D.Sc. * Gilberto F Alexandre

D.Sc Ian S M Martins

Alberto Luiz

An Interpretation of Secondary Consolidation for the Batiscan Clay

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Introduction

Secondary consolidation is a research topic that still attracts attention today as no agreement exist on the mechanism or its prediction. However, irrespective of its limitations, the C α /C c approach is the most used.

Another research line that is gaining attention is the Isotach approach. In this approach, the models intended for predicting secondary consolidation, as well as undrained creep and other rate eects, relate the stresses to strains by means of lines of same strain rate (isotach meaning same rate/speed). [START_REF] Taylor | Research on Consolidation of Clays[END_REF] was the rst to propose such a model, although nowadays many others have been proposed. Examples of Isotach models are the one proposed by [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] and the one by [START_REF] Martins | Fundamentals of a Behavioral Model for Saturated Clayey Soils[END_REF]. This paper deals with these two Isotach models and intends on presenting an alternative method for assessing secondary consolidation. Before discussing into detail the proposed method the two aforementioned isotach models will be briey presented.

2 The model proposed by [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] proposed a model for natural clays which was successfully applied to Champlain Sea clays. The model can be described by the two equations presented below:

σ p = f ( εv ) (1) σ v /σ p = g (ε v ) (2)
Where: σ p is the preconsolidation pressure; εv is the vertical strain rate; σ v is the vertical eective stress; and ε v is the vertical strain These equations were reached after [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] demonstrated based on their experimental ndings that the inuence of σ v is not signicant.

By combining Equations ( 1) and (2) it can be seen that the vertical eective stress is the product of a function of the vertical strain rate by a function of the vertical strain. More importantly, with Equations ( 1) and ( 2), it is possible to obtain a relationship between the vertical strain rate and the vertical strain for a given vertical eective stress. This relationship can then be used to predict the evolution of strain over time under a constant vertical eective stress. That is, this relationship can be used for predicting secondary consolidation. [START_REF] Watabe | Modelling and implementation of isotach concept for long-term consolidation behavior[END_REF] develop further Equation (1) by considering that the function f ( εv ) can be represented by a power law function which reaches a limit, called σ pL , at innitesimal strain rate ( ε = 0).

An isotach model that can provide an alternative interpretation of secondary consolidation is the one proposed by [START_REF] Martins | Fundamentals of a Behavioral Model for Saturated Clayey Soils[END_REF]. Although this model was originally developed for explaining rate eects such as creep and stress relaxation under undrained conditions in the triaxial apparatus, the modication proposed by Martins, which was conrmed by [START_REF] Thomasi | About the Existence of a Viscous Parcel on the Normal Eective Stress[END_REF], as well as the one proposed by [START_REF] Luiz | Contribution to the Understanding of the Undrained Creep[END_REF] allow the treatment of secondary consolidation. The model proposed by [START_REF] Martins | Fundamentals of a Behavioral Model for Saturated Clayey Soils[END_REF], as it was originally conceived, will be briey presented in the next section. solid bonds would result from the contact between the adsorbed water layers in the immediate vicinity of the clay particle, which, according to Terzaghi, would be in the solid state. The lm bonds would result from the contact between adsorbed water layers which would not be in the solid state but which would possess a viscosity higher than of the viscosity of the free water (by free water it should be understood the water that ows out of the pores of soil during seepage or consolidation).

Having this picture in mind, [START_REF] Martins | Fundamentals of a Behavioral Model for Saturated Clayey Soils[END_REF] assumed as a hypothesis that the shear strength of saturated normally consolidated clays has two components, the frictional resistance and the viscous resistance.

The frictional resistance would develop between Terzaghi's solid bonds and it would be a function of the shear strain. The viscous resistance would develop between Terzaghi's lm bonds and it would be a function of the strain rate.

The equation for the shear strength would then be:

τ = τ f + τ v = σ • tan φ mob + η (e) • ε (3)
Where:

τ f is the frictional resistance (the solid component of the shear stresses); τ v is the viscous resistance (the viscous component of the shear stresses); φ mob is the mobilized eective angle of internal friction; η (e) is the coecient of viscosity of the adsorbed water layer that surrounds the clay particles (a function of the void ratio for a normally consolidated clay); and ε is the strain rate.

The normal eective stress, σ , is taken as the dierence between the normal total stress, σ, and pore-pressure, u.

One problem in the model is that, because Equation (3) is written in terms of shear stresses, rate/time eects related to the normal eective stress such as the hydrostatic relaxation observed by [START_REF] Arulanandan | Undrained creep behavior of a coastal organic silty clay[END_REF], secondary consolidation under hydrostatic conditions and such cannot be macroscopically explained.

To overcome this problem it was suggested by Martins that Equation (3) could be generalized for the normal eective stresses. In this regard, [START_REF] Thomasi | About the Existence of a Viscous Parcel on the Normal Eective Stress[END_REF] carried out similar tests as [START_REF] Arulanandan | Undrained creep behavior of a coastal organic silty clay[END_REF], conrming the possibility of generalizing the normal eective stress equation as proposed by Martins. The generalized (3) Equation is:

σ = σ s (ε) + σ v (e, ε) (4)
Where:

σ is the normal eective stress; σ s is the normal eective stress due to the solid bonds; and σ v is the normal eective stress due to the lm bonds.

Equations similar to Equations ( 3) and (4) were proposed by [START_REF] Taylor | Research on Consolidation of Clays[END_REF] and (1948).

A state of stress that combines Equations ( 3) and ( 4) was proposed by [START_REF] Luiz | Contribution to the Understanding of the Undrained Creep[END_REF] for the case of axial-symmetry and is reproduced below:

σ s = σ 1s + σ 3s 2 + σ 1s -σ 3s 2 • cos (2 • α) (5) τ s = σ 1s -σ 3s 2 • sin (2 • α) (6) σ v = σ 1v + σ 3v 2 + σ 1v -σ 3v 2 • cos (2 • α) (7) τ v = σ 1v -σ 3v 2 • sin (2 • α) (8)
Where the indexes s and v represent the solid and the viscous components of the eective stresses and the indexes 1 and 3 represent the major and minor principle stresses. In the above equations and for a given plane, α is the angle between the direction perpendicular to the plane considered and the plane where σ 1 acts.

It can be seen that Equations ( 5) and ( 6) describe a circle for the solid eective stress state as well as Equations ( 7) and (8) constitute a circle for the viscous eective stress state. These two circles combined give the eective stress state at a point.

An important remark about this model is that when the strain rate is made equal to zero, the eective stress state decreases to a lower stationary value greater than zero, which is the solid eective stress state. If a real laboratory test could be carried-out with a strain rate equal to zero then the results of such a test would comprise only the solid component of the eective stress state.

However, as a test with a strain rate equal to zero is impossible to perform there will always be an additional component of the eective stress state, which is the viscous eective stress component.

The concepts introduced in this section were intended to provide the minimum information for the development of the study of the secondary consolidation mechanism, which will be presented in the next section. For a better understanding of this model the reader is referred to [START_REF] Martins | Fundamentals of a Behavioral Model for Saturated Clayey Soils[END_REF][START_REF] Martins | Behavioural Model of Saturated Clays[END_REF], [START_REF] Luiz | Contribution to the Understanding of the Undrained Creep[END_REF] and Santa [START_REF] Maria | Rheological behavior os soft clays[END_REF].

The mechanism of Secondary Consolidation

According to Equation ( 3) and ( 4), for the same strain, the higher the strain rate the higher the shear stress and the higher the eective normal stress. An example of this behavior, which is characteristic of Isotach models, can be seen in [START_REF] Vaid | Time-dependent behavior of undisturbed clay[END_REF] and another in [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] as reproduced below.

Figure 1: [START_REF] Vaid | Time-dependent behavior of undisturbed clay[END_REF] Figure 2: [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] Another way of understanding Equation ( 4) is thinking about a vertical line through a given eective stress, that is, to think what occurs during secondary consolidation. According to Equation (4) and considering that the strain rate decreases as the strain increases, then, the viscous eective stress should decrease and the solid eective stress should increase as the soil undergoes secondary consolidation. Therefore secondary consolidation is the transference of the viscous component of the eective stress to the solid component under constant eective stress. Physically, the transference occurs between the bonds, with the lm bonds becoming solid bonds with time. When the last lm bond becomes a solid bond, the strain rate drops to zero and secondary consolidation comes to an end. This mechanism of transference of eective stress components between bonds was originally conceived by [START_REF] Terzaghi | Undisturbed Clay Samples and Undisturbed Clays[END_REF].

If the relationship between strain rate and strain is known for a given constant eective stress, then, it is possible to assess settlement caused by secondary consolidation, at least between two known strain rate curves. To this point, the calculation of secondary consolidation using Martins's model is similar to the calculation using the model proposed by [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF]. However, according to Martins's model there will be a zero strain rate line where all the rate eects come to an end and all the eective stress is of a solid component nature.

In this way the model proposed by [START_REF] Martins | Fundamentals of a Behavioral Model for Saturated Clayey Soils[END_REF] allows for the assessment of the total secondary consolidation settlement, as long as it is possible to assess the zero strain rate line, or in other words, the End-of-Secondary (EOS) con-solidation line. The assessment of the EOS consolidation line will be addressed in the next section. The second method would be carrying out stress relaxation tests in the edometer apparatus once the soil reaches the end-of-primary consolidation,

where the excess pore-pressure is practically zero. In order to allow for a stress relaxation test in the consolidation apparatus, a load cell or a proving ring can be positioned in the edometric frame in a place that prevents further movements of the loading arm as the consolidation process proceeds. Some deformation that results from the loading of the load cell/proving ring always occurs and therefore these tests cannot be considered perfect. Therefore they are called imperfect stress relaxation tests. [START_REF] Garcia | Relationship Between Secondary Consolidation and Stress Relaxation of a soft clay[END_REF] carried out such tests on the same organic clay from Rio de Janeiro which Martins carried out the long term consolidation test mentioned above. The nine tests carried out by [START_REF] Garcia | Relationship Between Secondary Consolidation and Stress Relaxation of a soft clay[END_REF] lasted from 10,000 to 80,000 minutes. However, none of them have reached the EOS consolidation line, therefore imperfect stress relaxation tests are also very time consuming. Figure 5 below shows the stress relaxation test that lasted for 80,000 minutes, together with a prediction from [START_REF] Alexandre | Stress Relaxation Under Various Stress and Drainage Conditions[END_REF]. A third method consist in generating unloadings with dierent OCR's and observing the evolution of the rebound strain with time. This was carried out by [START_REF] Feijó | Relationship Between Secondary Compression, Over Consolidation Ratio and the At-rest Earth Pressure Coecient[END_REF] in the Sarapui river clay and is reproduced below. As can be seen in the gure above, the tests lasted to about 200,000 minutes.

The test carried-out with an OCR = 2 experienced a slight compression after the end of primary rebound and the test with OCR = 4 did not present any sign of compression after the end of primary rebound. Therefore, the EOS consolidation line should correspond to an OCR in between these two values.

According to his experience with other clays, Martins found that the EOS line is close to a line which corresponds to an OCR between 1.6 and 2.2 for some clays from Rio de Janeiro, Brazil.

Considering the time required for carrying out any of these tests, it is not economically feasible their execution on a routine basis. Alternatively, a mixed experimental-analytical method can be used.

Considering again Equation ( 4), for a given strain and time during secondary consolidation, what it is known is that the viscous component of the eective stress is been transferred to the solid component. However, their actual values are not known. In terms of measured quantities, all is known is the strain, the strain rate and the vertical eective stress which the soil is undergoing during secondary consolidation at a given time. Not even the shape of the viscous and solid curves are known. Therefore the unknowns at a given strain during secondary consolidation are the values of the solid eective stress, σ s and the viscous eective stress, which is a function of the strain rate. Assuming a particular function for representing the viscous component, characterized by n constants, the number of unknowns is n + 1. To solve for the n + 1 variables it is then required equal number of equations. These equations can be written if tests are carried out that allow the determination of n + 1 compression lines of same strain rate, or n + 1 isotachs. These n + 1 isotachs can be determined by any of the tests mentioned above, however, it is not required that these tests reach the EOS line, saving therefore considerable time and resources.

As the viscous function is not known it is necessary to search for the best tting one. Therefore the n + 1 variables will correspond to the viscous function with the higher number of constants, n. As this method applies to any rate eects under any drainage and stress condition, an example will be provided below using the tests results of conventional CIU tests with dierent strain rates carried out by [START_REF] Vaid | Time-dependent behavior of undisturbed clay[END_REF] shown in Figure 1.

According to [START_REF] Luiz | Contribution to the Understanding of the Undrained Creep[END_REF], a power law function of the strain rate was found to represent the viscous behavior with a correlation coecient of about R 2 = 0.99. Therefore, for assessing the zero strain rate deviatoric stress x strain curve, a power law function will be used. Considering Equations ( 5) to ( 8), the following equation can be written for the deviatoric stress:

σ d = σ ds + σ dv (9)
Where σ ds = σ 1s -σ 3s and σ dv = σ 1v -σ 3v . Assuming a power law function for the viscous component it can be written that σ dv = K • εn and therefore Equation ( 9) can be re-written as:

σ d = σ ds + K • εn (10)
In this case, the variables to be determined are σ ds , K and n. Therefore, the number of required equations are 3. Using the axial strain of 2.5% as the strain for this determination and the curves of the strain rates of 1.1, 1.4x10 -2 and 9.4x10 -4 %/min, one gets for their respective deviatoric stresses 0.66, 0.55 and 0.52 (normalized in respect to the consolidation stress). The non-linear system of equations is therefore:

0.66 = σ ds /σ 1c + K • 1.1 % min n (11) 0.55 = σ ds /σ 1c + K • 1.4x10 -2 % min n (12) 0.52 = σ ds /σ 1c + K • 9.4x10 -4 % min n (13)
The solution for the system presented above is σ ds /σ 1c = 0.485, K = 0.171 (in general [K] = F L -2 T n but in this case due to the normalization,[K ] = T n ) and n = 0.225. As there are 2 more curves with dierent strain rates, it is possible to set 2 more systems using these curves and the curves with strain rate equal to 1.1 % /min and 9.4x10 -4 % /min (the lower and upper bound strain rate curves). The solutions for these curves are σ ds /σ 1c = 0.484, K = 0.172, n = 0.224 and σ ds /σ 1c = 0.385, K = 0.272 and n = 0.101. The rst two sets of solutions agree well and the third seems too low. Regardless, by this method it is possible to assess lower and upper bounds for the unknowns. The value assessed by [START_REF] Luiz | Contribution to the Understanding of the Undrained Creep[END_REF] for σ ds /σ 1c , based on a graphical method and on the results of the undrained creep and undrained constant load tests was σ ds /σ 1c = 0.46. [START_REF] Vaid | Time-dependent behavior of undisturbed clay[END_REF], this value would be between 0.5, which is the creep test that did not fail within 3 weeks, and 0.518, which is the creep test that did fail.

Batiscan Clay

According to [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] Batiscan is situated on the north shore of the St. Lawrence river, about 110 km west of Quebec city. The clay tested was taken at a depth of 7.25-7.46 m. The water content is about 80 %, the plasticity index 21 % and the liquidity index 2.7. The preconsolidation pressure, as determined from conventional oedometer test, is 88 kPa.... Also according to the authors, the liquidity limit is 43 %, the clay fraction is 81 %, the undrained strenght, as determined by eld vane, is 25 kPa and the normal eective vertical stress at a depth of 7.3 m is 65 kPa.

Assessment of parameters

The method outlined in Section 5 was applied to the Bastican clay for vertical strains of 1, 5, 10, 15 and 20 % using the data provided in Figures 5 and7 from [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF]. As the power law function was successfully applied to the Haney clay, as shown in [START_REF] Luiz | Contribution to the Understanding of the Undrained Creep[END_REF], and on an organic clay from Rio de Janeiro-Brazil, as shown in [START_REF] Alexandre | Stress Relaxation Under Various Stress and Drainage Conditions[END_REF], the logical choice for this clay is also a power law function. For this determination 4 strain rates were chosen: 10 -5 s -1 , 10 -6 s -1 , 10 -7 s -1 and 10 -8 s -1 . As the number of variables requiring determination is 3, two systems of 3 equations were solved for each strain in order to provide a lower and upper bound range for the zero strain rate line. The rst consisted of the strain rates 10 -5 s -1 , 10 -6 s -1 , and 10 -8 s -1 while the second system consisted of the strain rates 10 -5 s -1 , 10 -7 s -1 and 10 -8 s -1 . Table 1 Note:

[K] = F L -2 T n
In fact, the procedure for determining the three unknowns, σ s , K and n, was applied as explained above but only the solid eective vertical normal stress, σ s was kept. A gure was then made with these points and a smooth curve was drawn by hand similar in shape to the curves from the CRS tests. The gure below presents these points as well as the assessed curve and the CRS test points. The lower bound curve for ε = 20% was assessed because of the scatter shown in Figure 5 of [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] and also because the pairs of values ε, σ s = (20%, 136.6kP a) and ε, σ s = (20%, 134.7kP a) are inconsistent with Figure 4 from [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF]. In this case only the lower bound solid eective stresses were used. Then, σ s was determined graphically for strains between 1 and 22 % at 1 % intervals using the smooth curve and each individual power law function was then determined using the CRS curves from Figure 1 of [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF]. The obtained values of σ s , K, n and the correlation coecient for each power law function are presented in the table below. A check regarding the consistency of the zero strain rate line was carried out using the power law functions presented in Table 2. In this regard, the viscous eective vertical normal stress were assessed based on the individual functions for the strain rate of the specic CRS test and then subtracted from the CRS curves. These curves were plotted together in the gure below: As can be seen on Figure 8, the zero strain rate line is in fact a band or a range. If it is assumed an average curve at the center of the band interval, then the dierence in σ s for the outer limits of the band and the average curve varies from about 3.5 to 5.5 kPa. Despite of the scatter, this band can be considered narrow, at least in comparison to the dierences between the CRS curves. Moreover, this narrow band shows the consistency of zero strain rate line assumption.

ε (%) σ s (kP a) K n R 2
One point that must be claried is that the procedure for assessing the zero strain rate line does not always give the correct estimate. Depending on the chosen ε, σ points, the , σ s values for the zero strain rate line are simply too low. In the present work the chosen points had a dierence in strain rate of one order of magnitude (10x faster or slower). Chosing points closer than one order of magnitude in strain rate led to values that the present authors believe not to be representative of the zero strain rate line. According to [START_REF] Brown | Slurry handling: design of solidliquid systems[END_REF], even non-linear least squares regression techniques present problems. In addition it is important to understand the geological history of the deposit and apply engineering judgment. In the case of the Batiscan clay, the creep tests presented in [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] give an indication that the zero strain rate line assessment is appropriate as comparisons of strains when the tests were interrupted are somehow consistent with the zero strain rate band.

Another important remark is that the test with a strain rate ε = 1.69x10 -8 s -1 was not used for the assessment of the parameters as it crosses the other test curves.

The reason for this behavior is not clear but [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] suspect of natural variability or thixotropic hardening.

A small note is appropriate at this point. Figure 8 can be seen to resemble the picture drawn by Bjerrum (1973) about secondary consolidation. The dierence is that instead of lines of equal times, in the present model these are lines of equal strain rates. In addition, in this model, the zero strain rate line represents the end of secondary consolidation.

Predictions

For a given test at and after the end of primary consolidation, when the excess pore-pressure is approximately negligible, it is possible to assess the strain rate for a given strain as below:

ε = σ -σ s (ε) K 1 n (14)
With the pairs (ε, ε) is then possible by numerical integration to assess the time for a given strain with the following equation:

t (ε) = ˆ dε dt -1 • dε (15)
Equations ( 14) and ( 15) were used in combination with the parameters assessed and shown in Table 2 to predict secondary consolidation. The results of the predictions as well as the experimental data are presented in the following gures. The dashed lines are the predictions and the circles are the experimental points. Predictions were made for ε ≤ 10 -6 s -1 as, according to [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF], based on their experience the eective stress is approximately equal to the applied stress for strain rates at or below this value.

Secondary Consolidation Predictions

As can be seen, the predictions are in general agreement with the experimental data both qualitatively and quantitatively for most of the tests which corroborates the decision of dividing the eective stresses in two components as suggested by Terzaghi himself. The predictions presented herein could have been obtained graphically, provided one could assess the zero strain rate curve and the remaining strain rate curves in between it and the experimentally determined ones. To this end, the proposed method is useful. However it does not go without mentioning that more research and analysis of eld cases, such as the one by [START_REF] Mcrostie | Canadian geotechnical research site no.1 at Gloucester[END_REF] are paramount in validating the proposed model and method.

Regarding the geological history of the deposit, according to the data in [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF], the OCR of the Batiscan clay at a depth of 7.3 m is about 1.35. However, assuming an average solid pre-consolidation stress, σ ps , of about 69.4 kPa, then the solid OCR, OCR s = σ ps/σ v , is about 1.07. Therefore in the present interpretation the clay from Batiscan is slightly over consolidated, being the dierence between these two denitions of OCR the aging of the deposit.

When considering the limits of the solid curve band, OCR s will be a little less or more than the indicated value, between about 1 (normally consolidated) and 1.14.

At this point it is important to discuss the applicability of the method for assessing secondary consolidation and its consequences.

As proposed, the equations for assessing secondary consolidation are only valid when the excess pore-pressure is small enough so it can be disregarded. In the case of Batiscan clay, according to [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF], this happens when the strain rate is equal or smaller than 10 -6 s -1 . However, that is not the same as to say that secondary consolidation comes only after primary consolidation.

The authors believe that during primary consolidation there is an interaction between excess pore-pressure and viscous eective stress that is dicult to assess however there will be a contribution of the viscous eective stress in the development of strains during this phase. Therefore, consolidation settlements should not, in a strict sense, be assessed as the sum of primary consolidation as given by Terzaghi's classical theory plus the proposed method outlined in this paper.

Despite of the limitations of the proposed method, it can be used for assessing secondary consolidation in the eld where the excess pore-pressure becomes small.

Regarding the models proposed by [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF] and by [START_REF] Martins | Fundamentals of a Behavioral Model for Saturated Clayey Soils[END_REF] it can be shown mathematically their similarity. In the model proposed by [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF], the eective stress is a product of a function f ( ε) and a function g (ε). On the other hand, in the model proposed my Martins (1992) σ = σ s (ε) + σ v (e, ε) where e is the void ratio. As the function f ( ε) cannot be zero when the strain rate is zero (otherwise the soil would behave like a uid), then it can be written that f ( ε) = [1 + j ( ε)] and therefore σ

= f ( ε) • g (ε) = [1 + j ( ε)] • g (ε) = g (ε) + g (ε)
• j ( ε) in the model proposed by [START_REF] Leroueil | Stressstrain-strain rate relation for the compressibility of sensitive natural clays[END_REF]. Comparison of this last expression with the one proposed by Martins (1992) reveals that σ s (ε) = g (ε) and that σ v (e, ε) = g (ε) • j ( ε) leading to the conclusion that both models are indeed mathematically similar.

When it comes to predicting the entire phenomena there are at least two know theories that can be somehow used. The rst is the one proposed by [START_REF] Taylor | Research on Consolidation of Clays[END_REF], known as Theory B, and the second is the one proposed by [START_REF] Barden | Consolidation of Clay with Non-Linear Viscosity[END_REF]. Both theories divide the eective stress in two components, one dependent on strain and the other dependent on strain rate but their functions are quite dierent from each other. To represent the viscous component Taylor suggested a linear function (a Newtonian viscous function) and was able to nd an analytical solution whereas Barden suggested a power law function just like the authors of this paper as a better approximation and solved the problem numerically. The problem with the theory proposed by Taylor is that the viscous function is highly non-linear as can be seen for example in Tables 1 and2 and therefore that will lead, most likely, to signicant dierences between predictions and the actual behavior. On the other hand, although Barden's theory incorporates a better function, the problem is solved only numerically.

In addition, he did not consider how the viscous function changes during secondary consolidation as can be seen once again in Tables 1 and2. That been said, the authors do not know of better alternatives for predicting (primary and secondary) consolidation.

Despite of its complexity, a possibility is the numerical solution of a dierential equation similar to the one proposed by Barden however using a power law function (or any other function) that changes with strain (void ratio). That is, considering that σ v = f (ε, ε).

Finally, the authors wonder if the process of structuration/desetructuration is related to the viscous component of the eective stress as noted by the ever changing character of the viscous function with strain development.

K 0 Schmertmann (1983) proposed what he called a simple question about secondary consolidation and more than 30 years later no denitive answer was reached.

However, the majority of the experimental results point in the direction of the increase of K 0 over time during secondary consolidation.

One interpretation for the increase of K 0 over time comes from Schertmann himself. In Schmertmann (2012), he wrote As previously discussed in 12.2, the secondary part of the consolidation process of a clay's structure adjusting to an increased load involves a time transfer from an unstable Φ´α to a stable Φ´β. As developed herein, Φ´α results from particle/AWL sliding and does not depend on particle interference eects, while Φ´β does so depend. Despite the one dimensional constraint for the Ko condition, Φ´α can dissipate with time because of its viscosity. But, a matching increase in Φ´β cannot fully develop with time because particles, on average, cannot move across, nor can shear strain develop to increase interference along the lateral boundary minor principal plane. This reducing Φ´α to Φ´β transfer ability reduces the clay's ability to sustain an increased shear loading and Ko increases with time to maintain equilibrium.

Another interpretation comes when using the model proposed by [START_REF] Martins | Fundamentals of a Behavioral Model for Saturated Clayey Soils[END_REF] or this model as modied by [START_REF] Luiz | Contribution to the Understanding of the Undrained Creep[END_REF]. Consider that in an inclined plane in a consolidation test specimen the shear stress is the sum of a frictional and a viscous component resisting compression. Writing the force equilibrium equation along the inclined plane will relate σ 3 (the lateral stress), σ 1 (the applied vertical stress) and the frictional and viscous components of the shear stresses. As the viscous shear stress resists compression, the eect of the viscous component will be to alleviate somehow the lateral stress. As at the end of secondary consolidation the strain rate and the viscous component have to drop to zero, then the lateral stress will have to increase. Therefore, K 0 should increase during secondary consolidation.

Following [START_REF] Alexandre | Stress Relaxation Under Various Stress and Drainage Conditions[END_REF], Equations ( 5) to ( 8) can be combined to assess the time dependent behavior of K 0 over time during secondary consolidation. Considering that K 0 is dened as:

K 0 = σ 3 σ 1 (16)
And that, on the other hand, σ 3 = σ 3s + σ 3v , σ 1 = σ 1s + σ 1v , it follows that:

K 0 = σ 3s + σ 3v σ 1s + σ 1v (17)
In addition, dening a solid coecient of earth pressure at rest as:

K 0s = σ 3s σ 1s (18)
Assuming that macroscopically there is no strain rate in the horizontal direction, then, the following can be written:

K 0 = K 0s • σ 1s σ 1s + σ 1v (19)
As during secondary consolidation the viscous component is transferred to the solid component, the denominator of Equation ( 19) remains unaltered while the numerator increases, therefore K 0 should increase over time. Assessing σ 1v and σ 1s at each strain allows for the assessment of the pattern of increase of K 0 over time. The igure below shows a sketch of this pattern. Although the pattern of variation of K 0 as shown in the gure above is not the same as the predicted by this model, K 0 does increase over time. Using a value of K 0s = 0.65, the equations mentioned above and the upper and lower bounds of the experimental data, the following gure can be made. The points represent the experimental data curve and the dashed line is the tting curve.

Conclusions

A new method for assessing secondary consolidation was proposed and its validity was assessed. Comparison between predictions and experimental results support the tentative conclusion of the validity of both model and method. Signicant savings in terms of cost and time can be achieved in making predictions of secondary consolidation by means of the present method. The limitations and applicability to eld cases are also presented in this paper.

5

  Experimental Determination of the zero strain rate consolidation lineThe rst experimental method for assessing the EOS consolidation line is to carry out long-term consolidation tests where the consolidation (primary + secondary) process comes to an end.Martins (1997) carried out 3 tests for a soil fabricated in the laboratory, a mix of caulin (80%) and bentonite (20%) that lasted to about 2 years and one test in an organic clay from Rio de Janeiro that lasted for about 5 years as shown in[START_REF] Andrade | Contribution to the Study of Soft Clays from the City of Santos[END_REF]. Figures3 and 4reproduced below present these results.

Figure

  Figure 3: Martins (1997)

Figure 5 :

 5 Figure 5: Edometric stress relaxation test result from[START_REF] Garcia | Relationship Between Secondary Consolidation and Stress Relaxation of a soft clay[END_REF] and a prediction from[START_REF] Alexandre | Stress Relaxation Under Various Stress and Drainage Conditions[END_REF]

Figure 6 :

 6 Figure6: Feijó (1991)

Figure 7 :

 7 Figure 7: Assessment of the end-of-secondary consolidation line (the zero strain rate line).

Figure 8 :

 8 Figure 8: Assessment of the zero strain rate band/range

Figure 16 :

 16 Figure 9: σ = 151kP a

Figure 18 :

 18 Figure 18: K 0 increase during secondary consolidation adapted from Mesri &Castro (1987)

Figure 19 :

 19 Figure 19: Fitting of the increase of K 0 over time during secondary consolidation.

Figure 20 :

 20 Figure 20: Hypothetical secondary consolidation stress path.

Table 1 -

 1 presents these determinations. Fist assessment of parameters.

	ε (%)	1 st System		2 nd System
		σ s (kP a)	K	n	σ s (kP a)	K	n
	1	63.5	218.1	0.13	74.3	374.13	0.19
	5	80.8	589.1	0.23	77.7	431.62	0.19
	10	89.1	539.2	0.21	84.9	400.8	0.18
	15	107.3	1786.6	0.32	104.8	1048.8	0.27
	20 (average curve)	136.6	1753.8	0.31	134.7	1221.7	0.27
	20 (lower bound curve)	123.0	3140.9	0.36	127.4	1520.1	0.30

Table 2 -

 2 Final assessment of parameters.

	1	69.4	237.0	0.15	0.96
	2	73.5	307.0	0.17	0.96
	3	76.7	414.2	0.19	0.97
	4	78.4	517.4	0.21	0.96
	5	79.4	550.0	0.22	0.97
	6	80.5	575.3	0.22	0.97
	7	82.4	624.4	0.22	0.97
	8	84.3	661.5	0.23	0.98
	9	86.5	772.4	0.24	0.98
	10	89.2	1050.5	0.27	0.98
	11	92.0	1116.1	0.27	0.98
	12	94.7	1277.5	0.28	0.98
	13	97.9	1772.9	0.31	0.97
	14	101.2	2245.2	0.33	0.97
	15	104.7	2479.2	0.34	0.97
	16	108.5	2894.5	0.35	0.96
	17	113.2	3429.2	0.36	0.95
	18	117.8	3031.7	0.34	0.97
	19	122.4	2510.3	0.33	0.96
	20	127.9	3101.9	0.34	0.92
	21	135.2	1822.5	0.29	0.92
	22	143.4	2156.8	0.30	1.00
	23	153.7	Not assessed
	23.55	160	Not assessed

The model proposed by[START_REF] Martins | Fundamentals of a Behavioral Model for Saturated Clayey Soils[END_REF] According to[START_REF] Terzaghi | Undisturbed Clay Samples and Undisturbed Clays[END_REF], the contact between clay particles would be of two types. Terzaghi called the contact types solid bonds and lm bonds . In his view, both contacts would be able to transmit eective stresses and would result from the adsorbed water layers that surround the clay particles. The

K0 Time (log scale)

Figure 17: Pattern of the increase of K 0 during secondary consolidation, after [START_REF] Alexandre | Stress Relaxation Under Various Stress and Drainage Conditions[END_REF] Although this pattern of increase in K 0 can be seen in [START_REF] Lacerda | Stress Relaxation and Creep Eects on Soil Deformation[END_REF], it can be said that the test carried out by Lacerda is not a true K 0 test as it was done in the triaxial apparatus. Regarding the Batiscan clay, a Figure adapted from Mesri & Castro (1987), which shows the variation of K 0 for the Batiscan clay can be seen below:


Philippe Charlot
email: ph.charlot@gmail.com

Vêlayoudom Marimoutou
email: velayoudom.marimoutou@univ-amu.fr

On the relationship between the Prices of oil and the Precious Metals: Revisiting with a Multivariate Regime-Switching Decision Tree ✩

Keywords: Multivariate GARCH, Dynamic correlations, Regime switching, Hidden Markov JEL Classification: C32, C51, G1, G0

This study examines the volatility and correlation and their relationships among the euro/US dollar exchange rates, the S&P500 equity indices, and the prices of WTI crude oil and the precious metals (gold, silver, and platinum) over the period 2005 to 2012. Our model links the univariate volatilities with the correlations via a hidden stochastic decision tree. The ensuing Hidden Markov Decision Tree (HMDT) model is in fact an extension of the Hidden Markov Model (HMM) introduced by Jordan et al. (1997). The architecture of this model is the opposite that of the classical deterministic approach based on a binary decision tree and, it allows a probabilistic vision of the relationship between univariate volatility and correlation. Our results are categorized into three groups, namely (1) exchange rates and oil, (2) S&P500 indices, and

(3) precious metals. A switching dynamics is seen to characterize the volatilities, while, in the case of the correlations, the series switch from one regime to another, this movement touching a peak during the period of the Subprime crisis in the US, and again during the days following the Tohoku earthquake in Japan. Our findings show that the relationships between volatility and correlation are dependent upon the nature of the series considered, sometimes corresponding to those found in econometric studies, according to which correlation increases in bear markets, at other times differing from them.

Introduction

The relationship between volatility and correlation has been widely studied. The overall conclusion of many studies, mainly based on international equity returns or indices, is that correlations are not only time-varying, but there is also a particular connection between the two statistical measures. Correlation appears indeed to be higher during Bear markets. That suggests that the degree of dependence between series rises as stock values fall. This asymmetric relationship between state of the market and correlations cannot be analyzed simply by comparing estimated values. A first set of studies have been undertaken using extreme value theory [START_REF] Longin | Extreme correlation of international equity markets[END_REF]), and a quantile correlation measure to a portfolio allocation [START_REF] Campbell | Increased correlation in bear markets[END_REF]) while [START_REF] Ang | International asset allocation with regime shifts[END_REF] estimate a Markov-switching multivariate-normal model. A second set of studies have proposed statistical methods to test the presence of asymmetric correlations (see [START_REF] Ang | Asymmetric correlations of equity portfolios[END_REF] and [START_REF] Hong | Asymmetries in stock returns: Statistical tests and economic evaluation[END_REF]).

In this paper, we examine the connections between univariate volatility and correlations between euro/US dollar exchange rate, S&P500 index, oil and precious metals from January 2005 to 19 October 2012, covering a period which includes the subprime crisis. In contrast to the previous studies, our method is based on a two level Hidden Markov Decision Tree (HMDT) proposed by [START_REF] Jordan | Hidden markov decision trees[END_REF]. The first level discriminates between low and turbulent volatility while the second determine the correlations. Then, a probabilistic relationship between the two levels is established.

The paper is organized as follows. Section 2 provides a brief review of the existing literature on the dynamics of exchange rates, S&P500 indices, and prices of oil and the precious metals.

In section 3 we present our model. Section 4 contains our findings, Section 5 presents our discussions and conclusions, and sets out areas for further research.

Review

The existing specialist literature on metals commodities and oil markets can be considered on two levels. At the first level, it consists of an analysis of the univariate characteristics of the series, while at the second level the analysis covers a multivariate setting, and attempts to explain the linkages between these variables.

Exchange rates and equity indices are the main field of application for volatility models. In that context, the autoregressive conditional heteroskedasticity (ARCH) model of [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation[END_REF] and its generalised counterpart (GARCH), proposed by [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF], are among the most widely used specifications in the empirical literature. The analysis of [START_REF] Lunde | A forecast comparison of volatility models: does anything beat a garch(1,1)[END_REF] points to GARCH(1,1) as the most favoured specification for a study of the volatility of exchange rates.

The litterature on oil price fluctuations is substantial. The main recent contributions regarding oil are listed in the introduction of the paper by [START_REF] Wang | Forecasting energy market volatility using garch models: Can multivariate models beat univariate models?[END_REF]. While these studies use GARCH type models for the volatility of oil prices, consensus about the dynamics seems hard to reach. The determinants of oil market price volatility are mainly due to macroeconomic factors but also speculative and exogenous geopolitical events (see [START_REF] Hamilton | Causes and consequences of the oil shock of 2007-08[END_REF]). Studies conclude that a simple GARCH(1,1) can reproduce oil series (see [START_REF] Cheong | Modeling and forecasting crude oil markets using arch-type models[END_REF], [START_REF] Agnolucci | Volatility in crude oil futures: A comparison of the predictive ability of garch and implied volatility models[END_REF]), while others consider that CGARCH model and FIGARCH [START_REF] Kang | Forecasting volatility of crude oil markets[END_REF]) or a Markovswitching GARCH model (see [START_REF] Nomikos | Forecasting petroleum futures markets volatility: The role of regimes and market conditions[END_REF]) are more suitable.

In recent years, numerous studies on the precious metals have been published, and they underline the importance of the dynamics of the volatility of their prices: time-dependence and GARCH effects in the spot price of gold and silver [START_REF] Akgiray | Conditional dependence in precious metal prices[END_REF], [START_REF] Schwartz | The stochastic behavior of commodity prices: Implications for valuation and hedging[END_REF]), non-linear models perform better than a random-walk model [START_REF] Escribano | Investigating the relationship between gold and silver prices[END_REF]), day effects and daily seasonality (Lucey and Tully (2006b)). In the case of platinum, [START_REF] Adrangi | The dynamics of palladium and platinum prices[END_REF] find the strong evidence of nonlinear dependence, which can be explained by ARCH-type processes.

More recently, some studies have investigated the potential existence of structural changes and long memory in precious metals prices [START_REF] Arouri | Long memory and structural breaks in modeling the return and volatility dynamics of precious metals[END_REF], [START_REF] Cochran | Volatility persistence in metal returns: A figarch approach[END_REF]). However, the distinction between a true switching model and the long-memory property is far from clear in the current econometric literature (see [START_REF] Diebold | Long memory and regime switching[END_REF] and [START_REF] Granger | Occasional structural breaks and long memory with an application to the s&p 500 absolute stock returns[END_REF] and also [START_REF] Cheung | Do gold market returns have long memory?[END_REF] for a discussion on long memory in gold returns).

At a multivariate level, co-movement of commodity prices was examinate by [START_REF] Deb | The excess co-movement of commodity prices reconsidered[END_REF] whose results suggest the presence of excess co-movement in the univariate setup, but not in the multivariate case. [START_REF] Pindyck | The excess co-movement of commodity prices[END_REF] found that unrelated commodities tend to move together, but this conclusion was subsequently disproved by [START_REF] Cashin | The myth of co-moving commodity prices. Reserve bank of new zealand discussion paper series[END_REF].

In the case of metals, [START_REF] Chen | Understanding world metals prices-returns, volatility and diversification[END_REF], investigated the relationship between precious and industrial metals from 1900 to 2007, and found the consequences of a shock on a metal's price depends on the category to which that metal belongs, and that any spill-over effects are only marginal. Among precious metals, gold and silver serve different industrial purposes, while at a financial level the two metals can be substituted one for the other. The two metals therefore are closely related. [START_REF] Escribano | Investigating the relationship between gold and silver prices[END_REF], [START_REF] Ciner | On the long run relationship between gold and silver prices: A note[END_REF], Lucey and Tully (2006a) and [START_REF] Sari | Dynamics of oil price, precious metal prices, and exchange rate[END_REF] find a strong cointegration relationship between the two precious metals, that suffered breaks at certain times.

The relationship between the prices of oil and the precious metals is far from simple. Due to its economic value, oil is the most traded among commodities, and its price is very volatile.

Precious metals are also traded for both industrial use and for their value in hedging strategies. The possible relationship between the variations in the prices of these two groups of commodities is an important issue in the econometric literature. [START_REF] Plourde | Crude oil prices between 1985 and 1994: how volatile in relation to other commodities?[END_REF] noted that oil price is more volatile than the prices of gold and silver. [START_REF] Baffes | Oil spills on other commodities[END_REF] found that precious metals exhibited a high pass-through, showing their high reactivity to crude oil price. [START_REF] Zhang | The crude oil market and the gold market: Evidence for cointegration, causality and price discovery[END_REF] found a strong relationship between oil and gold, with a unidirectional causality from oil to gold. [START_REF] Sari | Dynamics of oil price, precious metal prices, and exchange rate[END_REF] pointed out the existence of strong correlations among precious metals prices in the short run, but no cointegration in the long term. They also found that oil had a slight impact on precious metals. Oil and silver had a bidirectional relationship whereas the linkage between oil and gold was nearly unidirectional. The relationship between oil and platinum was weak in both directions.

Some recent literature has tended to focus on the impact that macroeconomic determinants such as business cycles, the monetary environment, and the financial markets might have on the volatility of commodity prices. It appears that precious metals are most of the time weakly sensitive to news (see [START_REF] Christie-David | Do macroeconomics news releases affect gold and silver prices[END_REF], [START_REF] Cai | What moves the gold market?[END_REF]), with the exception of gold, which assumed the role of a safe haven in the event of bad news [START_REF] Roache | The effects of economic news on commodity prices[END_REF], [START_REF] Hess | How do commodity futures response to macroeconomic news?[END_REF]). This finding is confirmed by [START_REF] Batten | The macroeconomic determinants of volatility in precious metals markets[END_REF], whose conclusion lends support to the view that gold and silver may not be considered to constitute a single group of assets. However, these findings are inconsistent with the results of [START_REF] Elder | Impact of macroeconomic news on metal futures[END_REF], who find metals have instantaneous but short responses to news. That also confirms the results of [START_REF] Sjaastad | The price of gold and the exchange rate[END_REF] and [START_REF] Sjaastad | The price of gold and the exchange rates: Once again[END_REF] that validate the market efficiency hypothesis for the international gold market. The question of how oil and the exchange rate are related has been widely discussed, both from a theoretical and an empirical point of view. Krugman (1983b) developed three oversimplified models for the various channels related to oil and affecting exchange rates (real factors, financial factors, and rational speculation). [START_REF] Krugman | Oil and the dollar[END_REF] and [START_REF] Golub | Oil prices and exchange rates[END_REF] provide a theoretical guide to an understanding of the effect of oil price as a determinant on the exchange rate of the US dollar. From an empirical point of view, it seems clear that the price of oil and exchange rate have a stable, long-run relationships with a Granger causality from oil prices to the dollar exchange rate (see [START_REF] Throop | A generalized uncovered interest parity model of exchange rates[END_REF], van Amano and Norden (1998) and [START_REF] Coudert | Oil price and the dollar[END_REF]).

Following Krugman (1983b) who proposes a structural model of oil and exchange rates, we intend to study in more detail the relation between real factors, financial factors, and rational speculation. Our study has three levels of analysis. Based on a tree-like structure, we examine univariate volatility dynamics for exchange rates, S&P500 price indices, and prices of oil and precious metals.

Methodology

The Hidden Markov Decision Tree

In this paper, we study the following problem: what is the probability of being in a regime of correlation rather than another given a volatility regime? Our approach differs from previous studies of the relationship between univariate volatility and correlations, such as those of Longin andSolnik (1995, 2001) or [START_REF] Ang | Asymmetric correlations of equity portfolios[END_REF], which provide tests of the conditional correlation structure. We specify a Hidden Markov Decision Tree (HMDT) to determine probabilistically the relationship between volatility and correlations. This model was first introduced by [START_REF] Jordan | Hidden markov decision trees[END_REF] and can be seen as an extension of a pure Hidden Markov model (HMM).

From this standpoint, the HMDT can be regarded both as a factorial HMM and as a coupled HMM.

The factorial hidden Markov model (FHMM) was proposed by [START_REF] Ghahramani | Factorial hidden Markov models[END_REF].

As an extension of the basic HMM, this model assumes that each state variable is factored into several state variables, each of them with its own independent Markovian dynamics, and that the output is the combination of the several processes that underlie the state variables (see figure 1(b)). Because the model has various independent hidden Markov models in parallel, the resulting state space of the model is the cartesian product of the parallel sub-processes. The coupled hidden Markov model (CHMM) (see [START_REF] Brand | Coupled hidden markov models for modeling interacting processes[END_REF]) is also an extension of the classical HMM and refers to a model with several HMMs whose Markov chains interact together (figure (a) HMM. Here, s t , s 1 t and s 2 t are variables follow an hidden Markov chain, ǫ t the observations and x t an (optional) variable in input.

1(c)).
s 2 t s 2 t-1 s 2 t+1 s 1 t s 1 t-1 s 1 t+1 ǫt ǫt-1 ǫt+1 (b) Factorial HMM. s 2 t s 2 t-1 s 2 t+1 ǫt ǫt-1 ǫt+1 s 1 t s 1 t-1 s 1 t+1 ǫt ǫt-1 ǫt+1 (c) Coupled HMM. s 2 t s 2 t-1 s 2 t+1 s 1 t s 1 t-1 s 1 t+1 ǫt ǫt-1 ǫt+1 xt xt-1 xt+1 (d) HMDT.
In this framework, an HMDT provides a factorized state space that leads to a state space decomposition by level of Markov chain. Then the constraint of a level on the following is done via a coupling transition matrix which produces the ordered hierarchy of the structure. As the links between decision states are driven with Markovian dynamics, and the switch from one level to the following one is done via a coupling transition matrix, this architecture gives a fully probabilistic decision tree.

Hidden Markov decision tree for correlations

Starting point

The model we propose in this paper is based on the dynamic correlation class of models introduced by [START_REF] Engle | Theoretical and empirical properties of dynamic conditional correlation multivariate garch[END_REF]). Given y t , a K dimensional time series of length T , the DCC class model assumes the form:

y t |F t-1 iid ∼ N (0, H t ) (1)
where F t-1 refers to the information set at time t -1. The conditional variance-covariance matrix of returns y t is expressed as follows:

H t = D t R t D t (2)
where R t is a K × K constant correlation matrix. The matrix D t is a K × K diagonal matrix containing univariate time-varying standard deviations:

D t = diag{h 1/2 i,t } (3)
for i = 1, ..., K. Getting the matrix D t is generally referred as the so-called degarching filtering required to construct standardized residuals expressed as:

ǫ t = D -1 t r t (4)
The conditional correlations are simply the expectation of the standardized residuals:

E t-1 [ǫ t ǫ ′ t ] = D -1 t H t D -1 t = R t (5)
Many formulations have been proposed for R t , and we refer to [START_REF] Bauwens | Multivariate garch models : a survey[END_REF] and [START_REF] Silvennoinen | Multivariate garch models[END_REF] for recent surveys in this field of research. The model we use in this study is a special case of the dynamic correlation model which allows one to classify the variances as low-regime or high-regime in a first step, to set up a rule to distinguish highcorrelation regimes from the low-correlation regimes in a second step, while in a third step it links the state of the volatilities to the state of the correlations probabilistically.

Hidden tree structure

As pointed out in Section 3.1, an HMDT can be viewed as a factorial and coupled HMM.

Decomposition of the factorial HMM is used to divide the space of the time series into low and high conditional variances for each series, and low and high for the sequences of correlations.

The transition matrix used in our model can be either static or time-varying, and can depend on either endogenous or exogenous variables 1 . Each time series has its own conditional variance 1 To introduce time-varying transition probabilities, one can use the specification proposed by [START_REF] Diebold | Regime switching with time-varying transition probabilities[END_REF]. Then, transition probabilities are assumed to follow a logistic function of an endogenous or exogenous variable, that is represented as the input in the figure 1(d).

driven by a transition matrix. As we partition the space of the univariate conditional variance in two subspaces, low and high variance, the transition matrix for the variance of the k th time series is of size 2 × 2 and can be expressed as follows:

P k vol =   p k 11 1 -p k 22 1 -p k 11 p k 22   (6)
Since all univariate volatility processes have the same Markovian dynamic specification, individual transition matrices can be aggregated to constitute the first level of the decision tree.

Individual HMMs are aggregated in a factorial HMM representation. The dynamic of the univariate volatility level can be represented in a general transition matrix P vol by the cross product of the transition matrices of univariate volatility models P k vol :

P vol = K i=1 P i vol (7)
This factorial representation allows a representation of all the dynamics of the K univariate volatilities containing 2 states with a single transition matrix of size 2 K × 2 K .

The same specification is used for the second level. This level discriminates between low and high correlations. Thus, the decision step is represented by a 2-by-2 transition matrix written as:

P corr =   p c 11 1 -p c 22 1 -p c 11 p c 22   (8)
whose elements can be both static and time-varying. Given the transition matrices of the first and the second levels, the partition of the space is represented by transition matrix P expressed as:

P = P vol ⊗ P corr (9) of size 2 K+1 × 2 K+1 .
Given this space partition the first and the second levels are then linked. This relation is fully probabilistic and attributes a weight to the decision related to the correlation given the decision of the univariate volatility. Formally, this link has been obtained with the use of a coupling matrix, which is the 2-by-2 transition matrix of an abstract Markov chain that does not directly emit observations:

P coupl =   c 11 1 -c 22 1 -c 11 c 22   (10)
To summarize the foregoing information on the probabilistic decision tree: the entire ordered hierarchy is based on a set of HMMs which partition the space. While the partitioning of space, as usually considered in binary tree models, is a recursive process following binary rules, our model has the sort which is static and defined a priori. The decision process occurs according to a cascade of ordered synchronous HMMs. The coupling process lacks the Markovian property, but can be treated as a vector of probabilities whose sum is equal to unity. By establishing a link between the HMMs, the coupling process captures the inter-process influences.

Specification for volatilities and correlations

In our decision tree, the first level distinguishes between low and high volatility. Thus, this decision step parameterizes a univariate regime-switching GARCH model (within the literature on time series, many specifications have been proposed, e.g. [START_REF] Hamilton | Autoregressive conditional heteroskedasticity and change in regime[END_REF], [START_REF] Gray | Modeling the conditional distribution of interest rates as a regime-switching process[END_REF], [START_REF] Dueker | Markov switching in garch processes and mean-reverting stock-market volatility[END_REF], [START_REF] Klaassen | Improving garch volatility forecasts with regime-switching garch[END_REF][START_REF] Haas | A new approach to markov-switching garch models[END_REF]). The second level discriminates between low and high conditional correlations and therefore one can use a model among those proposed in the literature (see [START_REF] Billio | Multivariate markov switching dynamic conditional correlation garch representations for contagion analysis[END_REF], [START_REF] Pelletier | Regime switching for dynamic correlations[END_REF], [START_REF] Haas | Multivariate regime-switching garch with an application to international stock markets[END_REF]).

For both volatilities or correlations there is no constraint on the choice of one specification rather than another, and only depends on the wishes of the modeller in terms of the dynamics, parsimony, etc.

Estimation

Estimation of the model is done with maximum likelihood and this can be done in one or in more steps. However, as often happens in the multivariate GARCH literature, because of the high number of parameters induced by the number of series, the one step estimation remains at best challenging, if not impossible from a numerical point of view.

One step estimation

With the assumption of normality, the log-likelihood can be written:

L = - 1 2 T t=1 K log(2π) + log(|H t |) + y ′ t H -1 t y t (11)
To be able to maximize the likelihood we need to make inferences on the state of the various Markov chains of the model. The strategy developed in our paper is to convert the complex dynamic of the factorial and coupled HMM into a simple regular HMM so that we can use standard tools for filtering and smoothing probabilities. A conversion relationship between coupled HMM and standard HMM has been proposed by [START_REF] Brand | Coupled hidden markov models for modeling interacting processes[END_REF]. Given P S|S and P S ′ |S ′ two transition matrices, P S|S ′ and P S ′ |S two coupling matrices, the relationship of [START_REF] Brand | Coupled hidden markov models for modeling interacting processes[END_REF] is given by:

(P S|S ⊗ P S ′ |S ′ ).R(P S ′ |S ⊗ P S|S ′ ) (12)
where ⊗ denotes the Kronecker product and R is a row permute operator swapping fast and slow indices. In our case, because we use only one coupling vector in a downward direction, the regular HMM representation P reg of our model can be expressed as:

P reg = (P vol ⊗ P corr ) • (P coupl ⊗ (ιι ′ )) ( 13
)
where • denotes the Hadamard product, with ι a vector of ones of length 2 K+1 . The conversion into regular HMM allows us to use the standard tools for the inference of Markov chains. Let ξ jt be the probability to be in regime j given the information set available at time t -1 and η jt the density under the regime j. The probability to be in each regime at time t given the observations set up to t, written ξt|t , can be calculated using the following expression of the so-called Hamilton filter:

ξt|t = ( ξt|t-1 • η t ) 1 ′ ( ξt|t-1 • η t ) (14)
and:

ξt|t+1 = P reg × ξt|t (15)
Each state of the regular representation corresponds to a combination of possible cases, e.g. the first series in a high or a low volatility period, the second in high or low, and so on, and similarly for the correlations.

Multi-step estimation

Multi-step estimation is in fact composed of three steps. The parameter space θ is split into three subsets: θ 1 for the parameters of the univariate volatility models, θ 2 for the parameters of the correlation model, and θ 3 for the parameters of the coupling matrix. Following [START_REF] Engle | Dynamic conditional correlation -a simple class of multivariate garch models[END_REF], the log-likelihood can be written as:

L(θ 1 , θ 2 , θ 3 ) = L v (θ 1 ) + L c (θ 1 , θ 2 ) + L m (θ 1 , θ 2 , θ 3 ) ( 16
)
where L v (θ 1 ) refers to the volatility term, L c (θ 1 , θ 2 ) the correlation term, and L m (θ 1 , θ 2 , θ 3 ) the coupling matrix term. Then the log-likelihood of the volatility component can be written as:

L v (θ 1 ) = - 1 2 T t=1 K log(2π) + log(|D t | 2 ) + y ′ t D -2 t y t (17)
The correlation term is:

L v (θ 1 , θ 2 ) = - 1 2 T t=1 log(|R t |) + ε ′ t R -1 t ε t -ε ′ t ε t (18)
and, finally, the coupling matrix term is:

L m (θ 1 , θ 2 , θ 3 ) = - 1 2 T t=1 K log(2π) + log(|H t |) + y ′ t H -1 t y t ( 19
)
For each step, we use Hamilton's filter. The multi-step estimation reduces a complex estimation problem into a sequence of simple estimations. Although it seems advisable to test several initial conditions, the multi-step estimation procedure greatly reduces the time needed and the numerical complexity for the estimation of the model compared to the one-step estimation.

Data and empirical results

Descriptive statistics

We use daily time series data (five working days per week) for WTI crude oil, euro/USD exchange rate, S&P500 index, gold, silver and platinum. All the data had been taken from The main statistical features of the returns are described in indicate more complex dynamics, with sudden extreme variation at both the beginning and the end of the period. Engle's LM1 test with five lags confirms the absence of serial correlation and the Engle's DCC test strongly rejects the hypothesis of constant correlations.

Univariate volatilities

In our decision tree, the first level distinguishes between low and high volatility. Thus, this decision step parameterizes a univariate regime-switching GARCH model. In the time series literature, many specifications have been proposed. Most of them have the drawback that they require approximation schemes to avoid the path dependency problem (see [START_REF] Gray | Modeling the conditional distribution of interest rates as a regime-switching process[END_REF] and [START_REF] Klaassen | Improving garch volatility forecasts with regime-switching garch[END_REF]). Apart from this numerical aspect, specifications with approximations can involves difficulties in the interpretation of the processes corresponding to each regime. These considerations led us to use model of [START_REF] Haas | A new approach to markov-switching garch models[END_REF], which can be expressed as follows:

     h 1,t . . . h N,t      =      ω 1 . . . ω N      +      α 1 . . . α N      y 2 t-1 +      β 1 . . . β N      •      h 1,t-1 . . . h N,t-1      (20)
where • stands for the Hadamard product. The stationarity condition implies α n + β n < 1 for each n = 1, ..., N . Besides its computational advantages, this approach has a clearcut interpretation. It assumes that the conditional variance can switch between N separate (G)ARCH models that evolve in parallel. Based on the same idea, we also use a simple Markovswitching ARCH(1) model:

     h 1,t . . . h N,t      =      ω 1 . . . ω N      +      α 1 . . . α N      y 2 t-1 (21)
which is stationarity if ω > 0 and 0 < α n < 1 for each n = 1, ..., N .

We first estimated the MS-GARCH(1,1) model for each series of returns. When this model did not give significant results, we applied the MS-ARCH(1) specification3 . Both models were estimated using maximum likelihood and four random vectors of starting points. In addition to the value of the likelihood, the choice between MS-GARCH( the GARCH has a conditional variance that is a linear function of the lagged squared values of the series and those of its own lags. In our case, the problem arose with the estimation of the parameter β, associated with the lagged variance. Typically, when applying MS-GARCH(1,1) for oil, gold, and silver, the estimated value of the parameter β is close to 0.9 in one regime and 0.4 in the other regime. This means that it is difficult to represent the turbulent period with a GARCH model due to the extreme variations that might occur during a short period. When the parameter β is near to unity, the impact of a shock on volatility appeared to be a persistent phenomenon. In the case where β is equal to zero (the ARCH model), the volatility can have a very sharp response to shocks. We can conclude from the foregoing that the volatility of oil, gold, and silver responded strongly to shocks, while shocks had a more persistent, lingering effect on the volatility of the exchange rate, S&P500, and platinum.

Another finding thrown up by the first level of our analysis relates to the periods during which changes in a regime occur. This information is provided by the smoothed probabilities.

Hereafter, and for the sake of simplicity, we shall consider a variable to be in a given regime when the smoothed probability is at least equal to 0.99. We shall also refer to the regime associated with low volatility as regime 1, and to the regime associated with turbulent volatility as regime 2. One can distinguish the three groups given the dating of the smoothed probabilities.

The first group included exchange rates and oil. For the exchange rates, the dating of the (19th April 2006). However, the duration of that phase of regime 2 is shorter in the case of gold than it is in the case of silver, ending respectively on 23rd October 2006 and 20th October 2006. In the case of platinum, the phase ended in late November.

The second phase of regime 2 appeared during the Subprime crash, its timing varying with the specific metal. As in the previous case, we observed that resistance to shocks was lower in the case of gold than it was in the case of the other two metals. The third phase of regime 2 ended on 12th January 2012 for gold, in late February 2012 for silver, and showed no end in the case of platinum. To sum up our results for the period under review, gold and silver showed themselves to be very reactive to economic shocks, while platinum responded slowly.

Correlations

The specification used for the correlation was that of [START_REF] Pelletier | Regime switching for dynamic correlations[END_REF], who proposes a Markovswitching structure for the correlation process by imposing constant correlations in each regime and establishing the switch of each other through a Markov chain of order one. Called Regime Switching for Dynamic Correlation (RSDC), the choice of the RSDC model was motivated by the idea of having an easily interpretable model from an economic point of view. Formally, the RSDC model assumes that the conditional correlation matrix R t has the following form:

R t = N n=1 1 {sn=i} R n ( 22
)
where {s t } t∈N is a sequence of a homogeneous first order Markov chain with N states. R n is a conditional correlation matrix of size K × K where R n = R n ′ for n = n ′ . In our analysis, we use two regimes that refer respectively to constant correlation matrices R 1 and R 2 .

The model has been estimated using maximum likelihood and four random vectors of starting points. Standardised residuals have been computed given the de-garching filtering of the last The first result of the correlation level is about the dating of the regimes. The sample begins in non-crisis regime up to the end of January 2009. In our analysis, the break in the correlations occurs just after the last quarter of 2008 which is considered as the peak of the Subprime crisis.

It is interesting to note that, for correlations, regime 2 began almost just when it was ending for the volatility of the univariate series of the exchange rates, oil prices, and the prices of gold and silver. There was a short interval between these occurrences of regime 2 depending on whether the analysis was conducted at a univariate level or at a multivariate level. Therefore, while all the variables had their own timing for correlations, which sometimes differed from the chronology of the Subprime crisis, the break in the correlations coincided exactly with the crash that came at the end of 2008. The correlation dynamics also experienced a break from the middle of March 2011 to the end of that month, more precisely, from 14th March 2011 to 28th March 2011. One might assume that the shock behind that break was Tohoku earthquake of 2011, and the tsunami which resulted which caused massive damage to Japan's economy. The effect of that happening is reflected in our series, but it might be interpreted as a statistical artifact. Equally interesting would be the thought that the tsunami might not have induced the break at the univariate level of analysis, but that it did have an effect at the multivariate level.

The second result concerns the level of the correlations. Our results clearly point to the distinction between two main types of econometric relation: correlations that are quasi-constant or constant in time throughout the period under study, and correlations that increase while switching from regime 1 to regime 2. We only had a single correlation outside of these two main groups, which showed a significant decrease when switching from regime 1 to regime 2. The first type of relation implies stability across time, in times of crisis as in normal times. The correlation value in our study did not remain unchanged between the two regimes, but the variation was too marginal to be considered significant. That was the case for the correlations between oil and gold (0.33 to 0.30) and silver (0.35 to 0.41). It was not surprising that the correlations of gold and silver were constant during the sample period (0.80 to 0.82). This finding tended to confirm that the two precious metals move together on the commodities market. It was more difficult to explain the steady correlation between silver and the exchange rates (0.49 to 0.43). The second group showed the behaviour one encounters in the literature on econometrics, namely, an increase in the correlation for platinum and gold and silver remained steady: an increase of 0.13 for gold, and 0.16 for silver. It is also notable that platinum was highly correlated in regime exchange rate, S&P500, oil, gold, silver and platinum.

R 1                  - -0.
- - - - - -                  R 2                  - 0.
1 with gold and silver, with a value of 0.58 for both the metals. One would thus have two subsets in this group: gold and silver in one subset, and platinum in the other. Oil showed an increase in the correlation with macroeconomic variables, S&P500 (0.01 to 0.57) and exchange rates (0.22 to 0.45). Platinum was the only precious metal on which oil showed an increased dependence (0.26 to 0.44). The upward variations of S&P500 correlations with gold and silver were very close (0.01 to 0.15 and 0.08 to 0.30) respectively. The third group represents correlations which decreased between regime 1 and regime 2. There was only one pair of correlations, between gold and the exchange rate, which fell (from 0.51 to 0.35). The decrease of EURO/USD is consistent with those of [START_REF] Ciner | Hedges and safe havens: An examination of stocks, bonds, gold, oil and exchange rates[END_REF] who argue that gold can act a safe haven against exchange rate movements.

The link between volatilities and correlations

The estimated results of the coupling transition matrix define the relationship between the first and the second levels, that is, the linkages between the univariate volatilities and the correlations. We used a coupling transition matrix, with one given regime of volatility associated with each regime of correlation. From the previous results, this vertical relation needed only a single 2-by-2 coupling matrix for estimation. The model was estimated using maximum likelihood and four random vectors as starting points. The smoothed probabilities for a variable to be in regime 1 for correlations given that the univariate volatilities are in low or turbulent regime are plotted in figure 6 and estimated parameters are in table 4.

The first results are related to all series. The link between the exchange rates, S&P500, and WTI was not identified when all the series had low volatilities. In the cases of turbulent volatilities, the probability of regime 1 for the correlation was equal to one from the beginning of the sample period to early 2009. This was also the case for the period covering the Tohoku earthquake (figure 6(d)).

Conclusion

We have studied the relationship between univariate volatilities and correlations among euro/US dollar exchange rate, S&P500, WTI crude oil, and three major precious metals, namely gold, silver and platinum, over the period 3 January 2005 to 19 October 2012. The contributions of our paper are of twofolds: methodological and empirical.

Results were obtained using a model based on the Hidden Markov Decision Tree (HMDT). Introduced by [START_REF] Jordan | Hidden markov decision trees[END_REF], HMDT is in fact an extension of the Hidden Markov Model (HMM) and is the opposite of the classical deterministic approach based on a binary decision tree. This specification allows one to quantify probabilistically the relationship inside a partitioned space. The model is fully probabilistic. The tree used in this paper has two levels:

the first discriminates between low and turbulent volatilities, the second between low and high correlations. At each level, the classification was carried out with a Markov-switching model.

The two levels were then linked such that one would have the probability to be in a regime of correlation given the regime of volatilities. As far as we know, unlike deterministic binary tree models, stochastic tree specifications have not been used up until now in the study of the linkages beween volatilities and correlations.

The empirical results highlight several elements of the dynamic of our dataset. First, we found that the volatility of oil, gold, and silver responded strongly to shocks. Shocks had a more persistent effect on the volatility of the exchange rates, S&P500 indices, and platinum prices. The exchange rates, and oil plunged into the turbulent volatility regime at the peak of the Subprime crisis in the fall of 2008 to the spring of 2009. The S&P500 switched into the turbulent regime in the fall of 2007, and remained there until the end of the sample. Precious metals showed nearly the same dynamics with three periods in the turbulent regimes. While all precious metals were sensitive to the same shocks, the lengths of their reaction were not the same. Secondly, our model showed that the correlations slipped into regime 2 just after the last quarter of 2008, coinciding with the Subprime crisis. The correlations switched a second time, though briefly, during the Tohoku earthquake. Third, the vertical probabilities highlight the relationship between the volatilities and the correlations. We found that low volatilities were associated with low correlations from the beginning of the period to the early 2009. The conclusion was the same with a turbulent regime of volatilities, that was linked to a low regime of correlations, which again was linked to a probability of one with low correlations during the first half the period and the Tohoku earthquake. In the case of precious metals, turbulent volatilities were clearly linked to the rise of the correlations. How precisely the low volatility regime of precious metals is linked to the state of the correlations, would requiel a somewhat lengthy and complicated explanation of the phenomenon. The results for gold and oil threw up one of the most surprising findings of our study, specifically that a low-volatility regime had a probability of one of being associated with high correlations throughout the sample period. The turbulent-volatility regime is itself linked to low correlations. However, our model was unable to identify the relation between the correlation regimes and low volatility in the case of the group consisting of the exchange rates, the equity indices, and the oil prices. Nevertheless, the model provided an indication that turbulent-volatility regimes were linked to low-volatility regimes during the first half of the sample period and particularly after the Japanese earthquake. Our results tend to contradict the findings of previous studies, specifically that correlations increase in bear markets, e.g. [START_REF] Ang | Asymmetric correlations of equity portfolios[END_REF], Longin and[START_REF] Longin | Extreme correlation of international equity markets[END_REF][START_REF] Campbell | Increased correlation in bear markets[END_REF], all of which had focused their attention on international equity returns or indices. A contradiction such as this could be a result of the choice of the series and the sample period. Our sample consisted of two macroeconomic variables and four commodities. Our results showed that within the group of commodities studied, oil and precious metals possessed their own distinct dynamics, and showed their own type of response to shocks.

The model developed in this paper could prove to be a useful tool with which to deepen our understanding of volatility and correlation. The specification that we used remains very simple, with only two levels and without introducing exogenous variables. It could be interesting to build complex Markovian tree-based structures to understand the determinant of correlations.

Figure 1 :

 1 Figure 1: Hidden Markov model (1(a)), Factorial HMM (1(b)), Coupled HMM (1(c)) and Hidden Markov Decision Tree (1(d)) as graphical models.

the

  Bloomberg database 2 . The sample contains 1961 observations and covers the period 3 January 2005 to 19 October 2012. Thus, our dataset covers the period where the subprime crisis appeared, which is interesting for a better understanding of the dynamics and the comovement of major commodities during a financial crisis. The six series are plotted in figure 2(a). We use 100 times the difference of the logarithm of each series minus the sample mean to filter the series. The returns are plotted in figure 2(b).

Figure 2 :

 2 Figure 2: Plots of EUR/USD, S&P500, WTI, Gold, Silver and Platinum series in level (subplot 2(a)) and in returns (subplot 2(b)).

Figure 3 :

 3 Figure 3: Estimated univariate volatilities (subplot 3(a)) and smoothed probabilities (subplot 3(b)).

  change in regime is very clear. The exchange rate started to fall into regime 2 on 9th July 2008. This occurrence is clear but relatively short, since the regime ended on 23rd September 2008 with the probability of being in regime 1 equal to 0.999. This result accords well with the timing of the exchange rate fluctuation. The value of one euro was 1.6038 US dollar on 15th July 2008, a value that was the currency's maximum for the year, but it suffered a considerable fall on 24th October the same year to 1.2530 US dollar. Thus regime 2 of the exchange rate was characterized by two attributes: a globally high value (the constant ω in the GARCH process larger in regime 2 than in regime 1 by a factor of 10), and very high volatility. The timing was about the same in the case of oil, regime 2 arriving on 18th September 2008, and ending on 6th May 2009. As has been said before, the volatility of oil is best taken into account by an MS-ARCH(1,1) model. Once again, regime 2 showed a high volatility component with variation ranging from 120.92 USD on 19th September 2008 to 31.41 USD on 19th December 2008. The second group had only one series, namely the S&P500, and the dynamic of its volatility was comparatively simple. The index tumbled into regime 2 around 5th September 2007, and remained there until the end of the period. During regime 2, the stock market index rose to its maximum on 8th October 2007 at 1565.2, and dropped to its minimum at 676.03 on 6th March 2009. The third group consisted of precious metals. The smoothed probabilities lent support to the notion of clubbing the metals into a single group, given their similar volatility patterns. Now, while the three precious metals exhibited three phases of regime 2 in their responses to shocks, the durations of these responses varied considerably. The chart of the smoothed probabilities shows the onset of the first appearance of regime 2 with one month's gap between gold (15th May 2006) and silver

Figure 5 :

 5 Figure 5: Smoothed probability for RSDC.

Figure 6 :

 6 Figure 6: Smoothed probabilities to be in regime 1 for correlations given univariate volatilites are in low or turbulent regime.

table 1 .

 1 The daily returns have extreme maxima and minima. At the same time, each of variables has a very low mean, close to zero. Given the value of the median, variance and standard deviation, one can conclude at the presence of extreme variation of returns at certain times. All series are slightly skewed and exhibit a significant kurtosis. The Jarque-Bera test shows us that all series have fat tails with a non-normal distribution. Starting from the beginning of 2005 to 2012, our all series clearly exhibit extreme variation in the middle of the period, corresponding to the beginning of the financial crisis in August 2007. According to these basic statistics, returns of precious metals

	1.8		EUR/USD			2000		S&P 500
	1.6					1500
	1.4
	1.2					1000
	03/01/05 1	15/12/06	01/12/08	11/11/10	19/10/12	03/01/05 500	15/12/06	01/12/08	11/11/10	19/10/12
	150		WTI			2000		Gold
	100					1500
						1000
	50					500
	03/01/05 0	15/12/06	01/12/08	11/11/10	19/10/12	03/01/05 0	15/12/06	01/12/08	11/11/10	19/10/12
	60		Silver			2500		Platinum
	40					2000
						1500
	20					1000
	03/01/05 0	15/12/06	01/12/08	11/11/10	19/10/12	03/01/05 500	15/12/06	01/12/08	11/11/10	19/10/12
					(a)
	4		EUR/USD			20		S&P 500
	2					10
	0
	-2					0
	03/01/05 -4	15/12/06	01/12/08	11/11/10	19/10/12	03/01/05 -10	15/12/06	01/12/08	11/11/10	19/10/12
	40		WTI			10		Gold
	20					5
						0
	0					-5
	03/01/05 -20	15/12/06	01/12/08	11/11/10	19/10/12	03/01/05 -10	15/12/06	01/12/08	11/11/10	19/10/12
	20		Silver			10		Platinum
	10					0
	0
	-10					-10
	03/01/05 -20	15/12/06	01/12/08	11/11/10	19/10/12	03/01/05 -20	15/12/06	01/12/08	11/11/10	19/10/12
					(b)

Table 2

 2

	: Estimated parameters for Markov-switching ARCH(1) and
	GARCH(1,1) models. The standard errors are in brackets.
	The results of the model specification appear to be quite clear-cut. The volatility of the ex-
	change rates, S&P500 index, and platinum was successfully modelled by the MS GARCH(1,1)
	specification, whereas the volatility of oil, gold, and silver was better modelled by the parsimo-
	nious MS ARCH(1) specification. By definition, the difference between the two models was that

  The turbulent period began on 18th March 2008 for gold, 29th August 2008 for silver, and 29th February 2008 for platinum. That period ended on 4th May 2009 for gold, 22nd June 2009 for silver, and 4th June 2010 for platinum. Clearly, platinum remained longer in regime 2 than did the other two metals. We observed a strong upward movement in precious metals with a delayed effect for silver and platinum in comparison with gold. Regime 2 appeared on 23rd August 2011 for gold, on 14th September 2011 for silver, and 28th October 2011 for platinum.

Table 3 :

 3 Estimated parameters correlation model RSDC. The standard errors are in brackets. The order of the columns (and rows) is as follows:

Table 4 :

 4 Estimated parameters for transition matrix linking the volatility and the correlations. The standard errors are in brackets.

The Bloomberg's ticker for WTI is USCRWTIC, that is the West Texas Intermediate (WTI) Cushing Crude Oil Spot Price (closing price). The tickers for exchange rates and equity index are respectively EUR Curncy and SPX. For precious metals, the tickers for gold, silver and platinum are XAUUSD, XAGUSD and XPTUSD.

The choice between MS-ARCH or MS-GARCH was not done using standard information criterions as AIC or BIC. As we must discriminate between two regimes, the selection was done by the ability of the model to correctly identify two regimes, with an order of preference from MS-GARCH to MS-ARCH.