C Javaux Léger

K Mishchik

O Dematteo-Caulier

S Skupin

B Chimier

G Duchateau

A Bourgeade

C Hoenninger

E Mottay

J Lopez

R Kling

Effects of burst mode on transparent materials processing

Keywords: ultrafast laser, burst, transparent material, welding, cutting, filamentation

We investigated the effect of burstmode with nanosecond (ns) time delay between subpulses on sodalime glass volume machining. We observed in tight focusing configuration that the use of burstmode with ns time delay between subpulses does not increase the absorption efficiency and does not bring a significant effect on the heat affected zone diameter with respect to single pulse mode. On the contrary in loose focusing configuration the use of burst mode allows increasing the aspect ratio of the heat affected zone without extra energy absorption. This effect is highly interesting for filamentation glass cutting applications.

INTRODUCTION

In recent years, transparent materials such as glass, quartz, sapphire and others are more and more used for mass markets such as smart-electronics, optoelectronics or watchmaking. However conventional mechanical glass processing techniques do not provide full satisfaction in terms of speed, throughput and quality. Indeed these techniques often introduce side and detrimental effects such as residual stresses or crack formation (chipping). Therefore, there is a room for improvement and lasers have a role to play, especially ultrafast lasers since they have the capacity to perform involume processing for waveguide inscription [START_REF] Davis | Writing waveguides in glass with a femtosecond laser[END_REF] , for welding [START_REF] Miyamoto | Internal modification of glass by ultrashort laser pulse and its application to microwelding[END_REF] , for cutting [START_REF] Kumkar | Comparison of different processes for separation of glass and crystals using ultrashort pulsed lasers[END_REF] or for midrodrilling [START_REF] Esser | Time dynamics of burst-train filamentation assisted femtosecond laser machining in glasses[END_REF][START_REF] Rezaei | Burst-train generation for femtosecond laser filamentation-driven micromachining[END_REF] for instance.

Glass welding by ultrafast laser irradiation is based on a localized melting at the interface between two glass substrates and a subsequent resolidification, which leads to a strong and permanent bonding. The bonding energy increases with the relative molten area [START_REF] Alexeev | Characterization of shear strength and bonding energy of laser produced welding seams in glass[END_REF][START_REF] Richter | Ultrashort pulse laser welding -a new approach for high-stability bonding of different glasses[END_REF] . This application usually requires tight focusing although the welding is also possible with a filament beam shaping [START_REF] Hélie | Reinforcing a Direct Bond between Optical Materials by Filamentation Based Femtosecond Laser Welding[END_REF] . In tight focusing conditions, the modification morphology has a comet-like shape with two embedded structures. The inner one corresponds to the laser photoexcited volume, meanwhile the outer one is directly related to the heat affected zone (HAZ) due to heat accumulation at high repetition rate, typically above 100 kHz [START_REF] Shimizu | Mechanism of heat-modification inside a glass after irradiation with high-repetition rate femtosecond laser pulses[END_REF] . Composition changes in both inner and outer regions have been also reported [START_REF] Shimizu | Formation mechanism of element distribution in glass under femtosecond laser irradiation[END_REF][START_REF] Sakakura | Condensation of Si-rich region inside soda-lime glass by parallel femtosecond laser irradiation[END_REF] . Extend of HAZ increases with pulse energy and repetition rate [START_REF] Miyamoto | Internal modification of glass by ultrashort laser pulse and its application to microwelding[END_REF] .

Burstmode is also an option that must be considered. However the effect of burstmode on laser-dielectric materials interaction highly depends on the delay between subsequent pulses in the burst (called thereafter subpulses). Indeed, if the delay between subpulses is shorter than the mean cooling time of photoexcited electrons (typically few tens of ps), the burstmode contributes to enhance the laser-matter coupling and enlarge the HAZ diameter as described by Sugioka et al [START_REF] Sugioka | Double-Pulsed Ultrafast Laser Welding of Glasses Toward Enhancement of Process Efficiency[END_REF] . On the other hand, if the delay between subsequent pulses in the burst is longer, the laser-matter interaction does not benefit from burstmode. Indeed on the ns time scale most of the photoexcited electrons have relaxed, and although the irradiated area has not yet returned to room temperature, the HAZ diameter has been shown to become smaller than in single pulse mode [START_REF] Sugioka | Double-Pulsed Ultrafast Laser Welding of Glasses Toward Enhancement of Process Efficiency[END_REF] .

Laser-based glass cutting is also a subject of high interest from an industrial point of view. Ablation cutting is quite slow and produces a lot of dust. Another technique is scribe and break but the main drawback is an uncontrolled path deviation along the sidewall (cut plan). In comparison to these previous techniques, filamentation cutting offers several advantages such as zerokerf, dustfree, smooth and regular cut plan, and rapid processing (several 100 mm/s). Filamentation cutting consists in weakening the glass by producing an elongated modification along the glass thickness as a precutting step, and then in releasing the two parts thanks to a mechanical pressure [START_REF] Kumkar | Comparison of different processes for separation of glass and crystals using ultrashort pulsed lasers[END_REF] . The filament is produced at high laser intensity, in specific focusing conditions, and results from equilibrium between self-focusing (Kerr effect) and defocusing when the intensity overcomes the critical power [START_REF] Esser | Time dynamics of burst-train filamentation assisted femtosecond laser machining in glasses[END_REF] . The filament may propagate the laser beam energy on mmlong distance. The effect of burstmode with ns-delay on filament morphology has been investigated by Esser and Rezaei [START_REF] Esser | Time dynamics of burst-train filamentation assisted femtosecond laser machining in glasses[END_REF][START_REF] Rezaei | Burst-train generation for femtosecond laser filamentation-driven micromachining[END_REF] . They have shown that burstmode contributes to create a filament, which may act as a waveguide for subsequent high aspect ratio microdrilling in dielectric materials.

In this paper we report on the influence of burstmode with a ns-delay on the heat affected zone morphology while focusing a femtosecond laser beam inside sodalime glass. The aim is to improve glass welding on one hand, and filamentation cutting on the other hand. The results will be compared to single pulse processing and discussed in terms of HAZ diameter, HAZ length and nonlinear absorptivity. We have demonstrated that burstmode with a ns-delay is useless for glass welding since it does not enlarge the HAZ, whereas it is useful for filamentation cutting since it contributes to extend the length of the filament for a same level burst energy.

EXPERIMENTAL SETUP

For our experiments we used a Satsuma Yb-doped fiber femtosecond laser from Amplitude Systemes, emitting at 1030 nm with pulse duration down to 300 fs and with repetition rate varying from 500 kHz to 2 MHz. Its output power is 5 W and the maximal output pulse energy varies from 2.5 µJ to 10 µJ. The laser can be used in single pulse mode as well as in burstmode (schematized on Fig. 1a). The delay between subpulses in the burst is defined by the oscillator repetition rate, which is of 40 MHz leading to a time delay of 25 ns. It is thereafter referred to as 25 ns-burstmode. The number of subpulses in the burst can be tuned from 1 (single pulse mode) to 10. It is commonly referred to as pulses per burst (PPB). The energy E p of the burst (sum of the energies of the subpulses) is equal to the energy of the pulse in single pulse mode. The experimental setup used for this study is presented in Figure 1b. It included the laser, a half-wave plate combined with a polarizer cube to control the pulse energy, a beam expander, an inverted microscope (Olympus, Eclipse) equipped with a microscope objective (Mitutoyo, M Plan APO NIR) to focalize the laser beam on the sample, and a XYZ motorized stages system (Aerotech, A3200) with a specific sample holder for positioning the sample under laser beam. We used three objectives successively in this study with a magnifying power of 20x, 10x and 5x respectively. Taking into account the beam diameter on the entrance pupil of the objectives, the effective used numerical apertures (NA) were 0.4, 0.26 and 0.08 respectively. When using 0.4 NA the setup is said to be in tight focusing configuration, and when using 0.26 or 0.08 NA it is said to be in moderate and loose focusing configurations respectively. The glass samples were commercially available 1 mm-thick sodalime microscope cover slides (Duran Group). We used the CCD camera to localize the sample surface and then the laser beam was focalized 250 µm deep in the glass volume.

We speak about experiments in static mode when the laser beam was focalized into the sample at a fixed position for 10,000 pulses. We speak about experiments in dynamic mode when the beam was scanned over the sample at constant speed. The number of equivalent pulses per point of irradiation is calculated taking into account the scanning velocity, the repetition rate and the interaction volume diameter. In order to be close to the static mode with 10,000 pulses per fixed position, we used a scanning speed of 5 mm•s -1 . The dynamic configuration allowed also the absorption measurements using a power meter measuring the time-averaged transmitted power through the glass sample during laser beam scanning the sample. Assuming reflection and scattering from the laser induced plasma are negligible the nonlinear absorptivity is given by [START_REF] Miyamoto | Internal modification of glass by ultrashort laser pulse and its application to microwelding[END_REF] , where Pi is the incident laser power, Pt the transmitted laser power through the glass sample and R the Fresnel reflectivity.

€ A = 1 - P t P i 1 (1 -R)
The HAZ in the glass after laser irradiation can be observed either from the laser incident direction (called top view), or perpendicularly to the incident laser beam such that the observed HAZ is composed of the superposition of the heat affected zones generated by the laser scanning (called side view). The sample characterization was made with an optical measurement microscope (Mitutoyo, MF-A1010H) coupled with a QM-Data 200 calculator, using 10x or 20x objectives.

RESULTS AND DISCUSSION

Burstmode influence on HAZ diameter for glass welding application

The bonding strength of a welding seam is linked to the absorption efficiency of the laser beam at the interface of the glass samples. When the absorption is more efficient, the HAZ is wider, and the strength of the joint is higher [START_REF] Alexeev | Characterization of shear strength and bonding energy of laser produced welding seams in glass[END_REF][START_REF] Richter | Ultrashort pulse laser welding -a new approach for high-stability bonding of different glasses[END_REF] . In order to evaluate the interest of burstmode on glass welding, we studied the evolution of the glass absorption efficiency when using the burstmode within tight focusing configuration (NA 0.4). To this aim we irradiated the sodalime glass sample in static mode varying the number of PPB from 1 to 5, for repetition rate varying from 2 MHz down to 250 kHz. The resulting material modifications observed by optical microscopy are reported on figure 2. ) and its incidence is perpendicular to the plane of the picture. Each HAZ was irradiated by 10,000 pulses in static configuration. The average power available on target was 500 mW for each repetition rate. The target pulse energy was 0.25, 0.5, 1 and 2 µJ for 2 MHz, 1MHz, 500 kHz and 250 kHz respectively.

Figure 2 shows the influence of the number of PPB on the HAZ diameter for repetition rates of 2 MHz, 1 MHz, 500 kHZ and 250 kHz. For each repetition rate the target power is kept constant and fixed at 500 mW. We observe that the HAZ is always composed of two concentric circular zones. The inner circular zone corresponds to the laser photoexcited volume and the outer zone to the molten HAZ as reported by Shimizu et al [START_REF] Shimizu | Mechanism of heat-modification inside a glass after irradiation with high-repetition rate femtosecond laser pulses[END_REF] . When the glass sample is irradiated in single pulse mode (1 PPB), the extent of the HAZ increases when the laser repetition rate decreases. Since the target power is kept constant when varying the repetition rate, a decrease of the repetition rate induces an increase of the target pulse energy. The results observed here are thus consistent with respect to previous published results [START_REF] Miyamoto | Internal modification of glass by ultrashort laser pulse and its application to microwelding[END_REF] . At each fixed number of PPB, from 2 to 5, the HAZ diameter also varies inversely to the repetition rate since the subpulses energy is higher when the repetition rate decreases. The measured values of the HAZ diameter with respect to the number of PPB for each repetition rate are reported on the graph of figure 3. At a fixed repetition rate, we observe a decrease of the diameter of the HAZ for higher numbers of PPB. This result is in accordance with the paper published by S. Wu et al [START_REF] Wu | Characterization and mechanism of glass microwelding by double-pulse ultrafast laser irradiation[END_REF] . As mentioned above, they demonstrated that the effect of doublepulse ultrasfast laser irradiation was linked to the delay time between the two subpulses. They reported that the HAZ becomes smaller than for single pulse mode for a time delay between sub-pulses beyond some tenth of ns, which is consistent with our 25 ns delay time burstmode.

We also measured the absorption of the laser beam induced by the laser-glass interaction while varying the number of PPB. Figure 4 reports the absorption measurements for 1 to 5 PPB and for repetition rates going from 500 kHz to 2 MHz. The average power available on target was still kept at 500 mW for each repetition rate. In single pulse mode, the absorption increases when the repetition rate decreases, in accordance with the HAZ size. Indeed the increase of pulse energy leads to stronger multiphoton absorption, and therefore to larger HAZ. For each repetition rate, the glass nonlinear absorptivity appears to decrease while increasing the number of PPB. This result is also consistent with the HAZ diameter measurements. When using the burstmode at constant pulse energy, the subpulses energy decreases with the number of PPB. Since after 15 ns most of the photoexcited electrons have relaxed, the use of burstmode only induces a decrease of the pulse energy and thus of the nonlinear absorption. A laser with ns-burstmode is therefore not adapted to address glass welding application. Figure 4. Absorption efficiency of the intravolume interaction versus the number of PPB. The measurements were done in dynamic configuration at 5 mm•s -1 . The average power available on target was 500 mW for each repetition rate. The target pulse energy was 0.25, 0.5, 1 and 2 µJ for 2 MHz, 1MHz, 500 kHz and 250 kHz respectively. The data are underlined with spline fits.

Burstmode influence on filament length for glass (pre)cutting application

Then we investigated the interest of using ns-burstmode for filamentation. Figure 5 shows the influence of PPB on the HAZ morphology for moderate (NA 0.26) and loose (NA 0.08) focusing configurations for static mode glass irradiation. a. The target power was 1.1 W, the pulse duration was 380 fs, the repetition rate was 500 kHz, and each HAZ was created by irradiation of 500 successive pulses. b. The target power was 1.5 W, the pulse duration was 300 fs, the repetition rate was 500 kHz, and each HAZ was created by irradiation of 1,000 successive pulses.

With 0.26 NA in single pulse mode, we observe that the material modification present a comet-like shape with a laser photoexcited zone surrounded with the molten HAZ zone as in the case of tight focusing [START_REF] Shimizu | Mechanism of heat-modification inside a glass after irradiation with high-repetition rate femtosecond laser pulses[END_REF] . The tail of the comet is localized on the geometrical focus position, and its front delimitation is positioned closer to the incident glass surface due to the optical Kerr effect. When increasing the PPB, the two embedded zones are still visible, however the aspect ratio of the HAZ increases. For 3 and 5 PPB the width of the HAZ decreases while its length increases. The front delimitation moves back towards the geometrical focus position for higher numbers of PPB because a diminution of the subpulses energy reduces the optical Kerr effect. The elongation of the HAZ follows the laser propagation direction, which indicates a better energy propagation efficiency.

In loose focusing configuration (0.08 NA), the material modification no longer has a comet shape, but rather looks like a filament. The HAZ is much thinner and longer with respect to higher NA. The filament is composed of a head, in front of the geometrical focus position, and a tail, which goes far beyond. We observe that the filament also lengthens when the number of PPB increases up to 3 and then decreases down to the size of the filament obtained in single pulse mode or even below. As for the moderate focusing configuration, the front delimitation of the glass modified zone appears deeper when the number of PPB increases up to 4, due to lower optical Kerr effect. For 5 PPB, we have no explanation concerning the position of the filament front delimitation. Figure 6 reports the observation of the filaments obtained in dynamic mode at velocity 5 mm•s -1 , in order to enhance the contrast of the tail of the filament. The length of the whole HAZ can thus be evaluated more precisely. The measured lengths are reported on the graph of figure 7. Figure 6. Side view optical microscopy observation of the HAZ created in the glass samples for moderate (NA 0.26) and loose (NA 0.08) focusing configurations in dynamic mode. The target power was 1.5 W, the pulse duration was 300 fs, the repetition rate was 500 kHz, and each HAZ was created with translation velocity 5 mm•s -1 .

Additional insights give us the influence of the pulse energies on the laser affected zone length in the burstmode irradiation. The results for four different target pulse energies are presented in figure 7. For pulse energies below 2.4 µJ the length of the generated filament decreases with the number of PPB. For higher energies available on target, the behavior alters. The use of burstmode interestingly generates longer filaments. The longest HAZ is obtained for 3 PPB for a burst energy of 3.1 µJ and 3.8 µJ. The filament reaches a length of around 950 µm and 1200 µm for 3.1 µJ and 3.8 µJ respectively, that is 200 µm longer compared with single pulse mode for both pulse energies. We then measured the absorption efficiency of the laser-glass interaction in loose focusing configuration for 1 to 5 PPB and for varying target energy. Figure 8 shows the influence of the number of PPB on the energy absorption in the sample for target energy varying from 1.6 µJ to 3.8 µJ. We observe that whatever the target energy the absorption efficiency decreases when the number of PPB increases. Consequently the use of burstmode in loose focusing configuration allows increasing the filamentation length and thus the energy propagation without extra energy absorption. This striking result is highly promising for filamentation glass cutting processes where the point is to generate filaments as long as possible. Single pass full cutting would require higher energy, but in present case the filament generation could be used as a precutting step. It would weaken the glass and generate mechanical stress along a predefined line, as required for cutting.

Transient waveguiding effect

Considering the results presented in the previous sections, we would like to focus on the particular filamentation regime realized in the burstmode. Despite the overall absorbed energy is lower, the filaments are longer. Moreover, longer filaments, which are caused by strongly nonlinear processes, are produced with the pulses having much lower peak intensity compared with convenient irradiation with single pulses. Apparently, the transient changes in electronic structure and structural modifications of the glass produced by the first subpulse strongly affect nonlinear propagation of the subsequent subpulses. Due to small time interval between the subpulses in the burst, relaxation processes have not finished yet and such transient changes are especially effective. Schematically these relaxation processes are illustrated on the Figure 9, where we present the characteristic times of the different steps of the light-matter interaction at high intensity in glasses. Among others we selected the factors, which are non-negligible when the next subpulse arrives and which can significantly influence propagation of the subsequent pulse as reported in the literature. These factors are residual electrons in the conduction band [START_REF] Wu | Characterization and mechanism of glass microwelding by double-pulse ultrafast laser irradiation[END_REF] , changes of nonlinear susceptibility directly causing nonlinear propagation [START_REF] Zoubir | Optical properties of infrared femtosecond laser-modified fused silica and application to waveguide fabrication[END_REF][START_REF] Royon | Evolution of the linear and nonlinear optical properties of femtosecond laser exposed fused silica[END_REF] , refractive index changes induced by the presence of the permanent defects [START_REF] Royon | Evolution of the linear and nonlinear optical properties of femtosecond laser exposed fused silica[END_REF][START_REF] Streltsov | Study of femtosecond-laser-written waveguides in glasses[END_REF] and thermally induced variation of the refractive index [START_REF] Saito | Absorption edge in silica glass[END_REF] . Most of these factors, however, lead to mainly light defocusing or negligibly affects nonlinear propagation. On the contrary, thermal-induced variation of refractive index can produce substantial index variation and deserves more attention.

In our case, when the laser works at 500 kHz, the absorbed energy has not completely dissipated before the following burst arrives. In the long run this causes heat accumulation and gradual pulse-to-pulse temperature rise [START_REF] Eaton | Heat accumulation effects in femtosecond laser-written waveguides with variable repetition rate[END_REF][START_REF] Richter | Ultrastable bonding of glass with femtosecond laser bursts[END_REF] . In the burst mode, the laser oscillator frequency is 40 MHz so the time delay between subpulses is 25 ns. At such short time scale heat diffusion processes have even less time to evacuate the absorbed energy. As a consequence the lattice temperature, which is subjected to sufficient local rise in the focal zone [START_REF] Richter | Ultrastable bonding of glass with femtosecond laser bursts[END_REF] , induces a bandgap collapse [START_REF] Saito | Absorption edge in silica glass[END_REF] . The main factor that determines the collapse of glass bandgap is the thermal vibrations rather than the structural disorder. Therefore the heatinduced bandgap changes follow the temperature increase almost instantly after 50-100fs (a time characteristic of lattice vibrations). This leads to a transient positive refraction index variation estimated at the order of 0.02 at the laser wavelength. On the other hand, the temperature increase of the material induces a volume expansion. This density variation leads to a decrease of the refractive index. The volume expansion, however, has different characteristic time, which is given by the viscoelastic relaxation τ s = η(T)/G, where η is the viscosity and G is the bulk modulus of sodalime glass. Considering a temperature close to the glass vitreous transition where the material is expected to be significantly modified, the volume expansion characteristic time lies in the range of hundreds of ns. Therefore we suggest that a competition between both the increase and decrease of the refractive index in the laser-material interaction zone may result in an overall positive index variation. This variation forms a transient focusing lens, leading to a transient waveguiding effect. The energy deposition zone thus elongates, favoring the filamentation phenomenon without extra energy absorption. This explanation is consistent with our experimental data, however it needs additional experimental confirmation supported by accurate theoretical simulations.

CONCLUSION

We studied the effect of burstmode with 25 ns time delay on sodalime glass volume machining. We investigated primarily the influence of numerical aperture by comparing material modifications induced in single pulse or in burstmode regime. In tight focusing configuration the use of burstmode did not show any positive effect. Ns-bursts lasers are not ideal candidates for welding processes. On the contrary, in loose focusing configuration the burstmode with nsdelay time proved to be highly interesting. It increases drastically the HAZ aspect ratio and therefore the filamentation efficiency without extra energy needed. We proposed a thermal effects-based explanation for this observation, named a transient waveguiding effect. Further experimental work such as pump-probe measurements would allow verifying this hypothesis. This effect is promising for glass cutting processes at least for the filament precutting step. The optimization of the subpulses delay time would probably even allow to increase this waveguiding effect and thus the filament length.

Figure 1 .

 1 Figure 1. a Schematic representation of the burstmode.. b. Schematic representation of the experimental setup. The energy control module is composed of a half-wave plate and a polarizing beam splitter cube.

Figure 2 .

 2 Figure 2. Top view optical microscopy observation of the HAZ created in the glass sample for 1 to 5 PPB and repetition rate from 2 MHz down to 250 kHz. The focusing laser beam has a numerical aperture of 0.4 (objective 20x) and its incidence is perpendicular to the plane of the picture. Each HAZ was irradiated by 10,000 pulses in static configuration. The average power available on target was 500 mW for each repetition rate. The target pulse energy was 0.25, 0.5, 1 and 2 µJ for 2 MHz, 1MHz, 500 kHz and 250 kHz respectively.

Figure 3 .

 3 Figure 3. Measurement of the diameter of the HAZ shown on figure 2 with respect to the number of PPB. The data are underlined with spline fits.

Figure 5 .

 5 Figure 5. Side view optical microscopy observation of the HAZ created in the glass samples for moderate (NA 0.26) and loose (NA 0.08) focusing configurations in static mode. The laser is incident on the glass sample from the left of the picture.a. The target power was 1.1 W, the pulse duration was 380 fs, the repetition rate was 500 kHz, and each HAZ was created by irradiation of 500 successive pulses. b. The target power was 1.5 W, the pulse duration was 300 fs, the repetition rate was 500 kHz, and each HAZ was created by irradiation of 1,000 successive pulses.

Figure 7 .

 7 Figure 7. Measurement of the filament length according to experiments in loose focusing configuration and dynamic mode reported on figure 6 with respect to the number of PPB, at 500 kHz. The data are underlined with spline fits.

Figure 8 .

 8 Figure 8. Measurement of the absorption efficiency versus the number of PPB at 500 kHz. The data are underlined with spline fits.

Figure 9 .

 9 Figure 9. Schematic representation of light-matter interactions at high intensity versus time in sodalime glass.

ACKNOWLEDGEMENTS

We acknowledge the European Commission, the French Ministry of Industry and the Aquitaine Regional Council for support and funding.


Anne Péguin-Feissolle

Bilel Sanhaji
email: bilel.sanhaji@univ-amu.fr

Testing the Constancy of Conditional Correlations in Multivariate GARCH-type Models (Extended Version with Appendix)

Keywords: Multivariate GARCH, Neural network, Taylor expansion JEL classification: C22, C45, C58 1

We introduce two multivariate constant conditional correlation tests that require little knowledge of the functional relationship determining the conditional correlations. The first test is based on artificial neural networks and the second one is based on a Taylor expansion of each unknown conditional correlation. These new tests can be seen as general misspecification tests of a large set of multivariate GARCH-type models.

We investigate the size and the power of these tests through Monte Carlo experiments. Moreover, we study their robustness to non-normality by simulating some models such as the GARCH -t and Beta -t -EGARCH models. We give some illustrative empirical examples based on financial data.

Introduction

During the last decade, there has been a rapid increase in the amount of literature on theoretical and empirical derivatives of multivariate GARCHtype modeling (see, for example, extensive surveys of Bauwens, Laurent and Rombouts 2006 and Silvennoinen and Teräsvirta 2009a). This econometric modeling is an important issue, particularly in finance. Indeed, it is largely accepted that empirical models are more relevant in a multivariate than an univariate framework. By taking into account the possible dependence between individual equations, one can determine for example option pricing, asset pricing, hedging and risk management or portfolio selection, as noted in [START_REF] Bauwens | Multivariate GARCH models: a survey[END_REF]. For instance, a crucial problem in financial studies is the correlation structure among different national stock returns because it determines the gains from international portfolio diversification. Although few tests are specific to multivariate models, it would be desirable to check ex ante the properties of the data especially since estimating multivariate GARCH (MGARCH) models is time-consuming. Moreover, it is worth noting that, if we don't test first the hypothesis of the constancy of conditional correlations, the estimation of some multivariate GARCH-type models could lead to estimation issues because some parameters could not be identified.

Most of the tests for the constancy of conditional correlations are based on the Lagrange Multiplier (LM) procedure. Some other tests are based on Likelihood Ratio procedure (see, among others, Longin and Solnik 1995 and [START_REF] Engle | Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH[END_REF]. But, as noted by [START_REF] Tse | A test for constant correlations in a multivariate GARCH model[END_REF], computing the LM test statistics requires only the estimation of the constant correlation model and is thus generally computationally convenient. [START_REF] Tse | A test for constant correlations in a multivariate GARCH model[END_REF] introduces a LM test for the constant correlation hypothesis in a multivariate GARCH model; he extends the constant correlation model to one in which the correlations are allowed to be time-varying and he tests for the zero restrictions on the key parameters. [START_REF] Bera | Testing constancy of correlation and other specifications of the BGARCH model with an application to international equity returns[END_REF] apply the [START_REF] White | Maximum likelihood estimation of misspecified models[END_REF] and [START_REF] Orme | The small sample performance of the information matrix test[END_REF] information matrix (IM) test to the constant correlation bivariate GARCH model and establish the linkage between the IM and score tests. (2000). In contrast to [START_REF] Silvennoinen | Modelling conditional correlations of asset returns: A smooth transition approach[END_REF], their model is bivariate and the variable controlling the transition between the extreme regimes is simply the time. [START_REF] Silvennoinen | Modeling Multivariate Autoregressive Conditional Heteroskedasticity with the Double Smooth Transition Conditional Correlation GARCH Model[END_REF] present a test for the constancy of correlations against the DSTCC-GARCH (double smooth transition conditional correlation GARCH) model, i.e. by allowing the conditional correlations to vary according to two transition variables that can be stochastic or deterministic (for example, exogenous variables or lagged elements of endogenous variables).

In this paper, we introduce two LM tests for constancy of the conditional correlations.

The first test is based on artificial neural networks (ANN) or, more precisely, on single hidden layer perceptrons (later on, we use the generic term of artificial neural networks, following the terminology employed in the econometric literature). This ANN-based test relies on a statistical technique proposed by Lee, [START_REF] Lee | Testing for neglecting nonlinearity in time series models[END_REF]. The ANN framework has been already used for some tests: see, for example, Lee, [START_REF] Lee | Testing for neglecting nonlinearity in time series models[END_REF] and [START_REF] Teräsvirta | Power of the neural network linearity test[END_REF] for linearity tests, [START_REF] Kamstra | A neural network test for heteroscedasticity[END_REF], [START_REF] Caulet | Un test d'hétéroscédasticité conditionnelle inspiré de la modélisation en termes de réseaux neuronaux artificiels[END_REF] and Péguin-Feissolle (1999) for conditional heteroscedasticity tests, [START_REF] Lebreton | Robust tests for heteroscedasticity in a general framework[END_REF] for heteroscedasticity test and Péguin-Feissolle and [START_REF] Péguin-Feissolle | A general framework for testing the Granger noncausality hypothesis[END_REF] for causality tests.

The second test we introduce is based on a Taylor expansion of each unknown conditional correlation around a given point in a sample space; it is a way of linearizing the testing problem by approximating the true relationships by a Taylor series expansion. Thus, because of the linearization of the unknown relationship determining each conditional correlation, this test is not computationally more difficult to carry out than traditional tests. This kind of test has already been introduced in order to test the causality (Péguin- Therefore, the two tests present four fundamental characteristics. First, they require little knowledge of the functional relationship determining the correlations. Secondly, they are easy to implement and perform well in our small-sample simulations; indeed, we show in the simulations that they are rel-evant in small samples and can be useful in investigating potential time-varying conditional correlation. Thirdly, they generalize well to high dimensions, i.e. with a high number of endogenous variables. Fourthly, given the properties of the ANN to be universal approximators and given that the Taylor expansion permits the linearization of an unknown relationship, a rejection of the null hypothesis of constancy of conditional correlations does not imply that the data have been generated from a model where the conditional correlations are specified as neural functions or specific functions; these tests can thus be seen as general misspecification tests of very different multivariate GARCH-type models. It is worthwhile to remark too that these tests could be used easily to test also the hypothesis of partially constant correlations.

Finite-sample properties of the two new tests are examined using Monte

Carlo methods by comparing them to two alternative conditional correlation tests: the tests of [START_REF] Tse | A test for constant correlations in a multivariate GARCH model[END_REF] and [START_REF] Silvennoinen | Modelling conditional correlations of asset returns: A smooth transition approach[END_REF]. Using a variety of different specifications for the MGARCH model, we show that they perform well in our small-sample simulations; they approximate quite well unknown specifications of the conditional correlations, even in the case where the normality hypothesis is not verified. Empirical illustrations using real data point out that these tests can be useful to reject the constant correlation hypothesis in multivariate modelling of conditional heteroscedasticity.

The paper is organized as follows. In Section 2, we introduce the ANNbased constant correlation test and the Taylor-expansion based test. Section 3 reports the results of a simulation study; we investigate the size and the power by Monte Carlo experiments in small samples. In Section 4, we study the robustness of the tests to non-normality by simulating some models such as the GARCH -t model of [START_REF] Bollerslev | A conditionally heteroskedastic time series model for security prices and rates of return data[END_REF], or the Beta -t -EGARCH model of [START_REF] Harvey | EGARCH models with fat tails, skewness and leverage[END_REF] and [START_REF] Harvey | Beta-t-(E)GARCH[END_REF].

Section 5 describes some illustrative empirical examples based on financial data. Section 6 contains some conclusions. Technical derivations of the test statistics presented in the paper can be found in the Appendix.

Testing the Constancy of Conditional Correlations

In order to fix the model that we consider and the notations, we present a general MGARCH model. Let {y t } be a multivariate time series, where y t is a N × 1 vector. Consider the following model defined by, for t = 1, ..., T :

y t = E[y t |Ω t-1 ] + ε t V ar [y t |Ω t-1 ] = H t (1)
where E[y t |Ω t-1 ] and V ar [y t |Ω t-1 ] are respectively the conditional expectation and the conditional covariance of y t with respect to Ω t-1 , the sigma-field generated by all the information until time t -1. The process y t is strictly stationary and ergodic. To simplify the discussion (but the test presented in this paper could be generalized easily to a large variety of time-varying structures of the conditional expectation of y t ), we assume that the observations are of zero means: for t = 1, ..., T, E t-1 [y t ] = 0 or

y t = ε t ; ( 2
)
ε t is defined by

ε t = H 1/2 t η t (3)
with η t ∼ iid (0, I N ). We assume for simplicity that the conditional variances follow a GARCH(1,1) process:

h iit = ζ i + α i ε 2 i,t-1 + β i h ii,t-1 i = 1, ...N (4)
where the standard positivity and covariance stationarity constraints are imposed, i.e. ζ i > 0, α i ≥ 0, β i ≥ 0, and α i + β i < 1 for i = 1, ..., N . Moreover, the conditional covariances are:

h ijt = ρ ijt h iit h jjt 1 ≤ i < j ≤ N (5)
where ρ ijt is the conditional correlation. Let P t = (ρ ijt ) i,j=1,...,N be the N × N conditional correlation matrix for the ε t ; we can write

H t = S t P t S t (6)
where S t is the N × N matrix given by

S t = diag( h 11t , ..., h N N t ). ( 7
)
We assume that the conditional correlation matrix P t is positive definite at each t; it guarantees the positive definiteness of H t . Moreover, we will assume the conditional normality of the ε t :

ε t |Ω t-1 ∼ N (0, H t ) (8)
which implies

z t |Ω t-1 ∼ N (0, P t ) , (9)
with z t = S -1 t ε t .

The ANN-based test

The first test we present is an artificial neural network (ANN) based LM-type test. In order to build the test, we extend here the ideas of the tests of constant correlation hypothesis of [START_REF] Tse | A test for constant correlations in a multivariate GARCH model[END_REF], Silvennoinen andTeräsvirta (2014, 2009b) and [START_REF] Berben | Comovement in international equity markets: A sectoral view[END_REF] Therefore, the test can be seen as a general misspecification test of a large set of multivariate GARCH-type models. Because the ANN are universal ap-proximators, a rejection of the null hypothesis of constancy of conditional correlations does not imply that the data have been generated from a model where the conditional correlations are specified as neural functions.

We are going to specify the time-varying structure of the conditional correlations. We assume that the conditional correlations ρ ijt are changing smoothly over time depending on a neural function as follows

ρ ijt = ρ ij + p m=1 δ ijm (1 + exp{-w ijt γ ijm }) -1 (10)
where 1 ≤ i < j ≤ N , p < ∞, w ijt and γ ijm are (2q + 1) × 1 vectors, and w ijt is given by

w ijt = (1, w ijt ) = (1, ε i,t-1 , ..., ε i,t-q , ε j,t-1 , ..., ε j,t-q ) . (11)
The conditional correlations ρ ijt should respect the conditions |ρ ijt | ≤ Following the definition [START_REF] Carroll | Construction of neural nets using the Radon transform[END_REF] of the conditional correlations, the null hypothesis of constant conditional correlation, i.e., ρ ijt = ρ ij , can be formulated as:

H 01 : δ ijm = 0, 1 ≤ i < j ≤ N , 1 ≤ m ≤ p. ( 12
)
Under H 01 , the γ ijm , for i = 1,...,N -1, j = i + 1,..., N and m = 1, ..., p, are not identified. For this reason, the conventional maximum likelihood theory for deriving the test procedure is not applicable, as noted in Lebreton and Péguin-Feissolle (2007). To solve this problem, we use the method given by Lee, White and Granger (1993), i.e. we draw the unidentified parameters randomly from a uniform distribution. Therefore, we generate the hidden unit weights, i.e. each γ ijm , for i = 1,...,N -1, j = i + 1,..., N and m = 1, ..., p, randomly from the uniform [-µ, µ] distribution. In our simulations, following Lee, White and Granger (1993), we choose µ = 2.

The null hypothesis can be tested using the LM procedure. Let θ the vector of all the parameters of the model, i.e. the 3 × 1 parameter vector for the conditional variances, ω i = (ζ i , α i , β i ) for i = 1, ..., N , the conditional correlation ρ ij with 1 ≤ i < j ≤ N and the parameters in the neural function,

δ ijm for 1 ≤ i < j ≤ N , 1 ≤ m ≤ p and p < ∞.
Under standard regularity conditions, the test statistic that we call N EU RAL is given by:

1 T T t=1 ∂l t (θ) ∂θ (θ) -1 T t=1 ∂l t (θ) ∂θ (13)
where (θ) is replaced by the consistent estimator:

(θ) = 1 T T t=1 E t-1 ∂l t (θ) ∂θ ∂l t (θ) ∂θ (14)
and the log-likelihood for the observation t, l t (θ), is given by:

l t (θ) = - N 2 ln (2π) - 1 2 ln (|P t |) - 1 2 N i=1 ln (h iit ) - 1 2 z t P -1 t z t . (15)
In practice (see [START_REF] Tse | A test for constant correlations in a multivariate GARCH model[END_REF], (θ) may be replaced using the negative of the Hessian matrix:

(θ) = - 1 T T t=1 E t-1 ∂ 2 l t (θ) ∂θ∂θ . (16)
The statistic N EU RAL has an asymptotic χ 2 distribution under the null with

p N (N -1)
2 degrees of freedom. Appendix presents the details of the technical derivations of the test statistic.

As Lee, White and Granger (1993) and Péguin-Feissolle and Teräsvirta (1999) pointed out, the elements of some matrices used to build the ANN-based test statistic can lead to collinearity problems when the number of hidden units p is large. Therefore, in our case, the p × 1 vectors g ijt defined as, for 1 ≤ i < j ≤ N , p < ∞:

g ijt =       g ijt,1 . . . g ijt,p       =       (1 + exp{-w ijt γ ij1 }) -1 . . . (1 + exp{-w ijt γ ijp }) -1       (17)
tend to be collinear among themselves especially when p is large. The condi-tional correlations ρ ijt given by ( 10) can be written as follows

ρ ijt = ρ ij + p m=1 δ ijm g ijt,m .
Let us define G ij the p × T matrix given by:

G ij = (g ij1 , ..., g ijT ) .
Thus The null hypothesis is now that the parameters associated to the main principal components are equal to zero in the following specification of the conditional correlations:

ρ ijt = ρ ij + p * ij m=1 δ ijm g * ijt,m .
The statistic N EU RAL will have in this case an asymptotic χ 2 distribution under the null with

N -1 i=1 N j=i+1
p * ij degrees of freedom.

The Taylor expansion-based test

We assume that the functional form f ij determining ρ ijt , for i, j = 1, ..., N and t = 1, ..., T , is unknown and is adequately represented by the following equation:

ρ ijt = f ij w ijt , θ * ij ( 18
)
where, as given in [START_REF] Castle | A low-dimension portmanteau test for non-linearity[END_REF], for each couple of indices (i, j), w ijt is a 2q × 1 vector given by

w ijt = (ε i,t-1 , ..., ε i,t-q , ε j,t-1 , ..., ε j,t-q ) (19)
and θ * ij is a r θ * ij × 1 unknown parameter vector. The conditional correlations ρ ijt should respect the conditions |ρ ijt | ≤ 1, 1 ≤ i < j ≤ N and t = 1, ...T , and the corresponding correlation matrix has to be positive semidefinite; but, like in the ANN-based test, we will simply assume that these restrictions are verified in a neighborhood of the null hypothesis.

The test is based on a finite-order Taylor expansion. Following Péguin-Feissolle and Teräsvirta (1999) and Péguin-Feissolle, Strikholm and Teräsvirta (2013), we assume that all the functions f ij have a convergent Taylor expansion at any arbitrary point of the sample space for every θ * ij ∈ Θ ij (the parameter spaces) in order to ensure that, when the order of the Taylor expansion increases, the remainder of the Taylor expansion converges to zero, for each couple (i, j).

In order to linearize now f ij in (18), we expand the function into a kth-order Taylor series around an arbitrary fixed point in the sample space. After approximating f ij , merging terms and reparametrizing, we obtain:

ρ ijt = ρ ij + q m=1 λ m ε i,t-m + q m=1 φ m ε j,t-m + q m 1 =1 q m 2 =m 1 λ m 1 m 2 ε i,t-m 1 ε i,t-m 2 + q m 1 =1 q m 2 =1 ψ m 1 m 2 ε i,t-m 1 ε j,t-m 2 + q m 1 =1 q m 2 =m 1 φ m 1 m 2 ε j,t-m 1 ε j,t-m 2 +... + q m 1 =1 q m 2 =m 1 ... q m k =m k-1 λ m 1 ...m k ε i,t-m 1 ...ε i,t-m k +... + q m 1 =1 q m 2 =m 1 ... q m k =m k-1 φ m 1 ...m k ε j,t-m 1 ...ε j,t-m k + R (k) t (20)
where

R (k) t
is the remainder; in the expansion [START_REF] Hornik | Approximation capabilities of multilayer feedforward networks[END_REF] we find all possible combinations of lagged values of ε i,t and ε j,t .

The assumption that the conditional correlation ρ ijt is constant means that all terms involving functions of elements of lagged values of ε i,t and ε j,t in [START_REF] Hornik | Approximation capabilities of multilayer feedforward networks[END_REF] must have zero coefficients, i.e. all the parameters except ρ ij are equal to zero.

Therefore, the null hypothesis of interest is:

H 02 :                                      λ m = 0, m = 1, ..., q φ m = 0, m = 1, ..., q λ m 1 m 2 = 0, m 1 = 1, ..., q, m 2 = m 1 , ..., q ψ m 1 m 2 = 0, m 1 = 1, ..., q, m 2 = 1, ..., q φ m 1 m 2 = 0, m 1 = 1, ..., q, m 2 = m 1 , ..., q . . . φ m 1 ...m k = 0, m 1 = 1, ..., q, m 2 = m 1 , ..., q, ..., m k = m k-1 , ..., q.
The number of parameters to be tested under the null hypothesis is:

N * = 2    k r=1    2q + r -1 r    - k r=1    q + r -1 r       . (21)
The null hypothesis can be tested based on the LM procedure as before in the case of the N EU RAL statistic. Under standard regularity conditions, the test statistic T AY LOR is given by formula ( 13) where θ is the vector of all the parameters of the model, i.e. the parameters for the conditional Following Péguin-Feissolle, Strikholm and Teräsvirta (2013), there are two practical difficulties when the order of the Taylor expansion k is increasing, first the regressors tend to be highly collinear, and second the dimension of the null hypothesis may become rather large because the number of these regressors increases rapidly with k. More precisely, the conditional correlations ρ ijt given in ( 20) can be written as

variances ω i = (ζ i , α i , β i ) for i = 1, ..., N , the conditional correlations ρ ij with 1 ≤ i < j ≤ N
ρ ijt = ρ ij + N * m=1 δ ijm d ijt,m + R (k) t
where the N * × 1 vectors d ijt correspond to all terms involving functions of elements of lagged values of ε i,t and ε j,t . We replace the matrix composed by the N * × 1 vectors d ijt by its largest principal components: like in the case of the ANN-based test, only the largest principal components that together explain at least 90% of the variation in the matrix are used, the number of principal components being determined automatically for each i and j. Instead of the N * × T matrix D ij composed by the d ijt vectors:

D ij = (d ij1 , ..., d ijT ) for 1 ≤ i < j ≤ N , we build the N * ij × T matrix D * ij of the N * ij chosen principal components, i.e. D * ij = d * ij1 , ..., d * ijT .
The null hypothesis will be that the parameters associated to the main principal components are equal to zero in the following relationship determining ρ ijt :

ρ ijt = ρ ij + N * ij m=1 δ ijm d * ijt,m .
The statistic T AY LOR will follow an asymptotic χ 2 distribution under the null with

N -1 i=1 N j=i+1
N * ij degrees of freedom.

Monte Carlo experiments

In this section, we investigate the small-sample performances of the ANNbased test and the Taylor expansion-based test for the constancy of conditional correlations in multivariate GARCH-type models. By using Monte-Carlo experiments, we compare both the sizes and the powers of these tests to two alternative conditional correlation tests: the tests of [START_REF] Tse | A test for constant correlations in a multivariate GARCH model[END_REF] and Silvennoinen and Teräsvirta (2015).

Simulation design

The data generating process (DGP) is based on the general multivariate GARCH model introduced before where the N × 1 vector of residuals is given by [Insert Table 1 here]

ε t = H 1/2 t η t with η t ∼ nid (0, I N ), t =
[Insert Table 2 here]
The performances of the two new tests are compared to two well-known tests:

Tse (2000) test: T SE denotes the LM statistic developed by [START_REF] Tse | A test for constant correlations in a multivariate GARCH model[END_REF] given by (we report the paper's notations, see Tse 2000 for further details):

T SE = s (S S) -1 s (22)
where s = S l, l is the T × 1 column vector of ones and S is the T × N matrix of which the rows are the partial derivatives ∂lt(θ) ∂θ for t = 1, .., T ; l t (θ) is given by [START_REF] Engle | Dynamic conditional correlation-a simple class of multivariate GARCH models[END_REF]. θ is evaluated at θ which is the maximum likelihood estimator of θ under the null hypothesis.

Silvennoinen and Teräsvirta (2015) test: these authors assume that the conditional correlation matrix P t is given by

P t = (1 -G t )P 1 + G t P 2
where P 1 and P 2 are positive definite correlation matrices; G t is a logistic transition function:

G t = 1 + e -γ(st-c) -1
where γ is the transition parameter (γ > 0), s t is the transition variable and is given by ( 13) where l t (θ) is given by [START_REF] Engle | Dynamic conditional correlation-a simple class of multivariate GARCH models[END_REF]. θ is evaluated at θ which is the maximum likelihood estimator of θ under the null hypothesis (see Silvennoinen and Teräsvirta 2014 for more details).

c

Size simulations

We present here the results concerning the size of the different tests, i.e. in the case where the data are generated under the null hypothesis using a constant conditional correlation model. Following [START_REF] Silvennoinen | Modelling conditional correlations of asset returns: A smooth transition approach[END_REF], the transition variable for the ST CC test will be generated in the simulations from an exogenous GARCH(1, 1) process such that s t = h 1/2 t z t where z t ∼ N (0, 1) and h t = 0.02 + 0.03s 2 t-1 + 0.94h t-1 .

We thus consider the following models. The first one is an extended CCC-GARCH model (ECCC-GARCH) defined by [START_REF] Jeantheau | Strong consistency of estimators for multivariate ARCH models[END_REF]. It is a generalization of the multivariate GARCH model with constant conditional correlations (CCC-GARCH) proposed by [START_REF] Bollerslev | Modelling the coherence in short run nominal exchange rates: a multivariate generalized ARCH model[END_REF] and Baillie and Boller-slev (1990); the individual conditional variance equations depend on the past squared returns and conditional variances of all series. Therefore, the elements of the conditional variance matrix are characterized for N = 2 by:

   h 11t h 22t    =    ζ 1 ζ 2    +    α 11 α 12 α 21 α 22       ε 2 1,t-1 ε 2 2,t-1    +    β 11 β 12 β 21 β 22       h 11,t-1 h 22,t-1    (23)
and

h 12t = ρ 12 h 11t h 22t (24)
where ρ 12 is the constant conditional correlation. This model will be denoted by ECCC. In the second model, called CCC, the off-diagonal parameters in 23) and [START_REF] Kamstra | A neural network test for heteroscedasticity[END_REF]. We also design an asymmetric model where ( 23) is completed by leverage coefficients such that

   h 11t h 22t    =    ζ 1 ζ 2    +    α 11 α 12 α 21 α 22       ε 2 1,t-1 ε 2 2,t-1    +    γ 11 γ 12 γ 21 γ 22       ν 2 1,t-1 ν 2 2,t-1    +    β 11 β 12 β 21 β 22       h 11,t-1 h 22,t-1    (25)
where ν i,t = I(ε i,t < 0)ε i,t for i = 1, . . . , N . The indicator function I() takes value one if the argument is true and zero otherwise. Thus, the leverage coefficients are applied to negative innovations, giving additional weights to negative changes. The fourth model denoted GJR is composed by two univariate independent GJR-GARCH models, i.e. α 12 = α 21 = 0, γ 12 = γ 21 = 0, β 12 = β 21 = 0 and ρ 12 = 0 in [START_REF] Kamstra | A neural network test for heteroscedasticity[END_REF] and [START_REF] Lebreton | Robust tests for heteroscedasticity in a general framework[END_REF].

To determine the optimal order of the Taylor expansion (k) and the optimal number of lags of residuals (q) used in both the new tests, we generate S = 1000 replications of the ECCC and the bivariate GARCH models. Tables 1 and2 summarize the 1%, 5% and the 10% rejection frequencies for various q lags of residuals and k orders of Taylor expansion for these models. The selection criterion is based on the size distortions of the both tests from the nominal sizes for different sample sizes. Regarding the simulation results, among other possibilities, the choice of q = 3 for N EU RAL seems to be the best choice. The test statistic N EU RAL is therefore simulated for a number of lags q in [START_REF] Castle | A low-dimension portmanteau test for non-linearity[END_REF] equal to 3. Moreover, the order of the Taylor expansion k = 3 accompanied by a number of lags in (19) q = 2 is likely to be a good choice, leading to N * = 50 before the main component analysis. Therefore, we summarize the different parameters for the two tests that will be used in the rest of the article as follows:

-for the test statistic N EU RAL: the number of hidden units is p = 20 and the number of lags of residuals is q = 3, -for the test statistic T AY LOR: the order of the Taylor expansion is k = 3 and the number of lags of residuals is q = 2.

Power simulations

To illustrate the behavior of all tests under the alternative hypothesis, we generate different time-varying conditional correlations multivariate GARCHtype models, chosen to represent a variety of situations. It is important to note that the performance of the ST CC test depends on the choice of the transition variable, but following Silvennoinen and Teräsvirta (2015), we define the transition variable for the ST CC test as a linear combination of lags of squared returns:

s t = (0.2, 0.2, 0.2, 0.2, 0.2) × ε 2 t-1 , ε 2 t-2 , ε 2 t-3 , ε 2 t-4 , ε 2 t-5
where ε 2 t is the mean of ε t over N of its squared elementwise. We consider the following models.

The BEKK(1, 1, 1) model (Engle and Kroner 1995) is defined by:

H t = Z Z + A ε t-1 ε t-1 A + B H t-1 B (26)
where Z, A and B are N ×N matrices and Z is upper triangular. With N = 2, we have:

Z =    ζ 11 ζ 12 0 ζ 22    , A =    α 11 α 12 α 21 α 22    and B =    β 11 β 12 β 21 β 22    (27)
Note that the conditional covariance matrices H t are positive definite by construction. This model will be denoted by BEKK.

An asymmetric scalar BEKK model is also tested [START_REF] Ding | Large scale conditional covariance matrix modeling, estimation and testing[END_REF]).

This model is a particular case of the BEKK model ( 27) and will be denoted by ASBEKK:

H t = ZZ + αε t-1 ε t-1 + γν t-1 ν t-1 + βH t-1 , (28)
where α, γ and β are scalars and

ν t = I(ε t < 0) ε t . I() is the N × 1
P t = diag q -1/2
11,t , ..., q

-1/2 N N,t Q t diag q -1/2
11,t , ..., q

-1/2 N N,t (29)
where Q t = (q ij,t ) is a symmetric positive definite N × N matrix given by

Q t = (1 -α -β) Q + αz t-1 z t-1 + βQ t-1 ; (30)
Q is the N × N constant matrix (see Aielli 2013 for more details on Q), z t = S -1 t ε t and α and β are non-negative scalar parameters satisfying α + β < 1.

Moreover, h iit is a simple GARCH(1, 1) model for i = 1, .., N .

The last model is the EDCC-GARCH model (denoted by EDCC); it is the extended specification of DCC-GARCH model, allowing for volatility spillovers, i.e. the individual conditional variance equations depend on the past squared returns and variances of all series (see formula [START_REF] Jeantheau | Strong consistency of estimators for multivariate ARCH models[END_REF]). In the case N = 2, we have for i, j = 1, 2:

h iit = ζ ii + α ii ε 2 i,t-1 + α ij ε 2 j,t-1 + β ii h ii,t-1 + β ij h jj,t-1 . (31)
We take different data generation processes for three of these models (BEKK

Robustness to non-normality

The tests developed in this paper have been studied in the above section with a DGP following a Gaussian distribution. In order to investigate the robust-ness of the different tests to non-normality, we simulate in this section some models with a non-normal conditional distribution: the multivariate versions of GARCH -t model of [START_REF] Bollerslev | A conditionally heteroskedastic time series model for security prices and rates of return data[END_REF] and Beta -t -EGARCH model of Harvey and Sucarrat (2014) and [START_REF] Harvey | Beta-t-(E)GARCH[END_REF]. In all the Monte Carlo simulations, the number of replications is S = 2000, the sample sizes are T = 1000, 1500 and 2500 and the number of endogenous variables is N = 2; we consider the nominal sizes of 1%, 5% and 10%.

Size simulations

We first study the empirical sizes of the different tests when the following two

y it = µ i + ε it exp λ i,t|t-1 , t = 1, . . . , T, (32)
where ε it has a t ν i -distribution and is serially independent and ν i , the number of degrees of freedom, is positive. Let us define the conditional score

u it = (ν i + 1)(y it -µ i ) 2 ν i exp(2λ i,t|t-1 ) + (y it -µ i ) 2 -1, -1 ≤ u it ≤ ν i , ν i > 0. ( 33
)
We consider the first order model given by

λ i,t|t-1 = δ i + φ i λ i,t-1|t-2 + κ i u i,t-1 ; (34)
with the stationarity condition |φ i | < 1. The first order Beta -t -EGARCH with a leverage effect is defined in Harvey and Sucarrat (2014) as follows

λ i,t|t-1 = δ i + φ i λ i,t-1|t-2 + κ i u i,t-1 + κ * i sign(-(y i,t-1 -µ i ))(u i,t-1 + 1). ( 35
)
As noted in [START_REF] Harvey | EGARCH models with fat tails, skewness and leverage[END_REF], if an increase in the absolute values of a standardized observation leads to an increase in volatility the restriction κ i ≥ κ * i ≥ 0 may be imposed (see [START_REF] Harvey | Beta-t-(E)GARCH[END_REF] Sucarrat 2014 for a complete definition of these models and their properties).

The generation parameters are given in Table 7. [Insert Table 7 here]

[Insert Table 8 here]

Power simulations

We study the empirical powers in small samples with the same models studied for the sizes except that the conditional correlations are no longer constant.

The conditional expectation model can be written, for t = 1, . . . , T, and i = 1, ..., N ,

y it = h iit ε it ( 36
)
where, for the GARCH(1, 1) models,

h iit = ζ i + α i ε 2 i,t-1 + β i h ii,t-1 (37)
and for the Beta -t -EGARCH models,

h iit = exp λ i,t|t-1 2 ( 38
)
following the formula (32); λ i,t|t-1 is determined as a first order model defined by [START_REF] Silvennoinen | Modeling Multivariate Autoregressive Conditional Heteroskedasticity with the Double Smooth Transition Conditional Correlation GARCH Model[END_REF] or [START_REF] Silvennoinen | Modelling conditional correlations of asset returns: A smooth transition approach[END_REF] respectively without or with a leverage effect. Moreover, the ε t have a conditional multivariate t ν -distribution with positive degrees of freedom ν:

ε t |Ω t-1 ∼ t ν (0, H t ) . ( 39
)
We use the same notations as in the DCC-GARCH model: H t is given by (6) (i.e. H t = S t P t S t ), the N × N conditional correlation matrix P t is written as in [START_REF] Orme | The small sample performance of the information matrix test[END_REF], S t has the following form:

S t = diag (σ
ε * = ε * i,t -ε * i 2 (T -1) -1 ,
and we obtain thus the conditions which ensure the existence of a unit variance with σii,t = σε * σ * ii,t . At last, Q t = (q ij,t ) is a symmetric positive definite N × N matrix given by ( 30) with Q a N × N constant matrix.

The generation parameters are given in Table 7. Table 9 shows the em- [Insert Table 9 here]

pirical
Everything considered, it is worthwhile noting here that the rejection prob- We compute the returns of these components of the Dow Jones index by the usual formula expressed for one day lag: 100 × log(P t /P t-1 ), where P t represents the daily closing price at time t, t = 1, . . . , T . To avoid a singularity issue of the Hessian matrix for the multivariate optimization and in order to have a comparable metric between the four tests, we removed the few peaks, caused by the market structure, by cutting them off at -10. Table 10 presents the summary statistics of the asset returns; medians, means, standard deviations, skewness and kurtosis take into account the cut off. The returns exhibit a positive excess of kurtosis.

[Insert Table 10 here]

We compute all the tests on the asset returns as bivariate and trivariate combinations in order to determine whether these series are conditionally correlated over time or not. Like in the simulation experiments, the number of hidden units is p = 20 and the number of lags of residuals is q = 3 for the test statistic N EU RAL; the order of the Taylor expansion is k = 3 and the number of lags of residuals is q = 2 for the test statistic T AY LOR. Following [START_REF] Silvennoinen | Modelling conditional correlations of asset returns: A smooth transition approach[END_REF], the ST CC test uses two different transition variables: the lagged absolute daily returns over seven days (ST CC 1 )

and the contemporaneous squared daily returns (ST CC 2 ), these returns being those of one or the other variable as appropriate. We choose in each case the transition variable maximizing the value of the statistic (or minimizing the corresponding p -value).

Table 11 shows the results for the different tests.

[Insert Table 11 here] For the trivariate case (JPM-KO-XOM), all the tests reject the null hypothesis at the 1% level of significance, except for ST CC 2 which rejects the constancy at the 5% level.

In
Therefore, if we had considered only the ST CC test in the first case, ST CC 2 in the second case and T SE in the third case, we would have accepted the constancy of conditional correlations while the other tests conclude to time-varying conditional correlations. These short illustrations can thus show that the tests developed in this article have nice properties. This implies that whether we want to determine that the correlation is constant or not, both our tests can be considered as general misspecification tests.

Concluding remarks

In this paper, we propose two tests for the constancy of conditional correlations in the MGARCH models: the first one is based on artificial neural networks and the second on a Taylor expansion of each unknown conditional correlation around given point in a sample space. The main practical findings in this paper are that these two new tests perform well in our small-sample simulations and that they keep very good performances in case of non-normality: the simulations show their robustness to non-normality when the DGP models are multivariate versions of GARCH -t and Beta -t -EGARCH models. They

show better performances with respect to the ST CC and T SE tests in most of cases probably because their original form is very general. Indeed, the first test is based on artificial neural networks that are universal approximators, and the second one comes from the linearization of an unknown relationship determining each conditional correlation.

In the empirical applications, both tests should be applied jointly because they have different finite-sample properties, as shown by the small sample simulations and the financial application. When we consider the powers, the N EU RAL test seems to have better properties when the variability of the conditional correlation coefficients is low; it is the opposite for the T AY LOR test. Therefore, it is important to apply both of them jointly. The idea is to use a more complex model than a constant conditional correlations model if at least one of the two tests rejects the null hypothesis of constant conditional correlations.

We can thus conclude that they approximate quite well unknown specifications of the conditional correlations and therefore are useful in investigating Each test statistic can be written (we simplify by T ):

ST AT IST IC = T t=1 ∂l t (θ) ∂θ (θ) -1 T t=1 ∂l t (θ) ∂θ . (A.1)
The vector θ of all the parameters of the model is given by: θ = (ω , ρ , δ ) ; ω = (ω 1 , ..., ω N ) is composed by the 3 × 1 parameter vector for the conditional variances ω i = (ζ i , α i , β i ) for i = 1, ..., N , ρ is the vector of the conditional correlations ρ ij with 1 ≤ i < j ≤ N and δ is the vector of parameters that are equal to zero under the null hypothesis. Therefore,

T t=1 ∂lt(θ) ∂θ is a (3N + (1 + r * ) N (N -1)
2

) × 1 vector as follows

T t=1 ∂l t (θ) ∂θ =       V ω V ρ V δ       (A.2)
and (θ) is a (3N + (1 + r * ) N (N -1)

2

) × (3N + (1 + r * ) N (N -1)

2

) matrix given by

(θ) =       M ωω M ρω M δω M ρω M ρρ M δρ M δω M δρ M δδ       . (A.3)
So we can write:

ST AT IST IC =       V ω V ρ V δ             M ωω M ρω M δω M ρω M ρρ M δρ M δω M δρ M δδ       -1       V ω V ρ V δ       (A.4) 1. Different sub-vectors of T t=1 ∂lt(θ) ∂θ V ω = T t=1 ∂lt(θ) ∂ω 1 • • • T t=1 ∂lt(θ) ∂ω N , V ρ = T t=1
∂l t (θ) ∂vecl(P) , (A.5) and

V δ = vecl   T t=1 ∂l t (θ) ∂δ ij i,j   , (A.6)
where V ω , V ρ and V δ are respectively 3N ×1, N (N -1)

2

×1 and r * N (N -1)

2

×1

vectors. vecl is an operator stacking the columns of the strict lower diagonal, i.e. excluding the diagonal elements of the matrix. Moreover, we have, for i = 1, ..., N and 1 ≤ i < j ≤ N :

           ∂lt(θ) ∂ω i = 1 2 1 h iit ∂h iit ∂ω i -1 + z it 1 i P -1 t z t ∂lt(θ) ∂vecl(P) = -1 2 U vec P -1 t -P -1 t ⊗ P -1 t (z t ⊗ z t ) ∂lt(θ) ∂δ ij = ∂lt(θ) ∂δ ji = -1 2 W ijt vec P -1 t -P -1 t ⊗ P -1 t (z t ⊗ z t ) (A.7)
where ∂lt(θ) ∂ω i , ∂lt(θ) ∂vecl(P) and ∂lt(θ) ∂δ ij are respectively 3×1, N (N -1)

2

×1 and r * ×1 vectors. Moreover, z t = S -1 t ε t and the matrix U is an N 2 × N (N -1) 2 matrix of zeros and ones whose columns are defined as

vec 1 i 1 j + 1 j 1 i i=1,...,N -1 j=i+1,...,N (A.8)
and the columns appear in the same order from left to right as the indices in vecl(P). The matrix W ijt is an r * × N 2 matrix given by

W ijt = diag (g ijt )                  1 . . . 1       ⊗ (vec 1 i 1 j + 1 j 1 i )            ; (A.9)
g ijt is a r * × 1 vector defined in the N EU RAL test as

g ijt =       g ijt,1
. . .

g ijt,r *       =       (1 + exp{-w ijt γ ij1 }) -1 . . . (1 + exp{-w ijt γ ijr * }) -1       (A.10)
and, in the T AY LOR test, it is replaced by d ijt , composed by all terms involving functions of elements of lagged values of ε i,t and ε j,t ; we have:

r * = 2    k r=1    2q + r -1 r    - k r=1    q + r -1 r       . (A.11)
2. Different elements of the matrix (θ)

• M ωω is the following 3N × 3N matrix:

M ωω = T t=1          M ωω,11 M ωω,12 • • • M ωω,1N M ωω,12 M ωω,22 • • • M ωω,2N . . . . . . . . . M ωω,1N M ωω,2N • • • M ωω,N N          (A.12)
where, for i = 1, ..., N and 1 ≤ i < j ≤ N , M ωω,ii and M ωω,ij are 3 × 3 matrices defined as

     M ωω,ii = E t-1 ∂lt(θ) ∂ω i ∂lt(θ) ∂ω i = 1 4 1 h 2 iit ∂h iit ∂ω i 1 i P -1 t 1 i + 1 ∂h iit ∂ω i M ωω,ij = E t-1 ∂lt(θ) ∂ω i ∂lt(θ) ∂ω j = 1 4 1 h iit h jjt ∂h iit ∂ω i ρ ijt 1 i P -1 t 1 j + 1 ∂h jjt ∂ω j (A.13) • M ρρ is the following N (N -1) 2 × N (N -1)
2 matrix:

M ρρ = T t=1 1 4 U P -1 t ⊗ P -1 t + P -1 t ⊗ I K(P -1 t ⊗ I) U (A.14)
• M ρω is a N (N -1)

2

× 3N matrix defined as

M ρω = T t=1 M ρω,1 M ρω,2 ... M ρω,N (A.15)
where each 3 × N (N -1) 2 matrix M ρω,i , for i = 1, ..., N , is:

M ρω,i = E t-1 ∂lt(θ) ∂ω i ∂lt(θ) ∂vecl(P) = 1 4 1 h iit ∂h iit ∂ω i 1 i 1 i ⊗ P -1 t + 1 i (1 i ⊗ I) K(P -1 t ⊗ I) U (A.16)
with M δω (i, j, n) a r * × 3 matrix defined by:

M δω (i, j, n) = E t-1 ∂lt(θ) ∂δ ij ∂lt(θ) ∂ω n = -1 4hnnt W ijt vec(P -1 t ) -(I ⊗ 1 n ) + P -1 t ⊗ I K(I ⊗ P t 1 n ) +vec(P -1 t )vec(1 n P t ) P -1 t 1 n ∂hnnt ∂ω n ; (A.22)
for each couple (i, j) taken in the same order as for V δ , take all the n = 1, 2, ..., N .

For the computation of all these vectors and matrices, we have:

∂h iit ∂ω i = ν i,t-1 + β i ∂h ii,t-1 ∂ω i . (A.23)
with ν it = 1, ε 2 it , h iit for i = 1, ..., N (we compute recursively ∂h iit ∂ω i ). Following [START_REF] Silvennoinen | Modelling conditional correlations of asset returns: A smooth transition approach[END_REF], we have for a model with general correlation matrix

P t            E t-1 [z t z t ⊗ z t z t ] = (P t ⊗ P t ) + (I ⊗ P t )K(I ⊗ P t ) + vecP t (vecP t ) E t-1 (z it z jt ⊗ z t z t ) = (1 i ⊗ I) E t-1 (z t z t ⊗ z t z t ) (1 j ⊗ I) E t-1 [z it z t ⊗ z t z t ] = (1 i ⊗ I) E t-1 [z t z t ⊗ z t z t ] (A.24)
with K a N 2 × N 2 matrix defined as

K =             1 1 1 1 • • • 1 i 1 1 • • • 1 N 1 1 . . . . . . . . . 1 1 1 j • • • 1 i 1 j • • • 1 N 1 j . . . . . . . . . 1 1 1 N • • • 1 i 1 N • • • 1 N 1 N             (A.25)
Following (A.4), and using the rules of the partitioned matrices and the null hypothesis, the test statistic can be written

ST AT IST IC = V δ    M δδ -M δω M δρ    M ωω M ρω M ρω M ρρ    -1 (A.26) ×    M δω M δρ       -1 V δ .
Table 1: Determination of the optimal order of the Taylor expansion k and the optimal number of lags of residuals q used in the ANNbased and Taylor expansion-based tests (ECCC-GARCH) T 1000 1500 2500 1% 5% 10% 1% 5% 10% 1% 5% 10% TAYLOR k = 1 q = 1 0.009 0.050 0.113 0.012 0.043 0.089 0.008 0.039 0.094 q = 2 0.016 0.063 0.123 0.013 0.055 0.097 0.006 0.059 0.124 q = 3 0.023 0.086 0.144 0.012 0.042 0.087 0.010 0.046 0.102 q = 4 0.017 0.070 0.131 0.010 0.059 0.110 0.006 0.044 0.088 TAYLOR k = 2 q = 1 0.083 0.159 0.225 0.081 0.166 0.231 0.007 0.052 0.102 q = 2 0.005 0.028 0.061 0.008 0.031 0.073 0.008 0.051 0.105 q = 3 0.022 0.058 0.122 0.006 0.035 0.079 0.012 0.060 0.114 q = 4 0.020 0.068 0.114 0.008 0.049 0.103 0.011 0.063 0.131 TAYLOR k = 3 q = 1 0.032 0.089 0.148 0.037 0.098 0.150 0.005 0.062 0.109 q = 2 0.012 0.042 0.083 0.014 0.053 0.097 0.007 0.049 0.106 q = 3 0.032 0.096 0.144 0.011 0.065 0.131 0.018 0.056 0.119 q = 4 0.024 0.086 0.151 0.022 0.093 0.155 0.021 0.067 0.126 TAYLOR k = 4 q = 1 0.049 0.095 0.133 0.036 0.073 0.111 0.009 0.056 0.104 q = 2 0.014 0.045 0.090 0.009 0.050 0.096 0.011 0.058 0.108 q = 3 0.026 0.083 0.146 0.008 0.056 0.117 0.016 0.061 0.122 q = 4 0.024 0.080 0.160 0.024 0.094 0.146 0.018 0.069 0.122 NEURAL q = 1 0.004 0.018 0.035 0.003 0.018 0.039 0.006 0.058 0.109 q = 2 0.009 0.027 0.055 0.007 0.026 0.057 0.013 0.058 0.107 q = 3 0.007 0.052 0.086 0.008 0.048 0.083 0.009 0.052 0.103 q = 4 0.009 0.036 0.080 0.007 0.035 0.072 0.009 0.052 0.108 Note: The number of replications is S = 1000. The generation parameters of the ECCC model are: 2: Determination of the optimal order of the Taylor expansion k and the optimal number of lags of residuals q used in the ANNbased and Taylor expansion-based tests (GARCH) T 1000 1500 2500 1% 5% 10% 1% 5% 10% 1% 5% 10% TAYLOR k = 1 q = 1 0.015 0.062 0.108 0.007 0.048 0.103 0.008 0.042 0.091 q = 2 0.018 0.067 0.110 0.012 0.045 0.098 0.012 0.057 0.118 q = 3 0.018 0.066 0.117 0.014 0.051 0.100 0.014 0.056 0.098 q = 4 0.016 0.062 0.130 0.011 0.049 0.105 0.008 0.049 0.106 TAYLOR k = 2 q = 1 0.010 0.049 0.085 0.006 0.038 0.088 0.011 0.055 0.095 q = 2 0.010 0.052 0.099 0.011 0.051 0.097 0.011 0.046 0.101 q = 3 0.013 0.062 0.111 0.005 0.045 0.098 0.017 0.058 0.111 q = 4 0.016 0.062 0.118 0.006 0.050 0.106 0.014 0.067 0.124 TAYLOR k = 3 q = 1 0.013 0.058 0.112 0.007 0.042 0.091 0.007 0.045 0.103 q = 2 0.011 0.041 0.091 0.011 0.052 0.101 0.011 0.052 0.109 q = 3 0.015 0.051 0.105 0.006 0.051 0.101 0.012 0.053 0.114 q = 4 0.017 0.068 0.119 0.014 0.044 0.092 0.010 0.046 0.093 TAYLOR k = 4 q = 1 0.011 0.041 0.077 0.010 0.050 0.095 0.012 0.046 0.119 q = 2 0.007 0.043 0.079 0.012 0.060 0.104 0.009 0.044 0.084 q = 3 0.014 0.058 0.092 0.009 0.042 0.088 0.015 0.056 0.105 q = 4 0.022 0.075 0.130 0.012 0.052 0.103 0.013 0.062 0.118 NEURAL q = 1 0.010 0.056 0.112 0.012 0.048 0.092 0.012 0.048 0.104 q = 2 0.013 0.047 0.097 0.014 0.053 0.107 0.011 0.059 0.103 q = 3 0.011 0.047 0.089 0.006 0.045 0.088 0.010 0.051 0.106 q = 4 0.015 0.047 0.093 0.014 0.053 0.105 0.010 0.044 0.100 Note: The number of replications is S = 1000. The generation parameters of the bivariate GARCH model are: Note: The number of replications is S = 2000. For N EU RAL the number of hidden units is p = 20 and the number of lags of residuals is q = 3, and for T AY LOR the order of the Taylor expansion is k = 3 and the number of lags of residuals is q = 2. The generation parameters are: N EU RAL 0.998 0.998 0.998 1.000 1.000 1.000 1.000 1.000 1.000 T AY LOR 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 T SE 0.832 0.986 0.995 0.989 0.998 0.999 1.000 1.000 1.000 ST CC 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 BEKK 2 N EU RAL 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 T AY LOR 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 T SE 0.983 0.987 0.990 0.995 1.000 1.000 1.000 1.000 1.000 ST CC 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 ASBEKK N EU RAL 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 T AY LOR 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 T SE 0.406 0.578 0.666 0.5335 0.720 0.800 0.739 0.876 0.919 ST CC 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Note: The number of replications is S = 2000. For N EU RAL the number of hidden units is p = 20 and the number of lags of residuals is q = 3, and for T AY LOR the order of the Taylor expansion is k = 3 and the number of lags of residuals is q = 2. 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

ζ 1 ζ 2 α
Note: The number of replications is S = 2000. For N EU RAL the number of hidden units is p = 20 and the number of lags of residuals is q = 3, and for T AY LOR the order of the Taylor expansion is k = 3 and the number of lags of residuals is q = 2. The generation parameters are: Note: The number of replications is S = 2000. For N EU RAL the number of hidden units is p = 20 and the number of lags of residuals is q = 3, and for T AY LOR the order of the Taylor expansion is k = 3 and the number of lags of residuals is q = 2. 0.91 0.339 (1) 2.92 0.087 (2) 2.44 0.118 (1) 24.4 2e-05 (1) ST CC 2 0.21 0.642 (1) 2.46 0.117 (1) 6.53 0.011 (1) 9.08 0.028 (3) Note: Stat. and p-v represent respectively the value of the test statistic and the associated p-value. For N EU RAL the number of hidden units is p = 20 and the number of lags of residuals is q = 3, and for T AY LOR the order of the Taylor expansion is k = 3 and the number of lags of residuals is q = 2.

The transition variables used in ST CC are the lagged absolute daily returns over seven days for ST CC 1 , and the contemporaneous squared daily returns for ST CC 2 . We choose in each case the transition variable maximizing the value of the statistic (or minimizing the corresponding p-value): (1), ( 2) or (3) for the first, the second or the third one (see [START_REF] Silvennoinen | Modelling conditional correlations of asset returns: A smooth transition approach[END_REF] for the definition of the transition variable).

  Silvennoinen and Teräsvirta (2015) derive an LM test for the constancy of correlations against the Smooth Transition Conditional Correlation (STCC) GARCH model in the multivariate case with an endogenous or exogenous transition variable. Berben and Jansen (2005) introduce a bivariate GARCH model with smoothed time-varying correlations based on logistic function, called the Smooth Transition Correlation GARCH (STCC-GARCH) model, and derive a new test for constant correlation, building on the LM test developed by Tse

Feissolle

  and Teräsvirta 1999 and Péguin-Feissolle, Strikholm and Teräsvirta 2013), the heteroscedasticity (Lebreton and Péguin-Feissolle 2007) and the conditional heteroscedasticity (Caulet and Péguin-Feissolle 2000 and Péguin-Feissolle 1999).

  and the N * × 1 parameters that are equal to zero under the null hypothesis. When the remainder R (k) t ≡ 0, the statistic T AY LOR has an asymptotic χ 2 distribution with N * degrees of freedom under the null hypothesis. Appendix presents the details of the technical derivations of the test statistic.

  1, . . . , T . In other words, we generate several MGARCH-type models generally used in financial time series analysis, based on the normality assumption; we will relax the normality assumption in Section 4. For each simulation, η t thus follows a multivariate Gaussian distribution with zero mean vector and covariance matrix I N . For each model, the DGP is used to compute the empirical sizes and powers in order to compare the performances of both the tests we propose (ANN-based test and Taylor expansion-based test) to two well-known tests (Tse (2000) test and Silvennoinen and Teräsvirta (2015) test). For all the Monte Carlo simulations, the sample sizes are T = 1000, 1500 and 2500 and the number of endogenous variables is N = 2 (except in Table 6 where N = 3). Moreover, we remove the first observations in order to eliminate initialization effects. The number of replications is S = 2000 (except in Tables1 and 2). Concerning the artificial neural network based tests, following Lee,[START_REF] Lee | Testing for neglecting nonlinearity in time series models[END_REF], the number of hidden units is p = 20; moreover, we generate the hidden unit weights, i.e. the different γ ijm in[START_REF] Carroll | Construction of neural nets using the Radon transform[END_REF], for 1 ≤ i < j ≤ N and 1 ≤ m ≤ p, randomly from the uniform [-µ, µ] distribution. Following the same authors, we choose µ = 2. For a fixed number of lags of residuals in the case of the artificial neural network based test and for a fixed number of lags of residuals and a fixed order of the Taylor expansion in the case of the Taylor expansion-based test, only the largest principal components that together explain at least 90% of the variation in the corresponding matrices are used, the number of principal components being determined automatically.

( 23 )

 23 are equal to zero i.e. α 12 = α 21 = 0 and β 12 = β 21 = 0. The third model, called GARCH, will be composed of two univariate independent GARCH(1,1) models, i.e. α 12 = α 21 = 0, β 12 = β 21 = 0 and ρ 12 = 0 in (

For

  all the tests, we can observe that sizes converge towards the nominal sizes when the number of observations T increases, for most of the tests. More precise conclusions can be given. In the ECCC case, for the different sample sizes, the T SE test statistic over-rejects the null hypothesis while a multivariate constant conditional correlation model is generated. The N EU RAL, the T AY LOR and the ST CC test statistics show similar performances and clearly outperform the T SE test. In the CCC case, the sizes seem close to the nominal sizes for all the test statistics when the sample sizes reach T = 2500. The bivariate GARCH case permits to study the possibility of no correlation at all, i.e. the nullity of the conditional covariances. The rejection frequencies show clear signs of over-rejections for all the sample sizes for the T SE test statistic; the N EU RAL, T AY LOR and ST CC test statistics clearly outperform the T SE test statistics, which shed light on their good size properties. The bivariate GJR case allows positive and negative shocks to have an asymmetric effect in the conditional variances. The N EU RAL, the T AY LOR and the ST CC tests show good properties for T = 2500, while the T SE test tends to over-reject.

  models are generated with null correlations. The first model corresponds to two independent univariate Student t-GARCH models; they are characterized by α 12 = α 21 = 0, β 12 = β 21 = 0 and ρ 12 = 0 in (23), and thus h 12t = 0. The second model consists of two independent univariate Beta -t -EGARCH with and without a leverage effect. We use the same notations as Harvey and Sucarrat (2014) who developed a more general framework than Harvey and Chakravarty (2008). Each Beta -t -EGARCH model without a leverage effect is defined as follows, for i = 1, 2:

  powers of the different tests, i.e. the rejection probabilities of the null hypothesis of constant correlations. For the DCC -t -GARCH model, all the tests reject the null hypothesis of constancy of conditional correlation, but the ST CC and T SE tests present less good results and the T AY LOR test the best one for each sample size. For the DCC -Beta -t -EGARCH model with and without leverage effects, the performances of the tests present the same characteristics: the two new tests show the best empirical powers, even for T = 1000 or 1500, and the T SE test again seems the least powerful.

  abilities of the null hypothesis increase again with the number of observations T . To conclude about the simulation results in the case of time-varying conditional correlation and non-normality assumptions, the tests showing the best performances are the T AY LOR and N EU RAL tests. On the other hand, the tests presenting very poor performances are the ST CC and especially the T SE tests. 5 Correlations between three asset returns We illustrate the applicability of the two constant correlation tests that we propose, that is the ANN-based test and the Taylor-expansion based test, by comparing them to Tse's (2000) test and Silvennoinen and Teräsvirta's (2015) test in an empirical example. We consider the asset returns of three of "bluechip" US daily stocks used to compute the Dow Jones Industrial Average index: JPMorgan Chase & Co. (JPM), The Coca-Cola Company (KO) and Exxon Mobil Corporation (XOM) from January 03, 2000 to October 12, 2012 (T = 3217 daily observations). The series correspond to closing prices adjusted for dividends and splits and are obtained from Yahoo Finance.

  the case of the relationship between JPMorgan Chase (JPM) and Coca-Cola returns (KO), the ANN-based test and the Taylor-expansion based test reject the null hypothesis of the constancy of conditional correlations at the 1% level (the p -value are respectively 0.001 and 0.005) whereas T SE rejects it only at the 10% level (the p -value is equal to 0.062). It is important to remark that ST CC 1 and ST CC 2 do not reject the null hypothesis considering that conditional correlations are not time-varying. In the case of the JPMorgan Chase (JPM) and Exxon Mobil (XOM) returns, the N EU RAL test rejects the constant correlation hypothesis at the nominal size of 10% (p -value = 0.067) like the T AY LOR test at the 1% level (p -value = 8 × 10 -8 ), the T SE test at the 5% level (p -value = 0.037) and ST CC 1 at the 10% level (p -value = 0.087). ST CC 2 does not reject the constancy of conditional correlations. When we consider Exxon Mobil (XOM) and The Coca-Cola Company (KO) returns, the test based on artificial neural network rejects the hypothesis of constancy of conditional correlation at the nominal size of 5% (p -value = 0.015). The test based on a Taylor expansion rejects the null at the 1% level (p -value = 3 × 10 -6 ) whereas the T SE and ST CC 1 tests do not reject the constancy. ST CC 2 rejects the hypothesis of constancy of conditional correlation at the nominal size of 5% (p -value = 0.011).

  potential time-varying conditional correlations between economic or financial time series. APPENDIX: DERIVATION OF THE STATISTICS OF THE ARTIFICIAL NEURAL NETWORK TEST AND THE TAYLOR EXPANSION-BASED TEST In order to compute the N EU RAL and the T AY LOR statistics, we use broadly Abadir and Magnus (2005), Anderson (2003) and Lütkepohl (1996) for the derivations based on the rules of matrix algebra. Let us call r * the number of parameters that are equal to zero under the null hypothesis, i.e. r * = p for the N EU RAL statistic and r * = N * for the T AY LOR statistic.

ζ 1 ζ 2 α

 2 11 α 12 α 21 α 22 β 11 β 12 β 21 β 22 ρ 12 0.02 0.02 0.01 0.02 0.02 0.01 0.85 0.02 0.02 0.85 0.60 Table

ζ 1 ζ 2 α

 2 11 α 22 β 11 β 22 0.02 0.01 0.04 0.03 0.95 0.96

  we conduct the test using the main principal components of each G ij

	matrix. Only the largest principal components that together explain at least
	90% of the variation in this matrix are used; the number of principal com-
	ponents are determined automatically for each i and j (see Péguin-Feissolle
	and Teräsvirta 1999, Lebreton and Péguin-Feissolle 2007 and Péguin-Feissolle,
	Strikholm and Teräsvirta 2013; see also Castle and Hendry 2010 for discus-
	sions on using principal components to solve the collinearity problem). More
	precisely, instead of G ij , we build the p * ij × T matrix G * ij of the p * ij principal
	components chosen according to the preceding rule, i.e.
	G * ij = g * ij1 , ..., g * ijT .

  is the location parameter. Using a first-order Taylor approximation, P t can

	be approximated as P * t = P * 1 -s t P * 2 where P * 1 and P * 2 are depending on γ,

P 1 and P 2 . The null hypothesis can be written vecl(P *

2 ) = 0, where the vecl operator stacks the columns of the strict lower diagonal of the square argument matrix (i.e. by excluding the diagonal elements). The LM statistic, ST CC,

Table 3

 3 shows the empirical sizes of the different tests, i.e. the rejection probabilities under the null hypothesis of constant conditional correlations, assuming nominal sizes of 1%, 5% and 10% and different sample sizes (T = 1000, 1500 and 2500). Moreover, we consider models with different constant correlations: high with ρ 12 = 0.60 for the ECCC-GARCH, medium with ρ 12 = 0.30 for the CCC model, and of course ρ 12 = 0 for the GARCH and GJR models. The DGP models correspond to different persistences, for example α 11 + β 11 = α 22 + β 22 = 0.99 for the CCC-GARCH model, and α 11 + β 11 = 0.95 and α 22 + β 22 = 0.90 in the bivariate GARCH model.

[Insert Table

3 here]

  [START_REF] Abadir | Matrix Algebra[END_REF] and BEKK 2 , DCC 1 and DCC 2 , EDCC 1 and EDCC 2 ), characterized by different generation parameters and different variabilities of conditional correlation coefficients. Table4gives the parameter values used to generate the different models; following[START_REF] Tse | A test for constant correlations in a multivariate GARCH model[END_REF], we give in the second part of the Ta-DCC 1 and DCC 2 , are characterized by parameters implying very different ranges of variability of the conditional correlation coefficients: the intervals are very large in the case of the DCC 2 model. Nevertheless, the T AY LOR test presents the best performance in both DGP models Concerning the advantages of the Neural test with respect to the Taylor test, the Neural test seems to have better properties when the variability of the conditional correlation coefficients is low (EDCC 2 ); it is important to note that in this last case, the Neural test is the only test that can detect the non constancy of conditional correlations. At the opposite, the Taylor test seems better in the case where the variability of the conditional correlation coefficients is high (DCC 2 and EDCC 1 ). Therefore, we can conclude that even if one or the other of the new tests is not the best each time, it is important to apply jointly both of them because at least one of them can detect the time-varying conditional correlations. The idea is that, if at least one of the two tests rejects the null hypothesis of constant conditional correlations, we can try a more complex model with time-varying conditional correlations.We also generalize the N EU RAL, T AY LOR, T SE and ST CC tests to the case of three endogenous variables (N = 3). Table6summarizes the results of the small sample sizes (GARCH 2 ) and powers (BEKK 3 ). All the tests,

	ertheless, even if most of them present good properties with a rejection prob-
	ability sometimes equal to 1, the T SE test seems to have less good results,
	especially for T = 1000.
	The DCC-GARCH models,
	T .
	[Insert Table 4 here]
	[Insert Table 5 here]
	For the three BEKK-GARCH specifications, BEKK 1 , BEKK 2 and ASBEKK,
	the tests reject the null hypothesis of constancy of conditional correlation; nev-

ble 4 the variability of the conditional correlation coefficients by calculating the range (maximum -minimum) of these coefficients in each simulated sample. The variability is high with DCC 2 and EDCC 1 , moderate with BEKK 2 and DCC 1 and low with BEKK 1 , ASBEKK and EDCC 2 . Table 5 summarizes the empirical powers of the ANN-based test, the Taylor expansion-based test, the T SE test and the ST CC test. In other words, it shows the rejection probabilities of the null hypothesis of constant conditional correlations assuming nominal sizes of 1%, 5% and 10% and different sample sizes (T = 1000, 1500 and 2500). For all the tests, we can observe in Table 5 that the rejection probabilities of the null hypothesis increase with the number of observations followed by the N EU RAL test in the first one and the T SE test in the second one. On the other hand, the tests performing worse than the other tests are the T SE and ST CC tests in the DCC 1 model and the N EU RAL and ST CC tests in the DCC 2 model. Concerning the extended DCC-GARCH models, EDCC 1 and EDCC 2 , the tests showing the best performances are the T AY LOR and T SE tests in the EDCC 1 case and the N EU RAL test in the EDCC 2 case. In the first case, the N EU RAL, T AY LOR and T SE tests outperform the ST CC test. In the second DGP model, the performances are altogether poor for each test. Whatever the variability is, at least one of the new tests seems to present good quality in order to conclude to the non constancy of the conditional correlations. except the T SE test for the small sample powers, show good results because the sizes are close to the nominal sizes (see GARCH 2 ) and the powers are high (see BEKK 3 ). [Insert Table 6 here] All the multivariate GARCH-type models have thus been tested satisfactorily using the ANN-based test and the Taylor expansion-based test compared to T SE and ST CC tests. This finding suggests that whatever the GARCHtype model, both tests developed in this paper show relevant properties to identify time-varying conditional correlations. To summarize our results, the ANN-based and Taylor expansion-based tests permit to generalize the acceptance or rejection of the hypothesis of constant correlations in a general multivariate GARCH framework.

Table 8

 8

shows the small sample sizes of t-GARCH and Beta-t-EGARCH. When the DGP follows a Student distribution, the results of t -GARCH model highlight the relative robustness to non-normality of the T AY LOR, N EU RAL and ST CC tests compared to the T SE test, even for small numbers of observations. When the observations are simulated with a first order Beta -t -EGARCH model with or without leverage effects, the T SE test over-rejects the null hypothesis of constant conditional correlations. The N EU RAL, T AY LOR and ST CC tests are very close to the nominal sizes and show again better results. Generally, the T AY LOR, N EU RAL and ST CC tests outperform thus the T SE test.

  11,t , . . . , σNN,t ) .To precise the different diagonal elements of S t , we refer toHarvey and

	Chakravarty (2008): they defined σ * ii,t =	√	h iit as the time-varying scale pa-
	rameter, which is not necessarily equal to the standard deviation. We resample
	the variance of residuals ε * i,t ∼ IID(0, σ 2 ε * ) which does not necessarily have unit
	variance by the sample variance of residuals σ2
			(40)

Table 3 :

 3 Small sample sizes of the different constant conditional

	correlation tests
	T		1000	1500	2500
		1%	5% 10% 1%	5% 10% 1%	5% 10%
	ECCC
	N EU RAL 0.008 0.052 0.087 0.009 0.049 0.085 0.011 0.054 0.105
	T AY LOR 0.012 0.042 0.083 0.014 0.053 0.097 0.009 0.053 0.104
	T SE	0.027 0.084 0.134 0.023 0.080 0.139 0.086 0.144 0.201
	ST CC	0.011 0.054 0.107 0.012 0.042 0.094 0.011 0.049 0.104
	CCC
	N EU RAL 0.018 0.077 0.134 0.017 0.061 0.118 0.008 0.056 0.104
	T AY LOR 0.021 0.083 0.141 0.017 0.070 0.133 0.010 0.046 0.105
	T SE	0.018 0.060 0.117 0.015 0.062 0.107 0.017 0.052 0.105
	ST CC	0.015 0.059 0.112 0.013 0.059 0.106 0.008 0.047 0.098
	GARCH
	N EU RAL 0.014 0.053 0.107 0.015 0.057 0.113 0.011 0.055 0.106
	T AY LOR 0.013 0.053 0.105 0.015 0.056 0.103 0.013 0.058 0.111
	T SE	0.093 0.193 0.264 0.089 0.189 0.265 0.089 0.192 0.264
	ST CC	0.006 0.045 0.101 0.011 0.046 0.095 0.008 0.054 0.105
	GJR
	N EU RAL 0.013 0.061 0.119 0.008 0.052 0.099 0.010 0.061 0.109
	T AY LOR 0.015 0.065 0.123 0.010 0.043 0.103 0.014 0.052 0.108
	T SE	0.033 0.099 0.153 0.030 0.088 0.148 0.028 0.091 0.151
	ST CC	0.008 0.043 0.082 0.007 0.050 0.104 0.008 0.049 0.098

Table 4 :

 4 11 α 12 α 21 α 22 β 11 β 12 β 21 β 22 γ 11 γ 22 ρ 12 MGARCH parameters for the generation of the DGP for the small sample powers BEKK 1 BEKK 2 DCC 1 DCC 2 EDCC 1 EDCC 2 ASBEKK

	ECCC	0.02 0.02 0.01 0.02 0.02 0.01 0.85 0.02 0.02 0.85 --0.60
	CCC	0.02 0.02 0.01 0.00 0.00 0.01 0.98 0.00 0.00 0.98 --0.30
	GARCH 0.40 0.20 0.15 0.00 0.00 0.20 0.80 0.00 0.00 0.70 --0.00
	GJR	0.02 0.01 0.05 0.00 0.00 0.04 0.91 0.00 0.00 0.93 0.02 0.01 0.00

  1.000 1.000 DCC 1 N EU RAL 0.392 0.536 0.612 0.410 0.546 0.619 0.433 0.585 0.659 T AY LOR 0.394 0.534 0.616 0.446 0.602 0.694 0.525 0.648 0.722

	T SE	0.060 0.149 0.241 0.100 0.234 0.344 0.191 0.375 0.487
	ST CC	0.197 0.329 0.416 0.213 0.350 0.442 0.241 0.383 0.466
	DCC 2
	N EU RAL 0.144 0.265 0.359 0.163 0.303 0.397 0.243 0.409 0.501
	T AY LOR 0.970 0.988 0.992 0.999 0.999 0.999 1.000 1.000 1.000
	T SE	0.829 0.943 0.966 0.967 0.991 0.997 0.997 0.999 1.000
	ST CC	0.237 0.369 0.454 0.232 0.388 0.463 0.249 0.397 0.456
	EDCC 1
	N EU RAL 0.693 0.784 0.830 0.711 0.797 0.835 0.756 0.832 0.870
	T AY LOR 0.923 0.956 0.969 0.969 0.981 0.986 0.994 0.996 0.996
	T SE	0.799 0.865 0.895 0.874 0.909 0.920 0.911 0.917 0.929
	ST CC	0.454 0.566 0.631 0.509 0.604 0.654 0.615 0.708 0.747
	EDCC 2
	N EU RAL 0.410 0.512 0.573 0.418 0.527 0.593 0.503 0.605 0.676
	T AY LOR 0.023 0.088 0.163 0.031 0.104 0.178 0.127 0.188 0.247
	T SE	0.022 0.081 0.138 0.025 0.094 0.158 0.131 0.199 0.262
	ST CC	0.064 0.156 0.238 0.069 0.165 0.243 0.154 0.250 0.335

Table 6 :

 6 Small sample sizes and powers of the different constant conditional correlation tests when the number of endogenous variables is N = 3 N EU RAL 0.008 0.039 0.086 0.013 0.047 0.091 0.010 0.046 0.082 T AY LOR 0.011 0.048 0.089 0.010 0.048 0.089 0.008 0.045 0.085 T SE 0.015 0.050 0.093 0.017 0.049 0.086 0.012 0.053 0.096 ST CC 0.013 0.052 0.104 0.009 0.049 0.098 0.011 0.054 0.104 BEKK 3 N EU RAL 0.993 0.996 0.996 0.996 0.996 0.996 0.999 0.999 0.999 T AY LOR 0.994 0.994 0.995 0.995 0.996 0.996 0.996 0.996 0.996 T SE 0.344 0.504 0.590 0.432 0.609 0.687 0.615 0.765 0.836 ST CC

	T	1000	1500	2500
	1%	5% 10% 1%	5% 10% 1%	5% 10%
	GARCH 2

Table 9 :

 9 Small sample powers of constant conditional correlation GARCH N EU RAL 0.283 0.450 0.536 0.374 0.526 0.619 0.490 0.643 0.727 T AY LOR 0.511 0.739 0.636 0.772 0.834 0.788 0.885 0.922 T SE 0.113 0.223 0.329 0.183 0.338 0.436 0.245 0.432 0.530 ST CC 0.246 0.461 0.301 0.427 0.493 0.339 0.455 0.529 DCC -Betat -EGARCH N EU RAL 0.601 0.736 0.806 0.68 0.807 0.857 0.779 0.862 0.901 T AY LOR 0.792 0.886 0.928 0.878 0.942 0.961 0.965 0.986 0.994 T SE 0.107 0.256 0.376 0.188 0.349 0.471 0.284 0.463 0.567 ST CC 0.459 0.548 0.595 0.594 0.663 0.704 0.591 0.669 0.720 DCC -Betat -EGARCH * N EU RAL 0.676 0.801 0.851 0.762 0.855 0.897 0.854 0.917 0.943 T AY LOR 0.856 0.931 0.959 0.917 0.967 0.978 0.982 0.994 0.997 T SE 0.156 0.251 0.342 0.193 0.292 0.386 0.224 0.338 0.430 ST CC 0.569 0.643 0.690 0.658 0.729 0.759 0.753 0.813 0.842

	tests under nonnormality
	T	1000	1500	2500
	1%	5% 10% 1%	5% 10% 1%	5% 10%
	DCC -t -

Table 11 :

 11 Constant conditional correlation tests of the daily asset returns

		KO-JPM	JPM-XOM	XOM-KO JPM-KO-XOM
		Stat. p-v	Stat. p-v	Stat. p-v	Stat. p-v
	N EU RAL 21.8 0.001	11.8 0.067	10.9 0.015	53.8 6e-05
	T AY LOR 39.7 0.005	73.7 8e-08	63.8 3e-06	126 2e-06
	T SE	3.48 0.062	4.36 0.037	0.00 0.985	16.8 7e-04
	ST CC 1

matrix given by

where each r * × N (N -1)

2 matrix:

with M δδ (i, j, n, m) a r * × r * matrix given by:

for each couple (i, j) taken in the same order as for V δ , take the couple (n, m) in the same order.

• M δω is a r * N (N -1)

2

× 3N matrix given by: Note: The number of replications is S = 2000. For N EU RAL the number of hidden units is p = 20 and the number of lags of residuals is q = 3, and for T AY LOR the order of the Taylor expansion is k = 3 and the number of lags of residuals is q = 2.