Yannick Baraud
email: baraud@unice.fr

ESTIMATION OF THE DENSITY OF A DETERMINANTAL PROCESS

Keywords:

We consider the problem of estimating the density Π of a determinantal process N from the observation of n independent copies of it. We use an aggregation procedure based on robust testing to build our estimator. We establish non-asymptotic risk bounds with respect to the Hellinger loss and deduce, when n goes to infinity, uniform rates of convergence over classes of densities Π of interest.

Introduction

The starting point of this work goes back to 2007 when Persi Diaconis visited our Laboratory Jean-Alexandre Dieudonné in Nice. At that time, he explained that determinantal processes were emerging in many areas and that there was no statistical procedure to estimate their distributions. Almost five years later, it still seems to be the case. The aim of this paper is therefore to contribute to the study of these processes. Our aim is not only to focus on statistical estimation but also to discuss some related problems. For example, how the class D of all determinantal densities can be parametrized? Is there an identifiable way of doing it? Another natural question, at least for a Statistician, is to understand how the elements of D can be approximated. Are there some specific parametric sets that should be used to approximate the densities lying in D? If so, what can be said about the approximation properties of these sets? Finally, given n independent copies of a determinantal process N , we propose an estimator of the density Π of N . We establish non-asymptotic risk bounds for our estimator and deduce uniform rates of convergence over classes of Π of interest. It turns out that our estimation strategy is robust with respect to the assumption that N is a determinantal process. This means that the risk bounds we get are not only valid when Π belongs to the class D but also when Π is close enough to it (in the Hellinger distance). Our approach is based on T -estimation as introduced by [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF]. More precisely, we start with a suitable family of models, which typically consists of compact sets of densities, and the role of which is to provide a good approximation of the elements of D. Then, we discretize these models.

This results in a family of points (Π m ) m∈M of D and we finally use the data in order to select a suitable point among the Π m . The way we select this point, which provides our estimator of Π, is based on robust testing and aims at finding an element among the Π m which is as close as possible to the target density Π. We establish non-asymptotic risk bounds for our estimator and show how they depend on the approximation properties of the models we started from. Under a posteriori assumptions on Π and for a suitable choice of the models, we specify this bounds and derive rates of convergence.

For an introduction to determinantal processes, we refer the interested reader to [START_REF] Lyons | Determinantal probability measures[END_REF], [START_REF] Hough | Determinantal processes and independence[END_REF] and the books by [START_REF] Anderson | An introduction to random matrices[END_REF] and [START_REF] Hough | Zeros of Gaussian analytic functions and determinantal point processes[END_REF] as well as the references therein. Part of the popularity of determinantal processes comes from the fact that they naturally arise in the study of the eigenvalues of large random matrices. Recently, [START_REF] Borodin | On adding a list of numbers (and other one-dependent determinantal processes)[END_REF] showed that these processes are also involved in the process of "caries" when adding a column of numbers.

The paper is organized as follows. In Section 2 we settle the probabilistic background as well as our main notations and conventions. We introduce determinantal processes in Section 3 and tackle the problem of estimating of their densities in Section 4. Finally, Section 5 is devoted to the proofs.

2. The background 2.1. Notations and conventions. Throughout this paper we use the conventions ∅ = 0 and ∅ = 1 and set N * = N \ {0} and R * + = (0, +∞). Given a finite set A, |A| denotes the cardinality of A and for z ∈ C, ℜ(z), z and |z| denote the real part, conjugate and modulus of z respectively. We denote by P the class of all finite subset J of N * and set P * = P \ {∅}. Moreover, we set

Λ = {λ ∈ [0, 1] N * , |λ| 2 = j≥1 λ 2 j < +∞}.
All along, we consider a metric space (X , d) which we endow with its Borel σ-field B(X ) and a σ-finite measure µ. Roughly speaking, a point process on (X , B(X )) will correspond to a random choice of a family of distinct points among X . One should typically think of X as {1, . . . , p}, N, R or R p for some positive integer p. When X is not finite, we denote by H the Hilbert space of measurable and complex-valued functions φ on (X , B(X )) satisfying

φ 2 = ˆX |φ| 2 dµ < +∞.
We endow H with the Hermitian inner product defined for φ, ψ ∈ H by φ, ψ = ˆX φψdµ.

For conveniency, we adopt the convention that ., . is linear with respect to the second argument and not the first one, as usually the case. In order to keep our notation as simple as possible, when X is finite, say X = {1, . . . , p}, we embed X into N * and use

H = ℓ 2 (N * ) = {φ ∈ C N * , i≥1 |φ(i)| 2 < +∞}.
More precisely, a mapping φ on X = {1, . . . , p} with values in C will be viewed as a sequence (φ(i)) i≥1 ∈ ℓ 2 (N * ) with φ(i) = 0 for all i > p. Since H is an infinite dimensional Hilbert space (whatever X ), we may define U as the set of all orthonormal sequences Φ = (φ j ) j≥1 in H. For J ∈ P * and Φ ∈ U, we set Φ J = (φ j ) j∈J and given an (ordered) finite subset α of X , denote by Φ α,J the |α| × |J|-matrix Φ α,J = (φ j (x)) x∈α,j∈J .

We extend this notation for rectangle matrices A with entries in C:

A α,J = (A i,j ) i∈α,j∈J . Moreover, A * denotes the transpose of the conjugate of A, that is, if A = (A i,j ) i=1,...,k,j=1,...,k ′ , A * = (A j,i ) j=1,...,k ′ ,i=1,...,k .
Finally, we recall that the Hellinger distance h between two densities p, q on a measured space (E, E, ν) is defined by the formula

h 2 (p, q) = 1 2 ˆE ( √ p - √ q) 2 dν.
For the sake of simplicity, we shall keep the same notation h throughout this paper even though the measured space (E, E, µ) may vary.

2.2. The probabilistic background. In this section, our aim is to introduce the probabilistic background we shall use throughout this paper. We denote by X the class of all finite subsets of X and for k ∈ N, denote by X k the class of those subsets with cardinality k. By convention, X 0 = {∅}. We identify X with the set of finite measures of the form

α = x∈α δ x with α ∈ X
and denote the same way α and α so that for all B ∈ B, α(B) means |α ∩ B|.

We equip X with the smallest σ-field B(X) for which the mappings

X → N M B : α → α(B)
are measurable for all B ∈ B(X ). In particular, the subsets X k = M -1 X (k) are measurable for all k ∈ N. We endow (X, B(X)) with the measure L defined for all measurable functions f from X into R + by

ˆX f (α)dL(α) = f (∅) + k≥1 1 k! ˆX ∨k f ({x 1 , . . . , x k })dµ(x 1 ) . . . dµ(x k )
where X ∨k is the set of all k-uplets (x 1 , . . . , x k ) ∈ X k with distinct coordinates. If X is finite, say X = {1, . . . , p}, and if µ is the counting measure on X then L is merely the counting measure on X.

Throughout this paper, a point process N on (X , B(X )) is a random variable defined on a probability space (Ω, A, P) with values in (X, B(X), L).

Introduction to determinantal processes

In this section, our aim is to define a determinantal process on (X , B(X )). To do so, we adopt the point of view developed in [START_REF] Hough | Determinantal processes and independence[END_REF]. In particular, we start with the simpler case of determinantal projection processes.

3.1. Determinantal projection processes.

Definition 1. Given J ∈ P * and Φ ∈ U, a determinantal projection process N of rank |J| with parameter Φ J = (φ j ) j∈J is a point process with density (with respect to L) given by

(3.1) Π Φ J (α) = |det [Φ α,J ]| 2 ½ X |J | (α) for all α ∈ X. When J = ∅, by convention Π Φ ∅ = δ ∅ .
If the matrix Φ α,J = (φ j (x)) x∈α,j∈J depends on an ordering on the set α, |det [Φ α,. ]| does not and we shall therefore omit to specify one. It follows from the definition of Π Φ J that with probability 1, |N (X )| = |J|. Hence, a determinantal projection process N of rank |J| consists of |J| distinct points of X . The location of these points depends on the geometry of the φ j for j ∈ J. If the φ j are realvalued, a configuration α = {x 1 , . . . , x k } of points is all the more likely that the volume of the parallelepiped based on the |J| vectors ((φ j (x 1 ), . . . , φ j (x k )) j∈J is large.

The fact that Π Φ J is a density on X might not be clear at first sight. In fact, when X = {1, . . . , p} this comes the the Cauchy-Binet formula: if A, B are k × p and p × k matrices respectively with p ≥ k, the Cauchy-Binet formula asserts that

(3.2) det [AB] = α∈X k det A {1,...,k},α det B α,{1,...,k} .
Again, note that this formula is independent of the choice of an ordering on α. By using the Cauchy-Binet formula with B = (φ j (x)) x∈X ,j∈J = Φ X ,J , A = B * and by using the fact the family (φ j ) j∈J is orthonormal we get

ˆXk Π Φ J (α)dL(α) = α∈X k det [Φ α,J ] det [Φ α,J ] = α∈X k det Φ * J,α det [Φ α,J ] = det Φ * J,X Φ X ,J = det φ i , φ j i,j∈J = 1.
When X is no longer finite, the Cauchy-Binet formula can be extended by using the identity below from which we can deduce in a similar way as above that Π Φ J is a density.

Proposition 1. Let Φ = (φ 1 , . . . , φ k ) and Ψ = (ψ 1 , . . . , ψ k ) be two elements of H k . We have that det ( φ i , ψ j ) i,j=1,...,k = ˆXk det Φ * {1,...,k},α det Ψ α,{1,...,k} dL(α).

This identity is known for a long time, especially when X is a compact interval of R and µ the Lebesgue measure on X (see de [START_REF] De Bruijn | On some multiple integrals involving determinants[END_REF]). The general form of this identity can be found in [START_REF] Baik | Algebraic aspects of increasing subsequences[END_REF].

3.2. The general case. As proved in [START_REF] Hough | Determinantal processes and independence[END_REF], the distribution of a (finite) determinantal process N can be viewed as a mixture of densities of some determinantal projection processes. More precisely, a determinantal process can be defined as follows.

Definition 2. Let Φ ∈ U and λ ∈ Λ. A determinantal process N with parameters (Φ, λ) is a point process with density

(3.3) Π Φ,λ = J∈P p λ J Π Φ J where p λ J = j∈J λ 2 j j ∈J
(1 -λ 2 j ) for all J ∈ P.

We use the convention

Π Φ ∅ = δ ∅ .
Since λ j ∈ [0, 1] for all j ≥ 1 and

(3.4) j≥1 λ 2 j < +∞, the numbers p λ J are nonnegative and well defined (the infinite product j ∈J (1λ 2 j ) converges for all J ∈ P). Besides,

J∈P p λ J = j≥1 (λ 2 j + (1 -λ 2 j )) = 1.
Consequently, Π Φ,λ is indeed an (at most countable) mixture of densities. Given J ∈ P * , it is not difficult to see that for the particular choice λ = λ J = (½ j∈J ) j≥1 , the density Π Φ,λ is that of a determinantal projection process with parameter Φ J : indeed, for J ′ = J, p λ J ′ = 1 and for J ′ = J, p λ J ′ = 0. As explained in [START_REF] Hough | Determinantal processes and independence[END_REF], another way of defining a determinantal process is as follows. First simulate a sequence (Z j ) j≥1 of independent Bernoulli random variables with respective parameters (λ 2 j ) j≥1 . Consider the subset J of those indices j ≥ 1 such that Z j = 1. Finally choose N according to a determinantal projection process of rank | J | with parameter Φ J . With such a description, Condition (3.4) is easy to understand: together with the Borel-Cantelli lemma, it ensures that J is finite almost surely. It is also clear that the distribution of a determinantal process remains unchanged if we change the labelling of the pairs ((λ j , φ j )) j≥1 . That is, for all bijection σ on N * , the parameters ((φ j ) j≥1 , (λ j ) j≥1 ) and ((φ σ(j) ) j≥1 , (λ σ(j) ) j≥1 ) lead to the same determinantal distribution. In particular, with no loss of generality, we may assume that the sequence λ = (λ j ) j≥1 is non-increasing with respect to j.

In the literature, one usually associates to a determinantal process N a square integrable kernel K on X 2 which defines a self-adjoint compact operator on H by the formula

H → H T K : φ → x → ˆX K(x, y)φ(y)dµ(y) . (3.5)
The sequences (λ 2 j ) j≥1 and Φ = (φ j ) j≥1 mentioned above correspond then to the eigenvalues and associated eigenvectors of T K . Conversely, given the sequences λ = (λ j ) j≥1 and Φ = (φ j ) j≥1 and provided that µ(X ) < +∞, the kernel K can be obtained by the fomula (Mercer's Theorem)

(3.6) K(x, y) = j≥1 λ 2 j φ j (x)φ j (y)
where the series converge absolutely for almost every (x, y) ∈ X 2 . When X = {1, . . . , p}, K is merely (any) p × p Hermitian matrix with eigenvalues in [0, 1].

Interestingly, the kernel K can be related to the distribution of N by the following formula which holds for all measurable functions f from

X into R + E α⊂N f (α) = ˆX f (α) det[K α,α ]dL(α) where K α,α = (K(x, y)) x∈α,y∈α .
The mapping α → det[K α,α ] determines the distribution of N and is called the correlation function. When X = {1, . . . , p}, this formula simply says that for all α ⊂ X P

[α ⊂ N ] = det[K α,α ].
3.3. Hellinger distance and determinantal process. In the previous section, we have seen that the distribution of a determinantal process can be parametrized by a pair (Φ, λ) in U × Λ and that, conversely, any choice of such a pair allows to define a determinantal process. The aim of this section is to relate the Hellinger distance between the distributions of two determinantal processes associated to two distinct pairs (Φ, λ) and (Ψ, γ) to some distance between these pairs. Again, we start with the simpler case of a determinantal projection process.

3.3.1. Case of a determinantal projection process. Let Φ = (φ j ) j≥1 and Ψ = (ψ j ) j≥1 be two elements of U and J, J ′ two elements of P.

If |J| = |J ′ |, the supports of Π Φ J and Π Ψ J ′ are disjoint (the densities are supported by X |J| and X |J ′ | respectively) and hence h 2 (Π Φ J , Π Ψ J ′ ) = 1. If J = J ′ = ∅, Π Φ J = Π Ψ J ′ and therefore h 2 (Π Φ J , Π Ψ J ′ ) = 0.
Consequently, the only case we need to consider is the one where |J| = |J ′ | ≥ 1. In fact, as already mentioned, we may re-index one of the two sequences, say Ψ, in order to have J = J ′ without changing the distribution Π Ψ . By doing so, the following result holds.

Proposition 2. Let J ∈ P * and Φ, Ψ ∈ U. We have,

h 2 (Π Φ J , Π Ψ J ) = 1 - ˆXk |det [Φ α,J ]| |det [Ψ α,J ]| dL(α) (3.7) ≤ 1 -det ( φ i , ψ j ) i,j∈J . Moreover, h 2 (Π Φ J , Π Ψ J ) ≤ 5 2 j∈J φ j -ψ j 2 .
Up to constants, the last inequality is sharp. For example, if J = {1}, Π Φ {1} and Π Ψ {1} correspond to the two densities on (X , B(X )) given by

(3.8) Π Φ {1} (x) = |φ 1 (x)| 2 and Π Ψ {1} (x) = |ψ 1 (x)| 2 for all x ∈ X and hence, if φ 1 , ψ 1 are two nonnegative real-valued functions on (X , B(X )), h 2 Π Φ {1} , Π Ψ {1} = 1 2 ˆX Π Φ {1} -Π Ψ {1} 2 dµ = 1 2 φ 1 -ψ 1 2 .
Clearly, this equality is no longer true when the nonnegativity assumption on φ 1 and ψ 1 is violated. Nevetheless, Proposition 2 says that the inequality remains true (up to a constant). The proof of this proposition is postponed to Section 5.1.

3.3.2. The general case. Since Π Φ,λ and Π Ψ,γ are mixtures, the problem of bounding the Hellinger distance between these two densities amounts to understanding how, more generally, the Hellinger distance behaves with respect to mixtures of densities. More precisely, let p, q be two densities on the measured space (T, T , m) and (P t ) t∈T and (Q t ) t∈T two families of densities on a measured space (E, A, ν). What can we say about the Hellinger distance between the two mixtures

P = ˆT P t p(t)dm(t) and Q = ˆT Q t q(t)dm(t)
when we known how far p is from q and the P t from the Q t ? The following result gives an answer.

Proposition 3. If m and ν are both σ-finite,

h 2 (P, Q) ≤ 2h 2 (p, q) + 2 ˆT h 2 (P t , Q t )q(t)dm(t).
The proof of this result is postponed to Section 5.2.

We may apply Proposition 3 with the choices T = P (m being the counting measure on P), (E, A, ν) = (X, B(X), L), p = p λ the density defined on P by p λ (J) = p λ J for all J ∈ P, q = p γ defined analogously and for t = J ∈ P, P t = Π Φ J and Q t = Π Ψ J . We obtain the following result the proof of which is detailed in Section 5.3. Proposition 4. Let Φ, Ψ ∈ U and λ, γ ∈ Λ and set

λ = 1 -1 -λ 2 j j≥1 and γ = 1 -1 -γ 2 j j≥1
.

The following inequalities hold

h 2 (p λ , p γ ) ≤ |λ -γ| 2 + λ -γ 2 (3.9) J∈P p γ J h 2 (Π Φ J , Π Ψ J ) ≤ 5 2 j≥1 γ 2 j φ j -ψ j 2 . (3.10)
In particular,

(3.11) h 2 (Π Φ,λ , Π Ψ,γ ) ≤ 2 |λ -γ| 2 + λ -γ 2 + 5 j≥1 γ 2 j φ j -ψ j 2 .

Statistical estimation

Throughout this section, we consider a point process N on (X , B(X )) with density Π with respect to L. Given n independent copies N 1 , . . . , N n of N , our aim is to estimate Π. One may naturally think of N as being a determinantal process which means that Π belongs to the set D of all determinantal distributions. Nevertheless, our result is robust with respect to such an assumption in the sense that Π may not belong to D. In this case, one may rather consider D as an approximation set for Π. Before turning to the estimation of Π, we shall first discuss some identifiability issues which are of independent interest and may therefore be skipped.

4.1. Identifiability and exterior algebra. When N is a determinantal process, we may write Π = Π Φ,λ for some pair (Φ, λ) ∈ U × Λ or, alternatively, define Π from some kernel K on X 2 as in (3.6). If these two approaches provide a parametrization of D, none is identifiable. More precisely, two distinct pairs in U × Λ or two distinct kernels may parametrize the same determinantal distribution. This lack of identifiability is already true if one restricts to the simpler class of determinantal projection processes. A simple counter-example can be obtained from (3.8) with (X , B(X

), µ) = ([0, 1], B([0, 1], dx) by taking φ 1 (x) = e ix and ψ 1 (x) = e 2ix .
In this case, the corresponding kernels K 1 (x, y) = e i(x-y) and K 2 (x, y) = e 2i(x-y) are distinct but both parametrize the uniform distribution on [0, 1]. It is also clear from this counter-example that there is no hope to estimate φ 1 , which is not identifiable either.

Consequently, a question arises. How can we define a one-to-one parametrization of D? As we shall see, this problem is rather difficult. In fact, we shall partially answer this question by restricting ourself to the case where X = {1, . . . , p} and by focusing on the class D p,k of all determinantal projection distributions of rank k with k ∈ {1, . . . , p -1} and p ≥ 2. It follows from Definition 1 that for each element Π ∈ D p,k , there exists an orthonormal family φ 1 , . . . , φ k (which is certainly not unique) such that for all α ∈ X k (4.1)

Π(α) = det Φ α,{1,...,k} 2 .
Let us now consider the exterior algebra E = k C p consisting of the sums of k-blades φ 1 ∧ . . . ∧ φ k with φ 1 , . . . , φ k ∈ C p . Since we shall only use the algebraic properties of these objects and more specifically their connections with determinants, we shall not define them and rather refer the interested reader to Mac Lane and Birkhoff (1988) (Chapter XVI, Section 7). Denoting by e 1 , . . . , e p the canonical basis of C p , this exterior algebra E can be viewed as a C-linear space, a basis of which being given by the k-blades of the form

e α = e i 1 ∧ . . . ∧ e i k where α = (i 1 , . . . , i k ) (with 1 ≤ i 1 < i 2 . . . < i k ≤ p) varies among X k .
This linear space can be equipped with an Hermitian inner product [., .] for which the elements (e α ) α∈X k provide an orthonormal family of E. Besides, for a k-blade

φ 1 ∧ . . . ∧ φ k (4.2) [e α , φ 1 ∧ . . . ∧ φ k ] = det Φ α,{1,...,k} for all α ∈ X k .
Let S E be the unit sphere of (E, [., .]), G the subset of S E gathering the elements of the form φ 1 ∧ . . . ∧ φ k for φ 1 , . . . , φ k being an orthonormal family of C p and G + be the subset of S E defined by

G + = {g + = α∈X k |[e α , g]| e α g ∈ G}.
It follows from (4.1) and (4.2) that the mapping

G + → D p,k g + → Π g+ : α → |[e α , g]| 2 = |[e α , g + ]| 2
is surjective. It is also clearly one-to-one and provides thus an identifiable parametrization of the elements of D p,k by those of G + . In fact, if ∆ denotes the Hermitian distance on E defined for g, g

′ ∈ E by ∆ 2 (g, g ′ ) = [g -g ′ , g -g ′ ], (G + , ∆) and (D p,k , √ 2h) are isometric: by (3.7), for all g + , g ′ + ∈ G + , ∆ 2 (g + , g ′ + ) = α∈X k [g + , e α ] -g ′ + , e α 2 = 2   1 - α∈X k |[g, e α ]| g ′ , e α   = 2h 2 (Π g+ , Π g ′ + ).
The metric dimension (in the sense given in [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF]) of a set of densities is usually closely related to the minimax rate of estimation over this set. Roughly speaking, if the metric dimension of the set is D, one can expect that the minimax rate be of order D/n. The above isometry shows that the metric dimension D p,k of (D p,k , h) is the same as that of the subset G + ⊂ E for the Hermitian distance. In particular D p,k is not larger than the dimension of E (in the usual sense, viewed as a linear space on R), that is D p,k ≤ 2 p k . This upper bound is unfortunately very crude and we shall see that the minimax rates can be much faster. We believe that the metric dimension of G + is actually of order kp.

4.2. The main result. Let us now turn to the statistical part of this paper.

As already mentioned, our aim is to estimate the density Π of a point process N from the observation of n independent copies of it. Our estimation strategy is based on T -estimation. More precisely, we start with an at most countable family {Π m , m ∈ M} of candidate determinantal densities, the choice of which will be explained below, and we use a test possessing robustness properties in view of selecting the closest element to Π among the Π m . We shall not detail the statistical procedure here and rather refer the reader to [START_REF] Birgé | Model selection via testing: an alternative to (penalized) maximum likelihood estimators[END_REF] (Theorem 9) or [START_REF] Baraud | Estimator selection with respect to Hellinger-type risks[END_REF]. Nevertheless, in order to give an brief account of the estimation strategies described there, let us merely say that they allow to endow {Π m , m ∈ M} with a (random) binary relation ∝ by means of a statistical test based on the observations. Given a pair (Π m , Π m ′ ) of distinct candidate densities, we either have Π m ∝ Π m ′ or Π m ′ ∝ Π m , the test being build in such way that the former (respectively the latter) relation is likely to occur when h(Π, Π m ) is small compared to h(Π, Π m ′ ) and vice-versa. If the relation ∝ were a total order on {Π m , m ∈ M}, a natural idea would be to define the estimator of Π as the minimal element of ({Π m , m ∈ M}, ∝). Unfortunately, this is not the case since ∝ fails to be transitive in general. A nice idea, which is actually due to Birgé, is to define the estimator as the element Π of {Π m , m ∈ M} minimizing the quantity

Π m → crit(Π m ) = sup{h(Π m , Π m ′ ), Π m ′ ∝ Π m }.
The property of the test ensures that the value of criterion at Π m is likely to be large when Π m is far from Π (provided that there exist some elements Π m ′ which are closer to Π) while it is likely to be small as soon as Π m lies in a small enough neighborhood of Π. With such an estimation strategy, we can design an estimator possessing the following property.

Proposition 5. Let Π = {Π m , m ∈ M} be an at most countable family of densities on (X, B(X), L) and π a sub-probability on M, that is m∈M π(m) ≤ 1 and π(m) ≥ 0 for all m ∈ M.

There exist a universal constant C > 0 and an estimator Π = Π(Π, π) solely based on N 1 , . . . , N n such that whatever the density Π,

CE h 2 (Π, Π) ≤ inf m∈M h 2 (Π, Π m ) + log(1/π(m)) n .
Proof. Proposition 3 page 363 of [START_REF] Baraud | Estimator selection with respect to Hellinger-type risks[END_REF] (Example 1, Density Estimation) ensures that the random measure n -1 n i=1 δ N i with intensity Π satisfies the assumption of Corollary 5 page 373 of [START_REF] Baraud | Estimator selection with respect to Hellinger-type risks[END_REF]. The result follows by applying this corollary with s = Π, Λ = M, s λ = Π m and ∆(Π m ) = log(1/π(m)) for all m ∈ M, the summation condition (4) of [START_REF] Baraud | Estimator selection with respect to Hellinger-type risks[END_REF] being satisfied under the assumption that π is a sub-probability.

Before turning to the choice of the family {Π m , m ∈ M}, let us comment on the role of π in our result. When π is a probability, it can be interpreted as a prior on the family {Π m , m ∈ M} and gives thus a bayesian flavor to our approach. Intuitively, our procedure tends to advantage densities Π m associated to values of π(m) which are not too small. We design our family {Π m , m ∈ M} in view of possessing good approximation properties with respect to the elements of the class D. Inequality (3.11) tells us that one can approximate a determinantal density Π Φ,λ (with respect to the Hellinger distance) by suitably approximating the sequence λ and the functions φ j of Φ corresponding to those indices j for which λ j is large enough. To do so, we introduce compacts subsets of Λ and H respectively defined as follows. Concerning λ = (λ j ) j≥1 , with no loss of generality, we may assume the sequence is non-increasing with respect to j and it is therefore natural to introduce compact sets of the form

Λ j = {γ ∈ Λ, γ j ′ = 0 for all j ′ > j}
for different values of j ≥ 1. This amounts to approximating λ by the truncated sequence keeping the j first entries of λ, the others being turned to 0. In order to approximate the φ j , we introduce an at most countable family H = (H m ) m∈M of compact subsets of (H,

). Examples of such compacts sets will be given in Section 4.3 for the purpose of providing rates of convergence. Given a compact subset H of (H,

) and some positive number η, we denote by H[η] a maximal η-separated subset of H, that is, any subset H ′ ⊂ H of maximal cardinality satisfying the property: for all φ, φ ′ ∈ H ′ with φ = φ ′ , φ -φ ′ > η. The maximality of H[η] implies that for all φ ∈ H, there exists φ ′ ∈ H[η] such that φ -φ ′ ≤ η. This means that H[η] is an η-net for the compact set H. By applying Proposition 5 to a suitable discretization of the compact sets Λ j and H m , we deduce the result below. Its proof is detailed in Section 5.4.

Theorem 1. Let H = (H m ) m∈M be an at most countable families of compact subsets H m of (H, ) and let π be a sub-probability on M. There exists a density estimator Π such that whatever the density Π on (X, B(X), L),

CE h 2 (Π, Π) ≤ inf Φ∈U,λ∈Λ   h 2 (Π, Π Φ,λ ) + inf j≥1   j j ′ =1 O(H, π, φ j ′ ) + j ′ >j λ 2 j ′    
where for all j ≥ 1,

O(H, π, φ j ) = inf m∈M inf ψ∈Hm φ j -ψ 2 + 1 n log |H m [1/ √ n]| n π(m)
and C is a positive universal constant.

Let us now comment on this risk bound. The term h 2 (Π, Π Φ,λ ) corresponds to the approximation of Π by an element of D. It expresses the fact that our estimation procedure is robust with respect to the assumption that Π belongs to D. The quantity j ′ >j λ 2 j ′ is the bias term that we get for approximating λ by the elements of Λ j . Given j ≥ 1 and m ∈ M, inf ψ∈Hm φ j -ψ 2 corresponds to the best approximation of φ j by some element of the compact set H m . Enlarging H m (for the inclusion) makes this term smaller but may increase the quantity n -1 log (|H m [1/

√ n]| n/π(m)) which measures in some sense the massiveness of H m . The quantity O(H, π, φ j ) corresponds to the best trade-off that can achieved between these two terms among the family H. It is typically the bound we would get for estimating the function φ j alone by a model selection procedure among H (up to possible extra logarithmic factors). The sum j j ′ =1 O(H, π, φ j ′ ) is therefore the risk bound we get for estimating the j first elements φ 1 , . . . , φ j of Φ = (φ j ′ ) j ′ ≥1 .

In order to specify these quantities, let us turn to the following typical situation. Let (S m ) m∈M be a family of finite-dimensional subspaces of H with respective dimension D m ≥ 1 (viewed as a linear space on R) and for m ∈ M, let us take H m = S ∩ S m where S denotes the unit sphere of H. The following results hold. Proposition 6. For all n ≥ 1,

(4.3) log H m [1/ √ n] ≤ D m log(2 √ n + 1).
Besides, for all φ ∈ S (4.4)

inf ψ∈Hm φ -ψ ≤ 4 inf ψ∈Sm φ -ψ .
The first inequality gives a control of the maximal size of a 1/ √ n-separated subset of H m . The second one shows that H m and S m share similar approximation properties with respect to the elements of S. The proof of the proposition is delayed to Section 5.5. With such a result, we deduce from Theorem 1 the following corollary.

Corollary 1. Let S = (S m ) m∈M be an at most countable family of finite dimensional subspaces of (H, ) with dimensions D m ≥ 1 and let π be a subprobability on M. There exists a density estimator Π = Π(S, π) such that whatever the density Π on (X, B(X), L),

CE h 2 (Π, Π) ≤ inf Φ∈U,λ∈Λ   h 2 (Π, Π Φ,λ ) + inf j≥1   j j ′ =1 O(S, π, φ j ′ ) + j ′ >j λ 2 j ′    
where for all j ≥ 1,

O(S, π, φ j ) = inf m∈M inf ψ∈Sm φ j -ψ 2 + D m log n + log(1/π(m)) n
and C is a positive universal constant.

For illustration, let us consider the elementary situation where X = {1, . . . , p} and assume that Π ∈ D p,k is a determinantal projection process on X of rank k as in Section 4.1. In this case, one can choose S = {S 0 } where S 0 is the linear subspace of dimension p of H = ℓ 2 (N * ) gathering the elements of the form (u 1 , . . . , u p , 0, . . .) with (u 1 , . . . , u p ) ∈ C p and π the Dirac mass at 0. Whatever Π ∈ D p,k there exists Φ ∈ U with φ 1 , . . . , φ k ∈ S 0 ∩ S such that Π = Π Φ,λ with λ 1 = . . . = λ k = 1 and λ j = 0 for j > k. Since O({S 0 }, π, φ j ′ ) ≤ p log n/n for all j ′ ∈ {1, . . . , k} and j ′ >k λ 2 j ′ = 0, by applying Corollary 1 we derive the risk bound

CE h 2 (Π, Π) ≤ kp log n n for all Π ∈ D p,k .
This inequality shows that the minimax rate of estimation over D p,k is not larger than kp log n/n. Since we expect that the metric dimension of D p,k is of order kp, we believe that the logarithmic factor could probably be dropped.

Rates of convergence.

In this section, we assume that X = [0, 1] k for some integer k ≥ 1. Our aim is to deduce from Corollary 1 some rates of convergence towards Π when it is of the form Π Φ,λ for some parameter (Φ, λ) ∈ U × Λ.

To do so, we make some a posteriori smoothness assumptions on Φ = (φ j ) j≥1 .

More precisely, we assume that the φ j are real-valued and belong to classes B β p,p ([0, 1] k ) of (possibly) anisotropic real-valued Besov functions indexed by a number p ∈ (0, +∞] and a smoothness parameter

β = (β i ) i=1,...,k ∈ (0, +∞) k . When p = +∞, B β ∞,∞ ([0, 1] k ) is merely the class of anisotropic β-Hölderian functions on [0, 1] k , which means that a function in B β ∞,∞ ([0, 1] k ) is β i -Holderian on [0, 1]
when we keep all the coordinates fixed expect the i-th. For a more precise definition of these smoothness classes we refer to [START_REF] Hochmuth | Wavelet characterizations for anisotropic Besov spaces[END_REF], at least when k = 2. The definition there can easily be generalized to larger values of k. Denoting by |φ| β,p,p the Besov semi-norm of a function φ in B β p,p ([0, 1] k ), we set for any R > 0

U β p,p (R) = (φ j ) j≥1 ∈ U φ j ∈ B β p,p ([0, 1] k ), |φ j | β,p,p ≤ R, ∀j ≥ 1 and for j 0 ∈ N * , U β p,p (R, j 0 ) = (φ j ) j≥1 ∈ U φ j ∈ B β p,p ([0, 1] k ), |φ j | β,p,p ≤ R, ∀j = 1, . . . , j 0 .
In order to approximate the elements of such class, we use the following result of [START_REF] Akakpo | Estimation adaptative par selection de partitions en rectangles dyadiques[END_REF].

Proposition 7. Let p > 0, k ∈ N * and r ∈ N. There exist a collection of linear spaces (S m ) m∈M k,r with M k,r = D≥1 M k,r (D) and a positive number C k,r such that for all positive integer D,

(4.5) |M k,r (D)| ≤ e C r,k D , sup m∈M k,r (D) dim(S m ) ≤ C r,k D and (4.6) inf m∈M k,r (D) inf ψ∈Sm φ -ψ ≤ C(k, r, p)|φ| β,p,p D -β/k for all φ ∈ B β p,p ([0, 1] k ) and β satisfying (4.7) sup 1≤i≤k β i < r + 1 and β = 1 k k i=1 1 β i -1 > k p -1 -2 -1 ∨ 0 .
Hereafter, k being fixed, we consider the family of linear spaces S = (S m ) m∈M indexed with M = r≥0 M k,r (omitting thus the dependency with respect to k) and endow M with the sub-probability π defined by

π(m) = inf e -(1+C r,k )D-r (r, D) ∈ N × N * , m ∈ M k,r (D) .
By using the first part of (4.5),

m∈M π(m) ≤ r≥0 D≥1 m∈M k,r (D) e -(1+C r,k )D-r ≤ r≥0 e -r D≥1 e -D ≤ 1
and hence π is a sub-probability on M.

By applying our Corollary 1 with the family (S m ) m∈M and this sub-probability π, we deduce the following uniform rates of convergence over the classes of densities of determinantal projection processes of rank j 0 ≥ 1 and parameter Φ belonging to U β p,p (R, j 0 ). The proof of the result is delayed to Section 5.6.

Proposition 8. There exists an estimator Π such that for all

j 0 ≥ 1, R > 0, β ∈ (0, +∞) k and p ∈ (0, +∞] such that β > k p -1 -2 -1 ∨ 0 , we have sup Φ∈U β p,p (R,j 0 ) E h 2 (Π Φ {1,...,j 0 } , Π) ≤ Cj 0 log n n 2β 2β+k
, where C denotes some positive number depending on k, R, p and β only.

When j 0 = 1, Π Φ {1} is merely a density on (X , B(X )) of the form |φ 1 | 2 for some function φ 1 of unit norm belonging to B β p,p ([0, 1] k ). Note that Π Φ {1} = |φ 1 | 2 also belongs to B β p,p ([0, 1] k
) and up to the logarithmic factor, the rate we get is the usual one for estimating a density in B β p,p ([0, 1] k ). Let us now establish uniform rates of convergence towards more general classes of determinantal densities. To do so, we also need to make a posteriori assumptions on λ. More precisely, we assume that it belongs to classes of the form

Λ (a) α (A) =    λ ∈ Λ j ′ >j λ 2 j ′ ≤ Aj -α , ∀j ≥ 1    or Λ (g) α (A) =    λ ∈ Λ j ′ >j λ 2 j ′ ≤ Ae -αj , ∀j ≥ 1   
for some A, α > 0. These sets contain sequences of (λ j ) j≥1 which are decreasing polynomially and exponentially fast respectively. We get the following result the proof of which is delayed to Section 5.6.

Proposition 9. There exists an estimator Π such that for all A, α, R > 0,

β ∈ (0, +∞) k and p ∈ (0, +∞] such that β > k p -1 -2 -1 ∨ 0 , we have sup (Φ,λ)∈U β p,p (R)×Λ (a) α (A) E h 2 (Π Φ,λ , Π) ≤ C log n n 2αβ (2β+k)(1+α) (4.8) sup (Φ,λ)∈U β p,p (R)×Λ (g) α (A) E h 2 (Π Φ,λ , Π) ≤ C (log n) 2+k/(2β) n 2β 2β+k (4.9)
where C denotes some positive number depending on k, A, R, p, α and β only.

Ψ J equals ρ(Π Φ J , Π Ψ J ) = ˆXk Π Φ J (α)Π Ψ J (α)dL(α) = ˆXk |det [Φ α,J ]| |det [Ψ α,J ]| dL(α).
For the second part we use Proposition 1 and get

1 - ˆXk |det [Φ α,J ]| |det [Ψ α,J ]| dL(α) ≤ 1 - ˆXk det Φ * J,α det [Ψ α,J ] dL(α) = 1 -det ( φ i , ψ j ) i,j∈J .
Let us now prove the last inequality and set a = j∈J φ j -ψ j 2 . If a > 2/5 then the result is true since the Hellinger distance is bounded by 1. We may therefore assume that a ≤ 2/5. In the remaining part of the proof we consider the linear space M J×J (C) of |J| × |J| matrices indexed by J with entries in C.

We endow M J×J (C) with the Hilbert-Schmidt norm defined by

A =   i∈J j∈J |A i,j | 2   1/2
.

It is well-known that this norm is sub-multiplicative in the sense that for all A, B ∈ M J×J (C), AB ≤ A B and it also satisfies

(5.1) |Tr(AB)| ≤ A B .
One can decompose the matrix A = ( φ i , ψ j ) i,j∈J as A = D + B where D is diagonal with entries D i,i = φ i , ψ i and

B = A -D. Since φ i = ψ i = 1 for all i (5.2) |D i,i | ≥ ℜ(D i,i ) = 1 - φ i -ψ i 2 2 ≥ 1 - a 2 > 0 ∀i = 1, . . . , k
and hence, D is non-singular. We may therefore write

det A = det D det(I + M ) with M = D -1 B and since for all i, j∈J | φ i , ψ j | 2 ≤ φ i 2 = 1, M 2 = i∈J 1 |D i,i | 2 j =i | φ i , ψ j | 2 ≤ i∈J 1 -|D i,i | 2 |D i,i | 2 = ∆ 2 J
and by using (5.2) with the fact that u → u -2 (1-u 2 ) is decreasing on (0, +∞),

(5.3) M 2 ≤ ∆ 2 J ≤ 1 -a/4 (1 -a/2) 2 j∈J φ j -ψ j 2 ≤ 9 16 < 1.
The matrix I + M is therefore non-singular and we may write I + M = e L for some matrix L ∈ M J×J (C). In fact,

L = M - M 2 2 + p≥3 (-1) p-1 M p p
where the series converge normally in (M J×J (C), ). Moreover, L) .

det(I + M ) = e Tr(
Since the mapping L → Tr(L) is linear and continuous on (M J×J (C), ) and since Tr(M ) = 0,

Tr(L) = - Tr(M 2 ) 2 + p≥3 (-1) p-1 Tr(M p ) p .
By using (5.1) and the sub-multiplicative property of the Hilbert-Schmidt norm, we get

Tr(L) + Tr(M 2 ) 2 ≤ p≥3 M p p ≤ M 3 3(1 -M )
and thus, by using that M ≤ ∆ J ,

ℜ(Tr(L)) ≥ -ℜ Tr(M 2 ) 2 - M 3 3(1 -M ) ≥ - M 2 2 1 + 2 M 3(1 -M ) ≥ - ∆ 2 J 2 1 + 2∆ J 3(1 -∆ J )
.

This inequality together with the fact that log u ≥ -(1 -u)/u for all u > 0 leads to

|det A| = |det D| e ℜ(Tr(L)) ≥ exp 1 2 i∈J log |D i,i | 2 - ∆ 2 J 2 1 + 2∆ J 3(1 -∆ J ) ≥ exp - ∆ 2 J 2 - ∆ 2 J 2 1 + 2∆ J 3(1 -∆ J ) = exp -c(∆ J )∆ 2 J with c(u) = 1 + u/[3(1 -u)]
for u ≥ 0. By using the facts that a ≤ 2/5, c is increasing and (5.3), we get

h 2 (Π Φ J , Π Ψ J ) ≤ 1 -|det A| ≤ 1 -exp -c a(1 -a/4) (1 -a/2) 2 ∆ 2 J ≤ c a(1 -a/4) (1 -a/2) 2 (1 -a/4) (1 -a/2) 2 j∈J φ j -ψ j 2 ≤ 5 2 j∈J φ j -ψ j 2 .
5.2. Proof of Proposition 3. Let us set R = ˆT P t q(t)dm(t).

Since h 2 (P, Q) ≤ 2h 2 (P, R) + 2h 2 (R, Q), it remains to bound each of those terms from above. The measures ν and m being σ-finite, we may apply Fubini-Tonnelli theorem. By using the Cauchy-Schwarz inequality, we bound the first term as follows.

2h 2 (P, R) = ˆE √ P - √ R 2 dν = ˆE (P -R) 2 ( √ P + √ R) 2 dν ≤ ˆE ´T P t (p(t) -q(t))dm(t) 2 P + R dν ≤ ˆE ´T √ P t ( p(t) -q(t)) √ P t ( p(t) + q(t))dm(t) 2 P + R dν ≤ ˆE ˆT P t p(t) -q(t) 2 dm(t) × ´T P t ( p(t) + q(t)) 2 dm(t) P + R dν ≤ 2 ˆT ˆE P t (x) p(t) -q(t) 2 dν(x) dm(t) = 4h 2 (p, q).
Let us now turn to the second term. By using similar arguments,

2h 2 (R, Q) ≤ ˆE ´T (P t -Q t )q(t)dm(t) 2 R + Q dν = ˆE ´T ( √ P t - √ Q t ) q(t) × ( √ P t + √ Q t ) q(t)dm(t) 2 R + Q dν ≤ ˆE ˆT P t -Q t 2 q(t)dm(t) × ´T ( √ P t + √ Q t ) 2 q(t)dm(t) R + Q dν ≤ 4 ˆT h 2 (P t , Q t )q(t)dm(t).
We conclude by adding these two upper bounds.

5.3. Proof of Proposition 4. Inequality (3.11) derives from (3.9), (3.10) and Proposition 3. Hence, it remains to prove (3.9) and (3.10).

Let us prove (3.9). To do so, we set a -1 = e/[2(e -1)] < 1 and prove the stronger inequality

h 2 (p λ , p γ ) ≤ a -1 |λ -γ| 2 + λ -γ 2 .
If there exists j ≥ 1 such that |λ j -γ j | 2 + λj -γj 2 > a then the result is

clear since h 2 (p λ , p γ ) ≤ 1. Otherwise, h 2 (p λ , p γ ) = 1 - J∈P j∈J λ j γ j j ∈J λj γj = 1 - j≥1 λ j γ j + λj γj = 1 - j≥1 1 - 1 2 |λ j -γ j | 2 + λj -γj 2 = 1 -exp   j≥1 log 1 - 1 2 |λ j -γ j | 2 + λj -γj 2   and by using that log(1 -u) ≥ [2a -1 log(1 -a/2)]u for all u ∈ [0, a/2], we get h 2 (p λ , p γ ) ≤ -log(1 -a/2) a j≥1 |λ j -γ j | 2 + λj -γj 2 = 1 a |λ -γ| 2 + λ -γ 2
with our choice of a.

Let us now prove (3.10). By using Proposition 2 we have that for all J ∈ P,

J = ∅ (5.4) h 2 (Π Φ J , Π Ψ J ) ≤ 5 2 j∈J φ j -ψ j 2 .
With the convention ∅ = 0, this inequality remains true when J = ∅ since in this case Π Φ J = Π Ψ J = δ ∅ and thus h 2 (Π Φ J , Π Ψ J ) = 0. We may therefore write

J∈P p γ J h 2 (Π Φ J , Π Ψ J ) ≤ 5 2 J∈P p γ J j∈J φ j -ψ j 2 ≤ 5 2 j≥1 φ j -ψ j 2 J∈P,j∈J p γ J ≤ 5 2 j≥1 γ 2 j φ j -ψ j 2 J∈P,j∈J j ′ ∈J,j ′ =j γ 2 j ′ j ′ ∈J (1 -γ 2 j ′ ) = 5 2 j≥1 γ 2 j φ j -ψ j 2 j ′ ≥1,j ′ =j (γ 2 j ′ + (1 -γ 2 j ′ )) = 5 2 j≥1 γ 2 j φ j -ψ j 2 as claimed.
5.4. Proof of Theorem 1. The proof is based on Proposition 5 with suitable choices of (Π m ) m∈M and sub-probability π ′ on M.

Let j be some positive integer. We set

Λ j [1/n] = {γ ∈ Λ| γ ℓ ∈ {i/n, i = 1, . . . n} if ℓ ≤ j, γ ℓ = 0 otherwise}, and
U j = {Φ {1,...,j} | Φ ∈ U} ⊂ H j .
We endow H j with the distance d(•, •) defined for (φ j ) ℓ=1,...,j and (ψ j ) ℓ=1,...,j in H j by d 2 ((φ j ) ℓ=1,...,j , (ψ j ) ℓ=1,...,j ) =

j ℓ=1 φ ℓ -ψ ℓ 2 .
For all m 1 , . . . , m j ∈ M and Φ ∈ U, there exist

φ 1 ∈ H m 1 [1/ √ n], . . . , φ j ∈ H m j [1/
√ n] such that for all ℓ ∈ {1, . . . , j}, and hence, by using (5.5)

(5.5) φ ℓ -φ ℓ ≤ inf ψ∈Hm ℓ φ ℓ -ψ + 1/ √ n since H m ℓ [1/ √ n] is a 1/ √ n-
d 2 (Φ {1,...,j} , Φ {1,...,j} ) ≤ 8 j ℓ=1 φ j -φ j 2 + 2 n ≤ 16 j ℓ=1 inf ψ∈Hm ℓ φ ℓ -ψ 2 + 16j + 2 n . (5.6) For all λ ∈ Λ, let us set λ = (λ ℓ ½ ℓ≤j ) ℓ≥1 ∈ Λ. Since for all ℓ ≥ 1, λ ℓ ∈ [0, 1], we have λ -λ 2 + λ - ˇ λ 2 = ℓ≥1 λ ℓ -λ ℓ 2 + 1 -λ 2 ℓ -1 -λ 2 ℓ 2 ≤ ℓ>j λ 2 ℓ + 1 -1 -λ 2 ℓ 2 ≤ ℓ>j   λ 2 ℓ + λ 4 ℓ 1 + 1 -λ 2 ℓ 2    ≤ 2 ℓ>j λ 2 ℓ . (5.7)
By construction of Λ j [1/n], there exists λ ∈ Λ j [1/n] such that λ ℓ -λ ℓ ≤ 1/n for all ℓ ≥ 1 and by using that for all u, v ∈ [0, 1], and hence, π ′ is a sub-probability on M. By using Proposition 5 with the family of densities (Π m ) m∈M and the sub-probability π ′ on M, we obtain an estimator Π for which E h 2 (Π, Π) is, up to a universal constant C > 0, not larger than h 2 (Π, Π m ) + log(1/π ′ (m)) n ≤ 2h 2 (Π, Π Φ,λ ) + 2h 2 (Π Φ,λ , Π m ) + log(1/π ′ (m)) n whatever (Φ, λ) ∈ U × Λ and m ∈ M. In particular by using (5.10), for any choices of j ≥ 1 and m 1 , . . . , m j ∈ M, there exists some m = (j, m 1 , . . . , m j , Φ, λ) ∈ M such that

√ 1 -u 2 - √ 1 -v 2 ≤ 2|u -v|,
h 2 (Π Φ,λ , Π m ) + log(1/π ′ (m)) n = h 2 (Π Φ,λ , Π Φ,λ ) + 1 n j ℓ=1 log 2 |H m ℓ [1/ √ n]| n π(m ℓ ) ≤ 102   ℓ>j λ 2 ℓ + j ℓ=1 inf ψ∈Hm ℓ φ ℓ -ψ 2 + 1 n log 2 |H m ℓ [1/ √ n]| n π(m ℓ ) + 1 n   .
Finally, we get the result from the fact that Φ, λ, j, m 1 , . . . , m j are arbitrary. Otherwise, there exists ψ ∈ S m such that φ -ψ < a. Hence, ψ ≥ φφ -ψ ≥ 1 -a > 0. In particular, ψ = 0 and we may set ψ ′ = ψ/ ψ ∈ H m . Since φ = 1, we have

φ -ψ ′ = φ - ψ ψ = φ - φ ψ + φ ψ - ψ ψ ≤ ψ -φ ψ + 1 ψ φ -ψ ≤ 2 φ -ψ ψ ≤ 2 φ -ψ 1 -a .
We get the result by choosing a = 1/2. 5.6. Proofs of Propositions 8 and 9. By using the collections of linear spaces S and our choice of π, and by using some classical optimization with respect to m ∈ M, we get that for all j ≥ 1 O(S, π, φ j ) ≤ C(log n/n) 2β/(2β+k)

where C is a positive constant depending on R, k, β and p. Up to the logarithmic factor, this bounds correspond to the usual estimation rate over B β p,p ([0, 1] k ). When λ = (½ j≤j 0 ) j≥1 , Π Φ,λ = Π Φ {1,...,j 0 } and Corollary 1 leads to Proposition 8. For all λ ∈ Λ (a) α (A), we get from Corollary 1 that

C ′ E h 2 (Π, Π) ≤ inf j≥1 j(log n/n) 2β/(2β+k) + Aj -α .
The minimum is achieved for j of order (n/ log n) 2β/[(2β+k)(1+α)] , which leads to the rate (log n/n) 2αβ/[(2β+k)(1+α)] as claimed. The other rate is obtained by arguing similarly and by choosing j of order 2β α(2β + k) log n.

  Proof of Proposition 2. The first equality is clear since by (3.1) the Hellinger affinity between Π Φ J and Π

  for such an element of Λ j [1/n],By combining (3.11) with (5.6) and (5.9), for all Φ ∈ U and λ ∈ Λ, there exists(Φ, λ) ∈ U(j, m 1 , . . . , m j ) × Λ j [1/n] such that h 2 (Π Φ,λ , Π Φ,λ ) ≤ 2 λ -λ 2 + λ -× U(j, m 1 , . . . , m j ) × Λ j [1/n]and for m = (j, m 1 , . . . , m j , Ψ, γ) ∈ M, Π m = Π Ψ,γ andπ ′Since for all j ≥ 1 and m 1 , . . . ,m j ∈ M, |U(j, m 1 , . . . , m j )| ≤ j ℓ=1 |H m ℓ [1/ √ n]| and |Λ j [1/n]| ≤ n j , m∈M π ′ (m) ≤j≥1 m 1 ,...,m j ∈M Ψ∈U(j,m 1 ,...,m j ) γ∈Λ j [1/n]

5. 5 .

 5 Proof of Proposition 6. For all m ∈ M, H m [1/ √ n] is a 1/ √ n-separated subset of the unit ball of a finite-dimensional linear space S m on R of dimension D m . Consequently, for all m ∈ M log H m [1/ √ n] ≤ D m log(2 √ n + 1). Let a < 1. If inf ψ∈Sm φ -ψ ≥ a, then by the triangular inequality, for all ψ ′ in H m φ -ψ ′ ≤ 2 ≤ 2 a inf ψ∈Sm φ -ψ .

  {1,...,j} , Φ {1,...,j} ) ≤ d(Φ {1,...,j} , ( φ j ) ℓ=1,...,j ) + inf

	Ψ∈U	d(( φ j ) ℓ=1,...,j , Ψ {1,...,j} ) +	1 √ n
	≤ 2d(Φ {1,...,j} , ( φ j ) ℓ=1,...,j ) +	1 √ n

net for H m ℓ . For such an element ( φ j ) ℓ=1,...,j ∈ H j , there exists Φ ∈ U such that d(( φ j ) ℓ=1,...,j , Φ {1,...,j} ) ≤ inf Ψ∈U d(( φ j ) ℓ=1,...,j , Ψ {1,...,j} ) + 1 √ n .

Since Φ only depends on φ 1 , . . . , φ j , the cardinality of the set U(j, m 1 , . . . , m j ) gathering such Φ when φ 1 , . . . , φ j vary among

H m 1 [1/ √ n], . . . , H m j [1/ √ n]

respectively is not larger than j ℓ=1 |H m ℓ [1/ √ n]|. Besides, by using that Φ ∈ U d(Φ

Acknowledgments I would like to thank Michel Merle for the very stimulating discussion we had about the identifiability problem raised in Section 4.1.


Jonathan Benchimol

Money in the production function: a new Keynesian DSGE perspective 3

Keywords: Money in the production function, DSGE, Bayesian estimation. JEL Classication Number: E23, E31, E51

This paper proposes a New Keynesian Dynamic Stochastic General Equilibrium (DSGE) model where real money balances enter the production function. By using a Bayesian analysis, our model shows that money is not an omitted input to the production process and rejects the decreasing returns to scale hypothesis. Our simulations suggest that money plays a negligible role in the dynamics of output and ination, despite its inclusion in the production function. In addition, we introduce the exible-price real money balances concept.

Introduction

The theoretical motivation for empirical implementations of money in the production function originates from monetary growth models of [START_REF] Levhari | The role of money in a simple growth model[END_REF], [START_REF] Friedman | The optimum quantity of money and other essays[END_REF], [START_REF] Johnson | Inside money, outside money, income, wealth and welfare in monetary theory[END_REF] and [START_REF] Stein | Monetary growth theory in perspective[END_REF], which include money directly in the production function. Firms hold money to facilitate production, on the grounds that money enables them to economize the use of other inputs, and spares the cost of running short of cash [START_REF] Fischer | Money and the production function[END_REF].

Real cash balances are at least in part a factor of production. To take a trivial example, a retailer can economize on his average cash balances by hiring an errand boy to go to the bank on the corner to get change for large bills tendered by customers. When it costs ten cents per dollar per year to hold an extra dollar of cash, there will be a greater incentive to hire the errand boy, that is, to substitute other productive resources for cash. This will mean both a reduction in the real ow of services from the given productive resources and a change in the structure of production, since di¤erent productive activities may di¤er in cash-intensity, just as they di¤er in labor -or land -intensity.

Milton [START_REF] Friedman | The optimum quantity of money and other essays[END_REF] In an old article, [START_REF] Sinai | Real money balances: an omitted variable from the production function ?[END_REF] present a very interesting test of the hypothesis that money enters the production function, and they suggest that real balances could be a missing variable that contributes to the attribution of the unexplained residual to technological changes. [START_REF] Ben-Zion | Money in the production function: an interpretation of empirical results[END_REF] conclude that money as a factor of demand seems to play an important role in explaining induced technological changes. [START_REF] Short | A new look at real money balances as a variable in the production function[END_REF] develops structural models based on [START_REF] Cobb | A theory of production[END_REF] and generalized translog production functions, both of which provide a more complete theoretical framework for analyzing the role of money in the production process. The empirical results obtained by estimating these two models indicate that the relationship between real cash balances and output, even after correcting for any simultaneity bias, is positive and statistically signicant. The results suggest that it is theoretically appropriate to include a real cash balances variable as a factor input in a production function in order to capture the productivity gains derived from using money. [START_REF] You | Money, technology, and the production function: an empirical study[END_REF] nds that the unexplained portion of output variation virtually vanishes with real balances included in the production function. Besides labor and capital, real money balances turn out to be an important factor of production, especially for developing countries. The results in [START_REF] Khan | Real money balances in the production function of a developing country[END_REF] are consistent with the hypothesis that real money balances are an important factor of production. [START_REF] Sephton | Money in the production function revisited[END_REF] shows that real balances are a valid factor of production within the connes of a CES production function. [START_REF] Hasan | Is money an omitted variable in the production function ? Some further results[END_REF] also support the hypothesis that money is an important factor in the production function and that there are potential supply side e¤ects of a change in the interest rate.

Recent developments in econometrics regarding co-integration and error correction models provide a rich environment in which the role of money in the production function can be reexamined. In a co-integrated space, [START_REF] Moghaddam | Co-integrated money in the production functionevidence and implications[END_REF] presents empirical evidence indicating that di¤erent denitions of money play an input role in the [START_REF] Cobb | A theory of production[END_REF] production function.

At the same time, [START_REF] Clarida | The science of monetary policy: a new Keynesian perspective[END_REF], [START_REF] Woodford | Interest and prices: foundations of a theory of monetary policy[END_REF] and [START_REF] Galí | Monetary policy, ination and the business cycle: an introduction to the new Keynesian framework[END_REF] develop New Keynesian Dynamic Stochastic General Equilibrium (DSGE) models to explain the dynamics of the economy. However, no studies use money as an input in the production function in New Keynesian DSGE models.

This article departs from the existing theoretical and empirical literature by specifying a New Keynesian DSGE model where money enters the production function. This feature generates a new ination dynamics where money could play a signicant role. Following [START_REF] Galí | Monetary policy, ination and the business cycle: an introduction to the new Keynesian framework[END_REF], we introduce the new concept of exible-price real money balances in order to close the model. We also analyze the dynamics of the economy by using Bayesian estimations and simulations to conrm or reject the potential role of money in the dynamics of the Eurozone. Moreover, this paper intends to solve the now-old controversial hypothesis about constant returns to scale of money in the production function initiated by [START_REF] Sinai | Real money balances: an omitted variable from the production function ?[END_REF].

Notice that in this paper, because data are available, we voluntary do not distinguish money for productive and nonproductive use as in [START_REF] Benhabib | Monetary policy and multiple equilibria[END_REF].

After describing the theoretical set up in Section 2, we calibrate and estimate two models (constrained i.e. constant returns to scale, and unconstrained i.e. decreasing returns to scale) of the Euro area using Bayesian techniques in 3. Impulse response functions and variance decomposition are analyzed in Section 4, and we solve the choice of the returns to scale hypothesis by comparing the two models of this paper in Section 5. Section 6 concludes, and Section 7 presents additional results.

The model

The model consists of households that supply labor, purchase goods for consumption, and hold money and bonds, and rms that hire labor and produce and sell di¤erentiated products in monopolistically competitive goods markets. Each rm sets the price of the good it produces, but not all rms reset their respective prices each period. Households and rms behave optimally: households maximize the expected present value of utility, and rms maximize prots. There is also a central bank that controls the nominal interest rate. This model is inspired by [START_REF] Smets | An estimated Dynamic Stochastic General Equilibrium model of the Euro area[END_REF], [START_REF] Galí | Monetary policy, ination and the business cycle: an introduction to the new Keynesian framework[END_REF], and Walsh (2010).

Households

We assume a representative innitely-lived household, seeking to maximize

E t " 1 X k=0 C k U t+k # (1)
where U t is the period utility function, and C < 1 is the discount factor.

We assume the existence of a continuum of goods represented by the interval [0; 1]. The household decides how to allocate its consumption expenditures among the di¤erent goods. This requires that the consumption index, C t , be maximized for any given level of expenditures. 8t 2 N, and conditionally on such optimal behavior, the period budget constraint takes the form

P t C t + M t + Q t B t B t01 + W t N t + M t01 (2)
where P t is an aggregate price index, M t is the quantity of money holdings at time t, B t is the quantity of one-period nominally riskless discount bonds purchased in period t and maturing in period t + 1 (each bond pays one unit of money at maturity and its price is Q t , so that the short term nominal rate i t is approximately equals to 0 log Q t ), W t is the nominal wage, and N t is hours of work (or the measure of household members employed).

The above sequence of period budget constraints is supplemented with a solvency condition, such as 8t lim

n0!1 E t [B n ] 0.
Preferences are measured with a common time-separable utility function. Under the assumption of a period utility given by

U t = e " p t C 10 t 1 0 + De " m t 1 0 M t P t 10 0 N 1+ t 1 + ! (3)
consumption, labor supply, money demand and bond holdings are chosen to maximize (1) subject to (2) and the solvency condition. This MIU utility function depends positively on the consumption of goods, C t , positively on real money balances, Mt Pt , and negatively on labor N t . is the coe¢cient of relative risk aversion of households or the inverse of the intertemporal elasticity of substitution, is the inverse of the elasticity of money holdings with respect to the interest rate, and is the inverse of the elasticity of work e¤ort with respect to the real wage (inverse of the Frisch elasticity of labor supply).

The utility function also contains two structural shocks: " p t is a general shock to preferences that a¤ects the intertemporal substitution of households (preference shock) and " m t is a money demand shock. D and are positive scale parameters.

This setting leads to the following conditions1 , which, in addition to the budget constraint, must hold in equilibrium. The resulting log-linear version of the rst order condition corresponding to the demand for contingent bonds implies that

c t = E t [c t+1 ] 0 1 (i t 0 E t [ t+1 ] 0 c ) 0 01 E t 2 1" p t+1 3 (4)
where the lowercase letters denote the logarithm of the original aggregated variables, c = 0 log (C), and 1 is the rst-di¤erence operator.

The demand for cash that follows from the households optimization problem is given by

" m t + c t 0 mp t 0 m = a 2 i t (5)
where mp t = m t 0 p t are the log-linearized real money balances, m = 0 log (D) + a 1 , a 1 and a 2 are resulting terms of the rst-order Taylor approximation of log (1 0 Q t ) = a 1 + a 2 i t . More precisely, if we compute our rst-order Taylor approximation around the steady-state interest rate, 1 C , we obtain a 1 = log 1 0 exp

0 1 C 0 1 C e 1 C 01
, and

a 2 = 1 e 1 C 01
.

Real cash holdings depend positively on consumption, with an elasticity equal to =, and negatively on the nominal interest rate ( 1C > 1 which implies that a 2 > 0). Below, we take the nominal interest rate as the central banks policy instrument.

In the literature, due to the assumption that consumption and real money balances are additively separable in the utility function, cash holdings do not enter any of the other structural equations: accordingly, the above equation becomes a recursive function of the rest of the system of equations. However, as in [START_REF] Sinai | Real money balances: an omitted variable from the production function ?[END_REF], [START_REF] Subrahmanyam | Real money balances as a factor of production: some new evidence[END_REF], or [START_REF] Khan | Real money balances in the production function of a developing country[END_REF], because real money balances enter the aggregate supply, we will use this money demand equation (eq. 5) in order to solve the equilibrium of our model. See for instance [START_REF] Ireland | Moneys role in the monetary business cycle[END_REF] and [START_REF] Benchimol | Money and risk in a DSGE framework: a Bayesian application to the Eurozone[END_REF] for models in which money balances enter the aggregate demand equation without entering the production function.

The resulting log-linear version of the rst order condition corresponding to the optimal consumption-leisure arbitrage implies that

w t 0 p t = c t + n t 0 n ( 6
)
where n = 0 log ().

Finally, these equations represent the Euler condition for the optimal intratemporal allocation of consumption (eq. 4), the intertemporal optimality condition setting the marginal rate of substitution between money and consumption equal to the opportunity cost of holding money (eq. 5), and the intratemporal optimality condition setting the marginal rate of substitution between leisure and consumption equal to the real wage (eq. 6).

Firms

We assume a continuum of rms indexed by i 2 [0; 1]. Each rm produces a di¤erentiated good, but they all use an identical technology, represented by the following money-in-the-production function

Y t (i) = A t M t P t Bm N t (i) 10Bn (7)
where A t = exp (" a t ) represents the level of technology, assumed to be common to all rms and to evolve exogenously over time.

All rms face an identical isoelastic demand schedule, and take the aggregate price level, P t , and aggregate consumption index, C t , as given. As in the standard [START_REF] Calvo | Staggered prices in a utility-maximizing framework[END_REF] model, our generalization features monopolistic competition and staggered price setting. At any time t, only a fraction 1 0 of rms, with 0 < < 1, can reset their prices optimally, while the remaining rms index their prices to lagged ination.

Price dynamics

Lets assume a set of rms that do not reoptimize their posted price in period t. As in [START_REF] Galí | Monetary policy, ination and the business cycle: an introduction to the new Keynesian framework[END_REF], using the denition of the aggregate price level and the fact that all rms that reset prices choose an identical price, P 3 t , leads to P t = 2 P 10" t01 + (1 0 ) (P 3 t ) 10" 3 1 10" . Dividing both sides by P t01 and log-linearizing around

P 3 t = P t01 yields t = (1 0 ) (p 3 t 0 p t01 ) (8)
In this set up, we do not assume inertial dynamics of prices. Ination results from the fact that rms reoptimizing their price plans in any given period, choose a price that di¤ers from the economys average price in the previous period.

Price setting

A rm reoptimizing in period t chooses the price P 3 t that maximizes the current market value of the prots generated while that price remains e¤ective. We solve this problem to obtain a rst-order Taylor expansion around the zero ination steady state of the rms rst-order condition, which leads to

p 3 t 0 p t01 = (1 0 C) 1 X k=0 (C) k E t 2 c mc t+kjt + (p t+k 0 p t01 ) 3 (9)
where c mc t+kjt = mc t+kjt 0 mc denotes the log deviation of marginal cost from its steady state value mc = 0, and = log ("= (" 0 1)) is the log of the desired gross markup.

Equilibrium

Market clearing in the goods market requires Y t (i) = C t (i) for all i 2 [0; 1] and all t. Aggregate output is dened as

Y t = R 1 0 Y t (i) 10 1 " di " "01 ; it fol- lows that Y t = C t must
hold for all t. One can combine the above goods market clearing condition with the consumers Euler equation to yield the equilibrium condition

y t = E t [y t+1 ] 0 01 (i t 0 E t [ t+1 ] 0 c ) 0 01 E t 2 1" p t+1 3 (10)
Market clearing in the labor market requires

N t = R 1 0 N t (i) di. Using (7) leads to N t = Z 1 0 0 @ Y t (i) A t Mt Pt Bm 1 A 1 10Bn di = 0 @ Y t A t Mt Pt Bm 1 A 1 10Bn Z 1 0 P t (i) P t 0 " 10Bn di (11)
where the second equality follows from the demand schedule and the goods market clearing condition. Taking logs leads to

(1 0 B n ) n t = y t 0 " a t 0 B m mp t + d t where d t = (1 0 B n ) log R 1 0 Pt(i) Pt 0 " 10Bn di
, and di is a measure of price (and, hence, output) dispersion across rms. Following [START_REF] Galí | Monetary policy, ination and the business cycle: an introduction to the new Keynesian framework[END_REF], in a neighborhood of the zero ination steady state, d t is equal to zero up to a rst-order approximation.

Hence, one can write the following approximate relation between aggregate output, employment, real money balances and technology as

y t = " a t + (1 0 B n ) n t + B m mp t ( 12
)
An expression is derived for an individual rms marginal cost in terms of the economys average real marginal cost. With the marginal product of labor,

mpn t = log @Y t @N t = log A t M t P t Bm (1 0 B n ) N t 0Bn = " a t + B m mp t + log (1 0 B n ) 0 B n n t
and the marginal product of real money balances,

mpmp t = log @Y t @ Mt Pt ! = log A t B m M t P t Bm01 N t 10Bn ! = " a t + log (B m ) + (B m 0 1) mp t + (1 0 B n ) n t we obtain an expression of the marginal cost mc t = (w t 0 p t ) 0 mpn t 0 mpmp t = w t 0 p t + 2B n 0 1 1 0 B n y t + 1 0 B m 0 B n 1 0 B n mp t 0 1 1 0 B n " a t 0 log (B m (1 0 B n ))
for all t, where the second equality denes the economys average marginal product of labor, mpn t , and the economys average marginal product of real money balances, mpmp t , in a way that is consistent with (12). Using the fact that mc t+kjt = (w t+k 0 p t+k ) 0 mpn t+kjt ,

mc t+kjt = (w t+k 0 p t+k ) + 2B n 0 1 1 0 B n y t+kjt + 1 0 B m 0 B n 1 0 B n mp t+k 0 1 1 0 B n " a t+k 0 log (B m (1 0 B n )) = mc t+k + 2B n 0 1 1 0 B n 0 y t+kjt 0 y t+k 1 = mc t+k 0 " 2B n 0 1 1 0 B n (p 3 t 0 p t+k ) (13)
where the second equality follows from the demand schedule, C t (i) =

Pt(i) Pt 0"
C t , combined with the market clearing condition (y t = c t ).

Substituting ( 13) into ( 9) and rearranging terms yields

p 3 t 0 p t01 = (1 0 C) 1 X k=0 (C) k E t c mc t+k 0 " 2Bn01 10Bn (p 3 t 0 p t+k ) + (p t+k 0 p t01 ) p 3 t 0 p t01 = (1 0 C) 2 1 X k=0 (C) k E t [ c mc t+k ] + 1 X k=0 (C) k E t [ t+k ] (14)
where 2 = 10Bn 10Bn+"(2Bn01) 1. Finally, combining (8) in ( 14) yields the ination equation

t = CE t [ t+1 ] + mc c mc t ( 15
)
where mc = 2 (10)(10C)

is strictly decreasing in the index of price stickiness, , in the measure of decreasing returns, B n , and in the demand elasticity, ".

Next, a relation is derived between the economys real marginal cost and a measure of aggregate economic activity. Notice that, independent of the nature of price setting, average real marginal cost can be expressed as

mc t = (w t 0 p t ) 0 mpn t 0 mpmp t = (y t + n t 0 n ) + 2B n 0 1 1 0 B n y t + 1 0 B m 0 B n 1 0 B n mp t 0 1 1 0 B n " a t 0 log (B m (1 0 B n )) = (1 0 B n ) + + 2B n 0 1 1 0 B n y t + 1 0 (1 + ) B m 0 B n 1 0 B n mp t (16) 0 1 + 1 0 B n " a t 0 log (B m (1 0 B n )) 0 n
where derivation of the second and third equalities makes use of the households optimality condition (6) and the (approximate) aggregate production relation ( 12). Knowing that > 0, B n 1, and 1, it is obvious that (1 0 B n ) + + 2B n 0 1 > 0. However, the inequality 1 0 (1 + ) B m 0 B n > 0 coming from (16) appears unusual. In fact, it conrms some studies from Sinai and Stokes [START_REF] Sinai | Real money balances: an omitted variable from the production function ? A reply[END_REF][START_REF] Sinai | Real money balances as a variable in the production function: reply[END_REF][START_REF] Sinai | Real money balances in the production function: a comment[END_REF][START_REF] Sinai | Money balances in the production function: a retrospective look[END_REF]) concluding that the weight on labor is more important than the weight on money (or real money balances).

Furthermore, and as shown previously, under exible prices the real marginal cost is constant and given by mc = 0. Dening the natural level of output, denoted by y f t , as the equilibrium level of output under exible prices

mc = (1 0 B n ) + + 2B n 0 1 1 0 B n y f t + 1 0 (1 + ) B m 0 B n 1 0 B n mp f t (17) 0 1 + 1 0 B n " a t 0 log (B m (1 0 B n )) 0 n thus implying y f t = a " a t + m mp f t + c (18)
where

y a = 1 + + + (1 0 ) B n 0 1 + B n y m = B n + B m (1 + ) 0 1 + + (1 0 ) B n 0 1 + B n y c = (1 0 B n ) (log (B m (1 0 B n )) + n 0 ) + + (1 0 ) B n 0 1 + B n
From (10), we obtain an expression for the natural interest rate,

i f t = c + E t h 1y f t+1 i (19)
Then, by using ( 5) and ( 19), we obtain an expression of exible-price real money balances

mp f t = 0 a 2 E t h 1y f t+1 i + y f t 0 a 2 c + m + 1 " m t (20)
Subtracting ( 17) from ( 16) yields

c mc t = (1 0 B n ) + + 2B n 0 1 1 0 B n y t 0 y f t + 1 0 (1 + ) B m 0 B n 1 0 B n mp t 0 mp f t (21)
where c mc t = mc t 0 mc is the real marginal cost gap, y t 0 y f t is the output gap, and mp t 0 mp f t is the real money balances gap. By combining eq. ( 15) and eq. ( 21) we obtain our rst equation relating ination to its next period forecast, output gap and real money balances gap

t = CE t [ t+1 ] + x y t 0 y f t + m mp t 0 mp f t ( 22
)
where x = (10)(10C)((10Bn)++2Bn01)

(10Bn+"(2Bn01))
and m = (10)(10C)(10(1+)Bm0Bn)

(10Bn+"(2Bn01))
. The second key equation describing the equilibrium of the New Keynesian model is obtained from (10):

y t = E t [y t+1 ] 0 01 (i t 0 E t [ t+1 ] 0 c ) 0 01 E t 2 1" p t+1 3 (23)
Henceforth ( 23) is referred to as the dynamic IS equation.

The third key equation describes behavior of the real money balances. Rearranging (5) yields

mp t = y t 0 a 2 i t 0 m + 1 " m t (24)
The last equation determines the interest rate through a standard smoothed Taylor-type rule,

i t = (1 0 i ) ( t 0 3 ) + x y t 0 y f t + i i t01 + " i t ( 25
)
where and x are policy coe¢cients reecting the weight on ination and on the output gap, and the parameter 0 < i < 1 captures the degree of interest rate smoothing. " i t is an exogenous ad hoc shock accounting for uctuations of nominal interest rate.

All structural shocks are assumed to follow a rst-order autoregressive process with an i.i.d. normal error term, such as 8k 2 fp; m; i; ag, " k t = k " k t01 + ! k;t , where ! k;t N (0; k ).

Results

DSGE model

Our model consists of six equations and six dependent variables: ination, nominal interest rate, output, exible-price output, real money balances and its exible-price counterpart. Flexible-price output and exible-price real money balances are completely determined by shocks: exible-price output is mainly driven by technology shocks (uctuations in the output gap can be attributed to supply and demand shocks), whereas the exible-price real money balances are driven by money shocks and exible-price output.

y f t = y a " a t + y m mp f t + y c ( 26
)
mp f t = m y+1 E t h 1y f t+1 i + m y y f t + m c + 1 " m t (27) t = CE t [ t+1 ] + x y t 0 y f t + m mp t 0 mp f t ( 28
)
y t = E t [y t+1 ] 0 01 (i t 0 E t [ t+1 ] 0 c ) 0 01 E t 2 1" p t+1 3 ( 29
)
mp t = y t 0 a 2 i t 0 m + 1 " m t ( 30
) i t = (1 0 i ) ( t 0 3 ) + x y t 0 y f t + i i t01 + " i t ( 31
)
where

y a = 1+ ++(10)Bn01+Bn x = (10)(10C)((10Bn)++2Bn01) (10Bn+"(2Bn01)) y m = Bn+Bm(1+)01 ++(10)Bn01+Bn m = (10)(10C)(10(1+)Bm0Bn) (10Bn+"(2Bn01)) y c = (10Bn)(log(Bm(10Bn))+ n 0log( " "01 )) ++(10)Bn01+Bn m = 0 log (D) + a 1 m y+1 = 0 a 2 n = 0 log () m y = c = 0 log (C) m c = 0 a 2 c + m a 1 = log 1 0 e 0 1 C 0 1 C e 1 C 01 a 2 = 1 e 1 C 01

Euro area data

To make output and real money balances stationary, we use rst log di¤erences, as in [START_REF] Adolfson | Evaluating an estimated new Keynesian small open economy model[END_REF]. In our model of the Eurozone, t is the log-linearized ination rate measured as the yearly log di¤erence of GDP Deator between one quarter and the same quarter of the previous year, ŷt is the log-linearized output measured as the yearly log di¤erence of GDP between one quarter and the same quarter of the previous year, and {t is the short-term (3-month) nominal interest rate. These data are extracted from the Euro area Wide Model (AWM) database of [START_REF] Fagan | An Area-Wide Model (AWM) for the Euro area[END_REF]. c mp t is the log-linearized real money balances measured as the yearly log di¤erence of real money between one quarter and the same quarter of the previous year, where real money is measured as the log di¤erence between the money stock and the GDP Deator. We use the M 3 monetary aggregate from the Eurostat database. ŷf t , the exible-price output, c mp f t , the exibleprice real money balances, and {f t , the exible-price (natural) interest rate, are completely determined by structural shocks.

80Q1 84Q4 89Q4 94Q4 99Q4 04Q4 09Q4 -2 0
We deal with four historical variables (Fig. 1) described latter and four shocks: a preference shock (" p t ), a money demand shock (" m t ), a technology shock (" a t ) and a monetary policy shock (" i t ).

Calibration and estimations

We calibrate our model following [START_REF] Galí | Monetary policy, ination and the business cycle: an introduction to the new Keynesian framework[END_REF], such that C = 0:99 (discount factor), = 1 (inverse of the Frisch elasticity), and " = 6 (elasticity of demand of households for consumption goods). These parameters are calibrated while other parameters are estimated with Bayesian techniques2 .

Calibration and estimation of structural parameters The estimation of the implied posterior distribution of the parameters (Table 1) is conducted using the Metropolis-Hastings algorithm3 (10 chains, each of 100000 draws).

The real money balances parameter (B m ) of the augmented production function is estimated to 0.064. This result is in line with [START_REF] Sinai | Real money balances: an omitted variable from the production function ?[END_REF]. They obtain a value of 0.087 for the same parameter and they also considering M3. The prior and posterior distributions are in Appendix 7.C.1 and estimates of the macro-parameters (aggregated structural parameters) are provided in Appendix 7.E.

As in Table 1, we use Bayesian techniques to estimate our model with money in the production function and a supplementary restriction. This restriction is adopted from [START_REF] Short | A new look at real money balances as a variable in the production function[END_REF] The resulting log marginal density for the model without constant returns to scale (-512.93) and for the model with constant returns to scale (-557.52) indicates that, if we admit that money enters the production function, this production function should have decreasing returns to scale. Robustness diagnosis about the numerical maximization of the posterior kernel are also computed and indicates that the optimization procedure leads to a robust maximum for the posterior kernel. The convergence of the proposed distribution to the target distribution is satised. A diagnosis of the overall convergence for the Metropolis-Hastings sampling algorithm is provided in Appendix 7.D. In response to a preference shock, the ination rate, the output, the output gap, the real money balances, the nominal and the real interest rates rise; real money growth displays a little overshooting process in the rst periods, then returns quickly to its steady-state value.

Simulations

Impulse response functions

After a technology shock, the output gap, the ination rate, the nominal and the real interest rates decrease, whereas output as well as real money balances and real money growth rise.

Following a money shock, ination, output, real and nominal interest rates and the output gap dynamics di¤ers depending on the model. The model of decreasing returns to scale displays more coherent results than that of constant returns to scale.

In response to an interest rate shock, the ination rate, the nominal interest rate, the output and the output gap fall. The real interest rate rises. A positive monetary policy shock induces a fall in interest rates due to a low enough degree of intertemporal substitution (i.e., the risk aversion parameter is high enough), which generates a large impact response of current consumption relative to future consumption. This result has been noted in, inter alia, [START_REF] Jeanne | Nominal rigidities and the liquidity e¤ect[END_REF] and [START_REF] Christiano | Sticky price and limited participation models of money: a comparison[END_REF].

Variance decompositions

We analyze in two di¤erent ways the forecast error variance of each variable following exogenous shocks: using constant returns to scale and decreasing returns to scale. The analysis is conducted rst via an unconditional variance decomposition (Table 3) and second via a conditional variance decomposition (Fig. 3 Adding the constant returns-to-scale restriction gives a larger role to real money demand in explaining the variance of output. Moreover, its radically changes the variance decomposition of output and its exible-price counterpart, whereas the variance decomposition of other variables are almost unchanged.

For the two models, most of the outputs variance comes from the technology shock, about a quarter of the variance of output results from the interest rate shock (31.29% for the decreasing returns case, and 19.08% for the constant returns case) and the remaining quarter from the other shocks. For the decreasing returns case, if money demand plays a role, its role is rather minor (an impact of less than 2.1%) and insignicant.

Moreover, Table 3 shows that assuming constant returns to scale gives money (demand) a role. As in [START_REF] Moghaddam | Co-integrated money in the production functionevidence and implications[END_REF], the variance decomposition indicates that money supply is capable of explaining some of the forecasting error variance of real output. This is the case only for the constant returns to scale model.

As Table 3 shows, the money shock contribution to the business cycle depends on the returns to scale hypothesis. If approximately half of the variance of output is still explained by the productivity shock, the role of the preference shock decreases notably, whereas the impact of the interest rate shock increases over time. Figure 3 also conrms the signicant role of money (demand) in the dynamics of output, and its increasing role over periods, under the constant returns to scale hypothesis.

A look at the conditional and unconditional ination variance decompositions shows the overwhelming role of the interest rate shock which explains more than 96% of the ination rates variance. As there is no signicant change over the two models, we dont represent this decomposition. The variance of the nominal interest rate is dominated by the direct shock to the interest rate. The relative importance of each of these shocks changes through time (Fig. 4). Over short horizons, the preference shock explains almost 70% of the nominal interest rate variance, whereas the interest rate shock explains less than 20%. For longer horizons, there is an inversion: the nominal interest rate shock explains close to 70% of the variance, and the preference shock explains a bit more than 20%.

Table 3 as well as the conditional variance decomposition of real money balances shows that real money balances are mainly explained by the real money balances shock and the technology shock. As there is no signicant change over the two models, we dont represent this decomposition. It is also interesting to note that the same type of analysis applies to the exible-price output variance decomposition (Fig. 5). Productivity, with a weight greater than 85%, is the main explanatory factor, while other shocks play minor roles.

As the exible-price real money balances variance is mainly explained by the money shock, with a signicant impact of the productivity shock, the impact of each of these shocks does not change through time, so we dont represent this decomposition either.

Interpretation

The constant returns-to-scale hypothesis gives money (demand) a more important role than does the decreasing returns to scale hypothesis. Following the log-marginal density criteria, the decreasing returns to scale hypothesis is preferred to the constant returns to scale hypothesis. This result disproves the hypothesis of [START_REF] Short | A new look at real money balances as a variable in the production function[END_REF], [START_REF] Startz | Can money matter ?[END_REF], [START_REF] Benzing | An update on money in the production function[END_REF][START_REF] Chang | Examining the long-run e¤ect of money on economic growth: an alternative view[END_REF] of constant returns to scale for money in the production function and conrms the hypothesis of [START_REF] Khan | Real money balances in the production function of a developing country[END_REF] (decreasing returns).

This criterion gives no signicant role to money (demand) in the dynamics of the variables, despite its introduction in the production function.

The simulation results are close to those obtained in the Galí (2008) baseline model and provide interesting results about the potential e¤ect of money on output and exible-price output under the constant returns to scale hypothesis. Interestingly, and even if money enters the ination equation, the variance decomposition of ination with respect to shocks is una¤ected under the two hypotheses.

Conclusion

One of the most unsettled issues of the postwar economic literature involves the role of money as a factor of production. The notion of money as a factor of production has been debated both theoretically and empirically by a number of researchers in the past ve decades. The question is whether money is an omitted variable in the production process.

However, empirical support for money as an input along with labor (and capital) has been mixed and, thus, the issue appears to be unsettled. Recent developments involve a reexamination of the role of money (demand) in the production function. One of these development is the New Keynesian DSGE theory mixed with Bayesian analysis.

We depart from existing theoretical (and empirical) New Keynesian literature by building a New Keynesian DSGE model à la [START_REF] Galí | Monetary policy, ination and the business cycle: an introduction to the new Keynesian framework[END_REF] that includes money in the production function, displaying money in the ination equation. Closing the model leads to the new concept of exible-price real money balances.

Despite their inclusion in the production function, and at least in the unconstrained estimation, real money holdings do not play a signicant role in the dynamics of the system. The only way to ascribe a role for real money demand in the dynamics of the system is to assume constant returns to scale to factors of production, which is a strong and controversial hypothesis.

Moreover, we conrm that the model with decreasing returns to scale is better than the model with constant returns to scale. Under decreasing returns to scale, real money holdings do not play a signicant role in the dynamics of the economy. We also show that adding a money component to the system does not necessarily create a role for it.

In this model, money demanded by households is also used (by them, as rms owners) as an input in production. But an issue of rival use (in facilitating transactions) would emerge. That is why for further research, as in [START_REF] Benhabib | Monetary policy and multiple equilibria[END_REF], a distinction between money for productive and nonproductive use seems warranted. This perspective could enlarge our money channel consideration.

Appendix

A Optimization problem

Our Lagrangian is given by

L t = E t " 1 X k=0 C k U t+k 0 t+k V t+k # where V t = C t + M t P t + Q t B t P t 0 B t01 P t 0 W t P t N t 0 M t01 P t and U t = e " p t C 10 t 1 0 + De " m t 1 0 M t P t 10 0 N 1+ t 1 +

!

The rst order condition related to consumption expenditures is given by

t = e " p t C 0 t ( 32
)
where t is the Lagrangian multiplier associated with the budget constraint at time t.

The rst order condition corresponding to the demand for contingent bonds implies that

t Q t P t = CE t t+1 P t+1 (33)
The demand for cash that follows from the households optimization problem is given by

De " p t e " m t M t P t 0 = t 0 CE t t+1 P t P t+1 (34)
which can be naturally interpreted as a demand for real balances. The latter is increasing in consumption and is inversely related to the nominal interest rate, as in conventional specications.

e " p t N t = t W t P t (35)
We obtain from eq. ( 32)

t = e " p t C 0 t , U c;t = e " p t C 0 t (36)
where

U c;t = @U k;t @C t+k C C C k=0
. Eq. ( 36) denes the marginal utility of consumption.

Hence, the optimal consumption/savings, real money balances and labor supply decisions are described by the following conditions:

F Combining (32) with (33) gives

Q t = CE t " e " p t+1 C 0 t+1 e " p t C 0 t P t P t+1 # , Q t = CE t U c;t+1 U c;t P t P t+1 (37)
where

U c;t+1 = @U k;t @C t+k C C C k=1
. Eq. ( 37) is the usual Euler equation for intertemporal consumption ows. It establishes that the ratio of marginal utility of future and current consumption is equal to the inverse of the real interest rate.

F Combining eq. ( 32) and eq. ( 34) gives

D e " m t C 0 t M t P t 0 = 1 0 Q t , U m;t U c;t = 1 0 Q t ( 38
)
where 38) is the intertemporal optimality condition setting the marginal rate of substitution between money and consumption equal to the opportunity cost of holding money.

U m;t = @U k;t @(M t+k =P t+k ) C C C k=0 . Eq. (
F And combining eq. ( 32) and eq. ( 35) gives

N t C 0 t = W t P t , U n;t U c;t = 0 W t P t ( 39
)
where

U n;t = @U k;t @N t+k C C C k=0
. Eq. ( 39) is the condition for the optimal consumption-leisure arbitrage, implying that the marginal rate of substitution between consumption and labor is equal to the real wage.

B Calibration

We estimate all parameters, except the discount factor (C), the inverse of the Frisch elasticity of labor supply (), and the elasticity of demand of households for consumption goods (").

Following standard conventions, we calibrate beta distributions for parameters that fall between zero and one, inverted gamma distributions for parameters that need to be constrained to be greater than zero and normal distributions in other cases.

As our goal is to compare two models, we adopt the same priors in the two models with the same calibration, except for the return to scale parameter. The calibration of is inspired by [START_REF] Rabanal | Comparing new Keynesian models of the business cycle: a Bayesian approach[END_REF] and [START_REF] Casares | Monetary policy rules in a new Keynesian Euro area model[END_REF]. They choose, respectively, a risk aversion parameter of 2:5 and 1:5. In line with these values, we consider that = 2 corresponds to a standard risk aversion, as in [START_REF] Benchimol | Money and risk in a DSGE framework: a Bayesian application to the Eurozone[END_REF].

As in [START_REF] Smets | An estimated Dynamic Stochastic General Equilibrium model of the Euro area[END_REF], the standard errors of the innovations are assumed to follow inverse gamma distributions, and we choose a beta distribution for shock persistence parameters (as well as for the backward component of the Taylor rule, scale parameters, D and , price stickiness index, , and output elasticities of labor, B n , and of real money balances, B m , of the production function) that should be lesser than one.

The calibration of B, C, , and " comes from [START_REF] Casares | Monetary policy rules in a new Keynesian Euro area model[END_REF] and [START_REF] Galí | Monetary policy, ination and the business cycle: an introduction to the new Keynesian framework[END_REF]. The smoothed Taylor rule ( i , , and x ) is calibrated following Gerlach-Kristen (2003), with priors analogous to those used by [START_REF] Smets | An estimated Dynamic Stochastic General Equilibrium model of the Euro area[END_REF]. In order to take into consideration possible behaviors of the central bank, we assign a higher standard error for the Taylor rule coe¢cients.

All the standard errors of shocks are assumed to be distributed according to inverted Gamma distributions, with prior means of 0.02. The latter law ensures that these parameters have a positive support. The autoregressive parameters are all assumed to follow Beta distributions. All these distributions are centered around 0.75 and we take a common standard error of 0.1 for the shock persistence parameters, as in [START_REF] Smets | An estimated Dynamic Stochastic General Equilibrium model of the Euro area[END_REF].

C Priors and posteriors

D Robustness checks

Each graph represents specic convergence measures with two distinct lines that show the results within (red line) and between (blue line) chains [START_REF] Geweke | Using simulation methods for Bayesian econometric models: inference, development, and communication[END_REF]. Those measures are related to the analysis of the model parameters mean (interval), variance (m2) and third moment (m3). For each of the three measures, convergence requires that both lines become relatively horizontal and converge to each other in both models 4 . (1 0 i )

D.1 Model with decreasing returns to scale

1.681117 1.461802

x (1 0 i ) 0.948134 0.806101 i 0.475001 0.543219

Figure 1 :

 1 Figure 1: Euro area data (source: AWM and Eurostat)

Fig. 2

 2 Fig. 2 presents the response of key variables to structural shocks. The thin solid line represents the decreasing return-to-scale model responses and the dashed line represents the constant return-to-scale model responses.In response to a preference shock, the ination rate, the output, the output gap, the real money balances, the nominal and the real interest rates rise; real money growth displays a little overshooting process in the rst periods, then returns quickly to its steady-state value.After a technology shock, the output gap, the ination rate, the nominal and the real interest rates decrease, whereas output as well as real money balances and real money growth rise.Following a money shock, ination, output, real and nominal interest rates and the output gap dynamics di¤ers depending on the model. The model of decreasing returns to scale displays more coherent results than that of constant returns to scale.In response to an interest rate shock, the ination rate, the nominal interest rate, the output and the output gap fall. The real interest rate rises. A positive monetary policy shock induces a fall in interest rates due to a low enough degree of intertemporal substitution (i.e., the risk aversion parameter is high enough), which generates a large impact response of current consumption relative to future consumption. This result has been noted in, inter alia,[START_REF] Jeanne | Nominal rigidities and the liquidity e¤ect[END_REF] and[START_REF] Christiano | Sticky price and limited participation models of money: a comparison[END_REF].

Figure 2 :

 2 Figure 2: Impulse response functions to structural shocks

Figure 3 :

 3 Figure 3: Forecast error variance decomposition of output

Figure 4 :

 4 Figure 4: Forecast error variance decomposition of interest rate

Figure 5 :

 5 Figure 5: Forecast error variance decomposition of exible-price output

4D

  Robustness analysis with respect to calibrated parameters is available upon request.28

Table 1 :

 1 Bayesian estimation of the model

			(decreasing return to scale)
		Priors	Posteriors
		Law	Mean Std.	Mean t-stat	Std.	5%	95%
	B n	beta	0.50 0.15	0.6430 12.85 0.0494 0.563 0.721
	B m	beta	0.50 0.15	0.0646 2.156 0.0256 0.022 0.105
		beta	0.66 0.05	0.8029 29.26 0.0278 0.755 0.848
	v	normal	2.00 0.10	2.1395 22.66 0.0941 1.987 2.294
		normal	2.00 0.10	1.8545 19.09 0.0970 1.692 2.010
	D	beta	0.25 0.10	0.2507 2.096 0.1041 0.085 0.406
		beta	0.25 0.10	0.2565 2.220 0.1020 0.095 0.410
	i	beta	0.50 0.10	0.4750 9.323 0.0513 0.388 0.557
		normal	3.00 0.20	3.2021 16.40 0.1937 2.886 3.523
	x	normal	1.50 0.20	1.8060 10.27 0.1764 1.507 2.094
	a	beta	0.75 0.10	0.9251 37.53 0.0247 0.886 0.964
	p	beta	0.75 0.10	0.9135 64.30 0.0142 0.889 0.936
	i	beta	0.75 0.10	0.9914 301.7 0.0033 0.985 0.997
	m	beta	0.75 0.10	0.9412 54.10 0.0174 0.912 0.969
	a invgamma	0.02 2.00	0.0072 14.09 0.0005 0.006 0.008
	i invgamma	0.02 2.00	0.0065 8.092 0.0008 0.005 0.007
	p invgamma	0.02 2.00	0.0992 6.943 0.0138 0.075 0.122
	m invgamma	0.02 2.00	0.0230 13.57 0.0017 0.020 0.025

Table 2 :

 2 and involves the hypothesis of constant returns to scale of the production function. Then, we assume that B n = B m and we test our model with this hypothesis. Bayesian estimation of the model with constant return to scale

		Calibration and estimation of structural parameters
			(constant return to scale)
			Priors		Posteriors
		Law	Mean Std.	Mean t-stat	Std.	5%	95%
	B n	beta	0.50 0.15	0.5744 17.63 0.0320 0.519 0.625
		beta	0.66 0.05	0.8388 35.52 0.0239 0.799 0.878
	v	normal	2.00 0.10	2.3026 25.88 0.0887 2.158 2.450
		normal	2.00 0.10	1.7821 18.06 0.0986 1.619 1.944
	D	beta	0.25 0.10	0.2493 2.096 0.1041 0.085 0.403
		beta	0.25 0.10	0.2998 2.529 0.1086 0.134 0.463
	i	beta	0.50 0.10	0.5432 10.74 0.0513 0.458 0.629
		normal	3.00 0.20	3.2002 16.27 0.1955 2.876 3.515
	x	normal	1.50 0.20	1.7647 10.09 0.1756 1.473 2.053
	a	beta	0.75 0.10	0.9169 32.79 0.0280 0.874 0.960
	p	beta	0.75 0.10	0.9056 58.20 0.0156 0.880 0.931
	i	beta	0.75 0.10	0.9907 271.7 0.0037 0.984 0.997
	m	beta	0.75 0.10	0.9480 57.74 0.0165 0.921 0.974
	a invgamma 0.02 2.00	0.0076 13.91 0.0005 0.006 0.008
	i invgamma 0.02 2.00	0.0059 7.101 0.0008 0.004 0.007
	p invgamma 0.02 2.00	0.0966 7.034 0.0132 0.073 0.118
	m invgamma 0.02 2.00	0.0238 13.75 0.0017 0.021 0.026

  to 5) to compare the two models dynamics of variance decomposition over time.

			Unconditional variance decomposition (%)
		with decreasing returns to scale	with constant returns to scale
		" p t	" i t	" m t	" a t	" p t	" i t	" m t	" a t
	y t	10.92 31.29	2.07 55.72	6.5 19.08 15.06 59.37
	t	0.9 99.08	0	0.02	0.9 99.07	0	0.02
	i t	31.08 68.45	0.01	0.46	31.83	67	0.12	1.05
	mp t	0.47	0.64 77.14 21.76	0.34	0.52 80.94	18.2
	y f t mp f t	0 0	0 0 76.45 23.55 3.5 96.5	0 0	0 19.37 80.63 0 80.12 19.88

Table 3 :

 3 Variance decomposition of the model

For helpful comments and suggestions, I would like to thank André Fourçans, Christian Bordes, Laurent Clerc, Marc-Alexandre Sénégas, Gunther Capelle-Blancard, and two anonymous referees.O PhD Program, ESSEC Business School, Avenue Bernard Hirsch, 95021 Cergy Pontoise Cedex 2, France, and CES-University Paris 1 Panthéon-Sorbonne, 106-112 Boulevard de lHôpital, 75647 Paris Cedex 13, France. Email: jonathan.benchimol@essec.edu

See Appendix 7.A.

See Appendix 7.B for the detailed calibration exercise.

See for instance[START_REF] Smets | An estimated Dynamic Stochastic General Equilibrium model of the Euro area[END_REF],[START_REF] Smets | Shocks and frictions in US business cycles: a Bayesian DSGE approach[END_REF],[START_REF] Adolfson | Bayesian estimation of an open economy DSGE model with incomplete pass-through[END_REF], and[START_REF] Adolfson | Evaluating an estimated new Keynesian small open economy model[END_REF].