Simona Perotto

Michele Giuliano Carlino

Francesco Ballarin

Model reduction by separation of variables: a comparison between Hierarchical Model reduction and Proper Generalized Decomposition

Hierarchical Model reduction and Proper Generalized Decomposition both exploit separation of variables to perform a model reduction. After setting the basics, we exemplify these techniques on some standard elliptic problems to highlight pros and cons of the two procedures, both from a methodological and a numerical viewpoint.

Introduction

This paper is meant as a first attempt to compare two procedures which share the idea of exploiting separation of variables to perform model reduction, albeit with different purposes. Proper Generalized Decomposition (PGD) is essentially employed as a powerful tool to deal with parametric problems in several fields of application [START_REF] Ammar | Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions[END_REF][START_REF] Niroomandi | Realtime simulation of biological soft tissues: a PGD approach[END_REF][START_REF] Signorini | Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems[END_REF]. Parametrized models characterize multi-query contexts, such as parameter optimization, statistical analysis or inverse problems. Here, the computation of the solution for many different parameters demands, in general, a huge computational effort, and this justifies the development of model reduction techniques.

For this purpose, projection-based techniques, such as Proper Orthogonal Decomposition (POD) or Reduced Basis methods, are widely used in the literature [START_REF] Hesthaven | Certified Reduced Basis Methods for Parametrized Partial Differential Equations[END_REF]. The Simona Perotto MOX -Modeling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy e-mail: simona.perotto@polimi.it Michele Giuliano Carlino Inria Bordeaux Sud-Ouest and Institut de Mathématiques de Bordeaux, University of Bordeaux, 351, cours de la Libération, F-33405 Talence e-mail: michele-giuliano.carlino@inria.fr Francesco Ballarin mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy e-mail: francesco.ballarin@sissa.it idea is to project the discrete operators onto a reduced space so that the problem can be solved rapidly in the lower dimensional space. PGD adopts a completely different way to deal with parameters. Here, parameters are considered as new independent variables of the problem, together with the standard space-time ones [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer[END_REF]. Although the dimensionality of the problem is inevitably increased, PGD transforms the computation of the solution for new values of the parameters into a plain evaluation of the reduced solution, with striking computational advantages. Hierarchical-Model (HiMod) reduction has been proposed to improve one-dimensional (1D) partial differential equation (PDE) solvers for problems defined in domains with a geometrically dominant direction, like slabs or pipes [START_REF] Ern | Hierarchical model reduction for advection-diffusionreaction problems[END_REF][START_REF] Perotto | Hierarchical local model reduction for elliptic problems: a domain decomposition approach[END_REF]. The main applicative field of interest is hemodynamics, in particular the modeling of blood flow in patient-specific geometries. Purely 1D hemodynamic models completely drop the transverse dynamics, which, however may be locally important (e.g., in the presence of a stenosis or an aneurism). HiMod aims at providing a numerical tool to incorporate the transverse components of the 3D solution into a conceptually 1D solver. To do this, the driving idea is to discretize main and transverse dynamics in a different way. The latter are generally of secondary importance and can be described by few degrees of freedom using a spectral approximation, in combination, for instance, with a finite element (FE) discretization of the mainstream.

The parametric version of HiMod (namely, HiPOD) is a more recent proposal [START_REF] Baroli | Hi-POD solution of parametrized fluid dynamics problems: preliminary results[END_REF][START_REF] Pasini | HiPOD: Hierarchical model reduction driven by a Proper Orthogonal Decomposition for parametrized advection-diffusion-reaction problems[END_REF]. On the other hand, PGD is not so widely employed in a non-parametric setting, despite its original formulation [START_REF] Ladevèze | The LATIN multiscale computational method and the proper generalized decomposition[END_REF]. Nevertheless, for the sake of comparison, in this paper we consider the non-parametric as well as the parametric versions of both the HiMod and PGD approaches. The goal is to begin a preliminary comparative analysis between the two methodologies, to highlight the respective weaknesses and strengths. The main limit of PGD remains its inability to deal with non-Cartesian geometries without losing the computational benefits arising from the separability of the spatial coordinates. HiMod turns out to be more flexible from a geometric viewpoint. On the other hand, PGD turns out to be extremely effective for parametric problems thanks to the explicit expression of the PGD solution in terms of the parameters, while HiPOD can be classified as a projection-based method with all the associated drawbacks. In perspective, the ultimate goal is to merge HiMod with PGD to emphasize the good features and mitigate the intrinsic limits of the two methods taken alone.

The HiMod approach

Hierarchical Model reduction proved to be an efficient and reliable method to deal with phenomena charaterized by dominant dynamics [START_REF] Guzzetti | Hierarchical model reduction for incompressible fluids in pipes[END_REF]. In general, the computational domain itself exhibits an intrinsic directionality. We assume Ω ⊂ R d (d = 2, 3) to coincide with a d-dimensional fiber bundle, Ω = x ∈Ω 1D {x}×γ x , where Ω 1D ⊂ R denotes the supporting fiber aligned with the main stream, while γ x ⊂ R d-1 is the transverse fiber at x ∈ Ω 1D , parallel to the transverse dynamics. For the sake of sim-plicity, we identify Ω 1D with a straight segment, (x 0 , x 1 ). We refer to [START_REF] Perotto | Hierarchical model (Hi-Mod) reduction in non-rectilinear domains[END_REF][START_REF] Perotto | HIGAMod: a hierarchical isogeometric approach for model reduction in curved pipes[END_REF] for the case where Ω 1D is curvilinear. From a computational viewpoint, the idea is to exploit a map, Ψ : Ω → Ω, transforming the physical domain, Ω, into a reference domain, Ω, and to make explicit computations in Ω only. Typically, Ω coincides with a rectangle in 2D, with a cylinder with circular section in 3D. To define Ψ, for each x ∈ Ω 1D , we introduce the map, ψ x : γ x → γd-1 , from fiber γ x to the reference transverse fiber, γd-1 , so that the reference domain coincides with Ω = x ∈Ω 1D {x} × γd-1 . The supporting fiber is preserved by map Ψ, which modifies the lateral boundaries only.

We consider now the (full) problem to be reduced. Due to the comparative purposes of the paper, we focus on a scalar elliptic equation, and, in particular, on the associated weak formulation,

find u ∈ V : a(u, v) = F (v) ∀v ∈ V, (1)
where V ⊆ H 1 (Ω), a(•, •) : V ×V → R is a continuous and coercive bilinear form and F (•) : V → R is a continuous linear functional. To provide the HiMod formulation for problem (1), we introduce the hierarchical reduced space

V m = v m (x, y) = m k=1 ṽk (x)ϕ k (ψ x (y)), with ṽk ∈ V h 1D , x ∈ Ω 1D , y ∈ γ x (2)
for a modal index m ∈ N + , where

V h 1D ⊆ H 1 (Ω 1D ) is a discrete space of dimension N h associated with a partition T h of Ω 1D , while {ϕ k } m
k=1 denotes a modal basis of functions orthogonal with respect to the L 2 ( γd-1 )-scalar product. Index m sets the hierarchical level of the HiMod space, being V m ⊂ V m+1 , for any m. Concerning V h 1D , we adopt here a standard FE space, although any discrete space can be employed (see, e.g., [START_REF] Perotto | HIGAMod: a hierarchical isogeometric approach for model reduction in curved pipes[END_REF], where an isogeometric discretization is used). Functions in V h 1D have to include the boundary conditions on {x 0 } × γ x 0 and {x 1 } × γ x 1 ; analogously, the modal functions have to take into account the boundary data along the horizontal sides. In Sect. 4 further comments are provided about the selection of the modal basis and of the modal index m. The HiMod formulation for problem (1) thus reads

find u HiMod m ∈ V m : a(u HiMod m , v m ) = F (v m ) ∀v m ∈ V m . (3)
To ensure the well-posedness of formulation (3) and the convergence of the HiMod approximation, u HiMod m , to the full solution, u, we endow the HiMod space with a conformity and a spectral approximability hypothesis, and we introduce a standard density assumption on the discrete space V h 1D (see [START_REF] Perotto | Hierarchical local model reduction for elliptic problems: a domain decomposition approach[END_REF] for all the details). The HiMod solution can be fully characterized by introducing a basis, {θ l } N h l=1 , for the space V h 1D . Actually, each modal coefficient, ũk , of u HiMod m can be expanded in terms of such a basis, so that, we obtain the modal representation

u HiMod m (x, y) = m k=1 N h l=1 ũk,l θ l (x)ϕ k (ψ x (y)). (4)
The actual unknowns of problem (3) become the mN h coefficients { ũk,l } m, N h k=1,l=1 . With reference to the Poisson problem, -∆u = f , completed with full homogeneous Dirichlet boundary data, the corresponding HiMod formulation, after exploiting (4) in [START_REF] Ammar | Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions[END_REF] and picking v m (x, y) = θ i (x)ϕ j (ψ x (y)) with i = 1, . . . , N h and j = 1, . . . , m, reduces to the system of mN h 1D equations in the mN h unknowns { ũk,l } m, N h k=1,l=1 ,

m k=1 N h l=1 ũk,l Ω 1D r1,1 jk (x) dθ l dx (x) dθ i dx (x) + r1,0 jk (x) dθ l dx (x)θ i (x)+ + r0,1 jk (x)θ l (x) dθ i dx (x) + r0,0 jk (x)θ l (x)θ i (x) dx = Ω 1D f j (x)θ i (x) dx, where r a,b jk (x) = γd-1 r a,b jk (x, ŷ)| J | d ŷ with a, b = 0,1, J = det D -1 2 (x, ψ -1 x ( ŷ)) with D 2 = D 2 (x, ψ -1 x ( ŷ)) = ∇ y ψ x , r 0,0 jk (x, ŷ) = ϕ k ( ŷ)ϕ j ( ŷ) D 2 1 + D 2 2 , r 0,1 jk (x, ŷ) = ϕ k ( ŷ)ϕ j ( ŷ)D 1 , r 1,0 jk (x, ŷ) = ϕ k ( ŷ)ϕ j ( ŷ)D 1 , r 1,1 jk (x, ŷ) = ϕ k ( ŷ)ϕ j ( ŷ), with D 1 = D 1 (x, ψ -1 x ( ŷ)) = ∂ψ x /∂ x, and f j (x) = γd-1 f (x, ψ -1 x ( ŷ))ϕ j ( ŷ)| J | d ŷ.
Information associated with the transverse dynamics are lumped in the coefficients { r a,b jk }, so that the HiMod system is solved on the supporting fiber, Ω 1D . Collecting the HiMod unknowns, by mode, in the vector u HiMod m ∈ R mN h , such that

u HiMod m = [ ũ1,1 , ũ1,2 , . . . , ũ1,N h , ũ2,1 , . . . , ũm,1 , . . . , ũm,N h ] T , (5)
we can rewrite the HiMod system in the compact form

A HiMod m u HiMod m = f HiMod m , (6)
where A HiMod m ∈ R mN h ×mN h and f HiMod m ∈ R mN h are the HiMod stiffness matrix and right-hand side, respectively, with [f HiMod m ] ji = Ω 1D f j (x)θ i (x)dx, and

[A HiMod m ] ji,kl = 1 a,b=0 Ω 1D r a,b jk (x) d a θ l dx (x) d b θ i dx (x)dx.
According to [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer[END_REF], for each modal index j, between 1 and m, the nodal index, i, takes the values 1, . . . , N h . Thus, HiMod reduction leads to solve a system of order mN h , independently of the dimension of the full problem (1).

The PGD approach

To perform PGD, we have to introduce on problem (1) a separability hypothesis with respect to both the spatial variables and the data [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer[END_REF][START_REF] Pruliere | On the deterministic solution of multidimensional parametric models using the proper generalized decomposition[END_REF]. Thus, domain Ω ⊂ R d coincides with the rectangle

Ω x × Ω y if d = 2, with the parallelepiped Ω x × Ω y × Ω z (total separability) or with the cylinder Ω x × Ω y (partial separability) if d = 3,
for Ω x , Ω y , Ω z ⊂ R and Ω y ⊂ R 2 , being y = (y, z). In the following, we focus on partial separability, since it is more suited to match HiMod reduction with PGD. Analogously, we assume that the generic problem data, d = d(x, y, z), can be written as d = d x (x)d y (y). The separability is inherited by the PGD space

W m = w m (x, y) = m k=1 w x k (x)w y k (y), with w x k ∈ W x h , w y k ∈ W y h , x ∈ Ω x , y ∈ Ω y , (7) where
W x h ⊆ H 1 (Ω x ) and W y h ⊆ H 1 (Ω y ; R d-1 ) are discrete spaces, with dim(W x h ) = N x h and dim(W y h ) = N y h ,
W m = W x h ⊗ W y h ⊆ H 1 (Ω x ) ⊗ H 1 (Ω y ; R d-1
). Index m plays the same role as in the HiMod reduction, setting the level of detail for the reduced solution (see Sect. 4 for possible criteria to choose m). PGD exploits the hierarchical structure in W m to build the generic function w m ∈ W m . In particular, w m is computed as in the test function. This yields, for any X ∈ W x h ,

w m (x, y) = w x m (x)w y m (y) + m-1 k=1 w x k (x)w y k (y), (8)
Ω x u x, p m X dx Ω y u y, p-1 m 2 dy + Ω x u x, p m X dx Ω y u y, p-1 m 2 dy = Ω x f x X dx Ω y f y u y, p-1 m dy -m-1 k=1 Ω x u x k X dx Ω y u y k u y, p-1 m dy -m-1 k=1 Ω x u x k X dx Ω y u y k u y, p-1 m dy, (9)
where the separability of f is exploited (the dependence on the independent variables, x and y, is omitted to simplify notation). Successively, we compute u y, p m , after setting u x m to u x, p m and choosing function X as to u

x, p m in the test function, so that we obtain, for any

Y ∈ W y h , Ω x u x, p m 2 dx Ω y u y, p m Y dy + Ω x u x, p m 2 dx Ω y u y, p m Y dy = Ω x f x u x, p m dx Ω y f y Y dy -m-1 k=1 Ω x u x k u x, p m dx Ω y u y k Y dy -m-1 k=1 Ω x u x k u x, p m dx Ω y u y k Y dy. (10)
The algebraic counterpart of ( 9) and ( 10) is obtained by introducing a basis, B x =

{θ x α } N x h α=1 and B y = {θ y β } N y h
β=1 , for the space W x h and W y h , respectively, so that u

q j (q) = N q h i=1 ũq ji θ q i (q), u q,s m (q) = N q h i=1
ũq,s mi θ q i (q), with q = x, y, s = p, p-1, j = 1, . . . , m-1, and, likewise,

X (x) = N x h α=1 xα θ x α (x) and Y (y) = N y h β=1 ỹβ θ y β (y).
Thanks to these expansions and to the arbitrariness of X and Y , we can rewrite ( 9) and [START_REF] Guzzetti | Hierarchical model reduction for incompressible fluids in pipes[END_REF] as

u y, p-1 m T M y u y, p-1 m K x + u y, p-1 m T K y u y, p-1 m M x u x, p m = u y, p-1 m T f y f x -m-1 k=1 u y, p-1 m T M y u y k K x + u y, p-1 m T K y u y k M x u x k , (11) and
u x, p m T K x u x, p m M y + u x, p m T M x u x, p m K y u y, p m = u x, p m T f x f y -m-1 k=1 u x, p m T K x u x k M y + u x, p m T M x u x k K y u y k , (12)
respectively, where vectors u

q j , u q,s m ∈ R N q h collect the PGD coefficients, being u q j i = ũq ji , u q,s m i = ũq,s mi and i = 1, . . . , N q h , K x , M x ∈ R N x h ×N x h and K y , M y ∈ R N y h ×N y h
are the stiffness and mass matrices associated with xand y-variables, with

K x αl = Ω x θ x α θ x l dx, K y βs = Ω y θ y β θ y s dy, M x αl = Ω x θ x α θ x l dx, M y βs = Ω y θ y β θ y
s dy, and where [START_REF] Hesthaven | Certified Reduced Basis Methods for Parametrized Partial Differential Equations[END_REF] and ( 12) are solved at each ADS iteration, so that the computational effort characterizing PGD is the one associated with the solution of two systems of order N x h and N y h , respectively, for each ADS iteration. When a certain stopping criterion is met (see the next section for more details), ADS procedure yields vectors u x m and u y m which identify the enrichment functions u x m and u y m .

f x ∈ R N x h , f y ∈ R N y h , with f x l = Ω x f x θ x l dx, f y s = Ω y f y θ y s dy, for α, l = 1, . . . , N x h , β, s = 1, . . . , N y h . Systems

HiMod reduction versus PGD

Both HiMod reduction and PGD exploit the separation of variables and, according to [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer[END_REF], belong to the a priori approaches, since they do not rely on any solution to the problem at hand. Nevertheless, we can easily itemize features which distinguish the two techniques. The most relevant ones concern the geometry of Ω, the selection of the transverse basis and of the modal index, and the numerical implementation of the two procedures. Pros and cons of the two methods are then here highlighted.

Domain geometry

HiMod reduction and PGD advance precise hypotheses on the geometry of the computational domain.

According to the HiMod approach, Ω is expected to coincide with a fiber bundle and to be mapped into the reference domain, Ω, by a sufficiently regular transformation. Actually, map Ψ is assumed differentiable, while map ψ x is required to be a C 1 -diffeomorphism, for all x ∈ Ω 1D [START_REF] Perotto | Hierarchical local model reduction for elliptic problems: a domain decomposition approach[END_REF]. These hypotheses introduce some constraints, in particular, on the lateral boundary of Ω which, e.g., cannot exhibit kinks. Additionally, geometries of interest in many applications, such as bifurcations or, more in general, networks are ruled out from the demands on ψ x and Ψ. An approach based on the domain decomposition technique is currently under investigation as a viable way to deal with such geometries. The isogeometric version of HiMod (i.e., the HIgaMod approach) will play a crucial role in view of HiMod simulations for the blood flow modeling in patient-specific geometries [START_REF] Perotto | HIGAMod: a hierarchical isogeometric approach for model reduction in curved pipes[END_REF].

The constraints introduced by PGD on the geometry of Ω are more restrictive. The separability hypothesis leads to consider essentially only Cartesian domains. This considerably reduces the applicability of PGD to practical contexts. Some techniques are available in the literature to overcome this issue. For instance, in [START_REF] González | Recent advances on the use of separated representations[END_REF] a generic domain is embedded into a Cartesian geometry, while in [START_REF] Ghnatios | First steps in the space separated representation of models defined in complex domains[END_REF] the authors introduce a parametrization map for quadrilateral domains.

Overall, HiMod reduction exhibits a higher geometric flexibility with respect to PGD, in its straightforward formulation. As discussed in Sect. 5, this limitation can be removed when considering a parametric setting.

Modeling of the transverse dynamics

In the HiMod expansion, y-components, ϕ k (ψ x (y)), are selected before starting the model reduction. This choice, although coherent with an a priori approach, introduces a constraint on the dynamics that can be described, so that hints about the solution trend along the transverse direction can be helpful to select a representative modal basis. In the original proposal of the HiMod procedure, sinusoidal functions are employed according to a Fourier expansion [START_REF] Ern | Hierarchical model reduction for advection-diffusionreaction problems[END_REF][START_REF] Perotto | Hierarchical local model reduction for elliptic problems: a domain decomposition approach[END_REF]. This turns out to be a reasonable choice when Dirichlet boundary conditions are assigned on the lateral surface, Γ lat = {x} × ∂γ x , of Ω. Legendre polynomials, properly modified to include the homogeneous Dirichlet data and orthonormalized, are employed in [START_REF] Perotto | Hierarchical local model reduction for elliptic problems: a domain decomposition approach[END_REF] as an alternative to a trigonometric expansion. Nevertheless, Legendre polynomials require high-order quadrature rules to accurately compute coefficients { r a,b jk }. In [START_REF] Aletti | HiMod reduction of advection-diffusion-reaction problems with general boundary conditions[END_REF], the concept of educated modal basis is introduced to impose generic boundary conditions on Γ lat . The idea is to solve an auxiliary Sturm-Liouville eigenvalue problem on the transverse reference fiber γd-1 , to build a basis which automatically includes the boundary values on Γ lat . The eigenfunctions of the Sturm-Liouville problem provide the modal basis. A first attempt to generalize the educated-HiMod reduction to three-dimensional (3D) cylindrical geometries is performed in [START_REF] Guzzetti | Hierarchical model reduction for incompressible fluids in pipes[END_REF], where the Navier-Stokes equations are hierarchically reduced to model the blood flow in pipes. This generalization is far from being straightforward due to the employment of polar coordinates. To overcome this issue, we are currently investigating the HIgaMod approach [START_REF] Perotto | HIGAMod: a hierarchical isogeometric approach for model reduction in curved pipes[END_REF], which allows us to define the transverse basis as the Cartesian product of 1D modal functions, independently of the considered geometry. Additionally, we remark that any modal basis can be precomputed on the transverse reference fiber before performing the HiMod reduction, thanks to the employment of map Ψ. This considerably simplifies computations.

When applying PGD, y-components are unknown as the ones associated with x. This leads to the nonlinear problems ( 9)-( 10), thus loosing any advantage related to a precomputation of the HiMod modal basis. On the other hand, PGD does not constrain the transverse dynamic to follow a prescribed (e.g., sinusoidal) analytical shape as HiMod procedure does. The educated-Himod reduction clearly is out of this comparison, since the modal basis strictly depends on the problem at hand.

Finally, we observe that HiMod modes are orthonormal with respect to the L 2 ( γd-1 )-norm. This property is not ensured by PGD.

Concerning the selection of the modal index m in ( 2) and ( 7), as a first attempt, both HiMod reduction and PGD resort to a trial-and-error approach, so that the modal index is gradually increased until a check on the accuracy of the reduced solution is satisfied. For instance, in [START_REF] Ern | Hierarchical model reduction for advection-diffusionreaction problems[END_REF][START_REF] Perotto | Hierarchical local model reduction for elliptic problems: a domain decomposition approach[END_REF] a qualitative investigation of the contour plot of the HiMod approximation drives the choice of m. Concerning PGD, the check on the relative enrichment

u x m u y m L 2 (Ω) u x 1 u y 1 L 2 (Ω) ≤ TOL E , (13)
is usually employed, with TOL E a user-defined tolerance [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer[END_REF]. An automatic selection of index m can yield a significant improvement. In [START_REF] Perotto | Coupled model and grid adaptivity in hierarchical reduction of elliptic problems[END_REF][START_REF] Perotto | Space-time adaptive hierarchical model reduction for parabolic equations[END_REF], an adaptive procedure is proposed for HiMod, based on an a posteriori modeling error analysis. In particular, the estimator in [START_REF] Perotto | Coupled model and grid adaptivity in hierarchical reduction of elliptic problems[END_REF] is derived in a goal-oriented setting to control a quantity of interest, and exploits the hierarchical structure (i.e., the inclusion V m ⊂ V m+d , ∀m, d ∈ N + ) typical of a HiMod reduction. A similar modeling error analysis is performed in [START_REF] Ammar | An error estimator for separated representations of highly multidimensional models[END_REF] for PGD, although no adaptive algorithm is here set to automatically pick the reduced model. Paper [START_REF] Perotto | Space-time adaptive hierarchical model reduction for parabolic equations[END_REF] generalizes the a posteriori analysis in [START_REF] Perotto | Coupled model and grid adaptivity in hierarchical reduction of elliptic problems[END_REF] to an unsteady setting, providing the tool to automatically select m together with the partition T h along Ω 1D and the time step. Finally, HiMod allows to tune the modal index along the domain Ω, according to the local complexity of the transverse dynamics. In particular, m can be varied in different areas of Ω or, in the presence of very localized dynamics, in correspondence with specific nodes of the partition T h . We refer to these two variants as to piecewise and pointwise HiMod reduction, in contrast to a uniform approach, where the same number of modes is adopted everywhere [START_REF] Perotto | A survey of hierarchical model (Hi-Mod) reduction methods for elliptic problems[END_REF][START_REF] Perotto | Hierarchical model reduction: three different approaches[END_REF]. This flexibility in the choice of m is currently not available for PGD. Adaptive strategies to select the modal index are available for the three variants of the HiMod procedure [START_REF] Perotto | Coupled model and grid adaptivity in hierarchical reduction of elliptic problems[END_REF][START_REF] Perotto | Space-time adaptive hierarchical model reduction for parabolic equations[END_REF].

Computational aspects

From a computational viewpoint, HiMod reduction and PGD lead to completely different procedures. Indeed, for a fixed value of m, we have to solve the only system (6) of order mN h when applying HiMod, in contrast to PGD which demands a multiple solution of systems ( 11)-( 12) of order N x h and N y h , respectively because of the fixed point and the enrichment algorithms. Thus, the direct solution of a single system, in general of larger order, is replaced by an iterative solution of several and smaller systems. This heterogeneity makes a computational comparison between PGD and HiMod not so meaningful. We verify the reliability of the HiMod and PGD procedures on a common test case, by choosing in (1)

V = H 1 0 (Ω) with Ω = (0, 5) × (0, 1), a(u, v) = Ω µ∇u • ∇v + b • ∇u dΩ for µ = 0.24, b = [-5, 0] T , and F (v) = Ω f vdΩ with f (x, y) = 50 exp -(x -2.85)/0.075 2 -(y -0.5)/0.075 2 + exp -(x -3.75)/0.075 2 -(y -0.5)/0.075 2 .
For both the methods, we uniformly subdivide Ω 1D into 285 subintervals. We set the PGD discretization along y as well as the PGD and the HiMod index m in order to ensure the same accuracy, TOL, on the reduced approximations with respect to a reference FE solution, computed on a 2500 × 500 structured mesh. In particular, for TOL = 8 • 10 -3 , we have to subdivide interval (0, 1) into 20 uniform subintervals, and to set m to 6 and to 9 in the PGD and the HiMod discretization, respectively. Sinusoidal functions are chosen for the HiMod modal basis. The ADS iterations are controlled in terms of the relative increment, as

u x, p m u y, p m -u x, p-1 m u y, p-1 m L 2 (Ω) u x, p m u y, p m L 2 (Ω) ≤ TOL FP , (14)
with TOL FP = 10 -2 . Fig. 1 shows the reduced approximations (which are fully comparable with the FE one, here omitted). The contourplots are very similar. The coarse PGD y-discretization justifies the slight roughness of the PGD contourlines. Another distinguishing feature between HiMod and PGD is the domain discretization. Indeed, HiMod requires only the partition T h along Ω 1D , independently of the dimension of Ω. No discretization is needed in the y-direction, although we have to carefully select the quadrature nodes to compute coefficients { r a,b jk }. This task becomes particularly challenging when dealing with polar coordinates [START_REF] Guzzetti | Hierarchical model reduction for incompressible fluids in pipes[END_REF]. With PGD to benefit of the computational advantages associated with a 1D discretization, we are obliged to assume the full separability of Ω; actually, a partial separability demands a 1D partition for Ω x , and a two-dimensional partition of Ω y . As explained in Sect. 5, non-Cartesian domains require a 3D discretization of Ω.

Finally we analyze the interplay between the enrichment and the ADS iterations in the PGD reduction. We investigate the possible relationship between TOL FP in [START_REF] Niroomandi | Realtime simulation of biological soft tissues: a PGD approach[END_REF] and TOL E in [START_REF] Pasini | HiPOD: Hierarchical model reduction driven by a Proper Orthogonal Decomposition for parametrized advection-diffusion-reaction problems[END_REF], to verify if a small tolerance for the fixed point iteration improves the accuracy of the PGD approximation, thus reducing the number of enrichment steps. To do this, we adopt the same test case used above. Table 1 gathers the number of ADS iterations, #IT FP , the number, m, of enrichment steps, and the CPU time1 (in seconds) demanded by the PGD procedure, for two different values of TOL E and three different choices of TOL FP . In particular, in column #IT FP we specify the number of ADS iterations required by each enrichment step. As expected, there exists a link between the two tolerances, namely, when a higher accuracy constrains the fixed point iteration, a smaller number of enrichment steps is performed to ensure the accuracy TOL E .

Table 1 Quantitative analysis for PGD in terms of fixed point iterations and enrichment steps

TOL E = 2 • 10 -2 TOL E = 8 • 10 -3 #IT FP m CPU [s] #IT FP m CPU [s] TOL FP = 10 -1 {2, 2, 2} 3 0.099640 {2, 2, 2, 2, 2} 5 0.337861 TOL FP = 10 -2 {4, 3} 2 0.046756 {4, 3, 2, 2, 4} 5 0
.358555 TOL FP = 10 -3 {5, 5} 2 0.077958 {5, 5, 2, 7} 4 0.341748

HiMod reduction and PGD for parametrized problems

The actual potential of PGD becomes more evident when considering a parametric setting, i.e., when problem (1) is replaced by the formulation

find u(µ) ∈ V : a(u(µ), v; µ) = F (v; µ) ∀v ∈ V, (15)
with µ a parameter, which may represent any data of the problem, e.g., the coefficients of the considered PDE, the source term, a boundary value or the domain geometry.

The technique adopted by PGD to deal with the parametric dependence in ( 15) is very effective. Parameter µ is considered as an additional independent variable which varies in a domain Ω µ [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer[END_REF]. Thus, the PGD space [START_REF] Ghnatios | First steps in the space separated representation of models defined in complex domains[END_REF] changes into the new one

W µ m = w m (x, y, µ) = m k=1 w x k (x)w y k (y)w µ k (µ), with w x k ∈ W x h , w y k ∈ W y h , w µ k ∈ W µ h , x ∈ Ω x , y ∈ Ω y , µ ∈ Ω µ , (16)
1 The computations have been run on a Intel Core i5 Dual-Core CPU 2.7 GHz 8GB RAM MacBook.

with W µ h a discretization of the space L 2 (Ω µ ; R Q ), being Q the length of vector µ. Generalizing the enrichment paradigm in [START_REF] Golub | Matrix Computations[END_REF], at the m-th step of the PGD approach applied to problem [START_REF] Perotto | Hierarchical model (Hi-Mod) reduction in non-rectilinear domains[END_REF] we have to compute three unknown functions, u x m , u y m and u µ m , by picking the test function as in the test function. This leads to a linear system which generalizes [START_REF] Hesthaven | Certified Reduced Basis Methods for Parametrized Partial Differential Equations[END_REF], namely

X (x)Y (y) Z (µ), with X ∈ W x h , Y ∈ W y h , Z ∈ W µ h . Functions u x m , u y m , u
u µ, p-1 m T M µ u µ, p-1 m u y, p-1 m T M y u y, p-1 m K x + u y, p-1 m T K y u y, p-1 m M x u x, p m = u y, p-1 m T f y u µ, p-1 m T f µ f x -m-1 k=1 u µ, p-1 m T M µ u µ k u y, p-1 m T M y u y k K x + u y, p-1 m T K y u y k M x u x k , (17) where
M µ ∈ R N µ h ×N µ h
is the mass matrix associated with the parameter µ, with

M µ i j = Ω µ µθ µ i θ µ j dµ for i, j = 1, . . . , N µ h and B µ = {θ µ γ } N µ h γ=1 a basis for the space W µ h , f µ ∈ R N µ h with f µ l = Ω µ f µ θ µ l dµ for l = 1, . . . , N µ
h after assuming the separability f = f x f y f µ for the source term f , and where we employ the same notation as in ( 11)-( 12 14) is set to 10 -2 . Fig. 2 compares the PGD approximation for µ = 1 and µ = 2.5 with a reference full solution coinciding with a linear FE approximation computed on a 300 × 100 structured mesh. The qualitative matching between the corresponding solutions is significant. From a quantitative viewpoint, the L 2 (Ω)-norm of the relative error associated with the PGD approximation does not vary significantly by increasing m, whereas a slight error reduction is detected by increasing µ.

The parametric counterpart of the HiMod reduction, known as HiPOD, merges HiMod with POD [START_REF] Baroli | Hi-POD solution of parametrized fluid dynamics problems: preliminary results[END_REF][START_REF] Pasini | HiPOD: Hierarchical model reduction driven by a Proper Orthogonal Decomposition for parametrized advection-diffusion-reaction problems[END_REF]. HiPOD pursues a different goal with respect to PGD. Indeed, for a new value, µ * , of the parameter, PGD provides an approximation for the full solution u(µ * ), while HiPOD approximates the HiMod solution associated with µ * . The offline/online paradigm of POD is followed also by HiPOD. The peculiarity is that the offline step is now performed in the HiMod setting to contain the computational burden typical of this stage and by relying on the good properties of HiMod in terms of reliability-versus-accuracy balance. Thus, we choose P different values, µ = µ i with i = 1, . . . , P, for parameter µ, and we collect the HiMod approximation for the corresponding problem (15) into the response matrix, S = u HiMod m (µ 1 ), u HiMod m (µ 2 ), . . . , u HiMod m (µ P ) ∈ R mN h ×P , according to representation [START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer[END_REF]. Successively, we define the null-average matrix

V = S - 1 P P i=1 u HiMod m (µ i ), u HiMod m (µ i ), . . . , u HiMod m (µ i ) ∈ R mN h ×P ,
and we apply the Singular Value Decomposition (SVD) to V, so that V = ΦΣΨ T , where Φ ∈ R (mN h )×(mN h ) and Ψ ∈ R P×P are the unitary matrices of the left-and of the right-singular vectors of V, respectively while Σ = diag (σ 1 , . . . , σ ρ ) ∈ R (mN h )×P denotes the pseudo-diagonal matrix of the singular values of V, being σ 1 ≥ σ 2 ≥ • • • ≥ σ ρ ≥ 0 and ρ = min(mN h , P) [START_REF] Golub | Matrix Computations[END_REF]. The POD basis is identified by the first l left singular vectors, φ i , of V, so that the reduced POD space is V l POD = span{φ 1 , . . . , φ l }, with dim(V l POD ) = l and l mN h . In the numerical assessment below, value l coincides with the smallest integer such that σ 2 l < ε, with ε a prescribed tolerance. The online phase of HiPOD approximates the HiMod solution to problem [START_REF] Perotto | Hierarchical model (Hi-Mod) reduction in non-rectilinear domains[END_REF] for a new value, µ * , of the parameter by exploiting the POD basis instead of solving system (6). This is performed via a projection step. After assembling the HiMod stiffness matrix and right-hand side, A HiMod m (µ * ) and f HiMod m (µ * ), associated with the new value of the parameter, we solve the POD system of order l

A POD (µ * )u POD (µ * ) = f POD (µ * ), (18)
where

A POD (µ * ) = (Φ l POD ) T A HiMod m (µ * ) Φ l POD and f POD (µ * ) = (Φ l POD ) T f HiMod m (µ *
) denote the POD stiffness matrix and right-hand side, respectively with Φ l POD = [φ 1 , . . . , φ l ] ∈ R (mN h )×l the matrix collecting the POD basis vectors. The HiMod solution is thus approximated by vector Φ l POD u POD (µ * ) ∈ R mN h , i.e., after solving a system of order l instead of mN h . Overall, HiPOD requires to solve P linear systems of order mN h during the offline phase, additionally to a system of order l in the online phase.

To check the performances of HiPOD, we adopt the test case used above for PGD, for the same values of the parameters, µ * = 1 and µ * = 2.5. The reference solution is the corresponding HiMod approximation computed by using m = 15 sinusoidal functions in the y-direction, and a linear FE discretization along the mainstream based on a uniform subdivision of Ω 1D into 50 subintervals. The same HiMod discretization is adopted to build the response matrix. Concerning the HiPOD approximation, we pick P = 100 by uniformly sampling the interval [START_REF] Aletti | HiMod reduction of advection-diffusion-reaction problems with general boundary conditions[END_REF][START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer[END_REF], and we select ε = 2.5 • 10 -15 . This choice sets the dimension of the POD space to l = 8, so that we have to solve a system of order 8 instead of 750. The contour plots in Fig. 3 qualitatively compare the HiMod solution with the HiPOD approximation for l = 1. The correspondence between the two approximations is good despite a single POD mode is employed (in such a case, system [START_REF] Perotto | HIGAMod: a hierarchical isogeometric approach for model reduction in curved pipes[END_REF] reduces to a scalar equation). We do not provide the HiPOD approximations for l = 8 since they qualitatively coincide with the corresponding HiMod solution. The left panels can be additionally compared with the FE solutions in Fig. 2 to verify the reliability of the HiMod procedure. Finally, the table in Fig. 3 gathers the L 2 (Ω)-norm of the relative error between HiMod and HiPOD solutions, for four different POD bases and for three choices of the viscosity (1, 2.5 and the average over a sampling of 30 random values of µ). The error monotonically decreases for larger and larger values of l, independently of the choice for µ. If we compare the values for µ = 1 and for µ = 2.5 (one of the endpoints and the midpoint of the sampling interval, respectively), we notice a higher accuracy (of about one order of magnitude) for the latter choice. This is rather standard in projection-based reduced order modeling [START_REF] Hesthaven | Certified Reduced Basis Methods for Parametrized Partial Differential Equations[END_REF]. Concerning the computational saving in terms of CPU time, HiPOD method requires on average O(10 -3 )[s] to be compared with O [START_REF] Guzzetti | Hierarchical model reduction for incompressible fluids in pipes[END_REF][s] demanded by HiMod, resulting in a speedup of 10 4 .

Although PGD and HiPOD are not directly comparable due to the different purpose they pursue, we highlight the main pros and cons of the two methods. The explicit dependence of the approximation on the parameters makes PGD an ideal tool to efficiently deal with parametric problems. For any new parameter, a direct evaluation yields the corresponding PGD approximation. On the other hand, HiPOD suffers of the drawbacks typical of the projection-based methods. The main bottleneck is the assembling of the HiMod arrays involved in A POD (µ * ) and f POD (µ * ). When PGD is applied to parametric problems, we recover the possibility to deal with any geometric domain. In such a case, a partial separability is applied to the problem, so that the space independent variables are kept together whereas parameters are separated. This approach clearly looses the computational advantages due to space separability. On the contrary, HiPOD inherits the geometric flexibility of the HiMod reduction, without giving up the spatial dimensional reduction of the problem.

where w x k and w y k are assumed known for k = 1 ,m

 1 . . . , m -1, so that the enrichment functions, w x m and w y m , become the actual unknowns. To provide the PGD formulation for the Poisson problem considered in Sect. 2, we exploit representation (8) for the PGD approximation, u PGD m , and we pick the test function as X (x)Y (y), with X ∈ W x h and Y ∈ W y h . The coupling between the unknowns, u x m and u y m, leads to a nonlinear problem, which is tackled by means of the Alternating Direction Strategy (ADS)[START_REF] Chinesta | The Proper Generalized Decomposition for Advanced Numerical Simulations: a Primer[END_REF]. The idea is to look for u x m and u y m , separately via a fixed point procedure. We introduce an auxiliary index to keep trace of the ADS iterations, so that, at the p-th ADS iteration we compute u for g = 1, . . . , p -1, following a two-step procedure. First, we compute u x, p m by identifying u y m with u y, p-1 m , and by selecting Y (y) = u y, p-1 m

Fig. 1

 1 Fig. 1 Qualitative comparison between a HiMod (left) and a PGD (right) approximations.

µm

  are computed by ADS, which now coincides with a three-step procedure. Thus, with reference to the Poisson problem, -∇ • µ∇u = f completed with full homogeneous Dirichlet boundary conditions and for µ ≡ µ, we first compute u by selecting Y (y) Z (µ) = u

Fig. 2

 2 Fig. 2 Qualitative comparison between the reference (left) and the PGD (right) solutions, for µ = 1 (top) and µ = 2.5 (bottom).

13 Fig. 3

 133 Fig. 3 Contour plots: comparison between the reference HiMod solution (left) and the HiPOD approximation with l = 1 (right), for µ * = 1 (top) and µ * = 2.5 (bottom). Table:relative error between HiMod and HiPOD solutions with respect to the L 2 (Ω)-norm.

  associated with partitions, T x h and T y h , of Ω x and Ω y , respectively. In general, W x h and W y h are FE spaces, although, a priori, any discretization can be adopted. It turns out that W m is a tensor function space, being

Acknowledgements

The authors thank Yves Antonio Brandes Costa Barbosa for his support in the HiMod simulations. This work has been partially funded by GNCS-INdAM 2018 project on "Tecniche di Riduzione di Modello per le Applicazioni Mediche". F. Ballarin also acknowledges the support by European Union Funding for Research and Innovation, Horizon 2020 Program, in the framework of European Research Council Executive Agency: H2020 ERC Consolidator Grant 2015 AROMA-CFD project 681447 "Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics" (P.I. G. Rozza).


Mohamed Traoré
email: mohamed.traore@etu.uca.fr

Zié Ballo
email: zieballo@hotmail.com

Catherine Araujo Bonjean

Keywords: Fiscal consolidation, Income inequality, Panel data, WAEMU. JEL Codes E62, D31, C23, O55

This paper examines the distributional effects of fiscal consolidation in WAEMU area, using Driscoll and Kraay fixed effects estimation method. Using a tighter fiscal stance, measured by changes in the cyclically-adjusted primary balance (CAPB), over the period 1990-2015, it shows income gap decreases after periods of fiscal adjustments in the region. These finding are contrary to the results of several works focusing on fiscal consolidation. Our results also suggest that when fiscal consolidation is achieved via revenue side, income inequality seems to be reduced further. However, reductions in primary expenditures do not seem to reduce the income gap.

1-Introduction

The last two decades have revealed a high level of inequality in developing countries and more precisely in West African countries. According to the AfDB (2018)1 , the Gini index in the region oscillated closely between 40 and 46 over the period 1985 and 2015. At the same time, West African countries, especially the resource-intensive ones have recorded a weakening of their economies due to collapsing in commodity prices due to their dependency on natural resource. One can add the waste of the public expenditure as well as civil conflicts. In addition to these factors explaining the poor economic performance of the economies of the Union, many have seen buffers erode and countries public debt and borrowing costs are rising. In order to cover permanent revenue losses, these countries have resorted to substantial fiscal austerity measures through a combination of spending cuts and tax hikes to ensure macroeconomic stability and sustainable growth. In this context, a relevant empirical question is: What is the effect of fiscal consolidation programs on income distribution? addressing such an issue seems necessary as poor income distribution could reduce the government's political support for implementing consolidation measures, but also because high levels of inequality could hurt long-term growth [START_REF] Berg | Inequality and Unsustainable Growth: Two Sides of the Same Coin?[END_REF]. Surprisingly, it must be noted there is little systematic analysis of the distributional effects of fiscal austerity in developing countries.

The choice of the WAEMU is motivated by the fact that the countries of the Union have the same characteristics of impoverishment, still presents a high rate of indebtedness and with some countries of West Africa offer the highest rates of inequality of the continent and the world.

Moreover, to our knowledge, there is no empirical investigations that study the relationship between fiscal adjustment and income inequality in WAEMU countries. Furthermore, the study provides guidance to the political and economic decision makers the best choice in term of fiscal austerities measures that are not going to compromise income distribution in WAEMU countries.

To carry out this study, we use the [START_REF] Driscoll | Consistent Covariance Matrix Estimation with Spatially Dependent Panel Data[END_REF] standard errors methodology as implemented by [START_REF] Hoechle | Robust Standard Errors for Panel Regressions with Cross-Sectional Dependence[END_REF], that deals with heteroscedasticity and cross-sectional dependence problems. In fact, cross-sectional dependence is caused by the presence of common factors, which are unobserved, but they are uncorrelated with the included regressors. In this case, the standard fixed effect is consistent, but not efficient and the standard errors that will be estimated will be biased.

Overall, we find that after periods of fiscal adjustments, income inequality significantly decreases. Moreover, differentiating between spending versus taxes-based adjustments, results suggest that when fiscal consolidation is achieved via tax-based, income inequality tends to be further reduced.

We organize the rest of the paper as follows. The next section presents the literature reviews.

Section 3 provides the stylized facts while section 4 describes the data and presents the empirical methodology used to examine the effects of fiscal consolidation episodes on income inequality. Section 5 describes the results. Finally, section 6 concludes.

2-Growth, inequality and fiscal Consolidation: Relation to the literature

There is a relatively large number of studies on the potential impact of austerity measures on economy. [START_REF] Guajardo | Expansionary austerity? International evidence[END_REF] analyze the effect of fiscal adjustment in short-term on economic activity in OECD countries, using the narrative approach of [START_REF] Ramey | Costly capital reallocation and the effects of government spending[END_REF] and [START_REF] Ramey | Identifying government spending shocks: it's all in the timing[END_REF]. Their results conclude that in a fiscal consolidation periods, there is a contraction of private domestic demand and GDP. Thereby, a 1 percent of fiscal consolidation leads to real GDP declines by 0.62 percent and reduces real private consumption by 0.75 percent. [START_REF] Yang | Macroeconomic effects of fiscal adjustment: A tale of two approaches[END_REF] focus also on the short-term effects of fiscal adjustment on economic activity from 1970 to 2009 in 20 OECD countries. They compare two approaches: the traditional approach based on changes in cyclically adjusted primary balance (CAPB) and the narrative approach based on historical records. Their finding highlights that in period of fiscal adjustments, economy activity always decline in the short term. In addition, the loss of output is less important in the period of spending-based adjustments relative to that of the taxbased adjustment. [START_REF] Mcdermott | An empirical analysis of fiscal adjustments[END_REF] argue that the success of debt ratio reduction depends on the fiscal austerity measure size and composition. Comparing the effect of spending-based adjustments and tax-based adjustment, authors show that it is more likely to be successful adjustments based on spending cuts. In addition, the chance of success of the adjustment progress in the same direction as its importance. So, when the adjustment is large, it follows a success. On the other hand, they show that fiscal adjustments are more likely to fail during a global recession. [START_REF] Alesina | The output effect of fiscal consolidation plans[END_REF] support previous conclusions, noting that tax adjustments produce deeper and longer recessions than those based on expenditures. [START_REF] Afonso | Fiscal consolidations in the Central and Eastern European countries[END_REF] assess fiscal consolidations in Central and Eastern European countries and suggest that expenditure-based adjustments have been more successful. [START_REF] Giudice | Non-Keynesian fiscal adjustments? A close look at expansionary fiscal consolidations in the EU[END_REF] conclude that fiscal consolidation is more likely to foster economic growth during periods of below potential output and when fiscal adjustment is based on spending cuts. [START_REF] Romei | Need for (the right) speed: The timing and composition of public debt deleveraging[END_REF] studies the issue of the optimal speed and composition of a fiscal consolidation, evaluating the impact of different speeds of adjustment and of variations in several fiscal instruments on aggregate welfare. The main finding of this work is that a fiscal consolidation should be done quickly and by cutting public expenditure.

It should be noted that most of the existing studies on the economic effects of fiscal consolidation have focused on the effects of fiscal adjustments on economic growth; the distributive effects of austerity have benefited relatively less from research efforts. Almost of the studies on the redistributive effect of fiscal consolidation are conducted on a sample of OCDE countries or a set of advanced countries. [START_REF] Mulas-Granados | Fiscal Adjustments and the Short-Term Trade-Off between economic growth and equality[END_REF] studies the short-term economic impact of alternative fiscal adjustment strategies, with an especial focus on their effect on economic growth and income distribution. Using a sample of 53 adjustment episodes occurred in the fifteen EU Member States over the period 1960-2000, the result supports evidence of a trade-off between growth and equality, mediated by fiscal consolidations. More specifically, while expenditure-based adjustments are more effective in terms of subsequent economic growth than revenue-based adjustments, they do so at the expense of increasing income inequality. Therefore, according to author, during fiscal adjustment, inequality tends to increase and especially when it is a spending-based adjustment. Moreover, the level of inequality is even higher in the case of reductions in social spending. Tax-based adjustments focusing on indirect taxes tend to exacerbate inequality. However, this result is not verified when increases in indirect taxes are combined with compensatory measures, such as direct measures for poor households. [START_REF] Agnello | Fiscal adjustments and income inequality: a first assessment[END_REF], investigating the distributional effects of fiscal consolidation in OECD countries, supports an equalizing effect of austerity measures. They find that "successful" fiscal adjustments tend to be strongly beneficial at achieving a more equal income distribution. In fact, the net income Gini index falls in statistically significant manner by when the fiscal consolidation succeeded in bringing public debt back to a lower level. [START_REF] Rawdanowicz | The Equity Implications of Fiscal Consolidation[END_REF] point out that fiscal consolidations could increase income inequality through multiple channels. An important channel could be an increase in unemployment that widens disparities in market incomes; Moreover, reductions in social transfers can affect households in the lower parts of the income distribution, and a decline in public programs benefiting the poor might also increase inequality in disposable income. [START_REF] Azevedo | Fiscal Adjustment and Income Inequality: Sub-national Evidence from Brazil[END_REF] analyze the relationship between sub-national fiscal policy and income distribution in Brazil, using household survey data. Their found that a stricter fiscal policy is not linked to a worsening in income gap. In addition, the results suggest that rising revenue in Brazilian states have not had an amplifying effect on inequality. Moreover, measures of inequality do not appear to be aggravated by reductions in primary spending. It should be noted that this conclusion is contrary to the results of several studies on the effects of fiscal consolidation on inequality using national data for OECD economies.

3-The challenge of equity in WAEMU countries: stylized facts

In this section, we give a comparison of trends in income inequality in the WAEMU countries.

Thus, we refer to the Gini index which is the measure of inequality broadly used and available for many years. We use the Gini net which is Gini index for disposable income (post-tax, posttransfer), Figure 1 terms of percentage of variations, Burkina Faso is the country that experienced the largest decline in the inequality with a -4.78 percentage, followed by Niger (-3.76 %), Mali (-2.12%) and Côte d'Ivoire (-0.39 %) during the same period. Source: Author, using SWIID database (SWIIDv6_2)

4-Data and Methodological Approach

4.1-Data

The dependent variable used in our regressions is the Gini coefficient for disposable income taken from the Standardized World Income Inequality Database (SWIID) introduced by Solt in 2008 to provide researchers with data on income inequality that maximizes comparability for the widest possible sample of countries and years [START_REF] Solt | Standardizing the World Income Inequality Database[END_REF]. More specifically, we use the Gini coefficient for disposable income from SWIID database (SWIIDv6_2),

The SWIID database is an attempt to address the problem of data scarcity. The data are based on extended imputations where, for countries that do not have the required data for a given year, observations from the same country in other years and other countries of the same year are used to impute both gross and net inequality indices. However, others have been very critical of the widespread use of imputations in general and the particular type of imputations used by SWIID, and caution against the use of the dataset in econometric work (see [START_REF] Jenkins | World income inequality databases: an assessment of WIID and SWIID[END_REF][START_REF] Ferreira | Appraising cross-national income inequality databases: An introduction[END_REF], All costs arising from its implementation must be considered alongside the potential benefits arising from broader coverage. The value of SWIID depends on the plausibility of the assumptions underlying the imputation model (potential bias problems) and the correct use of the multiplied-imputed data (precision problems), Even though the SWIID has some weaknesses, it is important to note that SWIID uses a transparent procedure to increase the comparability of available data on cross-national inequalities. Although it is not ideal for all research on economic inequality, its advantages over other transnational data sets will make it an invaluable resource for those interested in ascertaining the causes and effects of income inequality cross-nationally and over time.

Definition of fiscal adjustments

To assess the distributional effect of fiscal adjustments, we use the cyclically adjusted primary balance (CAPB) that represents an indicator government's fiscal stance. The interest of this fiscal measure is that it isolates discretionary policy action from effects resulting from economic activity notably change in real interest rate or inflation. Especially, we use primary fiscal variables that exclude interest payments because the fluctuations in interest payments cannot be considered discretionary. The cyclical correction is making by following the method proposed by [START_REF] Alesina | Large Changes in Fiscal Policy: Taxes versus Spending[END_REF] and [START_REF] Yang | Macroeconomic effects of fiscal adjustment: A tale of two approaches[END_REF], According to authors, such fiscal variable is simpler and more transparent than more complicated official measures such as those of the OECD and the IMF that use estimates of potential output and fiscal multipliers.

The underlying principle of this method, as mentioned [START_REF] Yang | Macroeconomic effects of fiscal adjustment: A tale of two approaches[END_REF], is that since public expenditure is negatively dependent on GDP as a result of unemployment benefits, and because revenue responds positively to GDP as a result of tax revenues, changes in cyclically-adjusted fiscal variables can be calculated from the difference between the predicted current-year value (which would prevail if unemployment had not changed since the previous year) and the actual value of the previous year.

Although CAPB is used to measure fiscal consolidations in several studies, however, this is not a perfect measure and can lead to biased results [START_REF] Romer | The Macroeconomic Effects of Tax Changes: Estimates Based on a New Measure of Fiscal Shocks[END_REF]. CAPB may consider non-policy factors that are correlated with exogenous variables affecting inequality or growth.

For instance, an expansion in the stock market leads to higher incomes and therefore higher tax revenues. This translates into an improvement in CAPB, although not being in response to fiscal adjustments. In addition, this will likely affect other variables, such as higher consumption and investment. Therefore, there is likely to have positive correlation between an increase in CAPB and the error term. This can lead to an upward biased estimate (IMF, 2011), CAPB may also suffer from measurement error if it omits periods in which fiscal adjustments have been accompanied by countervailing negative shocks.

The CAPB, however, can also be affected crucially by the commodity price volatility. Tax receipts in developing countries increase with rising prices for imported or exported products [START_REF] Ehrhart | Education or inflation?: the micro and macroeconomics of the brazilian income distribution during 1981-1995[END_REF]. For imported products, this increase in tax revenues is due to the fact that more tariffs being collected but, because of the numerous tax exemptions granted in times of high prices, the positive impact on tax revenues may not always occur. For large commodity-exporting economies, the increases in tax revenues due to soaring export prices come from an increase in profit tax and non-tax revenues, such as dividends or royalties levied on companies, which produce primary products. Then, commodity price instability, both for imported and exported products has an adverse effect on fiscal balance in developing countries.

Therefore, contrary to [START_REF] Yang | Macroeconomic effects of fiscal adjustment: A tale of two approaches[END_REF] that use a share price index as an additional variable determining the CAPB, we use the international price of cocoa. In fact, cocoa accounts for 15%

of Côte d'Ivoire's GDP and more than 50% of its export earnings2 . When considering cocoa price as a business cycle factor, it would be ideal to include other types of commodity prices such as oil price and coffee price. But we use only the price of cocoa as a business cycle factor due to its particular relevance to tax revenues, and we believe this index may be representative of the fluctuation in the price of others commodity.

First, to gather cyclically-adjusted expenditures as a share of GDP, we make a regression of primary spending on the unemployment rate (𝑈 𝑡 ) and a time trend for the country:

𝐷 𝑡 = 𝛾 0 + 𝛾 1 𝑇𝑟𝑒𝑛𝑑 + 𝛾 2 𝑈 𝑡 + 𝑒 𝑡 (1)
Then, with the estimated coefficients (𝛾 ̂1, 𝛾 ̂2) and the preceding-year unemployment rate (𝑈 𝑡-1 ), we calculate the value of primary spending adjusted for changes in unemployment:

𝐷 𝑡 * (𝑈 𝑡-1 ) = 𝛾 ̂0 + 𝛾 ̂1𝑇𝑟𝑒𝑛𝑑 + 𝛾 ̂2𝑈 𝑡-1 (2)
The discretionary expenditure change is calculated as: 𝐷 𝑡 * (𝑈 𝑡-1 ) -𝐷 𝑡-1 . We apply the same step to calculate the cyclically adjusted revenues. However, in this step, the cocoa price index is added to the regression.

𝑅 𝑡 = 𝜑 0 + 𝜑 1 𝑇𝑟𝑒𝑛𝑑 + 𝜑 2 𝑈 𝑡 + 𝜑 3 𝐶𝑜𝑐𝑜𝑎_𝑝𝑟𝑖𝑐𝑒 𝑡 + 𝜖 𝑡 (3) 𝑅 𝑡 * (𝑈 𝑡-1 , 𝐶𝑜𝑐𝑜𝑎 𝑝𝑟𝑖𝑐𝑒 𝑡-1 ) = 𝜑 ̂0 + 𝜑 ̂1𝑇𝑟𝑒𝑛𝑑 + 𝜑 ̂2𝑈 𝑡-1 + 𝜑 ̂3𝐶𝑜𝑐𝑜𝑎_𝑝𝑟𝑖𝑐𝑒 𝑡-1 (4)
Finally, the change in discretionary fiscal stance are gathered as follows:

∆𝐶𝐴𝑃𝐵 = [𝑅 𝑡 * -𝑅 𝑡-1 ] -[𝐷 𝑡 * -𝐷 𝑡-1 ] (5)
The literature identifies fiscal adjustment episodes as important and sustained changes in CAPB. The size and persistence criteria differ considerably from one study to the next. For instance, Alesina and Ardagna (2010) consider a period of fiscal consolidation when the change in CAPB is at least 1.5% of GDP in 1 year. Another example is the criteria defined by Alesina and Ardagna (2013) that deem the period of fiscal adjustment correspond at the time that the cumulative change in CABP is at least 2% p of GDP in 2 consecutive years and at least 3% of GDP in 3 or more years with the improvement of each year. [START_REF] Giavazzi | Non-Keynesian effects of fiscal policy changes: international evidence and the Swedish experience[END_REF], consider the episode of fiscal adjustment the period where the cumulative change in CAPB is at least 5, 4, 3% of GDP in respectively 4, 3, or 2 consecutive years, or 3% in 1 year.

The fiscal adjustment episodes considered in our study can be defined as follows:

A period of fiscal adjustment is a year in which the cyclically adjusted primary balance (CAPB)

improves by at least 1 per cent of GDP. This allows us not to consider the different periods when the change in CAPB could be due to exogenous changes in other variables rather than change in policy.

A tax-based fiscal adjustment corresponds to a period where CAPB improves by 1% or more and government revenues as % of GDP increase by 0.5% or more.

A spending-based fiscal adjustment corresponds a period where CAPB improves by 1% or more and government spending as % of GDP decreases by 0.5% or more.

Other variables are included in the baseline specifications. The institutional quality data are from ICRG. Fiscal Variables for the calculation of CAPB are from BCEAO online database.

All other data are obtained from World Bank database. Each variables of model have been presented briefly in Table (2), In advance we concisely explain about some variables of this model.

It has been argued that inflation generally worsens poverty and income inequality. First, inflation can reduce real income. Second, worker's real income will decline if their nominal wages increase less than the price of goods that they consume. Inflation lowers the overall purchasing power of households and tends to raise poverty [START_REF] Agénor | Unemployment-poverty Trade-offs[END_REF]. Romer and Romer (1998) studied the incomes of the poor and show that in the short run, higher inflation when following economic growth can sustain the incomes of the poor. However, in the long term, taking into account the economic uncertainty, it can harm both incomes and the incomes of the poor. Using large sample of household survey, [START_REF] Easterly | Inflation and the Poor[END_REF] found that the poor were more likely than the rich to mention inflation as a problem, and that inflation compromise the welfare of poor more than it does that of the rich. In the same line, Ferreira and Litchfield (2001) found in the case of Brazil that higher inflation leads to a lower share of income held by the poorest income group.

The population growth is expected worsen inclusive growth as faster growing societies experience faster growing demand for public services. This increases difficulty of governments to provide those services. Therefore, high population growth puts a considerable break on per capita growth prospects. It also contributes significantly to low achievement in poverty reduction and is associated with households moving into poverty.

The quality of institutions is very important. For example, the good socioeconomic conditions that is an assessment of the socioeconomic pressures at work in society that could constrain government action can contribute to better redistribution in income. By against, the weakness in institutional quality such as corruption is clearly a very serious problem that is detrimental to inclusive growth. Corruption robs the poor most of all and deprives people of access to service and of the opportunity to earn living. In some countries, it is a major obstacle to economic growth.

Others important variables that can affect the income inequality are, trade open, and social globalization.

Descriptive statistic of variables

The table 1 below provides a summary of the descriptive statistics analysis of the variables in our model. Generally, one can note that standard deviations are low meaning that the variances are minimal between the variable values.

4.2-Methodology of estimation

A standard model used to analyze the factors that affect the income inequality could be represent as follow:

𝑌 = 𝜃𝑋 + 𝜀 (6)
Where Y represents is the Gini index for income inequality, and X is a vector of macroeconomic variables including the GDP per capita growth rate at the previous year. Next, we expand our analysis by focusing on the relationship between fiscal adjustment and income inequality. To do this, we are expanding the model specification ( 6), including a fiscal consolidation indicator (𝐹𝐴 𝑡 ), More specifically, we control the episodes of fiscal adjustment by the cyclically-adjusted primary balance (CAPB), To estimate the distributional impact of fiscal consolidation episodes, we formally run the following equation:

𝑌 𝑡 = 𝛽 0 + 𝛽 1 𝐹𝐴 𝑡 + 𝛽 2 𝑋 𝑡 + 𝜀 𝑡 (7)
where 𝑌 denotes, one of our dependent variables (Gini index), 𝐹𝐴 represents the changes in the CAPB in percent of GDP in periods of fiscal adjustment and zero otherwise; 𝑋 is a vector of control variables and 𝜀 𝑡 is the error term.

Our data is a cross-sectional time series data set, and thus pooled OLS is the simple way for estimating since it does not consider non-observed effects. It is important to note that in our sample, fiscal adjustment episode as well as income inequality differs from one country to another. Hence it is possible that country effects can be relevant for explaining gap in income distribution. Therefore, the model in equation ( 7) is estimated employing fixed effect estimator (FE), The literature on panel data emphasizes that panel data sets are likely to exhibit substantial cross-sectional dependence, which may arise due to the presence of common shocks and unobserved components that become part of the error term ultimately, spatial dependent and due to idiosyncratic pair-wise dependence in the disturbances with no particular pattern of common components or spatial dependence [START_REF] Baltagi | Econometric analysis of panel data[END_REF]. Assuming that cross-sectional dependence is caused by the presence of common factors, which are unobserved (and as a result, the effect of these components is felt through the disturbance term) but they are uncorrelated with the included regressors, the standard FE is consistent, although not efficient and the estimated standard errors are biased. Using the tests developed by [START_REF] Frees | Assessing cross-sectional correlation in panel data[END_REF] and [START_REF] Pesaran | General Diagnostic Tests for Cross Section Dependence in Panels[END_REF], we found evidence for the presence of cross-sectional dependence. Also, the Breush-Pagan test and the Durbin Watson test showed the presence of panel heteroscedasticity and serial autocorrelation, as common for panel data with country-year observations. To correct these problems, the model (7) was estimated using the Driscoll and Kraay (1998) standard errors as implemented by [START_REF] Hoechle | Robust Standard Errors for Panel Regressions with Cross-Sectional Dependence[END_REF]. This estimation accounts for heteroscedasticity and cross-sectional dependence problems. Moreover, this option allowed us to correct the autocorrelation of any order. Some studies using GMM methods of reasons of a possible endogeneity of certain variables. Here, we decline to address the issue of endogeneity due limited number of observations. This should not have any effect on our results.

5-Empirical results

In this section, the results of the estimation of equation 1 are presented using fixed effects models that account state characteristics that are time invariant. To mitigate the likely problems of endogeneity, we use lagged values of the control variables and suppose that they are weakly exogenous. We also address cross-sectional dependence problems by including time effects in regressions and using standard errors corrected by [START_REF] Driscoll | Consistent Covariance Matrix Estimation with Spatially Dependent Panel Data[END_REF].

Table (2) presents the estimated coefficients of the changes in the CAPB on the income inequality measure by Gini index in our baseline model. The results show that that fiscal consolidations in previous years have a statistically significant negative effect on a current Gini coefficient. This result suggests that an improvement of 1% in previous years CAPB reduce income inequality in current period, with an overall impact of 0.856 on the Gini coefficient.

This means that income gap decreases after periods of fiscal consolidation. The result is in line to those obtained by [START_REF] Azevedo | Fiscal Adjustment and Income Inequality: Sub-national Evidence from Brazil[END_REF]. Their found a negative and significant relationship between in cyclically adjusted-primary balances and Gini coefficient, meaning that a tighter fiscal stance in Brazilian states does not seem to increase inequality over the period of study.

These results are contrary to those of several studies that have studied the impact of fiscal consolidations on inequality at the national level for OECD countries.

Spending versus taxes-based consolidation effects on equity

Does the composition of fiscal consolidation (spending versus taxes-based) matter for inequality? We attempt in this section to analyze the role played by the composition of fiscal adjustment in the distribution of income. Thus, we re-run our model by focusing on the two types of fiscal adjustments that are "tax-based adjustment" and "spending-based adjustment as defined above. Table (3) reports the results.

Results show that spending-based adjustments that mean cutting in current public expenditure appears to have no effect on income gap (Column 2), Interestingly, when fiscal consolidation is achieved via revenue side, income inequality seems to be reduced further (Column 1), These results are close to those obtained by [START_REF] Agnello | Fiscal adjustments and income inequality: a first assessment[END_REF]. According to these authors, while fiscal adjustments following the reduction of public expenditure seem to hurt income distribution, the increase in tax seems to reduce income inequality. This evidence suggests that policymakers could achieve their goal of equal income distribution from a properly designed tax-based consolidation plan. Furthermore, fiscal adjustment focusing on indirect taxes may improve equity [START_REF] Ciminelli | The composition effects of taxbased consolidations on income inequality[END_REF]. For authors, this outcome may be partly explained by the operation of a positive labor supply channel. Authors think that individual are more incentives to take part in the labor market with the establishment of indirect taxes. One explanation is that indirect taxes raise the price of consumer goods, leading to a negative income effect. Thereby, in developing economies, consolidation policies can be put in place to mitigate its negative impact on income redistribution. However, this must be accompanied by an improvement in the progressiveness of the tax and transfer systems of the global economy.

Robustness check

To check the robustness of our results, we perform several robustness checks, as calculating an alternative measure of fiscal consolidation as well as adding other control variables. For the construction of the new fiscal consolidation variable, we follow the same procedure developed above, but here, we use the output gap as a proxy for business cycle fluctuations instead of unemployment to cyclically adjust government spending and revenue. This allows us to calculate the alternative CAPB measure. Next we follow [START_REF] Alesina | Fiscal adjustments in OECD countries: composition and macroeconomic effects[END_REF] in order to

give a new definition of fiscal adjustment. Therefore, fiscal adjustment corresponds to period where the change in CAPB is at least 1.5% p of GDP in 1 year or at least 1.25% p of GDP per year in both 2 consecutive years.

Table (4) shows that the baseline results (columns 1) are robust to alternative CAPB definitions (columns 2) and alternative criteria for the definition of fiscal adjustment (columns 3), In each case, fiscal adjustment has a similarly negatively links to inequality meaning that after a period of fiscal austerity, there is a worsening in the distribution of income. In addition, almost all variables keep the right sign and their level of significance. However, our finding give evidence that international remittances tend to raise income inequality in WAEMU countries. In fact, households receiving international remittances are not poor at the first time and, with the receipt of remittances, they tend to improve their situation significantly. It should also be added that these receipts of remittances mainly benefit families whose members have emigrated, generating a gap in the household's income.

6-Conclusion

This paper examines the distributional effects of fiscal consolidation in WAEMU. Using a tighter fiscal stance, measured by changes in the cyclically-adjusted primary balance, over the period 1990-2015, it shows income gap decreases after periods of fiscal adjustments in the region. In other words, the results suggest there is no trade-off between growth and income inequality when implementing fiscal consolidations. These finding are contrary to the results of several works analyzing the effect of austerities measures on income inequality at the national level. Our results also suggest that when fiscal consolidation is achieved via revenue side (tax-based fiscal consolidations), income inequality seems to be reduced further. However, reductions in primary expenditures do not seem to reduce the income gap. These results could be explained by the fact that in developing countries, tax based-adjustment are generally a basebroadening measures. Therefore, this will contribute to strong tax revenue collections and play an important role in achieving higher, sustainable economic growth. Therefore, Fiscal adjustment in WAEMU economies should focus on tax revenue mobilization to achieve a sustainable goal. In this case,

  presents graph in income inequality by countries in WAEMU zone. For each country, the graph presents the mean income inequality over two sub-periods: 2000/2008 and 2009/2015. Generally, average income inequality remains high (above 30 points) for all countries over the two sub-periods. Over the period 2000-2008, that is before the 2008 economic crisis, Burkina Faso has had a relatively high Gini of over 39. Over the period 2009-2015, efforts have been made to improve income distribution in most countries. Of the eight WAEMU countries, just three saw their inequality index increase between 2009 and 2015 (Togo, Benin, and Guinea-Bissau) compared to the period 2000-2008. Niger is the country that registered a low level of income inequality in the area over the period 2009-2015. However, in

Figure 1 :

 1 Figure 1: Mean income inequality in WAEMU over the period 2000-2015.

Table 1 : Descriptive statistic

 1

	Variables	Obs	Mean	Std. Dev.	Min	Max
	Net Gini	150	37.348	2.375	31.5	42.9
	ΔCAPB	208	.3701	.484	0	1
	GDP pc growth	200	2.348	17.360	-43.164	169.902
	Inflation	208	87.453	26.879	4.687	147.543
	Social globalization	176	23.495	6.887	10.492	37.483
	Population growth	200	2.909	.749	1.926	9.879
	Corruption	168	2.171	.720	0	4
	External conflic	168	9.421	1.229	6	12
	Military in politics	168	2.135	1.231	0	5
	Development Aid (log)	208	19.986	.810	18.041	21.813
	Remittances	208	3.612	2.987	.226	12.600
	Trade open	208	30.176	8.917	14.187	62.516
	Source: Author estimates

Table ( 2

 ( ): Effects of fiscal adjustment on income inequality

	VARIABLES	Dependent variable: Gini coefficient
		(1)	(2)
		Pooled-OLS	Fixed-effects
		All adjust.	All adjust.
	ΔCAPB (t-1)	-0.758**	-0.856**
		(0.344)	(0.331)
	GDP pc growth (t-1)	0.000856	-0.00744***
		(0.00241)	(0.00239)
	Inflation (t-1)	-0.00818	0.00327
		(0.0111)	(0.00574)
	Social globalization (t-1)	-0.0529***	-0.234***
		(0.0109)	(0.0409)
	Population growth (t-1)	-0.188*	-0.190***
		(0.0975)	(0.0581)
	Corruption (t-1)	0.782***	0.236*
		(0.236)	(0.124)
	External conflic (t-1)	-0.582***	0.0597
		(0.0813)	(0.244)
	Military in politics (t-1)	-0.447**	-0.342***
		(0.210)	(0.0784)
	Constant	44.85***	43.08***
		(1.395)	(2.234)
	Number of observations	123	123
	Number of groups	7	7
	R-squared	0.2591	0.3692
	Source: Author estimates
	Note: Driscoll-Kraay robust t-statistics in parentheses. They are robust to very general forms of cross-sectional and temporal
	dependence. All regressions control for Country fixed effects, Time fixed effects, and a constant term are included in each
	regression. *** p<0.01, ** p<0.05, * p<0.1

Table ( 3

 ( ): Spending and tax-based adjustment effects on income inequality

	VARIABLES	Dependent variable: Gini coefficient
		Driscoll-Kraay Fixed-effects estimation
		(1)	(2)
		Tax-based	Spend.-based
	ΔCAPB (t-1)	-0.882***	-0.259
		(0.273)	(0.226)
	GDP pc growth (t-1)	-0.00851***	-0.00715**
		(0.00225)	(0.00279)
	Inflation (t-1)	0.00493	0.00363
		(0.00557)	(0.00518)
	Social globalization (t-1)	-0.232***	-0.242***
		(0.0422)	(0.0469)
	Population growth (t-1)	-0.275***	-0.225***
		(0.0429)	(0.0555)
	Corruption (t-1)	0.325***	0.149
		(0.0931)	(0.0935)
	External conflic (t-1)	0.143	0.136
		(0.266)	(0.324)
	Military in politics (t-1)	-0.362***	-0.377***
		(0.0776)	(0.0937)
	Constant	42.13***	42.63***
		(2.228)	(2.830)
	Number of observations	120	120
	Number of groups	7	7
	R-squared	0.3588	0.3209
	Source: Authors estimates

Note: Driscoll-Kraay robust t-statistics in parentheses. They are robust to very general forms of cross-sectional and temporal dependence. All regressions control for Country fixed effects, Time fixed effects, and a constant term are included in each regression. *** p<0.01, ** p<0.05, * p<0.1

Table 4 :

 4 Effects of fiscal adjustment on equity: alternative measures Kraay robust t-statistics in parentheses. They are robust to very general forms of cross-sectional and temporal dependence. All regressions control for Country fixed effects, Time fixed effects, and a constant term are included in each regression. *** p<0.01, ** p<0.05, * p<0.1 Next, we check the robustness of our findings to the inclusion of other variables to control for development aid, trade open as well as remittances received by countries. The results are present in the table 5. Here again, the results regarding our interest variable that measures fiscal adjustment are consistent with our previous estimates. Results also give evidence that trade open and development aid contribute to strengthening the good distribution of income.

	VARIABLES	Dependent variable: Gini coefficient
		(1)	(2)	(3)
			Alternative CAPB	Alternative criteria for
		Baseline result	definitions	fiscal adjust.
	ΔCAPB (t-1)	-0.856**	-0.849***	-0.844**
		(0.331)	(0.293)	(0.390)
	GDP pc growth (t-1)	-0.00744***	-0.00805***	0.000626
		(0.00239)	(0.00262)	(0.00227)
	Inflation (t-1)	0.00327	0.00656	-0.00820
		(0.00574)	(0.00601)	(0.0114)
	Social globalization (t-1)	-0.234***	-0.242***	-0.0576***
		(0.0409)	(0.0458)	(0.0109)
	Population growth (t-1)	-0.190***	-0.169***	-0.151
		(0.0581)	(0.0535)	(0.0950)
	Corruption (t-1)	0.236*	0.211*	0.813***
		(0.124)	(0.115)	(0.226)
	External conflic (t-1)	0.0597	0.129	-0.636***
		(0.244)	(0.267)	(0.0805)
	Military in politics (t-1)	-0.342***	-0.304***	-0.442*
		(0.0784)	(0.0805)	(0.218)
	Constant	43.08***	42.24***	45.33***
		(2.234)	(2.292)	(1.361)
	Number of observations	123	120	120
	Number of groups	7	7	7
	R-squared	0.3692	0.3657	0.273
	Source: Authors estimates
	Note: Driscoll-

Table 5 :

 5 Robustness with adding news variables

	VARIABLES	Dependent variable: Gini coefficient
		(1)	(2)	(3)
	ΔCAPB (t-1)	-0.831**	-1.063**	-0.839**
		(0.300)	(0.437)	(0.307)
	GDP pc growth (t-1)	-0.00510**	-0.0117***	-0.00606***
		(0.00241)	(0.00360)	(0.00199)
	Inflation (t-1)	0.00457	-7.34e-05	0.0143
		(0.00937)	(0.00497)	(0.0102)
	Social globalization (t-1)	-0.225***	-0.313***	-0.215***
		(0.0487)	(0.0401)	(0.0268)
	Population growth (t-1)	-0.169***	-0.259***	-0.291***
		(0.0538)	(0.0808)	(0.0736)
	Corruption (t-1)	0.344*	0.115	0.210
		(0.167)	(0.232)	(0.138)
	External conflic (t-1)	0.0695	0.0347	0.0455
		(0.239)	(0.179)	(0.205)
	Military in politics (t-1)	-0.256***	-0.171***	-0.388***
		(0.0685)	(0.0571)	(0.0870)
	Dev. Aid (t-1)	-0.667***
		(0.137)
	Remittances (t-1)		0.416***
			(0.0397)
	Trade open (t-1)			-0.114**
				(0.0493)
	Constant	55.50***	44.32***	45.57***
		(4.612)	(1.904)	(1.690)

West Africa Economic Outlook

-African Development BankÉtudes et Documents n°16, CERDI, 2018

It is important to remember that the Côte d'Ivoire's economy represents more than 40 percent of the area.Études et Documents n°16, CERDI, 2018

This work was supported by the LABEX IDGM+ (ANR-10-LABX-14-01) within the program "Investissements d'Avenir" operated by the French National Research Agency (ANR). at: https://cerdi.uca.fr/etudes-et-documents/