Sébastien Poncet
email: poncet@l3m.univ-mrs.fr

Roland Schiestel
email: schiestel@irphe.univ-mrs.fr

Romain Monchaux
email: romain.monchaux@cea.fr

Turbulence modeling of the Von Karman flow: viscous and inertial stirrings

Keywords: Von Kármán flow, turbulence modeling, LDV, viscous stirring, inertial stirring

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I. INTRODUCTION

The flow between two finite counter-rotating disks enclosed by a cylinder, known as the Von Kármán [START_REF] Kármán | Uber laminare und turbulente Reibung[END_REF] geometry, is of practical importance in many industrial devices. Counterrotating turbines may indeed be used to drive the counter-rotating fans in gas-turbine aeroengines. Moreover, this configuration is often used for studying fundamental aspects of developed turbulence and especially of magneto-hydrodynamic turbulence.

From an academic point of view, the laminar flow between two infinite disks has indeed justified many works since the beginning of the controversy between Batchelor [START_REF] Batchelor | Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow[END_REF] and Stewartson [START_REF] Stewartson | On the flow between two rotating coaxial disks[END_REF] on the flow structure. Batchelor [START_REF] Batchelor | Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow[END_REF] solved the system of differential equations relative to the steady rotationally-symmetric viscous flow between two infinite disks. In the exactly counter-rotating regime, the distribution of tangential velocity is symmetrical about the mid-plane and exhibits five distinct zones: two boundary layers developed on each disk, a transition shear layer at mid-plane, where the axial and tangential velocities change sign and two rotating cores on either side of the transition layer. The central cores rotate with a tangential velocity proportional to the disk velocities. The proportionality coefficient is always inferior to 1. This solution can be regarded as the connection of two Batchelor flows in the rotor-stator configuration. As stated by Batchelor [START_REF] Batchelor | Note on a class of solutions of the Navier-Stokes equations representing steady rotationally-symmetric flow[END_REF] himself, "this singular solution may not be realizable experimentally, of course", which supposes that another solution may exist. In 1953, Stewartson [START_REF] Stewartson | On the flow between two rotating coaxial disks[END_REF] found that the flow is divided into only three zones for large values of the Reynolds number Re H = ΩH 2 /ν > 100 based on the interdisk space H (Ω is the rotation rate of the disks and ν the kinematic viscosity of the fluid): one boundary layer on each disk separated by a zone of zero tangential velocity and uniform radial inflow. Lance and Rogers [START_REF] Lance | The axially symetric flow of a viscous fluid between two infinite rotating disks[END_REF] found numerically in the exactly counter-rotating regime a Stewartson solution for Re H = 1023. The existence at large Reynolds numbers of the Stewartson solution has been confirmed by the analysis of McLeod and Parter [START_REF] Mcleod | On the flow between two counter-rotating infinite plane disks[END_REF] in an infinite counter-rotating disk system. The Stewartson solution has also been obtained numerically by Pesch and Rentrop [START_REF] Pesch | Numerical solution of the flow between two counter-rotating infinite plane disks by multiple shooting[END_REF] at Re H = 2000. Kreiss and Parter [START_REF] Kreiss | On the swirling flow between rotating coaxial disks: existence and uniqueness[END_REF] have proved the existence and non-uniqueness of solutions at sufficiently large Reynolds numbers for the two-disk configuration. Thus, both Batchelor and Stewartson solutions are possible depending on the initial and boundary conditions but the Batchelor prediction has not been mentioned in the literature for the exact counter-rotating disk case. Pearson [START_REF] Pearson | Numerical solutions for time-dependent viscous flow between rotating coaxial disks[END_REF] obtained numerically a basic inviscid solution of the Von Kármán flow, which differs from both the Batchelor and Stewartson solutions: at high Reynolds number (Re H = 10 3 ), the solution is unsymmetrical and the main body of the fluid rotates faster than that of either disk. In the counter-rotating regime, Dijkstra and Van Heijst [START_REF] Dijkstra | The flow between two finite rotating disks enclosed by a cylinder[END_REF] showed numerically that the transition from the one cell to the two cell structure occurs for a given Reynolds number and corresponds to the appearance of a detached shear layer on the slower disk. Recently, Yang and Liao [START_REF] Yang | On the explicit, purely analytic solution of Von Kármán swirling viscous flow[END_REF] solved the Von Kármán swirling viscous flow using the homotopy analysis method. The reader is referred to the work of Holodniok et al. [START_REF] Holodniok | Computation of the flow between two rotating coaxial disks: multiplicity of steady-state solutions[END_REF] and to the review of Zandbergen and Dijkstra [START_REF] Zandbergen | Von Kármán swirling flows[END_REF] for a more extensive survey until 1987.

In the turbulent case, the Von Kármán flow is a model flow to study the turbulence characteristics on small scales. The main flow is axisymmetric and so offers an interesting intermediate situation between two-dimensional and three-dimensional flows. Fauve et al. [START_REF] Fauve | Pressure fluctuations in swirling turbulent flows[END_REF] reported measurements of pressure fluctuations in the turbulent Von Kármán flow.

They showed that the pressure probability function is strongly non-Gaussian and displays an exponential tail toward low pressure. Maurer et al. [START_REF] Maurer | Statistics of turbulence between two counterrotating disks in low-temperature helium gas[END_REF] used low-temperature helium gas to obtain high Reynolds numbers and well-defined scaling properties. They established the turbulence characteristics such as structure functions or the probability density function of the velocity differences and confirmed that turbulence on small scales has universal properties independent of the forcing. Mordant et al. [START_REF] Mordant | Characterization of turbulence in a closed flow[END_REF] investigated the dynamical behavior of the Von Kármán flow at moderate to high Reynolds numbers using spatially averaged measurements. Data of the power input and of pressure fluctuations at the wall are sufficient to calculate the main turbulence characteristics such as the velocity fluctuations or the typical length scales. Cadot et al. [START_REF] Cadot | Energy injection in closed turbulent flows: stirring through boundary layers versus inertial stirring[END_REF] measured the mean rates of energy injection and energy dissipation in steady regimes of turbulence in the flow between counter-rotating stirrers. The smooth stirrers are found to be less efficient in setting the fluid into motion than in the case of bladed disks. Pinton et al. [START_REF] Pinton | Power fluctuations in a closed turbulent shear flow[END_REF] measured the power consumption of the turbulent Von Kármán flow at constant Reynolds number and showed that power fluctuations occur and involve coherent fluid motions in the whole cell. Marié and Daviaud [START_REF] Marié | Experimental measurement of the scale-by-scale momentum transport budget in a turbulent shear flow[END_REF] performed full velocity measurements linking velocity fluctuations with the turbulent drag in this geometry. They showed especially that the turbulent drag is dominantly generated by coherent structures at the largest scales of the flow. Cadot and Le Maître [START_REF] Cadot | The turbulent flow between two rotating stirrers: similarity laws and transitions for the driving torques fluctuations[END_REF] considered the turbulent between two co-and counter-rotating stirrers. They measured the instanta-neous torques driving the flow and compared them to similarity laws having no dependence on the Reynolds number with a good agreement.

Ravelet et al. [START_REF] Ravelet | Multistability and memory effect in a highly turbulent flow: experimental evidence for a global bifurcation[END_REF][START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF] reported experimental evidence of a global bifurcation on a highly turbulent flow between two counter-rotating impellers. The transition between the symmetric and the unsymmetric solutions is subcritical and the system keeps a memory of its history. Monchaux et al. [START_REF] Monchaux | Properties of steady states in turbulent axisymmetric flows[END_REF] investigated the properties of the mean and most probable velocity fields in the same configuration. They showed that these two fields are described by two families of functions [START_REF] Leprovost | Dynamics and thermodynamics of axisymmetric flows: Theory[END_REF] depending on both the viscosity and the forcing. For large values of the Reynolds number, in some regions, the flow behaves like a Beltrami flow in which vorticity is locally aligned with velocity. Boroński [START_REF] Boroński | Méthode des potentiels poloïdal-toroïdal appliquée à l'écoulement de Von Kármán en cylindre fini[END_REF] simulated the laminar Von Kármán flow between two counter-rotating disks equipped or not by straight blades. For a rotational Reynolds number Re = ΩR 2 /ν, based on the disk radius R, equal to 500, the poloidal-to-toroidal ratio is increased from 13% in the smooth disk case to 51% in the bladed disk case.

A renewal of interest for the Von Kármán flow is born from the dynamo experiments.

The flow between counter-rotating impellers is considered as a possible candidate for the observation of a homogeneous fluid dynamo less constrained than the Riga and Karlsruhe devices. The flow needs to be highly turbulent in order for nonlinearities to develop in the magnetic induction. Numerous experimental [START_REF] Bourgoin | Magnetohydrodynamics measurements in the Von Kármán sodium experiment[END_REF][START_REF] Volk | Fluctuation of magnetic induction in Von Kármán swirling flows[END_REF] or numerical [START_REF] Bourgoin | An iterative study of time independent induction effects in magnetohydrodynamics[END_REF][START_REF] Ravelet | Toward an experimental Von Kármán dynamo: numerical studies for an optimized design[END_REF] studies have then been dedicated to magneto-hydrodynamics turbulence in the Von Kármán geometry. In the latter work, the flow has been optimized using a water model experiment, varying the driving impeller configuration, well described in [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF].

To our knowledge, only very few numerical works have been devoted to the characterization of the mean and turbulent flow properties in the Von Kármán geometry. Kilic et al. [START_REF] Kilic | Transitional flow between contra-rotating disks[END_REF] performed a combined numerical and experimental study of the transitional flow between smooth counter-rotating disks with a central hub for -1 ≤ Γ ≤ 0, Re = 10 5 and G = H/R = 0.12, where Γ is the ratio between the rotating speeds of the two disks and G is the aspect ratio of the cavity. They compared mean radial and tangential velocity measurements using a single-component laser Doppler anemometer with computed results either the low-Reynolds number k -ǫ turbulence model of Launder and Sharma [START_REF] Launder | Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc[END_REF] or a laminar elliptic code. For Γ = -1, the weakly turbulent flow is of Stewartson type, whereas the laminar computations and measurements produce a Batchelor type of flow. The tran-sitions from laminar to turbulent regime and from Batchelor to Stewartson flow structure occur for Γ = -0.4. A good agreement is obtained in the rotor-stator configuration (Γ = 0) and in the exactly counter-rotating regime (Γ = -1) but at intermediate values of Γ, the agreement is less satisfactory. The same authors [START_REF] Gan | Superposed flow between two discs contrarotating at differential speeds[END_REF] performed the same comparisons when a radial outflow of air is superimposed.

In this paper, we present comparisons between numerical predictions using a Reynolds Stress Model, denoted RSM, and velocity measurements performed at CEA for the turbulent flow between two counter-rotating disks. The main objective is to acquire a precise knowledge of both the flow structure and the turbulence properties of the high turbulent Von Kármán flow between smooth disks for a large range of the flow control parameters. A second objective is to propose an easy and efficient way to model impellers and to quantify their effect on the Von Kármán flow at high Reynolds number.

II. EXPERIMENTAL SET-UP

Velocity measurements using a laser Doppler velocimeter have been performed at CEA in the Von Kármán geometry during the PhD thesis of Ravelet [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF] and then by Romain Monchaux [START_REF] Monchaux | Properties of steady states in turbulent axisymmetric flows[END_REF] in two cases: viscous and inertial stirrings.

A. Geometrical configuration

We consider the Von Kármán flow generated by two counter-rotating disks fitted or not by straight blades in a cylindrical vessel, as illustrated in figures 1a,b. The geometrical parameters are fixed by the values studied experimentally by Ravelet [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF]. The cylinder and disk radii are respectively R c = 100 mm and R = 92.5 mm. The radius ratio R/R c is then fixed to 0.925. The distance between the inner faces of the disks H can vary between 1 and 180 mm. Disks 1 and 2 rotate respectively clockwise and counterclockwise with two rotation rates denoted Ω 1 and Ω 2 . The motor rotation rates can be varied independently in the range 0 -900 rpm, with |Ω 1 | ≥ |Ω 2 |. We use bladed disks (n blades of height h equal to 10 or 20 mm) to ensure inertial stirring or flat disks for viscous stirring. The impellers are driven by two independent brushless 1.8 kW motors, with speed servo loop control.

B. Measurement technique

Velocity measurements are performed using a laser doppler velocimetry (LDV). A basic acquisition of 190.000 randomly sampled values of one velocity component at one point of the flow lasts about two minutes. Due to geometry constraints, we can measure the axial V z and tangential V θ mean velocity components. From this raw data, one may compute the time-averaged flow at every point on a 11 * 15 grid.

C. Flow control parameters

In the smooth case, the mean flow is mainly governed by three control parameters: the aspect ratio of the cavity G, the rotational Reynolds number Re based on the cylinder radius and the ratio Γ between the two rotation rates, defined as follows:

0.01 ≤ G = H R c ≤ 1.8 2 × 10 5 ≤ Re = Ω 1 R 2 c ν ≤ 4 × 10 6 -1 ≤ Γ = - Ω 2 Ω 1 ≤ 0
where ν is the kinematic viscosity of water. In the exact counter-rotating regime and in the rotor-stator configuration, the ratio Γ is equal to -1 and 0 respectively. We define also the radial r * = r/R c and axial z * = 2z/H coordinates. Thus, r * = 0 is obtained at the center of the disks and r * = 1 on the outer cylinder for r = R c . In the same way, z * = -1 on the lower disk 1 and z * = 1 on the upper disk 2.

In the case of inertial stirring, the number of straight blades n and their dimensionless height h * = h/R c have also to be considered.

III. STATISTICAL MODELING

The predictions of the Reynolds Stress Model (RSM) used in the present work have already been validated in the rotor-stator configuration (Γ = 0) [START_REF] Elena | Modélisation de la turbulence inhomogène en présence de rotation[END_REF][START_REF] Elena | Turbulence modeling of rotating confined flows[END_REF][START_REF] Poncet | Batchelor versus Stewartson flow structures in a rotor-stator cavity with throughflow[END_REF][START_REF] Poncet | Écoulements de type rotor-stator soumis à un flux axial: de Batchelor à Stewartson[END_REF] for a wide range of aspect ratio G and Reynolds number Re. It showed that this level of closure is adequate in such flow configurations, while the usual k -ǫ model, which is blind to any rotation effect presents serious deficiencies. Thus, the purpose of this paper relying on a well established turbulent model is to extend its application to new flow conditions and to get a better insight into the dynamics of the highly turbulent Von Kármán flow.

A. The differential Reynolds Stress Model (RSM)

The flow studied here presents several complexities (high rotation rate, wall effects, transitional zone, shear layer), which are severe demands for turbulence modeling methods. Our approach is based on one-point statistical modeling using a low Reynolds number secondorder full stress transport closure derived from the Launder and Tselepidakis [START_REF] Launder | Application of a new second-moment closure to turbulent channel flow rotating in orthogonal mode[END_REF] model and sensitized to rotation effects [START_REF] Elena | Turbulence modeling of rotating confined flows[END_REF]. This approach allows for a detailed description of near-wall turbulence and is free from any eddy viscosity hypothesis. The general equation for the Reynolds stress tensor R ij can be written:

dR ij dt = P ij + D ij + Φ ij -ǫ ij + T ij (1)
where P ij , D ij , Φ ij , ǫ ij , and T ij respectively denote the production, diffusion, pressure-strain correlation, dissipation and extra terms.

The diffusion term D ij is split into two parts: a turbulent diffusion D T ij , which is interpreted as the diffusion due to both velocity and pressure fluctuations [START_REF] Daly | Transport equation for turbulence[END_REF] and a viscous diffusion D ν ij , which cannot be neglected in the low Reynolds number region. In a classical way, the pressure-strain correlation term Φ ij can be decomposed in three parts: a slow nonlinear return to isotropy modeled as a quadratic development in the stress anisotropy tensor and damped near the wall, a linear rapid part which includes cubic terms and a wall correction applied to the linear part which is modeled using the Gibson and Launder hypothesis [START_REF] Gibson | Ground effects on pressure fluctuations in the atmospheric boundary layer[END_REF]. In this last term, the widely adopted length scale k 3/2 ε -1 is replaced by the length scale of the fluctuations normal to the wall.

The viscous dissipation tensor has been modeled in order to conform with the wall limits obtained from Taylor series expansions of the fluctuating velocities [START_REF] Launder | Asymptotic near-wall stress dissipation rates in a turbulent flow[END_REF]. The extra term T ij accounts for implicit effects of the rotation on the turbulence field, it contains additional contributions in the pressure-strain correlation, a spectral jamming term, inhomogeneous effects and inverse flux due to rotation, which impedes the energy cascade [START_REF] Cambon | Spectral approach to non-isotropic turbulence subjected to rotation[END_REF]. A full description of the extra term T ij is given in [START_REF] Schiestel | Modeling of anisotropic turbulence in rapid rotation[END_REF].

The dissipation rate ε equation to solve is the one proposed by Launder and Tselepidakis [START_REF] Launder | Application of a new second-moment closure to turbulent channel flow rotating in orthogonal mode[END_REF]. The turbulence kinetic energy k equation which is redundant in a RSM model is still solved however, in order to get a more stable numerical convergence. It is verified that after convergence the turbulence kinetic energy k is exactly equal to 0.5R jj within 0.05% at each grid point.

B. Numerical method

The computational procedure is based on a finite volume method using staggered grids for mean velocity components with axisymmetry hypothesis in the mean. The computer code is steady elliptic and the numerical solution proceeds iteratively. It has been verified that a 120 × 120 mesh in the (r, z) frame is sufficient in smooth rotating disk cases to get grid-independent solutions. A refined mesh 160 × 160 is necessary to model flows with straight blades. It is to be compared to the 140 × 80 mesh used by Elena and Schiestel [START_REF] Elena | Modélisation de la turbulence inhomogène en présence de rotation[END_REF][START_REF] Elena | Turbulence modeling of rotating confined flows[END_REF] and Poncet et al. [START_REF] Poncet | Batchelor versus Stewartson flow structures in a rotor-stator cavity with throughflow[END_REF][START_REF] Poncet | Écoulements de type rotor-stator soumis à un flux axial: de Batchelor à Stewartson[END_REF] in rotor-stator systems. The calculation is initialized using realistic data fields, which satisfy the boundary conditions. About 20000 iterations (almost 20 hours on the bi-Opteron 18 nodes cluster of IRPHE) are necessary to obtain the numerical convergence of the calculation. The stress component equations are solved using matrix block tridiagonal solution to enhance stability using non staggered grids.

C. Boundary conditions

At the wall, all the variables are set to zero except for the tangential velocity V θ , which is set to Ω 1 r on disk 1, -Ω 2 r on disk 2 and zero on the stationary cylinder. The usual value ǫ = νk ,j k ,j /(2k) is imposed at the wall for the dissipation rate ǫ of the turbulence kinetic energy. At the periphery of the disks, for R ≤ r ≤ R c , V θ is supposed to vary linearly from zero on the stationary cylinder up to Ω 1 R on disk 1 and -Ω 2 R on disk 2 and the radial V r and axial V z velocity components are fixed to zero.

We can not implement real straight blades in our two-dimensional code. So we limit to modeling their most important effect, which is to increase the efficiency of the disks in forcing the flow. Thus, we add a volumic drag force f in the equation of the tangential velocity component V θ . If we consider n straight blades, the volumic drag force f can be written as:

f = n 2πr F = n 4πr ρC D |V rel |V rel ( 2
)
where F is the drag force of one blade, ρ the fluid density, C D the dimensionless drag coefficient and V rel = Ω i r -V θ the relative tangential velocity on disk i = 1, 2. The force is designed to make the fluid velocity closer to the local disk velocity near the disks. This form is close to the one proposed by Boroński [START_REF] Boroński | Méthode des potentiels poloïdal-toroïdal appliquée à l'écoulement de Von Kármán en cylindre fini[END_REF] for spectral code. For curved blades, the same approach can be used: a volumic lift force can be added in the equation of the radial velocity component V r . This will be the subject of a next study.

We can consider here that the height of the blades (h * = 0.1 or 0.2) is much larger than the boundary layer thickness δ/R c ≃ Re -1/2 ≤ 2 × 10 -3 (for Re = 2 × 10 5 ). In this case,

Blevins [START_REF] Blevins | Applied fluid dynamics handbook[END_REF] proposed some values for the drag coefficient (expected errors of ±20%) in an The reader is thus referred to [START_REF] Elena | Modélisation de la turbulence inhomogène en présence de rotation[END_REF][START_REF] Elena | Turbulence modeling of rotating confined flows[END_REF][START_REF] Poncet | Batchelor versus Stewartson flow structures in a rotor-stator cavity with throughflow[END_REF][START_REF] Poncet | Écoulements de type rotor-stator soumis à un flux axial: de Batchelor à Stewartson[END_REF] for more details about the RSM model and the numerical method.

IV. SMOOTH DISK CASE: VISCOUS STIRRING

In this section, we consider the turbulent flow between two counter-rotating flat smooth disks. Thus, we ensure a viscous stirring: the actuation is done by the setting in rotation of the smooth walls and the movement is communicated to the fluid by diffusion of the momentum through the boundary layers. We investigate the influence of the Reynolds number Re, the aspect ratio of the system G, and the ratio Γ between the two rotation rates on the mean and turbulent fields.

A. Flow structure in the exact counter-rotating regime

The structure of the mean flow in the exact counter-rotating regime is henceforth globally well known: it can be decomposed into two toroidal cells in the tangential direction θ (not modelled here because of the axisymmetry hypothesis in the mean) and into two poloidal recirculations in the (r, z) plane [START_REF] Ravelet | Toward an experimental Von Kármán dynamo: numerical studies for an optimized design[END_REF].

We focus here on the poloidal cells (fig. 5a): the fluid at the top and the bottom of the cavity is forced into two opposite rotation speeds, and is then entrained by the disks. Consequently, a shear layer develops in the equatorial plane. This is perceptible in figure 2 Nevertheless, the present velocity measurements performed on the same experimental set-up as [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF] confirm the numerical results. Compared to the previous measurements, an effect of Re is observed on the radial profiles of V θ at the periphery of the cavity. In fact, the critical Reynolds number for the laminar to turbulent state transition depends strongly on the boundary conditions and especially on the conditions imposed in the radial gap 0.925 ≤ r * ≤ 1. We recall that a linear profile is imposed in the numerical code for V θ , that does not take into account any recirculation zone and that could explain this difference.

This tendency for relaminarization of the RSM model has already been noticed by Poncet et al. [START_REF] Poncet | Batchelor versus Stewartson flow structures in a rotor-stator cavity with throughflow[END_REF][START_REF] Poncet | Écoulements de type rotor-stator soumis à un flux axial: de Batchelor à Stewartson[END_REF] in the rotor-stator configuration. As a conclusion, there is no significant effect of the Reynolds number on the mean flow for Re ≥ 10 5 , which confirms the results of Cadot et al. [START_REF] Cadot | Energy injection in closed turbulent flows: stirring through boundary layers versus inertial stirring[END_REF] and Ravelet [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF].

Figure 5 presents the corresponding streamline patterns. The mean flow is divided into two symmetric poloidal cells, whose size is equal here to 0.5H along the axial direction and independent of the Reynolds number. In the radial direction, the diameter of the largest eddies observed is of the order of the disk radius R, showing this scale is the order of the energy scale injection. Experimentally, Ravelet [START_REF] Ravelet | Toward an experimental Von Kármán dynamo: numerical studies for an optimized design[END_REF] observed a weak dissymmetry of the flow in the (r, z) plane, which disappears for increasing values of the Reynolds number.

The influence of the aspect ratio of the cavity G on the mean field has also been investigated for 0.01 ≤ G ≤ 1.8 (fig. 6) and a given Reynolds number Re = 1.3 × 10 6 . Note that the radial and most of all the axial velocity components are quite weak compared to the tangential one and to the disk velocity. For G = 1.8, the boundary layers are separated as already mentioned and the mean tangential velocity component is constant in the core of the flow. For G = 0.01, the flow is of torsional Couette type with merged boundary layers as V θ (fig. 6a) varies linearly in the median region of the flow. This is to be compared to the value G = 0.012 obtained in the rotor-stator configuration [START_REF] Poncet | Écoulements de type rotor-stator soumis à un flux axial: de Batchelor à Stewartson[END_REF]. For intermediate values G ≃ 0.4, both boundary layers interact. The transition between the two main regimes is continuous and not clear from the V θ -profile. Nevertheless, if we consider the V z -profile (fig. 6c), we can clearly see that the axial velocity component is almost zero whatever the value of G, expect for G = 0.4, where the fluid moves towards the upper and lower disks. The transition can also be characterized by considering the V r -profiles (fig. 6b), which exhibit the thinning of the boundary layers for increasing values of the aspect ratio.

B. Flow structure for -1 ≤ Γ ≤ 0

Another interesting feature in counter-rotating disk flows is the influence of the ratio Γ between the two rotating disk speeds (fig. 7). The Reynolds number and the aspect ratio of the cavity are respectively fixed to Re = 1.3 × 10 6 and G = 1.8. We focus on the counterrotating disk case for which -1 ≤ Γ ≤ 0.

In the exact counter-rotating regime (fig. 7a), the flow is symmetric and two cells with the same size 0.5H coexist. For small rotating speed differences, the structure of the mean flow is strongly dominated by the faster disk (fig. 7b). Varying the ratio Γ displaces the shear layer towards the slower disk. The cell close to the lower disk invades almost the whole interdisk spacing for Γ = -0.7 (fig. 7d). For Γ = -0.2 (fig. 7e), the flow structure resembles the one observed in the rotor-stator configuration [START_REF] Poncet | Écoulements de type rotor-stator soumis à un flux axial: de Batchelor à Stewartson[END_REF] with streamline patterns parallel to the rotating axis. This transition between the two cell and the one cell regimes can be seen also from figure 8. It presents the evolution with Γ of the dimensionless size S c /H of the smallest cell (along the upper disk) in the axial direction defined in figure 7b.

In the smooth disk case, we notice that S c decreases rapidly for decreasing values of |Γ| in the range -1 ≤ Γ ≤ -0.8 (see also fig. 7a-c) following S c /H ∝ -2.2Γ. For smallest values of |Γ|, the cell is reduced to a very thin region attached to the upper disk (fig. 7d), which disappears progressively along the external cylinder and so S c tends to zero.

In figure 8, our results are compared to the ones obtained by Kilic et al. [START_REF] Kilic | Transitional flow between contra-rotating disks[END_REF] and Gan et al. [START_REF] Gan | Superposed flow between two discs contrarotating at differential speeds[END_REF], who performed calculations for -1 ≤ Γ ≤ 0 and G = 0.12 using a classical k -ǫ turbulence model. Considering that the k -ǫ prediction of these authors is in relatively good agreement with the smooth disk case, the comparison given in figure 8 may be meaningful to get an idea of the effect of aspect ratio. For Re = 10 5 , Kilic et al.

C. Turbulence field in the exact counter-rotating regime

As already mentioned above, the influences of both the Reynolds number and the aspect ratio are relatively weak (compared to the effect of the ratio Γ between the rotation rates).

In the following, we focus on the exact counter-rotating regime Γ = -1 and Re and G are fixed respectively to Re = 6.28 × 10 5 and G = 1.8.

Figure 9 presents the axial profiles of the six components of the Reynolds stress tensor.

These components are normalized by the local disk 1 velocity Ω 1 r. For example, R * rr is defined as: R * rr = v ′ 2 r /(Ω 1 r) 2 . As in all rotating disk problems [START_REF] Poncet | Batchelor versus Stewartson flow structures in a rotor-stator cavity with throughflow[END_REF], turbulence is mainly concentrated in the boundary layers with the same turbulence levels in the upper and lower disk boundary layers. The main difference with the rotor-stator configuration is that turbulence is also generated in the median region of the interdisk spacing and is due to the shear, stretched by the recirculations. The Von Kármán arrangement is indeed known to produce an intense turbulence in a compact region of space [START_REF] Maurer | Statistics of turbulence between two counterrotating disks in low-temperature helium gas[END_REF]. The magnitudes of the three normal components (in principal axes) are almost the same in the equatorial plane. It means that turbulence is quasi isotropic in that region. The cross components are quite weak except for the R * rθ component, which behaves like the normal components with a bump at mid-plane. As expected, the maximum of the turbulence Reynolds number Re t = k 2 /(νǫ) = 5836 is located in the shear layer close to the periphery of the cavity, where the highest values of the local Reynolds number Re r = Ω 1 r 2 /ν are obtained (fig. 10a). This maximum is to be compared to the maximum value Re t ≃ 500 obtained by Poncet [START_REF] Poncet | Écoulements de type rotor-stator soumis à un flux axial: de Batchelor à Stewartson[END_REF] for Γ = 0 and Re ≃ 10 6 , which indicates the high turbulence level in that region.

Figure 10b shows the anisotropy invariant map for the Reynolds stress tensor in the whole interdisk spacing at r * = 0.51. The second A 2 and third A 3 invariants of the anisotropy tensor a ij of the second moments of the fluctuations are defined as: A 2 = a ij a ji and A 3 = a ij a jk a ki [START_REF] Lumley | Computational modeling of turbulent flows[END_REF], where

a ij = R ij /k -2 3 δ ij (δ ij the Kronecker symbol).
The results of the RSM model satisfy the realizability diagram of Lumley [START_REF] Lumley | Computational modeling of turbulent flows[END_REF]. Very close to the disks, the turbulence tends to follow the two-component behavior as the wall normal fluctuations are damped more effectively than fluctuations parallel to the disk. Outside the boundary layers and especially in the shear layer, the turbulence is fairly close to the isotropic case (A 2 = A 3 = 0), which confirms the results observed from figure 9. Note that very close to the mid-plane, the flow tends to the axisymmetric limit.

V. BLADED DISK CASE: INERTIAL STIRRING

To increase the efficiency of the disks in forcing the flow, we used n blades of height h * mounted on both disks. The stirring is called inertial because the fluid is set into motion thanks to areas of forcing perpendicular to the motion itself. In that case, Ravelet [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF] showed that all mean and turbulent quantities are independent of the Reynolds number in the range Re = [10 5 , 2 × 10 6 ]. Thus, we have chosen to fix the values of Re ≃ 2 × 10 5 and G = 1.8. In that case, the boundary layers are separated and the flow is found to be highly turbulent. Moreover, direct comparisons with the experiments of Ravelet [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF] can be performed. The purpose of this section is to propose an efficient way to model the effect of straight blades on both the mean and turbulent fields.

A. Flow structure in the exact counter-rotating regime

In the bladed disk case, the flow structure is completely different from the smooth disk case, where the velocity gradient are located in the boundary layers along the disks and decrease when the Reynolds number increases. For an inertially driven flow, the mean flow does not present any appreciable velocity gradient in the vicinity of the blades (fig. 11) and the gradients are distributed in the median region of the flow. The mean flow is divided into three main regions: a shear layer at mid-plane and two fluid regions close to each bladed disks. The intensity of the shear at mid-plane is increased compared to the viscous stirring case. This shear is due to the two recirculation cells. It induces a strong radial inflow (V r < 0) around z * = 0 and two opposite axial flows towards the disks. The magnitude of the mean axial and radial velocity components increase from the periphery (fig. 11c) to the rotation axis (fig. 11a). From the disk to the top of the blades, the tangential fluid velocity is fairly close to the local disk velocity. Moreover, a strong radial outflow is created along the bladed disks and goes with the impellers. At the top of the blades, there is a strong decrease of |V θ | interpreted as the wake of the blades. There is a very good agreement between the numerical predictions and the velocity measurements concerning the V θ -profiles. A small difference is observed in the shear layer, where the RSM model predicts a thinner layer than the one measured by Ravelet [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF]. This last author observed, for the same set of parameters, high energy levels for frequencies inferior to the injection frequency. This contribution is attributed to the appearance of strong coherent structures in the shear layer not observed in the smooth disk case and which may explain the weak discrepancies obtained.

B. Flow structure for -1 ≤ Γ ≤ 0
We perform the same analysis as in the smooth disk case by varying the ratio Γ between the two rotating disk speeds. Figure 8 presents comparisons between the smooth and bladed disk cases concerning the size S c of the cell along the slowest disk for -1 ≤ Γ ≤ 0. The same behavior is obtained but the transition between the two cell and the one cell structures (S c → 0) is slightly delayed. It occurs in the inertial stirring case for Γ ≃ -0.65, which is close to the experimental value obtained by Cadot and Le Maître [START_REF] Cadot | The turbulent flow between two rotating stirrers: similarity laws and transitions for the driving torques fluctuations[END_REF] in the same configuration Γ = -0.69 and the analytical one obtained by Dijkstra and Van Heijst [START_REF] Dijkstra | The flow between two finite rotating disks enclosed by a cylinder[END_REF] for Re → 0 in the smooth disk case Γ = -2/3. The measurements of Ravelet [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF] reveal a transition for Γ = -0.78. It confirms the similitude observed by [START_REF] Cadot | The turbulent flow between two rotating stirrers: similarity laws and transitions for the driving torques fluctuations[END_REF] between the smooth disk flow with a large viscosity and the mean flow in the inertial stirring case.

The transition from the two cell to the one cell structures can be seen also from figure 12. Compared to the smooth disk case, the cell along the slowest disk is larger for Γ = -0.8 (fig. 12b). For Γ = -0.7, only a small recirculation subsists along the upper disk and completely disappears for Γ = -0.6. For Γ ≥ -0.6, the same pattern is observed with streamlines parallel to the rotation axis.

C. Turbulence field in the exact counter-rotating regime

To enable direct comparisons with the viscous stirring case, figure 13 presents the axial profiles of the six components of the Reynolds stress tensor at the same radius r * = 0.81 and for the same values of G and Γ. The main difference between the smooth and the bladed disk configurations is that, in the latter case, the turbulence intensities vanish towards the disks. Apart from that, turbulence is also mostly generated at mid-plane because of the shear stretched by the recirculations. The blades induce a much stronger shear zone in the equatorial plane compared to the smooth disk case as already seen from the mean velocity profiles (fig. 11). Thus, the turbulence levels, regarding the normal Reynolds stress components (fig. 13a), are almost 20 times larger than for viscous stirring and quite comparable to the mean fluid velocity. It confirms the previous measurements of Cadot et al. [START_REF] Cadot | Energy injection in closed turbulent flows: stirring through boundary layers versus inertial stirring[END_REF] in steady regimes of turbulence in the Von Kármán geometry. They found that the fluid velocity fluctuations are close to the fluid mean velocity and 6 times larger in the bladed disk case than in the smooth disk case. In the present study, the R * rr component is much weaker than the two other normal components, which indicates the turbulence anisotropy in the core of the flow. The cross components are also stronger than in the smooth disk case. The level of the R * rθ component (fig. 13b) is of the order of R * rr . Note that the maximum of the R * 1/2 θθ component obtained at mid-plane (z * = 0) using the RSM model is in excellent agreement with the asymptotic value measured by Ravelet [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de Von Kármán turbulent[END_REF] for Re ≥ 10 4 (relative error inferior to 0.1%). Nevertheless, one must remark that only a single measurement point is available, and consequently it is hard to derive definite conclusions. Another point is that the periodic unsteadiness introduced by the blades is not exactly accounted for in the calculation and may explain small discrepancies in the mean velocity profiles.

To study the influence of the number n of blades and their height h * on the turbulent field, Nevertheless, in the whole flow, four blades seem to be sufficient to force the flow. On the other hand, the blade height h plays a more important role. The k * level is twice higher when the blades are twice higher too.

It is now established that all mean and turbulent quantities are independent of the Reynolds number in the range Re = [10 5 , 2 × 10 6 ]. The turbulent dissipation is indeed much stronger than the dissipation due to the boundary layers and hides the dependence on

Re. All these results can thus be extended to higher Reynolds numbers.

VI. CONCLUSION

We have performed some comparisons between numerical predictions using a RSM model and velocity measurements considering the turbulent flow between two flat or bladed counterrotating disks. This configuration known as the Von Kármán geometry is used to produce an intense turbulence in a compact region of space.

For viscous stirring, the flow is of Stewartson type close to the rotation axis and so exhibits three distinct regions: two boundary layers and one shear layer at mid-plane. When one approaches the periphery of the cavity, for r * ≃ 0.476, the flow gets of Batchelor type.

Turbulence is mainly concentrated in the boundary layers and in the transitional shear layer, where turbulence is almost isotropic. Turbulence intensities increase towards the outer cylinder. When one decreases the aspect ratio of the cavity until G ≤ 0.4, the boundary layers mixed and the flow is then of torsional Couette type for lower values of G. In the case of inertial stirring, the impellers are more efficient to force the flow. Thus, the transitional shear layer intensifies. Turbulence is so mainly concentrated around z * = 0 and vanish towards the disks. The turbulence intensities are almost 20 times larger than in the flat disk case. The height of the blades is found to be the preponderant parameter to increase the turbulence intensities more than the number of blades. In the flat and bladed disk cases, we have numerically verified the statement of Cadot et al. [START_REF] Cadot | Energy injection in closed turbulent flows: stirring through boundary layers versus inertial stirring[END_REF]: "smooth or rough, the efficiency of a given type of stirrer to set the bulk of the fluid in motion is independent of the Reynolds number". Moreover, we have characterized the transition between the two cell and the one cell regimes. For inertial stirring, it occurs for Γ ≃ -0.65 close to the values obtained by [START_REF] Dijkstra | The flow between two finite rotating disks enclosed by a cylinder[END_REF][START_REF] Cadot | The turbulent flow between two rotating stirrers: similarity laws and transitions for the driving torques fluctuations[END_REF].

The agreement between the numerical predictions and the LDV measurements is very satisfactory in both cases. For the first time, an easy and efficient way to model the main effect of straight blades has been proposed. Further experimental works are now required to provide more comparisons for the turbulent fields but also some calculations for curved blades.

• Fig. 1

  uniform flow. For a thin rectangular plate perpendicular to the mean flow, the value of C D is in the range [1.05 -1.9], depending on the size of the plate. As the flow is here not uniform along the blades and as the drag coefficient is supposed to decrease for increasing values of the Reynolds number, C D is expected to be lower than 2 depending on the flow and blade parameters. Some calculations have been performed for Γ = -1, G = 1.8, Re = 2 × 10 5 and straight blades (h * = 0.2, n = 8) to study the influence of the trailing coefficient C D . The differences on the extrema of the tangential velocity component are inferior to 0.5% for C D in the range [0.1 -2]. Thus, we have chosen to fix the value of C D equal to 0.5.

  figure 3 from the radial evolution of the boundary layer thickness δ for the same set of parameters. Very close to the rotation axis, the axial flow impinges the disks and creates very large boundary layers on both disks, whose size decreases with the local radius as expected[START_REF] Gauthier | Instabilities in the flow between co-and counter-rotating disks[END_REF]. The flow is then of Stewartson type. During the transition, δ increases as already observed by Poncet[START_REF] Poncet | Écoulements de type rotor-stator soumis à un flux axial: de Batchelor à Stewartson[END_REF] for rotor-stator flows (Γ = 0). For r * ≃ 0.47, the flow is clearly of

[ 29 ]

 29 found that the evolution of S c against Γ is non monotonous. It decreases more slowly from Γ = -1 to Γ = -0.2 than in our case. It is a combined effect of both the Reynolds number and the aspect ratio of the cavity. For Γ = -0.4, they observed a double transition: from laminar to turbulent flow and from Batchelor to Stewartson type of flow. The decrease of S c is much faster with Γ in the laminar case[START_REF] Kilic | Transitional flow between contra-rotating disks[END_REF]. For Re = 1.25×10 6 , Gan et al.[START_REF] Gan | Superposed flow between two discs contrarotating at differential speeds[END_REF] obtained streamline patterns different from the ones shown in figure7for Γ = [-0.8; -0.2] essentially because of the small value of G. A large cell along the slower disk is still observed for Γ = -0.4. This cell is trapped by the main flow due to the faster disk in the zone 0.3 ≤ r * ≤ 0.45.

figure 14 shows

 14 figure[START_REF] Maurer | Statistics of turbulence between two counterrotating disks in low-temperature helium gas[END_REF] shows radial profiles of the turbulence kinetic energy k * normalized by (Ω 1 R c ) 2 for various impeller configurations. These profiles are plotted at mid-plane where the maximum of k * prevails. As expected, k * increases towards the periphery of the cavity, it means for increasing local Reynolds number. Then, k * decreases for radial locations in the gap between the disks and the external cylinder. We can first notice the very weak level of turbulence kinetic energy in the smooth disk case compared to the other bladed disk cases. Secondly, the influence of the blade number n is quite weak for n = 4, 8 or 16. Only very close to the rotation axis, we can notice a different behavior in the configuration with 16 blades.

:•Fig. 2 :•Fig. 3 :•Fig. 4 :•Fig. 6 :•Fig. 7 :•Fig. 8 :•Fig. 9 :•Fig. 12 :•Fig. 13 :•Fig. 14 :

 2346789121314 Sketches of the cavity with (a) relevant notation in the smooth disk case and (b) straight blades. Axial profiles of the tangential velocity component for Γ = -1, Re = 6.28 × 10 5 and G = 1.8 at five radial locations: (a) r * = 0.346, (b) r * = 0.476, (c) r * = 0.605, (d) r * = 0.735, (e) r * = 0.865. Comparisons between the numerical results (-) and the experimental data (•) in the smooth disk case. Radial evolution of the boundary layer thickness δ/H in the smooth disk case for Γ = -1, Re = 6.28×10 5 and G = 1.8 (symbols: RSM, line: polynomial interpolation). Radial profiles of the tangential velocity component in the smooth disk case for Γ = -1, G = 1.8 and four Reynolds numbers at different axial locations: (a) z * = 0.91, (b) z * = 0.59, (c) z * = 0.02, (d) z * = -0.59, (e) z * = -0.92. • Fig.5: Computed streamlines between smooth disks for Γ = -1 and G = 1.8: (a) Re = 2 × 10 5 ; (b) Re = 7.8 × 10 5 ; (c) Re = 1.3 × 10 6 ; (d) Re = 4 × 10 6 . Axial profiles of the mean velocity components at r * = 0.81 for Γ = -1, Re = 1.3 × 10 6 and four values of G in the smooth disk case (RSM). Computed streamlines between smooth disks for Re = 1.3 × 10 6 and G = 1.8: (a) Γ = -1; (b) Γ = -0.9; (c) Γ = -0.8; (d) Γ = -0.7; (e) Γ = -0.2. Size S c /H of the smallest cell against Γ for G = 1.8 (RSM). Comparison between (-) the smooth disk case (Re = 1.3 × 10 6 ), (--) the bladed disk case (Re = 2 × 10 5 ) and previous numerical results of (•) Kilic et al. [29] and (⋄) Gan et al. [31] in the smooth disk case. Axial profiles of the six Reynolds stress tensor components at r * = 0.81 for Γ = -1, G = 1.8 and Re = 6.28 × 10 5 in the smooth disk case (RSM). • Fig.10: Γ = -1, G = 1.8 and Re = 6.28 × 10 5 in the smooth disk case (RSM): (a) Iso-turbulence Reynolds number Re t = k 2 /(νǫ) -(b) Anisotropy invariant map at r * = 0.51: (×) -1 ≤ z * ≤ 0, ( ) 0 ≤ z * ≤ 1. • Fig.11: Axial profiles of the mean velocity components for Γ = -1, Re = 2 × 10 5 , G = 1.8 and straight blades (n = 8, h * = 0.2) at three radial locations: (a) r * = 0.4, (b) r * = 0.5, (c) r * = 0.6. Comparisons between the predictions of the RSM model (lines) and the LDV measurements of Ravelet [21] (•). Computed streamlines for Re = 2 × 10 5 , G = 1.8 and straight blades (n = 8, h * = 0.2): (a) Γ = -1; (b) Γ = -0.9; (c) Γ = -0.8; (d) Γ = -0.7; (e) Γ = -0.6. Axial profiles of the six Reynolds stress tensor components at r * = 0.81 for Γ = -1, G = 1.8, Re = 2 × 10 5 and straight blades (n = 8, h * = 0.2)(RSM). (•) LDV data of Ravelet [21] for R * 1/2 θθ . Radial profiles of the turbulence kinetic energy k * = k/(Ω 1 R c ) 2 at z * = 0 for Γ = -1, G = 1.8, Re = 2 × 10 5 and different bladed disk configurations -comparison with the smooth disk case (Re = 6.28 × 10 5 ) (RSM).

Figure 1 :

 1 Figure 1: Poncet et al., submitted to Int. J. Heat Fluid Flow.

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: Poncet et al., submitted to Int. J. Heat Fluid Flow.

Figure 5 :

 5 Figure 5: Poncet et al., submitted to Int. J. Heat Fluid Flow.

Figure 6 :

 6 Figure 6: Poncet et al., submitted to Int. J. Heat Fluid Flow.

Figure 7 :

 7 Figure 7: Poncet et al., submitted to Int. J. Heat Fluid Flow.

Figure 8 :Figure 9 :Figure 10 :Figure 11 :

 891011 Figure 8: Poncet et al., submitted to Int. J. Heat Fluid Flow.

Figure 12 :Figure 13 :Figure 14 :

 121314 Figure 12: Poncet et al., submitted to Int. J. Heat Fluid Flow.


Mohamed El

Hedi Arouri
email: mohamed.arouri@univ-orleans.fr

Mondher Bellalah
email: mondher.bellalah@u-cergy.fr

Duc Khuong Nguyen
email: dnguyen@groupeisc.com

THE COMOVEMENTS IN INTERNATIONAL STOCK MARKETS: NEW EVIDENCE FROM LATIN AMERICAN EMERGING COUNTRIES

Keywords: JEL Classification: F37; G15 Stock market comovements, Latin American emerging markets, Multivariate GARCH models, Structural breaks

's structural break analysis technique is also employed to test for possibly changing nature of stock market comovements. Main findings of the paper are as follows. First, the degree of cross-market comovements changed over time and has significantly increased since 1994 and onwards, which is to the large extent informative of increasing market integration. Despite the significant interdependencies among the studied markets, room for international portfolio diversification nevertheless seems largely possible. Second, it is demonstrated that the cross-market comovements are subjected to various regime shifts due essentially to major stock market events. Finally, the purpose that stock markets move much more together in times of crisis than in normal times can not be rejected according to our empirical evidence.

INTRODUCTION

Studies on the time-variation and nature of international stock market comovements have recently gained ground in international finance literature. This increase of interests and motivations can be explained by various reasons, but the most relevant of all include the international portfolio diversification issues and the recurrence of financial crises that occurred in both developed and emerging countries during the 1990's decade.

As far as the portfolio diversification is concerned, we already acknowledge that, according to the basic lessons drawn from both the mean-variance framework of Markowitz (1958) and the Capital Asset Pricing Model (CAPM) developed by [START_REF] Sharpe | Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk[END_REF] and [START_REF] Lintner | The Valuation of Risk Assets and the Selection of Risky Investment in Stock Portfolio and Capital Budgets[END_REF], investors, either individual or institutional, can reduce the volatility of their portfolios through allocating their investments in various classes of financial instruments, industries and other categories of assets that would move in different ways in response to the same event. In other words, diversification benefits can be achieved because a portfolio's performance depends not only on the return and risk characteristics of the assets being held in the portfolio, but also on the correlation (or comovement) of these assets. The lower is the correlation between assets; the higher are the diversification gains. Based on the same rules, diversifying internationally the portfolios is beneficial to investors only if stock markets in different countries do not move together. Thus, the portfolio diversification issue naturally raises the question of investigating the relationship between international stock markets. The ongoing process of globalization, thanks to the market openings decisions in many emerging market countries in the late 1980s and the resulting cross-border capital flows, also contributes to further expand the literature on stock market comovements. Indeed, since emerging markets are now accessible to foreign investors and they become more integrated into the world stock market (e.g., [START_REF] Bekaert | Time-Varying World Market Integration[END_REF][START_REF] Gerard | Are the East Asian markets integrated? Evidence from the ICAPM[END_REF][START_REF] Carrieri | Characterizing World Market Integration through Time[END_REF], it is intriguing to question whether diversification benefits documented by [START_REF] Grubel | Internationaly diversified portfolios: welfare gains and capital flows[END_REF], Levy and Sarnat (1970), [START_REF] Lessard | World, National and Industry Factors in Equity Returns[END_REF][START_REF] Errunza | Gains from Portfolio Diversification into less Developed Countries' Securities[END_REF] have been significantly reduced.

The recurrence of financial crises during the 1990's decade, as we mentioned above, is another reason simulating the appetite of researchers for studying the comovement of inter-national stock markets. The rationale behind this appetite lies on the fact that financial institutions and policymakers would like to measure the intensity of the interdependences between national stock markets and thus to have a reliable management of contagion risk which results from potentially harmful spillovers of volatility across markets in time of distresses or crisis.

For instance, a number of theoretical and empirical studies have employed a wide variety of methods and data frequencies at both firm and country levels to model the comovement of international stock markets and searched for the reasons behind this phenomenon.

The focus was mostly on the correlations and the stock return and volatility spillovers between stock markets around the world. Since the literature is so abundant to be reviewed exhaustively here, we only provide a brief review of the most relevant works in the following paragraphs.

In an earlier paper that opens the debate on the potential benefits of international diversification, [START_REF] Grubel | Internationaly diversified portfolios: welfare gains and capital flows[END_REF] develops a model of internationally diversified portfolios where investors hold both domestic assets and assets denominated in foreign currency from eleven industrial countries. According to the author's findings, international diversification of portfolio constitutes a possible source of welfare gains for individual investors and this increased performance comes from the low correlations between selected countries. Levy and Sarnat (1970), [START_REF] Lessard | World, National and Industry Factors in Equity Returns[END_REF] and [START_REF] Errunza | Gains from Portfolio Diversification into less Developed Countries' Securities[END_REF] also examine the international diversification potential, but the analysis is more general due to the simultaneous treatment of developed and developing countries. Their empirical results show the superiority of portfolios which are not limited to only securities issued by developed countries.

In the spirit of the aforementioned papers, the literature focusing on the comovement of international stock markets has grown rapidly and includes, among others, [START_REF] Eun | International Transmission of Stock Market Movements[END_REF], [START_REF] Hamao | Correlations in Price Changes and Volatility across International Markets[END_REF], [START_REF] Karolyi | A Multivariate GARCH Model of International Transmissions of Stock Returns and Volatility: The case of the United States and Canada[END_REF], [START_REF] Richards | Comovements in National Stock Market Returns: Evidence of Predictability, but not Cointegration[END_REF], [START_REF] Harvey | Predictable Risk and Returns in Emerging Markets[END_REF], [START_REF] Forbes | No Contagion, only Interdependence: Measuring Stock Market Comovements[END_REF], Johnson and Soenen (2003), [START_REF] Hsin | A Multilateral Approach to Examining the Comovements among Major World Equity Markets[END_REF], [START_REF] Bekaert | International Stock Return Comovements[END_REF], [START_REF] Morana | Comovements in International Stock Markets[END_REF], [START_REF] Gilmore | The Dynamics of Central European Equity Market Comovements[END_REF]al. (2006), and[START_REF] Syriopoulos | Dynamic Linkages between Emerging European and Developed Stock Markets: Has the EMU any Impact?[END_REF]. We generally observe that in recent works, the concept of comovement covers not only the correlations, but also the spillovers of stock returns (or prices) and volatility across equity markets. As a result, empirical strategies such as [START_REF] Granger | Investigating Causal Relationships by Econometrics Models and Cross Spectral Methods[END_REF]'s causality test, Sims (1980)'s vector autoregressive models, [START_REF] Engle | Co-integration and Error Correction: Representation, Estimation and Testing[END_REF]'s cointegration test and ARCH/GARCH methodology are largely employed. For example, [START_REF] Hamao | Correlations in Price Changes and Volatility across International Markets[END_REF] explore the relationships between three marketplaces New York, London and Tokyo from an ARCH-type model and find significant spillovers of prices and price volatility. In Gilmore and al. (2007), dynamic cointegration and principal components methods are applied to investigate the short-term and long-term comovements between European developed markets and those of three Central European countries. Overall, main results from the majority of these studies can be summarized as follows: i) significant comovements are observed among world stock markets; ii) there is evidence of a positive relation between correlation and volatility, that is, correlations between international stock markets tend to be important in periods of high volatility or in the times of financial troubles. Some papers have empirically examined the time-varying feature of international stock market comovements and searched for their determinants (e.g., Longin and Solnik, 1995;[START_REF] Erb | Forecasting International Equity Correlations[END_REF][START_REF] Karolyi | Why Do Markets Move Together? An Investigation of U.S.-Japan Stock Return Comovements[END_REF][START_REF] Karolyi | Why Do Markets Move Together? An Investigation of U.S.-Japan Stock Return Comovements[END_REF][START_REF] Bae | A New Approach to Measuring Financial Contagion[END_REF]. For instance, using a GARCH model specification, Longin and Solnik (1995) document an increasing trend in the correlations among international equity markets and note that the correlation between these returns was dynamically changing in their research. [START_REF] Karolyi | Why Do Markets Move Together? An Investigation of U.S.-Japan Stock Return Comovements[END_REF] studied the United States and Japan indices. They discovered evidence of changing correlations in the daily returns of these countries.

The issue of equity market comovement in Latin America has been investigated by several studies. For instance, [START_REF] Choudry | Stochastic Trends in Stock Prices: Evidence from Latin American Markets[END_REF] employs unit root tests, cointegration tests and error correction models to examine the long-run relationship between six Latin American markets and the US market, and finds evidence of cointegration relationship and significant causality among these markets. In the same vein, [START_REF] Chen | Stock Market Linkages: Evidence from Latin America[END_REF] investigate the interdependencies of six equity markets in Latin America (Argentina, Brazil, Chile, Colombia, Mexico and Venezuela) and document that diversification benefits are limited when investing in these markets particularly due to their high comovement. [START_REF] Christofi | Correlation in Price Changes and Volatility of Major Latin American Stock Markets[END_REF] show evidence of significant cross-market linkages in five Latin American markets (Argen-tina, Brazil, Chile, Columbia and Mexico) from combining a vector autoregressive (VAR) model with a multivariate exponential GARCH process. Johnson and Soenen (2003) examine the cross-country comovement for eight American equity markets with the US market using the Geweke (1982)'s measure of contemporaneous feedback between return series.

Their sample includes equity markets from Argentina, Brazil, Chile, Mexico, Colombia, Peru, Venezuela, and Canada. They find a statistically significant linkage between eight equity markets of the Americas and the US stock market. Finally, in a recent paper, [START_REF] Fujii | Intra and Inter-regional Causal Linkages of Emerging Stock Markets: Evidence from Asia and Latin America In and Out of Crises[END_REF] studies the intra and inter-causal linkages among eight emerging stock markets in Asia and Latin America. Four Latin American markets included in the sample are Argentina, Brazil, Chile and Mexico. Using standardized residuals estimated from an autoregressive process and a GARCH(1,1) model to calculate the residual correlation functions for the first and second moments of the stock returns, the author finds the existence of significant causal linkages both within each region and across the two regions. In addition, the causal linkages are found to be stronger during the time of major financial crises.

Our paper joins the above literature by focusing on the aggregate stock market comovements. We empirically investigate the comovement issue in the context of Latin American emerging markets since they rank among the most mature markets within the universe of emerging countries and they actually attract a particular attention from global investors thanks to their great market openness. We contribute to the related literature in several aspects. First, instead of modeling the comovement by VAR and realized correlations as in past studies which capture the causal linkages but do not permit to quantify the comovement, this paper directly infer the cross-market linkage from the stock data using a multivariate Dynamic Conditional Correlation GARCH model (DCC-GARCH). Second, we are interested in dating the structural breaks in the time-paths of the conditional correlation indices to highlight whether the cross-market comovement encompasses significant changes in nature or not. Finally, our methodology enables the investigation of the differences in stock market comovements between normal and crisis periods.

Overall, our findings indicate a clear upward trend in correlation from 1994 and onwards as a result of market liberalization and increased globalization. Interestingly, there are sud-den increases in conditional correlation following the Asian and Brazilian financial crises in 1997-1998 and, to less extent, the stock market crash in 1987 and the Latin American markets crises in 1994 and 2001. The results from the stability test typically reveal that structural breaks are present in the time-path of the cross-market conditional correlation series and often coincide with major stock market events such as financial market liberalizations, ADR introduction and crisis. Finally, our findings support those of previous works because they evidence that the inter-market comovements were significantly higher during the crisis period than during the tranquil period.

The remainder of the article is organized as follows. Section 2 describes the data employed in the paper. Section 3 presents our empirical methodology. Section 4 reports and interprets the empirical results. We provide some concluding remarks in Section 5.

DATA

Given the aim of the article, the data used are on monthly basis and consist of the S&P's IFCG total return indices for the six main Latin American emerging markets and the MSCI World stock market index, sampled over the period January 1985 to August 2005. The Latin American emerging markets studied are: Argentina, Brazil, Chile, Colombia, Mexico, and Venezuela. All the indices are obtained from DataStream International and expressed in American dollars. They are converted to return using the log price relative filter. Descriptive statistics and stochastic properties of monthly returns are presented in Table 1.

[ [ [ [Please insert Table 1 about here] ] ] ]

Panel A reveals a number of interesting facts. Compared to World market, Latin American emerging markets have higher returns and risk. Chile has the highest monthly return followed by Colombia and Mexico. Argentina has the highest standard-deviation followed by Brazil and Venezuela.

The [START_REF] Engle | Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK Inflation[END_REF]'s test for conditional heteroscedasticity rejects the null hypothesis of no ARCH effects for Argentina, Brazil, Colombia and Mexico. Skewness is mostly negative and kurtosis is above three. The Jarque-Bera test statistic (JB) strongly rejects the hy-pothesis of normally distributed returns. These facts support our decision to use the quasimaximum likelihood (QML) approach of [START_REF] Bollerslev | Quasi Maximum Likelihood Estimation and Inference in Dynamic Models with time Varying Covariances[END_REF] to estimate and test the model.

The null hypothesis of no autocorrelation of order 12 is rejected for Chile, Colombia and Mexico. The first autocorrelation (reported in Panel B) is highly significant for these countries, which suggests that we need to include an auto-regressive correction in the mean equations.

Panel C reports the unconditional correlations among markets. As expected, there is a positive correlation between stock markets. The highest unconditional correlation is between Chile and Mexico (43.34%) and the lowest one is between Argentina and Colombia (7.00%). The values of the unconditional correlations are clearly low. This suggests that there are still benefits from diversification across Latin American emerging markets.

METHODOLOGY

Modeling dynamic conditional correlation

Finance literature on modeling the volatility of emerging stock market volatility has been extensively developed thanks to ARCH/GARCH-type models, which are respectively introduced by [START_REF] Engle | Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK Inflation[END_REF] and [START_REF] Bollerslev | Generalized Autoregressive Conditional Heteroscedasticity[END_REF]. The examples of application include, among others, Bekaert andHarvey (1997,2000), [START_REF] Kim | Stock Market Openings: Experience of Emerging Economies[END_REF], [START_REF] Miles | Financial Deregulation and Volatility in Emerging Equity Markets[END_REF] and [START_REF] Jayasuriya | Stock Market Liberalization and Volatility in the Presence of Favorable Market Characteristics and Institutions[END_REF]. As pointed out by past studies, GARCH models appear to successfully describe the stochastic properties of stock returns and correlation (i.e., time-variation and persistence).

In this paper we rely on the time-varying correlation coefficients estimated from a multivariate DCC-GARCH model developed by [START_REF] Engle | Dynamic Conditional Correlation: a New Simple Class of Multivariate GARCH Models[END_REF] to measure the comovements between Latin American emerging markets, and between them and the world stock market.

This model is obviously suitable to our problem because it allows us to directly infer the cross-market conditional correlations and to avoid the proliferation of parameters to be estimated in full parameterization of BEKK GARCH.

Assume that stock market returns from the k series are multivariate normally distributed with zero mean and conditional variance-covariance matrix H t , our multivariate DCC-GARCH model can be presented as follows:

   ≡ → + = - t t t t t t t t t t D R D H H N I r ) , 0 ( , 1 ε ε µ (1)
In these formulas, t r is the (k×1) vector of the returns on stock market indices; t ε is a (k×1)

vector of zero mean return innovations conditional on the information available at time t-1;

1 , 2 1 , 1 0 , - -+ + = t w i t i i i t i r r δ δ δ µ
for emerging market i and

) ... ( 2 / 1 2 / 1 11 kkt t t h h diag D = (2)
where iit h is chosen to be a univariate GARCH(1,1) process; The conditional correlation coefficient ij ρ between two markets i and j is then expressed by the following equation:

2 / 1 2 / 1 ) ( ) ( - - = t t t t diagQ Q diagQ R (3) where 1 ' 1 1 ) 1 ( - - - + + - - = t t t t Q u u Q Q β α β
( ) ( ) 2 / 1 1 , 2 1 , 2 / 1 1 , 2 1 , 1 , 1 , 1 , ) 1 ( ) 1 ( ) 1 ( - - - - - - - + + - - + + - - + + - - = t jj t j jj t ii t i ii t ij t j t i ij ij q u q q u q q u u q β α β α β α β α β α β α ρ (4)
In this formulation, ij q refers to the element located in the ith row and jth column of the symmetric positive definite matrix t Q . Accordingly, the estimation of the DCC-GARCH model as described above is can be estimated using a two-stage procedure. In the first stage, univariate GARCH(1,1) model is estimated for each time series. During the second stage, the transformed residuals from the first stage (i.e., the estimated residuals are standardized by their conditional standard deviations) are used to infer the conditional correlation estimators. The log-likelihood of the observations on t ε is given by ( )

∑ = - - - + + - = T t t t t t t t t t D R D D R D n L 1 1 1 1 ' log ) 2 log( 2 1 ε ε π Since t t t t t D h u ε ε 1 / - = =
, the log-likelihood function can be rewritten as follows:

( )

∑ = - + + + - = T t t t t t t u R u R D n L 1 1 ' log log 2 ) 2 log( 2 1 π
Note that a conditional variance term can be added into the mean equation to control for the risk-return tradeoffs in previous works. However, the majority of these studies found evidence of insignificant impact of conditional volatility on stock returns. [START_REF] Lundblad | The Risk Return Tradeoff in the Long Run: 1836-2003[END_REF] studied the same issue over the period from 1802 to today and concluded that the meanvariance tradeoffs are only positively significant over a long-term horizon. Accordingly, we intentionally do not model the in-mean effect of conditional volatility because we focus on a short-run comovement between stock and bond markets1 .

Detecting structural breaks in cross-market comovements

Over the last three decades, emerging markets in Latin American region have experienced significant reforms on capital markets and various periods of financial turbulences such as the Mexican crisis in 1994-1995, the Brazilian crisis in 1998 and Argentinean crisis in 2001. The cross-market comovements might be subjected to structural changes. Intuitionally, a higher degree of financial openness can strengthen the relationship between international stock markets, whereas the depth of financial crises seriously affected a country's economic and financial structure, which in turn leads to changes in the nature of its comovements with other countries. Since the changes in the cross-market comovements will impact the coordination of two countries' policymakers and the actions of portfolio managers, it is important to question the issue of structural changes.

In this study we refer to the bilateral dynamic conditional correlation as measure of cross-market comovements and use the [START_REF] Bai | Computation and Analysis of Multiple Structural Change Models[END_REF]'s testing procedure to investigate whether the time-varying cross-market comovements contain structural breaks. The idea here is to determine the number and location of breaks in a linear regression model.

t t t x y ε β + = T ) ,..., 1 ( n t = (5)
where t y is the estimated conditional correlation series at the time t,

T 1 ) , 1 ( - = t t y x
is the (2×1) vector of observations of the independent variables with the first component equal to unity, β is the (2×1) vector of regression coefficients, and t ε is assumed to be

) , 0 ( 2 σ iid .
The structural stability tests as presented above are concerned with testing the null hypothesis of 'no structural break' against the alternative that the regression coefficients vary over time. The breakpoint selection procedure is based on the Bayesian Information Criteria (BIC). Empirically, we set the maximum number of optimal breaks to be 5 and run the test. However, if the effective number of breaks is equal to 5, a higher number of breaks will be automatically chosen so that the testing procedure captures all possible breakpoints.

The optimal number of breaks corresponds to the one with the lowest BIC score.

EMPIRICAL RESULTS

This section reports and interprets the results obtained from the estimation of our empirical model. First, we present the estimation results of the empirical model and investigate the dynamics of conditional correlations between markets. Second, we take a close look into the results from structural break tests over the estimation period. Finally, as a checking exercise, we investigate whether our results validate or not the purpose that the comovement between stock markets is larger in crisis times.

Estimation Results

Table 2 contains parameter estimates and a number of diagnostic tests for DCC-GARCH model. The coefficients relating the return series to the one-lag local and world returns (Panel A) are insignificant, except for Colombia. The ARCH coefficients and GARCH coefficients reported in panel B are significant for all countries, except Venezuela. This is on line with previous results in the literature. The coefficients α are relatively small in size, which indicates that conditional volatility does not change very rapidly. However, the coefficients β are large, indicating gradual fluctuations over time. In addition, the estimates coefficients α and β satisfy the stationary conditions for all the variance and covariance processes.

[ [ [ [Please insert Table 2 about here] ] ] ]

Diagnostics of standardized residuals are provided in panel C. 2 One can remark that the indexes of kurtosis are often lower than those for the returns. However, the Bera-Jarque test statistics for normality indicates that the unconditional distribution of the conditionally normal GARCH process is not sufficiently fat-tailed to accommodate the excess kurtosis in the data. This result justifies the use of the QML procedures. We also compute the Lung-Box statistic to test the null hypothesis of absence of autocorrelation and the Engle (1982)'s test of absence of ARCH effects. The results indicate that the DCC-GARCH specification we use is flexible enough to capture the dynamics of the conditional covariance matrix.

[ [ [ [Please insert Figure 1 about here] ] ] ]

2 In the multivariate framework, the joint standardized residuals are given by

t t s t H ε ε 2 1 - = .
Dynamic conditional correlations within Latin American emerging markets and with the world market are plotted in Figure 1. These correlations are relatively low, 21.54% on average. The average time-varying correlations are quite similar to the unconditional correlations reported in Table 1. However, the conditional correlations vary considerably over time and from a couple of countries to another. The average of correlations between the studied Latin American markets and the world market is 25.30%. The average of correlations within the studied Latin American markets is only 20.04%. The highest conditional correlation is between Mexico and Chile (42.75%) and the lowest one is between Argentina and Colombia (6.76%). Interestingly, it is also observed that the evolution of these correlations witnesses some periods on negative values. This should mean high diversification gains from investing in Latin American emerging countries.

The most interesting patterns found is that there is a clear upward trend in correlation from 1994 and onwards as a result of market liberalization and increased globalization. The lowest average correlation between all studied markets can be found between 1985 and 1993. Interestingly, it can be seen that there are sudden increases in conditional correlation following the Asian and Brazilian financial crises in 1997-1998 and, to less extent, the stock market crash in 1987 and the Latin American markets crises in 1994 and 2001. The increased correlation in period of crises may be a symptom of international contagion. This point will be further investigated in section 4.3.

Do Structural Breaks Exist?

In this section, we investigate whether any structural change has occurred in conditional correlations between studied Latin American emerging markets and the world market over the estimation period. We apply the [START_REF] Bai | Computation and Analysis of Multiple Structural Change Models[END_REF]'s testing procedure to detect and date the structural breakpoints in our conditional correlations indices. In principle, a model's optimal number of breakpoints (m) is the one associated with the minimum BIC.

Figure 2 plots the BIC for models with m breakpoints for our sample markets. The selected optimal breakpoints for each market and their 95% confidence intervals are reported in Table 3.

[ [ [ [Please insert Figure 2 about here] ] ] ] [ [ [ [Please insert Table 3 about here] ] ] ]

The null hypothesis of stability is rejected for all the studied markets since the Bai-Perron's test detects breakpoints for the six markets. For Argentina, five significant breakpoints are obtained and for all the other studied countries four breakpoints are detected.

If we compare the estimated break dates in conditional correlations reported in Table 3 to the liberalization dates in Table 4, we find that the official liberalization dates fall into the 95% confidence intervals for the estimated break date in three markets: Brazil, Colombia and Mexico. As markets open up and become more liberalized and integrated with the rest of the world, one may expect that the correlation of a country with the world market increases. Our results confirm partially this opinion.

In Chile, the date where the first ADR was introduced is bounded by the 95% confidence interval of the first break date. In Argentina and Venezuela, none of the estimated break dates is related to market liberalization events. Finally, note the break date observed in all studied markets following the Asian and Brazilian financial crises in 1997-1998.

[ [ [ [Please insert Table 4 about here] ] ] ]

To sum up, major stock market events such as financial market liberalizations, ADR introduction and crisis have significant impacts on the Latin American emerging markets comovements. In general, the conditional correlations between these markets have increased in recent years as a result of the lifting of barriers.

Is the cross-market comovement higher in crisis period?

The majority of past studies claim that the comovement of stock markets is stronger during the crisis period than during the normal or tranquil ones. In this paper we test this purpose through investigating the differences in the level of conditional correlations between two 60-month subperiods. The first sub-period is a normal one starting in December 1989 and terminating in November 1994, whereas the crisis subperiod covers the period from December 1994 through November 1999 which contains the 1994-1995 Mexican and the 1997 Asian crises. The Asian crisis is included in our analysis because it spreads over quickly to Brazil causing the Brazilian crisis in 1998, and thus to the whole Latin American region.

Our method is simple and consists of calculating both the mean and the variance of the conditional correlation coefficients over two subperiods. A one-tail parametric test, called Z-test, is then applied to compare the means between two subsamples. Concretely, we test the null hypothesis that the mean of the conditional correlations in the crisis subperiod is equal to that in the tranquil period against the one-sided alternative that it is higher than the later one. The empirical z-statistic, assumed to follow a standard normal distribution under the null hypothesis is determined by the following expression:

) / ( ) ( n m x z σ ÷ - =
, where x refers to the mean of the crisis period, m is set to be the mean of the tranquil period, σ refers to the standard deviation of the crisis period and n is equal to 60.

[ [ [ [Please insert Table 5 about here] ] ] ]

Table 5 reports the obtained results. Overall, we observe that the inter-market comovements were significantly higher during the crisis period than during the tranquil period. The linkages between Latin American markets and the world market index follow the same schema, except for the case of Colombia where there was a significant decrease in correlation coefficient with the world market. Therefore, we are not able to reject the preposition according to which stock markets tend to move much more together during the crisis time than during a normal time.

DISCUSSIONS AND FUTURE EXTENSIONS

The purpose of this paper was to investigate empirically the comovements between Latin American emerging markets et between these markets and the World stock market over the 1985-2005 period. As pointed out by our empirical findings, there were upward trends in conditional correlations since 1994 and onwards in all sample markets. This evidence is strengthened by the fact that the cross-market correlation coefficients exceed 60% in recent years in many cases. As far as the issues of international market integration and portfolio diversification are concerned, there is still further room for global investors to get in Latin American emerging countries due to their low average correlations. At the same time, it is worth noting that the market liberalization process in selected emerging stock markets and the resulting increased globalization do increase the comovements of capital markets around the world, and thus reduce significantly the international diversification benefits.

However, the presence of frequent structural breaks in the time-path of cross-market correlation series as evidenced by our results typically adds that a reassessment of stock market comovements is necessary for implementing any investment strategy in Latin American, and so saying as a follow-up of major stock market events. Last but not least, higher comovement of sample stock markets in times of crisis can be considered as evidences of contagion effects highlighted by a number of previous studies.

For future perspectives, this study can be extended in some ways. First, the same methodology presented in this study can be applied to the issue of cross-country stock-bond return comovements to explore the relationships in different asset classes both at the country and international levels. This kind of study is of great importance because any portfolio optimization strategy hinges on the concept of correlation. Second, the empirical model presented in this paper is suitable to the investigation of the determinants of stock market comovements, of which macroeconomic announcements and economic and financial integration factors might be relevant candidates. Such research allows for example to determine the roots of stock market comovements and to answer the question of whether they result from world market integration or global bubbles. Notes: this table provides the differences in the means of conditional correlation coefficients between the 60month crisis period and the 60-month tranquil period. The number in parenthesis refers to the empirical zstatistics. The critical values of z at 95% and 99% confidence levels assuming normality of the series under consideration are equal to 1.96 and 2.33 respectively.

  time-path of the dependant variable, the problem of dating structural breaks turns to find the breakpoints ( resulting residual sum of squares based on the m regressions as shown by the following equation:

  Figure 1 Dynamic Conditional Correlations

  The test for Kurtosis coefficient has been normalized to zero. JB is the Jarque-Bera test for normality based on excess skewness and Kurtosis. Q(12) is the Ljung-Box test for autocorrelation of order 12.ARCH is the Engle (1982)'s test for conditional heteroscedasticity. *, ** and *** indicate significance of coefficients at the 1%, 5% and 10% respectively. +, ++ and +++ indicate rejection of the null hypotheses of no autocorrelation, normality and homocedasticity at the 1%, 5% and 1% levels of significance respectively for statistical tests.

				Table 1
			Descriptive Statistics
	Panel A: Summary Statistics
		Argentina	Brazil	Chile	Colombia	Mexico	Venezuela	World
	Mean	1.384	1.095	1.968	1.771	1.700	0.685	0.929
	(% per month)
	Std-Deviation	18.724	16.570	7.472	8.355	12.276	13.867	4.281
	(% per month) Skewness	0.379 **	-0.639 *	-0.338 **	0.493 *	-2.425 *	-0.921 *	-0.767 *
	Kurtosis	10.265 *	3.146 *	1.187 *	1.698 *	14.725 *	4.747 *	1.969 *
	JB	1090.425 + 118.673 +	19.219 +	39.669 +	2473.406 + 266.794 +	64.107 +
	Q(12)	15.571	10.718	22.025 ++	44.915 +	36.192 +	7.782	13.802
	ARCH(12)	49.042 +	25.773 ++	15.182	33.243 +	50.375 +	6.901	9.169
	Panel B: Autocorrelations
	Retard	Argentina	Brazil	Chile	Colombia	Mexico	Venezuela	World
	1	-0.054	-0.007	0.198 *	0.353 *	0.284 *	-0.005	0.0411
	2	-0.063	0.022	-0.045	0.102 ***	-0.074	0.107 ***	-0.033
	3	0.072	-0.087	-0.096	-0.004	-0.155 *	0.023	0.0213
	4	-0.097	-0.061	0.009	-0.014	-0.079	-0.022	-0.056
	5	-0.074	-0.011	0.077	0.108 ***	0.049	0.046	0.082
	6	0.089	0.003	-0.003	0.088	-0.079	-0.015	0.0245
	Panel C: Unconditional correlations of it r (in percentage)
		Argentina	Brazil	Chile	Colombia	Mexico	Venezuela	World
	Argentina	1.00	11.171	20.053	7.001	26.782	14.571	11.154
	Brazil		1.00	31.724	16.317	20.995	2.220	33.820
	Chile			1.00	23.777	43.345	9.518	35.762
	Colombia				1.00	13..765	20.772	14.942
	Mexico					1.00	9.164	40.883
	Venezuela						1.00	8.002
	MSCI World							1.00
	Notes:

Table 2 Estimation results

 2 The test for Kurtosis coefficient has been normalized to zero. JB is the Jarque-Bera test for normality based on excess skewness and Kurtosis. Q(12) is the Ljung-Box test for autocorrelation of order 12.ARCH is the Engle (1982)'s test for conditional heteroscedasticity. *, ** and *** indicate significance of coefficients at the 1%, 5% and 10% respectively. +, ++ and +++ indicate rejection of the null hypotheses of no autocorrelation, normality and homocedasticity at the 1%, 5% and 1% levels of significance respectively for statistical tests.

	Panel A -parameter estimates-mean equations
				Argentina	Brazil	Chile	Colombia	Mexico	Venezuela	World
	δ δ δ	0 1 2	0.024 * (0.009) 0.030 (0.086) (0.285) -0.163	0.023 * (0.010) 0.0299 (0.074) (0.287) -0.048	0.018 * (0.005) 0.041 (0.061) (0.108) 0.122	0.010 *** (0.005) 0.283 * (0.062) (0.106) 0.351 *	0.020 * (0.007) 0.089 (0.063) (0.143) -0.017	0.001 (0.013) 0.070 (0.092) (0.221) 0.093	0.010 * (0.003) -0.008 (0.067) -
	Panel B: parameter estimates-GARCH process
				Argentina	Brazil	Chile	Colombia	Mexico	Venezuela	World
	ϖ	0	0.001	0.0010	0.000	0.001 **	0.001 ***	0.019 *	0.000
	ϖ	1	(0.001) 0.218 **	(0.000) 0.167 *	(0.000) 0.045	(0.000) 0.145 *	(0.001) 0.231 *	(0.006) 0.221	(0.000) 0.079 *
	ϖ		2	(0.110) 0.771 *	(0.057) 0.811 *	(0.031) 0.876 *	(0.042) 0.703 *	(0.072) 0.695 *	(0.264) -0.097	(0.034) 0.885 *
	α	(0.085) 0.033 *	(0.041)	(0.058)	(0.089)	(0.059)	(0.103)	(0.039)
	β		(0.006) 0.961 *
				(0.005)
	Panel C -Robust tests for model standardized residuals
				Argentina	Brazil	Chile	Colombia	Mexico	Venezuela	World
			Mean	-0.052	-0.075	-0.020	-0.001	-0.033	0.014	-0.022
			Std-Deviation	1.033	1.018	1.016	0.967	1.038	0.970	0.996
			Skewness	0.601 *	-0.665 *	-0.318 **	0.288	-1.377	-0.998 *	-0.806 *
			Kurtosis	3.951 *	2.134 *	1.089 *	1.248 *	4.925 *	4.599 *	2.552 *
			JB Q(12)	174.803 * 2.937	64.820 * 6.654	16.308 * 17.235	19.356 * 9.496	326.38 * 18.974 ***	257.691 * 9.281	93.397 * 13.286
			ARCH(12)	6.462	10.262	14.842	5.303	3.708	7.628	5.804
	Notes: Bollerslev and Wooldridge (1992)'s robust standard errors are given in parentheses.	0 ϖ , 1 ϖ and	2 ϖ refer to
	the parameters of a GARCH(1,1) process.

Table 3 Results of the Bai-Perron's test for multiple structural breakpoints in the conditional correlations with the world market

 3

	Market	Number of breakpoints	Estimated break dates	95% confidence intervals
				for break dates
	Argentina	5	1989:07
			1992:07
			1995:07
			1998:09
			2002:08

Table 4 Stock market liberalization dates Market Official dates of liberalization Date of the first ADR introduction Date of the first Coun- try Fund introduction

 4

Table 5 Changes in conditional correlations: tranquil versus crisis periods

 5

		Argentina	Brazil	Chile	Colombia	Mexico	Venezuela
	Brazil	0.323
		(23.233)
	Chile	0.252	0.187
		(26.172)	(37.699)
	Colombia	0.120	0.073	0.140
		(18.622)	(14.899)	(5.433)
	Mexico	0.237	0.304	0.108	0.018
		(34.825)	(41.987)	(15.392)	(3.413)
	Venezuela	0.058	0.292	0.302	0.080	0.123
		(14.818)	(23.158)	(51.614)	(4.972)	(23.158)
	MSCI World	0.318	0.097	0.203	-0.031	0.065	0.210
		(26.044)	(14.722)	(19.622)	(-4.546)	(14.547)	(14.918)

We effectively attempted to estimate our model with the in-mean volatility effects in the return generating processes and the results are consistent with our intuition and Lundblad (2007)'s findings.