur'e system. The great feature of this new Lyapunov function is that the stability conditions were derived only with the help of the cone bounded sector conditions. The assumption about the nonlinearity bounding derivative (or variation) is not necessary, in contrast to the frameworks based on the Lur'e-type function. By considering the proposed function, LMI conditions for the global stability problem were formulated. Also, we have stated that the existence of such Lyapunov function is related to the strictly positive realness of an auxiliary system transfer matrix. Furthermore, in case of an input saturation for a given linear state and/or nonlinearity feedback, the proposed function is employed to derive an optimization problem under LMI constraints. The aim is to enlarge an estimate included in the basin of attraction. Several examples are given to highlight some special characteristics of this new Lyapunov function such as an estimate composed of disconnected and/or nonconvex sets. The examples illustrate that this estimate is less conservative than ellipsoidal domains related to quadratic functions.

Fig. 1 .

 1 Fig.1. Surfaces given by the proposed Lyapunov function V (x; ϕ(Cx)) and the bounding paraboloids, V (x) and V (x).

Fig. 2 .

 2 Fig. 2. Disconnected level sets LV (γ) and Inclusion (9) with the parametrization line x2 = x1 of Fig. (1).

Example 1 : 1 √ 2 , 1 √ 2 ]

 11212 Consider system (1)-(2) with n = 2; p = 1 andThe nonlinearity is the half-circle given in the motivation section. By taking Ω = a sector for ϕ(•) is defined and matrices A and A + F ΩC are Schur. By applying Theorem 2, the LMI (10) is feasible, with the following numerical results P = 0.9293 -0.1045 -0.1045 0.9462 and ∆ = 0.5477. In fact, one can conclude that the origin of the considered example is globally asymptotically stable for any nonlinearity ϕ(•) belonging to the sector [0;

Fig. 4 .

 4 Fig. 4. Disconnected estimate LV (1) (solid line) obtained by Theorem 10. Two trajectories for two initial conditions inside disconnected LV (1), x 1 0 and x 2 0. Bounding ellipsoids (red dot-dashed lines), the largest sphere E(µIn) (dashed lines) and the ellipsoid estimate E( P ) provided in[START_REF] Castelan | Absolute stabilization of discrete-time systems with a sector bounded nonlinearity under control saturations[END_REF] (black dot-dashed lines).

Fig. 5 . 2 k(bottom) generated from x 1 0 and x 2 0

 522 Fig. 5. Asymptotically stable trajectories x 1 k (top) and x 2 k (bottom) generated from x 1 0 and x 2 0 respectively -state components: x k,(1) with '+' and x k,(2) with '×'. matrices are

- 1 .

 1 2209 0.4497 and Γ = -0.7030 1.5212 . We apply Theorem 10 and one gets µ = 1.1433 as solution of the optimization problem. The numerical results are given by J 1 = -0.1893 0.4456 , J 2 = -0.0501 0.1971 ,

  , L V (1) and E( P ) are depicted in Fig.6. As in the previous example, L V (1) is nonconvex and disconnected. Numerically one gets A E( P ) = 7.69 and A L V (1) = 9.7299, which means an improvement of 26.53% of the estimate of B 0 with our approach. Moreover, due to the fact that Theorem 10 provides suf-Fig.6. Estimation set LV (1) (solid line) obtained by Theorem 10. The largest ball E(µIn) (dashed line) and E( P ) (dot-dashed line) are depicted. Markers '+' represent the initial conditions x0 = [x 0,(1) ; x 0,(2) ] ′ for which the associated trajectories are unstable.

			basin of attraction B 0
		leads to the estimate E( P )
	of B 0 , with P =	0.4146 0.4659 0.4659 0.9261	. The estimates of the

U A ′ cl Π

⋆ ⋆ -2T J ′ 2 U F ′ cl Π 2 ⋆ ⋆ ⋆ -2U -B ′ Π 2 ⋆ ⋆ ⋆

U F ′ cl C ′ Ω [W + ∆] ⋆ ⋆ -2U -B ′ C ′ Ω [W + ∆] ⋆ ⋆

Acknowledgements

This work was partially supported by ANR project ArHyCo, Programme "Systèmes Embarqués et Grandes Infrastructures" -ARPEGE, contract number ANR-2008 SEGI 004 01-30011459. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement no 257462: HYCON2 Network of Excellence "Highly-Complex and Networked Control Systems".


Jamal Ibrahim Haidar
email: jhaidar@worldbank.org

Allen Dennis

Ibrahim Elbadawi

Caroline Freund

Karim Ismail

Mustapha Aart Kraay

Kamel Nabli

TRADE AND PRODUCTIVITY: SELF-SELECTION OR LEARNING-BY-EXPORTING IN INDIA

Keywords: JEL codes: C12, F10, F20, F40, L1, L2, and L6 trade, learning-by-exporting hypothesis, self-selection hypothesis, total factor productivity, causality, heterogeneous firm model

Recent literature tried to explain the Indian growth miracle in different ways, ranging from trade liberalization to industrial reforms. Using data on Indian manufacturing firms, this paper analyzes the relationship between firm's productivity and export market participation during 1991-2004. While it provides evidence of the self-selection hypothesis by showing that more productive firms become exporters, the results do not show that entry into export markets enhances productivity. The paper examines the explanation of self selection hypothesis for total factor productivity differences across 33,510 exporting and non-exporting firms. It uses propensity score matching to test the learning-by-exporting hypothesis. In line with the prediction of recent heterogeneous firm models of international trade, the main finding of the paper is: more productive firms become exporters but it is not the case that learning by exporting is a channel fuelling growth in Indian manufacturing.

La littérature récente a essayé par différentes manières d'expliquer la miracle de la croissance indienne, allant de la libéralisation des échanges aux réformes industrielles. En utilisant les données sur les entreprises manufacturières indiennes, cet article analyse la relation entre la production des entreprises et la participation du marché d'exportation entre 1991 et 2004. Tandis que les résultats fournissent des épreuves sur l'hypothèse de l'auto-sélection en montrant que les entreprises les plus productives deviennent des exportateurs, les résultats ne prouvent pas que l'entrée sur les marchés d'exportation augmente la productivité. L'article examine l'hypothèse de l'auto-sélection pour les différences du facteur total de productivité entre 33,510 entreprises exportatrices et non exportatrices. Il utilise le score de propension pour tester l'hypothèse d'apprentissage par l'exportation. Conformément avec la prédiction des modèles récents des entreprises hétérogènes du commerce international, la principale conclusion de l'article est la suivante: les entreprises les plus productives deviennent des exportateurs, mais ce n'est pas le cas que l'apprentissage par l'exportation est un canal alimentant la croissance dans le secteur manufacturier Indien.

I. Introduction

Exporters tend to outperform non-exporters. The direction of causality -productivity increases exports or exports enhances productivitywithin this relationship is, however, still under discussion. Do more productive firms within an industry export? What are the determinants behind different trade patterns within an industry? How are these differences in trade behavior related to productivity differences among firms? This paper analyzes these questions empirically for a sample of firm data from the manufacturing industry in India, a country that has not yet been well investigated from this perspective.

There are two alternative, but not mutually exclusive, hypotheses on why exporters can be expected to be more productive than non-exporting firms (see [START_REF] Bernard | Exports and success in German manufacturing[END_REF][START_REF] Bernard | Exceptional exporter performance: cause, effect, or both?[END_REF]: self-selection or learning-by-exporting.

The first hypothesis points to self-selection (SS) of the more productive firms into export markets. The reason for this expectation is that there are additional costs of selling goods in foreign countries. The range of extra costs includes transportation costs, distribution or marketing costs, personnel with skills to manage foreign networks, or production costs in modifying current domestic products for foreign consumption. These costs provide an entry barrier that less successful firms cannot overcome. Firms face difficulties in foreign market, due to the existence of sunk costs associated with selling abroad and fiercer competition in international markets. For example, [START_REF] Roberts | The decision to export in Colombia: An empirical model of entry with sunk costs[END_REF] and [START_REF] Bernard | Export entry and exit by German firms[END_REF] find evidence for the existence of sunk costs in exporting.

In addition, competition could be fiercer outside home market, a feature that would again allow only the most productive firms to do well abroad. This explanation is Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 in line with the assumption made in the theoretical literature of international trade with heterogeneous firms that high-performing firms self-select themselves into foreign markets. [START_REF] Bernard | Exceptional exporter performance: cause, effect, or both?[END_REF] find that exporters have all their desirable characteristics before taking up exporting, and that the performance paths of exporters and non-exporters do not diverge following the launch of export activities by the former.

Cross-section differences between exporters and non-exporters, therefore, may in part be explained by ex ante differences between firms: According to the SS hypothesis, in the period prior to their entry, the productivity distribution of entering exporters should dominate the productivity distribution of non-exporters.

The second hypothesis points to the role of learning-by-exporting. Knowledge flows from international buyers and competitors help to improve the post-entry performance of export starters. Furthermore, firms participating in international markets are exposed to more intense competition and must improve faster than firms who sell their products domestically only. Exporting makes firms more productive.

Empirical papers have investigated the role of exports in promoting growth in general2 , and productivity in particular, using aggregate data for countries and industries for a long time. However, only recently have comprehensive longitudinal data at the firm level been used to look at the extent and causes of productivity differentials between exporters and their counterparts (which sell on the domestic market only).

For a decade following the seminal paper by [START_REF] Bernard | Exporters[END_REF], researchers all over the world used firm level data to investigate the relationship between exporting and productivity in micro-econometric studies. [START_REF] Wagner | Exports and productivity: A survey of the evidence from firm-level data[END_REF] surveyed the empirical strategies applied, and the results produced, in micro-econometric studies published between 1995 and 2004. In general, he found that the more productive firms self-select into export markets, while exporting does not necessarily improve productivity. Among the countries covered are industrialized countries (e.g., U.S., UK, Canada, Germany); Latin American countries (Chile, Colombia, Mexico); Asian countries (China, Korea, Indonesia, Taiwan); transition countries (Estonia, Slovenia); and least developed countries from sub-Saharan Africa. In particular, these findings include ones for Chile between 1990and 1996[START_REF] Alvarez | Exporting and performance: Evidence from Chilean plants[END_REF]; for China between 1988 and 1992 in [START_REF] Kraay | Exports and economic performance: evidence from a panel of Chinese enterprises[END_REF]for Colombia between 1981[START_REF] Wagner | Exports and productivity: A survey of the evidence from firm-level data[END_REF], Mexico between 1986and 1990, and Morocco between 1984[START_REF] Wagner | Exports and productivity: A survey of the evidence from firm-level data[END_REF][START_REF] Clerides | Is learning by exporting important? Microdynamic evidence from Colombia, Mexico, and Morocco[END_REF]; and for Indonesia between 1990 and 1996 in [START_REF] Blalock | Learning from exporting revisited in a less developed setting[END_REF].

However, India has not been explored in this literature. Nevertheless, it seems important to study the Indian context for two reasons. First, India is a large developing country, so it is useful to understand Indian exporting patterns. Second, India's growth in manufacturing and particularly in exporting has been slower than that of many other developing countries (notably China), so it is interesting to understand the drivers of Indian exports.

Hence, this paper adds to the empirical literature on the direction of causality between trade and firm productivity by studying a particularly important developing country, India. Using longitudinal micro data on Indian manufacturing firmsindustry accounts for 54.6% of the GDP and employs 17% of the workforce -this paper examines the validity of self selection and learning-by-exporting hypotheses for total factor productivity differences across 33,510 exporting and non-exporting Indian manufacturing Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 firms between 1991 and 2004. While it provides evidence of the self-selection hypothesis by showing that more productive firms become exporters, the paper does not find that entry into export markets enhances productivity. The paper uses propensity score matching to test the learning-by-exporting hypothesis. Firms which face foreign competition perform better than their domestic competitors years before they enter export markets. The changes in characteristics of exporting firms before they start exporting are not statistically different than those of the firms that serve only the domestic markets.

While there is weak evidence as to whether these exporting firms prepare themselves consciously for the international markets, the main result is robust.

The paper is organized as follows. Section II describes the data. Section III.A presents the export premium calculation methodology. Section III.B estimates total factor productivity. Sections III.C and III.D sketch the self selection hypotheses along with related empirical evidence as well as it applies propensity score matching to test the evidence for learning due to exposure to international markets. Section IV concludes.

II. Data

We use an Indian firm-level panel dataset of balance sheets and income statements spanning 14 years (1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)) throughout the analysis. The data comes from the Center for Monitoring the Indian Economy (CMIE) Prowess database. We limit the analysis to manufacturing firms because the main firm-level productivity measure used in the estimations is the total factor productivity (TFP), which is not an appropriate measure of productivity for non-manufacturing firms as these firms have a different structure of production than manufacturing ones. The dataset covers 33,510 domestically-owned Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 manufacturing companies categorized by sectors. The largest sectors, measured by the number of companies, are food products, textiles, chemicals, basic metals and machinery.

Table 1 provides descriptive statistics. The average percentage of exporters in total firms is 55 %. The firms that change their export status from non-export to export (entrant) and from export to non-export (quitter) constitute on average 5.5 % and 3.5 % of all firms, respectively across time. Exporting firms have on average larger sales, income and capital (Table 2). They spend more on raw materials, power and fuel expenses, and pay higher wages. Non-exporting firms tend to be younger than exporting firms. The TFP index is on average larger for exporters although the difference does not appear to be statistically significant.

However, the unbalanced nature of the sample, frequency of entry and exit behavior of firms, and missing observations make it difficult to interpret these results. A more formal and systematic analysis that takes into account the consistency of firms in terms of export behavior is required for a reliable comparison of exporters and nonexporters.

[Insert Table 1] [Insert Table 2]

III. Empirical model and analysis

The below four subsections provide estimation of export premium measurement and TFP as well as present empirical tests for the SS and LE hypothesis.

III.A Export Premium

Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 To document the differences between exporters and non-exporters, we measure the export premium (ceteris paribus percentage differences in firm characteristics between exporters and non-exporters) for each year during the 1991-2004 sample period.

The main firm characteristics of interest are productivity measure (TFP), capital, sales, and unit labor cost, which is obtained by dividing total labor cost (salaries and wages) by the value of real output. Following [START_REF] Bernard | Exceptional exporter performance: cause, effect, or both?[END_REF], we estimate the export premium for each firm i in each year by regressing the firm characteristics on an export dummy and a set of control variables. More specifically, the export premia is estimated from a regression of the following form:

(1) where represents the firm characteristics of interest (productivity measure -TFP, capital, sales, and unit labor cost -salaries and wages/value of real output); is a dummy for the current export status (1 if firm is an exporter, 0 otherwise); i Industry dummy includes the 2-digit NIC codes ; and is a vector of firm-specific controls (in logs except for size dummy), which include different combinations of firm characteristics such as a firm size dummy, firm age and capital. 3 . Table 3 presents the results of the export premium regressions.

[Insert Table 3]

III.B Total Factor Productivity estimation

3 We classify each firm as large, medium or small in size, following [START_REF] Topalova | Trade liberalization and firm productivity: The case of India[END_REF], depending on its average sales over the span of the data. We include the firm in the "large size" category if average sales over the entire period are in the top 1 percent of the distribution; in the "medium size" category if sales are greater than the median, excluding the top 1 percentile of the distribution; or in the "small size" category if average sales over the period are less than the median. The standard measure in the literature for firm size is the employment level. The ordinary least squares estimation of TFP -as the difference between actual and predicted output -leads to omitted variables bias since the firm's choice of inputs is likely to be correlated with any unobserved firm-specific productivity shocks. Adding firm fixed effects into the estimation could solve the simultaneity problem if productivity is assumed to be time-invariant. However, this strategy is not appropriate since we are interested in changes in firm-level productivity.

i i i i i Control Industry Export X           ln
The consistent firm-level measure of TFP, which we use in this paper, is constructed based on the methodology of [START_REF] Levinsohn | Estimating production functions using inputs to control for unobservables[END_REF]. Assuming a Cobb Douglas production function, this methodology uses firm's raw material inputs to correct for the simultaneity in the firm's production function.

(2) where y denotes output, l denotes labor, e denotes electricity consumption, m denotes raw material inputs, k denotes capital, and w denotes the unobservable part of the productivity shock that is correlated with the firm's inputs. All variables are expressed in natural logarithm. 4 We rewrite (2) as:

(3) where 4 All variables that enter into TFP estimation are deflated using appropriate deflators from India's National Account Statistics. Value of output is deflated using the corresponding industry deflators. Energy and fuel expenses are deflated by a fuel and energy deflator. Salaries and wages as well as material expenses are deflated by the wholesale price index. Finally, gross fixed assets are converted to real terms by a capital goods deflator. Energy and fuel consumption is used as the intermediate input as a proxy for unobserved productivity shocks. We estimate equation ( 2) following the approach of [START_REF] Robinson | Semi-parametric econometrics: A survey[END_REF]. The goal is to obtain the estimates on the coefficients of inputs that enter (2) linearly (i.e.

t i t i t i k t i m t i p t i l t i w k m e l y , , , , , , ,              t i t i t i t i m t i l t i e k m l y , , , , , , ) , ( 
) , m l   . Then, we define

t i m t i l t i t i m l y V , , , ,       
and estimate the following equation:

t i t i t i k t t i k t i e k g k V , , 1 , 1 , , ) (            (5)
where (.) g is a function of lagged values of  and k . We approximated this function by a high-order polynomial expression in 1  t  and 1  t k . We conducted the estimation using 2-digit National Industrial Classification codes (due to small number of companies in some of the 4-digit level industries) and over two time periods: a period of high-growth (before 1996) and a period of low-growth (after 1996). Having obtained consistent coefficients on the production inputs, we estimated the TFP using the initial production function.

Finally, after obtaining TFP measures, we created a TFP index in order to make the estimated TFP comparable across industries. 5 The resulting TFP index serves as the dependent variable in all the regressions.

III.C Self-selection hypothesis

Exports and firm productivity can be linked via at least two different ways, as explained above. This subsection addresses the self-selection hypothesis: better firms are more likely to enter into export markets.

The scale of operations differs significantly between exporters and non-exporters.

Exporters produce on average 3.75 times more than non-exporters. After controlling for capital endowment, this average difference drops down to 9%. Moreover, exporters employ more capital and have lower unit labor cost than non-exporters. Even after controlling for firm size, exporters employ on average 97% more capital than nonexporters. The differences in sales and capital endowment become more noticeable after 1997. The average unit labor cost is 12.4 % lower for exporters. Controlling for capital endowment in addition to firm size leads to a slightly larger average gap (-13.4 %) for the unit labor cost, as shown in the average coefficients in Table 3 andTable 4. The statistical significance of the difference increases when capital is included in the controls.

The regression results reported so far confirm the previous robust findings for other countries. Exporters have significantly different characteristics and exhibit superior performance in terms of productivity, sales, capital endowment and unit labor cost during each year in the sample.

The above cross-section analysis documents the different characteristics of exporters and non-exporters. However, this test is not sufficient to identify whether the firms with desirable characteristics self-select into export markets. To address this issue, we compare the performance of export-starters with non-exporters several years before entry.

Similar studies defined export-starters in different ways in the literature. An export-starter is defined in [START_REF] Bernard | Exports and success in German manufacturing[END_REF] as a plant that exports for the first time after at least three years in the sample. Accordingly, the subsample includes only plants that have at least four consecutive annual observations and do not export in any of their first three annual observations. Bernard and Jensen (1999) follow a similar approach. Serti and Tomasi (2008) define export-starters as firms that do not export at Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 least for two years and continue to export subsequent to their entry. Undoubtedly, these definitions above as well as others not mentioned here are influenced by data restrictions.

The approach of Serti and Tomasi (2008) has the clear advantage of identifying continuous export behavior. Defining a firm as an export-starter based on whether the firm exports for the first time after a few years might be unsatisfactory in that the firm in question could stop exporting after one year or so. If the export behavior is not consistent and continuous over time for the export-starters, these firms would not be exposed to the benefits of participating in the international markets, if any, to a measurable extent. Thus, it would be problematic to draw a sound conclusion from this type of analysis.

[Insert Table 4] Hence, we divide the sample into two sub-periods 1991-1997 and 1998-2004 in order to have consistent export behavior data and a reasonable number of observations in the analysis. The firms that do not export for the first three years in the sample, and export starting 1994 until 1997 are defined as the export starters for the first period. The firms that do not export for the first three years in the sample, and start export starting 2001 until 2004 are defined as the export starters for the second period. In each sample period, we define non-exporters as firms that did not export in any of the years in the selected sample periods. We follow the same regression in equation ( 1), but with different export status variable, to measure the systematic differences of plant characteristics between export-starters and non-exporters:

(6)
where is a dummy for the export-entry status (1 if firm is an export-starter in year (1994 or 2001), 0 if it is a non-exporter), represents the firm characteristic of 6) reflects industry and year interacted fixed effects controls which should capture most of these changes as well as macroeconomic consequences of these reforms that are experienced by all firms.

i it i iT it Control Industry Export X           ln iT Export i T it X
Table 5 shows the results from equation ( 6). It displays the differences between characteristics of export-starters and non-exporters 1-3 years before entry into international market during each sample period. Export-starters are more productive than non-exporters years before they export, controlling for firm size, and productivity measure results are stronger than previous cross-section results. For instance, TFP of export-starters is on average 30% higher than for non-exporters during 1991-1993 and the productivity gap in favor of export-starters is 33.4 % during 1998-2000 6 . TFP gap increases after controlling for capital; in this case, the results are statistically significant at 1 % confidence interval level. Moreover, export-starters have further pre-entry desirable characteristics. These findings imply that more productive firms decide to engage in export markets.

The above subsection addressed the self-selection hypothesis; it showed that better firms make it into export markets. Table 3 and Table 4 report the results. The coefficients in those Tables show export status from separate OLS regressions. Although these results are an important proof of superior performance of exporters, the crucial firm characteristic of interest is the productivity measure. After adjusting for industry and size 6 The 30% and 33.4% figures are averages of LnTFPindex coefficients of 1991-1993 and 1998-2000, respectively, in effects and using a TFP index, the export premium results indicate that exporters have higher level of productivity. Table 3 shows that exporters are on average 14.8 % more productive than non-exporters during 1991-1997 and 9.3% more productive than non exporters during 1998-2004. 7 [Insert Table 5]

III.D.1 Learning-by-exporting (LE) hypothesis:

This subsection looks at the learning-by-exporting (LE) hypotheses, which suggests that firm productivity increases after export market entry. Does firm productivity increase post export market entry? This subsection aims to answer this question.

To answer the above central question, we first try to answer a sub-question: How do export starters compare, in terms of firm productivity changes, to non-exporters post export market entry? We measure post export market entry growth rate premium by equation ( 7):

i i i iT i iT T Control Industry Export T X X X                 0 0 1 1 1 ln ln % (7)
where iT Export , it X and i Industry are defined the same way as before. Export-starters display no empirical evidence that they count on pre exportmarket entry superior characteristics. Table 6 shows that the economic magnitudes of total factor productivity, sales, capital, and (surprisingly) unit labor cost growth rate premiums are statistically insignificant at all conventional levels although they have positive coefficients between [START_REF] Wagner | Exports and productivity: A survey of the evidence from firm-level data[END_REF]and 1993. During 1998and 2000, the difference , the difference between exporters and non-exporters sales growth is 33%, and capital accumulation differential was 11%. Also, the coefficient of the unit labor cost has a negative (expected) but statistically insignificant sign. Moreover, the differential between exporters and nonexporters productivity growth is not statistically significant although the stock of capital increased. These observations allow concluding that non-exporters are inferior to export starters before the latter enter export markets. During 1998-2000, the stock of capital grows faster for export starters, probably because of sales growth. Nevertheless, during pre-entry years, export starters do not display significant firm productivity improvements compared to non-exporters.

Table 6 displays the results. During 1991-1993, the growth rate premia for TFP, sales and capital are all positive, but statistically insignificant. Surprisingly, the coefficient of unit labor cost also has a positive sign; however, it is not significant. Thus, during the initial pre-entry period, there is no evidence that exporter-starters build upon their already-superior characteristics. During 1988-2000, exporters accumulate around 11-12 % more capital than non-exporters. This is accompanied by a sales growth differential of 33 %. In spite of this increase in capital stock, exporters' productivity growth is not statistically different than non-exporters' productivity growth. The unit labor cost has the expected sign in this period although it is insignificant. To sum up, the export-starters already have the competitive advantage over non-exporters before they enter export markets. However, they do not experience any major productivity Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 improvements compared to non-exporters during the pre-entry periods. Only during 1998-2000, the capital stock appears to grow more for export-starters, possibly due to positive sales growth during this period.

[Insert Table 6] Export starters experience higher capital, sales, and TFP growth rates than nonexporters, suggesting that export activity enhances firm performance. Controlling for firm size, the capital accumulation growth rate of export-starters is 9-16% higher than nonexporters. And, between 1994 and 1996, the growth rate of TFP is 0.7% higher than nonexporters. Over the following 3 years this differential stays positive but with no statistical significance. However, between 2001 and 2004, the TFP differential (+1.38%) becomes statistically significant. Controlling for capital, growth rate of sales is 8-15% higher for exporters. Moreover, the coefficient of unit labor cost, during all years, holds the expected sign but remains statistically significant only during the first year (two years) after entry during 1991-1997 (1998-2004). For these periods, the export-starters' ULC growth rate is 3.83 % (6.65 %) lower than non-exporters' ULC growth.

A theoretically-reliable method to identify the effect of exporting and the pros of exporting on export-starters would require data on the performance of export-starters if they do not engage in export market. Given this data is not observable, the above indication carries a problematic implicit assumption. Recent literature addressed this obstacle within the heterogeneous firm model. The heterogeneous ingredients of firm productivity between exporters and non-exporters -i.e. [START_REF] Melitz | The impact of trade on Iintra-industry reallocations and aggregate industry productivity[END_REF], Head and Reis (2003), as well as Helpman, [START_REF] Helpman | Export versus FDWE with heterogeneous firms[END_REF] -can act as an objection to the assumption of [START_REF] Bernard | Exceptional exporter performance: cause, effect, or both?[END_REF] that compares export-starters with all non-Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 exporters, considering all non-exporting firms can provide a counter-factual. The propensity score matching method, which provide a more precise control for comparison groups (i.e. exporters and non-exporters) differentials, tackles this heterogeneity issue.

The next subsection summarizes the theoretical implications of matching techniques, and it shows the empirical results of its application.

III.D.2 Propensity Score Matching and Learning Effects

Technically, if identifies the change in TFP (or another firm characteristic) and represents an indicator of whether firm exported for the first time at time , then is the change in TFP at time ( ) following entry. A systematic measurement of effects of entry into export markets requires a counterfactual.

Thus, the export entry causal effect on firm at time can be written as where denotes the outcome for export-starters had they never entered export markets. The main problem is that this outcome is not observable. Following [START_REF] Heckman | Matching as an Econometric Evaluation Estimator: Evidence from Evaluating a Job Training Programme[END_REF], we define the average effect of exporting on export-starters as:

The appropriate identification of a counterfactual for the latter term in the above equation is a key determinant of the measurement quality. Assuming that a vector of observable firm characteristics can capture all the differences between export-starters and firms in this control group, we estimate this counterfactual by the corresponding average value of a control group of firms. Non-exporters, which perform comparably to exportstarters before export-market entry, are a common choice in the related literature for this control group. Thus, the appropriate counterfactual is . The goal of this matching technique is to identify a non-exporting firms group for which the variables distribution affecting the export decision is comparable to the export-starters relevant distribution. We adopt the propensity score (estimated probability of a firm to export given its characteristics) matching technique of [START_REF] Rosenbaum | The Central Role of the Propensity Score in Observational Studies for Causal Effects[END_REF] given matching non-exporters with export-starters on n-dimensional vector of characteristics is unfeasible. The propensity score matching technique facilitates comparison between firms and makes matching feasible by summarizing pre-treatment characteristics of each subject into a single index variable, namely, propensity score.

y    1 , 0  it Expdummy i t 1 , s t i y   s t  0  s i s t  1 , s t i y   0 , s t i y   0 , s t i y         1 1 1 0 , 1 , 0 , 1 ,               it s t i
As an initial step, we find the propensity score for all export-starters and nonexporters for 1991-1997 and 1998-2004 using a probit specification for each period as follows. We have used several different combinations of controls and earlier periods (t-2 and t-3) in the F(.) function. The resulting specification in equation ( 6) is the outcome of the algorithm described below.

( The results from the radius matching method are almost identical. With radius matching, each treated unit is matched only with the control units whose propensity scores fall in a predefined neighborhood (the radius determines the length of the neighborhood) of the treated unit's propensity score. Therefore, unlike the nearest neighborhood matching, the radius matching can potentially result in unmatched treated units. In the present case, several choices of reasonably small radii did not leave any unmatched treated units.

We follow [START_REF] Becker | Estimation of average treatment effects based on propensity scores[END_REF]'s algorithm to confirm that the probit specification is valid and that the optimal number of groups of firms in which the propensity scores and the means of company characteristics do not differ for the treated (export-starter) and the control (non-exporter) units8 . Initially, the nearest neighbor matching method eliminates substantially different non-exporting firms, and matches 98 (91) non-exporting firms to 110 (93) export-starters during 1991-1997 (1998-2004). The second step is to divide these updated samples into equally spaced intervals such that within each interval, the average propensity scores of the treatment and control group do not differ statistically. 1 Before the second step, after identifying the appropriate matches, we estimated the probit specification again with only the matched sample to update the propensity scores of the remaining control and treated units. This resulted in similar scores for both groups. The first group during 1991-1997 (1998-2004) includes 54 (47) export-starters and 50 (45) non-exporters. The second group during 1991-1997 (1998-2004) includes 56 (46) export-starters and 48 (46) non-exporters.

In both periods, splitting the sample into two intervals satisfies this condition.

Subsequently, we run a simple t-test of difference of means for the pre-entry period 1  t to see if the mean characteristics of firms in all these four groups do not differ statistically between the treated and control units, i.e., we test the samples for balancing hypothesis.

The results in table 7 indicate that this constraint is satisfied and the identified two groups for each period consist of appropriately matched firms. All the P-values are greater than 0.1, thus, the hypothesis that the means of these variables are equal for export-starters and non-exporters is not rejected at any confidence level.

[Insert Table 7] Having assured that the subgroups of firms include very similar control and treatment units, the final step is to estimate the differences in changes of firm characteristics in these four groups during the post-treatment period, i.e. export market entry. That is, we estimate regression equation ( 6) for these matched samples. Table 8 presents the results for these four different groups. Matching leads to substantially different post-entry results especially in terms of TFP growth premia from those in section III.D.1 with non-matched samples.

During the first 3 years of post export market entry, sales for export-starters increased, on average 11.4% faster, compared to their domestic rivals, across the four groups. Also, the growth premium of capital follows the same trend as for sales but with less statistically significant coefficient. For instance, the capital accumulation in both groups, between 1991 and 1997, shows that export-starters experience faster capital accumulation in the second year after export market entry -and the growth is continuous (3-4%) in the long run. Moreover, between 1998 and 2004, the capital accumulation grew faster -7% -for export-starters during the first year after export market entry. Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 The coefficients of unit labor cost (ULC) premium are mainly statistically insignificant although they have the expected negative sign except for the 3-year growth premium in group 1. Table 8 shows that export-starters experience a statistically significant reduction (1.54 % and 1.17 %) in their ULC relative to non-exporters only during the period 1998-2004, and specifically during the first post market entry year. In addition, during 1998-2004, export-starters experience TFP gains (1.03 % and 1.34 %) only during the first post market entry year while the coefficient of TFP growth premium is positive but statistically insignificant for both groups during 1991-1997.

In a nutshell, the above empirical evidence fails to make the case that access to foreign markets via the exports channel enhances learning and productivity for Indian manufacturing firms. TFP of Export-starters did not enhance significantly, compared to TFP of non-exporters, in the long run. However, sales and capital accumulation scaled up when firms were able to engage in foreign markets along with their engagements in their domestic markets.

[Insert Table 8]

IV. Conclusion:

Economists have argued that "openness to trade" increases productivity and stimulates growth. They viewed participation in export markets as a prerequisite for economic growth in developing countries. However, neither the theoretical studies nor the empirical cross-country analyses have reached a consensus on the channels through which exports enhances economic growth This paper examines the question of whether firms self-select into the export market using an Indian firm-level panel dataset of balance sheets and Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 income statements spanning 14 years (1991)(1992)(1993)(1994)(1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004) and covering 33,510 domesticallyowned manufacturing companies categorized by sectors. The analysis takes place in three primary steps. First, we estimate exporter premia, which measure the extent to which exporters outperform non-exporters in terms of TFP, capital, sales and unit labor cost.

We find that Indian exporters are larger (in terms of sales), employ more capital, have lower unit labor costs and higher productivities than non-exporters. Second, we examine whether firms that become exporters already have their desirable characteristics prior to entering the export market. We find that firms that will ultimately become exporters perform better than non-exporters. Third, we examine (i) whether firms prepare for exporting by consciously choosing to undertake productivity-increasing activities and (ii) whether firm productivity enhances following export market participation. We do not find evidence for this preparation or improvement hypotheses.

The self-selection hypothesis implies that more productive firms become exporters as they can cover high fixed costs of entering and serving foreign markets, including costs related to networking and adapting to new quality standards. The other hypothesis is the learning-by-exporting. The paper applies matching techniques to target the Learning-by-exporting hypothesis. The aim is to improve the identification quality of post-entry comparative analysis of exporters and non-exporters. In few words, matching decreases selection-bias as it helps identify certain non-exporters group with similar pre market entry productivity as exporters.

The paper provides evidence that more successful firms are more likely to enter export markets, giving support to the self-selection hypothesis. The results are robust, mainly that firms that engage in foreign competition perform better than their domestic Documents de Travail du Centre d'Economie de la Sorbonne -2012.46 competitors years before they enter export markets. However, there is weak evidence as to whether these exporting firms prepare themselves consciously for the international markets. On the learning-by-exporting front, the paper fails to show that exporters improve their productivity performance after they enter export markets. While the paper finds export benefits (mainly in sales and capital accumulation), it fails to establish positive evidence for the learning-by-exporting hypothesis.

On economic and trade policy levels, this paper shows that export policies targeting the less productive domestic firms in an economy -i.e. via export subsidy -may not lead to trade openness, especially if the only channel of trade openness lies within firm productivity growth.

Needless to mention, as discussed in the paper, that export market entry may increase sales, reduce unemployment, reduce probability of firm exist, and enhance firms learning opportunities (i.e. depending on export destinations). These further issues -along with the reasoning for why the magnitudes of the Indian estimated exporter premium are different than elsewhere in the literature -are left as topics for future research. Moreover, further research can examine the conditions under which learning from exporting may occur. For instance, it is commonly argued that firms will learn more from exporting to industrialized countries than they can from poorer countries, and they will learn more from exporting to dynamic and fast changing environments than from exporting to weak and static environments. In other words, whether most of the Indian exporters export to countries or environments that contained relatively few opportunities for learning is a question for further research. 1994-1995 1994-1996 1994-1997 1994-1995 1994-1996 1994-1997 2001-2002 2001-2003 2001-2004 2001-2002 2001-2003 2001-2004 Log

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  -specific controls (in logs) in the base year (1991 or 1998) including a firm size dummy, firm age and capital. Thus, equation (7) estimates the growth rate premia of export-starters for certain firm characteristics during1991-1993 and 1998-2000 based on their initial firm characteristics.

Table 5 .

 5

	t	t 	T

it Control t i Industry Documents de Travail du Centre d'Economie de la Sorbonne -2012.46

Table 1 : Export patterns of manufacturing firms

 1 Documents de Travail du Centre d'Economie de la Sorbonne -2012.46

	Year	Number of firms	Exporters (%)	Entrants (%)

Table 2 : Descriptive statistics for exporters and non-exporters

 2 Documents de Travail du Centre d'Economie de la Sorbonne -2012.46

		Observations Mean	St. Dev.	Minimum Maximum
	Variables (exporters)
	Sales	16,857	315.97	2,685.43	0.00	159,984.40
	Total income	16,857	323.46	2,736.81	-7.40	162,755.20
	Raw material expenses	16,857	116.80	971.51	-6.77	55,826.18
	Power and fuel expenses	16,857	13.78	72.22	0.00	3,389.74
	salaries and wages	16,857	19.87	113.45	0.00	5,176.53
	Capital	16,857	218.61	1,686.69	0.00	90,204.68
	company age	16,134	28.82	75.14	0.00	181.00
	TFP index	16,207	0.24	0.69	-5.27	8.84
	Variables (non-exporters)
	Sales	16,653	54.61	349.64	0.00	26,966.30
	Total income	16,653	56.28	359.79	-18.76	27,388.14
	Raw material expenses	16,653	23.81	198.32	-5.95	19,645.06
	Power and fuel expenses	16,653	3.80	23.49	0.00	1,551.34
	Salaries and wages	16,653	4.61	38.58	0.06	1,625.04
	Capital	16,653	59.31	558.95	0.00	46,231.60
	Company age	16,653	24.90	94.87	0.00	172.00
	TFP index	16,653	0.22	0.73	-4.99	7.79

Table 3 : Export Premium OLS regression of log values of firm characteristics on export status

 3 : P-values are reported in parenthesis below the estimates. ** and *** refer to 5% and 1% statistical significance levels, respectively. The reported export premium estimates are the exact percentage differentials given by (e^β-1)*100 where β is the export dummy coefficient from regression equation (2). All regressions include industry dummies and company age (in log). The TFP and unit labor cost regressions include a firm size dummy.

		1991	1992	1993	1994	1995	1996	1997	1998	1999	2000	2001	2002	2003	2004
	LnTFP	0.186 *** (0.000)	0.165 *** (0.000)	0.168 *** (0.000)	0.155 *** (0.000)	0.157 *** (0.000)	0.111 *** (0.000)	0.095 *** (0.000)	0.059 ** (0.031)	0.078*** (0.003)	0.129 *** (0.000)	0.115 *** (0.000)	0.090 *** (0.003)	0.088 *** (0.003)	0.090 *** (0.003)
	LnCapital	1.253 *** (0.000)	1.118 *** (0.000)	1.542 *** (0.000)	1.861 *** (0.000)	1.925 *** (0.000)	1.987 *** (0.000)	2.179 *** (0.000)	2.455 *** (0.000)	2.525 *** (0.000)	2.557 *** (0.000)	2.807 *** (0.000)	3.221 *** (0.000)	3.085 *** (0.000)	3.396 *** (0.000)
	LnSales	1.734 *** (0.000)	1.735 *** (0.000)	2.066 *** (0.000)	2.422 *** (0.000)	3.056 *** (0.000)	2.827 *** (0.000)	3.219 *** (0.000)	3.38 *** (0.000)	4.163 *** (0.000)	4.632 *** (0.000)	5.18 *** (0.000)	6.120 *** (0.000)	5.80 *** (0.000)	6.215 *** (0.000)
		-	-		-	-	-	-
	LnUnitLaborCost	0.068	0.055	-0.094 **	0.190 ***	0.187 ***	0.159 ***	0.128 ***	3.38 ***	4.163 ***	4.632 ***	5.180 ***	6.120 ***	5.80 ***	6.215 ***
		(0.127)	(0.214)	(0.027)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Observations	1986	2050	2110	2724	3285	3478	3459	3465	3728	3946	3902	3949	4058	3864

Note

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Table 4 : Export Premium OLS regression of log values of firm characteristics on export status and firm-specific controls (in logs)

 4 Note: P-values are reported in parenthesis below the estimates. ** and *** refer to 5% and 1% statistical significance levels, respectively. The reported export premium estimates are the exact percentage differentials given by (e^β-1)*100 where β is the export dummy coefficient from regression equation (2). All regressions include industry dummies and company age (in log). The TFP controls for capital stock in addition to firm size. The capital regressions control for firm size and the sales regressions control for capital stocks. (Firm size dummy is left out since it is constructed based on sales). The unit labor cost regressions include both a firm size and capital.

		1991	1992	1993	1994	1995	1996	1997	1998	1999	2000	2001	2002	2003	2004
	LNTFP	0.224 *** (0.000)	0.196 *** (0.000)	0.213 *** (0.000)	0.227 *** (0.000)	0.225 *** (0.000)	0.240 *** (0.000)	0.235 *** (0.000)	0.196 *** (0.000)	0.217 *** (0.000)	0.272 *** (0.000)	0.267 *** (0.000)	0.251 *** (0.000)	0.256 *** (0.000)	0.261 *** (0.000)
	LnCapital	0.731 *** (0.000)	0.526 *** (0.000)	0.763 *** (0.000)	0.972 *** (0.000)	0.887 *** (0.000)	0.867 *** (0.000)	0.955 *** (0.000)	1.055 *** (0.000)	1.036 *** (0.000)	0.974 *** (0.000)	1.071 *** (0.000)	1.182 *** (0.000)	1.135 *** (0.000)	1.363 *** (0.000)
	LnSales	0.54 *** (0.000)	0.519 *** (0.000)	0.567 *** (0.000)	0.634 *** (0.000)	0.865 *** (0.000)	0.645 *** (0.000)	0.693 *** (0.000)	0.670 *** (0.000)	0.953 *** (0.000)	1.169 *** (0.000)	1.222 *** (0.000)	1.284 *** (0.000)	1.392 *** (0.000)	1.392 *** (0.000)
		-	-	-	-	-	-	-
	LnUnitLaborCost	0.070	0.056	0.092 **	0.193 ***	0.184 ***	0.148 ***	0.121 ***	0.670 ***	0.953 ***	1.169 ***	1.222 ***	1.284 ***	1.392 ***	1.392 ***
		(0.123)	(0.209)	(0.030)	(0.000)	(0.000)	(0.000)	(0.001)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
	Observations	1986	2050	2110	2724	3285	3478	3459	3465	3728	3946	3902	3949	4058	3864

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Table 5 : Characteristics of export starters

 5 : P-values are reported in parenthesis below the estimates. *, **, and *** refer to 10%, 5% and 1% statistical significance levels, respectively. The reported export premium estimates are the exact percentage differentials given by (e^β-1)*100 where β is the export dummy coefficient from regression equation (6). All regressions include industry dummies and company age (in log). The TFP and unit labor cost regressions include a firm size dummy. Industry and year interacted fixed effects controls are included.

	Variables	1991	1992	1993	1998	1999	2000
	LnTFPindex	0.278 ** (0.012)	0.289 ** (0.017)	0.322 *** (0.007)	0.34 ** (0.013)	0.335 ** (0.021)	0.326 *** (0.039)
	LnCapital	1.476 *** (0.004)	1.686 *** (0.001)	1.809 *** (0.000)	1.595 *** (0.003)	2.41 *** (0.000)	2.76 *** (0.000)
	LnSales	3.232 *** (0.000)	2.552 *** (0.000)	2.493 *** (0.000)	2.91 *** (0.000)	4.027 *** (0.000)	5.565 *** (0.000)
	LnUnitLaborCost	-0.488 *** (0.000)	-0.373 *** (0.002)	-0.382 *** (0.001)	-0.409 ** (0.039)	-0.346 * (0.072)	-0.508 *** (0.005)
	Observations	350	350	352	576	580	582

Note

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Table 6 : Pre-entry export premium of growth rates

 6 -values are reported in parenthesis below the estimates. *** refer to 1% statistical significance level. The reported export premium estimates are the exact percentage differentials given by (e^β-1)*100 where β is the export dummy coefficient from regression equation (8). All regressions include industry dummies and company age (in log). The TFP and unit labor cost regressions include a firm size dummy in panel A. The TFP controls for capital stock in addition to firm size in panel B. The capital regressions control for firm size and the sales regressions control for capital stocks (firm size dummy is left out since it is constructed based on sales) and the unit labor cost regressions include both a firm size and capital. Industry and year interacted fixed effects controls are included.

			Panel A		Panel B
	Variables	1991-1993	1998-2000	1991-1993	1998-2000
	LnTFPindex	0.0182 (0.526)	0.0043 (0.934)	0.0113 (0.702)	0.0032 (0.956)
	LnCapital	0.0624 (0.329)	0.127 *** (0.004)	0.0585 (0.374)	0.1144 *** (0.009)
	LnSales	0.0301 (0.712)	0.3318 *** (0.006)	0.0904 (0.286)	0.3342 *** (0.005)
	LnUnitLaborCost	0.0717 (0.228)	-0.0944 (0.315)	0.018 (0.741)	-0.0973 (0.281)
	Observations	350	574	350	574

PDocuments de Travail du Centre d'Economie de la Sorbonne -2012.46

Figure 7:

Basic data characteristics of non-exporters and exporters in the matched samples

  The differences of means are calculated by subtracting the means of the relevant variables of non-exporters from the means of the corresponding variables of exporter-starters. Numbers in brackets refer to p-values. P-values refer to the t-tests performed for the equality of means, for which the null hypothesis is that the selected groups do not differ in population means.

		Matched sample		Matched sample
		(1991-1997)		(1998-2004)
		Group 1	Group 2	Group 1	Group 2
	Variable	Difference in Means	Difference in Means
	TFP	0.013	0.025	0.106	0.041
		(0.540)	(0.423)	(0.554)	(0.643)
	Capital	0.022	0.138	0.056	0.232
		(0.785)	(0.251)	(0.182)	(0.611)
	Sales	0.125	0.117	0.129	0.223
		(0.332)	(0.721)	(0.294)	(0.336)
	ULC	-0.241	-0.003	-0.247	-0.095
		(0.152)	(0.224)	(0.183)	(0.151)

Documents de Travail du Centre d'Economie de la Sorbonne -2012.46

Table 8 : Post-entry export premium of growth rates -Matched samples

 8

	Group 1 (1991-1997) Growth Rate	Group 2 (1991-1997) Growth Rate Group 3 (1998-2004) Growth Rate	Group 4 (1998-2004) Growth Rate
	Variables

Another line of research -i.e.[START_REF] Amin | Trade Facilitation and Country Size[END_REF] Haidar (2011), Haidar (2009),[START_REF] Haidar | The Impact of Business Regulatory Reforms on Economic Growth[END_REF] -looked at the impact of export facilitation and export costs on economic growth. Documents de Travail du Centre d'Economie de laSorbonne -2012.46

The productivity index is calculated as the logarithmic deviation of a firm in a particular industry from a reference firm's productivity in that same industry in a base year. The productivity of the reference firms in each industry is calculated from the respective industry's TFP regression, using the mean log output and mean log input level in 1988-89. Documents de Travail du Centre d'Economie de laSorbonne -2012.46

The 14.8% and 9.3% figures are averages of TFP coefficients of1991-1997 and 1998-2004, respectively. Documents de Travail du Centre d'Economie de laSorbonne -2012.46

The Stata codes (Pscore and attnd) for these applications are provided in[START_REF] Becker | Estimation of average treatment effects based on propensity scores[END_REF]. Documents de Travail du Centre d'Economie de laSorbonne -2012.46

All regressions include dummies and company age (in log). The above figures include a firm size dummy for TFP regressions and ULC regressions. P-values are in parentheses. ** and *** refer to 5% and 1% statistical significance levels, respectively. Documents de Travail du Centre d'Economie de laSorbonne -2012.46

conclusions expressed in this paper are entirely those of the author. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive