Emmanuel Fricain

AND Andreas Hartmann
email: andreas.hartmann@math.u-bordeaux1.fr

William T Ross
email: wross@richmond.edu

CONCRETE EXAMPLES OF H (b) SPACES

Keywords: 2010 Mathematics Subject Classification. 30J05, 30H10, 46E22 de Branges-Rovnyak spaces, non-extreme points, kernel functions, corona pairs

In this paper we give an explicit description of de Branges-Rovnyak spaces H (b) when b is of the form q r , where q is a rational outer function in the closed unit ball of H ∞ and r is a positive number.

Introduction

The purpose of this paper is to explicitly describe the elements of the de Branges Besides possessing a fascinating internal structure [START_REF] Sarason | Sub-Hardy Hilbert spaces in the unit disk[END_REF], H (b) spaces play an important role in several aspects of function theory and operator theory, most importantly, in the model theory for many types of contraction operators [START_REF] De | Canonical models in quantum scattering theory[END_REF][START_REF] De | Square summable power series[END_REF].

Despite the important role H (b) spaces play in operator theory, the exact contents of H (b) often remain mysterious. What functions belong to H (b)? Certainly the kernel functions k b λ , λ ∈ D, do (and have dense linear span). What else?

In this paper, we give a precise description of the elements of H (b) for certain relatively simple b, namely positive powers of rational outer functions. Our description needs the following set up. If b ∈ b(H ∞ ) is a non-extreme point of b(H ∞ ), equivalently, log(1 -|b|) ∈ L 1 (T, m) (where T := {ζ ∈ C : |ζ| = 1} and m Lebesgue measure on T normalized so that m(T) = 1), then there exists a unique outer function a ∈ b(H ∞ ), called the Pythagorean mate for b, such that a(0) > 0 and |a| 2 + |b| 2 = 1 almost everywhere on T. The pair (a, b) is said to be a Pythagorean pair.

Our first observation says that in certain situations H (b r ) does not depend on r > 0. (3) Statement (c) of the theorem says that b is a multiplier of H (b). We refer the reader to Sarason's book [START_REF] Sarason | Sub-Hardy Hilbert spaces in the unit disk[END_REF] for further information and references about multipliers of H (b).

(1.3) b is invertible in H ∞ . Remark 1.4. ( 1
(4) By Carleson's corona theorem [START_REF] Garnett | Bounded analytic functions[END_REF], the condition (1.2) is equivalent to existence of φ, ψ ∈ H ∞ so that aφ + bψ = 1 on D. Such a pair (a, b) satisfying this condition is called a corona pair.

When b is a rational outer function, or any positive power of a rational function (which is necessarily non-extreme (see Lemma 3.1)), we obtain the following complete description of H (b) involving the derivatives of the reproducing kernels. Indeed, when b = q r , where q is outer and rational and r > 0, we set

v ℓ r,λ (z) := d ℓ dλ ℓ k q r λ (z) = d ℓ dλ ℓ 1 -q r (λ)q r (z) 1 -λz ,
for any z ∈ D, λ ∈ D -, and ℓ 0. We let H 2 denote the classical Hardy space [START_REF] Garnett | Bounded analytic functions[END_REF]. By means of the Féjer-Riesz theorem (see Section 6), one can prove that if q is a rational function then so is its Pythagorean mate a. In this case, also notice that for ζ ∈ T we have |q(ζ)| = 1 if and only if a(ζ) = 0. Theorem 1.5. Suppose q ∈ b(H ∞ ) is a rational outer function and r is a positive real number. Then (1) H (q r ) = H (q) as sets.

(2) If a is the Pythagorean mate for q and a has distinct zeros ζ 1 , . . . , ζ n on T with corresponding multiplicities m 1 , . . . , m n , then (a) the functions v ℓ r,j := v ℓ r,ζ j are well-defined and belong to H (q r ) for 1 j n and 0 ℓ m j -1. Moreover, they are orthogonal to

aH 2 = n j=1 (z -ζ j ) m j H 2 . (b) H (q r ) is equal to n j=1 (z -ζ j ) m j H 2 ⊕ v ℓ r,j : 0 ℓ m j -1, 1 j n ,
where the orthogonal decomposition is in terms of the inner product in H (q r ).

Writing v ℓ j = v ℓ 1,j , the theorem above implies that H (q r ) is equal to

n j=1 (z -ζ j ) m j H 2 ∔ v ℓ j (z) : 0 ℓ m j -1, 1 j n ,
where the sum is no longer necessarily orthogonal.

It was shown in [START_REF] Costara | Which de Branges-Rovnyak spaces are Dirichlet spaces (and vice versa)?[END_REF], and rediscovered in [START_REF] Blandignères | Direct and reverse Carleson measures for H (b) spaces[END_REF], that

H (q) = n j=1 (z -ζ j ) m j H 2 ∔ P N -1 , (1.6)
where N = n j=1 m j , P N -1 is the N-dimensional vector space of polynomials of degree at most N -1, and the sum is an algebraic direct sum (not necessarily an orthogonal one). The novelty of our result is that we can precisely identify the orthogonal complement of

aH 2 = n j=1 (z -ζ j ) m j H 2 in H (q) without using (1.6).
In a recent preprint, Lanucha and Nowak [START_REF] Lanucha | De Branges spaces and generalized Dirichlet spaces[END_REF] examined when an H (b) space is isomorphic to a Dirichlet type space. Their discussion naturally leads to the situation when a is a polynomial with simple zeros on T and a similar description of H (b) for such a.

A key ingredient used to show statement (1) of Theorem 1.5, and an added bonus to our result, is that if a r is the Pythagorean mate for q r then the co-analytic Toeplitz operators T a and T ar on H 2 have the same range, namely H (q).

Preliminaries

There are several equivalent definitions of the de Branges-Rovnyak space H (b). We can, for instance, define it in the standard way [START_REF] Paulsen | An Introduction to the theory of reproducing kernel Hilbert spaces[END_REF] as the reproducing kernel Hilbert space associated with the (positive definite) reproducing kernel The space H (b) can also be defined as the range space (I -

k b λ (z) := 1 -b(λ)b(z) 1 -λz , λ, z ∈ D.
T b T b ) 1/2 H 2 equipped with the norm which makes (I -T b T b ) 1/2 a partial isometry.
Here T ϕ is the Toeplitz operator on H 2 with symbol ϕ ∈ L ∞ (T) defined by

T ϕ f = P + (ϕf ), f ∈ H 2 ,
where P + is the orthogonal projection of L 2 (T) onto H 2 . The book [START_REF] Sarason | Sub-Hardy Hilbert spaces in the unit disk[END_REF] is the classic reference for H (b) spaces.

When b ∞ < 1, H (b) turns out to be a renormed version of H 2 while if b is an inner function, then H (b) turns out to be one of the classical and well-studied model spaces

H 2 ⊖ bH 2 .
When b is non-extreme and a is its Pythagorean mate, two important (not necessarily closed) vector spaces of functions in H (b) are

M (a) := T a H 2 and M (a) := T a H 2 .
It follows from the Douglas factorization theorem and the operator inequalities

(2.1) T a T a T a T a and T a T a = I -T b T b I -T b T b that M (a) ⊂ M (a) ⊂ H (b) (see [9, p. 24]).
For technical reasons, we will make use of the space H (b) which, for any b ∈ b(H ∞ ), is defined similarly as with H (b) but as the range space (I -

T b T b ) 1/2 H 2 . The operator inequalities from (2.1) show that H (b) is contractively contained in H (b).

Corona pairs

This following lemma is well-known but we record it here along with a proof for the sake of completeness and for the discussion of the examples in Section 6.

Lemma 3.1. Suppose q ∈ b(H ∞ ) is rational and not inner. Then q is non-extreme and, if a is the Pythagorean mate for q, then a is also rational.

Proof. Since q is rational then q = p 1 /p 2 where p 1 and p 2 are analytic polynomials and p 2 has no zeros on D -. We can, of course, choose p 2 such that p 2 (0) > 0. Since q ∈ b(H ∞ ), we see that 1 -|q(e iθ )| 2 0 for all θ and so |p 2 (e iθ )| 2 -|p 1 (e iθ )| 2 is a non-negative trigonometric polynomial. Furthermore, |p 2 (e iθ )| 2 -|p 1 (e iθ )| 2 is not the zero function since we are assuming that q is not an inner function. By the Féjer-Riesz theorem, 2 , where p is an analytic polynomial which is zero free in D and p(0) > 0.

|p 2 (e iθ )| 2 -|p 1 (e iθ )| 2 = |p(e iθ )|
Let a = p/p 2 . Note that a is rational and zero free in D, hence outer. Moreover, a(0) > 0.

Furthermore, on T we have

|a| 2 = p p 2 2 = |p 2 | 2 -|p 1 | 2 |p 2 | 2 = 1 - p 1 p 2 2 = 1 -|q| 2 .
This means that (a, q) is a Pythagorean pair which, in particular, implies that q is non-extreme. Proof. Since

(3.3) 1 -x r 1 -x ≍ 1, x ∈ [0, 1), we see that 1 -|b| r ≍ 1 -|b| when b ∈ b(H ∞ )
, from which we deduce the first part of the Lemma.

Now observe that |a| 2 |a r | 2 = 1 -|b| 2 1 -|b 2 | r ≍ 1,
and since a and a r are outer, Smirnov's theorem (which says that if the boundary function for the quotient of two outer functions is bounded on are strictly positive (or not) simultaneously. Indeed, if there is a sequence {z n } n 1 in D such that one expression goes to 0 then, since both a(z n ) and b(z n ) go to zero, the other expression will go to zero as well.

T, then f ∈ H ∞ ), shows that a/a r is invertible in H ∞ .
A special situation where b forms a corona pair with its Pythagorean mate is when b is rational.

Lemma 3.4. Suppose q ∈ b(H ∞ ) is rational and not inner. If a is the Pythagorean mate for q, then (a, q) is a corona pair.

Proof. According to the proof of Lemma 3.1, we know that a is rational, a = p/p 2 , where p and p 2 are polynomials, p 2 has no zeros in D -and p is zero free in D. In particular, a is analytic in an open neighborhood of D -and thus has a finite number of zeros on T, say {ζ 1 , . . . , ζ n }. Note that, due to the identity |a| 2 + |q| 2 = 1 on T, the zeros of a (on T) must lie where q is unimodular on T.

Let D j be disjoint open disks with center at the zeros ζ j of a and let

F = D -\ n j=1 D j .
By making the disks smaller, one can, by using the continuity of |q| on D -, arrange things so that |q| 1 2 on each D j ∩ D -. Notice that F is closed and omits all of the zeros of a in D -and so (1) (a, b) is a corona pair;

(2)

H (b) = M (a).
The second is the following. Proposition 3.6. If a, a 1 ∈ H ∞ are two outer functions such that a/a 1 and a 1 /a belong to L ∞ , then M (a) = M (a 1 ).

Proof. Again, by Smirnov's theorem, we know that a/a 1 and a 1 /a belong to H ∞ . Thus T a/a 1 , and hence T a/a 1 , are invertible operators on H 2 . From here we get

M (a) = T a H 2 = T a 1 T a/a 1 H 2 = T a 1 H 2 = M (a 1 ).

The contents of H (b)

We now can give the proof of Theorem 1.5. Indeed, statement (1) of the theorem follows from Lemma 3.4 and Theorem 1.1. Let us consider statement (2).

In [START_REF] Fricain | Integral representation of the n-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel[END_REF] it was shown, for an outer function b, that if ζ ∈ T and (5.1)

T | log |b(w)|| |w -ζ| 2n+2 dm(w) < ∞,
then every function in H (b), as well as its derivatives up to order n, has a finite non-tangential limit at ζ.

Recalling the notation v ℓ r,λ for the ℓ-th derivative in the variable λ of the reproducing kernel in H (q r ), the results of [START_REF] Fricain | Integral representation of the n-th derivative in de Branges-Rovnyak spaces and the norm convergence of its reproducing kernel[END_REF] also show that v ℓ r,ζ ∈ H (q r ), 0 ℓ n, and

f (ℓ) (ζ) = f, v ℓ r,ζ q r , f ∈ H (q r ), 0 ℓ n.
Let us check condition (5.1) for our situation. Since q is rational its Pythagorean mate a is also rational and can be written as

(5.2) a(z) = s(z) n j=1 (z -ζ j ) m j ,
where s is a rational function whose poles and zeros lie on the complement of D -. Pick w = e it near one of the zeros ζ j = e iθ j of a. Then

| log |q r (e it )|| ≍ | log |q(e it )| 2 | = | log(1 -|a(e it )| 2 )| ≍ |a(e it )| 2 ≍ |e it -e iθ j | 2m j
This means that for t near θ j we have | log |q r (e it )|| |e it -e iθ j | 2(m j -1)+2 ≍ 1 and so, by (5.1), every function in H (b) as well as its derivatives up the order m j -1 admits non-tangential limits at ζ j , and v ℓ r,ζ j ∈ H (q r ) for all 0 ℓ m j -1.

The following interesting observation will be very useful in the proof of our main theorem. Lemma 5.3. Suppose a(z) = n j=1 (z -ζ j ) m j , where ζ j ∈ T and m j is the corresponding multiplicity. If the non-tangential limits if an f = T a g ∈ M (a), along with the non-tangential limits of its derivatives up to order m j -1, vanish at every point ζ j , j = 1, . . . , n, then

(5.4) g(0) = g(1) = • • • = g(N -1) = 0,
where N = n j=1 m j .

Proof of Lemma. We prove (5.4) as follows. Consider the kernels

k λ,ℓ (z) = c ℓ z ℓ (1 -λz) ℓ+1 ,
where c ℓ is adjusted so that these are the reproducing kernels for ℓ-th derivatives at point λ ∈ D in the Hardy space H 2 , that is to say,

f (ℓ) (λ) = f, k λ,ℓ H 2 = T f (ζ)k λ,ℓ (ζ)dm(ζ), f ∈ H 2 .
Observe, for 1 j n and 0 ℓ m j -1, that

a(z)k tζ j ,ℓ (z) = c ℓ z ℓ (z -ζ j ) m j (1 -tζ j z) ℓ+1 k =j (z -ζ k ) m k = c ℓ z ℓ (z -ζ j ) m j -(ℓ+1) z -ζ j 1 -tζ j z ℓ+1 k =j (z -ζ k ) m k . Writing z -ζ j 1 -tζ j z = -ζ j 1 -ζ j z 1 -t 1 -tζ i z ,
we see that a(z)k tζ j ,ℓ (z) is uniformly bounded in z ∈ D and t ∈ [0, 1), and moreover

z -ζ j 1 -tζ j z → -ζ j , t → 1,
for every z. Thus, by the dominated convergence theorem,

ak t,ℓ → cz ℓ (z -ζ j ) m j -(ℓ+1) k =j (z -ζ k ) m k
in the norm of H 2 , where c is some non zero constant depending on ℓ and j.

Choose any function f = T a g ∈ M (a) with (T a g) (ℓ) (ζ j ) = 0 for all 1 j n, 0 ℓ m j -1. Recall that M (a) ⊂ H (b) and so f , as well as all its derivatives up to order m j -1, admits non-tangential limits at ζ j for all 1 j n. Then 0 = (T a g) (ℓ)

(ζ j ) = lim t→1 -(T a g) (ℓ) (tζ j ) = lim t→1 -T a g, k tζ j ,ℓ H 2 = lim t→1 -g, ak tζ j ,ℓ H 2 = c g, cz ℓ (z -ζ j ) m j -(ℓ+1) k =j (z -ζ k ) m k H 2 .
In order to prove the lemma, it suffices to show that the set

ϕ j,ℓ (z) := z ℓ (z -ζ j ) m j -(ℓ+1) k =j (z -ζ k ) m k ,
where j = 1, . . . , n and ℓ = 0, . . . , m j -1, is a basis for the space of polynomials of degree at most N -1. Clearly each ϕ j,ℓ is a polynomial of degree N -1 and there are N -1 functions ϕ j,ℓ . It remains to show that the elements of this family are linearly independent. Obviously, for fixed 1 r n and 0 k m r -1, we have

ϕ (k) j,ℓ (ζ r ) = 0, j = r, 0 k m r -1, and
ϕ (k) r,ℓ (ζ r ) = 0, 0 k m r -(ℓ + 2). (5.5)
In particular, if j,ℓ α j,ℓ ϕ j,ℓ = 0, then, for fixed r and 0 k m r -1,

j,ℓ α j,ℓ ϕ (k) j,ℓ (ζ r ) = 0 which reduces to ℓ α r,ℓ ϕ (k) r,ℓ (ζ r ) = 0. Writing ϕ j,ℓ (z) = (z -ζ j ) m j -(ℓ+1) p j,ℓ (z)
, where p j,ℓ does not vanish at ζ j , Leibniz's formula gives

ϕ j,ℓ (z) m j -(ℓ+1) (z) = m j -(ℓ+1) k=0 m j -(ℓ + 1) k × (m j -(ℓ + 1))! (m j -(ℓ + 1) -k)! (z -ζ j ) m j -(ℓ+1)-k p (m j -(ℓ+1)-k) j,ℓ (z).
Evaluating this expression at ζ j makes all terms of the sum vanish except for k = m j -(ℓ + 1), and thus

ϕ j,ℓ (z) m j -(ℓ+1) (ζ j ) = (m j -(ℓ + 1))!p j,ℓ (ζ j ) = 0.
This together with (5.5) generates a triangular system of linear equations with non-zero diagonal entries. Thus α r,ℓ = 0, 0 ℓ m r -1.

We are now in a position to prove Theorem 1.5.

Our arguments so far yield (5.6) H (q r ) = M (a r ) and

(5.7) M (a r ) = M (a) ⊃ M (a)+ {v ℓ r,ζ j : 1 j n, 0 ℓ m j -1}.
First we show that the sum is orthogonal in the H (q r ) inner product:

v ℓ r,ζ j ⊥ M (a), 1 j n, 0 ℓ m j -1.
Indeed, for each f ∈ H (q r ) the radial limits f (ℓ) (tζ j ) exist as t → 1 -. Since f (ℓ) (tζ j ) = f, v ℓ r,tζ j q r , we can apply the principle of uniform boundedness to see that v ℓ r,tζ j q r is uniformly bounded as t → 1 -. Since v ℓ r,tζ j converges pointwise to v ℓ r,ζ j as t → 1 -we see that v ℓ r,tζ j converges weakly to v ℓ r,ζ j . Thus, since v ℓ r,tζ j reproduces the ℓ-th derivative of H (q r )-functions at point tζ j , for any g ∈ H 2 , we have ag, v ℓ r,ζ j q r = lim t→1 -ag, v ℓ r,tζ j q r = lim t→1 (ag) (ℓ) (tζ j ) = lim t→1 ℓ p=0 ℓ p a (p) (tζ j )g (ℓ-p) (tζ j ).

Using the estimate |a (p) (tζ j )| (1 -t) m j -p along with the following standard H 2 estimate on the growth of the derivative of an H 2 function

|g (ℓ-p) (tζ j )| 1 (1 -t) (ℓ-p)+1/2 , we see that |a (p) (tζ j )g (ℓ-p) (tζ j )| (1 -t) m j -p-((ℓ-p)+1/2) .
But since 0 ℓ m j -1 we see that

m j -p -((ℓ -p) + 1/2) = m j -ℓ -1 2 1 2
and so lim t→1 -|a (p) (tζ j )g (ℓ-p) (tζ j )| = 0. Thus ag, v ℓ r,ζ j q r = 0 and v ℓ r,ζ j ⊥ M (a) in H (q r ), for all 0 ℓ m j -1. This upgrades (5.6) and (5.7) to (5.8) H (q r ) = M (a) ⊃ M (a)⊕ {v ℓ r,ζ j : 1 j n, 0 ℓ m j -1}, and orthogonality is with respect to the norm in H (q r ).

To show equality in (5.8), our second step is to show that if f ∈ M (a) and f ⊥ v ℓ r,ζ j for all 1 j n, 0 ℓ m j -1, then f ∈ M (a). Since M (a) = T a H 2 this is equivalent to prove that if g ∈ H 2 and 0 = (T a g) (ℓ) (ζ j ) = lim t→1 -(T a g) (ℓ) (tζ j ) for all 1 j n, 0 ℓ m j -1 then T a g ∈ M (a). To simplify matters a bit, let us recall the formula for a from (5.2). Since s is a rational function with zeros and poles outside D -then certainly the Toeplitz operators T 1/s and T 1/s are invertible, and so M (a) = M (a/s). We can therefore make the simplifying assumption that

a(z) = n j=1 (z -ζ j ) m j .
We will show that (5.9) (T a g) (ℓ) (ζ j ) = 0, 1 j n, 0 ℓ m -1 =⇒ T a g ∈ aH 2 .

With N = n j=1 m j , one can verify the identify

a(ζ) = ζ N a(ζ) n j=1 (-ζ j ) m j , ζ ∈ T.
Thus

T a g = n j=1
(-ζ j ) m j P + (aζ N g).

By Lemma 5.3, we have g(0

) = g(1) = • • • = g(N -1) = 0, which shows that ζ N g ∈ H 2 and so T a g = n j=1 (-ζ j ) m j P + (aζ N g) ∈ aH 2 .
This completes the proof.

Examples

Example 6.1. Consider the function

q(z) = 1 2 (1 + z)
and notice that q is outer and q ∞ = 1. One can easily guess the Pythagorean mate for q to be a(z) = 1 2 (1 -z). The function a(z) has one zero of order 1 at z = 1 and a computation reveals that

v 0 1,1 (z) = 1 -q(1)q(z) 1 -z = 1 2 .
In this case H (q) = (z -1)H 2 ⊕ C. Moreover, for any r > 0 we get H (q r ) = H (q) and

H (q r ) = (z -1)H 2 ∔ C = (z -1)H 2 ⊕ C 1 - 1 + z 2 r 1 -z .
For more general q we need to review the proof of the Féjer-Riesz theorem which says that if

w(e iθ ) = n j=-n c j e ijθ and s(z) = z 2 w(z) = z 4 4 + z 2 2 + 1 4 = 1 4 (z -i) 2 (z + i) 2 .
Notice how the zeros occur in pairs i = 1/i and -i = 1/-i as guaranteed by the above proof of the Féjer-Riesz theorem. Thus the Pythagorean mate a for q is of the form a(z) = c(z -i)(z + i) for some c adjusted so that a(0) > 0 and 1 -|q(e iθ )| 2 = |a(e iθ )| 2 . One can check by direct calculation that c = 1/2 works and so a(z) = 1 2 (z -i)(z + i). Of course the exact value of c is not important for our calculations since we only need to identify the zeros of a along with their multiplicities.

The zeros of a are at z = i and z = -i and each has order one. Thus

H (q) = (z -i)(z + i)H 2 ⊕ {v 0 1,i , v 0 1,-i },
where the kernels can be computed directly as

v 0 1,i (z) = 1 2i (z + i), v 0 1,-i (z) = 1 2i (z -i).
Again, as in the previous example, H (q r ) = H (q) and so

H (q r ) = (z -i)(z + i)H 2 ∔ {z + i, z -i}. Example 6.3. Consider the function q(z) = 1 4 (z + 1) 2
and note that q is outer and belongs to b(H ∞ ). Following our Fejer-Riesz computations as in the previous example, note that Notice how these roots occur in the pairs α, 1/α. The function a is then a(z) = c(z -1)(z + 3 + 2 √ 2) for some appropriate constant c. There is one zero of a at z = 1 with multiplicity one and so H (q) = (z -1)H 2 ⊕ Cv 0 1,1 (z). The kernel can be computed to be

1 -|q(e it )| 2 = - e -it 4 - e it 4 - 1
v 0 1,1 (z) = z + 3 4 .
As in our previous examples, note that H (q r ) = (z -1)H 2 ∔ C(z + 3).

Observe that the q from this example is the square of the q from Example 6.1 and thus the corresponding spaces should be the same. Indeed, a little algebra will show that (z -1)H 2 ∔ C = (z -1)H 2 ∔ C(z + 3).

Example 6.4. Reversing the roles of a and q in the preceding example:

a(z) = 1 4 (z + 1) 2 , q(z) = c(z -1)(z + 3 + 2 √ 2),
with suitable c so that q ∞ = 1 (the maximum modulus on D -being attained at -1, one has c = (4(1 + √ 2)) -1 , and q(-1) = -1 corresponding to the normalization q(0) > 0), we obtain a function a with double zero, and so H (q) = (z + 1) 2 H 2 ⊕ {v 0 1,-1 , v 1 1,-1 } where v 0 1,-1 (z) = 1 -q(-1)q(z)

1 -(-1)z = 1 + q(z) 1 + z .

Using the facts that q(-1) = -1, q ′ (z) = c(2 + 2 √ 2), and q ′ (-1) = -1/2, we obtain

v 1 1,-1 (z) = 1 2 q(z)(1 + z) + z(1 + q(z)) (1 + z) 2 = 1 2
q(z)(1 + 3z) + 2z (1 + z) 2 .

  -Rovnyak space H (b) for certain b ∈ b(H ∞ ). Here H ∞ denotes the space of bounded analytic functions on the open unit disk D normed by f ∞ := sup z∈D |f (z)|, and b(H ∞ ) := {g ∈ H ∞ : g ∞ 1} is the closed unit ball in H ∞ and, for b ∈ b(H ∞ ), the de Branges-Rovnyak space H (b) is the reproducing kernel Hilbert space of analytic functions on D whose kernel is k b λ (z) := 1 -b(λ)b(z) 1 -λz , λ, z ∈ D.

( 1 )

 1 Suppose b ∈ b(H ∞ ) is outer. The following are equivalent: (a) For any r > 0 we have H (b r ) = H (b) as sets. (b) H (b 2 ) = H (b) as sets. (c) bH (b) ⊂ H ((b).

( 2 )( 3 )

 23 If b is non-extreme, i.e., log(1 -|b|) ∈ L 1 (T), with Pythagorean mate a, then conditions (a), (b), and (c) are equivalent to the condition (1.2) inf{|a(z)| + |b(z)| : z ∈ D} > 0. If b extreme, i.e., log(1 -|b|) ∈ L 1 (T), then conditions (a), (b), and (c) are equivalent to the condition

  ) Since b is outer, it has no zeros on D and so we can define b r by taking any logarithm of b. Note that b r ∈ b(H ∞ ).

( 2 )

 2 Statement (a) of Theorem 1.1 says that H (b r ) = H (b) as sets. Though the norms on H (b r ) and H (b) are different, one sees from the closed graph theorem that they are equivalent.

  By definition, f (λ) = f, k b λ b for all f ∈ H (b) and λ ∈ D, where •, • b represents the scalar product in H (b).

Lemma 3 . 2 .

 32 Suppose b ∈ b(H ∞ ) is outer and r is a positive real number. Then b and b r are simultaneously non-extreme. Moreover, if a r is the Pythagorean mate for b r , the pairs (a, b) and (a r , b r ) are simultaneously corona.

  Thus both expressions inf z∈D (|a(z)| + |b(z)|) and inf z∈D (|a r (z)| + |b r (z)|)

  inf z∈F |a(z)| = δ > 0. Thus inf z∈D (|a(z)| + |q(z)|) min( 1 2 , δ) > 0 concluding the proof. The first statement of Theorem 1.1 depends on the following two results. The first is from Sarason's book [9, p. 62]. Proposition 3.5. For b ∈ b(H ∞ ) and non-extreme, the following are equivalent:

4 .

 4 H (b r ) = H (b) as sets We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. The implication (a) =⇒ (b) is trivial. To show (b) =⇒ (c) note from [9, I-10] we have H (b 2 ) = H (b) + bH (b). But since we are assuming that H (b 2 ) = H (b) is follows that bH (b) ⊂ H (b). For the implication (c) =⇒ (1.2), we use the fact that b is nonextreme and [9, VIII-1, VIII-7] to see that b being a multiplier of H (b) is equivalent to (a, b) being a corona pair. To show that (1.2) =⇒ (a), we proceed as follows. By Lemma 3.2 we know that since (a, b) is a corona pair, then so is (a r , b r ). Thus from Proposition 3.5 we see that H (b) = M (a) and H (b r ) = M (a r ). As in the proof of Lemma 3.2 a/a r and a r /a belong to H ∞ so, by Proposition 3.6, we get M (a) = M (a r ). Putting this all together we get the desired set equality H (b r ) = H (b). For the implication (c) =⇒ (1.3), we use the fact that b is extreme and [9, VIII-1, VIII-5] to see that b being a multiplier of H (b) is equivalent to b being an invertible element of H ∞ . It remains to show (1.3) =⇒ (a). Assuming that b is invertible in H ∞ , we use, once again, [9, VIII-1] to see that H (b) = H ( b). But, since b is invertible in H ∞ , then so is b r and we thus also have H (b r ) = H (b r ). Remember that 1 -|b| 2 1 -|b r | 2 ≍ 1 and thus there are two constants c 1 , c 2 > 0 such that c 1 (I -T b T b ) I -T b r T b r c 2 (I -T b T b ). The Douglas factorization theorem implies that H ( b) = H (b r ) which concludes the proof. Remark 4.1. In Theorem 1.1 we see from the above proofs that one can add the condition H (b) = H (b) to the list of equivalent conditions.

  z) 2 (1 + 6z + z 2 ).The zeros of s are atz = -1, z = -1, z = -3-2 √ 2 ≈ -5.82843, z = -3+2 √ 2 ≈ -0.171573.

Question 6 . 5 .

 65 So far we have computed the exact contents of H (b) when b outer, rational, and non-extreme. Can one compute the contents of H (b) when b is outer and extreme. For example if b is the outer function corresponding to the outer function which satisfies |b(e iθ )| = 1 for 0 θ π and |b(e iθ )| = 1 2 for π < θ < 2π, can one describe the functions in H (b)?

This work was initiated while the first two authors were staying at the University of Richmond. These authors would like to thank that institution for the great hospitality. Work supported by Labex CEMPI (ANR-11-LABX-

is a non-zero trigonometric polynomial which assumes non-negative values for all θ, then there is an analytic polynomial p(z) = n j=0 a j z j so that w(e iθ ) = |p(e iθ )| 2 . Since the proof gives us the algorithm for computing p, we give a quick sketch. Indeed, as a function of the complex variable z, we see that if

is a polynomial of degree 2n and the roots of s occur in the pairs α, 1/α of equal multiplicity. It follows that

for some positive constant c and where α 1 , . . . , α n satisfy |α j | 1 for 1 j n. The desired polynomial p is

Note that p is zero free in D and we can multiply p by a unimodular constant so that p(0) > 0.

Recall from the proof of Lemma (3.1) that if q = p 1 /p 2 is rational then the Pythagorean mate a for q is given by a = p/p 2 , where p is the analytic polynomial (guaranteed by the Féjer-Riesz theorem) which satisfies |p(e iθ )| 2 = w(e iθ ) = |p 2 (e iθ )| 2 -|p 1 (e iθ )| 2 0, and p is chosen to that a(0) > 0.

Example 6.2. Consider the function

and note that q ∈ b(H ∞ ) and is outer. A computation shows that


Pham Thi

Hong Hanh

Sovereign Bond Yields in emerging Asia: New evidence

Keywords: JEL: E43, E44, G18, H63, Debt market, Sovereign bond yields, Emerging Asia, Panel analysis

This paper studies the determinants of sovereign bond yields in nine emerging Asian countries over the period 1994-2012. In the long-run, we first reveal that sovereign bond yields weakly and negatively depends on the changes in public debt. This result is not consistent with the theoretical hypothesis that rising government debt may foster sovereign bond yields through the default risk. Second, we fail to find out a long-run relationship between potential economic growth and sovereign borrowing costs in emerging Asia. Lastly, this paper evidences the preliminary interventions of emerging Asian authorities in separating government debt management from monetary management.

Introduction

After the 1997 Asian crisis, due to multiple bank failures and the vulnerability of a heavy reliance on foreign currency borrowing, emerging Asia has needed other alternatives to bank credit and foreign currency debt as sources of funding. Moreover, calling for other capital sources could allow emerging Asia to finance its accelerating growth and to avoid currency mismatches. In this circumstance, the choice of emerging Asia has been to develop a wellfunctioning local currency bond market, including government bonds, corporate bonds and mortgage bonds. A well-functioning local bond market, which is considered as an alternative to the banking sector brings a number of benefits to emerging Asia. First, the development of bond markets not only increases the supply of available financial assets, but also fosters a rise in competition on the financing market and contributes to lower borrowing and intermediation costs for households and companies. Second, the deepening of local bond market also results in the design of more sophisticated financial products (derivatives, securitization…) and the risk diversification. In particular, government bonds provide a benchmark yield curve for the issue of private sector bonds and thus facilitate the pricing.

Third, in a strengthened local bond markets, the central banks dispose of more sophisticated instruments for their monetary policy and can apply interest rate policies instead of direct credit control policies. Lastly, a well-functioning yield curve provides efficient information about market expectations and allows central banks to use market-oriented instruments, notably interest rates, to manage their monetary policies instead of manipulating private credit or bank reserve requirements.

To foster the development of local bond markets, Asian authorities have had to face to a number of challenges. One of the key challenges is to shed light on determinants of bond yields. Resolving this issue also allows explaining a tremendous variation in the interest rates across countries and over time, which governments or companies pay on their external debt. In the literature, borrowing costs depend on the fundamental conditions in the economy, and especially the fiscal accounts. However, we should distinguish between longrun and short-run determinants of bond yields. The main reason is that the long-run linkage between borrowing costs and macroeconomic fundamentals can break down in the shortrun, in particular during the periods of financial turmoil. In light of this meaning, the present paper tends to examine the long-run and short-run determinants of bond yields for a sample of emerging Asian countries -one of the most dynamic emerging financial markets. On the other hand, this paper only focuses on determinants of sovereign bond yields due to the fact that information on local currency bonds in emerging countries is lacking. We first employ the General Method of Moments (GMM) estimator developed by [START_REF] Arellano | Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations[END_REF] for a dynamic heterogeneous panel will be used to assess explicitly the transmission channels through which the macroeconomic fundamentals can affect sovereign bond yields.

We also use the Instrumental Variable (IV) estimator that allows controlling for simultaneity bias and reverse causality running from explicative variables to sovereign bond yields.

The rest of this paper is structured as follows. Section 2 reviews the existing literature.

Section 3 documents recent trends in local bond yields in emerging Asia. Section 4 describes the data and details the chosen empirical method. Section 5 presents and discusses the empirical findings. Concluding remarks are in Section 6.

Literature review

As mentioned in the introductory section, the existing literature distinguishes between the long-run and short-run determinants of sovereign bond yields. In the long term, the first determinant of sovereign bond yields is potential output growth. According to [START_REF] Laubach | New Evidence on the Interest Rate Effects of Budget Deficits and Debt[END_REF], a useful benchmark model, which explains this relationship, is the Ramsey model of optimal growth. Combined with a representative household with CES utility, the Ramsey model implies that the deterministic steady state of the real rate of return on capital is determined by:

= + (1)
where r is the rate of return, g denotes the net growth rate of per capita consumption, σ is the coefficient of relative risk aversion, and θ is the household's rate of time preference. The rate of return positively depends on all three of these parameters.

The second long-run macroeconomic determinant of sovereign bond yields is the government debt. The link between the government debt and real bond yields can be illustrated by the fact that fiscal expansion may reduce private investment. [START_REF] Engen | Federal Government Debt and Interest Rates[END_REF] suggest that a decline in private investment leads to a lower steady-state capital stock, which in turn stimulates marginal product of capital and consequently increases real interest rate. Additionally, by developing an early-warning model of sovereign debt crises, [START_REF] Manasse | Predicting Sovereign Debt Crises[END_REF] imply that rising government debt may foster sovereign bond yields through the default risk premium.

In addition to the long-run determinants, in the short-run, government bond yields may be also affected by a number of macroeconomic variables. First of all, monetary policy rate can affect nominal interest rates through its impact on inflation expectations. As shown in neoclassical models with nominal rigidities (Woodford, 2003), the monetary authority has to control short-term real interest rates and reinforce the role of stabilization policy in order to affect real activity. The government control over short-term real interest rates can be written by the following equation [START_REF] Laubach | New Evidence on the Interest Rate Effects of Budget Deficits and Debt[END_REF]:

= + * + + * (2)
Where is expected inflation in the period (t+1), and the natural rate * is a function of the factors shown in Equation 1 and presents the value of the real interest rate with the presence of nominal rigidities. From Equation 2, the real interest rate between two periods t and t+1 is defined as = -.

Together with monetary policy, financial openness has been considered as a short-run determinant of sovereign bond yields, especially in the case of emerging economies which have experienced comparatively large capital inflows over recent years. For instance, foreign investors can be an important source of demand for local debt securities and thus help lower bond yields (IMF, 2005). Foreign participation can also help increase the liquidity of government bond markets. However, foreign participation could lead to greater rate volatility. Differing from this meaning, [START_REF] Prasad | A Pragmatic Approach to Capital Account iberalization[END_REF] find that foreign participation, supported by the necessary institutional and regulatory framework, may minimize volatility. Lastly, temporary changes in fiscal position and variations of output gap are also considered as the key factors that lead to temporary changes in sovereign bond yields.

To shed light on the theoretical hypotheses, a large number of recent empirical works have been developed. Most of empirical studies have focused on the case of the U.S. and the key role of fiscal imbalance on determining sovereign bond yields. Using a sample of US states , [START_REF] Bayoumi | Do Credit Markets Discipline Sovereign Borrowers? Evidence from U.S. States[END_REF] find that sovereign debt levels affect state spreads versus New Jersey (the benchmark) in a non-linear way. At the mean level of debt, an increase of one percentage point in the ratio of debt to state product can raise 23 basis points in interest rate.

The non-linear relationship between fiscal imbalance and bond yields is also found in [START_REF] Conway | The GIRM: A Global Interest Rate Model[END_REF] for the major OECD countries, in which the impacts of fiscal factors on interest rate are non-linear and tend to be greater at higher levels of indebtedness. By contrast, employing a VAR approach, [START_REF] Plosser | Fiscal Policy and the Term Structure[END_REF] and [START_REF] Evans | Interest Rates and Expected Future Budget Deficits in the U.S[END_REF] do not support the impact of unexpected changes in fiscal variables on government bond yields. 2 On the other hand, several recent studies tend to show that sovereign borrowing costs could depend on expected rather than current fiscal variables. Using predicted values of U.S. fiscal variables as determinants of sovereign bond yields, [START_REF] Laubach | New Evidence on the Interest Rate Effects of Budget Deficits and Debt[END_REF] finds that a 1 percentage point increase in the expected government debt-to-GDP ratio raises real long-term government bond yield by about 2-5 basis points. Instead of the case study of the U.S., [START_REF] Chinn | The Euro Area and World Interest Rates[END_REF] use time series data for the 1988-2004 period to study separately the cases of five European countries (France, Germany, Italy, Spain, and the U.K.) and compare with the U.S.

The author argues that the impact of a 1 percentage point increase in the government debtto-GDP ratio on real long-term government bond yields varies slightly across countries. The impact is stronger in European countries, ranging from 5-8 basis points (Germany) to 10-16 basis points (France, Italy, the U.K., and Spain), compared to the U.S., where the impact is 5 basis points.

While the single-country studies mostly explore a static specification or the dynamic impacts of fiscal variables on government borrowing costs, the panel data studies typically employ the Fixed -effects (FE) estimation to resolve the question of interest. Together with fiscal variables measured by the debt-to-GDP ratio, another potential long-run determinants of sovereign bond yields, notably GDP growth, is also introduced in the cross-country studies. [START_REF] Kinoshita | Government Debt and Long-Term Interest Rate[END_REF] develops a dynamic general equilibrium model linking government bond yields to government debt. The author also provides numerical simulations by using a panel of 19 industrial economies. Kinoshita finds that the simulated and estimated interest rate effects of government debt tend to be small, while an increase in government consumption and debt leads to a considerably larger effect. Precisely, a 1 percentage point increase in the government debt-to-GDP ratio raises the real long-term government bond yield by about 2-5 basis points.

Similarly, basing on a panel of 16 OECD countries over 1960-2002[START_REF] Ardagna | Fiscal Discipline and the Cost of Public Debt Service: Some Estimates for OECD Countries[END_REF] examines the impact of fiscal deficit and debt on long-term government bond yields through two different econometric approaches. In a simple static specification, they find that a one percentage point increase in the primary deficit relative to GDP increases contemporaneous long-term interest rates by about 10 basis points. On the other hand, in a VAR approach, the same shock leads to a cumulative increase of almost 150 basis points after 10 years. The authors also confirm nonlinearities in the impact of government debt: debt affects the interest rate only in the case of countries with above-average levels of debt. In another study basing on a sample of ten euro area countries over 1979-2002[START_REF] Faini | Fiscal Policy and Interest Rates in Europe[END_REF] finds no significant impact of government debt on long-term government bond yields in individual country regressions while the positive impact of public debt becomes significant for the 10 euro area countries as a whole. Similarly to [START_REF] Ardagna | Fiscal Discipline and the Cost of Public Debt Service: Some Estimates for OECD Countries[END_REF][START_REF] Faini | Fiscal Policy and Interest Rates in Europe[END_REF] also concludes a nonlinear relationship between sovereign debt and sovereign borrowing costs.

Recently, [START_REF] Poghosyan | Long-run and Short-run determinants of Sovereign Bond Yields in Advanced Economies[END_REF] analyzes determinants of sovereign bond yields in 22 advanced economies over the 1980-2010 period by employing the panel cointegration techniques, which allows the author to distinguish between long-run (debt-to-GDP ratio, potential growth) and short-run (inflation, short-term interest rates, etc.) determinants. The main finding is that in the long-run, government bond yields increase by about 2 basis points due to a 1 percentage point increase in government debt-to-GDP ratio and by about 45 basis points due to a 1 percentage point increase in potential growth rate. In the short-run, sovereign bond yields deviate from the level determined by the long-run fundamentals, but about half of the deviation adjusts in one year.

Overall, the existing recent empirical literature on determinants of sovereign bond yields can be subdivided into two strands: single-country studies and panel data studies. On the other hand, to the best of our knowledge, the existing empirical studies seem to only focus on the case of advanced countries, except the contribution of Baldacci and Kumar (2010) using a sample of 31 advanced and emerging economies during the pre-crisis period 1980-2008.

According to Baldacci and Kumar, higher deficits and public debt result in a significant increase in long-term interest rates. Moreover, the magnitude of this impact strongly dependents on initial fiscal, institutional and other structural conditions, as well as spillovers from global financial markets. However, the findings of this paper should be taken with caution given the very heterogeneous levels of economic development between advanced economies and emerging economies. The aim of our paper is, therefore, to provide a complementary contribution to the literature as asking for the question "What factors affect the sovereign bond yields in emerging Asia?", which has not yet deal with in earlier studies.

Recent trends in emerging Asia's sovereign bond yields

As shown in Figure 1, sovereign bond yields in emerging Asia vary across countries and over the time. Since 1994, a downward trend in government bond yields has been observed in emerging Asia. In the case of Indonesia and the Philippines, the decline has been particularly significant, due to their structurally higher interest rates level. The main reason explaining this trend is the inflation decline and the improvement of economic fundamentals in emerging Asia. Besides, before the subprime crisis period, lower public debt in emerging Asia and abundant global capital source lead to a sharp declining in sovereign bonds' issuance. This favorable macroeconomic circumstance encouraged both foreign and domestic investors to buy local bonds and then push sovereign bond yields lower.

<Insert Figure 1> In Second, domestic factors in emerging Asia, including healthy banking systems, are also strengthening the bond markets in general and the sovereign bond markets in particular.

Data and empirical methodology

As mentioned above, basing on a sample of 9 emerging Asian countries: China, Hong Kong -China, India, Indonesia, Korea, Malaysia, Philippines, Singapore, and Thailand, this empirical study aims to investigate the long-run and short-run determinants of sovereign bond yields. Given this aim, our empirical strategy endeavors to make maximum use of both the time and cross-country dimensions of available data sets, which dictates using data at a quarterly frequency in the estimation. Our empirical specification partially adopted from Pogoshyan (2012) takes the following form:

= + + + + ! + " !# × % + & + ' (3)
where i and t indices denote country and time, and the dependent variable is real sovereign bond yields ( . Equation 3 includes two long-run determinants (LRit-1): the potential growth rate and the debt-to-GDP ratio. In light of the existing literature, we also introduce in the empirical model five short-run determinants *+ ( : real money market rate (monetary policy effect), changes in inflation (nominal shocks), fiscal balance ratio (short-term fiscal policy), economic growth rate (cyclical fluctuations), and foreign participation in the local bond markets (captured by the stock of foreign holdings of local currency bonds). We also include lagged dependent variable in the right hand side of Equation 3. , ( is a disturbance term assumed to satisfy the Gauss-Markov conditions. A trend termhas been introduced

to allow for a shift of the intercept over time. [START_REF] Csonto | Determinants of sovereign bond spreads in emerging markets: Local fundamentals and Global factors vs. Ever-changing misalignments[END_REF], as lower Fed funds rate is assumed to be associated with higher liquidity, it is expected to have a positive relationship the variations of sovereign bond yields. Second, a dummy variable takes the value of 1 during the crisis period and of 0 in other cases. This dummy allows us to resolve the conundrum of government bond yields' volatility and high fiscal imbalances observed in the aftermath of the crisis.

On the other hand, it is noteworthy that the magnitude of fiscal factors' impact on sovereign bond yields varies across countries. [START_REF] Baldacci | Fiscal Deficits, Public Debt, and Sovereign Bond Yields[END_REF] argue that the overall impact of higher deficits on interest rates could be greater in countries with the following characteristics: weak initial fiscal conditions, weak or inadequate institutions, weak structural factors (such as low domestic savings), and limited access to global capital. For this reason, our empirical model also allows for an additional interactive term between the fiscal balance /0 ( and a dummy variable 1 ( reflecting countries' characteristics listed above.

However, several econometric problems may arise from Equation 3:

i)

The independent variables are assumed to be endogenous. Because causality may run in both directions -from independent variables to dependent variable and also these regressors may be correlated with the error term.

ii)

Time-invariant individual characteristics (fixed effects) can be correlated with the explanatory variables.

iii)

Introducing lagged dependent variable gives rise to the correlation between the regressors and the error term. Because of this correlation, dynamic panel data estimation of Equation 3 suffers from the [START_REF] Nickell | Biases in dynamic models with fixed effects[END_REF] bias, which disappears only if T tends to infinity.

In addition, heteroscedasticity is expected to be present because, in the panel data, heterogeneous errors might exist with different panel members. To resolve these problems, the GMM method developed by [START_REF] Arellano | Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations[END_REF] seems to produce more efficient and consistent estimators compared with other procedures. This method also eliminates any endogeneity that may be due to the correlation of these country specific effects and the right hand side regressors. If we first difference Equation 3, we get:

∆ = ∆ + ∆ + ∆ + ∆ ! + " ∆!# × % + & + ∆' (4)
In equation 4, we have removed group effects and time trend. We now turn to the two key tests of the GMM estimator. The first one is the serial correlation test derived by [START_REF] Arellano | Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations[END_REF], in which the null hypothesis assumes no serial correlation in error term. [START_REF] Arellano | Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations[END_REF] introduce the tests for serial correlation, often labelled "m1" for first-order and "m2" for second-order serial correlation. We expect to find the first-order serial correlation in the first differenced residuals. The key problem arises if there is the second or a higher order serial correlation, suggesting that some of the moment conditions are invalid. The second one is the Sargan test to assess the model specification and overidentifying restrictions, whether the instruments, as a group, appear exogenous. This test is also known in the GMM context as the Hansen's J test.

Data setting

Our empirical analysis uses the sample consists of quarterly data on real sovereign bond yields and their fundamental determinants for the period 1994-2012 for nine emerging Asian countries. 10-year benchmark government bond yields available in Datastream will be used as a measure of sovereign borrowing costs Data on fundamental macroeconomic variables was obtained from the International Financial Statistics (IFS) and other international and national sources (see Appendix 1). Exceptionally, the data on potential economic growth rate is not available for the case of emerging and developing countries. We therefore use the average value of growth rates during ten previous periods as a potential growth rate.

Appendix 1 and Table 1 provide the definition and source of all key variables, their units of measurement, means, standard deviations (overall, between and within countries), and minimum and maximum values. Table 2 reports the correlation coefficients between all key variables which aid the modelling and help to confirm the choice of instruments. As shown in Table 1, almost correlation coefficients are significant that aids the modelling and help to confirm the choice of dependent variables. Besides, the magnitudes, the statistical significance even the sign of correlation coefficient have been more or less altered. Thus, we should not be surprised to see different empirical results on the impacts of macroeconomic variables on sovereign bond yields.

Empirical findings

Table 3 reports the estimation results for different models: the baseline model (Model 1) and the models with the interaction terms between fiscal balance and dummy variables accounting for nonlinear impact of fiscal policy on bond yields. In general, we observe that the baseline model's results are similar to those of other models, in which we control for countries' different characteristics.

As reported in Table 3, the potential growth enters in all estimated regressions with a negative but insignificant coefficient. This result seems to be not consistent with the theoretical hypotheses discussed above. The negative sign of the potential economic growth coefficient does not support the hypothesis that faster growing economies pay higher borrowing costs. This negative sign should be explained by the fact that in a wealthy economy leads to a decrease in sovereign debt and then a decrease in sovereign borrowing costs. However, this long-run relationship is not statistically significant, implying that in emerging Asia, sovereign bond yields are not determined by potential economic growth.

Second, differing from earlier empirical works, we find evidence of a negative relationship between public debt and sovereign bond yields. This finding, once again, does not support the consideration of [START_REF] Manasse | Predicting Sovereign Debt Crises[END_REF] that rising government debt may foster sovereign bond yields through the default risk premium. The negative but pretty small relationship between public debt and sovereign bond yields in emerging Asia may be explained in two ways. First, emerging Asia has experienced a number of supportive factors including strong underlying growth, conservative fiscal management, and financial repression that keep interest rates low even in a context of a rising sovereign debt. Second, comparing with advanced countries particularly debt-laden Europe, the balance sheets of most Asian economies remain strong (see Figure 3). External debt-to-GDP ratios have either declined significantly since 2009, as in the cases of Indonesia and the Philippines, or have stayed at manageable levels. For this reason, in emerging Asia, the reaction of sovereign bond yields to public debt is so small: sovereign bond yields go down by about 0.5 basis points due to 100 points increase in public debt.

<Insert Figure 3> Third, most of the short-run determinants enter in estimated regression with significant coefficients. The results suggest that bond yields increase significantly when the fiscal balance deteriorates: an increase in the fiscal deficit of 100 points pushes down bond yields by 5.8 basis points. Similarly, for each 100 points increase in inflation rate, real sovereign bond yields decrease about 3.2 basis points. Additionally, inflation has an additional effect on yields through its potential impact on volatility, which in turn could complicate the conduct of fiscal policy and lead to higher deficits [START_REF] Baldacci | Fiscal Deficits, Public Debt, and Sovereign Bond Yields[END_REF]) that once again lessen the value of sovereign bond yields. In general, inflation expectation plays an important role, either direct or indirect, in explaining sovereign bond yields' variation.

Unlike the impact of inflation rate, that of short-run changes in Central Bank's rate is not confirmed. This finding evidences the preliminary interventions of emerging Asian authorities in separating government debt management from monetary management. The separation of debt and monetary managements may bring a set of benefits for emerging Asia. First, this separation would offer the central bank a necessary independence in monetary management and an environment to pursue an inflation target, if assigned by the government. Second, this separation would also provide focus to the task of asset-liability management of government liabilities, undertake risk analysis and allow prioritizing government expenditure through higher awareness of interest costs.

Forth, economic growth is also significantly related to yields. In detail, higher growth leads to a compression in yields. This may be explained by the fact that higher growth results in higher tax revenues and smaller public social expenditures, which in turn reduce risk of fiscal slippage and fiscal vulnerability, and thus lower risk premium. Similarly, the relationship between foreign participation and bond yields is also negative. This finding seems to be consistent with some theoretical considerations meaning that foreign investors can be an important source of demand for local debt securities and thus allow lowering bond yields. In fact, foreign capital inflows enable emerging Asia to finance investment and foreign participation in local bond markets is seen as a way to accelerate the development of domestic financial infrastructure. Even though a growing demand of foreign investors for sovereign bonds lead to a decrease in bond yields, the emerging Asian authorities still tend to maintain an attractive level of sovereign bond yields in order to avoid the possible retire of foreign investors especially in times of global financial instability. This issue is verified by such a small fall about 0.1 basis points in sovereign bond yield following 100 points increase in foreign participation.

Firth, relating to the impacts of global factors, on one hand, we find that the financial turmoil slightly drives sovereign bond yields up. On the other hand, the coefficient on the U.S.

Federal funds rate is positive and significant. This result suggests that the U.S. monetary policy decisions at least partially reflect global risk aversion, which also influences sovereign bond markets in emerging Asia. Consequently, a short-run changes in the U.S. monetary policy results in a temporary rising trend in Asian real bond yields. Lastly, the empirical results find evidence that sovereign bond yields in emerging Asia do not depend on the initial level of macroeconomic characteristics such as fiscal condition, institutional quality, domestic savings and capital account openness.

We now turn our attention to the empirical results in the alternative models (Model 2-5) in which the fiscal balance is interacted with dummy variables accounting for several specific characteristics of each country (listed in the previous section). Overall, the empirical results are robust to controlling for the potential endogeneity of the fiscal balance due to the changes in some structural factors. As reported in Table 3, most of interactive terms between fiscal balance and macroeconomic characteristics enter in estimated model with an insignificant coefficient, except the case of Model 3, in which we control for the potential endogeneity of fiscal balance by using the initial level of public debt as an instrumental variable. Precisely, the interactive term between fiscal balance and initial debt level plays a negative and significant role in determining sovereign bond yields. This evidence reinforces the negative link between fiscal position and sovereign bond yields in emerging Asia. Overall, the empirical results allow us to conclude that the nature of short-run relationship between sovereign bond yields and fiscal balance does not depend on the initial level of macroeconomic fundamentals as suggested by [START_REF] Baldacci | Fiscal Deficits, Public Debt, and Sovereign Bond Yields[END_REF].

Sensitivity of empirical results

This sub-section carries out a set of robustness checks to examine the results' sensitivity to alternative estimation strategies. The first robustness check involves using an alternative estimation method. We report in Table 4 the results of estimating the sovereign bond yields using the Fixed-effects FE estimator, in which country and time dummies are included. The empirical results are qualitatively very similar to those obtained using the Dynamic GMM estimator that are reported in Table 3. While the magnitude of estimated coefficients is changed, most of coefficients have the same sign and are highly significant as those obtained with the GMM estimator.

We also estimated the same equation using Instrumental Variable (IV) estimator. In this step, instead of including in empirical model the interaction terms between fiscal balance and some specific characteristics of country, we estimate the model by directly instrumenting the fiscal balance with a set of structural macroeconomic variables. Moreover, the fundamental macroeconomic variables are now measured in percentage of GDP (not explained in dummy), including the initial level of: fiscal deficit, debt, institutional quality, domestic savings, and inward FDI. We use the average value over the 1990-1993 period as a proxy of the initial level of five variables listed above. The IV allows correcting the country-specific and time-specific effects and getting rid of any endogeneity in explanatory variables.

Above all, we test for the robustness of each IV estimated equation by performing four econometric tests, which results are reported in the lower part of Table 4. The first one is the under-identification test under the null hypothesis that the equation is under-identified. As reported in Table 4, the Kleibergen-Paap rk LM statistics are such important, meaning that the null hypothesis is rejected and the model is identified. Second, in the weak instrument test, the Kleibergen-Paap rk Wald F statistics, which are superior to the critical value of 5% maximal IV size proposed by [START_REF] Stock | Testing for weak instruments in linear IV regression[END_REF], also allow us to argue that weak instrument is not present in our models. Third, the Hansen J test checks the overidentification of all instruments. According to the empirical results, we cannot reject the null hypothesis of Hansen test, meaning that the instruments are valid instruments, notably uncorrelated with the error term, and that the excluded instruments are correctly excluded from the estimated equation. Lastly, the results of endogeneity test reported in the last line allow us to accept the null hypothesis that the specified endogenous regressors can actually be treated as exogenous.

We now look at the IV estimator's main results reported in the last column of Table 4. The IV results are, by and large, similar to those of the GMM estimator in terms of sign and significance, but the magnitudes are different as would be expected. Therefore, we conclude that the qualitative nature of our results is robust to alternative estimation methods.

Conclusion

The question about the fundamental factors, which can affect government borrowing costs, has been reviewed by a number of studies, either theoretical or empirical. However, this issue is less documented for emerging economies, especially for emerging Asian countries.

To fill this knowledge gap, this present paper attempts to shed light on the macroeconomic determinants of sovereign bond yields for a sample of nine emerging Asian countries over the period 1994-2012. Our empirical findings on the link between the long-run and short-run determinants and sovereign bond yields are, by and large, insensitive to a range of datasets and estimation methods.

We find that both country-specific fundamentals and global factors are important determinants of sovereign bond yields. Our empirical findings, however, provide only partial support to the concerned literature. In the long-run, we first reveal that the relationship between public debt and sovereign bond yields in emerging Asia is significant but weakly negative. This evidence is not consistent with the theoretical consideration suggesting that rising government debt may push up sovereign bond yields. Second, we fail

to confirm the role of potential growth in explaining the changes in sovereign bond curve.

In the short-run, the empirical results suggest that emerging Asia's sovereign bond yields negatively vary in function of the changes in economic growth rate, fiscal balance, inflation expectation, and foreign participation in the local bond markets. On the other hand, the variation of bond curve in emerging Asia depends on global factors, such as the U.S.

monetary policy decision and global financial stability. This paper also finds out a rising independence between government debt management and the central bank's monetary control.

To conclude, the present study is complementary to the existing literature that has seemed to only focus on the determinants of sovereign bond yields in advanced economies. Moreover, our results offer an important blessing for emerging Asia's authorities aspiring to manage their sovereign bond curves by considering the changes in macroeconomics fundamentals such as public debt, economic growth, and fiscal balance and so on. [START_REF] Stock | Testing for weak instruments in linear IV regression[END_REF].

Les Cahiers de la Chaire Finance rassemblent les documents de travail du LEMNA s'inscrivant dans des projets de recherche qu'elle soutient.

  2008, following the beginning of global economic recession, Asian sovereign bond yields went through two different phases. First, Asian yields surged in early 2008 due to an important increase in commodity prices, which in turn led to the back of inflationary concerns. Another main factor behind this figure is an important increase in central bank rate hikes as shown in Figure 2 (for example, 219bp in Indonesia, 42bp in India). The second phase began in the last half of 2009 when sovereign bond yields started to decline. This downturn could be explained by the global credit crisis (notably with the Lehman Brothers failure) and the subsequent lower growth and inflation expectations. A remarkable downfall is observed from the last quarter of 2009, when Asian central banks started a cycle of monetary easing and aggressively cutting rates. <Insert Figure 2> Despite local and global concerns, sovereign bond markets in Asia staged a recovery in the last quarter of 2011, with most yield curves flattening. So that, emerging Asian bond markets have been largely viewed as a safe haven for domestic and foreign investors for two following reasons. First, the low debt positions of most emerging East Asian economies make them attractive to investors compared with debt-laden Europe. On one hand, the balance sheets of most Asian economies remain strong. On the other hand, the external debt-to-GDP ratios have either declined significantly since 2009, or have stayed at manageable levels.

Figure 1 :Figure 2 :

 12 Figure 1 : 10 -year sovereign bond yields in emerging Asia

N° 2012 -

 2012 06 Are Islamic Indexes more Volatile than Conventional Indexes? Evidence from Dow Jones Indexes CHARLES Amélie, DARNÉ Olivier et POP Adrian N° 2012-07 Large Shocks in the Volatility of the Dow Jones Industrial Average Index: 1928-2010 CHARLES Amélie et DARNÉ Olivier N° 2012-08 The Quality of Private Monitoring in European Banking: Completing the Picture POP Adrian et POP Diana N° 2012-09 Effets socioéconomiques de la crise financière : implications pour le Vietnam LAUZANAS Jean-Marc, PERRAUDEAU Yves et POP Adrian N° 2012-31 Efficiency Gains from Narrowing Banks: A Search-Theoretic Approach TRIPIER N° 2012-32 Volatility Persistence in Crude Oil Markets CHARLES Amélie et DARNÉ Olivier N° 2012-40 La modélisation en équilibre général et stochastique des cycles économiques en Afrique Sub-saharienne : une revue de la littérature NOUASSI Claude Francis et TRIPIER Fabien Les opinions exposées dans ce document n'engagent que les auteurs. Ceux-ci assument la responsabilité de toute erreur ou omission. La Chaire Finance est une initiative de la Banque Populaire Atlantique, la Caisse d'Epargne Bretagne-Pays de Loire, la Caisse des Dépôts, du Crédit Maritime et de l'Université de Nantes, sous l'égide de la Fondation de Projets de l'Université de Nantes. Site web : http://www.univ-nantes.fr/fondation/chairefinance Contact Chaire Finance, Banque Populaire -Caisse d'Epargne IEMN-IAE, Chemin de Censive du Tertre -BP 52231 44322 Dynamics of Gasoline Prices: Evidence from Daily French Micro Data GAUTIER Erwan et LE SAOUT Ronan N° 2014-07 Stock Exchange Mergers and Market Efficiency CHARLES Amélie, DARNÉ Olivier, KIM Jae et REDOR Etienne N° 2014-08 On the Relationship between the Prices of Oil and the Precious Metals: Revisiting with a Multivariate Regime-Switching Decision Tree CHARLOT Philippe et MARIMOUTOU Vêlayoudom N° 2014-17 Precious metals shine? A market efficiency perspective CHARLES Amélie, DARNÉ Olivier et KIM Jae N° 2014-18 Sovereign bond yields in emerging Asia: New evidence PHAM Thi Hong Hanh Les Cahiers de la Chaire Finance Working Paper Series

Table 1 : Descriptive Statistics

 1

	Variable	Obs	Mean	Std. Dev.	Min	Max
	Sovereign bond yield (SY)	653	0.350	0.868	-0.989	10.150
	Public debt (PD)	684	46.005	25.458	6.137	108.300
	Potential GDP growth rate (PoGDP)	684	3.562	3.627	-11.738	14.755
	Real money market rate (RATE)	673	1.311	4.592	-25.840	44.620
	Inflation rate (INF)	680	5.153	5.791	-0.600	90.800
	Fiscal balance (FB)	684	0.136	4.589	-6.686	20.488
	GDP growth rate (GDP)	679	3.626	4.157	-17.930	16.116
	Foreign participation (FP)	684	43.175	49.912	-18.865	231.569
	The U.S. Federal fund rates (FED)	684	3.225	2.237	0.075	6.520

Table 2 : Bivariate Correlation coefficients

 2

		SY	PD	PoGDP RATE INF	FB	GDP	FP	FED
	SY	1.00
	PD	-0.075* 1.00
	PoGDP	-0.060 0.025	1.00
	Money Rate 0.070* -0.064* 0.229*	1.00
	INF	-0.365* 0.018	0.029	-0.179* 1.00
	FB	-0.228* 0.185* -0.137* -0.027 -0.113* 1.00
	GDP	-0.026 0.059	0.706*	0.149* -0.153* -0.147* 1.00
	FP	0.004	0.058	-0.366* -0.287* -0.206* 0.305* -0.331* 1.00
	FED	0.015	-0.118* 0.040	0.361* 0.148* 0.218* 0.014	-0.216* 1.00

Note: * indicates statistical significance at least the 10% level

Table 3 : GMM estimator's results

 3

	Independent variables	Model 1	Model 2 1	Model 3 1	Model 4 1	Model 5 1
	Sovereign bond yield	0.597*** (0.104)	0.611*** (0.103)	0.609*** (0.102)	0.610*** (0.103)	0.611*** (0.102)
	Public debt	-0.005** (0.002)	-0.004** (0.002)	-0.005** (0.002)	-0.004** (0.002)	-0.004** (0.002)
	Potential GDP growth rate	-0.011 (0.013)	-0.018 (0.016)	-0.017 (0.018)	-0.015 (0.014)	-0.017 (0.017)
	Real money market rate	-0.002 (0.007)	-0.005 (0.007)	-0.006 (0.006)	-0.005 (0.007)	-0.005 (0.007)
	Inflation rate	-0.032** (0.015)	-0.033** (0.015)	-0.034** (0.015)	-0.033** (0.015)	-0.033** (0.015)
	Fiscal balance	-0.058*** (0.009)	-0.057*** (0.009)	-0.046*** (0.011)	-0.067** (0.029)	-0.055*** (0.016)
	GDP growth rate	-0.026*** (0.009)	-0.030*** (0.009)	-0.031*** (0.009)	-0.029*** (0.010)	-0.030*** (0.009)
	Foreign participation	-0.001** (0.000)	-0.002*** (0.001)	-0.002*** (0.001)	-0.002*** (0.001)	-0.002*** (0.001)
	The U.S. Federal fund rates	0.034** (0.011)	0.029** (0.012)	0.029** (0.013)	0.030** (0.013)	0.029** (0.012)
	Financial crisis	0.124* (0.073)	0.121* (0.068)	0.124* (0.066)	0.124* (0.067)	0.122* (0.068)
	Dummy variable
	High initial fiscal deficit	0.005 (0.081)	0.004 (0.025)
	High initial debt	0.061 (0.096)		-0.020* (0.011)
	Quality of institutions	-0.233 (0.187)			0.011 (0.024)
	High FDI	-0.061 (0.067)				-0.002 (0.011)
	Constant	0.717*** (0.188)	0.638*** (0.158)	0.675*** (0.200)	0.604*** (0.154)	0.632*** (0.168)
	Number of observations	623	623	623	623	623
	Serial Corr. (m1) P-value	0.1284	0.1305	0.1317	0.1317	0.1310
	Serial Corr. (m2) P-value	0.4443	0.4397	0.4408	0.4428	0.4402
	Values in parentheses are robust standard errors. *** (**; *): Significant at 1%, 5% and 10% level respectively.

1 : Model with the interaction of fiscal balance with dummy variables for countries' characteristics.

Table 4 : Results of FE and IV estimators Independent variables FE estimator IV estimator Model 1 Model 2 1 Model 3 1 Model 4 1 Model 5 1

 4 Values in are robust standard errors. Values in brackets are *** *): Significant at 1%, 5% and 10% level respectively. 1 : Model with the interaction of fiscal balance with dummy variables for countries' characteristics in bold. Values in parentheses are robust standard errors.()

	Sovereign bond yield	0.612*** (0.112)	0.624*** (0.111)	0.625*** (0.108)	0,620*** (0,110)	0,624*** (0,108)	0,683*** (0,135)
	Public debt	-0.006** (0.002)	-0.005** (0.002)	-0.005** (0.002)	-0,005** (0,002)	-0,005** (0,002)	-0,005*** (0,001)
	Potential GDP growth	-0.007	-0.012	-0.013	-0,010	-0,011	0,005
	rate	(0.009)	(0.011)	(0.013)	(0,010)	(0,012)	(0,007)
	Real money market rate	-0.002 (0.007)	-0.004 (0.007)	-0.005 (0.006)	-0,004 (0,007)	-0,004 (0,007)	-0,003 (0,005)
	Inflation rate	-0.031* (0.015)	-0.031* (0.015)	-0.032** (0.015)	-0,031** (0,015)	-0,031** (0,015)	-0,026*** (0,009)
	Fiscal balance	-0.051*** (0.010)	-0.048*** (0.009)	-0.045*** (0.012)	-0,078** (0,037)	-0,055** (0,018)	-0,015*** (0,006)
	GDP growth rate	-0.024** (0.010)	-0.028** (0.010)	-0.028** (0.011)	-0,026** (0,010)	-0,027** (0,010)	-0,018*** (0,006)
	Foreign participation	-0.001** (0.000)	-0.002** (0.001)	-0.002** (0.001)	-0,001** (0,001)	-0,002** (0,001)	-0,001** (0,000)
	The U.S. Federal fund	0.034***	0.030**	0.030**	0,032**	0,030**	0,022**
	rates	(0.010)	(0.011)	(0.011)	(0,013)	(0,011)	(0,010)
	Financial crisis	0.169** (0.080)	0.146** (0.074)	0.144** (0.070)	0,147µµ (0,072)	0,145** (0,072)	0,123** (0,052)
	Dummy variables
	High initial fiscal deficit	0.017 (0.087)	-0.007 (0.027)
	High initial debt	0.101** (0.047)		-0.009 (0.008)
	Quality of institutions	-0.201 (0.121)			0,029 (0,028)
	High FDI	-0.047 (0.066)				0,007 (0,012)
	Constant	0.679*** (0.207)	0.578*** (0.176)	0.614*** (0.202)	0,542*** (0,173)	0,580*** (0,185)	0,186** (0,081)
	Number of observations	638	638	638	638	638	638
				Underidentification test		186.746
			(Kleibergen-Paap rk LM statistic)	[0.000]
				Weak identification test		(148.342) a
			(Kleibergen-Paap rk Wald F statistic)
			Overidentification test of		3.920
				(Hansen J statistic)		[0.4169]
				Endogeneity		2.671 [0.1022]

a , Critical value of 5 percent maximal IV size proposed by

See further[START_REF] Haugh | What Drives Sovereign Risk Premiums? An Analysis of Recent Evidence from the Euro Area[END_REF] for a review of empirical literature.

Acknowledgment: The author gratefully acknowledges financial support from the Chair Finance of the University of Nantes Research Foundation. Woodford, M., 2003. "Interest and Prices: Foundations of a Theory of Monetary Policy", Princeton, NJ: Princeton University Press.

The author gratefully acknowledges financial support from the Chair Finance of the University of Nantes Research Foundation.