Emanuele Alesci
email: alesci@fis.uniroma3.it

Eugenio Bianchi
email: 2e.bianchi@sns.it

Elena Magliaro
email: 3elena.magliaro@gmail.com

Claudio Perini
email: claude.perin@libero.it

Asymptotics of loop quantum gravity fusion coefficients

Introduction

The recent construction of a class of spinfoam models [START_REF] Engle | The loop-quantum-gravity vertex-amplitude[END_REF][START_REF] Engle | Flipped spinfoam vertex and loop gravity[END_REF][START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF][START_REF] Freidel | A New Spin Foam Model for 4d Gravity[END_REF][START_REF] Livine | A new spinfoam vertex for quantum gravity[END_REF] compatible with loop quantum gravity (LQG) [START_REF] Rovelli | A general covariant quantum field theory and a prediction on quantum measurements of geometry[END_REF][START_REF] Rovelli | Quantum gravity[END_REF][START_REF] Ashtekar | Background independent quantum gravity: A status report[END_REF][START_REF] Thiemann | Modern canonical quantum general relativity[END_REF] has opened the possibility of consistently defining the LQG dynamics using spinfoam techniques [START_REF] Baez | Spin foam models[END_REF][START_REF] Oriti | Spacetime geometry from algebra: Spin foam models for non-perturbative quantum gravity[END_REF][START_REF] Perez | Spin foam models for quantum gravity[END_REF][START_REF] Alesci | Spin-Foam Models and the Physical Scalar Product[END_REF]. In this paper we focus on the Engle-Pereira-Rovelli-Livine (EPRL) spinfoam model for Riemannian gravity introduced in [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF]. For given Immirzi parameter γ, the vertex amplitude is defined as follows: it is a function of five SO(3) intertwiners i a and ten spins j ab (with a, b = 1, . . , 5 and a < b) given by

W (j ab , i a ) = i L a i R a {15j} N |1 -γ|j ab 2 , i L a {15j} N (1 + γ)j ab 2 , i R a a f ia i L a i R a (j ab ) . (1)
The functions {15j} N are normalized 15j-symbols, namely the contraction of five normalized 4-valent SU (2) intertwiners according to the pattern of a 4-simplex, and the f ia i L a i R a are fusion coefficients from SO(3) to SU (2) L × SU (2) R introduced in [START_REF] Engle | LQG vertex with finite Immirzi parameter[END_REF] and defined below. Such coefficients play a key role in the definition of the model. Indeed the model differs from the one introduced by Barrett and Crane [START_REF] Barrett | Relativistic spin networks and quantum gravity[END_REF] only for the structure of these coefficients. In this paper we study the large spin asymptotics of the EPRL fusion coefficients.

A careful analysis of the asymptotics of fusion coefficients is a step needed for the study of the semiclassical properties of the model. In fact, we have already used the results that we present in this paper in order to understand the features of the wavepacket evolution. The propagation of semiclassical wavepackets was introduced in [START_REF] Magliaro | Numerical indications on the semiclassical limit of the flipped vertex[END_REF] as a new way to test the semiclassical limit of a spinfoam model. A spinfoam model has a good semiclassical behavior if semiclassical wavepackets (peaked on a classical 3-geometry) follow the trajectories predicted by the classical equations of motion. In [START_REF] Magliaro | Numerical indications on the semiclassical limit of the flipped vertex[END_REF] this new technique was implemented in the EPR flipped vertex model to study the propagation of intertwiner wavepackets. In [START_REF] Alesci | Intertwiner dynamics in the flipped vertex[END_REF] we developed a more efficient numerical algorithm (using techniques similar to [START_REF] Khavkine | Evaluation of new spin foam vertex amplitudes[END_REF]) and applied the asymptotic analysis presented here. This kind of study is, in the general context of spinfoam models, complementary to the semiclassical analysis based on the calculation of n-point functions.

In [START_REF] Rovelli | Graviton propagator from background-independent quantum gravity[END_REF][START_REF] Bianchi | Graviton propagator in loop quantum gravity[END_REF], a strategy for recovering graviton correlations from a background-independent theory was introduced. The idea was tested on the Barrett-Crane model at the "single-vertex" level. At this level, correlations of geometric operators can be checked against perturbative Regge-calculus with a single 4-simplex [START_REF] Bianchi | The perturbative Regge-calculus regime of Loop Quantum Gravity[END_REF]. Given the fact that the Barrett-Crane model gives trivial dynamics to intertwiners, the analysis was restricted to the spin degrees of freedom -namely to area correlations only. On the other hand, the new models are consistent with the LQG kinematics and allow the computations of semiclassical correlations of geometric observables as the area, the angle, the volume or the length [START_REF] Rovelli | Discreteness of area and volume in quantum gravity[END_REF][START_REF] Ashtekar | Quantum theory of geometry. I: Area operators[END_REF][START_REF] Ashtekar | Quantum theory of geometry. II: Volume operators[END_REF][START_REF] Major | Operators for quantized directions[END_REF][START_REF] Thiemann | A length operator for canonical quantum gravity[END_REF][START_REF] Bianchi | The length operator in Loop Quantum Gravity[END_REF]. At the single-vertex level, the semiclassical correlations for two local geometric operators Ô1 , Ô2 are simply given by Ô1 Ô2 q = j ab ia W (j ab , i a ) Ô1 Ô2 Ψ q (j ab , i a )

j ab ia W (j ab , i a ) Ψ q (j ab , i a ) , (2)
where W (j ab , i a ) is the vertex-amplitude introduced in (1) and Ψ q (j ab , i a ) is a boundary semiclassical state peaked on a configuration q of the intrinsic and the extrinsic geometry of the boundary of a region of space-time. The appropriate dependence on spins and intertwiners of the state Ψ q (j ab , i a ) is discussed in [START_REF] Alesci | The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex[END_REF][START_REF] Alesci | The complete LQG propagator: II. Asymptotic behavior of the vertex[END_REF] and uses the semiclassical tetrahedron state of [START_REF] Rovelli | A semiclassical tetrahedron[END_REF]. Moreover, in order to guarantee that the appropriate correlations are present, in [START_REF] Alesci | The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex[END_REF][START_REF] Alesci | The complete LQG propagator: II. Asymptotic behavior of the vertex[END_REF] a specific form of the large spin asymptotics for the vertex amplitude was conjectured (see [START_REF] Alesci | Tensorial Structure of the LQG graviton propagator[END_REF]). In order to show that the EPRL vertex amplitude satisfies this conjecture, an analysis of the asymptotics of the fusion coefficients is needed. The region of parameter space of interest is large spins j ab and intertwiners i a of the same order of magnitude of the spins. As a result, the fusion coefficients for the node ā,

f iā i L ā i R ā (j āb )
, can be seen as a function of the two bare variables i L ā , i R ā , of the fluctuation of the intertwiner i ā and of the fluctuation of the four spins j āb . In this paper we focus on this analysis. For different approaches to the semiclassical limit, see [START_REF] Bianchi | Semiclassical regime of Regge calculus and spin foams[END_REF] and [START_REF] Conrady | On the semiclassical limit of 4d spin foam models[END_REF].

The paper is organized as follows: in section 2 we show a simple analytic expression for the EPRL fusion coefficients; in section 3 we use this expression for the analysis of the asymptotics of the coefficients in the region of parameter space of interest; in section 4 we show that the fusion coefficients map SO(3) semiclassical intertwiners into SU (2) L × SU (2) R semiclassical intertwiners. We conclude discussing the relevance of this result for the analysis of the semiclassical behavior of the model. In the appendix we collect some useful formula involving Wigner coefficients.

Analytical expression for the fusion coefficients

The fusion coefficients provide a map from four-valent SO(3) intertwiners to four-valent SO(4) intertwiners. They can be defined in terms of contractions of SU (2) 3j-symbols. In the following we use a planar diagrammatic notation for SU (2) recoupling theory [START_REF] Yutsin | Mathematical Apparatus of the Theory of Angular Momentum[END_REF]. We represent the SU (2) Wigner metric and the SU (2) three-valent intertwiner respectively by an oriented line and by a node with three links oriented counter-clockwise1 . A four-valent SO(3) intertwiner |i can be represented in terms of the recoupling basis as

|i = √ 2i + 1 j 2 j 1 j 3 j 4 + + i ( 3
)
where a dashed line has been used to denote the virtual link associated to the coupling channel. Similarly a four-valent SO(4) intertwiner can be represented in terms of an SU (2

) L × SU (2) R basis as |i L |i R .
Using this diagrammatic notation, the EPRL fusion coefficients for given Immirzi parameter γ are given by

f i iLiR (j 1 , j 2 , j 3 , j 4 ) =(-1) j1-j2+j3-j4 (2i + 1)(2i L + 1)(2i R + 1)Π 4 n=1 (2j n + 1) × (4)
× + + + + + + - - i L i i R j 1 j 2 j 3 j 4 |1-γ|j 1 2 |1-γ|j 2 2
|1-γ|j 3 2 |1-γ|j 4 2 (1+γ)j 1 2 (1+γ)j 2 2 (1+γ)j 3 2 (1+γ)j 4 2
.

These coefficients define a map

f : Inv[H j1 ⊗ . . . ⊗ H j4 ] -→ Inv[H ( |1-γ|j 1 2 , (1+γ)j 1 2 ) ⊗ . . . ⊗ H ( |1-γ|j 4 2 , (1+γ)j 4 2 ) ] ( 5 )
from SO(3) to SO(4) intertwiners. Using the identity = [START_REF] Rovelli | A general covariant quantum field theory and a prediction on quantum measurements of geometry[END_REF] where the shaded rectangles represent arbitrary closed graphs, we have that the diagram in (4) can be written as the product of two terms

f i iLiR (j 1 , j 2 , j 3 , j 4 ) = (-1) j1-j2+j3-j4 (2i + 1)(2i L + 1)(2i R + 1)Π n (2j n + 1) q i iLiR (j 1 , j 2 ) q i iLiR (j 3 , j 4 ) (7)
where q i iLiR is given by the following 9j-symbol

q i iLiR (j 1 , j 2 ) = + + + - - + i L i R i j 1 j 2 |1-γ|j 1 2 |1-γ|j 2 2 (1+γ)j 2 2 (1+γ)j 1 2 =      |1-γ| 2 j 1 i L |1-γ| 2 j 2 1+γ 2 j 1 i R 1+γ 2 j 2 j 1 i j 2      . ( 8
)
From the form of q i iLiR we can read a number of properties of the fusion coefficients. First of all, the diagram in expression (8) displays a node with three links labelled i, i L , i R . This corresponds to a triangular inequality between the intertwiners i, i L , i R which is not evident from formula (4). As a result we have that the fusion coefficients vanish outside the domain

|i L -i R | ≤ i ≤ i L + i R . (9)
Moreover in the monochromatic case, j 1 = j 2 = j 3 = j 4 , we have that the fusion coefficients are nonnegative (as follows from ( 7)) and, for i L + i R + i odd, they vanish (because the first and the third column in the 9j-symbol are identical).

As discussed in [START_REF] Freidel | A New Spin Foam Model for 4d Gravity[END_REF][START_REF] Livine | A new spinfoam vertex for quantum gravity[END_REF], the fact that the spins labeling the links in (4) have to be half-integers imposes a quantization condition on the Immirzi parameter γ. In particular γ has to be rational and a restriction on spins may be present. Such restrictions are absent in the Lorentzian case. Now notice that for 0 ≤ γ < 1 we have that 1+γ 2 + |1-γ| 2 = 1, while for γ > 1 we have that 1+γ 2 -|1-γ| 2 = 1 (with the limiting case γ = 1 corresponding to a selfdual connection). As a result, in the first and the third column of the 9j-symbol in [START_REF] Ashtekar | Background independent quantum gravity: A status report[END_REF], the third entry is either the sum or the difference of the first two. In both cases the 9j-symbol admits a simple expression in terms of a product of factorials and of a 3j-symbol (see appendix A). Using this result we have that, for 0 ≤ γ < 1, the coefficient q i iLiR (j 1 , j 2 ) can be written as

q i iLiR (j 1 , j 2 ) = (-1) iL-iR+(j1-j2) i L i R i |1-γ|(j1-j2) 2 (1+γ)(j1-j2) 2 -(j 1 -j 2 ) A i iLiR (j 1 , j 2 ) (10)
with A i iLiR (j 1 , j 2 ) given by

A i iLiR (j 1 , j 2 ) = (j 1 + j 2 -i)! (j 1 + j 2 + i + 1)! (2j 1 + 1)! (2j 2 + 1)! × (11)
× (|1 -γ|j 1 )! (|1 -γ|j 2 )! |1-γ|j1 2 + |1-γ|j2 2 -i L ! |1-γ|j1 2 + |1-γ|j2 2 + i L + 1 ! × × ((1 + γ)j 1 )! ((1 + γ)j 2 )! (1+γ)j1 2 + (1+γ)j2 2 -i R ! (1+γ)j1 2 + (1+γ)j2 2 + i R + 1 ! .
A similar result is available for γ > 1. The Wigner 3j-symbol in expression [START_REF] Baez | Spin foam models[END_REF] displays explicitly the triangle inequality (9) among the intertwiners. Notice that the expression simplifies further in the monochromatic case as we have a 3j-symbol with vanishing magnetic indices. The fact that the fusion coefficients (4) admit an analytic expression which is so simple is certainly remarkable. The algebraic expression ( 7),( 10), [START_REF] Oriti | Spacetime geometry from algebra: Spin foam models for non-perturbative quantum gravity[END_REF] involves no sum over magnetic indices. On the other hand, expression (4) involves ten 3j-symbols (one for each node in the graph) and naively fifteen sums over magnetic indices (one for each link). In the following we will use this expression as starting point for our asymptotic analysis.

Asymptotic analysis

The new analytic formula [START_REF] Rovelli | Quantum gravity[END_REF], [START_REF] Baez | Spin foam models[END_REF], [START_REF] Oriti | Spacetime geometry from algebra: Spin foam models for non-perturbative quantum gravity[END_REF] is well suited for studying the behavior of the EPRL fusion coefficients in different asymptotic regions of parameter space. In this paper we focus on the region of interest in the analysis of semiclassical correlations as discussed in the introduction. This region is identified as follows: let us introduce a large spin j 0 and a large intertwiner (i.e. virtual spin in a coupling channel) i 0 ; let us also fix the ratio between i 0 and j 0 to be of order one -in particular we will take i 0 = 2 √ 3 j 0 ; then we assume that • the spins j 1 , j 2 , j 3 , j 4 , are restricted to be of the form j e = j 0 + δj e with the fluctuation δj e small with respect to the background value j 0 . More precisely we require that the relative fluctuation δje j0 is of order o(1/ √ j 0 );

• the SO(3) intertwiner i is restricted to be of the form i = i 0 + δi with the relative fluctuation δi i0 of order o(1/ √ j 0 );

• the intertwiners for SU (2) L and SU (2) R are studied in the region close to the background values

i 0 L = |1-γ| 2 i 0 and i 0 R = 1+γ 2 i 0 .
We study the dependence of the fusion coefficients on the fluctuations of these background values assuming that the relative fluctuations δi L /i 0 and δi R /i 0 are of order o(1/ √ j 0 ).

A detailed motivation for these assumptions is provided in section 4. Here we notice that, both for 0 ≤ γ < 1 and for γ > 1, the background value of the intertwiners i L , i R , i, saturate one of the two triangular inequalities [START_REF] Thiemann | Modern canonical quantum general relativity[END_REF]. As a result, we have that the fusion coefficients vanish unless the perturbations on the background satisfy the following inequality

δi ≤ δi L + δi R 0 ≤ γ < 1 ( 1 2 ) δi R ≤ δi + δi L γ > 1 . (13)
In order to derive the asymptotics of the EPRL fusion coefficients in this region of parameter space we need to analyze both the asymptotics of the 3j-symbol in (10) and of the coefficients A i iLiR (j 1 , j 2 ). This is done in the following two subsections.

Asymptotics of 3j-symbols

The behavior of the 3j-symbol appearing in equation [START_REF] Baez | Spin foam models[END_REF] in the asymptotic region described above is given by Ponzano-Regge asymptotic expression (equation 2.6 in [START_REF] Ponzano | Semiclassical limit of Racah coeffecients[END_REF]; see also appendix B):

i L i R i |1-γ|(j1-j2) 2 (1+γ)(j1-j2) 2 -(j 1 -j 2 ) ∼ (14)
∼ (-1) iL+iR-i+1 √ 2πA cos (i L + 1 2 )θ L + (i R + 1 2 )θ R + (i + 1 2 )θ + |1-γ|(j1-j2) 2 φ --(1+γ)(j1-j2) 2 φ + + π 4 .
The quantities A, θ L , θ R , θ, φ -, φ + admit a simple geometrical representation: let us consider a triangle with sides of length i L + 1 2 , i R + 1 2 , i + 1 2 embedded in 3d Euclidean space as shown below

i L + 1 2 i R + 1 2 i + 1 2 h + |1-γ|(j 1 -j 2 ) 2 h + (1+γ)(j 1 -j 2 ) 2 h -(j 1 -j 2 ) (15)
In the figure the height of the three vertices of the triangle with respect to a plane are given; this fixes the orientation of the triangle and forms an orthogonal prism with triangular base. The quantity A is the area of the base of the prism (shaded in picture). The quantities θ L , θ R , θ are dihedral angles between the faces of the prism which intersect at the sides i L , i R , i of the triangle. The quantities φ -, φ + are dihedral angles between the faces of the prism which share the side of length h + |1 -γ|(j 1 -j 2 )/2 and the side of length h + (1 + γ)(j 1 -j 2 )/2, respectively. For explicit expressions we refer to the appendix. In the monochromatic case, j 1 = j 2 , we have that the triangle is parallel to the plane and the formula simplifies a lot; in particular we have that the area A of the base of the prism is simply given by Heron formula in terms of i L , i R , i only, and the dihedral angles θ L , θ R , θ are all equal to π/2. As a result the asymptotics is given by

i L i R i 0 0 0 ∼ 1 √ 2πA 1 + (-1) iL+iR+i 2 (-1) i L +i R +i 2 . ( 16
)
Notice that the sum i L + i R + i is required to be integer and that the asymptotic expression vanishes if the sum is odd and is real if the sum is even. Now, the background configuration of i L , i R and i we are interested in corresponds to a triangle which is close to be degenerate (close to a segment). This is due to the fact that (1-γ) 2 i 0 + (1+γ) 2 i 0 = i 0 for 0 ≤ γ < 1, and (γ+1) 2 i 0 -(γ-1) 2 i 0 = i 0 for γ > 1. In fact the triangle is not degenerate as an offset 1 2 is present in the length of its edges. As a result the area of this almost-degenerate triangle is non-zero and scales as i 3/2 0 for large i 0 . When we take into account allowed perturbations of the edge-lengths of the triangle we find

A =    1 4 1 -γ 2 i 3/2 0 1 + 2(δi L + δi R -δi) + o(i -3/4 0 ) 0 ≤ γ < 1 1 4 γ 2 -1 i 3/2 0 1 + 2(δi + δi L -δi R ) + o(i -3/4 0 ) γ > 1 . (17)
This formula holds both when the respective sums δi L +δi R -δi and δi+δi L -δi R vanish and when they are positive and at most of order O( √ i 0 ). As a result we have that, when δi L + δi R -δi, or δi + δi L -δi R respectively, is even the perturbative asymptotics of the square of the 3j-symbol is

|1-γ| 2 i 0 + δi L (1+γ) 2 i 0 + δi R i 0 + δi 0 0 0 2 ∼ (18) ∼        2 π 1 √ 1-γ 2 i -3/2 0 √ 1+2(δiL+δiR-δi) θ(δi L + δi R -δi) 0 ≤ γ < 1 2 π 1 √ γ 2 -1 i -3/2 0 √ 1+2(δi+δiL -δiR) θ(δi + δi L -δi R ) γ > 1 . (19)
The theta functions implement the triangular inequality on the fluctuations. In the more general case when j 1 -j 2 is non-zero but small with respect to the size of the triangle, we have that the fluctuation in δj e can be treated perturbatively and, to leading order, the asymptotic expression remains unchanged.

Gaussians from factorials

In this subsection we study the asymptotics of the function A i iLiR (j 1 , j 2 ) which, for 0 ≤ γ < 1, is given by expression [START_REF] Oriti | Spacetime geometry from algebra: Spin foam models for non-perturbative quantum gravity[END_REF]. The proof in the case γ > 1 goes the same way. In the asymptotic region of interest all the factorials in [START_REF] Oriti | Spacetime geometry from algebra: Spin foam models for non-perturbative quantum gravity[END_REF] have large argument, therefore Stirling's asymptotic expansion can be used:

j 0 ! = 2πj 0 e +j 0 (log j 0 -1) 1 + N n=1 a n j -n 0 + O(j -(N +1) 0 ) for all N > 0, ( 20
)
where a n are coefficients which can be computed; for instance a 1 = 1 12 . The formula we need is a perturbative expansion of the factorial of (1 + ξ)j 0 when the parameter ξ is of order o(1/ √ j 0 ). We have that

(1 + ξ)j 0 ! = 2πj 0 exp + j 0 (log j 0 -1) + ξj 0 log j 0 + j 0 ∞ k=1 c k ξ k × (21) × 1 + N n=1 M m=1 a n b m j -n 0 ξ m + O(j -(N + M 2 +1) 0 ) ( 22
)
where the coefficients b m and c k can be computed explicitly. We find that the function A i iLiR (j 1 , j 2 ) has the following asymptotic behavior

A i0+δi |1-γ|i 0 2 +δiL , (1+γ)i 0 2 +δiR (j 0 + δj 1 , j 0 + δj 2 ) ∼ A 0 (j 0 ) e -H(δiL,δiR,δi,δj1,δj2) (23)
where A 0 (j 0 ) is the function evaluated at the background values and H(δi L , δi R , δi, δj 1 , δj 2 ) is given by

H(δi L , δi R , δi, δj 1 , δj 2 ) = 1 2 (arcsinh √ 3) δi L + δi R -δi + ( 2 4 ) + √ 3 2 (δi L ) 2 |1 -γ|i 0 + √ 3 2 (δi R ) 2 (1 + γ)i 0 - √ 3 4 (δi) 2 i 0 + - 1 2 δi L + δi R -δi i 0 (δj 1 + δj 2 ) + O( 1 √ j 0 )
for 0 ≤ γ < 1, while for γ > 1 it is given by

H(δi L , δi R , δi, δj 1 , δj 2 ) = 1 2 (arcsinh √ 3) δi + δi L -δi R + ( 2 5 ) + √ 3 2 (δi L ) 2 |1 -γ|i 0 + √ 3 2 (δi R ) 2 (1 + γ)i 0 - √ 3 4 (δi) 2 i 0 + - 1 2 δi + δi L -δi R i 0 (δj 1 + δj 2 ) + O( 1 √ j 0 ) .

Perturbative asymptotics of the fusion coefficients

Collecting the results of the previous two subsections we find for the fusion coefficients the asymptotic formula

f i0+δi |1-γ|i 0 2 +δiL , (1+γ)i 0 2 +δiR (j 0 + δj e ) ∼ f 0 (j 0 ) 1 1 + 2(δi L + δi R -δi) θ(δi L + δi R -δi) × (26) × exp -arcsinh( √ 3) (δi L + δi R -δi) × × exp - √ 3 (δi L ) 2 |1 -γ| i 0 - √ 3 (δi R ) 2 (1 + γ) i 0 + √ 3 2 (δi) 2 i 0 × × exp 1 2 δi L + δi R -δi i 0 (δj 1 + δj 2 + δj 3 + δj 4 )
for 0 ≤ γ < 1, and

f i0+δi |1-γ|i 0 2 +δiL , (1+γ)i 0 2 +δiR (j 0 + δj e ) ∼ f 0 (j 0 ) 1 1 + 2(δi + δi L -δi R ) θ(δi + δi L -δi R ) × (27)
× exparcsinh(

√ 3) (δi + δi L -δi R ) × × exp - √ 3 (δi L ) 2 |1 -γ| i 0 - √ 3 (δi R ) 2 (1 + γ) i 0 + √ 3 2 (δi) 2 i 0 × × exp 1 2 δi + δi L -δi R i 0 (δj 1 + δj 2 + δj 3 + δj 4 )
for γ > 1, where f 0 (j 0 ) is the value of the fusion coefficients at the background configuration. As we will show in next section, this asymptotic expression has an appealing geometrical interpretation and plays a key role in the connection between the semiclassical behavior of the spin foam vertex and simplicial geometries.

Semiclassical behavior

In [START_REF] Magliaro | Numerical indications on the semiclassical limit of the flipped vertex[END_REF] the propagation of boundary wave packets was introduced as a way to test the semiclassical behavior of a spinfoam model. In particular, the authors considered an "initial" state made by the product of four intertwiner wavepackets; this state has the geometrical interpretation of four semiclassical regular tetrahedra in the boundary of a 4-simplex of linear size of order √ j 0 . Then this state was evolved (numerically) by contraction with the flipped vertex amplitude to give the "final" state, which in turn is an intertwiner wavepacket. While in [START_REF] Magliaro | Numerical indications on the semiclassical limit of the flipped vertex[END_REF] only very small j 0 's were considered, in [START_REF] Alesci | Intertwiner dynamics in the flipped vertex[END_REF] we make the same calculation for higher spins both numerically and semi-analitically, and the results are clear: the "final" state is a semiclassical regular tetrahedron with the same size as the incoming ones. This is exactly what we expect from the classical equations of motion.

The evolution is defined by

i1...i5 W (j 0 , i 1 , . . . , i 5 ) ψ(i 1 , j 0 ) . . . ψ(i 5 , j 0 ) ≡ φ(i 5 , j 0 ), (28)
where

ψ(i, j 0 ) = C(j 0 ) exp - √ 3 2 (i -i 0 ) 2 i 0 + i π 2 (i -i 0 ) (29)
is a semiclassical SO(3) intertwiner (actually its components in the base |i ), or a semiclassical tetrahedron, in the equilateral configuration, with C(j 0 ) a normalization constant, and W (j 0 , i 1 , . . . , i 5 ) is the vertex (1) with γ = 0 evaluated in the homogeneous spin configuration (the ten spins to j 0 ). In [START_REF] Alesci | The complete LQG propagator: II. Asymptotic behavior of the vertex[END_REF], if we want to make the sum over intertwiners, for fixed j 0 , then we have to evaluate the function g defined as follows:

g(i L , i R , j 0 ) = i f i iL iR (j 0 ) ψ(i, j 0 ) . (30)
The values of g are the components of an SO(4) intertwiner in the basis |i L |i R , where |i L is an intertwiner between four SU (2) irreducible representations of spin j L 0 ≡ |1-γ| 2 j 0 , and |i R is an intertwiner between representations of spin j R 0 ≡ 1+γ 2 j 0 . We show that EPRL fusion coefficients map SO(3) semiclassical intertwiners into SU (2) L × SU (2) R semiclassical intertwiners. The sum over the intertwiner i of the fusion coefficients times the semiclassical state can be computed explicitly at leading order in a stationary phase approximation, using the asymptotic formula [START_REF] Bianchi | The length operator in Loop Quantum Gravity[END_REF] [START_REF] Alesci | The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex[END_REF]. The result is

i f i iL iR (j 0 ) ψ(i, j 0 ) ≈ α 0 f 0 (j 0 ) C(j 0 ) × exp - √ 3 (i L -|1-γ| 2 i 0 ) 2 |1 -γ| i 0 ± i π 2 (i L -|1-γ| 2 i 0 ) × (31) × exp - √ 3 (i R -(1+γ) 2 i 0 ) 2 (1 + γ) i 0 + i π 2 (i R -(1+γ) 2 i 0 )
where

α 0 = k∈2N e -arcsinh( √ 3)k √ 1 + 2k e ∓i π 2 k 0.97 ; (32)
the plus-minus signs both in [START_REF] Bianchi | Semiclassical regime of Regge calculus and spin foams[END_REF] and [START_REF] Conrady | On the semiclassical limit of 4d spin foam models[END_REF] refer to the two cases γ < 1 (upper sign) and γ > 1 (lower sign). The r.h.s. of [START_REF] Bianchi | Semiclassical regime of Regge calculus and spin foams[END_REF], besides being a very simple formula for the asymptotical action of the map f on a semiclassical intertwiner, is asymptotically invariant under change of the coupling channels for the virtual spins i L and i R (up to a normalization N ). For example, under a joint change of coupling for i L an i R , we have where the brackets are 6j-symbols. The result [START_REF] Bianchi | Semiclassical regime of Regge calculus and spin foams[END_REF] holds because each of the two exponentials (r.h.s.) is of the form

iL iR dim(i L )dim(i R )(-1) iL+kL+iR+kR |1-γ| 2 j 0 |1-γ| 2 j 0 i L |1-γ| 2 j 0 |1-γ| 2 j 0 k L × (33)
× 1+γ 2 j 0 1+γ 2 j 0 i R 1+γ 2 j 0 1+γ 2 j 0 k R g(i L , i R ) ≈ N (j 0 ) g(k L , k R , j 0 ) (a) (b)
exp - √ 3 2 (k -k 0 ) 2 k 0 ± i π 2 (k -k 0 ) , (34)
which represents a semiclassical equilateral tetrahedron with area quantum numbers k 0 ; it follows that g is (asymptotically) an SO(4) semiclassical intertwiner. The formula (31) can be checked against plots of the exact formula for large j 0 's; a particular case is provided in fig. 1.

In addition, we can ask whether the inverse map f -1 has the same semiclassical property. Remarkably, the answer is positive: f -1 maps semiclassical SO(4) intertwiners into semiclassical SO(3) intertwiners. The calculation, not reported here, involves error functions (because of the presence of the theta function) which have to be expanded to leading order in 1/j 0 .

A final remark on our choice for the asymptotic region is needed. The goal we have in mind is to apply the asymptotic formula for the fusion coefficients to the calculation of observables like (2) in the semiclassical regime. If the classical geometry q over which the boundary state is peaked is the geometry of the boundary of a regular 4-simplex, then the sums in (2) are dominated by spins of the form j ab = j 0 + δj ab and intertwiners of the form i a = i 0 + δi a , with i 0 = 2j 0 / √ 3, where the fluctuations must be such that the relative fluctuations δj/j 0 , δi/j 0 go to zero in the limit j 0 → ∞. More precisely, the fluctuations are usually chosen to be at most of order O( √ j 0 ). This is exactly the region we study in this paper. As to the region in the (i L , i R ) parameter space, the choice of the background values |1-γ| 2 i 0 , 1+γ 2 i 0 and the order of their fluctuations is made a posteriori both by numerical investigation and by the form of the asymptotic expansion. It is evident that the previous considerations hold in particular for the function g analyzed in this section.

The case γ = 1

When γ = 1 we have that j L ≡ |1-γ| 2 j = 0 and we can read from the graph (4) that the fusion coefficients vanish unless i L = 0. Furthermore it is easy to see that for γ = 1 the fusion coefficients vanish also when i R is different from i. This can be seen, for instance, applying the identity

= 1 dim i δ i,iR (35)
to the graph (4) with i L = 0. As a result, we have simply

f i iL iR (j 1 , j 2 , j 3 , j 4 ) = δ iL,0 δ iR,i (36)
and the asymptotic analysis is trivial. The previous equation can be also considered as a normalization check; in fact, with the definition (4) for the fusion coefficients, the EPRL vertex amplitude (1) reduces for γ = 1 to the usual SO(3) BF vertex amplitude.

Conclusions

We summarize our results and give some outlook in a few points.

• We have shown a simple analytic formula for the LQG fusion coefficients, as defined in the EPRL spinfoam model.

• We have given a large spin asymptotic formula for the coefficients; specifically, we made a perturbative asymptotic expansion around a background configuration dictated by the kind of boundary state considered.

• The picture coming out from our analysis is promising: the fusion coefficients not only give nontrivial dynamics to intertwiners at the quantum level, but they seem to behave very well at semiclassical level, in fact they map semiclassical SO(3) tetrahedra into semiclassical SO(4) tetrahedra. This is to us a highly non-trivial property which, in turn, makes the semiclassical analysis of dynamics less obscure. A first application of the asymptotic formula can be found in [START_REF] Alesci | Intertwiner dynamics in the flipped vertex[END_REF].

• Our analysis is a step needed for the study of the full asymptotic expansion of the EPRL vertex, which is part of our work in progress.

A Properties of 9j-symbols

The 9j-symbol with two columns with third entry given by the sum of the first two can be written as

These formula can be derived from [START_REF] Yutsin | Mathematical Apparatus of the Theory of Angular Momentum[END_REF][START_REF] Varshalovich | Quantum Theory of Angular Momentum[END_REF].

B Regge asymptotic formula for 3j-symbols

The asymptotic formula of 3j-symbols for large spins a, b, c and admitted magnetic numbers, i.e. m a + m b + m c = 0, given by G. Ponzano and T. Regge in [START_REF] Ponzano | Semiclassical limit of Racah coeffecients[END_REF] is

Figure 1 :

 1 Figure 1: (a) Interpolated plot of the modulus of g(i L , i R , j 0 ) for j 0 = 20 and γ = 0 computed using the exact formula of the fusion coefficients. (b) Top view of the imaginary part.

  b -d -(a + b -(c + d)) × (37) × (2a)!(2b)!(2c)!(2d)!(a + b + c + d -h)!(a + b + c + d + h + 1)! (2a + 2b + 1)!(2c + 2d + 1)!(a + c -f )!(a + c + f + 1)!(b + d -g)!(b + d + g + 1)! .An analogous formula for the 9j-symbol with two columns with third entry given by the difference of the first two can be obtained from the formula above noting that  in the previous case. The 3j-symbol with vanishing magnetic numbers has the simple expression

4 2 ) 2 ) 2 + (a + 1 2 ) 2 -(b + 1 2 ) 2 (a + 1 2 ) 2 -m 2

 4222222222 c + m a φ a -m b φ b + π 2 m c + m a (c + 1

A minus sign in place of the + will be used to indicate clockwise orientation of the links.

Acknowledgments

We thank Carlo Rovelli for numerous discussions. E. Bianchi and E. Alesci gratefully acknowledge support by Fondazione Della Riccia.


Niklas Potrafke
email: niklas.potrafke@uni-konstanz.de

Labor market deregulation and globalization: Empirical evidence from OECD countries

Keywords: labor market (de)regulation, globalization, panel data JEL Classification: F57, F16, J58, J88, C23

 on labor market institutions in OECD countries and the KOF index of globalization. The data set covers 20 OECD countries in the 1982-2003 period. The results suggest that globalization did neither influence the unemployment replacement rate, the unemployment benefit length, public expenditures on ALMP, the tax wedge, union density nor overall employment protection. In contrast, protection of regular employment contracts was diminished when globalization was proceeding rapidly. In fact, domestic aspects, such as unemployment and government ideology are more important determinants of labor market institutions and deregulation processes in OECD countries than globalization. For this reason, working conditions of unskilled workers are not likely to deteriorate and the jobs of unskilled workers are not likely to disappear in the course of globalization. All this is, of course, not to insinuate that globalization has any benign influence on labor market institutions.

Introduction

Commentators believe that globalization plays a significant part in shaping labor market institutions. Advocates of the skeptical view on the role of globalization fear that working conditions for unskilled workers will deteriorate and many jobs of unskilled workers will disappear in the course of globalization. They therefore encourage a more stringent role of government in the domestic economy. In contrast, advocates of the market-oriented view argue in favor of a less regulated labor market because labor market regulation comes at a cost for employers, raises labor costs and unemployment. 1 A final verdict on the two views of globalization can only be derived from an empirical analysis.

Several recent studies have focused on the relationship between globalization and labor market (de)regulation. In his intriguing paper Boulhol (2009a, p. 223), for example, presents a theoretical model which "incorporates labor market rigidities … of footloose capital in order to study how globalization might affect the trade-offs generated by labor market regulation and put pressure on labor market in institutions." Boulhol (2009a, p. 223) identifies two transmission channels of this process: first, "capital mobility triggers a re-allocation of resources, which trade integration amplifies, away from the high-rent / highly unionized sector. Second, the threat of costly relocations encourages labor market deregulation. The latter channel is more efficient because it avoids sub-optimal sectoral specialization". These predictions on the influence of globalization on labor market (de)regulation need to be evaluated empirically. Implementing this task, however, raises the question of how to measure labor market (de)regulation and globalization.

Scholars have investigated various aspects of labor market (de)regulation such as, for example, unemployment benefits, employment protection and deunionization and various facets of globalization. Empirical studies suggest mixed results about the relationship between labor market 1 For empirical investigations how labor market deregulation affects unemployment see, for example, [START_REF] Blanchard | The role of shocks and institutions in the rise of European unemployment: the aggregate evidence[END_REF], [START_REF] Baccaro | Institutional determinants of unemployment in OECD countries: Does the deregulatory view hold water?[END_REF] and [START_REF] Feldmann | The unemployment effects of labor regulation around the world[END_REF]. See also [START_REF] Boulhol | Unemployment and relative labor market institutions between trading partners[END_REF] who shows how foreign labor market institutions affect a country"s unemployment rate through the trade channel. institutions and globalization. In a nutshell, globalization does not appear to have a systematic influence on various aspects of labor market (de)regulation. I will discuss the different approaches and empirical findings in more detail below.

Many economic indicators are associated with labor market (de)regulation. [START_REF] Nickell | Unemployment and labour market rigidities: Europe versus North America[END_REF], Blanchard and Wolfers (2000: C19f.) and [START_REF] Bassanini | Employment patterns in OECD countries: reassessing the role Of policies and institutions[END_REF], for example, distinguish between eight labor market institutions: Three measures of different dimensions of the unemployment insurance system: the replacement rate, benefit length, and a measure of active labor policy.

One measure of employment protection.

The tax wedge.

Three measure aspects of collective bargaining: union contract coverage, union density and (union and employer) coordination of bargaining.

An encompassing empirical analysis investigating the influence of globalization on labor market (de)regulation needs to address all these potential channels. Moreover, globalization is a multifaceted concept that cannot be captured by single economic indicators such as trade openness and foreign direct investment. Therefore, all-embracing globalization indicators have been developed over the last years. The KOF index of globalization is a case in point (see [START_REF] Dreher | Does globalization affect growth? Evidence from a new index of globalization[END_REF]Dreher et al. 2008a).

In this paper, I employ the annual data set on labor market institutions by [START_REF] Bassanini | Employment patterns in OECD countries: reassessing the role Of policies and institutions[END_REF] and the KOF index of globalization in order to empirically investigate whether globalization has induced labor market deregulation in OECD countries in the 1982-2003 period.

My analytical design takes advantage of a compatible data set on labor market institutions. The results suggest that globalization did not have a systematic influence on labor market deregulation.

In fact, domestic aspects, such as such as unemployment and government ideology are more important determinants of labor market institutions and their deregulation processes in OECD countries than globalization.

The paper is organized as follows: section 2 presents the data on labor market institutions and globalization. Section 3 sets up the dynamic panel data model and describes the empirical strategy. Section 4 empirically investigates the relationship between the single measures for labor market institutions and globalization: in each scenario, I will first briefly describe the political economic reasoning and results of related empirical studies (if available), specify a hypothesis to be tested and then turn to discuss my own regression results for every single labor market institution indicator. Section 5 concludes.

Data

Labor market institutions

I use the data set on labor market institutions for OECD countries by [START_REF] Bassanini | Employment patterns in OECD countries: reassessing the role Of policies and institutions[END_REF]. 2 The data are available from 1982-2003 and cover 20 OECD countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, the Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, the United Kingdom, and the United States (unbalanced panel).

The average unemployment benefit replacement rate refers to "two income situations (100% and 67% of APW earnings), three family situations (single, with dependent spouse, with spouse in work) and three different unemployment durations (1 st year, 2 nd and 3 rd years, and 4 th and 5 th years of unemployment)" (Bassanini and Duval 2006: 106). Figure 1(a) depicts the average development of the replacement rate for the 20 OECD countries in the 1982-2003 period. The average replacement rate has increased from 25.5 % in 1982 to 32.1% in 1999 (maximum), and slightly decreased to 31.4 % in 2003. Compared across countries, the replacement rates were high on average in countries such as Denmark (54.9) and Belgium (41.1), and low in countries such as Japan (9.9) and the United States (12.8).

2 These data have been employed, for example, by [START_REF] Bassanini | Job protection legislation and productivity growth In OECD countries[END_REF], who examine the influence of job protection legislation on productivity growth and are available at http://bassax.freeyellow.com/

The unemployment benefit duration (in years) is defined as the ratio of average to initial unemployment benefit replacement rate. Figure 1 (b) shows that the average benefit duration was about 0.65 years, and that it has increased from 0.60 years in 1982 to its maximum of 0.70 in 1995.

The benefit length, however, varied between the individual OECD countries: in Australia and New Zealand, it was about one year and the benefit duration was nearly constant over time. In Sweden, it was about 0.33 years and nearly constant over time. In Italy, the benefit duration dramatically increased in the beginnings of the 1990s, but then also immediately decreased again.

Public expenditures on active labor market expenditures (ALMP) are measured as a share of GDP and cover five different subcategories: public employment services and administration, labor market training, youth measures (such as special programs for unemployed and disadvantaged youth), subsidized employment and measures for the disabled. The tax wedge refers to the combined labor and consumption tax rate derived from national accounts. It represents a significant indicator because some labor market institutions do not have much effect on unemployment as such, but on wages. For this reason, it matters how taxes affect the ratio of after-tax unemployment benefits to after-tax wages (Blanchard and Wolfers 2000: C13). On average, the tax wedge was 28.75% indicating that employers and consumers pay about one third more than workers and producers receive (Figure 1e). There are two peaks: one in 1987 with 30.1% and one in 1995 with 29.8%. Since 1995, the tax wedge has decreased to 26.8% in 2003. Tax wedges differed between countries such as, for example, Australia (15.3%) and Japan (16.8%) and Italy (41.3%) and Sweden (42.3%). Over time, the tax wedge decreased in countries such as Ireland, the United Kingdom and the United States but increased in countries such as Canada and Japan.

The union density rate measures the share of workers affiliated to a trade union in percent. Union contract coverage and (union and employer) coordination of bargaining display timeinvariant variables in the data set by [START_REF] Bassanini | Employment patterns in OECD countries: reassessing the role Of policies and institutions[END_REF]. For this reason, I cannot examine these two labor market institutions indicators in a dynamic panel data model.

The KOF index of globalization

I use the KOF index of globalization. Globalization is a multi-faceted concept that cannot be entirely captured by a single economic indicator such as international trade (as a share of GDP), foreign direct investment or capital account restrictions. The KOF index [START_REF] Dreher | Does globalization affect growth? Evidence from a new index of globalization[END_REF]Dreher et al. 2008a) represents an attempt to measure globalization in the broad sense that has been accepted in the recent empirical literature. 5 The index covers 123 countries and includes 23 variables and portrays the economic, social and political dimension of globalization. Each of these three dimensions has further subdimensions. For example, economic globalization is described by actual flows (trade, foreign direct investment, portfolio investment and income payments to foreign nationals, each measured as a percentage of GDP) and restrictions (hidden import barriers, mean tariff rate, taxes on international trade and capital account restrictions). Social globalization covers, among others, items such as international tourism, number of internet hosts and users, as well as the number of McDonald"s restaurants and the number of IKEA shops (per capita). Political globalization is measured by the number of foreign embassies, membership in international organizations and the participation in U.N. Security Council missions (see Dreher et al. 2008a: 43 ff. for further details). In this study, I use the updated 2009 KOF index of globalization which measures globalization on a scale of 1 to 100, where higher values represent higher levels of globalization.

Time series properties

A significant point concerns the time series properties of the labor market institution indicators and the KOF globalization indicators. Panel unit root tests show that the variables are non-stationary in levels, but stationary in growth rates. 6 For this reason, the growth rates of the labor market institutions indicators need to be regressed on the growth rates of the KOF globalization indicators in order to avoid spurious regression.

The empirical model

The basic estimated dynamic panel data model has the following form: …, 20; j=1,…,6; t=1,…,21 (1)

Δ ln Labor market deregulation indicator ijt = α Δ ln Globalization it + β Government Ideology it γ Δ ln Labor market deregulation indicator ijt-1 + η i + ε t + u ijt with i = 1,
where the dependent variable "Δ ln Labor market deregulation indicator ijt " denotes the growth rates of the six respective labor market indicators. "Δ ln Globalization it " describes the growth rates of the KOF globalization indicators. I distinguish between a group of control variables that I include in every model describing labor market deregulation and specific variables to take into account the respective individual characteristics and differences between labor market deregulation indicators such as ALMP expenditures or EPL. I follow related studies and always include a government ideology indicator because we expect higher labor market regulation under leftwing governments (for a discussion of this issue see, for example, [START_REF] Botero | The regulation of labor[END_REF]). The variable "Government Ideology it " is measured by the indicator presented by [START_REF] Potrafke | Did globalization restrict partisan politics? An empirical evaluation of social expenditures in a panel of OECD countries[END_REF] which is based on the coding of [START_REF] Budge | Political data 1945-1990. Party government in 20 democracies[END_REF], whose index of governments" ideological positions has been updated by [START_REF] Woldendorp | Party government in 20 democracies: an update 1990-1995[END_REF][START_REF] Woldendorp | Party government in 48 democracies 1945-1998: composition, duration, personnel[END_REF]. This index places the cabinet on a left-right scale with values between 1 and 5. It takes the value 1 if the share of governing rightwing parties in terms of seats in the cabinet and in parliament is larger than 2/3, and 2 if it is between 1/3 and 2/3. The index is 3 if the share of centre parties is 50%, or if the leftwing and rightwing parties form a coalition government not dominated by one side or the other. The index is symmetric and takes the values 4 and 5 if the leftwing parties dominate. Potrafke"s (2009) coding is consistent across time but does not attempt to capture differences between the party-families across countries. 7 I include the government ideology variable in levels. In fact, this implies that leftist and rightwing governments implement their preferred policies incrementally. 8 The variable "Δ ln Labor market deregulation indicator ijt-1 " describes the lagged dependent variable to tackle the persistence of the deregulation indicators. Lastly, "η i " represents a fixed country effect, "ε t " is a fixed period effect and "u ijt " describes an error term. Table A1 in the appendix shows descriptive statistics of all variables included.

I now turn to discussing my choice of the panel data estimation method. In the context of dynamic estimation, the common fixed-effect estimator is biased. The estimators taking into account the resulting bias can be broadly grouped into a class of instrumental estimators and a class of direct bias corrected estimators (see [START_REF] Behr | A comparison of dynamic panel data estimators: Monte Carlo evidence and an application to the investment function[END_REF], for example, for a discussion). In accordance with large sample properties of the GMM methods, e.g., the estimator proposed by [START_REF] Arellano | Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations[END_REF] will be biased in my econometric model with N=20. For this reason, bias corrected 7 Years, in which the government changed, are labeled according to the government that was in office for a longer period, e.g. when a rightwing government followed a leftwing government in August, this year is labeled as leftwing.

8 This is a significant point because politicians implement their preferred policies step by step during the legislative periods.

estimators are more appropriate. I apply Bruno"s (2005a[START_REF] Bruno | Estimation and inference in dynamic unbalanced panel data models with a small number of individuals[END_REF] bias corrected least squares dummy variable estimator for dynamic panel data models with small N.9

Results

4.1 Replacement rate [START_REF] Blanchard | The role of shocks and institutions in the rise of European unemployment: the aggregate evidence[END_REF] distinguish between three main benefit components of the unemployment insurance system: the replacement rate, benefit length, and measures of active labor policy. Political economic reasoning emphasizes these aspects. One group of political economic models focuses on the determinants of unemployment benefits, and, thus explains the replacement rate and the benefit length. [START_REF] Goerke | A positive theory of the earnings relationship of unemployment benefits[END_REF], for example, examine the political economy of the earnings relationship of unemployment benefits. Their model predicts that unemployment compensation is stronger under rightwing governments and weaker when unions are influential and that deepening international economic integration has ambiguous effects on unemployment compensation. The empirical evaluation with a panel of 19 OECD countries in the 1960-2003 period suggests that leftwing governments have indeed had higher unemployment compensation, whereas trade openness has had a negative influence on the earnings relationship. [START_REF] Gaston | Structural change and the labor-market effects of globalization[END_REF] examine the effects of trade liberalization (their measure of globalization) on labor market outcomes via its direct effects on wage bargaining and indirect effects working through redistributive polices such as unemployment benefits. Their empirical results with a panel of OECD countries suggest a positive relationship between trade openness and unemployment benefits, but a negative relationship between trade openness interacted with the budget deficit and unemployment benefits. This finding indicates that an increase in the government debt (as a share of GDP) lowers the response of the benefit replacement rate to trade openness. Gaston and Nelson"s (2004) result notwithstanding, I will examine the following hypothesis:

H1: The replacement rate of unemployment benefits decrease in the course of globalization.

Table 1 shows the regression results and indicates that globalization did not influence the replacement rate. The coefficients of the KOF indices of globalization do not turn out to be statistically significant in all specifications. This finding does not depend on the inclusion of any of the control variables. I ran several additional regressions, which included further potential explanatory variables such as the growth rates of the working-age population share, total population, total central government debt (following [START_REF] Gaston | Structural change and the labor-market effects of globalization[END_REF], the lagged unemployment rate, product market regulation. 10 These variables all do not turn out to be statistically significant. For this reason, one does not need to be concerned with omitted variable bias that could lead to different inferences with respect to the globalization variables.

The regression results in Table 1 illustrate that unemployment replacement rates were highly persistent over time: the lagged dependent variable is statistically significant at the 1% level and corroborates a point estimate of around 0.16. Moreover, an "F-Test" on the joint insignificance of the fixed period effects can be strongly rejected. Government ideology does not turn out to be statistically significant. This result is thus not in line with [START_REF] Goerke | A positive theory of the earnings relationship of unemployment benefits[END_REF], who found significant ideology effects. Possible explanations for this discrepancy are that [START_REF] Goerke | A positive theory of the earnings relationship of unemployment benefits[END_REF] estimate their model in levels while I employ growth rates, that they consider a longer observation period, and that their dependent variables is a broader measure of unemployment compensation.

10 (I employed the data by [START_REF] Bassanini | Employment patterns in OECD countries: reassessing the role Of policies and institutions[END_REF].

Benefit duration

The benefit duration is the second important component of unemployment benefits. The political economic explanations of benefit duration are similar to the explanations of the replacement rate. 11 I will examine the following hypothesis:

H2: The benefit duration decreases in the course of globalization.

Table 2 shows the regression results indicating similar inferences as with respect to the replacement rate: globalization did not influence the benefit duration. The lagged dependent variable and fixed period effects (fixed period effects not shown in Table 2) turn out to be important explanatory variables. I have again tested for further potential control variables as named above, which do not turn out to be statistically significant. To include these variables does not change the inferences regarding the globalization variables.

Active labor market expenditures

Globalization is believed to have ambiguous effects on the welfare state: economic reasoning either suggests that the welfare state collapses (via the supply side or efficiency effect) or that the welfare state is extended (via the demand side or compensation effect). For an encompassing portrait of the globalization-welfare state nexus see, for example, Schulze and [START_REF] Schulze | Globalisation of the economy and the nation state[END_REF] and [START_REF] Ursprung | Globalisation and the welfare state[END_REF]. One strand of this literature deals with the structure of welfare state spending 12 . Decomposing social expenditures focusing on labor, for example, in order to elucidate potential compensating effects, has enjoyed remarkable popularity in the literature.

Recent studies also focus on spending on (active) labor market policies: the theoretical model by [START_REF] Gaston | The rise (and fall) of labour market programmes: domestic vs. global factors[END_REF], for example, predicts that in times of deepening globalization workers 11 I will sketch the globalization-induced responses of the welfare state in the next subsection. 12 Dreher et al. (2008b) and [START_REF] Gemmell | Foreign investment, international trade and the size and structure of public expenditures[END_REF], for example, empirically investigate the influence of globalization on the budget composition.

unexposed to the threat of unemployment may prefer public spending on active labor market programs to passive spending such as unemployment benefits. [START_REF] Gaston | The rise (and fall) of labour market programmes: domestic vs. global factors[END_REF] empirically investigate their theoretical predictions in a panel vector autoregressive model (panel VAR) of 16 OECD countries in the 1980-1999 period. They use trade openness, foreign direct investment, portfolio investment and migration to measure globalization and do not find that globalization has had an influence on active and passive labor market policies. In any event, I will examine the following hypothesis:

H3: Active labor market expenditures decreases in the course of globalization.

Table 3 reports the regression results. Similar to the two previous labor market institution indicators, the results again clearly suggest that globalization did not influence ALMP spending (as a share of GDP). In contrast to the two previous labor market institution indicators, however, ALMP spending appears to be driven by important domestic economic variables. I have followed related studies on the globalization-welfare state nexus to include the growth rate of the working age population (share of the 15-64 years old of total population) and the lagged unemployment rate.

The unemployment rate in period t-1 is statistically significant and displays the expected positive sign. It shows that ALMP spending (as a share of GDP) increased by about 0.14% when the unemployment rate in period t-1 increased by 1%. In contrast, government ideology (as in Gaston and Rajaguru 2008) and the working-age population variable do not turn out to be statistically significant. In sum, the results in Table 3 show that including different control variables does not affect the inferences drawn from the globalization variables.

Employment protection

Globalization is expected to have an influence on employment protection13 . How globalization affects employment protection is, however, unclear. The political economic model by [START_REF] Dimitrova | Globalization and labour markets deregulation[END_REF], for example, examines the influence of globalization, e. g. an increase in international capital mobility on labor market regulations and predicts that labor market Besides globalization, other political economic determinants appear to influence employment protection: the model by [START_REF] Neugart | The choice of insurance in the labor market[END_REF], for example, predicts a political economic equilibrium in which voters, who are not part of the labor force but receive relatively high intrahousehold transfers, will demand high unemployment protection and low unemployment benefits.

The empirical results by [START_REF] Algan | Job protection: the macho hypothesis[END_REF] suggest that religious values have an influence on employment protection. They employ previous OECD EPL data for the 1970-1999 period (decadal averages) and find that compared to Catholics, Protestants tend to decrease the level of employment protection, while Muslims increase it. In the following, I will examine the hypothesis:

H4: Employment protection decreases in the course of globalization.

The results reported in Table 4 show that globalization did not influence EPL. In fact, the empirical model performs quite poorly. I have again included further other control variables which do not turn out to be statistically significant, and which I therefore do not report. The inclusion of these variables does not affect the inferences with respect to the globalization variables at all.

The results somewhat change when EPL subindicators are used: globalization has diminished EPL protection of regularly employed persons (Table 5). This finding is in line with the results by [START_REF] Fischer | Globalization and protection of employment[END_REF], although the empirical approaches differ: [START_REF] Fischer | Globalization and protection of employment[END_REF] estimate their model in levels, whereas I employ growth rates. However, my results reported in Table 6 do not suggest that globalization has had a positive influence on EPL of temporarily employed workers as the results by [START_REF] Fischer | Globalization and protection of employment[END_REF] suggest.

In particular, my results in Table 5 indicate that social globalization was the main driving force of EPL deregulation for regularly employed workers. This finding appears to be in line with the finding by [START_REF] Dreher | Has globalisation really had no effect on unions?[END_REF] that it is not economic, but social globalization that fosters labor market deregulation. [START_REF] Friedman | The lexus and the olive tree[END_REF], for example, puts globalization at a level with "Americanisation". Following this argument, Dreher and Gaston (2007: 166) conclude: "if globalisation implies institutional convergence to some common (U.S.) benchmark, then developed country labour markets are in the process of becoming less unionized and less regulated".

Tax wedge

The tax wedge represents an additional cost for enterprises. For this reason, increasing economic internationalization and competition is likely to influence the tax wedge. However, I am not aware of any political economic studies that explicitly examine the influence of globalization on the tax wedge.14 I advance the following hypothesis.

H5: The tax wedge decreases in the course of globalization.

Table 7 reports the regression results, which indicate that globalization did not influence the tax wedge. I have also included the growth rate of the working-age population and of central government debt (as a share of GDP) in period t-1 because these domestic concerns are likely to influence the tax wedge. For example, a rising share in the working-age population tends to reduce the wedge because the working-age population carries the burden of redistribution via higher taxes.

An increase in the tax wedge tends to result from rising public debt in the past. The working-age population and the lagged central public debt, however, do not turn out to be statistically significant in Table 7. It is important to note that the lack of statistical significance of the working-age population and of lagged central government debt (as a share of GDP) is a matter of the econometric specification. Excluding the lagged dependent variable turns the working-age population variable statistically significant with a negative coefficient and the lagged central public debt variable statistically significant with a positive coefficient as expected (see also Table 9 in section 4.7). The lagged dependent variable is statistically significant at the 10% level in column

(1) and displays the expected positive sign. Government ideology does not turn out to be statistically significant. In addition, I have examined whether globalization has had an influence on the narrowly defined labor tax wedge (not including consumption taxes): it does not.

Union density

Globalization is likely to influence union organization. [START_REF] Dreher | Has globalisation really had no effect on unions?[END_REF], for example, investigate whether globalization has affected union membership in OECD countries in the 1980-1997 period. The novelty of their approach was to use the KOF indices of globalization.

Their results suggest that, overall, economic and political globalization did not influence deunionization. In contrast, they find that social globalization was important and fostered deunionization. I will examine the following hypothesis:

H6: Union density decreases in the course of globalization.

The results in Table 8 show that globalization did not influence union density.

Deunionization, however, increased under rightwing governments. Following [START_REF] Dreher | Has globalisation really had no effect on unions?[END_REF], I have also included the growth rate of the population density as well as inflation (growth rate of the GDP deflator). Both variables do not turn out to be statistically significant and do not affect the result that globalization did not influence union density and that leftwing governments had higher union densities. Relating my results to [START_REF] Dreher | Has globalisation really had no effect on unions?[END_REF], it is important to note that I employ annual data whereas they use five-year averages; they regress the level of the globalization indices on the change of union density and also consider a slightly different sample.

Robustness of the results

I have also estimated the model with feasible generalized least squares (FGLS) with fixed and random country effects and excluded the lagged dependent variable. I have implemented 9 suggest that excluding all individual country effects does not change the inferences at all.

It is conceivable that the reported effects could depend on idiosyncratic circumstances in the individual countries. I have therefore tested whether the results are sensitive to the inclusion/exclusion of particular countries in all the previous econometric models. In neither case, excluding one particular country turns the globalization variable statistically significant.

The influence of globalization on labor market institutions could differ between regions, such as East and West (see, for example, Saint-Paul 2007) or due to legal origins. In the analyzed OECD panel, differences due to legal origin may well play a significant role. Hence, I have included dummy variables that take on the value one for German (reference category), French, British, and Scandinavian legal origins (La [START_REF] Porta | The quality of government[END_REF]. Including these dummies does not change the main result at all (results not shown).

A general caveat with panel data models concerns endogeneity of the dependent variable. It is, however, if at all, individual aspects of economic globalization such as trade openness or foreign direct investment that may have been affected by labor market institutions because investors will choose the most appropriate investment locations. In contrast, labor market institutions are not likely to have an influence on the overall globalization process. In any event, to address the potential endogeneity issue technically, I ran all regressions with lagged globalization variables (t-1, t-2). In almost all cases, the lagged KOF indices of globalization do not turn out to be statistically significant. One exception, however, is the model on ALMP spending: the overall KOF index of globalization (t-2) has a negative influence on ALMP expenditures. This effect, however, is driven by political globalization. All the KOF indices of globalization in period t-1 do not turn out to be statistically significant (results not shown).

Politicians may not implement their desired labor market reforms incrementally but try to influence labor market (de)regulation shortly after they are elected in office. I have therefore replaced the ideology variable by lagged first differences of the ideology variables (t-1, t-2, t-3).

The lagged first differences of the ideology variables do not turn out to be statistically significant and the inclusion of these variables does not change the inferences of the globalization variables. 15A different econometric approach is not employing annual data, but five-year averages to address long-run effects. I have also estimated the models with five-year averaging (similar to column 1 in Tables 1 to 8). The data set by [START_REF] Bassanini | Employment patterns in OECD countries: reassessing the role Of policies and institutions[END_REF]) covers the 1982[START_REF] Behr | A comparison of dynamic panel data estimators: Monte Carlo evidence and an application to the investment function[END_REF] period. Hence, taking five-year averages of the variables results in four data points per country, I investigate the 1982-2001 period with five-year averages. It is important to note that the data on public expenditures on ALMP are unbalanced so that taking five-year averages is not possible. The results in Table 10 show that the five-year average of the overall KOF index of globalization does not turn out to be statistically significant.

Contract coverage and (union and employer) coordination of bargaining

The data collected by [START_REF] Bassanini | Employment patterns in OECD countries: reassessing the role Of policies and institutions[END_REF] do not allow to empirically investigate whether globalization influenced union contract coverage and (union and employer) coordination of bargaining in a dynamic panel data model. For this reason, I will briefly discuss the empirical results of the related studies that choose different empirical approaches.

Globalization appears to have an ambiguous influence on markups and union bargaining power. [START_REF] Abraham | The effect of globalization on union bargaining and price-cost margins of firms[END_REF], for example, use a panel of Belgian manufacturing firms and find that import competition puts pressure on price-cost margins and union bargaining power. In contrast, the results by [START_REF] Brock | Has international trade affected workers" bargaining power?[END_REF], who also employ a micro data set of Belgian manufacturing firms, suggest that trade openness and inward foreign direct investments have influenced workers" bargaining power only very little, if at all. [START_REF] Dumont | Does internationalization affect union bargaining power? An empirical study for five EU countries[END_REF] investigate the influence of international trade on union bargaining power in five EU countries. Their results suggest a negative influence of internationalization on union bargaining power that is comparable in newly industrialized countries and OECD countries. In a similar vein, [START_REF] Boulhol | Imports as product and labor market discipline[END_REF] use data on manufacturing firms in the United Kingdom in the 1998-2003 period and find that imports from developed countries have significantly contributed to the decrease in both mark-ups and workers" bargaining power. Further related studies focus on wage, employment and income effects of globalization. [START_REF] Cuyvers | Wage and employment effects in the EU of international trade with the emerging economies[END_REF], for example, analyze how wages and employment in the EU is affected by international trade with emerging economies. 16 In any event, potential influence of globalization on (union and employer) coordination and bargaining require a more encompassing empirical analysis with time-variant macro data.

Conclusion

Globalization did not have a systematic influence on labor market institutions in OECD countries in the 1982-2003 period.17 Employing the data set by [START_REF] Bassanini | Employment patterns in OECD countries: reassessing the role Of policies and institutions[END_REF] my results suggest that globalization did neither influence the unemployment replacement rate, the unemployment benefit length, public expenditures on ALMP, the tax wedge, union density nor overall employment protection. In contrast, protection of regular employment contracts was diminished when globalization was proceeding rapidly. In fact, domestic aspects, such as unemployment and government ideology are more important determinants of labor market institutions and their deregulation processes in OECD countries than globalization. For this reason, working conditions of unskilled workers are not likely to deteriorate and the jobs of unskilled workers are not likely to disappear in the course of globalization. All this is, of course, not to insinuate that globalization has any benign influence on labor market institutions.

My findings indicate that other explanations than globalization are required to portray the development of labor market institutions. I will briefly discuss two intriguing political economic determinants: honesty indicators and, although touched upon in this paper, government ideology.

Moreover, the threat of international outsourcing is likely to play an important role that could be addressed in future research.

Civic virtue may well play a significant part in explaining the design of unemployment benefits and employment protection [START_REF] Algan | Civic virtue and labor market institutions[END_REF] and distrust may create public demand for regulation [START_REF] Aghion | Regulation and distrust[END_REF]). The results by [START_REF] Heinemann | Choosing from the reform menu card -Individual determinants of labour market policy preferences[END_REF] Other studies suggest that parties do matter: leftwing government extended the role of government in the labor market. 18 These results are remarkable because Cukierman"s and Tommasi"s "When a

Nixon goes to China"-Argument does not appear to apply to labor market reforms: Cukierman and Tomassi (1998) have argued that leftwing governments may well have more political credibility to convince the electorate of the need for reform and, thus, labor market deregulation should have appeared under leftwing governments.

18 [START_REF] Botero | The regulation of labor[END_REF], for example, examine labor market deregulation in 85 countries and find that leftwing governments have been associated with more stringent labor regulations than rightwing governments. Di [START_REF] Di Tella | The determination of unemployment benefits[END_REF] examine unemployment benefits in OECD countries in the 1971-1989 period and find that leftwing governments have provided more generous unemployment benefits than rightwing governments. Overall, however, economic variables such as unemployment and interest rates appear to be more important determinants of unemployment benefits than political variables. [START_REF] Johansen | Politics in wage setting: does government colour matter?[END_REF] investigate whether government ideology influenced wage setting in Norway. Their results suggest that changing from a conservative to a social democratic government significantly reduces manufacturing wages and makes wages more responsive to unemployment. Vaubel"s (2008: 462) case study evidence, however, suggests that labor market deregulation in the EU, did not appear to be related to government ideology.

The threat of international outsourcing may also influence labor market institutions for several reasons. First, globalization might operate via the threat of outsourcing. 19 Second, globalization may increase the incentives for international outsourcing [START_REF] Lommerud | Can deunionization lead to international outsourcing[END_REF].

Third, outsourcing is likely to have an influence on wages for both skilled and unskilled domestic workers. 20 As such, outsourcing will have backfiring effects on unemployment and domestic labor market institutions. I acknowledge that all these potential concerns have not been addressed in my econometric models. Considering the threat of international outsourcing as a cause for reforms of labor market institutions and attempts to measure the threat of international outsourcing certainly remain as worthwhile endeavors for future research.

  Figure 1(c) clearly indicates that ALMP has increased till the mid 1990s on average: its share of GDP was about 1% in 1994 and has declined to 0.76% in 2001. It is important to note that these data are not available for all 20 OECD countries from 1982-2003 in this data set. 3 ALMP spending (as a share of GDP) differs between low-spending countries, such as the United States, Canada and the United Kingdom and the highspending Scandinavian countries. Employment protection is measured by the OECD summary indicator of the stringency of employment protection legislation (EPL). It can be obtained for three main areas: (i) employment protection of regular workers against individual dismissal; (ii) specific requirements for collective dismissals; and (iii) regulation of temporary forms of employment. The OECD has developed a procedure with several steps for constructing cardinal summary indicators of EPL strictness that allow meaningful comparisons to be made, both across countries and between different years. The indicators are normalized to range from 0 to 6, with higher scores representing stricter regulation.

Figure 1

 1 Figure 1(d) illustrates that employment protection has become less strict over time: the indicator

Figure 1

 1 Figure 1(f) shows that the union density has decreased over time: it was 46.3 % in 1982 and 34.6%

  institutions are not necessarily scaled down in the course of globalization. In fact, the direction of the globalization-induced policy response is determined by the relative strength of the politically active groups. The number of empirical studies on the relationship between globalization and employment protection is quite limited:[START_REF] Fischer | Globalization and protection of employment[END_REF], for example, investigate whether globalization has lowered employment protection of workers in 28 OECD countries in the 1985-2003 period. Their results suggest that globalization (measured by the KOF indices of globalization) has weakened protection of regularly employed, whereas it has tightened the protection of temporarily employed.

  heteroscedastic and autocorrelation consistent (HAC) Newey-West type[START_REF] Newey | A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix[END_REF] standard errors and variance-covariance estimates, because the Wooldridge test(Wooldridge 2002: 176-177) for serial correlation in the idiosyncratic errors of a linear static panel-data model implies the existence of arbitrary serial correlation. Employing these panel data estimators does not change the inferences regarding the globalization variables at all. Taking growth rates of the dependent variables eliminates time-invariant fixed effects in levels. But in case of individual time trends in each country, computing growth rates just eliminates the time-invariant country effects, but not the individual time components. For this reason, I have included fixed country effects in all the previous regressions. One might argue, however, that given the pattern of the labor market regulation indicators, the time trends capture a great share of the variance, thereby leaving little chance to detect a significant influence of globalization. I have therefore excluded any individual country effects and estimated the model with ordinary least squares (OLS) regression with a common constant. The results in Table

  suggest that self-interest is a major determinant for individual assessment of labor market reforms. Future research could investigate whether social capital influences labor market deregulation. An empirical analysis in a panel with macro-data would require time-variant indicators on honesty issues. Developing such indicators emerges as a worthwhile endeavor.The influence of government ideology on labor market institutions deserves further examinations for two important reasons. First, empirical studies indeed have mixed results. My findings merely suggest that deunionization was proceeding under rightwing governments whereas government ideology did not appear to influence the other five labor market institution indicators.

Table 1 :

 1 Regression Results. Dependent variable: Δ ln Replacement rate. Dynamic bias corrected estimator.

	(1)	(2)	(3)	(4)	(5)

Notes: Absolute value of t-statistics in parentheses ***, ** and * indicate significance at the level of 1, 5 and 10%, respectively

Table 3 :

 3 Regression Results. Dependent variable: Δ ln Public Expenditures on ALMP.

	Dynamic bias corrected estimator.
			(2)	(3)	(4)	(5)
	Δ ln KOF index of globalization (overall)	0.3401	0.3241
		(0.70)	(0.69)
	Δ ln KOF index of globalization (economic)			0.0366
				(0.12)
	Δ ln KOF index of globalization (social)				0.1322
					(0.34)
	Δ ln KOF index of globalization (political)					0.1441
						(0.64)
	Ideology (leftwing)		0.0090	0.0090	0.0093	0.0089
			(0.82)	(0.82)	(0.84)	(0.82)
	Δ ln Working-age population		-1.7379	-1.3968	-1.4603	-1.6274
			(0.35)	(0.28)	(0.29)	(0.33)
	Δ ln Unemployment rate (t-1)		0.1345***	0.1338**	0.1358***	0.1391***
			(2.68)	(2.57)	(2.72)	(2.70)
	Lagged dependent variable	0.2182***	0.1848***	0.1844***	0.1847***	0.1849***
		(3.15)	(2.66)	(2.63)	(2.65)	(2.63)
	Fixed country effects	Yes	Yes	Yes	Yes	Yes
	Fixed period effects	Yes	Yes	Yes	Yes	Yes
	Observations	293	293	293	293	293
	Number of n	20	20	20	20	20
	Notes: Absolute value of t-statistics in parentheses ***, ** and * indicate significance at the level
	of 1, 5 and 10%, respectively

Table 6 :

 6 Regression Results. Dependent variable: Δ ln Employment protection of temporarily employed workers Dynamic bias corrected estimator.

	(1)	(2)	(3)	(4)	(5)

Notes: Absolute value of t-statistics in parentheses ***, ** and * indicate significance at the level of 1, 5 and 10%, respectively

Table 7 :

 7 Regression Results. Dependent variable: Δ ln Tax wedge. Dynamic bias corrected estimator.

	(1)	(2)	(3)	(4)	(5)

Notes: Absolute value of t-statistics in parentheses ***, ** and * indicate significance at the level of 1, 5 and 10%, respectively

Table 8 :

 8 Regression Results. Dependent variable: Δ ln Union density. Dynamic bias corrected estimator.

	(1)	(2)	(3)	(4)	(5)

Notes: Absolute value of t-statistics in parentheses ***, ** and * indicate significance at the level of 1, 5 and 10%, respectively

Table 9 :

 9 Regression Results. Robustness Checks. Dependent variables: Growth rates of the labor market deregulation indicators. Ordinary least squares (OLS) with robust standard errors. Absolute value of t-statistics in parentheses ***, ** and * indicate significance at the level of 1, 5 and 10%, respectively

					Δ ln	Δ ln
	Δ ln Replacement rate	Δ ln Benefit duration	Δ ln Public expenditures on ALMP	Δ ln Employment protection	Employment protection of employed regularly	Employment protection of employed temporarily	Δ ln Tax wedge	Δ ln Union density
					workers	workers
	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)

In the 1985[START_REF] Choi | Unit root tests for panel data[END_REF] period, data are missing for Denmark in 1985, for Japan in 1986, for Ireland in 1992[START_REF] Budge | Political data 1945-1990. Party government in 20 democracies[END_REF][START_REF] Feenstra | Globalization, outsourcing, and wage inequality[END_REF][START_REF] Nickell | Unemployment and labour market rigidities: Europe versus North America[END_REF][START_REF] Feenstra | Integration of trade and disintegration of production in the global economy[END_REF][START_REF] Friedman | The lexus and the olive tree[END_REF] and 2000, for Portugal in 2001.

Moreover, the data allow distinguishing between two types of EPL: protection of regularly and temporary employed workers.

On the measurement of globalization see also, for example,[START_REF] Edwards | Measuring global and regional trade integration in terms of concentration of access[END_REF].

In order to test for stationarity of the time series, I apply a battery of panel unit root tests. The advantage of the panel unit root tests compared to the univariate counterparts is their greater statistical power. It is important to note, however, that the tests to a panel also relate to asymptotic theory and therefore loose power in small samples (see, for example, the survey on unit roots and cointegration in panels by[START_REF] Breitung | Unit Roots and cointegration in panels[END_REF]. I applied the[START_REF] Levin | Unit root tests in panel data: Asymptotic and finitesample properties[END_REF],[START_REF] Im | Testing for roots in heterogenous panels[END_REF],[START_REF] Breitung | The local power of some unit root tests for panel data[END_REF] and the Fisher tests referring to[START_REF] Maddala | A comparative study of unit root tests with panel data and a new simple test[END_REF] and[START_REF] Choi | Unit root tests for panel data[END_REF]. The results were obtained using Eviews 6. Regarding the first three tests, maximum lag lengths are automatically selected based on the Schwarz Information Criterion. The remaining two tests use the Bartlett kernel for the Newey-West bandwidth selection. The probabilities for the Fisher tests are computed using an asymptotic Chi-square distribution. All other tests assume asymptotic normality. The test regressions in levels include a constant and a linear deterministic trend; the test regressions in growth rates include a constant but no linear deterministic trend. The results of different unit root tests demonstrate that one mostly cannot reject the null hypothesis of a unit root in levels, but one can always reject the null hypotheses of a unit root in growth rates. For this reason, the time series in growth rates are stationary.

I choose the[START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] estimator as the initial estimator in which the instruments are collapsed as suggested by[START_REF] Roodman | How to do xtabond2: An introduction to "Difference" and "System" GMM in Stata[END_REF]. This procedure makes sure to avoid using invalid and too many instruments (see[START_REF] Roodman | How to do xtabond2: An introduction to "Difference" and "System" GMM in Stata[END_REF] 2009 for further details). Following[START_REF] Bloom | Demographic change, social security systems, and savings[END_REF] I undertake 50 repetitions of the procedure to bootstrap the estimated standard errors. Bootstrapping the standard errors is common practice applying this estimator. The reason is that Monte Carlo simulations demonstrated that the analytical variance estimator performs poorly for large coefficients of the lagged dependent variable (see Bruno 2005b for further details). The results do not qualitatively change with more repetitions such as

100, 200 or 500 or when the[START_REF] Arellano | Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations[END_REF] estimator is chosen as initial estimator.

See Blanchard and Wolfers (2000: C14ff.), for example, for a brief discussion of the history of unemployment protection in Europe and[START_REF] Deakin | The evolution of labour law: calibrating and comparing regulatory regimes[END_REF], for example, for an encompassing survey on the evolution of labor law in France, Germany, India, the United Kingdom and the United States.

[START_REF] Goerke | The wedge[END_REF], for example, presents a theoretical model on employment effects of changes in the composition of the tax wedge.

There are two exceptions: the first difference of the ideology variable in period t-3 has a positive influence on the growth rate of ALMP spending and is statistically significant at the 10% percent level. Including the first difference of the ideology variable in period t-3 turns the growth rate of the overall KOF index of globalization to be statistically significant at the 10% level in the union density equation.

On the effects of trade, trade policy and domestic factors in union wage determination see, for example,[START_REF] Gaston | Union wage sensitivity to trade and protection: theory and evidence[END_REF].

This finding perfectly corresponds with research on product market deregulation (e.g.,[START_REF] Heinemann | The drivers of deregulation in the era of globalization[END_REF][START_REF] Potrafke | Does government ideology influence product market deregulation? Empirical evidence from OECD countries[END_REF] and economic reforms (e.g.,[START_REF] Gassebner | The inverse domino effect: are economic reforms contagious?[END_REF]).

Acknowledgments

I thank Harmen Lehment, Heinrich Ursprung and an anonymous referee for helpful comments, hints and suggestions. 19 See Feenstra andHanson 1996 andFeenstra 1998, for contributions on globalization and outsourcing. 20 See, for example, Munch and Skaksen 2009 for an empirical analysis of Danish manufacturing industries.