Qian Xie
email: qxie@stat.fsu.edu

Anuj Srivastava

An Improved Estimator of GRID Model for Representing Large Diffeomorphic Deformations

Keywords: Large deformation, GRID model, parameter estimation

The growth by random iterated diffeomorphisms (GRID) model seeks to decompose large deformations, caused by growth, anomaly, or anatomical differences, into smaller, biologically-meaningful components. These components are spatially local and parametric, and are characterized by radial deformation patterns around randomly-placed seeds. A sequential composition of these components, using the group structure of diffeomorphism group, models the cumulative deformation. The actual decomposition requires estimation of GRID parameters from observations of large growth, typically from 2D or 3D images. While past papers have estimated parameters under certain simplifying assumptions, including that different components are spatially separated and non-interacting, we address the problem of parameter estimation under the original GRID model that advocates sequential composition of arbitrarily interacting components. Using a gradient-based approach, we present an algorithm for estimation of GRID parameters by minimizing an energy function and demonstrate its superiority over the past additive methods.

Introduction

The mathematical and statistical modeling of diffeomorphic deformations over time is an important problem with a variety of applications ranging from medical diagnostics to evolutionary biology. The use of medical images, especially the MRI images of human parts, in studying anatomical structures is a growing area of research by itself. Here one uses 2D and/or 3D images taken across time, species, or specimens to compare to extract salient differences in anatomical structures, and to analyze and model their variations both within and across biological classes. These differences may result from standard biological growth, abnormalities, inter-specimen variability, or other reasons. In terms of imagebased analysis of anatomical structures, the study of shapes of anatomical parts has become a central idea. For instance, one can use longitudinal image data for tracking biological growth [START_REF] Ling | Entropic analysis of biological growth models[END_REF][START_REF] Kansal | Simulated brain tumor growth dynamics using a three-dimensional cellular automaton[END_REF][START_REF] Swanson | Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion[END_REF][START_REF] Clatz | In silico tumor growth: application to glioblastomas[END_REF][START_REF] Chaplain | Mathematical modeling of tumor growth[END_REF] in fetus brains and evaluating tumor growth. A major difficulty in solving such problems is the high dimensionality of image data. The diffeomorphic deformation when estimated from image sequences can be very high dimensional and not amenable to standard tools from multivariate statistics. Some current methods simplify this analysis by using simplistic measures like lengths, sizes, or areas as indicators of overall shape changes. Some others use relatively simple geometrical models, such as spheres or ellipsoids, to represent shapes in parametric forms and to study the evolution of parameters during growth.

We start with the basic question of how to represent large deformations in a mathematical framework. There is a large body of work on representing differences in imaged objects using deformations of background space [START_REF] Arsigny | Polyaffine transformations: a new class of diffeomorphisms for locally rigid or affine registration[END_REF][START_REF] Miller | Mathematical textbook of deformable neuronatomies[END_REF][START_REF] Grenander | Computational anatomy: an emerging discipline[END_REF][START_REF] Trouvé | Diffeomorphisms groups and pattern matching in image analysis[END_REF][START_REF] Andresen | Non-rigid registration by geometry-constraint diffusion[END_REF][START_REF] Miller | Group actions, homeomorphisms, and matching: a general framework[END_REF]. This approach utilizes diffeomorphisms of the underlying coordinate systems to represent and measure shape and other differences. Let an image be I : [0, 1] d → IR, where d = 1, 2, 3. A deformation is then a mapping Φ : [0, 1] d → [0, 1] d , with the resulting deformed image is I • Φ : [0, 1] d → IR. Thus, a point on an anatomical landmark is always observed with the same color intensity; it simply moves to a different location under the deformation. The goal is to use Φ to model, understand and analyze large deformations. These deformations are typically very high dimensional and do not permit standard statistical analysis directly. Therefore, some tools for reducing dimensionality become important. One can apply some standard dimension reduction algorithms, such as PCA, but it is difficult to interpret the resulting representation in biological terms. Durrleman et al. [START_REF] Durrleman | Optimal data-driven sparse parameterization of diffeomorphisms for population analysis[END_REF] proposed a parametric way of representing large diffeomorphisms by forcing the instantaneous velocity fields to take a parametric form. One starts with a finite number of so-called control points and for each of them specifies a vector that defines the deformation at that point. The vector field over the whole domain is obtained using a Kernel-based interpolation. This approach provides a data-driven sparse parametric method to estimate the large diffeomorphic deformation.

Motivated by the need for biologically-interpretable decompositions of large deformations, Grenander [START_REF] Grenander | A pattern-theoretic characterization of biological growth[END_REF] introduced the Growth as Random Iterated Diffeomorphisms (GRID) model. It highlights the role of gene control in biological growths and uses a combination of local, structured deformations to form the large composite deformations. This model has been studied extensively, but mostly from a perspective of synthesis and asymptotics. Some authors proposed a "thermodynamic limit equation" that approximates the growth pattern in a macroscopic way [START_REF] Portman | Existence and uniqueness of solutions to the grid macroscopic growth equation[END_REF][START_REF] Portman | Grid macroscopic growth law and its application to image inference[END_REF]. Portman et al. [START_REF] Portman | Direct estimation of biological growth properties from image data using the "grid" model[END_REF] further developed the GRID model by analyzing the growth patterns at microscopic levels. In addition to synthesis, one is also interested in the inverse problem where we want to decompose large biological growth into smaller biologically-interpretable units. Grenander et al. [START_REF] Srivastava | Maximum-likelihood estimation of biological growth variables[END_REF][START_REF] Grenander | A pattern-theoretic characterization of biological growth[END_REF] studied this inverse problem albeit in a limited context. The estimation of growth components was done in two steps: [START_REF] Andresen | Non-rigid registration by geometry-constraint diffusion[END_REF] estimate the full deformation between a pair of images that represents biological growth, (2) estimate parameters for growth components under the GRID model, with a major simplification that different components are spatially local and do not interact with each other. With this assumption, the cumulative growth becomes a simple superimposition of different components and one can use standard projection procedures to esti-mate component parameters. In this paper we seek a solution to the problem of parameter estimation under the original GRID model, without assuming spatial independence of components. This model is different from Durrleman et al. [START_REF] Durrleman | Optimal data-driven sparse parameterization of diffeomorphisms for population analysis[END_REF] in the sense that it is the individual diffeomorphisms that take the parametric forms, rather than the instantaneous velocity fields. This results in different local deformations around the focal points. For example, in our method the diffeomorphism around a seed is restricted to be radial while in Durrleman et al. [START_REF] Durrleman | Optimal data-driven sparse parameterization of diffeomorphisms for population analysis[END_REF] there is no such structure.

The estimation follows the two steps as Grenander et al. [START_REF] Grenander | A pattern-theoretic characterization of biological growth[END_REF]. In the first step, the full deformation Φ is estimated using the shape matching technique of [START_REF] Kurtek | Parameterization-invariant shape comparisons of anatomical surfaces[END_REF]. As for the second step, since there is a concatenation of deformation associated with different components, the time-ordering of the components becomes important. Due to the nonlinear effect of compositions, it is not possible to solve for GRID parameters using linear methods. This general estimation problem is posed as an optimization problem with a gradient-based minimization of the cost function. The difficulty of getting trapped in local solutions is handled using clever initializations of the gradient algorithm.

Grenander's GRID Model

We start by describing the general GRID model as introduced by Grenander [START_REF] Grenander | On the mathematics of growth[END_REF]. In this model the overall large deformation is modeled as a composition of a sequence of local, elementary deformations. At time t, the elementary growth is a diffeomorphism φ t : [0, 1] d → [0, 1] d such that the point x moves to φ t (x). The full deformation is then expressed in the form of the composition of iterated diffeomorphisms.

Φ tn t1 ≡ φ tn • • • • • φ t2 • φ t1 (1)
The next step in the GRID model is to simplify diffemorphic components by expressing each φ t in a parametric form. Here the elementary deformation φ t is:

(1) assumed to be centered at a point of activation called a growth seed x seed , and (2) the growth around the seed is assumed to be radial. Therefore, it is easier to express this local deformation using polar coordinates centered at the seed x seed : (r, τ ) -→ (ρ (r, τ ) , τ) with r = 0 denoting the seed. Furthermore, the model assumes that the change in radial distance can be decomposed into two independent parts: ρ (r, τ ) = r + R (r) A (τ ). Here, A : S 1 → R is called the angular deformation function (ADF) and R : IR + → IR + is called the radial deformation function (RDF). These individual deformation functions are allowed to take the following forms:

1. The radial deformation function RDF can be one of the following two types:

R (r) = re -r 2 /c 2 , r≥ 0, c > 0 (r/c) p-1 e -(r/c) , r ≥ 0, p, c > 0 . ( 2
)
In both cases the deformation is zero at the seed (r = 0), increases steadily with r, reaches a peak, and then decreases for a further increase in r. The "zone" of influence of a seed is determined by the parameter c.

2. Similarly, the angular deformation function ADF can also take many forms, including:

A (τ ) = ⎧ ⎪ ⎨ ⎪ ⎩ a, τ ∈ S 1 , a ∈ IR ae κ cos(τ -τ0)-κ , τ ∈ S 1 , a ∈ IR, κ ≥ 0, and τ 0 ∈ S 1 α sin 2πτ σ , τ ∈ S 1
, α ∈ IR, and σ > 0.

(

) 3
The first case provides an isotropic deformation, the second provides a unimodal deformation with a well-defined growth/decay direction, while the last one provides a sinusoidal variation.

In this paper we will use R(r) = re -r 2 /c 2 and will study two choices for ADFs:

(1) A(τ ) = a, and ( 2) A(τ ) = ae κ cos(τ -τ0) . It has been shown that the resulting φ t is a diffeomorphism as long as -1 < A (τ ) < 2.2408 [START_REF] Grenander | A pattern-theoretic characterization of biological growth[END_REF]. Each such φ t is now characterized by the following set of parameters: θ = [ξ, a, c, κ, τ 0 ] ∈ IR 6 , where ξ ∈ IR 2 is the seed location.

Problem Statement: Having chosen the model, the estimation problem can be described as follows. Let Φ represent the observed deformation associated with a growth experiment, observed over a time interval [0, T ]. The goal now is to estimate n, the number of diffeomorphism components and the associated parameters θ j ∈ IR 6 for each j = 1, 2, . . . , n. This problem has been studied by several papers in the past. However, a common simplifying assumption in the past papers is that different seeds are placed away from each other so that there is no or negligible interaction between the corresponding deformations. In this case, the total displacement field Ψ (x) = Φ(x)x can be written as a superposition of the displacements resulting from individual seeds:

Ψ tn t1 ≡ ψ tn + • • • + ψ t2 + ψ t1 , ( 4
)
where ψ t (x) = φ t (x)x. This is a very restrictive assumption and reduces the efficacy of the GRID model. The additive model has several problems, including the fact that the set of diffeomorphisms is not a group under the additive model. In the context of biological growth, it is difficult to interpret growth components under the assumption that there is no spatial interaction between them. Also, as shown in Fig. 1, the results of these different models, composite versus additive models are quite different for the same components. It is also illustrated that, for the composite model, the ordering of components is also important in determining the cumulative deformation. In this paper, we study the problem of parameter estimation under the composite model (Eqn. 1) and compare the results with those obtained under the additive model (Eqn. 4).

Our Approach

In this section we formulate the problem of parameter estimation as minimization of a certain objective function. The goal is to estimate difeomorphic components In the remaining panels, we show Φ -Φ (4th panel), and Φ -Φ (5th panel) as vector fields.

φ 1 , φ 2 , . . . , φ n such that their order composition is as closed to the given Φ as possible. In order words, we can use a distance between Φ and (φ n • • • • • φ 1 ) as the objective function. Although the choice of a geodesic distance in the space of diffeomorphisms (under a suitable Riemannian metric) seems like a natural choice, the use of the L 2 distance simplifies the problem, by forming an energy

E = [0,1] 2 Φ(x) -(φ n • • • • • φ 1 )(x) 2 dx . ( 5
)
We justify the use of L 2 distance, over the geodesic distance, with the argument that minimization under one distance often leads to a minimizer under the other. It is rather difficult to solve for all the parameters (for all the seeds) simultaneously. Indeed, the expression for cumulative deformation with just two local deformations gets complicated. Therefore, we take a sequential approach and add one local deformation to the model at a time. Let

φ (k) = φ k • φ k-1 • • • • • φ 1
be the cumulative deformation generated by first k seeds. Define two energy functions associated with this partial inference problem:

E (k+1) = Φ(x) -(φ k+1 • φ (k) )(x) 2 dx and Ẽ(k+1) = Φ(x) -φ (k) (x) -ψ k+1 (x) 2
dx .

E (k+1) denotes the energy under the composite model for k + 1 seeds while Ẽ(k+1) denotes a similar energy except that the contribution from the last seed is considered additive. (Since this last seed is additive, it is relatively easier to solve for its parameters by minimizing Ẽ(k+1) .) Our iterative approach is to solve for the parameters of φ k+1 to minimize E (k+1) , for k = 1, 2, . . . , n, and we will do so using a gradient approach. Similar to any gradient-based solution, the initialization of parameters becomes very important. For the purpose of improving initialization, we will solve for the parameters of φ k+1 under Ẽ(k+1) first and use these values as initial conditions in optimization of E (k+1) . We summarize the iterative procedure for estimating GRID parameters.

Algorithm 1 Set k = 0.

1. Given the current estimated parameters for the k-seed composite model, {θ j } k = {(ξ j , a j , c j , κ j , τ 0,j ) | j = 1, . . . , k}, compute the cumulative deformation φ (k) . 2. Find θ k+1 = (ξ k+1 , a k+1 , c k+1 , κ k+1 , τ 0,k+1 ) for the (k + 1) th seed by minimizing Ẽ(k+1) . Use these values as initial condition for parameters of the (k + 1) th seed. 3. For each possible permutation group of the set {1, 2, . . . , k + 1}, perform the following. Update each set of parameters {θ j } k+1 using the gradient method to minimize E (k+1) . Finally, choose the permutation/parameters that result in the minimum E (k+1) . 4. Test the significance of the (k + 1) th seed. If it is found significant, set k = k + 1 and go to step 1; if not, stop.

Note that even though the complexity of the composed deformation φ (k) and thus E (k) increases with the number of seeds (or local elementary deformations), we still have analytical expressions for the gradients using the chain rule. It is important to note that all the previous seeds are re-estimated/updated as new seeds are added to the deformation. Thus, although this process is iterative, it is not incremental.

We have studied two cases for estimating ADFs:

1. Constant ADF: In the first case, we simplify the discussion by first assuming that A(τ ) = a for all τ ∈ S 1 . In this model, the partial derivatives of E (k) with respect to different parameters of φ j for j = 1, • • • , k are given by:

∂E (k) ∂ξ r j = n i=1 ∂E (k) ∂φ (k) • ∂φ (k) ∂φ (k-1) • ∂φ (k-1) ∂φ (k-2) • • • • • ∂φ (j+1) ∂φ (j) • ∂φ (j) ∂ξ r j , r = 1, 2 , ∂E (k) ∂a j = n i=1 ∂E (k) ∂φ (k) • ∂φ (k) ∂φ (k-1) • ∂φ (k-1) ∂φ (k-2) • • • • • ∂φ (j+1) ∂φ (j) • ∂φ (j) ∂a j , ∂E (k) ∂c j = n i=1 ∂E (k) ∂φ (k) • ∂φ (k) ∂φ (k-1) • ∂φ (k-1) ∂φ (k-2) • • • • • ∂φ (j+1) ∂φ (j) • ∂φ (j) ∂c j .
The different terms needed in these expressions, including ∂E (j) ∂φ (j) , ∂φ (j) ∂φ (j-1) , ∂φ (j) ∂ξj , ∂φ (j) ∂aj , and ∂φ (j) ∂cj , for j = 1, . . . , k, are given in the appendix. Similar expressions can also be derived for the gradient of Ẽ(k) wrt the GRID parameters and those expressions are, as expected, simpler compared to the gradient given above. 2. Non-isotropic ADF: In the general case where the ADF is non-isotropic, we represent it using a scaled von-Mises density (the second term in Eqn.

3) and is parameterized by (a, κ, τ 0 ). For every deformation φ j , the two additional parameters κ j and τ 0,j are initialized in two steps. First, the ADF is estimated non-parametrically by integrating the deformation along each direction, as was done in [START_REF] Grenander | A pattern-theoretic characterization of biological growth[END_REF]. Then, the parameters κ and τ 0 are estimated from the nonparametric estimate using moment matching. The estimation is similar to estimating the parameters of von-Mises density from a sample by treating the non-parametric estimation as a weighted sample [START_REF] Sra | A short note on parameter approximation for Von Mises-Fisher distributions and a fast implementation of is[END_REF]. After these parameters are initialized, they are estimated using the gradient method, similar to the other parameters, with the gradient expression given in the appendix.

The last remaining item in Algorithm 1 is the test of significance of an incremental local deformation. In order to test the significance of the model with one more seed, several general methods for model selection may apply. Possible methods includes partial F test, AIC or BIC, and adjusted R 2 . In this paper, a model is selected based on the adjusted R 2 . It is a modification of R 2 , which denotes the coefficient of determinant, that adjusts for the number of model parameters. Given any two estimated models, the model with the larger value of adjusted R 2 is preferred. In the experiments, we add one more seed if the improvement of adjusted R 2 is larger than a small cutoff value.

Since this method is based on a gradient search, it is difficult to claim a global solution. In principle, the solution obtained in the parameter space is a local one. However, there are some advantages to using this approach. Firstly, since the gradients of the energy function are available analytically, the gradient iterations are computationally fast. Secondly, for relatively small number of seeds in the model, the search over different orderings is efficient and gets us out of several local solutions.

Experimental Illustrations

Here we demonstrate the use of our framework for decomposing large cumulative deformations into smaller, parametric components using Algorithm 1. We will use both the simulated and real data to illustrate the estimation process, and will compare our results with those obtained using the additive model.

Synthetic Data

In order to validate estimation method for the composite model, we perform two experiments on the following types of synthetic data: (1) a 2D deformation with constant ADF, and (2) 2D deformation with non-constant ADF. In these experiments, a cumulative deformation is simulated using Eqn. 1 for an arbitrary number of local deformations, each with arbitrary parameters, and a white Gaussian noise is added at the end to form the observed deformation Φ.

Example 1: Fig. 2 shows the estimation results for the isotropic model. It shows the true underlying deformation (a), made up of n = 2 seeds, its noisy observation Φ (b) and several different GRID estimates from the observed Φ (ce). Firstly, we estimate GRID components as described in Algorithm 1, with the result shown as Φc in (e). Then, we reverse the order of two estimated seeds and try to optimize their parameters, with the result shown as Φ c in (c). The result of estimated deformation under the additive model is shown as Φa in (d). To highlight the differences between different estimated deformations we also show their differences in the remaining panels. The Table 1 provides a quantification of estimation performance. It compares the energy E and estimated parameters for the three models, with the true values. Since the energy for Φc is same as that for true underlying parameters, it shows the superiority of that estimation process described in Algorithm 1.

1 2 1 2 1 2 (a) (b) (c) (d) 1 2 (e) (f ) (g) (h)
(2) 2 a2 c2 True 0.0334 0.6500 0.4500 1.0000 0.1500 0.5000 0.6000 0.8000 0.1000 Estimated Φc 0.0334 0.6500 0.4502 1.0045 0.1498 0.4995 0.6006 0.7947 0.0998 Reverse order Φ c 0.0576 0.5070 0.5930 0.7072 0.0956 0.6478 0.4525 1.0108 0.1527 Additive Φa 0.0548 0.6496 0.4507 1.0262 0.1501 0.5172 0.5827 0.7352 0.1004

Estimation of GRID Component for Image Data

This method is applied to analysis of differences in human brains observed using MRI scans. In this case we perform the following experiment. We take two image scans as I and J, which are two slices of MRI of the same z coordinates from different subjects, and use a shape-based technique [START_REF] Kurtek | Parameterization-invariant shape comparisons of anatomical surfaces[END_REF] to estimate a cumulative deformation Φ from I to J. This Φ is in the set of diffeomorphisms from [0, 1] 2 to itself and minimizes a certain cost function involving I • Φ and J. Then, we use Algorithm 1, to estimate components of this deformation under the GRID model. We present two examples of this idea.

Example 1: Fig. 5 shows an example of images I and J and the deformation Φ that deforms I to match with J. In addition to plotting the maps x → Φ(x) as a surface mesh, one can also plot the displacement vector field Ψ (x) = Φ(x)x for better visualization. Shown in the remaining panels are the estimated displacement vector Ψ , showing the ordered sequence of displacement fields with the GRID components that were found using Algorithm 1, and a couple of ways of comparing the estimated deformations with the true deformations. Example 2: Fig. 6 shows another example of estimating GRID parameters for deformation estimated from MRI images.

Conclusion

In this paper we have proposed a method to decompose and estimate the parameters in GRID based decomposition of anatomical deformations. The method preserves the iterative structure of the GRID model and gives an analytical form of the gradient for parameter estimation. Experimental results show that impacts from faraway seeds can be approximated by additive seeds model and composite model can not add much to it; however, as for seeds that are close to each other and have interaction, our method improves the estimation for large deformation as well as the model parameters.

A Gradient Expressions

The terms, ∂E (j) ∂φ (j) ∈ R 1×d , ∂φ (j) ∂φ (j-1) ∈ R d×d , ∂φ (j) ∂ξj ∈ R d×d , ∂φ (j) ∂aj , and ∂φ (j) ∂cj ∈ R d×1 , that are required in gradient computation are given as follows. ∂E (j) ∂φ (j) = 2 φ (j) -Φ , ∂φ (j) ∂φ (j-1) = a j-1 • expφ (j-1)ξ j-1 2 /c 2 j-1 -2a j-1 • (φ (j-1)ξ j-1 ) 2 /c 2 j-1 • expφ (j-1)ξ j-1 2 /c 2 j-1 + 1 , ∂φ (j) ∂ξ j = a j • expφ (j-1)ξ j 2 /c 2 j

• 2 φ (j-1)ξj 2 /c 2 j -1 , ∂φ (j) ∂a j = (φ (j-1)ξ j ) • expφ (j-1)ξ j 2 /c 2 j , ∂φ (j) ∂c j = 2a j • expφ (j-1)ξ j 2 /c 2 j

• φ (j-1)ξ j 3 /c 3 j .

When the ADF takes parametric forms of the second type, the gradient of the energy function with respect to the two parameters κ j and τ 0,j for j = 1, • • • , k is calculated in the similar way as the other parameters. The partial derivatives from the one time deformation is shown as below. ∂φ (j) ∂κ j = a j • exp (κ j cos (ττ 0,j )κ j ) • φ (j-1)ξ j • expφ (j-1)ξ j 2 /c 2 j

• (cos(ττ 0,j ) -1) , ∂φ (j) ∂τ 0,j = a j • exp (κ j cos (ττ 0,j )κ j ) • φ (j-1)ξ j • expφ (j-1)ξ j 2 /c 2 j • κ j • sin (ττ 0,j ) .

Fig. 1 :

 1 Fig. 1: Cumulative deformation for composite models with: (a) Φ = φ 2 • φ 1 , (b) Φ = φ 1 • φ 2 , and additive model (c) Φ = x + ψ 1 + ψ 2 , for the same φ 1 and φ 2 .In the remaining panels, we show Φ -Φ (4th panel), and Φ -Φ (5th panel) as vector fields.

Fig. 2 :

 2 Fig. 2: Experiment with isotropic ADF: (a) synthetic deformation; (b)noisy observation Φ, and different estimated deformations (c) Φ c ; (d) Φa , and (e) Φc . The differences (f) Φc -Φ, (g) Φ c -Φ, and (h) Φa -Φ

Example 2 :

 2 Similarly, Fig.3(a)shows an experiment involving seeds with nonisotropic ADFs. In this case we show the synthetic deformation (a) , its noisy version Φ (b), and the estimated deformation under the composite model Φc (c). For further evaluation of this estimation, we show the true displacement Ψ (x) = Φ(x)x (d) and the estimated displacement Ψ (e).

Fig. 3 :

 3 Fig. 3: Experiment with non-isotropic ADF: (a)synthetic deformation; (b) noisy observation Φ; (c) estimated deformation map Φ; (d)true displacement field Ψ ; (e)estimated displacement field Ψ

Ensemble Results:Fig. 4 :

 4 Fig. 4: (a) Relative errors under two models:Φa-Φ -Φc-Φ Φa-Φ

Fig. 5 :Fig. 6 :

 56 Fig. 5: Estimation of GRID components for deformation between MRI images.

Table 1 :

 1 Estimation results for Example 1.

Table 2 :

 2 Estimation Results for Example 2.

			1	a1	c1	ξ	(1) 2	ξ	(2) 2	a2	c2
	True	0.0330 0.6700 0.4000 1.0000 0.2000 0.3200 0.6000 1.0000 0.2000
	Composite 0.0333 0.6677 0.4014 1.0004 0.1975 0.3233 0.5980 1.0248 0.1948
		κ1	τ0,1			κ2	τ0,2
	True	2.0000 2.3562			5.0000 5.4978
	Composite	1.9200 2.3379			4.9050 5.4865


Sabri Boubaker
email: sabri.boubaker@groupe-esc-troyes.com

Hind Sami
email: hind.sami@univ-lyon2.fr

Multiple Large Shareholders and Earnings Informativeness

Keywords: Multiple large shareholders, earnings informativeness, excess control

Purpose -The purpose of this paper is to add to our understanding of the monitoring role of multiple large shareholders by examining their impact on the informativeness of firms' earnings. Design/methodology/approach -We use regression models that relate earnings to stock returns for a sample of 402 French publicly traded firms covered during 2003-2007. Findings -We show that earnings informativeness is significantly positively related to the owner's ultimate cash flow rights. Consistent with the alignment effect, stock ownership aligns management and shareholders interests which reduces managers' incentives to manipulate accounting information. We also find that earnings informativeness is significantly negatively related to the excess control of the ultimate controlling shareholder. This result supports the entrenchment effect and suggests that controlling shareholders have greater incentives to obscure accounting figures when expropriation is likely. Finally, control contestability of the largest controlling shareholder mitigates information asymmetry problems thereby enhancing earnings informativeness. Limitations/implications -Our findings stress the importance of MLS in enhancing internal monitoring and mitigating agency costs. Because France is characterized by a weak legal system, highly concentrated ownership structures and excess control, our results provide valuable insights to mitigate extreme agency problems. Originality/value -The paper adds to the literature on corporate governance and the quality of accounting information by investigating strategic interactions between various blockholders and their impact on earnings informativeness. The study complements prior studies on the monitoring role of MLS by demonstrating that both their presence and control size translate into significantly greater earnings informativeness.

Introduction

Information is the lifeblood of financial markets. It is well recognized that the quality of corporate disclosure influences to a great extent the quality of investment decisions. Potential benefits of more disclosure include lower cost of capital [START_REF] Diamond | Disclosure, Liquidity, and the Cost of Capital[END_REF], agency cost reduction [START_REF] Leftwich | Voluntary Corporate Disclosure: The Case of Interim Reporting[END_REF] and improved share price [START_REF] Gelb | Corporate Disclosure Policy and the Informativeness of Stock Price[END_REF][START_REF] Lang | Voluntary Disclosure and Equity Offerings: Reducing Information Asymmetry or Hyping the Stock?[END_REF]. To the extent that management has discretion over the quality of reported accounting information, a large literature on corporate governance has focused on the relation between managerial ownership and the quality of accounting information [START_REF] Warfield | Managerial Ownership, Accounting Choices and Informativeness of Earnings[END_REF].

Departing from [START_REF] Berle | The Modern Corporation and Private Property[END_REF] seminal work on widely diffused corporate ownership, recent studies have focused on the conflicts of interest between controlling shareholders and minority shareholders [START_REF] Villalonga | How do family ownership, control and management affect firm value?[END_REF][START_REF] Ali | Corporate disclosures by family firms[END_REF]. Specifically, when minority shareholders face possible expropriation by large controlling shareholders, the contest to the control is a key issue (See, e.g., [START_REF] Shleifer | A Survey of Corporate Governance[END_REF][START_REF] Porta | Corporate Ownership around the World[END_REF][START_REF] Faccio | The Ultimate Ownership of Western European Corporations[END_REF][START_REF] Maury | Multiple Large Shareholders and Firm Value[END_REF].

The present paper contributes to prior research by further examining the impact of the distribution of ownership and the contest to the control on earnings informativeness in a concentrated ownership setting. More specifically, this study seeks to add to the understanding of the monitoring role of multiple large shareholders (MLS) by examining their impact on the informativeness of accounting earnings. Our study is carried out on a sample of 402 French publicly traded firms covered during the 2003-2007 period. France is characterized by a weak legal system and poor investors' protections laws which may in turn give rise to minority expropriation [START_REF] Porta | Corporate Ownership around the World[END_REF]. Indeed, French listed firms have highly concentrated ownership structures and control is enhanced via devices such as double voting shares and pyramiding. The control of French firms is often ensured through a relatively small proportion of cash flow rights compared to that of control rights [START_REF] Faccio | The Ultimate Ownership of Western European Corporations[END_REF][START_REF] Boubaker | Ownership Structure, Corporate Governance and Analyst Following: A Study of French Listed Firms[END_REF]. These characteristics increase the potential for extreme agency problems between the large controlling shareholders and minority shareholders. In this paper, we investigate strategic interactions between various blockholders in order to examine the effects of multiple large shareholders on firms' quality of earnings information.

First, focusing on agency problems between controlling and minority shareholders we explore whether having more than one large shareholder in the firm plays as a disciplining device which in turn may translate into greater informativeness of accounting earnings. Because we believe that control of ultimate owners is more relevant in assessing the outcome of agency conflicts we focus on the distribution of ultimate control rights. [START_REF] Warfield | Managerial Ownership, Accounting Choices and Informativeness of Earnings[END_REF] focus on the ownership incentive effects and find that earnings quality is generally increasing in the level of managerial ownership. Because stock ownership aligns management's interests with those of shareholders, it hence reduces or eliminates managers' incentives to manipulate accounting information. [START_REF] Fan | Corporate Ownership Structure and the Informativeness of Accounting Earnings in East Asia[END_REF], in turn, examine a sample of East Asian firms. They suggest that concentrated ownership is associated with low quality earnings since controlling shareholders may report selfinterestedly and may have incentives to disclose as little proprietary information as possible. We contribute to the unresolved debate by empirically examining the relation between the presence of large controlling shareholders and firm's earnings informativeness.

Second, we study the interaction between main large shareholders in concentrated ownership context by examining the effects of voting size and voting distribution among them on the quality of firm earnings. We thus contribute to the discussion of firm's agency costs stemming from the misalignment between control and ownership rights of ultimate shareholders (see e.g. [START_REF] Claessens | Disentangling the Incentive and Entrenchment Effects of Large Shareholdings[END_REF]. [START_REF] Fan | Corporate Ownership Structure and the Informativeness of Accounting Earnings in East Asia[END_REF] show that due to the entrenchment effect, the credibility of the firm's accounting information is reduced which in turn lowers the stock price informativeness of the earnings. This loss of earnings credibility is thus increasing in the degree of divergence between the cash flow rights and voting rights [START_REF] Francis | Earnings and Dividend Informativeness when Cash Flow Rights are Separated from Voting Rights[END_REF]. We extend previous works by providing evidence on the governance impact of MLS on nature and the strength of the relationship between accounting earnings and stock return. More precisely, considering the ownership and control divergence (or equivalently excess control), we analyze whether the presence of MLS may alleviate agency problems with minority shareholders thereby improving earnings informativeness.

Finally, we investigate the impact of shared control on the informativeness of the firm's earnings. Corporate governance literature agrees that the existence of large shareholders (beyond the largest controlling shareholder) can achieve a valuable internal monitoring function. Empirical studies find that the presence of multiple controlling shareholders might be value enhancing [START_REF] Lehmann | Does the Governed Corporation Perform Better? Governance Structures and Corporate Performance in Germany[END_REF][START_REF] Volpin | Governance with Poor Investor Protection: Evidence from Top Executive Turnover in Italy[END_REF][START_REF] Maury | Multiple Large Shareholders and Firm Value[END_REF]. Higher control contestability prevents opportunistic self-serving behavior thereby lessening minority expropriation [START_REF] Gomes | Sharing of Control as a Corporate Governance Mechanism[END_REF], reduces the appropriation of private benefits of control [START_REF] Bloch | Multiple Shareholders and Control Contests[END_REF]Gutiérrez and Tribo, 2004), reduces rent extraction (Bennedson and Wolfenzon, 2000), enhances the value of firm's excess cash holdings [START_REF] Laeven | Complex Ownership Structures and Corporate Valuations[END_REF][START_REF] Attig | Do Multiple Large Shareholders Play a Corporate Governance Role? Evidence from East Asia[END_REF] and is associated with a lower cost of equity capital for firms [START_REF] Guedhami | Excess Control, Corporate Governance, and Implied Cost of Equity: International Evidence[END_REF]. We extent this line of research by identifying a channel through which high control contestability of the largest controlling shareholder (or equivalently the presence of MLS) might enhance a firm's information quality and thus greater earnings informativeness.

Overall, our results consolidate the findings of who provide compelling evidence that control contestability may limit the negative effects associated with the presence of a single large shareholder, such as earnings manipulation and extraction of private benefits. More specifically, our study of how the presence of multiple large shareholders affects the informativeness of firms' accounting earnings generates several valuable insights. First, we find that earnings informativeness measured by the cumulative net-of-market 12-month stock returns (CAR), is significantly positively related to the ultimate cash flow rights. This evidence is consistent with the alignment effect whereby stock ownership aligns large shareholder's interests with those of minority shareholders, which in turn encourages transparency and reduces incentives to manipulate accounting information. We also find that earnings informativeness is significantly negatively related to the divergence between ultimate cash flow rights and control rights. This result is consistent with arguments advocating that excess control of the ultimate controlling shareholder is conducive to self-serving behavior (see e.g. [START_REF] Ball | Incentives versus Standards: Properties of Accounting Income in Four East Asian Countries, and Implications for Acceptance of IAS[END_REF][START_REF] Francis | Earnings and Dividend Informativeness when Cash Flow Rights are Separated from Voting Rights[END_REF][START_REF] Attig | Effects of Large Shareholding on Information Asymmetry and Stock Liquidity[END_REF]. Our result lends support to the entrenchment effect suggesting that controlling shareholders have greater incentives to obscure financial figures using, among other earnings management, untimely disclosure, etc., to hide their diversion activities when the expropriation likelihood is substantial (i.e. excess control is high). Our findings also show that the presence of multiple large shareholders and the contestability of control are related to significantly greater earnings informativeness. This result is consistent with the monitoring role of MLS and supports the findings of Bennendsen and Wolfenzon (2000), [START_REF] Gomes | Sharing of Control as a Corporate Governance Mechanism[END_REF], [START_REF] Attig | Multiple Large Shareholders, Control Contests, and Implied Cost of Equity[END_REF], and [START_REF] Guedhami | Excess Control, Corporate Governance, and Implied Cost of Equity: International Evidence[END_REF]. Indeed, we find that control contestability of the largest controlling shareholder mitigates information asymmetry problems thereby enhancing the informational environment of the firm.

The remainder of the paper is organized as follows. Section 2 details the development of our hypotheses. Section 3 describes data and sample selection procedure. Section 4 presents the methodology. Section 5 reports the empirical results. The penultimate section offers robustness checks and the last section concludes the paper.

Development of hypotheses

Ultimate cash flow rights

The effect of ownership structures on firm value has been extensively discussed. In publicly listed corporations, the separation between ownership and control may lead to conflicts of interest between financiers and managers who run the company [START_REF] Fama | Separation of ownership and control[END_REF][START_REF] Jensen | Theory of the Firm: Managerial Behavior, Agency Costs, and Ownership Structure[END_REF]. These agency conflicts can be mitigated by monitoring. Investors with large ownership have strong incentives and enough control over the firm's assets to monitor the owner manager thereby pursuing firm's value maximization [START_REF] Berle | The Modern Corporation and Private Property[END_REF][START_REF] Demsetz | The Structure of Corporate Ownership: Causes and Consequences[END_REF][START_REF] Shleifer | Large Shareholders and Corporate Control[END_REF][START_REF] Admati | Large Shareholder Activism, Risk Sharing, and Financial Market Equilibrium[END_REF]. [START_REF] Faure-Grimaud | Public Trading and Private Incentives[END_REF] argue that large shareholders have incentives to engage in value-increasing activities since stock prices convey information to outsiders through public trading. Shareholders' incentives to pursue value-increasing activities translate in turn into higher firm valuation [START_REF] Claessens | Disentangling the Incentive and Entrenchment Effects of Large Shareholdings[END_REF]. La [START_REF] Porta | Investor protection and corporate valuation[END_REF] also support the incentive effect by showing that higher cash flow ownership by the controlling shareholder increases firm value as measured by the Tobin's Q. Thus, large investors provide a solution to the free-rider problem (see e.g. [START_REF] Grossman | Takeover Bids, the Free-rider Problem, and the Theory of the Corporation[END_REF][START_REF] Maug | Large Shareholders as Monitors: Is there a Trade-off between Liquidity and Control?[END_REF]. [START_REF] Gomes | Going Public Without Governance: Managerial Reputation effects[END_REF] argues that reputation concerns of the controlling shareholder plays as a credible commitment device not to expropriate minority shareholders. As ownership stakes increase the large shareholders' interests become aligned with those of small investors. In addition to high incentives to help overcome the conflicts of interest between shareholders and managers, large shareholders may coordinate their actions and put pressure on managers since voting power is not split among a highly segmented group of investors. Therefore, large ownership levels not only give incentive to decrease agency costs, but also the power to do so [START_REF] Shleifer | A Survey of Corporate Governance[END_REF][START_REF] Kaplan | Appointments of Outsiders to Japanese Boards: Determinants and Implications for Managers[END_REF]. The presence of large controlling shareholders translates into strong incentives to monitor and discipline managers thereby replacing legal constraints on managerial opportunism [START_REF] Porta | Corporate Ownership around the World[END_REF][START_REF] Boubakri | Investor Protection, Corporate Governance and Privatization[END_REF]. Drawing on the above arguments, we argue that large controlling shareholders have strong incentives to encourage management to report high quality earnings. [START_REF] Warfield | Managerial Ownership, Accounting Choices and Informativeness of Earnings[END_REF] focus on the incentive effects and find that earnings quality is generally increasing in the level of managerial ownership while the magnitude of discretionary accounting accrual adjustments decreases. Because stock ownership aligns management's interests with those of shareholders, this reduces or eliminates managers' incentives to manipulate accounting information. The relation between earnings informativeness and managerial ownership is not linear. The relation is increasing in small to medium levels of managerial ownership while decreasing in high levels of ownership (see e.g. [START_REF] Yeo | Corporate Ownership Structure and the Informativeness of Earnings[END_REF]. Studying a sample of Japanese firms, [START_REF] Yafeh | Large Shareholders and Banks: Who Monitors and How?[END_REF] uncover evidence that large ownership levels are associated with lower discretionary spending. Extending their reasoning we expect that owning a large fraction of firm shares should increase earnings informativeness. The previous discussion provides an opportunity to further explore this relation through the following hypothesis:

Hypothesis 1: The informativeness of accounting earnings increases with the ultimate cash flow rights of the controlling shareholder.

Divergence between ownership and control

The ownership structure of French listed companies is characterized by the presence of large controlling shareholders who possess more control than their equity ownership indicates. Control concentration enhanced beyond ownership stakes may depart the interests of controlling shareholders from those of minority investors (see e.g. La [START_REF] Porta | Investor protection and corporate governance[END_REF]. Extensive research documents that excess control built through dual-class shares, pyramidal structures or cross-holding among firms enables controlling shareholders to accrue private benefits thereby expropriating minority shareholders [START_REF] Porta | Corporate Ownership around the World[END_REF][START_REF] Claessens | The Separation of Ownership and Control in East Asian Corporations[END_REF][START_REF] Bertrand | Ferreting out Tunneling: An Application to Indian Business Groups[END_REF][START_REF] Faccio | The Ultimate Ownership of Western European Corporations[END_REF]. As such, the contest to the control of the dominant shareholders stems from the conflict between controlling and non-controlling shareholders [START_REF] Ali | Corporate disclosures by family firms[END_REF].

The likelihood of minority shareholder expropriation is particularly high if large investors hold voting rights in excess of cash flow rights [START_REF] Faccio | Who Disciplines Management in Poorly Performing Companies?[END_REF]. In these cases, they have an incentive to pay out a larger proportion of company cash flows to themselves instead of evenly distributing funds among all shareholders. [START_REF] Francis | Earnings and Dividend Informativeness when Cash Flow Rights are Separated from Voting Rights[END_REF] find that earnings in dual class firms (i.e. firms characterized a separation of cash flow rights from voting rights) are less informative than earnings in single class firms, suggesting that the separation of control right and ownership is negatively related to informativeness of accounting earnings. [START_REF] Fan | Corporate Ownership Structure and the Informativeness of Accounting Earnings in East Asia[END_REF] argue that East Asian firms with a discrepancy between voting rights and cash flow rights are perceived to provide controlling shareholders incentives to take self-interested actions causing the reported earnings to lose credibility to outside investors. In other words, there is a credibility-reducing effect of entrenchment which is increasing in the degree of divergence between cash flow rights and voting rights. Evidence on the effect of the divergence between ownership and control is provided, among many others, by [START_REF] Claessens | Disentangling the Incentive and Entrenchment Effects of Large Shareholdings[END_REF]. Studying East Asian economies, they find a positive and significant relationship between the cash flow ownership of the largest shareholder and firm value and interpret this result as a positive incentive effect of large shareholdings. In light of the above arguments, we draw the following hypothesis: Hypothesis 2: Earnings informativeness decreases with the separation between ultimate cash flow rights and control rights.

Multiple controlling shareholders

Private benefits of control imply conflicts of interests that might be alleviated by relying on mechanisms capable of protecting the rights of minority shareholders. A large body of literature has emphasized the monitoring role that may be played by outside shareholders (see e.g., [START_REF] Shleifer | Large Shareholders and Corporate Control[END_REF][START_REF] Kahn | Ownership Structure, Speculation, and Shareholder Intervention[END_REF][START_REF] Bloch | Multiple Shareholders and Control Contests[END_REF][START_REF] Noe | Investor Activism and Financial Market Structure[END_REF]. Outside investors may lack incentives to monitor controlling owners, since they would bear all the monitoring costs but capture only a small portion of the monitoring gains (see [START_REF] Grossman | Takeover Bids, the Free-rider Problem, and the Theory of the Corporation[END_REF]. [START_REF] Burkart | Large Shareholders, Monitoring, and the Value of the Firm[END_REF] and [START_REF] Bolton | Blocks, Liquidity and Corporate Control[END_REF] argue, however, that outside investors have incentives to increase the liquidity costs without offering compensating advantages in monitoring, due to shareholders liquidity-control trade-off. Because the monitoring of investment decisions by outside investors fails to solve conflicts of interest, related themes of multiple large shareholders have also been visited as alternative mechanisms that may protect minority shareholders. More specifically, an ownership structure with multiple controlling shareholders may play as a commitment device that allows reaching an optimal monitoring intensity [START_REF] Pagano | The Choice of Stock Ownership Structure: Agency Costs, Monitoring, and the Decision to Go Public[END_REF].

The presence of several large shareholders may then be associated with valuable monitoring, in particular when their share investments are evenly distributed. In other words, multiple blocks commit the firm to protect minority investors. Yet, a number of papers argue that multiple blockholders are unlikely to emerge. In [START_REF] Zwiebel | Block Investment and Partial Benefits of Corporate Control[END_REF] general equilibrium model, investors are sorting such that only one of them holds a block in any given firm, precisely because they want to eschew the sort of competition over benefits. [START_REF] Winton | Limitations of liability and the ownership structure of the firm[END_REF] emphasizes the free-rider problem in monitoring efforts among multiple large shareholders.

Contrariwise, some authors contend that the presence of multiple large shareholders may ensure the eschewal of expropriation. In a context where multiple large shareholders compete to gather minority votes as the latter is pivotal to seize control, [START_REF] Bennedsen | The Balance of Power in Closely Held Corporations[END_REF] show that it is optimal to have as much shareholders as possible in a controlling coalition. This coalition with the lowest possible cash flow stake will have the largest group of non-controlling shareholders from whom to expropriate which is considered as a bad outcome in terms of efficiency. Furthermore, the greater the cash flow possessed by this coalition, the more the coalition internalizes the costs of private benefits extraction and imposes limits on minority expropriation. [START_REF] Gomes | Sharing of Control as a Corporate Governance Mechanism[END_REF] consider the trade-off that may exist between an inefficient monitoring of outside investors and the bargaining costs associated with the existence of multiple controlling shareholders. Their theoretical model allows solving this trade-off and shows that sharing control among large shareholders may increase efficiency in firms with investment opportunities that are hard to evaluate by outsiders. The presence of multiple large shareholders is seen as a device capable of limiting rent extraction, and securing the vote of minority shareholders when the different large shareholders' shares are fairly similar [START_REF] Bloch | Multiple Shareholders and Control Contests[END_REF]. Hence, all large shareholders perform a monitoring role and compete for the votes of minority shareholders by committing to reduce their private benefits.

In line with these arguments, empirical studies were conducted to derive evidence on the impact of the presence of multiple large shareholders. [START_REF] Zwiebel | Block Investment and Partial Benefits of Corporate Control[END_REF] considers a setting in which ownership structure is defined by shareholders who allocate their wealth across firms in order to receive a larger share of private benefits and shows that the control benefits will be divided among the multiple large shareholders depending on the relative size of their respective shares. [START_REF] Maury | Multiple Large Shareholders and Firm Value[END_REF] argue that the presence of several large shareholders should have a positive effect on minority investors and firm's efficiency. Contestability of the controlling coalition's power might increase firm value and limit the expropriation of minority shareholders. Moreover, the marginal cost of stealing increases with the number of coalition partners thereby discouraging the extraction of private benefits and diversion of profits. Indeed, high control contestability of the largest controlling shareholder might enhance a firm's information quality and thus greater earnings informativeness. This result is consolidated by the findings of [START_REF] Attig | Do Multiple Large Shareholders Play a Corporate Governance Role? Evidence from East Asia[END_REF] who provide compelling evidence that the distribution of control rights among multiple large shareholders alleviate agency costs of firm's liquid assets. Their results indicate that an increase in the contestability of the largest controlling shareholder's control -as a result of the higher relative voting power of the second and third largest shareholders or the pivotal voting power of the minority interests-may enhance the value of firm's excess cash holdings, since it will be harder to convert liquid assets into private benefits. In line with [START_REF] Laeven | Complex Ownership Structures and Corporate Valuations[END_REF], this result indicates that a firm with uneven distribution of voting rights among the controlling shareholders is likely to have more serious agency problems that may lead to cash diversion. In turn, the existence of large shareholders (beyond the largest controlling shareholder) can achieve valuable internal monitoring [START_REF] Guedhami | Excess Control, Corporate Governance, and Implied Cost of Equity: International Evidence[END_REF]. [START_REF] Attig | Multiple Large Shareholders, Control Contests, and Implied Cost of Equity[END_REF] document that when other large shareholders intervene in the preparation of financial information concealing or manipulating earnings by the largest shareholder becomes costly and more difficult. A high control contestability of the largest controlling shareholder is then a device to protect minority shareholders plausibly because it alleviates the diversion of firm's resources for private benefits. The intuition behind this result is similar to that in Gutiérrez and Tribo (2004) who find that the presence of more than one controlling shareholder substantially decreases private benefit extraction because of the bargaining problem that occurs among large shareholders who are forced to share control.

Overall, control contestability may downgrade the negative effects of a single large shareholder, such as the presence of earnings manipulation, low disclosure quality and extraction of private benefits. Based on these arguments we propose the following hypothesis: Hypothesis 3: Earnings informativeness increases when control is contestable.

Data Sources and sample selection procedure

The initial sample is restricted to all French firms available in the Worldscope database for the 2003-2007 period. Following previous studies, we remove regulated utilities (SIC code 4900-4999) and financial firms (SIC code 6000-6999) due to the specificities of their informational environment which is more likely to be affected by legal and regulatory motives rather than by agency concerns. We discard all unlisted firms, those listed since less than one year and those with incomplete financial and/or return data. We also exclude firms for which we were unable, due to missing ownership data, to trace control chains back to the ultimate controlling owners. Applying the above-mentioned criteria, we were left with a sample of 402 firms corresponding to 2,384 firmyear observations (See Table 1, for detailed definition of all variables used in the analysis and Table 2 for more details on sample selection criteria and on distribution of firms on a year-by-year basis).

Financial data and stock price related data are retrieved from the databases Worldscope and Datastream, respectively. Data on ownership structures were excerpted from firms' annual reports which provide the list of all shareholders with more than five percent of the ownership or voting stake. Annual reports are available from the AMF's website (Autorité des Marchés Financiers).

Methodology

The effect of ownership structure

To test the effect of multiple large shareholders on earnings informativeness, we adopt the following regression model:

CAR it = α 0 + α 1 (NI it ) + α 2 (NI it * UCF it ) + α 3 (NI it * Excess Control it ) + α 4 (NI it * MLS Variables it ) + α 5 (NI it * Firm Size it ) + α 6 (NI it * Q it ) + α 7 (NI it * Leverage it ) + α 8 (NI it *Industry diversification it ) + (Fixed Effects) + ǫ it
where, for sample firm i, CAR it is the cumulative net-of-market 12-month stock returns ending at the fiscal year-end t (winsorized at 1st and 99th percentile); NI it is the net earnings at the end of the fiscal year t scaled by the market value of equity at the beginning of the fiscal year t (winsorized at 1st and 99th percentile); UCF is the ultimate cash flow rights at the 10% threshold. It is measured as the sum of products of direct cash flow rights along the different ownership chain; Excess Control is a proxy for the separation between ultimate cash flow and control rights. It is computed as the difference between the ultimate control stake (UCO) and the ultimate cash flow stake (UCF), all divided by the ultimate control stake ((UCO-UCF)/UCO). UCO is the ultimate control rights at the 10% threshold. It is the sum of weakest links along the different control chains. MLS Variables are various proxies for the presence and relative voting power of large controlling shareholders beyond the dominant owner. They are included one at once depending on the specification. Among the MLS Variables we include the following measures. First, MLDS is the multiple large shareholder dummy variable. It equals one if there are at least two large shareholders who hold each more than 10% of the control rights, and zero otherwise. Second, VOTE21 is the ratio of control rights of the second largest shareholder to control rights of the largest shareholder, VOTE2/VOTE1.

Table 1

Definitions and sources of variables

Variable

Definition Source

CAR

The cumulative net-of-market 12-month stock returns. The annual returns are continuously compounded from weekly stock returns starting from 52 weeks (12 months) before the latest date that the firm discloses its annual report.

Datastream

NI

The net earnings at year t divided by the market value of equity at the beginning of the fiscal year.

Wordscope UCF Ultimate Cash Flow rights (at the 10% threshold): the sum of the products of direct cash flow rights along the different ownership chains.

Authors' calculations based on annual reports

Excess control

Excess Control (at the 10% threshold) is the difference between the LCS's ultimate control rights (UCO) and ultimate cash flow rights (UCF), all divided by his/her ultimate control rights (UCO) (i.e., (UCO -UCF) / UCO). UCO is the sum of the weakest links along the different control chains (at the 10% threshold)

As above MLSD Multiple large shareholder dummy variable equals one if there are at least two large shareholders who hold each more than 10% of the control rights, and zero otherwise.

As above

VOTE21

Ratio of control rights of the second largest shareholder to control rights of the largest shareholder, VOTE2/ VOTE1.

As above

VOTE231

Ratio of voting rights of the second and third largest shareholders to voting rights of the largest shareholder, (VOTE2+ VOTE3)/ VOTE1.

As above

HERFDVOTE

The natural logarithm of the Herfindhal index of the differences in the voting rights of two successive large shareholders, (VOTE1-VOTE2) 2 +(VOTE2-VOTE 3) 2 .

As above

SIZE

The natural logarithm of the market value of equity (ln(MVE)) in millions of Euros at the beginning of the fiscal year.

Wordscope

Q

The market value of equity divided by the book value of total assets at the beginning of the fiscal year.

As above

Leverage Total liability divided by total assets at the beginning of the fiscal year.

As above

Industry diversification

Number of distinct business segment(s) at the two-digit SIC code level in which the firm operates.

As above

Industry dummies Industry dummies following [START_REF] Campbell | Understanding risk and return[END_REF] We introduce a set of variables as surrogates of firm characteristics to control for observed variations in the relation between earnings and returns that are likely due to causes other than those related to firm ownership structure. We rely on previous research in considering the following variables that may affect firms' earnings informativeness. Firm size was found by [START_REF] Bamber | Unexpected Earnings, Firm Size and Trading Volume around Quarterly Earnings Announcements[END_REF] and [START_REF] Freeman | The Association between Accounting Earnings and Security Returns for Large and Small Firms[END_REF] to be positively associated with the informational content of earnings announcements. We include it in our regressions to control for its potential effect on earnings informativeness. One explanation for this positive association is that large firms have more informative security prices due to more timely, alternative information being available as compared to small firms [START_REF] Atiase | Predisclosure Information, Firm Capitalization and Security Price Behavior around Earnings Announcements[END_REF]. Thus, we include the natural logarithm of the market value of equity in millions Euros at the beginning of the fiscal year t (Firm size) to control for the effects of firm size on the earnings-return relation.

We also use, Q, the ratio of the market value of equity divided by the book value of total assets at the beginning of the fiscal year t (winsorized at 1st and 99th percentile) to control for the impact of growth prospects on the earnings-return relation. This effect is ambiguous at least theoretically. On the one hand, high company growth opportunity translates into larger expected earnings growth. As investors stock price responses towards current earnings is significant, the stock price informativeness of firms' earnings increases. On the other hand, the market-to-book ratio is tied to firm risk. The risk associated with high growth firms and/or fast growing firms may translate into less informative earnings. Empirically, [START_REF] Collins | An Analysis of Intertemporal and Cross-sectional Determinants of Earnings Response Coefficients[END_REF] show a positive relation between investment opportunities and earnings transparency. [START_REF] Smith | The Investment Opportunity Set and Corporate Financing, Dividend and Compensation Policies[END_REF] show that the issuance of diluted securities is more frequent in firms with higher growth opportunities hence earnings information will be impounded better on the firm's stock prices.

We also include Leverage as a control for firms' default risk. Leverage negatively impacts bond assessment [START_REF] Dhaliwal | The Effect of the Default Risk of Debt on the Earnings Response Coefficient[END_REF] and earnings informativeness [START_REF] Billings | Revisiting the Relation between the Default Risk of Debt and the Earnings Response Coefficient[END_REF].

Therefore, we expect that highly levered firms have less informative earnings. Leverage is measured as the ratio of the book value of total liabilities divided by the book value of total assets at the beginning of the fiscal year t.

Finally, we include Industry diversification as the number of distinct business segments at the two-digit SIC code level in which the sampled firm operates. Prior studies show that a company's industry diversification has a detrimental effect on financial analysts' ability to forecast earnings [START_REF] Denis | Agency Problems, Equity Ownership and Corporate Diversification[END_REF]. Hence, a company's unrelated and related industry diversification may have negative impact on earnings informativeness. present both large dispersion across sampled firms with a median value of -5.41% and 5.49%, respectively. Summary statistics of ownership variables are in accordance with those provided by [START_REF] Faccio | The Ultimate Ownership of Western European Corporations[END_REF] and [START_REF] Boubaker | Ownership-Control Discrepancy and Firm Value: Evidence from France[END_REF]. They show that the control of French firms is often ensured through a relatively small proportion of cash flow rights compared to that of control rights. In fact, ultimate controlling owners hold, on average, 41.00% of the ultimate cash flow rights and also control significantly more control rights than those determined by their corresponding ownership stakes leading to an average excess control of 21.24%. This separation between ultimate cash flow rights and control rights is prevalent in roughly 81.08% of the sampled firms (1933 firm-year observations). The voting power of the largest controlling shareholder is, on average, four times higher than that of the second largest shareholders (VOTE21 = 25%). The Herfindahl index measures the dispersion of voting rights among main large shareholders. It shows, as other MLS proxies, systematic differences across sampled firms and ranges from a minimum of 0 (equal voting power between largest shareholders) to a maximum of 98% (one dominant shareholder). Firms that have more than one significant shareholder (MLSD=1) represent 42.20% of the sampled firms (1006 firmyear observations). Regarding size and leverage, firms vary from small-sized (ln(MVE) = 3.34) to large-sized (ln(MVE) =18.99) and from unleveraged (Leverage = 0.00%) to highly leveraged (Leverage = 97.91%). The median value of Q equals 0.72 suggesting that more than half of the sampled firms destroy value. As for the degree of industry diversification, the mean (median) of diversification is 2.82 (2), suggesting that on average French firms operate in three industries. Sample firms present different degrees of diversification, from specialized firms (singe-industry firms) to highly diversified firms. Note that we have winsorized variables whose values do not vary within reasonable ranges to avoid that our estimates will be unduly influenced by outlier observations.

Basic regression

As a starting point, we first estimate a basic simple linear regression model that relates earnings to stock returns in France:

CAR it = α 0 + α 1 (NI it ) + (Industry dummies) +ǫ it
where, the CAR it is the cumulative net-of-market 12-month stock returns ending at the fiscal year-end t. The annual returns are continuously compounded from weekly stock returns starting from 52 weeks (12 months) before the latest date that the firm discloses its annual report; NI it is the net earnings at the end of the fiscal year t scaled by the market value of equity at the beginning of the fiscal year t. Both variables are winsorized at 1st and 99th percentile. Industry dummies are constructed following [START_REF] Campbell | Understanding risk and return[END_REF] classification. ǫ it the error term at year t. Similar to prior research, the ordinary least squares regression results show that the estimated coefficients of earnings are positive and statistically significant at a 1% threshold level irrespective of whether the equation is estimated on a year by year basis or on the pooled sample of all years from 2003 to 2007. This implies that earnings have substantial informational content for French listed firms (See, Table 5).

Empirical results

The largest controlling shareholder and earnings informativeness

Following in the footsteps of [START_REF] Claessens | Disentangling the Incentive and Entrenchment Effects of Large Shareholdings[END_REF], we include ultimate cash flow rights (UCF) in all regressions to control for the incentive alignment effect of the largest shareholder and ultimate control rights in excess of cash flow rights (Excess Control) as to control for the extent of his/her entrenchment. Consistent with Hypotheses 1 and 2, the coefficients on UCF (α 2 ) and on Excess Control (α 3 ) are expected to be significantly different from zero. A positive estimate on α 2 will be evidence that a higher interest stake of the largest controlling shareholder is associated with more informative earnings which is consistent with the incentive effect. A negative estimate on α 3 will provide evidence that the earnings of firms exhibiting larger separation between ultimate cash flow rights and control rights are less informative that others. The empirical results are portrayed in Table 6. The coefficient estimates are from the pooled ordinary least squares regressions using the CAR as the dependent variable where the t-statistics are based on standard errors with [START_REF] White | A heteroskedastic-consistent Covariance Matrix Estimator and a Direct Test of Heteroskedasticity[END_REF] correction for heteroscedasticity and adjusted for clustering for observations at the firm level. In conformity with the first hypothesis, the coefficient UCF is positive and statistically significant at 1% confidence level. This suggests that higher cash flow stakes of the large controlling shareholder are associated with more convergence between his/her interests with those of minority shareholders.

Large shareholders are more inclined to opt for higher financial reporting quality and for more disclosure of material information in a timely and convenient manner which leads to greater informativeness of accounting earnings.

Consistent with our second hypothesis and in accordance with the entrenchment effect argument, the coefficient of Excess control is negative and statistically significant at the 1% confidence level. Higher separation between cash flow rights and control rights makes the market expect more entrenchment of the controlling shareholder who may prefer to establish delayed or misleading disclosure practices that hide his/her potential expropriation activities. This market perception reduces reliance on accounting earnings reports which weakens earnings informativeness.

Overall, results provide strong evidence that earnings are more informative in firms where ultimate owners hold large cash flow rights and less informative when cash flow right are separated from control rights which supports our first two hypotheses. All specifications in Table 6 are significant at least at 0.001 level. The adjusted R 2 values are about 0.07 and are comparable with those of relevant studies. The effects of growth opportunities on the relation between accounting earnings and market returns are controlled through the use of Q. The coefficient on this variable is positive indicating that higher growth potentials strengthen the earnings-return relation which is consistent with [START_REF] Jung | Ownership Structure and Earnings Informativeness: Evidence from Korea[END_REF]. Leverage is negatively associated with earnings informativeness. High leverage often implies higher likelihood of default risk suggesting loosely relationship between return and earnings. The coefficient on Industry diversification is negative suggesting that earnings are more informative in focused than in diversified firms. One possible explanation is that earnings in focused firms are generated through a relatively less complex process. Firm size, proxied by the natural logarithm of the market value of equity, does not seem to affect the extent of earnings informativeness in France.

Multiple controlling shareholders and earnings informativeness

The main focus of this study is to investigate the impact of multiple large shareholders on the informativeness of the firms' earnings. To the extent that multiple controlling shareholders play an efficient-monitoring role and limit the agency costs associated with the presence of a dominant owner, we expect that earnings informativeness increases with the presence of more than one controlling shareholder (MLSD=1), when the power of the remaining large shareholders (Vote21 and Vote231) is substantial and when the concentration of differences in voting rights of two successive large shareholders (Herfdvote) is low. Lower values of Herfdvote correspond to situations where controlling shareholders have comparable voting stakes. We expect that more contestable control (i.e., lower Herfdvote) reduces the likelihood of private benefits extraction, increases the commitment to a more consistent disclosure policy and thereby enhances the informative content of firm earnings.

Table 6 displays regression results of multivariate regression with MLS-related proxies as independent variables and the CAR as the dependent variable. The coefficient on net income (NI) in specification 1 (Table 6) is 0.1590 and significant at p < 0.05. The coefficient on our key test variable (NI*MLSD) is 0.1149 with a p-value less than 0.01. This suggests that, on average, the earning response coefficient of firms with multiple large shareholders is 72.26% (0.1149/0.1590 = 0.7226) higher than that for firms controlled by a unique large shareholder. This result is economically significant using the 10% rule of thumb for materiality. MLS seem to play an important governance role in limiting rent-seeking behavior. Their presence enhances the quality of accounting earnings and increases confidence on reported figures which in turns strengthens the relationship between stock prices and accounting of earnings.

In the second specification, we use a continuous measure of the relative power of the second largest controlling shareholder versus the largest shareholder (Vote21). Consistent with the precedent result, the coefficient on this variable (NI*VOTE21) is 0.1989 with a p-value less that 0.01, which suggests that earnings are more informative in firms where control is contestable. If the relative power increases by 1%, the earnings response coefficient will be 1.2975% (0.1989%/0.1533 = 1.2975%) higher every thing else being equal. Results remain qualitatively similar but significant at only 5% when we consider the relative power of the second and third largest shareholders to voting rights of the largest shareholder (Vote 231).

The results from the last specification in Table 6 confirm our previous conclusions. The coefficient on (NI*Herfdvote) is negative and statistically significant at the 1% level. This shows that the presence of controlling shareholders with comparable distribution of voting power curbs the extraction of private gains, hence reducing the accounting earnings management needed to veil potential opportunistic behavior and leading to better earnings information.

Overall, the results in specifications 1 through 4 show that the presence and voting size of large controlling blockholders beyond the dominant owner are associated with higher-quality earnings. These findings lend support to the argument that multiple large shareholders play a valuable governance role in mitigating the agency costs that arise between the controlling shareholder and minority shareholders.

Check of robustness

This section is dedicated to check the robustness of our results by conducting several sensitivity tests. Only the results of some of them are reported for sake of brevity. First, we rerun all regressions by interacting MLS related variables with Excess control high which is a dummy variable that takes on a value of 1 for all observations where the value of Excess control is greater than the median of the sample, and zero otherwise. By doing so, we test whether multiple controlling shareholders are efficient in alleviating extreme agency problems that may arise from the conflict of interest between the largest controlling shareholder and minority shareholders and hence whether their presence strengthens the relation between accounting earnings and stock prices.

Consistent with previous findings, the coefficient on the interaction terms between Excess control high on one hand and MLSD, VOTE21 and Vote231 from the other hand are positive and statistically significant at least at 10% threshold level whereas the coefficient on the interaction term with HERFDVOTE is negative and statistically significant at the 10% threshold level (See Table 7). These findings show that the negative effect of the separation between ultimate cash flow rights and control rights on earnings informativeness turns to be positive in the presence of multiple controlling shareholders or when control is contestable. Hence, the presence of multiple controlling shareholders does matter in the presence of a large divergence between the interests of the largest controlling shareholder and those of minority shareholders. Second, six percent of the sample firms are with atomistic ownership structure where the relevant conflict of interest is that between shareholders and professional managers. Empirically, we have assigned the value of zero to the Excess control of these widely held firms. Excluding them from the analysis do not seem to qualitatively change our results. Third, to be considered as a controlled entity, a firm must have a controlling owner who maintains at least 10% of the control rights. We re-estimate our regression using a tighter control threshold (20% of the control rights). The results are qualitatively similar.

Fourth, we rerun regressions using cumulative abnormal returns calculated differently to check the sensitivity of our results to alternative dependent variables. The conclusions remain virtually the same when (1) we use cumulative raw stock returns instead of the cumulative net-of-market stock returns or (2) we compute the CAR over the 39 (26) weeks prior to and 13 (26) weeks after the current fiscal year end. Finally, results do not vary to the use of the change in earnings (∆NI) instead of earnings (NI) or to the inclusion of additional controlling variables such as firm age, affiliation to a pyramid or loss firms which is a dummy variable that take one for firms with negative net income.

Conclusion

Prior research provides evidence that there is a relationship between ownership structure and the quality of firm earnings (see e.g., [START_REF] Fan | Corporate Ownership Structure and the Informativeness of Accounting Earnings in East Asia[END_REF]. Using data on ultimate owners of French listed firms, this paper complements prior findings with respect to the effect of ultimate cash flow rights and excess control on the quality of firms' accounting information. It also extends this line of research to shed light on the effects of the presence of multiple controlling shareholders on firms' earnings informativeness.

The current study documents several interesting findings. First, we show that earnings informativeness is significantly positively related to the ultimate cash flow rights of the owner. This evidence is consistence with the alignment effect whereby stock ownership aligns management's interests with those of shareholders, which in turn reduces managers' incentives to manipulate accounting information. Second, we find that earnings informativeness is significantly negatively related to the excess control of the ultimate controlling shareholder. This result supports the entrenchment effect argument and suggests that controlling shareholders have greater incentives to undertake activities that hide their egregious behaviour when the potential expropriation of minority investors is substantial. Finally, and more importantly, we document the monitoring role of MLS and show that the presence and control size of multiple large shareholders translate into significantly greater earnings informativeness. Control contestability of the largest controlling shareholder mitigates information asymmetry problems thereby enhancing firms' earnings information.

  classification. As above Third, VOTE231 is the ratio of voting rights of the second and third largest shareholders to voting rights of the largest shareholder, (VOTE2+VOTE3)/VOTE1. Finally, HERFDVOTE is the Herfindhal index of the differences in the voting rights of two successive large shareholders, (VOTE1-VOTE2) 2 +(VOTE2-VOTE3) 2 .

Table 2 Sample selection criteria of French listed firms Panel A: Number of firm-years by sample selection criteria

 2

	Selection criteria

Table 3

 3

analyzes the identity of the controlling shareholders of French listed firms depending on whether the control is maintained by one or multiple controlling shareholders. The results show that concentrated ownership is a ubiquitous feature of these firms. Only 6% of the French listed firms do not exhibit a controlling shareholder at a control threshold of 10%. Following previous relevant literature, we categorize ultimate owners into five classes namely (1) family; (2) widely held firm; (3) widely held financial institution; (4) State; and (5) miscellaneous (i.e., a charity, a cooperative, employees, etc.). As a result, we find that family firms dominate the corporate landscape in France (83.31%). The State, widely held corporations and widely held financial institutions control only 4.51%, 3.53% and 4.77% of the French listed firms, respectively.

Multiple large shareholders are fairly common (44.89% of controlled listed firms). They are more present in firms controlled by widely held financial institutions (67.29%) and by widely held firms (55.70%) than in family firms (43.06%) and in state-owned firms (43.56%).

Table 3 Sample characteristics

 3

		All sample firms	MLS firms	Non MLS firms
		(2384)		(1006)		(1235)
		Number	Percentage Number Percentage	Number	Percentage
	Widely held firms	143	6.00%
	Controlled firms	2241	94.00%	1006	44.89%	1235	55.11%
	All sampled firms	2384	100.00%
	Identity of the ultimate controlling shareholder		Number Percentage	Number	Percentage
					within class		within class
	Family-controlled firms	1867	83.31%	804	43.06%	1063	56.94%
	State controlled firms	101	4.51%	44	43.56%	57	56.44%
	Widely held corporation	79	3.53%	44	55.70%	35	44.30%
	Controlled firms
	Widely held financial institution	107	4.77%	72	67.29%	35	32.71%
	controlled firms
	Miscellaneous	87	3.88%	42	48.28%	45	51.72%
	Total of controlled firms	2241	100.00%	1006	51.80%	1235	42.20%

Table 4

 4 provides summary statistics of all variables used in the empirical analysis. The CAR and NI

Table 4 Summary statistics of regression variables

 4

		Minimum	First quartile	Median	Mean	Third quartile	Maximum.	Std. Dev.
	CAR (%)	-103.33	-26.18	-5.41	-4.25	14.80	110.55	39.09
	NI (%)	-462.30	0.72	5.49	-0.18	9.03	426.98	48.18
	LCS related variables
	UCF (%)	0.30	19.71	39.52	41.00	60.31	99.49	25.15
	Excess control (%)	-29.48	2.30	18.08	21.24	32.53	96.47	20.87
	MLS related variables
	MLSD	0.00	0.00	0.00	0.42	1	1	0.49
	VOTE21	0.00	0.00	0.42	0.25	0.45	1.00	0.35
	VOTE231	0.00	0.00	0.15	0.37	0.64	2.00	0.46
	HERFDVOTE	0.00	0.05	0.19	0.28	0.46	0.98	0.26
	Control variables
	Firm size	3.34	10.44	11.68	12.05	13.28	18.99	2.22
	Q	0.004	0.40	0.72	1.22	1.32	20.86	2.09
	Leverage (%)	0.00	49.39	64.62	82.63	80.41	97.91	2.05
	Industry Diversification	1	2	2	2.82	4	8	1.50

Table 5 Year-by-year regressions of stock returns on earnings

 5 This table presents the results of the estimation of the following model: CAR it = α 0 + α 1 (NI it ) + (industry dummies) + ǫ it . The tstatistics, based on the White's heteroscedasticity-consistent robust standard errors, are between parentheses below the estimated coefficients. The last regression uses all observations (pooled) and controls for clustering at the firm level. The symbols a, b and c denote statistical significance at the 1%, 5% and 10% level, respectively.

		Intercept	NI	Adjusted R²	F de Fisher	Number of
						observations
	2003	-0.1284	0.1460	7.82%	4.86 a	501
		(-1.8486) c	(5.5756) a
	2004	-0.0500	0.1678	6.94%	3.13 a	474
		(-0.6765)	(4.9656) a
	2005	-0.1110	0.2121	5.02%	3.20 a	460
		(-1.5749)	(4.6102) a
	2006	-0.0373	0.1199	3.18%	3.27 a	469
		(-2.2114) b	(2.7064) a
	2007	-0.0363	0.2977	4.65%	3.12 a	480
		(-2.1179) b	(4.9137) a
	Pooled observations	-0.0605	0.1573	4.08%	3.76 a	2384
		(-2.1573) b	(4.8647) a

Table 6 Multiple large shareholders and earnings informativeness

 6

		Specification 1	Specification 2	Specification 3	Specification 4
		MLSD	VOTE21	VOTE231	HERFDVOTE
	Variables	Coef.	t-value	Coef.	t-value	Coef.	t-value	Coef.	t-value
	NI	0.1590 b	(2.5307)	0.1535 b	(2.4198)	0.1619 b	(2.3453)	0.2457 a	(4.6503)
	NI*UCF	0.0582 a	(5.6641)	0.0554 a	(5.1643)	0.0573 a	(4.6744)	0.0687 a	(5.3637)
	NI*Excess control	-0.0791 a	(-9.6302)	-0.0764 a	(-8.6746)	-0.0749 a	(-8.1587) -0.0648 a (-6.1455)
	NI*MLSD	0.1149 a	(3.0822)
	NI*VOTE21			0.1989 a	(2.9919)
	NI*VOTE231					0.1085 b	(2.0072)
	NI*HERFDVOTE							-0.2669 a (-2.7712)
	Control Variables
	NI*Firm size	0.0022	(0.6727)	0.0003	(0.1264)	0.0001	(0.0155)	0.0040	(1.2993)
	NI*Q	0.0010 b	(2.0348)	0.0003 b	(2.1248)	0.0004 b	(2.1483)	0.0006 b	(2.2136)
	NI*Leverage	-0.0023 a	(-2.6982)	-0.0009 b	(-2.0158)	-0.0010 c	(1.8486)	-0.0013 b (-2.4436)
	NI*Industry diversification	-0.0181 b	(-2.4821)	-0.0091 b	(-2.0956)	-0.0079 b	(-2.0467) -0.0152 b (-2.1754)
	Industry dummies	YES		YES		YES	YES
	Constant	-0.0490 c	(-1.7527)	-0.0537 c	(-1.9333)	-0.0562 b	(-2.0181) -0.0536 c (-1.9346)
	F-Value	13.83 a	16.75 a	13.58 a	15.96 a
	Adjusted R-squared	7.31%	7.18%	6.87%	7.03%
	Number of	2384		2384		2384	2384
	observations

Table 6

 6 estimates the parameters of the pooled time-series cross-section regression model of multiple large shareholders on earnings informativeness. The dependent variable is earnings informativeness proxied by the cumulative net-of-market 12-month stock returns (CAR). See Table1for the definition if the remaining variables. The t-statistics are based standard errors with[START_REF] White | A heteroskedastic-consistent Covariance Matrix Estimator and a Direct Test of Heteroskedasticity[END_REF] correction for heteroscedasticity and adjusted for clustering for observations at the firm level. The symbols a, b and c denote statistical significance at the 1%, 5% and 10% level, respectively.

Table 7 Multiple large shareholders, excess control and earnings informativeness

 7

		Specification 1	Specification 2	Specification 3	Specification 4
		Interaction with	Interaction with	Interaction with	Interaction with
		MLSD =1	VOTE21	VOTE231	HERFDVOTE
	Variables	Coef.	t-value	Coef.	t-value	Coef.	t-value	Coef.	t-value
	NI	0.1840 a	(2.8981)	0.2025 a	(3.2963)	0.2114 a	(3.4248)	0.1783 a	(2.6055)
	NI*UCF	0.0603 a	(4.5142)	0.0550 a	(3.9583)	0.0524 a	(3.4193)	0.0602 a	(4.0889)
	NI*Excess control	-0.0799 a	(-10.9288)	-0.7802 a	(-9.5363)	-0.0764 a	(-7.9615) -0.0705 a (-7.2175)
	NI*Excess control high *MLSD =1	0.1840	(1.9655)
	NI*Excess control high *VOTE21			0.1383 b	(2.0123)
	NI**Excess control high*VOTE231					0.0623 c	(1.9012)
	NI*Excess control high*HERFDVOTE							-0.2212 c (-1.7118)
	Control Variables
	NI*Firm size	-0.0005	(-0.1814)	-0.0006	(-0.2238)	-0.0001	(-0.0352)	0.0003	(0.1123)
	NI*Q	0.0007 b	(1.9931)	0.0005 c	(1.8934)	0.0005 c	(1.9125)	0.0007 b	(1.9978)
	NI*Leverage	-0.0015 b	(-2.1532)	-0.0011 c	(-1.9398)	-0.0010 c	(-1.7025) -0.0015 b (-2.0670)
	NI*Industry diversification	-0.0055 b	(-2.3125)	-0.0076 b	(-2.2178)	-0.0109 b	(-2.2756) -0.0080 b (-2.1106)
	Industry dummies	YES		YES		YES	YES
	Constant	-0.0556 b	(-1.9932)	-0.0573 b	(-2.0612)	-0.0581b (-2.0878) -0.0564 b (-2.0245)
	F-Value	15.31 a	12.48 a	9.55 a	41.81 a
	Adjusted R-squared	6.75%	6.60%	6.50%	6.64%
	Number of	2384	2384		2384	2384
	observations

Table 7

 7 tests the role of multiple controlling shareholders on earnings informativeness in firms exhibiting excess control. The dependent variable is earnings informativeness proxied by the cumulative net-of-market 12-month stock returns (CAR). Excess control high is a dummy variable that takes on a value of 1 for all observations where the value of Excess control is greater than the median of the sample, and zero otherwise. See Table1for the definition if the remaining variables. The t-statistics are based standard errors with[START_REF] White | A heteroskedastic-consistent Covariance Matrix Estimator and a Direct Test of Heteroskedasticity[END_REF] correction for heteroscedasticity and adjusted for clustering for observations at the firm level. The symbols a, b and c denote statistical significance at the 1%, 5% and 10% level, respectively.