Keywords: noyaux à valeurs opérateurs, estimation non-paramétrique d'opérateurs, analyse des données fonctionnelles, espace

Noyaux à Valeurs Opérateurs pour l'Estimation Non-paramétrique d'Opérateurs Résumé : Dans ce rapport de recherche, nous considérons le problème d'apprentissage supervisé dans le cas où les variables explicatives ainsi que les variables d'intérêts sont de dimension infinie et représentées par des fonctions. Nous développons une approche non-paramétrique pour l'estimation d'opérateurs dans un espace de Hilbert à noyau reproduisant de fonctions à valeurs fonctions. Nous introduisons une classe de noyaux à valeurs opérateurs et nous montrons comment résoudre le problème de minimisation correspondant sans discrétiser les variables explicatives et à expliquer. Finalement, nous appliquons l'approche développée à un problème de régression à réponse fonctionnelle.

Introduction

Data streams are more and more commonly encountered in data mining and machine learning; for instance, data streams are very common in imagery in neuroscience. A data stream may be either of discrete nature, or of continuous nature; in any case, it comes as a discrete series of objects (scalars, or more complex entities). In the case of continuous streams, the stream is really a function rather than a vector. If we wish to address such data in the sound framework of reproducing kernel Hilbert spaces (RKHS), we have to consider RKHS which elements are operators that map a function to an other function space, possibly source and target function spaces being different. Working in such RKHS, we are able to draw on the important core of work that have been performed on real RKHS, and multi-valued real RKHS. Such a functional RKHS framework has been introduced very recently [START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF]; the present paper aims at building on this early work, and addresses in particular the problem of exhibiting operator-valued kernels and the resolution of the related optimization problem. This study is valuable from a variety of perspectives. Our main motivation is the problem of nonparametric regression when both input and output data are curves. One of the simplest ways to handle these data is to treat them as multivariate vectors. However this method does not consider any dependencies of different values over subsequent time-points within the same functional datum and suffer when data dimension is very large or infinite. Therefore, we adopt a functional data analysis viewpoint [START_REF] Zhao | The functional data analysis view of longitudinal data[END_REF][START_REF] Ramsay | Functional Data Analysis[END_REF][START_REF] Ferraty | Nonparametric Functional Data Analysis[END_REF] in which multiple curves are viewed as functional realizations of a common function.

The problem of nonparametric function estimation continues to be a challenging research problem which significantly impacts machine learning algorithm performances. During the past decades, a large number of algorithms have been proposed to deal with that problem in the case of single-valued functions (e.g., binary-output function for classification or a real number output for regression). Recently, there has been considerable interest in estimating vector-valued functions (Micchelli and Pontil, 2005a). Much of this interest has arisen from the need to learn multiple tasks simultaneously, see [START_REF] Evgeniou | Learning multiple tasks with kernel methods[END_REF] and references therein. The primary focus of this paper is on extending existing kernel-based function estimation procedure to include large, even infinite-dimensional output functions. More precisely, since input and output data are infinite dimensional and then belong to a space of functions, we look for a solution of an operator estimation problem in a set of operators that belong to some reproducing kernel Hilbert space.

Reproducing kernels play an important role in statistical learning theory and functional estimation. Scalar-valued kernels are widely used to design nonlinear learning methods which have been successfully applied in several machine learning applications [START_REF] Schölkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF]. Moreover, their extension to matrix-valued kernels has helped to bring about additional improvements in learning vector-valued functions (Micchelli and Pontil, 2005b;[START_REF] Reisert | Learning equivariant functions with matrix valued kernels[END_REF]. The most common and most successful applications of matrix-valued kernel methods are in multi-task learning [START_REF] Evgeniou | Learning multiple tasks with kernel methods[END_REF]Micchelli and Pontil, 2005b), even though some successful applications in other areas, for example image processing, also exist [START_REF] Ha Quang | Image and video colorization using vectorvalued reproducing kernel Hilbert spaces[END_REF][START_REF] Reisert | Learning equivariant functions with matrix valued kernels[END_REF]. A basic question always present with repro-ducing kernels is how to build these kernels and what is the optimal kernel choice. This question has been studied extensively for scalar-valued kernels [START_REF] Schölkopf | Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond[END_REF][START_REF] Ong | Learning the kernel with hyperkernels[END_REF], however it has not been investigated enough in the matrix-valued case. In the context of multi-task learning, matrix-valued kernels are constructed from real kernels which are carried over to the vector-valued setting by a positive definite matrix (Micchelli and Pontil, 2005b;[START_REF] Caponnetto | Universal multi-task kernels[END_REF]. In this paper we consider the problem from a more general point of view. We are interested in the construction of operatorvalued kernels which are the generalization of matrix-valued kernels in infinite dimensional spaces.

Problem setting

In this subsection, we briefly review notions and properties of reproducing kernel Hilbert spaces with operator-valued kernels and show their connection to learning from multiple response data (multiple outputs ; see Micchelli and Pontil (2005a) for discrete data and [START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF] for continuous data). We first consider the problem of estimating a function f such that f (x i ) = y i when observed data (x i , y i ) i=1,...,n are assumed to be elements of infinite dimensional Hilbert spaces. In the following we denote by G x , and G y the domains of x i and y i respectively. X = {x 1 , . . . , x n } denotes the training set with corresponding targets Y = {y 1 , . . . , y n }. Since G x and G y are spaces of functions, the problem can be thought of as an operator estimation problem, where the desired operator maps a Hilbert space of factors to a Hilbert space of targets. We can define the regularized operator estimate of f ∈ F

f λ arg min f ∈F n i=1 y i -f (x i ) 2 Gy + λ f 2 F (1)
In this work, we are looking for a solution to this minimization problem in a reproducing kernel Hilbert space F of function-valued functions on some infinitedimensional input space G x . We start by introducing function-valued reproducing kernel Hilbert spaces and showing the correspondence between such spaces and positive operator-valued kernels. Bijection between scalar-valued kernel and RKHS was first established by [START_REF] Aronszajn | Theory of reproducing kernels[END_REF]. Then [START_REF] Schwartz | Sous-espaces hilbertiens d'espaces vectoriels topologiques et noyaux associés (noyaux reproduisants)[END_REF] shows that this was a particular case of a more general situation. More recently, interest has grown in exploring Hilbert spaces of vector random functions for learning vector-valued functions (Micchelli and Pontil, 2005a;[START_REF] Carmeli | Vector valued reproducing kernel Hilbert spaces of integrable functions and mercer theorem[END_REF][START_REF] Caponnetto | Optimal rates for the regularized least-squares algorithm[END_REF]. The function-valued RKHS approach extends the vector-valued case to infinite-dimensional output data (Wahba, 1992;[START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF].

Let G x and G y be infinite-dimensional Hilbert spaces and F a linear space of operators on G x with values in G y . We assume that F is a Hilbert space with inner product ., . F . Let L(G y ) be the set of bounded linear operators from G y to G y .

Definition 1 (function-valued RKHS)

A Hilbert space F of functions from G x to G y is called a reproducing kernel Hilbert space if there is a nonnegative L(G y )-valued kernel K F (w, z) on G x × G x such that: The proof is performed in two steps. The necessity is an immediate result from the reproducing property. For the sufficiency, we assume F 0 to be the space of all G y -valued functions f of the form f (.) = n i=1 K F (w i , .)α i where w i ∈ G x and α i ∈ G y , with the following inner product f (.), g(.) F0 = n i,j=1 K F (w i , z j )α i , β j Gy . We show that (F 0 , ., . F0 ) is a pre-Hilbert space. Then we complete this pre-Hilbert space via Cauchy sequences to construct the Hilbert space F of G y -valued functions. Finally, we conclude that F is a reproducing kernel Hilbert space, since F is a real inner product space that is complete under the norm . F defined by f (.) F = lim n→∞ f n (.) F0 , and has K F (., .) as reproducing kernel.

INRIA i. the function z -→ K F (w, z)g belongs to F , ∀z ∈ G x , w ∈ G x , g ∈ G y , ii. ∀f ∈ F , f, K F (w, .)g F = f (w), g Gy (reproducing property). Definition 2 (operator-valued kernel) An L(G y )-valued kernel K F (w, z) on G x is a function K F (., .) : G x × G x -→ L(G y ); furthermore: K F is Hermitian if K F (w, z) = K F (z, w) * , where K F (z, w) * is the adjoint operator of K F (z, w) K F is nonnegative on G x if

Related works

The problem of operator estimation arises in a number of different contexts. Here we briefly discuss two contexts: collaborative filtering (CF) and functional data analysis (FDA). The goal of collaborative filtering is to build a model to predict preferences of clients "users" over a range of products "items" based on information from customer's past purchases. In [START_REF] Abernethy | A new approach to collaborative filtering: Operator estimation with spectral regularization[END_REF], the authors show that several CF methods such as rank-constrained optimization, trace-norm regularization, and those based on Frobenius norm regularization, can all be cast as special cases of spectral regularization on operator spaces. Using operator estimation and spectral regularization as a framework for CF permit to use potentially more information and incorporate additional useritem attributes to predict preferences. A generalized CF approach consists in learning a preference function f (., .) that takes the form of a linear operator from a Hilbert space of users to a Hilbert space of items, f (x, y) = x, F y for some compact operator F .

From the FDA point of view, operator estimation problems are frequently encountered in functional regression analysis. FDA is an extension of multivariate data analysis suitable when the data are curves [START_REF] Ramsay | Functional Data Analysis[END_REF]. In this framework, data are assumed to be a single function observation rather than a collection of individual observations. A functional regression problem takes the form y i = f (x i ) + ǫ i where one or more of the components y i , x i and ǫ i are functions. Three subcategories of such models can be distinguished: predictors x i are functions and responses y i are scalars; predictors are scalars and responses are functions; both predictors and responses are functions. In the latter case which is usually referred to as general functional regression model, the function f is a compact operator between two infinite-dimensional Hilbert spaces. Most previous works on this model suppose that the relation between functional responses and predictors is linear. In this case, the functional regression model is an extension of the multivariate linear regression model

y(t) = α(t) + β(t)x(t) + ǫ(t) (2)
for a regression parameter function β. In this model, known as the concurrent model, the response y and the covariate x are both functions of the same argument t, and the influence of a covariate on the response is concurrent or point-wise in the sense that x only influences y(t) through its value x(t) at time t (see chapter 14 of [START_REF] Ramsay | Functional Data Analysis[END_REF]). An extended linear model in which the influence of a covariate x can involve a range of argument values x(s) takes the form (see chapter 16 of [START_REF] Ramsay | Functional Data Analysis[END_REF])

y(t) = α(t) + x(s)β(s, t)ds + ǫ(t) (3)
In this setting, an extension to nonlinear contexts can be found in [START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF] where the authors showed how Hilbert spaces of function-valued functions and functional reproducing kernels can be used as a theoretical framework to develop nonlinear functional regression methods.

Contribution and paper organization

The objective of this paper is to present and discuss a set of rigorously defined operator-valued kernels that can be valuably applied to nonparametric operator estimation in infinite-dimensional space for continuous input and output data [START_REF] Ramsay | Functional Data Analysis[END_REF]. Such kernels are not very well understood and deserve more investigation. The first step in accomplishing our objective will be to identify a working range of operators that are well suited to take into account the local properties of the input and output spaces. Using this class of operators, we then characterize algebraic transformations that preserve the positivity of operator-valued kernels. Since data, even if they are represented by functions, are often observed at discrete points, we discuss the construction of operator-valued kernels from positive scalar kernels. This has the benefit of both ensuring the nonnegativity of the operator-valued kernel and estimating the operator without discretizing input and output functions. The operator-valued kernel performs the mapping between a space of functions and a space of operators, while the scalar one establishes the link between the space of functions and the space of measured values. We also provide a discretization free solution of nonparametric estimation problems in infinite dimensional spaces. We illustrate the proposed procedure by giving examples of operator-valued kernels and showing their potential as a tool for estimating operators and extending functional linear models to nonlinear contexts.

INRIA

The remainder of this paper is organized as follows. Section 2 discusses the construction of operator-valued kernels. Section 3 presents the nonparametric operation estimation procedure. Experimental setup and results are provided in Section 4. Section 5 concludes this paper.

Operator-valued kernels

In this section, we describe the construction of operator-valued kernels in the case where input data {x i } i=1,...,n ∈ G x and output data {y i } i=1,...,n ∈ G y are infinite dimensional. It is well known that a number of difficulties arise when standard machine learning techniques are applied to high-dimensional data. Contrary to multivariate methods, we adopt a different data modeling formalism by considering each x i and y i as a single observation represented by a smooth function. It is true that the data measurement process often provides a vector rather than a function, but the vector is a discretization of a real attribute which is a function. In this setting, it is possible to build an L(G y )-valued kernel from an operator T associated with a function in the space G y [START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF]. A simple and familiar example of operator generated by a function is the composition operator T φ which is defined by T φ f = f • φ. This class of operator was widely used in various problems of operator theory, system theory and interpolation [START_REF] Alpay | Algorithme de Schur, espaces à noyau reproduisant et théorie des systèmes[END_REF][START_REF] Dym | J contractive matrix functions, reproducing kernel spaces and interpolation[END_REF]. We suggest to construct operator-valued kernels using the reproducing kernel of the space G y . This has the benefit of facilitating the design of operator estimation algorithms as we show in section 3. Using the reproducing property of the operator-valued kernel allows us to compute a dot product in a space of operators by a dot product in a space of functions which can also be calculated from the scalar kernel. The operator-valued kernel performs the mapping between a space of functions and a space of operators, while the scalar one establishes the link between the space of functions and the space of measured values. G y is the space of functions containing output data y i . Assuming that a functional datum for replication i arrives as a set of discrete measured values, y i1 , . . . , y ip , the first task is to convert these values to a function y i with values y i (t) computable for any desired argument value t. If the discrete values are assumed to be errorless, then the process is interpolation, but if they have some observational error that needs removing, then the conversion from discrete data to functions may involve smoothing [START_REF] Ramsay | Functional Data Analysis[END_REF]. This seems to confirm the suitability of considering G y to be a real reproducing kernel Hilbert space.

Combinations of positive operator-valued kernels

It is known that there is a bijection between positive operator-valued kernels and function-valued reproducing kernel Hilbert spaces. So, as in the scalar case, it will be helpful to characterize algebraic transformations that preserve the positivity of operator-valued kernels. The theorem 2 gives some building rules to obtain a positive operator-valued kernel from combinations of positive existing ones constructed as described above. Similar results for the case of matrix-valued kernels can be found in [START_REF] Caponnetto | Universal multi-task kernels[END_REF]; [START_REF] Reisert | Learning equivariant functions with matrix valued kernels[END_REF] and for a more general context we refer the reader to section 3 of [START_REF] Carmeli | Vector valued reproducing kernel Hilbert spaces and universality[END_REF]. In our setting, assuming that K 1 = T φ1 1 and K 2 = T φ2 2 be two kernels RR n 7607 constructed using operators T 1 and T 2 and functions φ 1 and φ 2 , we are interested in constructing a kernel K = T φ = Ψ(T 1 , T 2 ) ϕ(φ1,φ2) from K 1 and K 2 using operators combination Ψ and functions combination ϕ.

Theorem 2 Let K 1 : G x × G x -→ L(G y ) and K 2 : G x × G x -→ L(G y ) two non- negative kernels such that K 1 (x 1 , x 2 ) = T φ1(x1,x2) 1 and K 2 (x 1 , x 2 ) = T φ2(x1,x2) 2 i. T φ1 1 + T φ2 2 is a nonnegative kernel, ii. If T φ1 1 (T φ2 2 ) = T φ2 2 (T φ1 1 ) then T φ1 1 (T φ2 2 ) is a nonnegative kernel. iii. C ϕ(x2) T φ1(x1,x2) 1 C ϕ(x1) * is a nonnegative kernel for any C ϕ ∈ L(G y )
Obviously (i) follows from the linearity of the inner product. (ii) can be proved by induction using sequences of linear operators. For the proof of (iii), we observe that

i,j K F (w i , w j )u i , u j = i,j C ϕ(wj ) T φ1(wi,wj ) 1 C ϕ(wi) * u i , u j = i,j T φ1 1 C ϕ(wi) * u i , C ϕ(wj ) * u j
which implies the nonnegativity of the kernel since T 1 is nonnegative.

Examples of positive operator-valued kernels

Examples in this section all deal with operator-valued kernels constructed following the above described scheme and assuming that G y be a real RKHS included in a L 2 space. Motivated by extending functional linear models (see equations ( 2) and ( 3)) to nonlinear context, the first two examples deal with operator-valued kernels constructed from multiplication and Hilbert-Schmidt integral self-adjoint operators. The third example based on the composition operator shows how to build such kernels from non self-adjoint operators. It also illustrates the kernel combination defined in theorem 2(iii).

Multiplication operator:

In [START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF], the authors attempted to extend the widely used Gaussian kernel to functional data domain using a multiplication operator and assuming that input and output data belong to the same space of functions. Here we consider a slightly different setting, where the characteristic function of the operator-valued kernel may be the reproducing kernel of G y .

A multiplication operator on G y is defined as follows:

T h : G y -→ G y y -→ T h y ; T h y (t) h(t)y(t)
The operator-valued kernel K F (., .) is the following: ).,x2(.).)

K F : G x × G x -→ L(G y ) x 1 , x 2 -→ T k(x1(.

INRIA

where k(., .) ∈ G y is a scalar kernel. It easy to see that T h x, y = x, T h y , then T h is a self-adjoint operator. Thus K F (x 2 , x 1 ) * = K F (x 2 , x 1 ) and

K F is Hermitian since K F (x 1 , x 2 ) = K F (x 2 , x 1 ).
Moreover, we have i,j

K F (x i , x j )y i , y j Gy = i,j
k(x i (.)., x j (.).)y i (.), y j (.)

Gy = i,j k(x i (t)t, x j (t)t)y i (t)y j (t)dt ≥ 0
since k is a positive definite scalar-valued kernel. Therefore K F is a nonnegative operator-valued kernel.

Hilbert-Schmidt integral operator:

A Hilbert-Schmidt integral operator on G y associated with a kernel h(., .) is defined as follows:

T h : G y -→ G y y -→ T h y ; T h y (t) h(s, t)y(s)ds
In this case, an operator-valued kernel K F is a Hilbert-Schmidt integral operator associated with the scalar kernel k and it takes the following form

K F (x 1 , x 2 ) : G y -→ G y f -→ g
where g(t) = k(x 1 (s)s, x 2 (t)t)f (s)ds.

The Hilbert-Schmidt integral operator is self-adjoint if k is Hermitian. This condition is verified and then it is easy to check that K F is also Hermitian. K F is nonnegative since

i,j K F (x i , x j )y i , y j Gy = i,j k(x i (s)s, x j (t)t)y i (s)y j (t)dsdt
which is positive because of the positive definiteness of the kernel k.

Composition operator:

Let ϕ be an analytic map. The composition operator associated with ϕ is the linear map

C ϕ : f -→ f • ϕ
First, we look for an expression of the adjoint of the composition operator C ϕ acting on G y in the case where G y is a real RKHS of functions on Ω y and ϕ an analytic map of Ω y into itself. For any f in the space G y associated with the real kernel k ,

f, C * ϕ k t (.) = C ϕ f, k t = f • ϕ, k t = f (ϕ(t)) = f, k ϕ(t)
This is true for any f ∈ G y and then C * ϕ k t = k ϕ(t) . In a similar way, C * ϕ f can be computed at each point of the function f :

(C * ϕ f )(t) = C * ϕ f, k t = f, C ϕ k t = f, k t • ϕ
Once we have expressed the adjoint of a composition operator in a reproducing kernel Hilbert space, we consider the following operator-valued kernel

K F : G x × G x -→ L(G y ) x 1 , x 2 -→ C ψ(x1) C * ψ(x2)
where ψ(x 1 ) and ψ(x 2 ) are analytic maps of Ω y into itself. It is easy to see that the kernel K F is Hermitian. Using theorem 2 we obtain the nonnegativity property of the kernel.

Operator estimation

In this section we turn our attention to the problem of minimizing a regularized functional in the context of function-valued functions (see equation ( 1)) within the RKHS framework described above. First, we remind the operator version of the representer theorem Theorem 3 (representer theorem) Let K F a positive operator-valued kernel and F its corresponding function-valued reproducing kernel Hilbert space. The solution f * λ ∈ F of the regularized minimal norm interpolation

f * λ = arg min f ∈F n i=1 y i -f (x i ) 2 Gy + λ f 2 F
has the following form:

f * λ (.) = n i=1 K F (x i , .)β i ; β i ∈ G y
Using the representer theorem and the reproducing property of F , the minimization problem (1) becomes:

β * λ = arg min β∈(Gy) n n i=1 y i - n j=1 K F (x i , x j )β j 2 Gy +λ n i,j K F (x i , x j )β i , β j Gy (4)
From this general case and unlike the procedure proposed in [START_REF] Kadri | Nonlinear functional regression: a functional rkhs approach[END_REF], we derive an estimation of the operator f * λ without discretizing the covariate, target and "weight" functions x i , y i and β i . The idea is to consider the output space G y as a scalar-valued RKHS and then minimizing the problem (4) over the functions βi , approximation of β i in G y , rather than minimizing it over the discrete values β i (t 1 ), . . . , β i (t p ). In the FDA literature, a similar idea has been adopted by [START_REF] Ramsay | Functional Data Analysis[END_REF] and, Prchal and Sarda (2007) who

INRIA

expressed not only the regression functional parameters but also the observed input and output data in a basis functions specified a priori (eg, Fourier basis or B-spline basis).

Furthermore, we pay more attention to the multiplication and integral operatorvalued kernels. We think that these kernels could provide an interesting alternative to extend functional linear models for functional responses proposed by [START_REF] Ramsay | Functional Data Analysis[END_REF] to nonlinear context. Indeed, these linear models are based on the multiplication and the Hilbert-Schmidt integral operators, as shown in Equations ( 2) and (3). In section 2 we have argued that it is possible to consider the space G y as a real reproducing kernel Hilbert space associated with the kernel k and then each function β i can be expressed (approximated) by a finite linear combination of the real-valued kernel function βi = p k=1 α ik k(s k , .). In this setting and from equation ( 4), the original functional estimation problem is reduced to the following multivariate problem:

min α∈R np n i=1 y i - n j=1 p l=1 α jl K F (x i , x j )k(s l , .) 2 Gy +λ n i,j p k,l α ik K F (x i , x j )k(s k , .), α jl k(s l , .) Gy
Using the reproducing property of G y , we can re-express this expression via the matrix formulation:

min α∈R np trace(J) -2 trace(J 1 α) +trace(J 2 αα T ) + λ trace(J 3 αα T ) ( 5
)
where J is the n × n matrix which elements (J ij ) 1≤i,j≤n = y i , y j . α = (α k ) 1≤k≤p is the np × 1 matrix such that α k = (α ik ) 1≤i≤n . The block matrices J 1 , J 2 and J 3 depend on the choice of the operator-valued kernel. With the multiplication operator based kernel, we obtain J 1 = CY where the np

× np matrix C = (C k ) 1≤k≤p is block diagonal with C k = {k(x i (s k )s k , x j (s k )s k )} 1≤i,j≤n and the np × 1 matrix Y is defined by Y = (Y k ) 1≤k≤p with Y k = (y i (s k )) 1≤i≤n .
The np × np matrix J 3 = (J 3kl ) 1≤k,l≤p is a block matrix where each

J 3kl = {k(x i (s k )s k , x j (s k )s k ) k(s l , s k )} 1≤i,j≤n is a n × n matrix. The matrix J 2 is equal to J T 3 C.
For more details see Appendix A. Using function formulation, the matrices J 1 , J 2 , and J 3 , in the case of the multiplication operator-valued kernel, can be written as

   J 1 = CY J 2 = J T 3 C J 3 = δK T ⊗ c(s, t) dt ds (6) with        C = diag(δ(s -s k )δ(t -s k )) p k=1 ⊗ c(s, t)dt ds Y = (δ(s -s k )) 1≤k≤p ⊗ (y i (s)) 1≤i≤n ds δ(s, t) = (δ(s -s k )δ(t -s k )) 1≤k≤p K(s) = (k(s, s l )) 1≤l≤p
where c(s, t) is the n × n matrix of functions c(s, t) = (k(x i (s)s, x j (t)t) 1≤i,j≤n , δ(s) is the Dirac delta function and ⊗ is the Kronecker product. This formulation is suited for the two kernels and would help the understanding of their effect on functional data. With the Hilbert-Schmidt integral operator, we obtain the following result for the matrices J 1 , J 2 , and J 3 :

   J 1 = C(s)Y (s)ds J 2 = (HK T ⊗ c(s, t)) T C(s)ds dt J 3 = δK T ⊗ c(s, t) dt ds (7) with        C(s) = diag(δ(t -s k )) 1≤k≤p ⊗ c(s, t)dt ds Y (s) = (H(s)) 1≤k≤p ⊗ (y i (s)) 1≤i≤n ds δ(s, t) = (δ(t -s k )) 1≤k≤p K(s) = (k(s, s l )) 1≤l≤p
where H(s) is the Heaviside step function and H the p-dimensional vector such that H k = H(s). Now we compute the derivative of ( 5) with respect to matrix α and set the result to zero. We find that α satisfies the matrix system of linear equations

(J 2 + J T 2 + λ(J 3 + J T 3 ))α = 2J 1 (8)
4 Experiments

We study the effectiveness of the proposed nonparametric operator estimation method through experiments, comparing it with the functional linear regression (FLR) estimate, see equation ( 3). In all of our experiments, we use an implementation in Matlab according to [START_REF] Ramsay | Functional Data Analysis with R and Matlab[END_REF]. For the experiments with FLR parametric approach based on basis expansion, we use B-spline basis of order 4 with 10 equispaced knots and a penalty term involving the second partial derivatives of β(s, t). Experiments with our RKHS approach are performed using the two proposed kernels based on multiplication and Hilbert-Schmidt integral operators, since functional linear models are based on these two operators. The scalar kernel used to construct these operator-valued kernels is the the gaussian RBF kernel. We apply the operator estimation methods to curve prediction problems. We use the lip-EMG1 functional dataset introduced by [START_REF] Ramsay | Applied Functional Data Analysis[END_REF] to study the dependence of the acceleration of the lower lip in speech on neural activity, as measured by electromyographical (EMG) recording. As illustrated by Figure 1, the dataset consists of 32 records of the movement of the center of the lower lip when a subjet was repeatedly required to say the syllable "bob", embedded in the phrase, "Say bob again" and the corresponding EMG activities of the primary muscle depressing the lower lip, the depressor labii inferior (DLI). More information about the data collection process can be found in chapter 10 of [START_REF] Ramsay | Applied Functional Data Analysis[END_REF]. The goal is to predict lip acceleration from EMG activities. We use the residual sum of squares error of the full data set, defined by Equation ( 9), as the evaluation criterion for curve prediction where y i (t) is the prediction of the lip acceleration curve y i (t).

RSSE = i {y i (t) -y i (t)} 2 dt (9) 0
Table 1 provides the prediction accuracy of the lip acceleration curves from information of EMG curves and shows the RSSE values for FLR and RKHS methods. These results are obtained with Gaussian RBF kernel parameter σ and regularization parameter λ equal to 0.03 and 0.1, respectively. These coefficients could also be chosen using cross-validation. A comparison of RSSE of the two estimation methods demonstrates significant overall improvement when using RKHS with operator-valued kernels. This could be due to the fact that the relationship between EMG and lip acceleration data are not linear in the input space and then it would be advantageous to model them by a nonlinear procedure. Moreover, the best prediction result is obtained when using RKHS with the Hilbert Schmidt integral operator based kernel. This can be explained by observing that the RKHS approach using the integral operator is the extension of the functional linear model (3) to nonlinear context, while using the multiplication operator corresponds to the nonlinear extension of the concurrent model (2). Using the integral operator, the corresponding operatorvalued kernel permits to predict a lip acceleration value at time t by taking EMG information of a range of argument values and not only the EMG information at time t and this has the advantage of improving prediction accuracy.

Conclusion

We have presented a kernel-based approach to nonparametric estimation in infinite-dimensional space of operators which map a function space of covariates to a function space of targets. We have proposed a set of rigorously defined operator-valued kernels constructed from scalar kernels and we have showed their use for solving the problem of minimizing a regularized functional in the context of function-valued functions without the need to discretize covariate and target functions. The operator-valued kernel performs the mapping between a space of functions and a space of operators, while the scalar one establishes the link between the space of functions and the space of measured values. Experiments on predicting lip acceleration from EMG activity have shown that the function-valued RKHS approach achieves improvement over the functional linear regression method. However, a more extensive benchmark study remains to be pursued. In future we will explore more experiments, not only on functional datasets but also on time-series and longitudinal datasets, and compare our operator-valued kernel based approach with previous related methods for multiple output regression and multi-task learning, such as those in [START_REF] Breiman | Predicting multivariate responses in multiple linear regression[END_REF][START_REF] Hoover | Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data[END_REF][START_REF] Simila | Input selection and shrinkage in multiresponse linear regression[END_REF][START_REF] Evgeniou | Learning multiple tasks with kernel methods[END_REF]. Finally, it would be interesting to explore a nonlinear extension of operator-valued kernels, and to study sparse representation for functions.

Appendix A.

In this appendix we show how to solve the minimization problem (4) using the multiplication operator valued kernel. To simplify notation, we will write kqr(s, t) for k(xq(s)s, xr(t)t).

Solving eq. ( 4) is equivalent to solving: min where H(s) is the Heaviside step function and H the p-dimensional vector such that H k = H(s).

β i i yi 2 -2 i yi, j KF (
To find α ik solution of the matrix formulation of the problem (4), since each function βi can be expressed as a finite linear combination of the real-valued kernel function βi = p k=1 α ik k(s k , .), we need to compute the derivative with respect to matrix α and set the result to zero. Using the fact that the derivative of trace(A1XA2) with respect to matrix X is A2A1, we find that α satisfies the matrix system of linear equations (J2 + J T 2 + λ(J3 + J T 3 ))α = 2J1

RR n 7607

Figure 1 :

 1 Figure1: EMG and lip acceleration curves. The left panel displays EMG recordings from a facial muscle that depresses the lower lip, the depressor labii inferior. The right panel shows the accelerations of the center of the lower lip of a speaker pronouncing the syllable "bob" for 32 replications.

  , xj)βi, βj Gy

αα

  jl kiz(., .)kij (., .)k(s l , .), α zk k(s k , .) jk yi(s k )kij (s k , s k ) + i,j,z k,lα jk α zl kiz(s k , s k )kij(s k , s k )k(s l , s k ) +λ i,j k,l α jk α il kij(s k , s k )k(s l , s k ) ⇐⇒ min α trace(J) -2 trace(J1α) + trace(J2αα T ) +λ trace(J3αα T ) where J is the n × n matrix with elements (Jij ) 1≤i,j≤n = yi, yj . α = (α k ) 1≤k≤p is the np × 1 matrix such that α k = (α ik ) 1≤i≤n . J1 = CY where C is the np × np block-matrix defined by C k = {k(xi(s k )s k , xj (s k )s k )} 1≤i,j≤n . The np × 1 matrix Y = (Y k ) 1≤k≤p is defined by Y k = (yi(s k )) 1≤i≤n . The np × np matrix J3 = (J 3kl ) 1≤k,l≤p is a block matrix where each J 3kl = {k(xi(s k )s k , xj(s k )s k ) k(s l , s k )} 1≤i,j≤n is a n × n matrix. The matrix J2 is equal to J2 = J T 3Using function formulation, the matrices J1, J2 and J3 can be written as the following:ss k )δ(ts k )) p k=1 ⊗ c(s, t)dt ds Y = (δ(ss k )) 1≤k≤p ⊗ (yi(s)) 1≤i≤n ds δ(s, t) = (δ(ss k )δ(ts k )) 1≤k≤p K(s) = (k(s, s l )) 1≤l≤pwhere c(s, t) is the n × n matrix of functions c(s, t) = (k(xi(s)s, xj(t)t) 1≤i,j≤n , δ(s) is the dirac delta function and ⊗ is the Kronecker product.The procedure described above can be employed with any operator-valued kernel. We obtain a similar result for the Hilbert-Schmidt integral operator valued kernel with matrices: )Y (s)ds J2 = (HK T ⊗ c(s, t)) T C(s)ds dt J3 = δK T ⊗ c(s, t) dt ds with = diag(δ(ts k )) 1≤k≤p ⊗ c(s, t)dt ds Y (s) = (H(s)) 1≤k≤p ⊗ (yi(s)) 1≤i≤n ds δ(s, t) = (δ(ts k )) 1≤k≤p K(s) = (k(s, s l )) 1≤l≤p

  it is Hermitian and for every natural number r and all {(w i , u i ) i=1,...,r } ∈ G x × G y , the block matrix with ij-th entry K F (w i , w j )u i , u j Gy is nonnegative.

	Theorem 1 (bijection between function-valued RKHS and operator-valued ker-
	nel)
	A L(G y )-valued kernel K F (w, z) on G x is the reproducing kernel of some Hilbert
	space F , if and only if it is positive definite.

Table 1 :

 1 Evaluation of the prediction of lip acceleration curves from EMG curves

	Estimation method	RSSE
	FLR -B-spline	793.44
	RKHS -Multiplication operator	766.16
	RKHS -Hilbert-Schmidt integral operator 715.37

RR n 7607

Lip-EMG dataset is available from www.stats.ox.ac.uk/˜silverma/fdacasebook/lipemg.html INRIA


Nicolas Jacquemet
email: nicolas.jacquemet@univ-paris1.fr.

Constantine Yannelis
email: yannelis@stanford.edu.

Kevin Lang

Yannick L'horty

Phillip Oreopoulos

Amine Ouazad

Marie-Anne Valfort

Keywords: Correspondence testing, Racial discrimination. JEL Classification: J71, J64. Résumé Expérience de testing, Discrimination raciale

The extent of racial discrimination in the labor market is now clearly identified, but its nature largely remains an open question. This paper reports results from an experiment in which fabricated resumes are sent to help-wanted advertisements in Chicago newspapers. We use three groups of identical resumes: one with Anglo-Saxon names, one with African-American names, and one with fictitious foreign names whose ethnic origin is unidentifiable to most Americans. We find that resumes with Anglo-Saxon names generate nearly one half more call-backs than identical resumes with African-American or Foreign names. Resumes with non-Anglo-Saxon names, whether African-American or Foreign, show no statistically significant difference in the number of callbacks they elicit. We also find that discrimination is significantly higher in the Chicago suburbs -where ethnic homogenity is high -as opposed to the city proper. We take this as evidence that discriminatory behavior is part of a larger pattern of unequal treatment of any member of non-majority groups -ethnic homophily.

Introduction

Discrimination against specific racial and ethnic groups has been robustly substantiated in different countries [START_REF] Riach | Field Experiments of Discrimination in the Market Place[END_REF]. In the United States and other Western countries, affirmative action programs are in place in government, academic and even corporate institutions to counter the adverse effects of racial discrimination. For obvious practical reasons, programs and classifications limit themselves to traditional racial categorizations, focusing on established minority groups. 1 Such policies, however, are only well designed if discrimination is directed against specific minority groups due to historical, cultural and social factors. If discrimination is rather directed against all non-majority groups -what we call ethnic homophily, based on the similarity of this phenomenon with the identified driving forces of network formation (e.g., [START_REF] Currarini | An Economic Model of Friendship: Homophily, Minorities, and Segregation[END_REF] affirmative action programs will fail to accurately restore equality of opportunity, since a substantial proportion of those affected adversely by discrimination may not be receiving benefits from programs enacted to counter discrimination.

In both political science and economics, a growing strand of research highlights the implications of cultural heterogeneity on individual behavior and group ability to achieve efficient outcomes (e.g., [START_REF] Miguel | Ethnic diversity, social sanctions, and public goods in Kenya[END_REF][START_REF] Putnam | E Pluribus Unum: Diversity and Community in the Twenty-first Century[END_REF][START_REF] Habyarimana | Coethnicity: Diversity and the Dilemmas of Collective Action[END_REF]Alesina and Ferrara, 2005, for a survey). The primary finding from these studies is that it is ethnic diversity per se that drives interactions between members of different ethnic groups. Economic research on ethnic diversity in the labor market has focused on identifying and accounting for discrimination against pre-established groups with a history of discrimination, such as African-Americans in the United States or Asian immigrants in Canada [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF][START_REF] List | The Nature and Extent of Discrimination in the Marketplace: Evidence from the Field[END_REF]. Most of these studies use the same methodology -correspondence testing, which is conducted by sending resumes that differ only in the name of the applicant. Strikingly, several correspondence tests targeting diverse groups in different countries have found similar levels of discrimination. As noted by [START_REF] Bendick | Situation Testing for Employment Discrimination in the United States of America[END_REF], this may suggest observed discrimination arises because of employers' reluctance to hire people who differ from their "ideal", rather than hostility towards a specific group.

This paper reports results from an experiment designed so as to assess the extent of ethnic homophily in the labor market. We rely on a correspondence test involving three different groups of names: Anglo-Saxon, African-American and "Foreign" names with no clear ethnic association. The perceived ethnic origin of each set of names is established through a pre-experimental survey. We rule out potential gender bias by only using names that are clearly identified as female. Three resumes, that only differ in the identified ethnic origin of the applicant, are sent to employers posting vacancies in Chicago and its suburbs -allowing a precise comparison with [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF] results. Our results unambiguously support the existence of ethnic homophily.

1 For instance, the 2000 Census Equal Employment Opportunity data tool offers statistics on prescribed racial and ethnic groups, White, Black Hispanic, Asian, Native Hawaiian Other Pacific Islander, AIAN. The University of Michigan, a public institution, gave members of under-represented ethnic groups such as African-Americans, Hispanics, and Native Americans, a 20-point bonus out of 150 on their numeric admissions system before Gratz vs. Bollinger ruled this system unconstitutional.

We find that Foreign names are nearly as much discriminated against as are the African-American names, eliciting a third fewer call-backs than names identified as Anglo-Saxon.

2 Prior research on ethnic discrimination in the labor market

Economic Models of Discrimination

Economic models have traditionally focused on two mechanisms, taste-based discrimination and statistical discrimination.2 Taste-based discrimination posits that members of the majority prefer candidates of their own background. According to the original formulation due to [START_REF] Becker | The Economics of Discrimination[END_REF], white populations have a general "distaste" for minorities. Employers thus have some given preference for one group over another. The model is taste-based in the sense that no explanation is given for employers' inherent disutility for black labor. [START_REF] Goldberg | Discrimination, Nepotism, and Long-Run Wage Differentials[END_REF] reframes Becker's theory of taste-based discrimination as a theory of nepotism. Instead of employers having a disutility for hiring black workers, employers gain utility from hiring white workers. This disutility now measures the extent of nepotism -i.e. white employers' preference for hiring white workers.

The theory of statistical discrimination was initially proposed as an alternative to the theory of taste-based discrimination. Discrimination is statistical in the sense that differences between groups are used by employers to distinguish between individuals. [START_REF] Arrow | The Theory of Discrimination[END_REF]; [START_REF] Phelps | The Statistical Theory of Racism and Sexism[END_REF] formalize statistical discrimination as a response to imperfect information: employers use signals, such as race and ethnicity, to infer about other individual attributes such as productivity, educational attainment or language skills. Assuming a risk neutral employer, Arrow lists three assumptions necessary for employers to discriminate: (i) employers must be able to distinguish between workers of various races at a low-cost, (ii) employers must incur some cost before a worker's true productivity is determined, and (iii) employers must have some preconception of the distribution of productivity between the categories of workers. Since beliefs about the mean qualities of various groups are then used to make statistical inference about applicants' attributes on hiring decisions, discrimination arises. Note, employers are only concerned with hiring the best candidate, and do not have any intrinsic preference for one type of worker over another.

The existence of ethnic homophily would not necessarily lend support to the adoption of either a taste-based or a statistical model of discrimination. The phenomenon would rather provide interesting refinements of either model. The existence of general ethnic homophily supports one of two hypotheses: (i) in the framework of a taste-based discrimination model, the existence of taste based discrimination against non-majority groups or a taste based model in which nepotism is rewarded akin to that of [START_REF] Goldberg | Discrimination, Nepotism, and Long-Run Wage Differentials[END_REF] ; (ii) in a model of statistical discrimination, employers infer bad characteristics on applicants for whom the ethnic origin is uncertain.

Empirical evidence of discrimination in the labor market

Early literature used audit studies to attempt to measure racial discrimination. In audit studies, actors are thrown into real world social and economic situations, such as job interviews. Researchers observe the differences between two individuals who are nearly identical in all aspects with the exception of race, and then measure their performance. Most of these studies find that minority actors tend to perform worse in the hiring process (see [START_REF] Riach | Field Experiments of Discrimination in the Market Place[END_REF], for a survey of the literature). Audit studies have several major drawbacks. First, they are tremendously expensive. Second, due to the aforementioned factor, sample sizes tend to be small. Third, since the actors know that they are in a study, there is some speculation that minority participants in audit studies would have psychological motivation to underperform [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF]. Fourth, due to the nature of social interactions and infinite human variability, there will always be immeasurable differences between two actors other than race.

Correspondence testing, first developed in the United Kingdom by [START_REF] Jowell | Racial Discrimination and White-collar Workers in Britain[END_REF], overcomes most of these issues. The experiment consists of sending out written requests for job offers, with variations in the applicants' name. The method is much cheaper than audit testing and many laboratory experiments, thus allowing for larger sample sizes. It also gives the researcher nearly complete control over confounding factors in the experiment since the design relies only on paper applications which differ in highly controlled ways. The main weakness of correspondence testing is that differences in job offers cannot be measured directly since companies generally have multistage recruiting processes, which eventually require personal interviews. Thus correspondence tests may under or overstate the level of discrimination for actual job offers, since success in job search is measured by callbacks rather than actual offers. 3To our knowledge, the first major correspondence test in the United States on discrimination is due to [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF], which elicited a wealth of similar studies. Their field experiment attempts to measure racial discrimination against African-Americans in two major American cities, using names as a marker of ethnic origin and the percentage of callback offers received as a measure of discrimination. Resumes with African-American sounding names elicit 50% fewer callback offers than the resumes with white sounding names. Using data on social background and name specific call-back rates, they also find little evidence of inferring social class from names -pointing to racial discrimination rather than discrimination based on social class. Last, they find a "marginally significant" but "extremely small" relationship between employers being located in postal codes with a higher percentage of African-Americans and call-backs for African-Americans. It is worth noting that this relationship is statistically significant for their data in Chicago.

Other studies find similar results in alternative labor markets. In Australia, [START_REF] Booth | Does Racial Discrimination Vary Across Minority Groups? Evidence From Three Experiments[END_REF] rely on Anglo-Saxon, Indigenous, Italian, Chinese and Middle Eastern names, and find statistically significant evidence of discrimination. They also find significantly less prejudice against Italians, a more established immigrant group in Australia, as opposed to less established groups like Chinese and Middle Easterners. [START_REF] Bursell | What's in a name? A field experiment test for the existence of ethnic discrimination in the hiring process[END_REF] applies a similar methodology in the Swedish labor market, sending out resumes with either Arabic names or names from the Horn of Africa region, and finds that a person with similar characteristics but an Arabic or African name would have to apply to twice as many jobs in order to obtain a callback offer. [START_REF] Oreopoulos | Why Do Skilled Immigrants Struggle in the Labor Market? A Field Experiment with Six Thousand Resumes[END_REF] conducts a field experiment with over 6,000 resumes to examine discrimination against immigrants in the Canadian province of British Columbia. The resumes contrast English-named job-seekers with Canadian education and experience to similar candidates with Chinese, Indian, British and Pakistani names. A computer program randomizes various resume characteristics such as experience, time spent in Canada and the location where a bachelors degree was completed. This rich set of treatment variables leads to very detailed results relating discrimination to the social distance between applicants and the members of the dominant group. The four main results of the study are (i) English-named applicants with Canadian education are more than three times as likely to be offered an interview than Chinese, Indian or Pakistani-named applicants with foreign work experience, but there is no significant difference with British applicants, (ii) employers value experience in Canada much more than they value foreign experience, (iii) for resumes with substantial Canadian experience, whether the applicant has a Canadian degree or not has no effect on call-back offers, and (iv) Canadian applicants that differ only by name have substantially different call-back rates -English-named Canadians receive almost one and half times as many call-back offers as those with Chinese, Indian or Pakistani names. This rate is quite similar to that found in Bertrand and Mullainathan between whites and African-Americans in the US. [START_REF] Giuliano | Manager Race and the Race of New Hires[END_REF] study the effect of manager race on the race of new hires using data from a large U.S. retail firm. They find that black managers tend to hire more blacks and fewer whites than non-black managers, and that Hispanic managers in areas with large Hispanic populations tend to hire more Hispanic workers. Giuliano, Levine, and Leonard discuss several possible explanations for their results, including the possibility that some workers have tastebased racial preferences, or nepotism, for members of their own group. In the same spirit, [START_REF] Duguet | Discriminations a l'embauche: un testing sur les jeunes des banlieues d'ile de France[END_REF] observe that applicants with family names of North African origin with Gallicized first names face substantially less discrimination amongst French employers than applicants with both North African first and last names, almost eliminating the difference in call-backs.

Overall, field experiments show robust evidence for the existence of discrimination in the labor market against diverse ethnic groups in a wide range of countries. 4 Despite the different dates, ethnic groups used and geographic regions, most studies find strikingly similar differences in callback rates between majority and minority groups -from 30% to 50% fewer callbacks for the minority group in each case -and provide evidence that such discrimination is induced by race or ethnic origin rather than social class. Our experiment aims to test this hypothesis directly, by introducing a set of names which is clearly identified as non-majority, but is not associated with a particular ethnic group.

Experimental Design

We assess the existence and extent of ethnic homophily in the labor market, drawing heavily on the methodology of past experiments such as [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF] and [START_REF] Oreopoulos | Why Do Skilled Immigrants Struggle in the Labor Market? A Field Experiment with Six Thousand Resumes[END_REF]. We create a bank of resumes and use them to answer job advertisements in the greater Chicagoland area. We use call-back rates as a binary measure of discrimination.

Set of names

The experiment relies on three sets of names. As a benchmark, we used one set of Anglo-Saxon sounding names and one set of African-American sounding names. We use different data sources in choosing first and last names for these sets. The bank of Anglo-Saxon and African-American first names is built using the names from the [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF] study which received the highest call-back rates. The frequency data for first names was taken from Chicago, measuring names given to babies born into different ethnic groups between 1974 and 1979. For last names, we used frequency data from the US census indicating the most common last names among US racial groups to provide us with a bank of Anglo-Saxon and African-American family names. In assigning Anglo-Saxon last names we choose the most common family names for whites in the US according to the census. We also use US Census data indicating the most common African-American family names to assign African-American last names for our study, however we do not use any last names that are in the top 10 most common names for both white and African-American families. 5Our main treatment variable is the third set of names, which consists of names that are not clearly identifiable to Americans. A set of names is first created by randomly mixing first names from ethnic groups in the Balkans and Caucasus. We hypothesize that Chicago employers would be unable to recognize the origin of these names, and would simply determine them to be foreign and unfamiliar. In order to ascertain that the general population does not have any prior associations with these names, a survey was conducted using Illinois college students. Although a survey of members of human resource departments in various Chicago companies might be a more accurate measure of our sample's biases, we use college students for two reasons. The first is ease, as it is much simpler to arrange a survey of college students. The second is that we worry about biasing the results of our experiment -it is plausible that the members of human resource departments in Chicago will see the names in the survey again at later stages of the experiment once resumes are sent out. We thus assume that Chicago employers and Illinois students have similar abilities in determining the origin of names. Note. Origins elicited through the initial survey for the three sets of names included in the study. Upper part: share of correct guesses of both origin and gender for each name included in the Anglo-Saxon set and in the African-Amercian set.Bottom part: Results from the initial survey for the five names most often identified as from unknown origin. The first column shows the percentage of respondents leaving the field blank or filling in a question mark when asked to identify the ethnic origin of the given name, the second column provides the number of respondents who declared that the name was female. The ethnic group and frequency (computed among those who actually provided a guess) are displayed for the first, second and third most common guess respectively.

The survey mixes the Foreign sample of names with clearly identifiable Anglo-Saxon, African-American and other ethnic names. Survey respondents are asked to indicate to what group, if any, they associate the names. Respondents are also asked their guess for a gender associated to the name. One concern in identifying the names as foreign was to rule out random guesses of origins. We thus throw out for further use in the study any Foreign name for which more than a quarter of respondents identify the name with a particular group. Table 1 displays the results of the initial survey as regards the set of fifteen names (five for each origin) used in the study. The upper part of the Table displays both perceived origin and perceived gender of Anglo-Saxon names and African-American names. In both regards and for both ethnic origins, the very high rates of correct guesses confirms ex post the strategy used to choose names: their high frequency in their corresponding origin suggests very accurate perceptions.

The bottom part of the Table provides a more detailed report of answers elicited as regards the five names most often identified as from unknown origin. The first two columns show the gross results of the survey: the share of respondents that are unable to associate any origin to the name (leaving the field blank or putting a question mark) and the share that do not identify the name as female. 6 A potential hazard to our study is that in the set of foreign names, employers may be projecting their own beliefs on various names. While highly unlikely, it is not beyond the realm of possibility that in this case we are measuring extremely high rates of discrimination against other groups to which a minority of employers identify the names. To shed some light on the likelihood of this effect, the rightmost part of the table provides the guesses of respondents that did not classify the names as unknown. We observe highly diverse perceived origins for each name, with only one of them (Dziella Lousaper) being identified to the same origin by more than half the respondents who provide a guess. The Table also indicates that in general the names are not associated with non-white groups, making it unlikely that race was a factor in accounting for any observed discrimination. Based on this result, we consider that foreign names are not identified by the general Chicago population with any specific minority group.

Elicitation of employers' responses

The experiment was carried out between August 20 and February 28 in 2010 in Chicago, Illinois. We responded daily to job advertisements in the two major Chicago-area newspapers: the Chicago Sun-Times and the Chicago Tribune. Two screening rules are applied to answer advertisements in the study. First, we sought occupation categories that would not be heavily impacted by the 2008 financial crisis in order to have more call-backs. This led us to restrict offers to the fields of Healthcare, Accounting and Information Technology. We moreover responded to any advertisements that did not require personal appearances or telephone calls, and to which we had not already responded. We record the name of the employer, job title, the area of the company, any specific requirements indicated in the advertisement, whether or not the ad states that the company is an equal opportunity employer, whether or not the job is a government job, the date of the advertisement and where the company is based.

To design the applications, nine resumes (three for each field included in the study) of actual job seekers in the relevant geographic area -Chicago and surrounding suburbs, comprising the greater Chicagoland area in Illinois, Wisconsin and Indiana -are submitted to job vacancy ads online. 7 The resumes are modified to retain anonymity. We alter the names of various companies and educational institutions at which the job seeker has spent time, and we replace them with similar institutions in terms of standard qualitative rankings. Three resumes were sent to each advertisement: one with an Anglo-Saxon sounding name, one with an African-American sounding name and one with a Foreign name that should not be readily identifiable to the general populace. We cycle resumes -i.e., an equal number of each resume is associated with every set of namesto control for any biases in the perceived quality of the resumes. Resumes may be altered slightly to fit a particular advertisement, for example a particular skill may be added if it was requested in the advertisement. These alterations only occurred in the event that a specific skill was required in the job advertisements, and each resume sent out received the same alteration. Addresses were not included on the resumes to limit location-based inferences about the social class of the applicant. This should not lead employers to treat our applications differently, since initial correspondence for job offers is primarily done via phone and email. Note. Overview of academic studies using correspondence testing to measure ethnic discrimination. The first column indicates the author(s) of the study. The second column lists the ethnic group included. The next two columns provide the callback rates that the ethnic group received along with the ratio of callbacks between minority groups and the majority group -which is the one that appears first in the list of ethnic origins for each study. The last two columns describe the sectors the study examined and the location where the testing was performed.

Past experimental studies have indicated that substantial gender biases exist (see, e.g., [START_REF] Weichselbaumer | Is It Sex or Personality? The Impact of Sex Stereotypes on Discrimination in Applicant Selection[END_REF][START_REF] Booth | Does Racial Discrimination Vary Across Minority Groups? Evidence From Three Experiments[END_REF][START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF], female applicants being more likely to receive a callback offer. The effect is even more pronounced for minority applicants. To control for such biases, we seek to use only female names. However, this raises the obvious question that if the names we use are unfamiliar to Illinois residents, the same may well apply to the gender of these names. First, all of the resumes have some clear indicator of gender, such as a women's achievement award or membership in some sort of women's professional group. We also selected female first names of Albanian, Armenian and Georgian origin to create the Foreign sample of names. Last, we asked respondents of the initial survey to guess whether or not the name was male or female. For all of the names we used (Table 1), more than 90% of respondents either guessed that the name was female or left the field blank, indicating the names, although having unknown origins, appeared to be perceived as either female or unidentified. Note. Call-Backs elicited by first names from "identified" origins, ranked in increasing order. The data for the [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF] (B&M) study are taken from their Table 8.

Results

Table 2 contrasts the callback rates we elicit with those from similar experiments, along with the ratios of callback between the socially dominant group and various minority groups.8 Two results emerge. First, we observe discrimination against foreign sounding names, which are not identified with any minority group; and the callback ratio is very similar to the one we observe in the benchmark. Second, the difference in call-backs elicited between Anglo-Saxon sounding names and names associated with a clearly identified minority are in line with other experiments across the globe. In particular, our 1.41 callbacks ratio between black and white is very close to the one elicited by [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF], who found 1.49. Note, both studies focus on the same two ethnic groups, in the same geographic area. Results from our benchmark sets of names hence suggest discrimination did not evolve since the 2004 study of Bertrand and Mullainathan. We present a more precise comparison with existing results in Table 3. We rank the first names used in our study from the highest call-back rate to the lowest, along with an extract of [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF]'s results for the first names we used as well. Our overall callback rates are higher than those of [START_REF] Bertrand | Are Emily and Greg More Employable than Lakisha and Jamal? A Field Experiment on Labor Market Discrimination[END_REF], which can be attributed to the fact that we selected sectors in which there was high labor demand. Strikingly, when the same first names are used, the order in which names received callbacks is nearly the same in both studies. This holds true within both the white and black groups of names. This is even more significant given the fact that there is a statistically significant difference in the within group callbacks for several names.9 This may suggests that there exists some favouritism for certain given names beyond simple unequal treatment over ethnic groups. Note. The bottom part summarizes final results from the experiment for the Anglo-Saxon set of names (left-hand side), the African-American set of names (middle part) and the set of Foreign names (right-hand side). For each set, the first column provides the name of the applicant along with the number of applications, the second column shows the number and share of applications that received a request for further information or a meeting. The bottom part of the

Evidence of general ethnic homophily

Table 4 contains a more detailed summary of the results from our experiment, providing call-back rates for each name included in the study (sorted according to the guess rates provided in Table 1), organized by ethnic origin. Overall, Anglo-Saxon names generate far more callbacks than both African-American and Foreign names. Focusing on our baseline sets of names, resumes with Anglo-Saxon names have a 23% call-back rate compared to 16% for African-American. The bottom part of the Table displays the results of t-tests of two by two callback rate comparisons. We strongly reject (with p-value equal to 0.0009) the assumption that callbacks are the same as those elicited by Anglo-Saxon names, against the alternative that applications with Anglo-Saxon sounding names have better odds of obtaining a positive answer. African-American names are very likely to be identified as such by potential employers. Foreign names, by contrast, were intentionally chosen so that they are not associated with any identified group by employers, except for the fact they clearly sound exotic. Based on observed callbacks, we confirm our hypothesis that resumes with such names do face discrimination. On average, applications from this set of names obtain a positive call-back in 16.4% of cases. Again, we strongly reject the hypothesis of equal opportunities with Anglo-Saxon sounding names (with p-value equal to 0.0022): the experiment unambiguously confirms that some discrimination arises from general mistrust against, or uncertainty about non-majority sounding names.

The callbacks elicited by African-American and Foreign names moreover appear strikingly similar. As shown in the middle column of Table 4, the data are unable to reject the assumption of same mean callbacks between the two origins (p-value of 0.7674). Taken together, these two results are clear evidence of discrimination, since the design of the experiment guarantees that resumes are identical except for the perceived origin of the applicant's name: in order to generate the same Note. The Table reports how employers treated the three applications received from us: equal treatment (callbacks or no answer for all three origins), or favoritism towards one origin (middle part) -in the form of a call-back for only one application out of the three -or discrimination against only one origin (right-hand side) -in the sense that all names receives a callback except for one application. The first row refers to the number of employers, the second one to the share among all employers that received applications.

number of call-backs, potential applicants with African-American or Foreign names will have to send forty percent more resumes. While this cannot be translated directly into fewer job opportunities -since we have no data on interview success -this puts non-Anglo-Saxon job applicants at a distinct disadvantage in the labor market. The within group variance of callbacks for individual names moreover provide indirect support to variations due to familiarity. The variance appears much greater in the familiar Anglo-Saxon and African-American groups when compared to names in the Foreign group, with which employers are unfamiliar. 10Table 5 provides a complementary look at the output from the experiment, focusing on the extent, or lack thereof, of discrimination faced by the various types of applicants. Most employers treat all three applications they received from us in the same way, either calling back all three candidates or disregarding them all. Unequal treatment depending on the name of the applicant hence comes from 20% of all employers included in the study. Note exogenous variations in the quality of the resumes should induce a uniform distribution of employers across the cells of the Table. This is far from what we observe. There is some evidence of discrimination directed towards one group. Out of the three resumes sent that do not receive a callback, 23 employers exhibit such behavior: 4 exclude the Anglo-Saxon applicant, 10 exclude the African-American applicant and 9 exclude the Foreign applicant. This phenomenon is much more pronounced if we look at those employers that call back only one applicant out of the three: among discriminating employers, nearly two-thirds (26 out of 43) give a callback to the Anglo-Saxon name only and disregard any exotic sounding name. This represents three times the number of employers that favor only either the African-American or the Foreign application. Again, we observe equal treatment of African-American and Foreign sounding names.

Table 6 disaggregates the results according to the field of the job and the geographic location of the employer. Since we send three resumes to each ad, we have the same number of applications from each origin in each field (the number of applications appear in parenthesis in each column's title). Whatever the field, Anglo-Saxon applications receive the larger share of call-backs. Fields are ranked according to the discrimination ratio between the Anglo-Saxon and the non Anglo-Saxon Note. For each origin in raw, the first column recalls the overall call-back rates elicited by the corresponding applications.

The next three columns provides the call-backs by field of the job offer (total number of applications appears in brackets in the title of the column): accounting, nursing, programming. The next two columns distinguish call-backs based on the geographic location of the potential employer. The last three columns report the correlation between call-backs and the correct guesses of origin for the corresponding name we elicited in the survey.

sounding names. While the discrimination ratio is quite large in accounting, we observe almost no difference between ethnic origins for applications in programming occupations. This difference is very unlikely to be explained by differences in tightness across occupations-specific labor markets.11 One difference between the programming sector and both accounting and nursing is that employees in the programming sector are less likely to be in direct contact with customers.

To explore further potential heterogeneity in employer's behavior, the right-hand side of Table 6 distinguishes callbacks elicited by each origin depending on the location of the potential employer. Using a t-test, we cannot say that call-backs are significantly lower for Anglo-Saxons as opposed to other groups in the city of Chicago (the p-value for equal means with non-Anglo-Saxons is .1710), while we can say that call-backs are significantly higher for Anglo-Saxons than other groups in the suburbs -p-values of t-test of equal means are .0055 against non-Anglo-Saxons, .0032 against African-Americans and .0092 against foreign names. Last, we cannot definitively say whether or not there is a statistically significant difference between callbacks for members of the African-American sample and the exotic sample in either the city of Chicago or the Chicago suburbs -the respective p-values of equal means t-tests are .9859 for Chicago, .6623 for suburbs. Once we break down the results by geographic area, nearly all of the discrimination that we see hence comes from employers in the Chicago suburbs as opposed to the city center.

Such a correlation between suburbanization and discrimination is consistent with the sociology induced by the historical development of suburbs in the United States. The impetus for the expansion of many Chicago suburbs is that former city residents fled the city center once non-white ethnic groups moved into Chicago. This phenomenon of "white flight" -white families choosing to leave an area once black families begin to move in -is well documented in the sociological literature in the United States [START_REF] Crowder | The Racial Context of White Mobility: An Individual-Level Assessment of the White Flight Hypothesis[END_REF]. As a result, whites are a plurality with 37.6% of the population in the city of Chicago, whereas whites are 78.1% of Cook County excluding Chicago. 12 Whites are thus more prevalent in the Chicago suburbs when compared to the city of Chicago itself, which is extremely diverse and in which no racial group has an absolute majority. What is more, many residents of suburbs who moved there due to the "white flight" phenomenon live in the suburbs precisely because they wanted to live in communities made up primarily of their own ethnic group. Self-selection into geographic areas may hence reinforce favouritism towards the dominant ethnic group in the suburbs. 13 The last three columns of Table 6 provide spearman rank-order correlations of callbacks with the guesses elicited in the survey as regards the ethnic origin of the name. The correlation is positive for Anglo-Saxon sounding names. Note, however, that all guesses are very high for this set of names, so correlations for this group are computed on very thin variations. For the non Anglo-Saxon groups, by contrast, the correlations are negative: the higher the rate of respondents identifying the name as either African-American or Foreign, the lower the number of call backs the name elicits. Comparing these correlations according to the location, we again confirm strong differences between suburbs and the city proper: while correlations are quite similar across all ethnic groups in Chicago, there is a clear contrast between Anglo-Saxon and non Anglo-Saxon sounding names in the suburbs.

Summary of the results

We assess the overall significance of our results using Probit regressions on callbacks. Table 7 shows the results from several specifications. The main variables of interest are dummies for the origin of the applicant. In all estimated equations, we include controls for employers and job offers characteristics. We use dummy variables indicating whether a bachelors degree, certification or experience is mentioned as a necessary prerequisite for the job, and whether the vacancy advertisement states that the company is an equal opportunity employer. Last, we include a measure of the ethnic diversity of people living in the geographic location of the employer, through the percent of African-Americans living in the geographic area in which the job is located. The data was collected at the city or town level from the US Census American Community Survey in 2009.

The first model contrasts callbacks of Anglo-Saxon sounding names with the other two origins, and the second one estimates separately the effect of African-American sounding names and Foreign sounding names. Both regressions confirm our main result, with higher significance levels. Having an Anglo-Saxon name has a positive effect on eliciting a call-back and both other origins are negatively correlated (with comparable marginal effects) with callbacks. Models (3) and (4) explore the differences between locations, through the interaction with a dummy variable for employers in Chicago. Both estimation results are conditional on the ethnic diversity of the location, defined 12 Our source is the US Census. Note, these numbers include Hispanic whites, who are also considered a separate minority group. If we consider only non-hispanic whites, whites make up 30.6% of the city's population, being the second largest ethnic group after non-Hispanic African-Americans at 34.7% of the population.

13 It is worth noting that Bertrand and Mullainathan failed to find any relationship between zip-codes and the percentage of African-Americans in that zip-code. However, zip-codes may not be the best proxy to measure racial differences in areas as the postal zip codes do not really measure any real boundaries between community zones in Chicago.

This paper relies on a correspondence test in Chicago to measure the extent of ethnic homophily in the labor market, sending 990 resumes to different employers with three sets of names: Anglo-Saxon, African-American and Foreign names that are not clearly identified with any specific origin by the general population. The design of the experiment allows us to contrast discrimination faced by clearly identified minority groups with discrimination suffered by applicants for whom employers have no specific belief or taste -except for the fact that the name is clearly atypical. Based on callbacks elicited by each set of names, we find that resumes with African-American and Foreign names receive one third fewer callbacks than resumes with Anglo-Saxon names. This suggests discriminatory behavior is part of a larger pattern of ethnic homophily, i.e. either general distrust against individuals not belonging to the dominant group or a kind of nepotism in favor of members of the dominant group. The rate at which resumes with foreign names receive callbacks is moreover very similar to the rate obtained by African-American names. This result is further substantiated by the geographic pattern of discrimination: the largest part of observed discrimination comes from the suburbs, where whites are a clear majority, while we do not find any significant difference in the rate of callbacks in Chicago proper, an extremely diverse city where non-Hispanic whites are a minority group.

In the policy sphere, programs to counter discriminatory treatment such as affirmative action programs in the United States have traditionally focused on aiding specific minority groups. This implicitly relies on the assumption that these groups are specifically targeted by discrimination. By contrast, the general ethnic homophily substantiated in this paper suggests that many members of other unrecognised minority groups may face discrimination, and that current programs may ignore this fact when only focusing on recognized groups. Our results also suggest that any migration flows of people coming from geographic areas that are unknown to most residents face discrimination in the labor market. This should turn attention to programs that rule out as much name identification as possible in the job matching process, such as mandatory anonymity of resumes.

Table 1 :

 1 Origins guessed from the initial survey Most Com. 2 nd Most Com. 3 rd Most Com.

	Anglo-saxon names				Afro-American names
		Correct guess Male			Correct guess Male
	Kristen Smith	100	0	Ebony Williams		84	1
	Carrie Johnson	100	0	Latoya Brown		82	1
	Meredith Miller	99	0	Tanisha White		85	1
	Sarah Davis	100	0	Jasmine Washington		82	0
	Laurie Anderson	99	0	Lakisha Jones		85	1
			Foreign names
		Guesses		Perceived origin among guessing respondents
	Unknown Male 1 st Dziella Lousaper 86 1 French	55	German	9	Spanish	9
	Edona Sanatroug	85	3	German	27	Russian	10	Chinese	10
	Sofiko Kushtrim	78	23	Russian	36	Indian	7	Dutch	7
	Aferdita Dzaghig	75	11	Indian	17	African	6	Asian	6
	Gadarine Besnik	75	11	Russian	33	Polish	27	Turkish	20

Table 2 :

 2 Comparison with previously observed discrimination

		Ethnic	Call-Back Discrimination	Job offers
		Origin	rate	ratio	Sectors	Location
	This study	White	23.0	1.00	Accounting,	Chicago
		Black	15.8	1.46	Health,
		Foreign	16.4	1.41	IT
	Bertrand et al. (2004) White	9.9	1.00	Retail.	US Female
		Black	6.6	1.49	Sales
	Bertrand et al. (2005) White	8.1	1.00		Chicago
		Black	5.4	1.49
	Booth et al. (2009)	Anglo-Saxon	35.0	1.00	Waitstaff	Australia
		Indigenous	26.0	1.35	Cust. Service
		Chinese	21.0	1.67	Sales
		Italian	32.0	1.09	Data Entry
		Middle East	22.0	1.59	Admin. Finance
	Oreopoulos (2009)	English	15.8	1.00	Sales Prog.	Canada
		Indian	12.1	1.31	Retail
		Chinese	10.8	1.46
		Pakistani	11.1	1.42
		Foreign	11.3	1.14
	Duguet (2007)	French	16.6	1.00	Restaurants	France
		Moroccan	6.0	2.77	Hotels
		Fr. Surname &	10.7	1.55
		Mor. Lastname

Table 3 :

 3 Identified origins: first names ranked by call-backs

	Anglo-saxon names		African-American names
	This study	B&M (2004)	This study	B&M (2004)
	Kristen	39.2 Kristen	13.1	Ebony	25.3 Ebony	9.6
	Sarah	28.0 Carrie	13.1	Latoya	16.3 Latoya	8.4
	Meredith 21.3 Meredith 10.2	Tanisha 16.3 Tanisha 5.8
	Carrie	16.7 Sarah	9.8	Jasmine 12.1 -	-
	Laurie	12.9 Laurie	9.7	Lakisha	8.0	Lakisha 5.5

Table 4 :

 4 Experimental Results

	Anglo-saxon names		Afro-American names		Foreign names
	Applications	Callbacks	Applications	Callbacks	Applications	Callbacks
	Kristen Smith (51)	20 39.2%	Ebony Williams (79)	20 25.3%	Dziella Lousaper (55)	9	16.3%
	Carrie Johnson (54)	9	16.7%	Latoya Brown (55)	9	16.3%	Edona Sanatroug (86)	9	10.4%
	Meredith Miller (80)	17 21.3%	Tanisha White (55)	9	16.3%	Sofiko Kushtrim (89)	15 16.9%
	Sarah Davis (75)	21 28.0%	Jasmine Washington (66)	8	12.1%	Aferdita Dzaghig (55)	11 20.0%
	Laurie Anderson (70)	9	12.9%	Lakisha Jones (75)	6	8.0%	Gadarine Besnik (45)	10 22.2%
	Total (330)	76 23.0%	Total (330)	52 15.8%	Total (330)	54 16.4%
				Student t-tests
	H 0 : Anglo-saxon = Afro-American	H 0 : Afro-American = Foreign	H 0 : Anglo-saxon = Foreign
	H 1 : Anglo-saxon ≥ Afro-American	H 1 : Afro-American = Foreign	H 1 : Anglo-saxon ≥ Foreign
	t = 3.1321, p-value=0.0009	t = 0.2960, p-value=0.7674		t = 2.8736, p-value=0.0022

  Table provides t-tests testing the assumption H 0 of equality in mean callbacks elicited by different ethnic origins, against the alternative H 1 .

Table 5 :

 5 Discrimination intensity from employers

	Equal	One origin discriminated		One origin favored
	treatment	Anglo-Saxon Afro-American Foreign	Anglo-Saxon Afro-American Foreign
	264	4	10	9	26	8	9
	(80%)	(1%)	(3%)	(3%)	(8%)	(2%)	(3%)

Table 6 :

 6 Callbacks by field, location and guesses

			Field of the job		Employer location	Correlation with guesses
		Overall	Account. Nursing Prog.	Chicago Suburbs	Overall Chicago Suburbs
			(135)	(106)	(89)
	Anglo-Saxon	0.230	0.267	0.198	0.213	0.202	0.257	0.586	0.253	0.773
	Afro-American	0.158	0.156	0.151	0.191	0.159	0.156	-0.310	0.161	-0.368
	Foreign	0.164	0.178	0.113	0.180	0.159	0.168	-0.832	0.566	-0.350
	Overall	0.184	0.200	0.154	0.195	0.174	0.194	0.349	0.079	0.139

We restrict ourselves to a brief presentation of the literature. See[START_REF] Lang | Racial Discrimination in the Labor Market: Theory and Empirics[END_REF] for a detailed survey of both existing theories and their empirical relevance.

Documents de Travail du Centre d'Economie de la Sorbonne -2011.13

There are also ethical concerns, as sending out fake resumes or letters is a form of deception. See[START_REF] Riach | Deceptive Field Experiments of Discrimination: Are they Ethical?[END_REF] for a discussion.

Some recent studies substantiate the existence of ethnic discrimination beyond the labor market.[START_REF] Correll | The police officer's dilemma: Using ethnicity to disambiguate potentially threatening individuals[END_REF] use a videogame to demonstrate that subjects are quicker in deciding not to shoot an unarmed white target as opposed to an unarmed African-American target. Based on survey and audit data,[START_REF] Ayres | To Insure Prejudice: Racial Disparities in Taxicab Tipping[END_REF] show that African-American cab drivers receive smaller tips than their Anglo-Saxon counter-parts.[START_REF] List | The Nature and Extent of Discrimination in the Marketplace: Evidence from the Field[END_REF] uses evidence from the sports card trading market, and finds evidence of discrimination on both sides, although "consumer-side" discrimination is more pronounced than "seller-side" discrimination.6Documents de Travail du Centre d'Economie de laSorbonne -2011.13

We used the Cook County Municipalities and Demographics survey, available from the Census Bureau at www.census.gov.

The full results are available from authors upon request. 8 Documents de Travail du Centre d'Economie de la Sorbonne -2011.13

We use resumes posted on www.bestsampleresumes.com.

[START_REF] Booth | Does Racial Discrimination Vary Across Minority Groups? Evidence From Three Experiments[END_REF] also provide a detailed table of results from various studies not reported here, the results of which are broadly in line with other studies: the ratios range from 1.1 (Greeks in Australia in 1991) to 2.77 (Moroccans in France in 2007).

p-values of the joint chi squared-test of equal means are 0.009 for the Anglo-Saxon group, 0.0559 for the African-American group, and 0.4094 for the Foreign group

The p-value of the joint chi squared-test of equal means between names is 0.4094 within the Foreign group, while the differences are significant in both other groups.

According to the 2009 Occupational Employment Statistics from Bureau of Labor Statistics, the unemployment rate is 4% for computer programmers and interactive media developers, 2.9% for nurses and 2.2% for accountants, while it is 7% on the average for professional occupations.

Dummy variables are also added indicating: whether or not the vacancy advertisement indicates a bachelors degree, certification or experience as a necessary prerequisite for the job, and whether the add states that the company is an equal opportunity employer. The variable % Afro-American measures the percent of African-Americans living in the geographic area (defined at the city or town level) in which the job is located. The number of observations is N = 990 in all specifications.

at a narrower level than the suburb/city proper distinction. The results described previously are robust to this further conditioning: no discriminatory behavior can be found in the city proper, and all discrimination we observe comes from the suburbs. Last, Model (5 ) accounts for differences across fields in which jobs are offered. The p-values on estimated coefficients confirm that most discrimination against non-Anglo-Saxons comes from offers in accounting. The covariates included in the model seems to account for the differences observed above between the two other sectors: neither the fields by themselves nor the interaction with the ethnic origin of the name have any significant impact on call-backs.