with the mackey-glass(1977)’s model in nonlinear …we retain two speciﬁc equations:...

« Can the Lasota(1977)’s model compete

with the Mackey-Glass(1977)’s model

in nonlinear modelling of financial

time series? »

Rachida HENNANI

DR n°2015-09

Can the Lasota(1977)’s model compete with theMackey-Glass(1977)’s model in nonlinear modelling

of financial time series?

Rachida HENNANI∗Lameta (UMR CNRS 5474)University of Montpellier

Preliminary version

Abstract

The existence of nonlinear structures in the mean equation leads someauthors [43, 44] to model financial time series by a Mackey-Glass equation,which is a differential equation with delay. We propose, in this paper, tocompare the contributions of the [52]’s equation in the modelling of nonlinearstructures in the mean equation with that of [48], published the same yearbut which may lead to different results in finance. Theoretical results pointout that these two equations can describe mean dynamics’ of financial timeseries. These dynamics reflect the interaction between two types of agents,fundamentalists and chartists, that creates chaotic structures. To verify this,we apply these two models to two Europeans stock markets indices [CAC40 and DAX 30] on the period [2003-2011]. We show the adequacy of thesemodels, associated with a GARCH specification, to financial time series, com-paratively to the ARMA-GARCH model. Moreover, it seems that the [48]’smodel is more suitable than the [52]’s model for strongly leptokurtic financialtime series: these findings are based on the backtesting results’ conducted onVaR forecasts’.

Key words: nonlinearity; Lasota’s model; Mackey-Glass’s model; GARCH;chaos; Value-at-Risk

Codes JEL: C01; C22; C58

∗Corresponding author: Department of Economics, LAMETA (UMR CNRS 5474), University ofMontpellier 1, Avenue Raymond Dugrand, Site de Richter CS 79606, Montpellier Cedex 1, France.E-mail : [email protected]

1

2

Introduction’Give me only the equation of motion, and I will show you the future of the

universe’ [37].

This sentence, taken from Economics, Philosophy and Physics reflects the needto model the world around us. Understand, integrate and predict are the key wordsof this constantly search of the "good" model: understanding the changing dynamics,integrate existing structures in a model to achieve good predictions. The impera-tives related to the prediction are numerous: it must allow anticipating some events,it can guide the decisions of regulators, and it also reveals some information on thedata evolution. In finance, the prediction is essential because it helps agents toanticipate movements in the financial markets, including financial crises. It is withthis objective that models aimed at integrating the main structures of financial timeseries has been developed. Financial econometrics is marked by theoretical revivals:ARMA models are born of disinterest of structural Keynesian models but their in-ability to capture more complex structures, including nonlinear structures has led tothe development of ARCH models by [20]. The unprecedented interest generated bythese processes has led to the development of a variety of models able to take intoaccount some volatilities of financial time series. Furthermore, some linear modelsable to take into account such nonlinear features have been developed: this is thecase, for example, of regime-switching models. These different models are based onthe assumption of randomness which assumes that the data generating process is astochastic process while some authors question this assumption by referring to thepossibility of a deterministic generating process. The hypothesis of "pure" determin-ism is hardly acceptable for financial time series, so the idea of a chaotic generatorprocess has led many researchers to highlight structures of this type that may existin financial time series. This is the case of [1, 31–33, 46] among others. Anotherapproach, called stochastic chaos, combines the two conceptions. The identificationof chaotic structures in economic and financial time series is based on a complexprocedure which can lead to questionable results in economy, especially due to thenature of the used data (noisy, too short history ... for more information, see [21]).Nevertheless, a number of studies develop stochastic chaos models to take into ac-count the nonlinearities of financial time series. This is the case of [40, 41, 43–46].These authors use a Mackey-Glass model for the mean equation and a GARCH pro-cess for the variance equation. The Mackey-Glass model [52] was presented for thefirst time in 1977. It is a nonlinear differential equation with delay:

dx

dt= β

xτ1 + xnτ

− γx (1)

With τ, n > 0. According to the values taken by the parameters γ, β and n, thisequation can represent periodic orbits or chaotic dynamics. The same year and inan independent research, [48] proposed the following nonlinear differential equationwith delay:

dx

dt= βxnτ e

−xτ − γx (2)

1 THEORETICAL RESULTS 3

With τ, n > 0. These two equations have, whence their interest, a common featurein their nonlinear and identical non-monotonic nature. The popularity and infat-uation generated by the Mackey-Glass equation shaded the potential contributionsof Lasota equation. As there is a very little literature concerning it, we want toshow in this article, its ability, combined with a GARCH process, to account fornonlinearities of financial time series.We propose in a first point to revisit mathematical results of these equations. Weexplain in a second point, the contributions of these equations on the interpretationof interactions in financial markets. In a final point, we propose an empirical ap-plication to two European indices that allows us to compare these two equations.We evaluate the relevance of these models comparatively to ARMA-GARCH modelby backtesting tests conducted on VaR predictions, before making some concludingremarks.

1 Mackey-Glass and Lasota models: theoretical resultsFinancial time series dynamics’ reflect complex interactions that exist on the fi-nancial markets. Some authors choose to model only the volatility of financial timeseries. [24] explains that this choice is motivated by unjustified and practical reasons.Indeed, when one supposes that the mean returns is zero or is well described by alinear model, then one choose to ignore an important part of nonlinear structuresthat exists in financial time series. The structures considered by the mean equationare important: they represent a part of the interactions of agents on the market andcontain valuable information about the type of agents dominating the market. [53]pointed out the gap in economics to use models with delays, [36] constitute an ex-ception. The lack of economic literature on these topics is just as much surprisingas [56, 57, 62] use simultaneously the known delay between the start of the produc-tion and distribution of goods to analyze the cycle in a discrete time mathematicalframework, that would later become the Cobweb theory. In addition, a number ofresearchers have developed models based on differential equations to account for thebehaviour of economic cycles. This is the case of [28, 34, 47]. Finally, there is awide spectrum of dynamic behaviours that can be taken into account by differentialnonlinear equations with delays.We propose to analyze the characteristics of two differential equations with delays,of this form:

dxdt

= F (x(t);x(t− τ)) (3)

This type of equation incorporates a delay corresponding to the reaction time fol-lowing a shock. There is a multitude of equations of this form, but in this paper,we retain two specific equations: Mackey-Glass equation (1) and Lasota equation(2). These two equations were developed at the same time but have particularspecificities.

1.1 Mackey-Glass (1977)’s equation

[52]’s equation was introduced, for the first time, in the physiological control sys-tems. It is a nonlinear differential equation with delay able to describe many dy-


namics complex behaviours. [52] had two objectives:

• to show to theoreticians two medical examples of complex dynamics;

• to show that simple mathematical models are able to describe periodic andaperiodic dynamics, similar to those of medical studies.

The success encountered by this equation has led to the development of broad stud-ies in many disciplines. We return on the main results.[52] introduce 3 models allowing to describe respectively respiratory problems inadults with Cheyne-Stokes phenomenon (y(t)) (Model 1) and the fluctuations inwhite blood cells of patients with chronic granulocytic leukemia (Models 2 & 3).Model 1 :

dy

dt= λ− αVmy(t)yn(t− τ)

θn + yn(t− τ) (4)

Where Vm is the maximal respiration, θ and n are adjusted parameters to empiricalobservations, λ is the CO2 production rate, τ is the delay between blood oxygenationin the lungs and stimulation of chemoreceptor in the brainstem. α is a constant.Model 2 :

dy

dt= −γy(t) + F0θ

n

θn + yn(t− τ) (5)

Model 3:

dy

dt= −γy(t) + βθny(t− τ)

θn + yn(t− τ) (6)

Where F0, θ > 0, n and γ are constants. In these models, parameters are positivesand are determined by an empirical analysis. In the literature, these models andtheir variants have been studied1. Thus, [5–7,64,65,67] study a variant of model 3,the non-autonomous Mackey-Glass equation defined by :

dx

dt= −γ(t)y(t) + β(t)x(t− τ(t))

1 + xn(t− τ(t)) (7)

With γ, β ∈ C[ℝ, (0,∞)], τ ∈ C[ℝ+,ℝ+]. In this equation, all the parametersare time-dependent. Another specification derived from the model 3 is studied by[38,66,68]. It is described by the following equation:

dx

dt= −γ(t)x(t) + β(t)

∫ ∞0

P (τ)x(t− τ)dr1 + xn(t− τ) (8)

With γ, β ∈ C[ℝ+, (0,∞)], P : ℝ+ → ℝ+ is continuous.In financial econometrics, the studies on the Mackey-Glass model are based on adiscrete version of the model 3. By substituting x(t) by θy(t), we obtain:

dx

dt= −γx(t) + βx(t− τ)

1 + xn(t− τ) (9)

1To avoid any confusion, we note the time series x(t) instead of y(t)


With t > 0 and the initial condition φ for t ≥ 0 is given by x(t) = φ(t) whereφ ∈ C[[−τ, 0],ℝ+], φ(0) = 0. Many studies define the mathematical proprietiesof this model. It is the case of [29] which give the proprieties of attractiveness ofequilibriums and global asymptotic stability conditions of equilibriums . [5] definebounds and positivity, persistence and extinction conditions of solutions. [8] returnon the main results of [52]’s model and precise the stability of solutions of model 3.Stability conditions are important for identifying bifurcations of a chaotic system.These conditions allow defining a long-term equilibrium. In the financial framework,these conditions indicate the convergence of the dynamic to the equilibrium pricewhen the market is dominated by fundamentalists.Stability of solutions :The model described by the equation 9 have a trivial equilibrium. If the followinginequality β > γ > 0 is verified, the model has a positive equilibrium given by:

K = (βγ− 1)(1/n) (10)

We retain the definition of solutions stability’s, from [8] :

Definition 1 StabilityAn equilibrium solution of a model, given by the equation 10, is locally stable, if forall ε > 0, there is δ > 0, such that for each initial condition φ(t), the inequality|φ(t)−K| < δ for all t ≥ 0 implies |x(t) − K| < ε for the solution x. If, inaddition, for each solution limt→∞ x(t) = K then the solution x = K is locallyasymptotically stable (LAS). The equilibrium K is globally asymptoticallystable (GAS) for initial conditions in the open set Q0 ⊂ ℝ if K is an attractor forall solutions x(t) with initial conditions in the open set Q0 ⊂ ℝ ; i.e. limt→∞ x(t) =K and it is also locally stable.

To define the global attractivity theorem of the model described by the equation9, [8] use the following lemma:

Lemma 1 If f ′(0) < γ where f(x) = βX1+Xn then for all τ ≥ 0, the trivial equilibrium

of all equation with the form:

dx

dt= −γx(t) + f(x(t, τ)) (11)

where f(x(t, τ)) = f(x), is globally attractive.

From this lemma, we can define the following theorem [8]:

Theorem 1.1 Global attractivityIf n > 1 and β < γ then the trivial solution of the model given by the equation 10 isa global attractor.

[5] set the following theorem:


Theorem 1.2 Let the model defined by :dx

dt= −γ(t)x(t) + β(t)x(h(t))

1 + xn(h(t)) (12)

With the initial condition x(t) = φ(t), t ≤ 0, h(t) is a Lebesgue measurable func-tion which satisfies h(t) ≤ t and limt→∞ h(t) = ∞. The following hypothesis areassociated to the model :• n > 0, β(t) ≥ 0, γ(t) ≥ 0 are Lebesgue measurable functions, locally essentiallybounded.

• φ : (−∞; 0)→ ℝ is a Borel measurable bounded function, φ(t) ≥ 0, φ(0) > 0Under these hypothesis, if we have :

limt→∞

supβ(t)γ(t) = λ < 1∫ ∞

0γ(s)ds =∞

supt<0

x(t) <∞

then limt→∞ x(t) = 0 where x(t) is a solution of the equation 12.For details, see [5,8]. These results allow to define the stability theorem of solutions,independently of τ :

Theorem 1.3 Stability of solutionsThe positive equilibrium K of the model described by the equation 9 is LAS for all τif and only if one of the two following conditions is satisfied:

1. 0 < n ≤ 2 ;

2. n > 2 et βγ< n

n−2

This theorem allows to infer the global stability theorem independently of delay.Theorem 1.4 If one of the two conditions (1) or (2) is satisfied then the positiveequilibrium K of the model given by the equation 9 is GAS for all τ .Many studies have examined the sufficient conditions of global stability of the modeldescribed by the equation 9 [see [8] for details]. [55] had already concluded on similarresults for a class of Mackey-Glass equations. They showed that there can exist,depending on the values taken by the parameters, three stationary points given byx0 = 0, x0 = −1, x0 = 1. [50] provide mathematical proofs of the convergence of aset of solutions to 0 and the first intuitions about the equivalence between the localand global asymptotic stability.A discrete version of the equation 9 can be given by :

Xt = βXt−τ

1 +Xnt−τ

+ (1− δ)Xt−1 (13)

With β, n > 0 and (1−δ) = γ. It is interesting to note that the equation 13 describe,in a simple way, a complex system that can be interpreted as follow :


• Xt−τ1+Xn

t−τis the nonlinear part of the equation 13. The mathematical form of

this ratio is relatively close to that of a discount factor. In economics, thisnonlinearity can be interpreted as an update of the variable Xt−τ by using asdiscount factor 1

1+xnt−τ.

• Xt−1 is an autoregressive process of order 1.

The simplicity of the equation 13 includes complex dynamics which can be chaotic.The equation 13 has a special feature that allows the presentation of infinite dimen-sional systems. So, it is important to define previously the time interval whereinevolves the trajectory. [23] showed that increasing the parameter τ leads to an in-crease of the dimension of the attractor of 13 as in the chaotic systems.The interest of this model, in financial econometrics, is its ability to describe marketfluctuations when it is dominated by a stabilising force [43].

1.2 Lasota(1977)’s equation

[48] introduces a nonlinear differential equation with delay of the same form as thatof [52].

dxdt

= βxnτ e−xτ − γx (14)

Despite the lack of literature on this equation, it is possible to say, as [27] that thenonlinear character of the equation 14 has the same non-monotonic behaviour asequation 13. A discrete version of 14 can be given by:

Xt = βXnt−τe

−Xt−τ + (1− δ)Xt−1 (15)

With β, n > 0 and (1 − δ) = γ. The nonlinear part of 15 is interesting in severalrespects :

• it reflects nonlinear dynamics different from that of Mackey-Glass equationwhose applications to financial time series has led to conclusive results.

• the nature of the nonlinearity described by this equation is particularly inter-esting because it has an exponential term that can produce explosive dynamicsas those of financial time series in crisis periods.

The dynamics of this model are essentially based on the behaviour of the nonlinearpart:

βXnt−τe

−Xt−τ (16)[48] shows that for some values of β, there is, in [0,∞) a continuous, ergodic andinvariant measure for the equation 16. For β sufficiently large, the equation 16can describe turbulent and stationary trajectories in long run. Periodic and chaotictrajectories can be described by the equation 14. Also, if we retain τ = 1, δ =0.1, n = 8, it is possible to define two types of trajectories for the equation 13 :

• If X0 ≥ 2.5, Xt converge to a stationary solution X > 0

• If X0 ≤ 2.5, Xt converge to a stationary solution X = 0

2 CHAOS-STOCHASTIC MODELS 8

[48] shows that, under conditions τ = 1, δ = 0.4, n = 8, there is a 2-periodic solutionand all trajectories, with an initial condition X0 ≥ 3.4, are asymptotically periodic.However, for some trajectories with an initial condition X0 ≤ 3.4, the long runequilibrium is an unstable fixed point. Under the same conditions with X0 ≤ 3.3,the long run equilibrium is null. In the case where n > 0, [48] shows that, forpositives values of β, τ and γ, there is a continuous, ergodic and invariant measurefor the equation 14. The use of these deterministic differential equations for themean equation is interesting in several respects:

• these equations describe a long term dynamic that converges to an equilibriumX. If we disregard the high volatility that exist in the financial series andthus fall under the variance equation, these models can perfectly describe theevolution of the mean dynamics.

• in times of crisis, the volatility is very strong. It can also be the result of amean volatility that can be perfectly integrated in these equations to the extentthat some of the literature [42, 43] have shown that for some parameters, thedynamic described by these equations can be volatile.

• filter the nonlinearity present in financial time series by these equations notonly allows to capture the chaotic structures (since these equations are ableto describe the chaotic dynamics for certain parameters) but also periodicstructures, where the double interest of modelling financial time series by theseequations.

The interest to these equations in the modelling of financial time series lies mainlyin the ability of these models to capture the dynamics that can not be taken into ac-count by the ARMA-GARCH standard models. The presence of nonlinear structuresin the equations of the mean and variance makes this type of modelling (ARMA-GARCH) insufficient.

2 Interactions of heterogeneous agents in the financial markets:an approach by the chaos-stochastic models

The shortcomings of linear approaches in modelling financial time series lead to thedevelopment of analyses highlighting the nonlinearity and complexity of financialtime series. A number of studies had focused on nonlinearities in time series thatwere gradually incorporated into the modelling. These different results are mostlydeveloped in a stochastic approach. The recent financial crisis revealed the limitsof these contributions in the understanding, modelling and prediction of changes infinancial time series. Questioning existing models is accompanied by an invitationto researchers to be open to other disciplines to better detect, integrate, model andpredict the evolution of structures in financial time series. This need is even strongerthan understanding behaviour originally dynamic is improving and it constantly re-fer to potential contributions of chaos theory.The questioning of the neoclassical assumptions of financial markets is largely dueto the financial crises which succeed in these markets. The hypothesis of efficientcapital markets related to walking random of stock prices is according to [14] one


of the causes of the recent subprime crisis, in the sense that these assumptions arethe basis of a number models widely used by the financial world. However, theassumption of efficient capital markets is far from being observed and to overcomethese weaknesses, some propose to use complex and nonlinear stochastic models,stochastic or deterministic models or models that combine stochastic and chaoticstructures.If nonlinear stochastic models have aroused great interest among financial econo-metricians, chaotic and chaos-stochastic models have a weak interest despite theirnon-explosive nature. According to [30], these models have an attractor that can bedetected, in which the paths are evolving and providing valuable information. Thereconstruction of the attractor is therefore an important step which is increasinglyfacilitated by the development of appropriate statistical tools. The modern theoryof chaos known a significant progress so that it is now possible not only to recon-struct the attractor but also to estimate the invariants related to this attractor andmake predictions. The complexity and nonlinearities that it is possible to detect arethe result of the coexistence on the market of two antagonistic behaviours whoseinteraction creates chaotic dynamics. The analysis of chartists and fundamentalistsbehaviours was undertaken by many authors that showed that their simultaneouspresence on the market created chaotic structures. These complexities and non-linearities must be taken into account in the analysis of financial markets but thedevelopment of instruments related to these characteristics come from various dis-ciplines that allow to consider a transdisciplinary approach of financial markets.

2.1 Noisy chaotic models

The inadequacy of the deterministic approach to account for fluctuations in financialtime series leads us to consider a chaos-stochastic approach. This latter allows totake into account not only "moderate" fluctuations around a long-run equilibriumthat is the fundamental value of the asset but also the major disruptions that can becreated by speculative agents in the market. The ability of this approach to reportmore paths is to confront to conventional models, confined to a stochastic approachof financial markets and ignorant the presence of chaotic structures on mean. Itis true that the idea of a chaotic modelling can generate some reservations aboutthe final use that can be made: the forecast horizon is relatively small given thehigh sensitivity to initial conditions and the consideration of significant fluctuationsrequires the use of a chaos with high dimensions. These doubts are remedied in thechaos-stochastic approach for several reasons:• the use of a high-dimensional chaos is not always necessary since the non-

integrated fluctuations by the chaotic model can be fully integrated by thestochastic part.

• the forecast horizon is no longer linked to sensitivity to initial conditions: thelong-term equilibrium described by the chaotic part is quickly achieved sincethe empirical analysis is based on a number of important data. The strongfluctuations due to the presence of speculative agents are then integrated bythe stochastic part.


One must understand that the chaos-stochastic approach offers many models whichallows to take into account the different paths of financial time series. This diver-sity is not specific to the stochastic part of the model but is rather related to theplethora of trajectories that a chaotic equation can describe depending upon thechosen parameters.Many studies on noisy Mackey-Glass model show the interest of this approach forfinancial time series [including but not limited [18,40,41,43,45,46,54]]

2.2 Complexities of financial markets: an approach by the chaos theory

The nonlinearity in the financial time series, if it is detected, often reflects the com-plex dynamics that need to be analysed. These result from the interaction of twotypes of agents: chartists and fundamentalists. A relatively large part of the econo-metric literature justifies the presence of volatility clusters of financial time series bythe coexistence of these two categories of agents [9,22,25,49,51]. The fundamentalistapproach stipulates the existence of an intrinsic value of the asset around which theprice fluctuates. A price development around this fundamental value is the resultof a statistical noise fuelled by market rumours. The fundamentalists argue thatwhatever the market fluctuations, the price always comes back to the fundamentalvalue of the asset. In a market dominated only by fundamentalists, price fluctua-tions are zero mean and the fundamental value of assets reflects the intrinsic valueof the firm. A decrease of its fundamental value is induced by poor performance, bypoor financial results; i.e. by objective values. Speculation is not a winning strat-egy in a market dominated by fundamentalists. The technical analysis, used by 90% of traders, stipulates that the past evolution of the prices provides informationfor future evolution of prices. The central hypothesis of this approach is the repe-tition of the prices evolution which allow determining future prices based on pastprices. The use of graphical representations of prices allows to identify the markettrends for forecasting purposes. The renewal of trajectories allows to chartists toanticipate turning points. The phases of high volatility in financial time series aredue to endogenous phenomenon caused by the chartist dominance’s in the market.The behaviour of chartists is often called non-rational in the sense that they mayact contrary to market dynamics. These agents can also extrapolate the marketchanges, creating high volatility. The interaction of chartists and fundamentalistscan lead to chaotic nonlinear dynamics in financial markets. Indeed, [26, 43, 45, 46]explain that the interaction of heterogeneous agents leads to high complexity of fi-nancial markets. One reason of this complexity is given by [15, 63] which explainthat it is due to the reaction of the public and private information. [35] involve therole of investors which would induce a certain complexity. [26] justify the rejectionof the efficient market hypothesis by the stylised facts observed in empirical data.Volatility clusters that can be modelled by GARCH process are due to an interac-tion between chartists and fundamentalists. Endogenous explanation of volatilityclusters has led many authors (for example: [3, 9, 12, 26, 49]) to propose multi-agentsystems in which these two types of traders interact. If the construction of thesemulti-agent systems is profoundly different, the description of volatility clusters isthe same: the low volatility phases reflect market dominance by fundamentalists

3 EMPIRICAL ANALYSIS 11

while high volatility is due to market dominance by chartists that contribute to thedeviation of prices from their fundamental value. The proportions of chartist andfundamentalist traders are not static, they evolve depending on the type of agentsdominating the market. Thus, chartists tend to behave like the fundamentalists,abandoning their tools when the asset price’s converges to its fundamental value.In contrast, during periods of high volatility, fundamentalists prefer to entrust theirassets to chartists which extrapolate the observed volatility. The interaction be-tween the two types of agents lead to a chaotic nonlinear dynamics in the sense thatclusters of high or low volatility do not remain indefinitely, there is a turnaroundsituation due to the nonlinearity resulting of the link between market share andprofitability of different types of agents [4].

3 Empirical application to the French and German indexesThe conducted analysis covers the period from 11/28/2003 to 11/28/2011. Thischoice is explained by the wish to integrate both periods of calm (2003- mid 2007),crisis (2007-2011) and recovery (2011-2012). Furthermore, we propose a study bysub-periods which highlights the econometric characteristics of these stock marketindices over the periods [2004-2006] & [2007-2009]. It should be noted that theGerman index is the only one which integrates dividends, which may explain somedifferences in the following study.On the three samples, we estimate the following models:

• ARMA-GARCH (ARMAG) : it is a classical model that can include or not anARMA part.

• Mackey-Glass-GARCH (MGG) : This chaos-stochastic model need the fixationof the parameters τ and c. We give in the table 15 the selected values:

Xt = βXt−τ

1 +Xnt−τ− γXt−1 + εt

εt|It−1 ∼ N(0, ht),εt = zt

√ht et ht = α0 + α1εt−1

2 + β1ht−1

(17)

• Lasota-GARCH (LAG) : it is the second chaos-stochastic model and the se-lected values of τ and c are given in the table 15:

Xt = βXnt−τe

−Xt−τ − γXt−1 + εt

εt|It−1 ∼ N(0, ht),εt = zt

√ht et ht = α0 + α1εt−1

2 + β1ht−1

(18)

The assessment of these models is conducted on the predictive abilities within andout of sample. We propose to evaluate the predictions of the Value-at-Risk by 3


backtesting tests.We begin with a preliminary analysis on the three samples. We then show theobtained results for each model and assess the relevance of each to predict theValue-at-Risk.

3.1 Preliminary analysis and tests

The evolution of the indices over the period [2003-2011] is given in the figure 1. Inthese graphs, we can clearly identify a break at the end of 2008, during the subprimecrisis. Before this key date, prices are characterized by a more or less rise until mid-2007 before a decline until to the breaking point. After the break date, we note amodest recovery for the two indices. Prices are marked by a high volatility.A study of the stationary is conducted by applying the unit root tests of [16]2,of [19, 39, 69]3. The results of various tests are given in the tables 1, 2, 3, 4. Stan-dard tests (DF, KPSS and ERS) confirm the presence of a unit root for the twoindices. The Zivot-Andrews test indicates that all price series are non-stationary (at10%).So, we use the first difference of the logarithm of prices which approximate the finan-cial returns. The figure 1 represents the return series. It refers to two very differentsituations: the first is characterized by a low volatility, it relates to the pre-crisisperiod, while the second is marked by significant volatility clusters, symptomaticof a crisis. Unlike analyzes on prices, the third period of recovery is difficult todissociate from the crisis period as volatility in this period is important. We notepatterns low-high type volatility.An analysis of descriptive statistics on prices, given in the table 5 confirms the par-ticular amplitudes of the different indexes while statistics on returns in the table5 show common characteristics, namely a leptokurtic, asymmetric and not normaldistribution.The graphics 2 and 3 give respectively the evolution of returns over the periods[2007-2009] and [2004-2006]. One notable difference between the two periods con-cerns the homogeneity of variations: in times of crisis, the indices are marked byvery high volatility clusters while in quiet period, variability is nearly homogeneousover the whole period, there are not volatility clusters. These findings are confirmedby descriptives statistics, given in the table 10.In order to highlight the nature of the data generating process, we apply a linearitytest on the sample data [2003-2011]: the [10]’s test. This is justified by the need tohighlight the nonlinear nature of time series by tests whose the alternative hypoth-esis is as wide as possible 4.

2For applying the Dickey-Fuller test, we retain the procedure of Henin-Jobert.3If the first tests are standard unit root, the ERS test and especially the Zivot-Andrews tests

seem more suited to series which may include ruptures. The Seo (1999)’s test could have beenimplemented but the results of Charles and Darné (2009) show the ineffectiveness of such testagainst the Dickey-Fuller test when the sum of the estimated parameters α and β of a GARCHmodel is near to 1 and when β > α.

4We do not retain the nonlinearity tests designed to verify under the alternative hypothesis aspecific nonlinear model. This is the case of Tsay(1989) and Hansen (1996) tests, which, under


Prior to applying the BDS test, we filter all ARMA type structure. We give in thetable 6 the various filters selected for each series of returns. We vary ε as advocatedby [10]. The table 7 gives the results of the BDS test5, to compare to the valueof the standard normal distribution at 95 %, or 1.96. We note that in all cases,the z-statistic is greater than 1.96. Therefore the different series of returns are notindependently and identically distributed.Given these results, we can conclude to a nonlinear generating data process. Byfiltering all linear dependence (ARMA type), the hypothesis of a nonlinear processis emerging and many alternatives should be tested. The table 8 give ARCH testresults’ performed on each returns series on the [2003-2011] period. We varied thedelay (2, 5, 10, 15, 20) and it appears from this test that the selected indicators havean ARCH effect. This hypothesis is widely validated in the early delays. This indi-cates that these different series are characterized by conditional heteroscedasticity.The assumptions of heteroscedasticity and autocorrelations are confirmed on seriesof returns of the sub-periods [2004-2006] & [2007-2009]6. We propose to study theexistence of long memory structures in the returns through the GPH and GSP tests.We analyze by [59, 60]’s test, the presence of dependency structures in returns andvolatility by retaining as a proxy of volatility the squared-returns.An application of the GPH test requires to fix the parameter m. It is recommendedto retain a number m =

√T , where T is the total number of observations.

GPH test results, reported in the table 11 show that all estimated fractional pa-rameters are less than 0.5. The significance tests carried out on these parametersindicate that the null hypothesis of significance of the parameter is accepted. Beforeconfirming the existence of long memory structures in the different series, we applythe [59]’s test.The [59]’s test is applied to the fractional parameters determined by the local es-timator method of Whittle ( [61]). It is necessary to fix m and we follow the rec-ommendations of [59] by varying this parameter from 200 to 800. Furthermore, thenumber of sub-samples to build is recommended by [59] to b = 2 and b = 4. Wereported in the table 12, the estimated d̂, d and the calculated Wald statistic.Comparing the Wald statistics calculated on 2 and 4 sub-samples to the respectivevalues of χ2(1) = 3.84 and χ2(3) = 7.82 at the threshold α = 5%, we find thatthe null hypothesis of constancy of the parameter is accepted. The long memoryhypothesis is therefore rejected for the above indices.The results of [59]’s test on the volatility, reported in the table 13, reveal that theassumption of long memory structures is rejected.To highlight the existence of chaotic structures, we retain the [2]’s test7.[2] recommend to use the modified tests that consider the problem of embedding andthus to avoid any problem of spurious structures. For the purposes of these tests,we therefore retain an embedding dimension m = 1, which means that the test is

the alternative hypothesis, aim to detect a model with regime change. This constraint may leadoften to reject the null hypothesis of linearity in favour of the alternative hypothesis that is notnecessarily true.

5The z statistic is taken in absolute value.6The results are available on request.7We thank Teresa Aparicio for provides matlab code associated with these tests.


applied to returns series. Furthermore, the maximum norm is selected. Finally, thethreshold ε is determined based on the standard deviation of the time series. Wetherefore choose to retain a value of ε which corresponds to 80 % of the value of thestandard deviation of the time series8. The table 14 gives the results of [2]’s test,applied to the returns series. We report in this table indicators, statistical tests andp-values for the three tests. The first observation we can make is about the rejectionof the null hypothesis for all series. The existence of deterministic structures inthe two time series is confirmed. A more precise analysis shows that the statisticscalculated for each test are relatively high, so we can say that the null hypothesis ofpure randomness of returns is strongly rejected. Indeed, the percentage of determin-ism is more than 70 % for all considered time series, which suggests that there aredeterministic structures in these data. Given the particular behaviour of financialtime series, the existence of chaotic structures seems obvious9.

3.2 Estimation results

The table 15 gives estimation results of ARMA- , Lasota-, et MG-GARCH(1,1)models on the period [2003-2011]. Before to detail the results, we comment the choiceof ARMA models and of the parameters τ and c. For the ARMA-GARCH models,we retain for the mean equation an AR(1) process with constant for the french indexand only a constant for the DAX index. The selection of the different process is basedon statistics criterions: the significativity of parameters and the existence of peaks inthe FAC justify the choice of these processes. The selection of these processes is verystandard in modelling financial returns. The analysis of the variance equation revealsthe adequacy of a GARCH (1,1) process to account for the volatility of selectedindices, reinforced by a sum of the coefficients α1 and β1 close to 1. The table 15 alsogives the selection, according to Schwarz minimisation criteria, of the parameters τand c for LAG and MGG models. We find, in the case of the German index that theparameters (τ, c) are the same for the two models. The complexity that characterisesthe German index is relatively small since the dimension used is 1. However, in thecase of the French index, we note two different pairs (τ, c): the particularity of theLAG model, able to describe very complex structures using small dimensions, is tocompare to the Mackey-Glass model that requires a higher dimension to describecomplex chaotic structures. The relevance of the models should be confirmed onsubsamples characterized by different evolutions. The estimation results of the sub-periods are shown in the tables 16 and 17. Standards models allow to model themean structures of the CAC index over the period 2007-2009. The chaos-stochasticmodels are fit on:

8This choice is a consensus between:

• the need to retain a value of ε neither too big nor too small;

• a value of ε which guarantees a percentage of recurrence higher than 10%;

• a value of ε relatively large for detecting, in spite of everything, some recurring points.

9For obvious reasons related to the length of the samples used, it is unnecessary to apply longmemory and detection of chaotic structures tests to the sub-samples.


• the DAX index over the period 2004-2006;

• the DAX and CAC indexes over the period 2007-2009.

A more precise analysis of the parameters τ and c over the 2004-2006 period indicatesthat these models primarily filter chaotic structures in small dimensions. Over theperiod of stress, the dimensions are higher, especially in the case of MGG model.This reveals more complex dynamics in times of crisis that require a larger dimension.

3.3 Assessments of chaos-stochastic modelling: an approach by the Value-at-Risk

The variety of models available for reporting structures of a time series requiresto assess in a relevant and independent way the different models. This necessaryassessment is often conducted on the out of sample predictions. Nevertheless, therelevance of a model should also be judged on its ability to integrate the structuresdetected in the preliminary analysis: this is the whole point of a within sampleevaluation. If this latter is unable to provide information on the ability of a model toprevent future losses of a stock market index, it is a significant additional informationon the utility and the adequacy of the models to the results of the preliminary tests.In the context of risk management, the VaR is an essential element which reflectsmarket risk. Within the framework of this study, the parametric estimation and theforecasting of the VaR are a goal which should allow to identify the more suitablemodels to describe the dynamics of European indices. We evaluate the ability ofthe different models to account for the dynamics in the sample by calculating theVaR at a threshold of 99 % and test the within sample prediction by applying thebacktesting tests. We hold 3 backtesting tests10:

• a standard test : the [13]’s test. We verify the conditional hypothesis by a LRtest.

• a test based on a regression of the sequence of violations: the [17]’s test.The [17]’s test is based on the estimation of a nonlinear model of the violationsthat may take different forms. Among the 7 specifications proposed by theauthors, the first four include the lagged sequence of violations at orders 1, 2,3 (DB2, DB3, DB4). The specifications DB5, DB6, and DB7 introduce thelagged sequence of the VaR at orders 1, 2 and 3. The performance analysisconducted by the authors highlighted several elements:

– better results, for small samples in the case of the DB1 specificationcompared to other specifications

– the impossibility of calculating the maximum likelihood in the case of acomplete separation of the points. From this, it appears that only DB1,DB5 and DB6 specifications are, all time, defined. In comparison withprevious tests, the absence of violations led de facto to reject the nullhypothesis since the statistics could not be calculated. In the case of theDB test, the statistics can be determined even in the absence of violations.

10These various tests were applied using the codes available on the website runmycode.com


• a duration test : the GMM test of [11]. We retain 6 polynomials for this test.

We propose to retain a forecast year for each of the sub periods [2004-2006] and[2007-2009] and two years (2007 & 2009) for the [2003-2011] sample.

3.3.1 Within sample assessment

We find that the VaR predicted for 2007 by the chaos-stochastic models are quiteclose and they often overlap. There is a significant difference between the VaR basedon standard models and chaos-stochastic models, which reflects a reaction time anda different depth of the VaR. This reveals the impact of the mean equation in deter-mining the VaR but only evaluation tests can conclude about the usefulness or notof a chaotic model. A similar observation can be established for the year 2010 witha gap between standard VaR and chaos-stochastic VaR11.The tables 18 and 19 give the P-values associated with [13, 17]’s tests. We note,in 2007, that the LAG model is more suitable for the French index. We cannotdiscriminate chaos-stochastic models for the German index that appear both veryrelevant compared to the standard model. The results of the GMM test [11] aregiven in the table 20 to 23. We note that it is impossible to determine the statisticalJCC and JIND associated to the V aR99% which recorded no exception. This is thecase for VaR prediction of the french index provided by the classical model. We alsonote a rejection of 3 assumptions for the V aR99% of the french index provided bythe MGG model.In 2010, the classical model fails to provide a conditional coverage to the frenchVaR. These "failures" are related to the absence of violations, which does not allowto determine conditional coverage statistics. The VaR provided, even if they remainvalidate 12, appear too conservative and useless since they use capital that cannotbe used for other activities. In the same year, the MGG model does not validatethe conditional coverage hypothesis for the french V aR99%.This first test allows to highlight the weaknesses of some models, including stan-dards, to satisfy the validity assumptions’ of the VaR.The tables 22 to 23 give the results of backtesting tests applied to VaR forecasts forthe years 2005 and 2008.For 2005’s predictions, the [13]’s test, whose results are given in the table 24 vali-dates the VaR provided by each model.We give in the tables 24 and 25, the corresponding pvalues to the [17]’s test, con-ducted on the specifications DB1, DB5 and DB6. The validity of the VaR is acceptedfor both indices. The results of the GMM test [11] [⊳ Tables 22 to 23] conclude onthe validation of the VaR forecasts from the different models.

11These findings are based on a visual observation. Graphics are available on request12For a confidence level of 99 %, an absence of violations is possible and validates the uncondi-

tional coverage hypothesis for a sample of 255 days.


3.3.2 An out-of-sample assessment

The study of financial time series, including stock indices, shows the investor con-fidence’s in the national economy. It is a key indicator whose the results allow togauge the impact of economic policies on investors. Beyond the macroeconomic con-ception, the analysis of financial time series is part of a microeconomic frameworkthat help investors to understand and to predict changes in the prices of assets.These two facets of the analysis of financial time series are synthetized under theBasel agreements with one hand Basel II which introduces a microeconomic ap-proach to market risk and Basel III that envisages a macroeconomic framework ofrisks. Whatever the analytical framework in which the study is conducted, a crucialstep is to show the interest of the models used and their ability to satisfy the statedrequirements. Specifically, the study is of particular interest to the out of sampleforecast. This latter is particularly interesting because it inevitably leads to a blindchoice. Indeed, only the final performance of a model allows to judge the ability ofthis model to account and to predict the dynamic evolution and this, regardless ofthe complexity of the model.In this study, we retain the models used previously. For the out of sample predic-tion, we used a one-step-ahead prediction, that is to say a re-estimation on the samesample size. Specifically, we consider the different models on a sample size N withN = 1, ..., T then we predict the T + 1 observation. We then repeat the procedure,holding a sample of size N with N = 2, ..., T+1 and so on. Given the criteria specificto each model, the number of forecasts varies between 259 to 261 forecasts. In orderto judge the relevance of the different models to integrate and to predict the existingstructures in European indices, an assessment in terms of backtesting of the V aR99%is done. It allows highlighting the more suitable model for anticipating future loss.

• Sample [2003-2011] : the GMM test offers the ability to check the three as-sumptions (coverage and independence). For the forecasts of the VaR for theyear 2012, we note that the 2 chaos-stochastic models satisfy the assumptionsof the GMM test [⊳ Table 29]. It is difficult to rank the models from this testthat validates the 2 specifications but we note, given a lower FEV (Frequencyof Empirical Violations), the superiority of the LAG model for CAC and DAXindexes and of the MGG model for the German index. The tests DB (DB1,DB5 and DB6), shown in the tables 26 to 28, allow to refine these results:they indicate a rejection of the hypothesis of validity of the V aR99% providedby the ARMAG model for CAC and DAX indexes. It is the same case for theCAC provided by the MGG model. The LR tests do not allow a classificationof the 2 models and the only lesson to be drawn from these tests concerns thesuperiority of LAG model for the indices DAX and CAC. The MGG modelvalidates all the tests for the German index.

• Samples [2004-2006] & [2007-2009] : the V aR99% predictions for the 2007 arevalidated by the GMM test for the 3 models. These results are confirmed bythe DB and LR tests.

4 CONCLUDING REMARKS 18

It remains very difficult to discriminate the models because of the near valuesof FEV and of backtesting tests. Furthermore, all three models validate theGMM test for the CAC.

4 Concluding remarksThe evaluation of chaos-stochastic approaches allows to rule on the relevance of thesemodels in the analysis and the prediction of market risk. We conducted a study onthe predictive abilities of the models used in a particular context which is that of theassessment of market risk. This empirical part, essentially quantitative, highlightsthe particular performances of these models:

• within sample : it highlights a better performance of chaos-stochastic modelsover the period [2003-2011]. The LAG model is particularly suitable to theFrench index which is the more leptokurtic;

• out of sample: it confirms the superiority of the LAG model and the utility ofchaos-stochastic models for the German index.

The analysis, carried out in this article, highlights the utility to use models involvingnonlinear equations with delays. It seems that the superiority of the LAG modelrelatively to the MGG model is due to its abilities to take into account more lep-tokurtic dynamics. These results support the utility of a chaos-stochastic model forfinancial time series. They confirm the relevance of Lasota model to account, as wellas the Mackey-Glass model, the chaotic structures of financial time series. Theseresults are particularly visible in the long run: the detection of chaotic structuresis closely related to the need of long time series. Indeed, the main limitations ofdetection tests of chaotic structures are related to the lengths of time series. Also,it is not surprising to get as good results as the classical model for short periods.Beyond the econometric results, this study reveals the coexistence of chartists andfundamentalists that create frictions in European markets, the existence of a com-mon European process that can be considered by the equation of the variance andthe presence of chaotic structures resulting of national specificities. These resultscannot be generalized to all financial series but offer an initial assessment of theability of these models to account for nonlinearities in the financial time series.

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Index Selected model Dickey Fuller Stat (individualtest)

C.V. Dickey Fuller stat (joint test) C.V.

CAC 6 -1,502 -2,8626 1,12881755 [4,59; 4,61]DAX 2 -1,8627 -2,8626 1,73482286 [4,59; 4,61]

Table 1: Dickey-Fuller test

Index LM Stat (trend and constant) C.V. LM stat (Constant) C.V.CAC 0,7792 0,146 1,883 0,463DAX 0,56527 0,146 2,0887 0,463

Table 2: KPSS test

Index Trend and constant C.V. Constant C.V.CAC -1,205 -2,89 -1,159 -1,941DAX -1,7945 -2,89 -0,1313 -1,941

Table 3: ERS test

REFERENCES 25

Index Constant Breakpoint

Trend Breakpoint

Trend andconstant

Breakpoint

CAC -3,7394 19/05/2008 -2,6245 21/12/2005 -4,268 01/01/2008DAX -4,093 19/05/2008 -2,4078 26/01/2006 -4,2647 01/01/2008C.V. -4,8 -4,42 -5,08

Table 4: Zivot-Andrews test

Mean Std. Dev. Skewness Kurtosis Jarque-Bera Prob.Price series

CAC 4127,494 848,5337 0,635503 2,368146 197,1884 0DAX 5838,552 1185,432 -0,136274 1,956841 113,7762 0

Return seriesRCAC 1,11E-05 0,014634 0,062559 10,00529 4802,608 0RDAX 0,000287 0,014154 0,040548 9,915897 4679,975 0

Table 5: Descriptives statistics

Returns AR filterRCAC AR(1)RDAX AR(1)

Table 6: Filters for BDS test

Dimensions2 3 4 5

ε = 0.5RCAC 7.469484 12.49401 16.36752 20.35298RDAX 5.901510 9.518128 13.32398 16.44955

ε = 1RCAC 7.729220 12.84071 16.26604 19.45432RDAX 6.331004 10.18444 13.41021 15.76727

ε = 1, 5RCAC 8.263269 12.87020 15.41519 17.72100RDAX 8.178040 12.19330 14.91781 16.72442

ε = 2RCAC 9.206488 13.53248 15.60116 17.37087RDAX 8.692273 13.15251 15.59274 17.01750

Table 7: z-stat of BDS test

Lags 2 5 10 15 20RCAC 197,732349 405,170662 441,166492 484,282056 539,146917RDAX 185,841618 350,217862 425,901202 501,709951 588,620126C.V. 5,99146455 11,0704977 18,3070381 24,9957901 31,4104328

Table 8: ARCH test on return series

REFERENCES 26

Lags 2 5 10 15 20RCAC 10,3917478 35,9081562 45,0912167 50,1697812 59,6453811RDAX 3,01228482 14,0561971 17,4476793 22,5803168 36,4454136C.V. 5,99146455 11,0704977 18,3070381 24,9957901 31,4104328

Table 9: Q-stat on returns series

Stressed period [2007-2009]Mean Std. Dev. Skewness Kurtosis Jarque-Bera

RCAC -0,001242 0,021336 0,244632 7,378912 423,0688RDAX -0,00094 0,020353 0,342166 8,282776 618,3611

Calm period [2004-2006]Mean Std. Dev. Skewness Kurtosis Jarque-Bera

RCAC 0,000461 0,008126 -0,373872 4,407453 56,30428RDAX 0,000554 0,008814 -0,375897 3,940886 32,15185

Table 10: Descriptives statistics on the samples [2004-2006] & [2007-2009]

d obs tols tasy sigols sigasyRCAC 0,18651339 48 2,10890363 1,76710865 0,04030916 0,08370272RDAX 0,09277474 48 1,17411902 0,87898805 0,24626247 0,38387967

Table 11: GPH test on returns series

Note:The test consists of calculating the fractional parameter d, the ratio of Student(Tasy) and the asymptotic probability associated to the significance test on theparameter d, the Student ratio (Tols) and the associated probability of significancetest on a finished sample.

m d d Wc

CACb=2 b=4 b=2 b=4

200 -0,03535713 -0,04092773 -0,04941348 0,00041379 0,5322109400 -0,05697698 -0,0475909 -0,0619883 0,99640513 3,12585365600 -0,05767084 -0,04396204 -0,05093153 1,58383099 1,6087191800 -0,03557979 -0,02229209 -0,02138057 1,87141636 2,34605355

DAXb=2 b=4 b=2 b=4

200 -0,01520906 -0,02588777 -0,01571727 0,15478598 0,40186792400 -0,02506572 -0,01902166 -0,02141707 0,22351907 2,53536718600 -0,02282199 -0,00432163 -0,00028204 2,0643037 1,67576057800 0,00324824 0,02463886 0,03369628 3,66451335 3,98796834

Table 12: Shimotsu(2006)’s test on return series

REFERENCES 27

m d d Wc

CACb=2 b=4 b=2 b=4

200 0,40046338 0,33147404 0,31038763 3,54251235 5,19622288400 0,42768791 0,39321426 0,33706478 2,03966629 11,7854614600 0,2583818 0,29535996 0,26860893 5,78305947 7,17714553800 0,25967723 0,2542544 0,24236567 0,01044997 0,59206376

DAXb=2 b=4 b=2 b=4

200 0,42182525 0,31638256 0,32727798 10,6042167 9,58603888400 0,36210511 0,33118465 0,30661897 1,55691546 6,0026159600 0,28209692 0,30547084 0,28432714 2,61561411 5,57507304800 0,2607509 0,25042805 0,2494779 0,07380959 0,1622639

Table 13: Shimotsu(2006)’s test on volatility

Index %DET E(%DET) STAT %DET L E(L) STAT L ALL E(ALL) STAT ALL PvalueRCAC 77,2 77,13 7,28 4,03 3,18 474,21 2,21 2,09 −103,02 0,006RDAX 77,11 76,95 19,78 3,81 3,17 364,57 2,18 2,08 −81,3 0

Table 14: Aparicio et al.(2011)’s test results on returns

Parameters β γ α0 α1 β1 LL τ cLAG

DAX −1, 88∗ 1, 87∗ 2, 49e−06∗∗∗ 0, 097∗∗∗ 0, 89∗∗∗ 6323, 39 1 1CAC −1, 92∗ 1, 875∗ 1, 84e−06∗∗∗ 0, 1∗∗∗ 0, 89∗∗∗ 6318, 59 1 1

MGGDAX −1, 90∗ 1, 88∗ 2, 49e−06∗∗∗ 0, 097∗∗∗ 0, 89∗∗∗ 6323, 42 1 1CAC −0, 05∗∗ −0, 042∗ 1, 81e−06∗∗∗ 0, 09∗∗∗ 0, 89∗∗∗ 6311, 48 3 2

ARMAGParameters c φ α0 α1 β1 LL

DAX 0, 001∗∗∗ 2, 76e−06∗∗∗ 0, 105∗∗∗ 0, 88∗∗∗ 6326, 73CAC 0, 001∗∗∗ −0, 05∗∗ 1, 94e−06∗∗∗ 0, 09∗∗∗ 0, 89∗∗∗ 7064, 85

∗∗∗ significant at 1%; ∗∗ significant at 5%; ∗ significant at 10%LL is the likelihood value

Table 15: Estimation results of (chaos)-stochastic models [2003-2011]

REFERENCES 28


DAX −5.853072∗∗∗ 5.851273∗∗∗ 3.75e− 06∗∗∗ 0.069600∗∗∗ 0.883146∗∗∗ 1776.967 1 1CAC −5, 08 5, 13 3, 78e−06∗∗ 0, 07∗∗∗ 0, 87∗∗∗ 1824, 26 1 1

MGGDAX −5, 8∗∗ 5, 8∗∗ 3, 76e−06∗∗ 0, 07∗∗∗ 0, 88∗∗∗ 1776, 94 1 1CAC −5, 12 −5, 08 3, 78e−06∗∗ 0, 07∗∗∗ 0, 87∗∗∗ 1824, 24 1 1

ARMAGParameters c φ θ α0 α1 β1 LL

DAX 0, 001∗∗ 3, 59e−06∗ 0, 07∗∗∗ 0, 88∗∗∗ 1780, 99CAC 0, 001∗∗ 3, 34e−06∗∗∗ 0, 07∗∗∗ 0, 88∗∗∗ 1827, 47

∗∗∗ significant at 1%; ∗∗ significant at 5%; ∗ significant at 10%



DAX −0, 09∗ −0, 087 5, 01e− 06∗∗∗ 0, 13∗∗∗ 0, 87∗∗∗ 1380, 73 3 1CAC −0, 098∗∗ −0, 11∗∗ 7, 5e−06∗∗ 0, 12∗∗∗ 0, 87∗∗∗ 1345, 19 3 1

MGGDAX −13, 901e03∗∗∗ −13, 901e03∗∗∗ 5, 03e−06∗∗∗ 0, 12∗∗∗ 0, 87∗∗∗ 1384, 81 1 5CAC −0, 099∗∗ −0, 109∗ 7, 5e−06∗∗∗ 0, 12∗∗∗ 0, 87∗∗∗ 1345, 27 3 5

ARMAGParameters c φ α0 α1 β1 LL

DAX 5, 36e−06∗∗∗ 0, 12∗∗∗ 0, 87∗∗∗ 1385, 72CAC −0, 11∗∗ 7, 43e−06∗∗∗ 0, 11∗∗∗ 0, 87∗∗∗ 1348, 57

∗∗∗ significant at 1%; ∗∗ significant at 5%; ∗ significant at 10%


2007DB1 DB5 DB6 LR

ARMAG01CAC 0,95 0,16 0,18 0,04DAX 0,95 0,72 0,84 0,93

LAG01CAC 0,70 0,83 0,91 0,67DAX 0,95 0,73 0,84 0,93

MGG01CAC 0,07 0,11 0,15 0,02DAX 0,95 0,73 0,84 0,93

Table 18: DB, LR et DQ tests for within sample predicted VaR in 2007

REFERENCES 29

2010DB1 DB5 DB6 LR

ARMAG01CAC 0,08 0,28 0,40 NaNDAX 0,95 0,96 0,99 0,93

LAG01CAC 0,39 0,75 0,86 0,35DAX 0,95 0,96 0,99 0,93

MGG01CAC 0,07 0,09 0,14 0,03DAX 0,95 0,96 0,99 0,93

Table 19: DB, LR et DQ tests for within sample predicted VaR in 2010

2007 1% JCC JINDIndex FEV p=1 p=2 p=3 p=4 p=5 p=6 p=1 p=2 p=3 p=4 p=5 p=6

ARMAGCAC 0,00 0,57 0,67 0,84 0,86 0,78 0,78 0,13 0,09 0,40 0,64 0,68 0,72DAX 0,01 0,31 0,53 0,77 0,83 0,86 0,91 0,58 0,83 0,86 0,89 0,94 0,96

LAGCAC 0,01 0,48 0,42 0,49 0,53 0,43 0,41 0,54 0,34 0,22 0,21 0,21 0,23DAX 0,01 0,28 0,53 0,77 0,84 0,87 0,92 0,65 0,83 0,87 0,90 0,95 0,97

MGGCAC 0,02 0,06 0,10 0,11 0,18 0,26 0,30 0,87 0,73 0,55 0,49 0,57 0,64DAX 0,01 0,30 0,53 0,78 0,85 0,87 0,92 0,54 0,84 0,87 0,90 0,95 0,97

Table 20: P-values of GMM test for within sample V aR99% predictions of the 2007year


ARMAGCAC 0,00 1 1 1 1 1 1 1 1 1 1 1 1DAX 0,01 0,38 0,57 0,73 0,79 0,83 0,87 0,70 0,79 0,82 0,80 0,86 0,90

LAGCAC 0,02 0,10 0,09 0,09 0,14 0,22 0,28 0,89 0,95 0,95 0,97 0,83 0,61DAX 0,01 0,46 0,57 0,75 0,80 0,84 0,88 0,70 0,79 0,82 0,81 0,86 0,91

MGGCAC 0,03 0,02 0,01 0,01 0,01 0,01 0,02 0,96 0,99 0,96 0,70 0,46 0,39DAX 0,01 0,40 0,57 0,73 0,78 0,83 0,87 0,70 0,80 0,82 0,80 0,86 0,90

Table 21: P-values of GMM test for within sample V aR99% predictions of the 2010year

REFERENCES 30

2008 1% Tests JCC Tests JINDIndex FEV p=1 p=2 p=3 p=4 p=5 p=6 p=1 p=2 p=3 p=4 p=5 p=6

ARMAGCAC 0,0115 0,288 0,4223 0,5055 0,5297 0,3718 0,2491 0,4842 0,4839 0,2745 0,1841 0,1311 0,0983DAX 0,0154 0,1601 0,1966 0,3044 0,5011 0,5614 0,5814 0,8153 0,8913 0,8326 0,8126 0,8551 0,9098

LAGCAC 0,0155 0,1491 0,1648 0,2494 0,3433 0,4155 0,4286 0,98 0,9994 0,9646 0,461 0,3345 0,3653

MGGCAC 0,0155 0,1325 0,158 0,2398 0,3422 0,4227 0,4351 0,9804 0,9998 0,9631 0,476 0,3534 0,3793DAX 0,0154 0,2252 0,199 0,3072 0,5011 0,5651 0,5921 0,9826 0,9621 0,8797 0,9126 0,9583 0,9723

Table 22: GMM test results’ for the within sample predictions (1%)


ARMAGCAC 0,0115 0,2889 0,5777 0,7349 0,7974 0,8426 0,877 0,6419 0,7889 0,8059 0,817 0,8668 0,876DAX 0,0194 0,0797 0,116 0,1203 0,1997 0,2353 0,2458 0,8722 0,3047 0,1078 0,1828 0,2632 0,3758

LAGDAX 0,0194 0,0611 0,1226 0,1255 0,204 0,2334 0,2409 0,8581 0,2975 0,1064 0,1746 0,2575 0,3693

MGGDAX 0,0194 0,1061 0,1145 0,1188 0,1927 0,2319 0,2379 0,8744 0,2942 0,1078 0,1833 0,2667 0,3706

Table 23: GMM test results’ for the within sample predictions(1%)

DB1 DB5 DB6 LRARMA01

DAX 0,39 0,24 0,31 0,08CAC 0,95 0,17 0,20 0,06

LAG01DAX 0,39 0,24 0,32 0,08

MGG01DAX 0,39 0,24 0,32 0,08

Table 24: P-values of DB, LR and DQ tests on the within sample predictions (2005)

DB1 DB5 DB6 LRARMA01

CAC 0,69 0,84 0,92 0,67DAX 0,69 0,49 0,64 0,66

LAG01CAC 0,69 0,79 0,89 0,66

MGG01CAC 0,69 0,79 0,89 0,66DAX 0,69 0,51 0,66 0,66

Table 25: P-values of DB, LR and DQ tests on the within sample predictions (2008)

2012DB1 DB5 DB6 LR

ARMAGCAC 0,54 0 0,01 0,54DAX 0,54 0 0,01 0,54

MGGCAC 0,54 0 0,014 0,54DAX 0,07 0,27 0,4 NaN

LAGCAC 0,07 0,27 0,4 NaNDAX 0,07 0,27 0,4 NaN

Table 26: P-values of DB, LR and DQ tests on the out of sample predictions (2012)

2007DB1 DB5 DB6 LR

MGGDAX 0,6982 0,5197 0,6642 0,6618

LAGDAX 0,6982 0,5194 0,6639 0,6618

ARMAGCAC 0,9259 0,9922 0,9983 0,9207DAX 0,6982 0,5607 0,7026 0,6618


REFERENCES 31

2010DB1 DB5 DB6 LR

MGGCAC 0,1820 0,0887 0,1519 0,1591

LAGCAC 0,6982 0,1355 0,2201 0,6618

ARMAGCAC 0,5300 0,8663 0,9379 0,5333DAX 0,9259 0,2446 0,3639 0,9207


JCC JIND1% FEV p=1 p=2 p=3 p=4 p=5 p=6 p=1 p=2 p=3 p=4 p=5 p=6

ARMAGCAC 0,00389105 0,6332 0,3889 0,4327 0,5736 0,6486 0,6755 0,045 0,1358 0,525 0,5428 0,5385 0,5286DAX 0,00389105 0,5898 0,389 0,4353 0,5744 0,6453 0,6691 0,0805 0,1356 0,5257 0,5365 0,5276 0,5168

MGGCAC 0,00389105 0,6415 0,3933 0,434 0,5758 0,6441 0,6705 0,0863 0,1385 0,5237 0,5398 0,5303 0,52DAX 0 1 1 1 1 1 1 1 1 1 1 1 1

LAGCAC 0 1 1 1 1 1 1 1 1 1 1 1 1DAX 0 1 1 1 1 1 1 1 1 1 1 1 1

Table 29: P-values of the GMM test for the out of sample predictions (2012)

Tests JCC Tests JIND1% FEV p=1 p=2 p=3 p=4 p=5 p=6 p=1 p=2 p=3 p=4 p=5 p=6

MGGDAX 0,02 0,1637 0,2417 0,375 0,4257 0,4541 0,4527 0,8043 0,5424 0,1937 0,2537 0,3026 0,4256

LAGDAX 0,02 0,1596 0,2407 0,3723 0,4219 0,4473 0,4465 0,811 0,5379 0,1843 0,2492 0,3026 0,4246

ARMAGCAC 0,01 0,8501 0,9065 0,9202 0,9372 0,9612 0,96 0,3185 0,6318 0,7488 0,7833 0,8293 0,8881DAX 0,02 0,1376 0,2339 0,3674 0,4194 0,4436 0,4454 0,8003 0,5434 0,1847 0,2419 0,2945 0,4164


JCC JIND1% FEV p=1 p=2 p=3 p=4 p=5 p=6 p=1 p=2 p=3 p=4 p=5 p=6

MGGCAC 0,02 0,2266 0,1757 0,2684 0,4354 0,5313 0,5557 0,7726 0,928 0,9543 0,959 0,7809 0,5315

LAGCAC 0,02 0,247 0,1548 0,2405 0,3404 0,4295 0,4606 0,8216 0,9311 0,9195 0,8161 0,7525 0,7329

ARMAGCAC 0,02 0,1296 0,1002 0,104 0,1538 0,251 0,2985 0,88 0,7496 0,5898 0,6625 0,7238 0,7516DAX 0,01 0,4979 0,385 0,3967 0,26 0,1577 0,1257 0,6233 0,258 0,0669 0,0573 0,0535 0,0592


REFERENCES 32

2,000

3,000

4,000

5,000

6,000

7,000

2004 2005 2006 2007 2008 2009 2010 2011

CAC

3,000

4,000

5,000

6,000

7,000

8,000

9,000

2004 2005 2006 2007 2008 2009 2010 2011

DAX

-.10

-.05

.00

.05

.10

.15

2004 2005 2006 2007 2008 2009 2010 2011

RCAC

-.08

-.04

.00

.04

.08

.12

2004 2005 2006 2007 2008 2009 2010 2011

RDAX

Figure 1: Price and return series of french and german indexes

REFERENCES 33

2,000

3,000

4,000

5,000

6,000

IV I II III IV I II III IV

2008 2009

CAC

3,000

4,000

5,000

6,000

7,000

8,000

9,000


2008 2009

DAX

-.10

-.05

.00

.05

.10

.15


2008 2009

RCAC

-.08

-.04

.00

.04

.08

.12


2008 2009

RDAX

Figure 2: Return series on the sample [2007-2009]

REFERENCES 34

3,600

4,000

4,400

4,800

5,200

5,600


2005 2006

CAC

4,000

4,500

5,000

5,500

6,000

6,500


2005 2006

DAX

-.04

-.03

-.02

-.01

.00

.01

.02

.03


2005 2006

RCAC

-.04

-.03

-.02

-.01

.00

.01

.02

.03


2005 2006

RDAX

Figure 3: Return series on the sample [2004-2006]

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