Eugenio Azpeitia

Gabrielle Tichtinsky

Marie Le Masson

Antonio Serrano-Mislata

Jérémy Lucas

Veronica Gregis

Carlos Gimenez

Nathanaël Prunet

Etienne Farcot

Martin M Kater

Desmond Bradley

Francisco Madueño

Christophe Godin

Francois Parcy

Cauliflower fractal forms arise from perturbations of floral gene networks

Throughout development, plant meristems regularly produce organs in defined spiral, opposite or whorl patterns, called phyllotaxis. Cauliflowers present an unusual phyllotaxis with a multitude of spirals nested over a wide range of scales. How such a fractal self-similar organization emerges from developmental mechanisms has remained elusive. Combining experimental analyses in Arabidopsis thaliana cauliflower-like mutant with modeling, we found that curd selfsimilarity arises because the meristems fail to form flowers but keep the "memory" of their transient passage in a floral state. Additional mutations affecting meristem growth can induce the production of conical phyllotactic structures reminiscent of the conspicuous fractal Romanesco shape. This study reveals how fractal-like forms may emerge from the combination of key, defined perturbations of floral developmental programs and growth dynamics.

Main Text

Above-ground plant architectures arise from activity of shoot apical meristems (SAM), which are pools of stem cells that give rise to organs such as leaves, shoots or flowers. The arrangement of organs on stems is termed phyllotaxis. Plants with a spiral phyllotaxis usually form two families of organ spirals, visible on compact structures such as flower heads, pine cones or cacti (Fig. 1a-c). These two families of spirals turn in opposite directions, and come in two consecutive numbers of the Fibonacci series (Fig. 1a) (1). In cauliflowers, spiral families are visible not only at one but at several scales (Fig. 1d-f). This self-similar organization culminates in the Romanesco cultivar where the spirals appear in relief due to their conical shape at all scales, a geometrical feature conferring the whole curd a marked fractal-like aspect (Fig. 1g).

Cauliflowers (Brassica oleracea var. botrytis) were domesticated from cabbages [START_REF] Quiros | Genetics and Genomics of the Brassicaceae[END_REF]. The cauliflower inflorescence (the shoot bearing flowers) takes a curd shape because each emerging flower primordia never matures to the floral stage but instead generates more curd-shaped inflorescences [START_REF] Quiros | Genetics and Genomics of the Brassicaceae[END_REF][START_REF] Duclos | [END_REF]. In B. oleracea, the genetic modifications causing curd development are still debated and likely affect multiple genes [START_REF] Quiros | Genetics and Genomics of the Brassicaceae[END_REF][START_REF] Duclos | [END_REF](4)(5). However, cauliflower-like structures also exist in the model brassicaceae Arabidopsis thaliana and are caused by a double mutation in APETALA1 (AP1) and CAULIFLOWER (CAL) (Fig. 1h-i), two paralogous genes encoding MADS-box transcription factors (TF) promoting floral development (6,7). The Arabidopsis molecular regulators governing the development of shoots and flowers have been largely identified (8-10)(Table S1). Network models based on these regulators have been proposed to explain wildtype flower development (11)(12)(13)(14). However, whether variants of these networks are able to account for development of Arabidopsis ap1 cal curds is unknown.

To address this question, we first built a network of the main regulators involved in both flower and curd development. Then, we embedded this network within a 3D computational model of plant development to understand how mutations could transform wild-type (WT) inflorescences into curds.

The genetic basis of cauliflower curds

In Arabidopsis, flowers are initiated by the TF LEAFY (LFY) (Fig. 1j) (Table S1). LFY is upregulated by the SUPPRESSOR-OF-OVEREXPRESSION-OF-CO 1 (SOC1) and AGAMOUS-LIKE 24 (AGL24) MADS-box proteins (induced throughout the inflorescence meristem by environmental and endogenous cues) and by auxin phytohormone maxima that mark floral meristem initiation sites. LFY is expressed specifically in floral primordia because its induction in the SAM is repressed by the TFL1 inflorescence identity protein. In the floral primordium, LFY induces AP1 and CAL (AP1/CAL) that positively feedback on LFY and repress both SOC1/AGL24 and TFL1, thereby stabilizing the floral fate of the new meristem. In the ap1 cal cauliflower mutant, the AP1/LFY positive feedback is absent and TFL1 is not repressed by AP1/CAL in the nascent floral meristem. Consequently, young flower primordia cannot maintain LFY expression and start themselves expressing TFL1. As a result, they lose their floral identity and become inflorescence meristems (6). Whereas TFL1 repression in nascent flower primordia is well understood, the factors directly responsible for its upregulation in ap1 cal and inflorescence meristems are unknown.

To complete our network, we thus searched for direct positive regulators of TFL1, other than LFY (that induces TFL1 (15) but is not active in inflorescence meristems). TFL1 is indirectly regulated by day length (16): in long days (LD) TFL1 is up-regulated by CONSTANS (CO) and FT, two key upstream effectors of the LD pathway (11,(17)(18)(19) (Fig. S1). To search for direct regulators, we examined SOC1 and AGL24 that act downstream of CO and FT in the LD pathway (9). Loss-and gain-of-function experiments demonstrated that both SOC1 and AGL24 induce TFL1 (Fig. 2a-i) and Chromatin Immuno-Precipitation showed that these two TFs bind to the TFL1 regions that regulate its expression in the SAM (20) (Fig. 2j-l). These regions were sufficient to activate a TFL1 reporter construct by SOC1 and AGL24 in a transient assay (Fig. 2m-n) confirming that both MADS-box TFs are direct regulators of TFL1. Since XAANTAL2 (XAL2), a homolog of SOC1 and AGL24 also bound to and induced TFL1 (21), we aggregated the activities of SOC1, AGL24 and XAL2 into a SAX proxy acting as TFL1 positive regulator (Fig. 3a).

We thus created the SALT network (for SAX, AP1/CAL, LFY, and TFL1; Fig. 3a) made of these 4 regulator sets, auxin (22), and F, a flower inducing signal (a proxy for the FT florigen) that increases when the plant ages or is exposed to flower-inducing environmental conditions (23,24). We also added a short-lived transient early Repressor of TFL1 (eREP), as a proxy for TFL1 early repression in the young flower bud performed by the redundant activities of SOC1, AGL24, SHORT VEGETATIVE PHASE, and SEPALLATA4 (25). The steady states of the SALT network correspond to the gene expression patterns observed in wild-type vegetative (low SALT values), inflorescence (high TFL1/SAX, low AP1/CAL/LFY) and flower (low TFL1/SAX, high AP1/CAL/LFY) meristems (Fig. 3b,c, Fig. S2). Above an F threshold value, the network generates a flower or an inflorescence state depending on F and auxin values. Simulations of tfl1, lfy, ap1 cal mutants produce expected outputs consistent with experimentally reported gene expressions (6,16,26,27) (Fig. 3b,c). The simulated sax mutant did not reach a floral state, consistent with the late flowering behavior of the soc1 agl24 double mutant (28).

The modelled gene expression dynamics (Fig. 3d) illuminate the fundamental differences between WT and cauliflower meristems: in a WT flower primordium, F induces SAX. SAX and auxin induce LFY, that, together with F, induce AP1/CAL. AP1 positively feeds back on LFY and represses SAX (Fig. 3d). TFL1 expression, that could be induced by SAX and LFY in early floral stages, is constantly repressed, first by eREP and later by SAX plus AP1/CAL. High AP1/CAL and LFY with low TFL1 and SAX expression stabilize the floral fate. In contrast, in the ap1 cal flower primordia, the absence of AP1/CAL activity has two consequences: i) LFY expression is upregulated only transiently since AP1/CAL positive feedback is missing (Fig. 3d) and ii) SAX genes are not repressed by AP1 and thus induce TFL1 in nascent flower meristems. TFL1 represses LFY even further and the meristem returns to a shoot meristem state (Fig. 3d). Note that, the early LFY induction would likely be reinforced (while remaining transient) by incorporating the recently discovered direct induction of LFY by the F partner protein FD (29).

The SALT model predicts that SAX expression should extend over the entire cauliflower. We analyzed a SOC1-GFP reporter line and indeed observed expansion of its expression domain in ap1 cal as compared to WT (Fig. 3e,f). The SALT network thus recapitulates realistic gene expressions driving meristem fates.

However, a plant architecture does not only depend on meristem fates but also on morphodynamic parameters including molecular thresholds for fate decisions, organ growth rate, delay for meristems to start organ production and organ production rate which are independently regulated. Plant inflorescence architecture thus emerges from the complex interaction between the floral GRN and morphodynamic parameters. This is illustrated here by the lfy and ap1 cal mutants that have the same GRN outputs (Fig. 3c) but markedly different architectures (6,27).

To study how this interaction operates in Arabidopsis, we integrated the SALT GRN in a 3D plant computational model implemented as an L-system (see Supplementary materials Modeling Methods).

A multi-scale model generates Arabidopsis cauliflower structures

The 3D model is made of the 4 types of organs that shape plant above-ground architecture: meristems, internodes, leaves and flowers (Fig. 4a, Supplementary materials). Each meristem's identity (vegetative, inflorescence and floral) is determined by the GRN steady state, computed at each time step as a function of the meristem's previous state and external factors (auxin and F). The GRN model is implemented as single compartment ordinary differential equations (Supplementary materials Modeling Methods). We assume that the GRN dynamics is faster than growth and reaches its steady state within a time step. A set of growth rules defines meristem production: a vegetative meristem produces a compressed stem (non-elongated internodes) with rosette leaves; an inflorescence meristem produces an elongating internode, a cauline leaf and a new shoot meristem in the leaf axil; a floral meristem produces an internode terminating with a flower meristem, devoid of bracts (leaf-like organs subtending flowers) since they are repressed by LFY ( 6)). Each newly generated axillary meristem begins with maximal auxin level (22), SAX/LFY/AP1/CAL values inherited from the parent meristem, together with a fraction of the parent TFL1 value as, in the real plant, this non-cell autonomous protein is present in the primordia region (30). To match the wild-type plant architecture, indeterminate meristems at orders >2 (Fig. 4a) were kept quiescent, a likely effect of apical dominance (the inhibition of lateral meristem outgrowth) (Fig. S3a). The model also contains rules describing organ growth dynamics (internode and leaf elongation, flower growth, organ production rate, growth initiation delay). Simulated plants start with a single vegetative SAM and repeatedly produce new organs according to the GRN, the morphodynamic rules and an input value of F.

By adjusting the GRN and morphodynamic parameters within a range of plausible values (Supplementary materials), we successfully calibrated the model to produce realistic architectures for wild-type and lfy plants (Supplementary Movies 1-2), as well as for the tfl1 mutant (Fig. 4b-d) and a non-flowering phenotype for the sax mutant. However, our simulations could not generate a realistic ap1 cal mutant growing without bract/cauline leaves and displaying high order meristems (Fig. S3a-b) suggesting that the cauliflower phenotype involves additional regulations. We reasoned that laterally produced ap1 cal inflorescence meristems are different from those produced in other genotypes as, according to our GRN, they have been transiently exposed to LFY expression (Fig. 3d). Several pieces of evidence suggest that this transient LFY expression, already known to repress bracts (6), could also contribute to high-order meristem release. First, the outgrowth of otherwise inhibited axillary meristems in the rosette is stimulated by ectopic expression of LFY (or a LFY allele) (31,32). Second, it was established that the lfy ap1 cal triple mutant does not form cauliflowers (6) and we found that, in this mutant, the number of high-order meristems is significantly reduced as compared to ap1 cal (Fig. S3d-h), thus supporting our hypothesis.

We abstracted this critical molecular pathway, by introducing in the model a factor X upregulated when LFY exceeds a minimal threshold level. Upregulated factor X releases highorder meristem growth and suppresses the bract. This was sufficient to unlock the recursive growth of lateral meristems and to generate the ap1 cal curd structure that arises from the transient but irreversible exposure of meristems to the floral signal without any alteration of wild type growth dynamics (Fig. 4e,h, Supplementary Movie 3). Overall, our work shows that the ap1 cal and lfy architectures are different (Fig. 3c) because the molecular histories of their inflorescence meristems are different, thereby revealing the existence of a developmental hysteresis.

Growth dynamics define cauliflower and Romanesco curd structures

Our work in Arabidopsis offers a conceptual framework to explain how inflorescence architecture emerges from coupling a floral GRN to morphodynamic parameters. We wondered whether modifications affecting components of this framework could also explain the architecture of the cauliflowers that arose during domestication, namely the edible Brassica oleracea (Bo) var. botrytis (Bob) and its Romanesco variant. Whether similar genetic defects as in Arabidopsis are responsible for curd development in B. oleracea is still debated (4,5). To further investigate this point, we analysed RNA-seq data of Bob curds: we confirmed the previously identified mutation in the BobCAL gene (Fig. S4a) (4,5,7) and observed that the two AP1 paralogs (BobAP1-a and BobAP1-c) are expressed at much lower levels than in cabbage (Bo var. capitata) inflorescences (Fig. S4b). These functional proteins are induced with a delay only when the cauliflower elongates and start forming normal flowers [START_REF] Duclos | [END_REF]33). Comparing cauliflower and cabbage sequences, we identified differences in binding sites for candidate regulators of BoAP1 that could account for their delayed activation (Fig. S4d). The combination of BoCAL inactivation and BobAP1-a/c expression delay (heterochrony due to cis or trans mutations) thus likely participates to Bob curd development. Similar to Arabidopsis ap1 cal, cauliflowers have meristems of higher maximal order (n ≥ 7) than cabbages (n = 3-4) (Fig. S5). Nevertheless, the development of single massive cauliflower curds is not the exact equivalent of the Arabidopsis mutant [START_REF] Duclos | [END_REF]5) and involves additional multifactorial alterations of morphodynamics parameters (such as reduction of internode elongation and branches diameter increase).

The conical shapes appearing in Romanesco spirals at all scales (Fig. 1f) represent an additional geometric variation obtained through domestication that seems to be associated with a change in morphodynamic parameters. Indeed, several such parameters remain constant during cauliflower development but vary in Romanesco (34): i) the plastochron, the time between two successive meristem productions, ii) the number of visual spirals originating from a given meristem, iii) the time (measured in number of plastochrons) needed before a lateral primordium starts producing its own primordia (or lateral production onset delay), and iv) the size of the meristems. Whether some of these parameters are causal to the Romanesco phenotype remains unclear but phyllotaxis studies (1,35,36) indicate that the first three parameters are linked to the meristem size: an augmentation of the size of the meristem central zone should decrease the plastochron, which in turn increases the number of spirals, and the lateral production onset delay. We thus hypothesized that passing from a constant to a decreasing plastochron in meristems could change cauliflower into Romanesco morphologies. We first tested this in silico using a simplified, purely geometric model of curd growth, independent from the Arabidopsis GRN and specific growth dynamics (Supplementary materials). A decreasing plastochron was sufficient to produce Romanesco shapes (Fig. 4g) whereas constant values of this parameter produce cauliflower morphologies (Fig. 4f).

We then introduced the same change in the more complex GRN-based, Arabidopsis cauliflower architectural model, while keeping its organ growth dynamics as calibrated on the WT. Although not as complete as in the purely geometric model, the curd changed towards a "Romanesco-like" morphology with typical conical curd shapes (Fig. 4h,i). We then tested this hypothesis experimentally in Arabidopsis by altering the size of the meristem directly. We achieved this by introducing a mutation in the CLAVATA3 (CLV3) gene that controls meristem homeostasis and induces an increase of the meristem central zone during growth (37,38). As predicted by our analysis, introduction of a clv3 mutation in ap1 cal Arabidopsis mutant modified the curd shape, which lost its round morphology and acquired a more conical shape, with similar structures at different scales, features recognized as hallmarks of Romanesco curds (39) (Fig. 4l-m). Two additional pieces of evidence support the hypothesis that meristem homeostasis is perturbed in Romanesco curds: they occasionally show fasciation, a feature typical of meristem enlargement also observed in clv3 or ap1 cal clv3 mutants (Fig. 4n,o) (37). Moreover, the expression of CLV3 (and possibly two other genes acting in the same pathway) (38) are lower in Romanesco curds than in cauliflowers (Fig. S6). Altogether, these observations establish that meristem size regulates the final curd morphology through control of plastochron value.

These results reveal how fractal patterns can be generated through growth and developmental networks that alter identities and meristem dynamics. Our data, GRN and growth models now clarify the molecular and morphological changes over time by which meristems gain different identities to form the highly diverse and fascinating array of plant architectures found throughout nature and crops.

Figure 1 :

 1 Figure 1: Illustrations of phyllotactic spirals on plant inflorescences (a) Daisy capitulum: the two families of spirals are indicated in the close-up (13 blue spirals and 21 red). (b) Dahlia composite flower (c) Zingiber inflorescence. (d-f) Brassica oleracea var. botrytis cauliflower with (e) 8 counterclockwise (brown family) and (f) 5 clockwise (green family) main spirals. Dashed rectangles show families of spirals nested over several scales (g) Romanesco curd, (h) Arabidopsis wild-type inflorescence (h) and ap1 cal curd (i), Bar = 2 cm (ag), 500 µm (h-i). (j) Interactions between major floral regulators; arrows depict activation whereas barred lines indicate repression.

Fig. 2 :

 2 Fig. 2: AGL24 and SOC1 are direct positive regulators of TFL1. (a-c), TFL1p:GUS activity in WT (a), soc1-2 (b) and agl24-2 (c) inflorescence apices. (d-i), TFL1p:GUS activity (blue signal) in WT (d-f) and 35Sp:SOC1 (g-i) apices at vegetative (d,g) and flowering (e,f,h,i) stages. (f-i), longitudinal sections through flowering shoots. Arrows mark the SAM. Scale bars in (f) and (i), 40 μm. (j-l) Structure of TFL1 locus, with regions conserved in Brassicaceae (pink lines), regulatory regions (20) (blue boxes I-V), and fragments used in ChIP (black lines 1-6). ChIP experiments on plants expressing a tagged version of AGL24 (k,

  white bars) or the WT SOC1 protein (l, white bars) or on control plants (grey bars, see Material and Methods), show that AGL24 binds region IV (k, fragments 4-5) and SOC1 region V (l, fragment 6). A representative biological replicate is shown with the mean =/-SE for three technical replicates. (m,n) Transient assays showing transactivation of the LUCIFERASE (LUC) reporter driven by region IV (activation by 35Sp:AGL24) and region V (activation by 35Sp:SOC1). NGA3 is an unrelated TF used as negative control. Bars denote the mean and standard deviation of three independent biological replicates. P values are for the equality of means (Student's t-test).

Fig. 3 :

 3 Fig. 3: SALT Gene Regulatory Network model and experimental validation.

  (a) SALT GRN network structure (b) Known expression patterns of SAX, AP1/CAL, LFY, and TFL1 in the SAM and lateral primordia of WT and ap1 cal mutant. The question mark indicates a predicted expression pattern of the model. (c) WT, tfl1, ap1 cal and lfy steady states of the model at different F values in the SAM (low auxin) and in lateral meristems (high auxin). The genetic identity predicted for WT and all mutant meristems correspond to the experimentally observed phenotypes. (d) Temporal simulation of gene expression in lateral primordia with high F value. (e, f) Expression of the SOC1:GFP (white/light blue signal) reporter construct in WT (e) and in the ap1-7 cal-1 mutant (f) inflorescences. Asterisks mark the SAM. Bar = 50 µm.

Fig. 4 :

 4 Fig. 4: Simulation and assessment of a GRN-based plant development model. (a) Schematic representation of the multi-scale model of Arabidopsis development. Each meristem state is composed of signal levels (auxin, F) and a GRN steady state. At time t, the plant is made up of a collection of organs (left). At time t+ t (right) the model updates the signal levels and GRN state in each meristem. The steady state defines the identity of the meristems (vegetative, inflorescence or flower) used to compute meristem lateral productions. Green numbers indicate meristem order (b-e). Plant morphologies obtained in the WT (b), lfy (c), tfl1 (d) and ap1 cal (e) simulations. Simulated morphologies with constant (f,h) or increased meristem size (g,i) in a simplified (f,g) and the Arabidopsis model (h,i). Light micrographs (j,l,n) and s.e.m (k,m,o) of cauliflower structures in Arabidopsis ap1 cal (j, k), Arabidopsis ap1 cal

Acknowledgments

We thank Anne-Marie Chèvre, Richard Immink, Rüdiger Simon, Lars Ostergaard and Mariana Benitez for advice, Teva Vernoux, Chloe Zubieta and Hicham Chahtane for proofreading and useful feedback on the manuscript, Dominique Tardy, Eric Giraud, Renaud Dumas and Vincent Martin (OBS, France) for providing cauliflower samples and Lydia Bousset Vaslin for images and branch counting, Frédéric Boudon for help with L-Py, Richard Immink (Wageningen, Netherlands), C. Ferrándiz (IBMCP; Spain), George Coupland (MPIPZ, Germany), Miguel

Funding

This project received support from the INRAE Caulimodel project (FP and CG), Inria Project Lab Morphogenetics (CG, EA and FP), the ANR BBSRC Flower model project (FP and CG), the GRAL LabEX (ANR-10-LABX-49-01) with the frame of the CBH-EUR-GS (ANR-17-EURE-0003), EU H2020 773875 ROMI project funding (CG), the Spanish Ministerio de Ciencia Innovación and FE ER (grant no. PGC2018-099232-B-I00)(FM).

Data and Materials Availability

All data are in the main paper or the supplement.

Competing interests

The authors declare no competing interests.

All plant materials are available upon request.

The following secure token has been created to allow review of record GSE150627 while it remains in private status: khkjgckmdtkhpgb.

All source codes to run the simulations are available as supplementary archive file (description of installation and execution available as README.txt.

Supplementary Materials:

Materials and Methods

Figures S1 to S6


Bertrand Candelon

Gilbert Colletaz

Christophe Hurlin
email: christophe.hurlin@univ-orleans.fr

Sessi Tokpavi

Backtesting Value-at-Risk: A GMM Duration-Based Test

Keywords: Value-at-Risk, backtesting, GMM, duration-based test. J.E.L Classi…cation : C22, C52, G28

 to test for the distributional assumption (i:e: the geometric distribution) is applied to the case of the VaR forecasts validity. Using simple J-statistic based on the moments de…ned by the orthonormal polynomials associated with the geometric distribution, this new approach tackles most of the drawbacks usually associated to duration based backtesting procedures. First, its implementation is extremely easy. Second, it allows for a separate test for unconditional coverage, independence and conditional coverage hypothesis (Christo¤ersen, 1998). Third, feasibility of the tests is improved. Fourth, Monte-Carlo simulations show that for realistic sample sizes, our GMM test outperforms traditional duration based test. An empirical application for Nasdaq returns con…rms that using GMM test leads to major consequences for the ex-post evaluation of the risk by regulation authorities. Without any doubt, this paper provides a strong support for the empirical application of duration-based tests for VaR forecasts.

Introduction

The recent Basel II agreements have left the possibility for …nancial institutions to develop and apply their own internal model of risk management.

The Value-at-Risk (VaR thereafter), which measures the quantile of the projected distribution of gains and losses over a target horizon, constitutes the most popular measure of risk. Consequently, regulatory authorities need to set up adequate ex-post techniques validating or not the amount of risk taken by …nancial institutions. The standard assessment method of VaR consists in backtesting or reality check procedures. As de…ned by [START_REF] Jorion | Value-at-Risk[END_REF], backtesting is a formal statistical framework that consists in verifying if actual trading losses are in line with projected losses. This involves a systemic comparison of the history of model-generated VaR forecasts with actual returns and generally relies on testing over VaR violations (also called the Hit).

A violation is said to occur when ex-post portfolio returns are lower than VaR forecasts 2 . Christo¤ersen (1998) argues that a VaR with a chosen coverage rate of % is valid as soon as VaR violations satisfy both the hypothesis of unconditional coverage and independence. The hypothesis of unconditional coverage means that the expected frequency of observed violations is precisely equal to %. If the unconditional probability of violation is signi…cantly higher than %, it means that VaR model understates the portfolio's actual level of risk. The opposite …nding of too few VaR violations would alternatively signal an overly conservative VaR measure. The hypothesis of independence means that if the model of VaR calculation is valid then violations must be distributed independently. In other words, there must not have any cluster in the violation sequence. As noted by [START_REF] Campbell | A review of backtesting and backtesting procedures[END_REF], the unconditional coverage property places a restriction on how often VaR violations may occur, whereas the independence property restricts the ways in which these violations may occur. But both assumptions are essential to characterize VaR forecast validity: only hit sequences that satisfy each of these properties (and hence the conditional coverage hypothesis) can be presented as evidence of a useful VaR model.

Even if the literature about conditional coverage is quite recent, various tests on independence and unconditional coverage hypotheses have already been developed (see Campbell, 2007 for a survey). Most of them directly exploit the violation process3 . However another streamline of the literature uses the statistical properties of the duration between two consecutive hits. The baseline idea is that if the one-period ahead VaR is correctly speci…ed for a coverage rate ; then the durations between two consecutive hits must have a geometric distribution with a success probability equal to %: On these grounds Christo¤ersen and Pelletier (2004) proposed a test of independence. The gen-eral idea of their duration-based backtesting test consists in specifying a duration distribution that nests the geometric distribution and allows for duration dependence, so that the independence hypothesis can be tested by means of simple likelihood ratio (LR) tests. As noted by Haas (2007), this general duration-based approach of backtesting sounds very appealing. It is easy to apply and provides a clear-cut interpretation of parameters. Nevertheless, it must be note that one have to specify a particular distribution under the alternative hypothesis. Moreover, LR test turns out to su¤er from the relative scarcity of violations: even with one year of daily returns, the associated series of durations is likely to be short, in particular for a 1% coverage rate (the value recommended by supervision authorities). Consequently duration-based backtesting methods have relatively small power for realistic sample sizes (Haas, 2007) and it even often happens that standard LR duration-based statistics cannot be computed4 . For these reasons, actual duration-based backtesting procedures are not very popular among practitioners. However we show in this paper that it is possible to signi…cantly improve these procedures.

Relying on the GMM framework of [START_REF] Bontemps | Moment-based tests for discrete distributions[END_REF]Meddahi(2005,2006) we derive test statistics similar to J-statistics based on particular moments de…ned by the orthonormal polynomials associated with the geometric distribution. Also our duration-based backtest considers discrete lifetime distributions: we expect in particular that this leads to an improvement in the power and size of our test upon ones based on continuous approximations as for example in Christo¤ersen and Pelletier (2004). To sum up, the present approach appears to have several advantages. First, it provides an uni…ed framework in which we can investigate separately the unconditional coverage hypothesis, the independence assumption and the conditional coverage hypothesis. Second, the optimal weight matrix of our test is known and does not have to be estimated.

Third the GMM statistics can be numerically computed for almost all realistic backtesting sample sizes. Fourth, it bene…ts from a result of [START_REF] Bontemps | Moment-based tests for discrete distributions[END_REF] and appears to be robust to parameter uncertainty. Fifth, in contrast with the LR tests, it does not impose a particular distribution under the alternative. Finally, some Monte-Carlo simulations indicate that for realistic sample sizes, our GMM test have good power properties when compared to other duration-based backtests.

The paper is organized as follows. In section 2, we present the main VaR assessment tests and more particularly the duration-based backtesting procedures. Section 3 presents our GMM duration-based test. In section 4 we present the results of various Monte Carlo simulations in order to illustrate the …nite sample properties of the proposed test. In section 5 we realize an empirical application using daily Nasdaq returns. Finally the last section concludes.

Duration-Based Backtesting of Value-at-Risk

Let us denote r t the return of an asset or a portfolio of assets at time t.

The ex-ante VaR for a % coverage rate denoted V aR tjt 1 ( ); anticipated conditionally to an information set t 1 available at time t 1; is de…ned by: Pr[r t < V aR tjt 1 ( )] = where t = 1; :::; T:

(1)

Let I t ( ) be the hit variable associated to the ex-post observation of a % VaR violation at time t:

I t ( ) = 8 > > < > > : 1 if r t < V aR tjt 1 ( ) 0 else (2)
As suggested by Christo¤ersen (1998), VaR forecasts are valid if and only if the violation sequence fI t g T t=1 satis…es the following two assumptions:

The unconditional coverage (UC) hypothesis: the probability of a violation must be equal to the coverage rate:

Pr [I t ( ) = 1] = E [I t ( )] = : (3)
The independence (IND) hypothesis: VaR violations observed at two different dates for the same coverage rate must be independently distributed.

In other words, past VaR violations do not hold information about current and future violations.

The UC hypothesis is a straightforward one: if the frequency of violations observed over T periods, is signi…cantly lower (respectively higher) than the nominal coverage rate , then the VaR overestimates (respectively underestimates) the risk. Nevertheless, the UC hypothesis shades no light on the possible dependence of VaR violations and a model which does not satisfy the IND hypothesis could then lead to some clustering in violations and therefore does not o¤er a proper framework to appreciate the risk 5 .

When the assumptions of UC and IND hypotheses are simultaneously valid then VaR forecasts are said to have a correct conditional coverage (CC thereafter). Under the CC assumption, the VaR violation process becomes a martingale di¤erence:

E [ I t ( ) j t 1 ] = : (4)
The sequence fI t g T t=1 of VaR violations should then be a random sample from a Bernoulli distribution with a probability of violation equal to . It results that the process consisting in the numbers of periods between two violations has a geometric distribution with no memory. More precisely, it signi…es that when a violation occurs then the conditional probability distribution of the number of periods which are passed before a new hit occurs does not depend on how many violations were already observed.

5 An e¢ cient measure of risk should indeed adjust automatically and immediately to any new information so that the ex-ante probability of a violation for t+1 must be equal to the accepted nominal coverage rate whatever the value of the hit function at time t.

Formally, let us denote D i the duration between two consecutive violations as:

D i = t i t i 1 ; (5)
where t i denotes the date of the i th violation: Under CC hypothesis, the duration D i follows a geometric distribution with a probability equal to and a probability mass function given by:

f D (d; ) = (1 ) d 1 d 2 N : (6)
Exploiting this relation, it is straightforward to develop a likelihood ratio test for the IND and/or the CC hypotheses. The general idea of these durationbased backtesting tests consists in specifying a duration distribution nesting the geometric distribution f D (d; ) and allowing for duration dependence.6 .

Hence Christo¤ersen and Pelletier (2004) proposed the …rst duration-based test for which they used the exponential distribution, which is the continuous analogue of the geometric distribution, and has a probability density function de…ned as:

f D (d; p) = p exp ( pd) : (7)
With (7) we have E (d) = 1=p and, as the CC hypothesis implies a mean duration equals to 1= , it also implies the condition p = . With regard to the IND hypothesis, they postulate a Weibull distribution so that under the alternative the density of durations between successive hits is given by:

f D (d; p; b) = p b b d b 1 exp h (p d) b i : (8)
As the Exponential is a Weibull with a ‡at hazard function, i.e b = 1, the test for IND (Christo¤ersen and Pelletier, 2004) is then simply:

H 0;IN D : b = 1: (9)
In a recent work, [START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF] extended this approach to consider the CC hypothesis, that is:

H 0;CC : b = 1; p = : (10)
However even if duration-based backtesting tests relying on continuous function are attractive because of their elegance and simplicity, they are not entirely satisfying. In particular, [START_REF] Haas | Improved duration-based backtesting of Value-at-Risk[END_REF] Similar lack of power is also apparent in Hurlin and Tokpavi (2007).

Second, duration based tests turn out to have a rather limited feasibility.

Indeed, for realistic backtesting sample sizes (T around 250) and a coverage rate of 1%, it often happens that the LR duration-based statistics cannot be computed. This is because the implementation of the test requires at least one non-censored duration and an additional, possibly censored, duration (i.e. two violations). As observed by [START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF], this call for rather huge samples: when he considers one year of trading days (T = 250) then, under the null of conditional coverage with = 0:01, the Weibull LR test can be computed only in 6 cases out of 10. In other words, in 4 cases out of 10, the duration based statistics can not be used to test the VaR forecasts validity, whereas other backtesting approaches not based on durations7 can be used.

Third, duration-based tests do not allow formal separate tests for UC, IND

and CC within a uni…ed framework 8 . It seems however usefull to propose a backtesting strategy that could test (i) the unconditional coverage, the (ii) conditional coverage assumption and eventually (iii) the independence assumption.

A GMM Duration-Based Test

This paper proposes a new duration-based backtesting tests able to tackle these three issues. Extending the framework proposed by [START_REF] Bontemps | Moment-based tests for discrete distributions[END_REF]Meddahi, (2005, 2006) and [START_REF] Bontemps | Moment-based tests for discrete distributions[END_REF], it consists in using a GMM framework in order to test if durations of VaR violations are geometrically distributed. This approach presents several advantages. First, it is extremely easy to implement, as it consists in GMM moment condition test. Second, it requires only few constraints improving the feasibility of the backtesting tests. Third, it allows for an optimal treatment of the problem associated with parameter uncertainty. Finally the choice of moment conditions enables us to elaborate separate tests for the UC, IND and CC assumptions, which was not possible with the existing duration based tests. Finally, Monte-Carlo simulations will

show that this new test has relatively high power properties.

Orthonormal Polynomials and Moment Conditions

In the continuous case, it is well known that the Pearson family of distributions (Normal, Student, Gamma, Beta, Uniform..) can be associated to some particular orthonormal polynomials whose expectation is equal to zero. These polynomials can be used as special moments to test for a distributional assumption. For instance, the Hermite polynomials associated to the normal distribution are employed to test for normality [START_REF] Bontemps | Testing normality: A GMM approach[END_REF].

In the discrete case, orthonormal polynomials can be de…ned for distributions belonging to the Ord's family (Poisson, Binomial, Pascal, hypergeometric).

The orthonormal polynomials associated to the geometric distribution (6) are de…ned9 as follows:

De…nition 1 The orthonormal polynomials associated to a geometric distribution with a success probability are de…ned by the following recursive relationship, 8d 2 N :

M j+1 (d; ) = (1 ) (2j + 1) + (j d + 1) (j + 1) p 1 M j (d; ) j j + 1 ! M j 1 (d; ) ; (11)
for any order j 2 N , with M 1 (d; ) = 0 and M 0 (d; ) = 1: If the true distribution of d is a geometric distribution with a success probability then, it follows that:

E [M j (d; )] = 0 8j 2 N ; 8d 2 N : (12)
The duration GMM backtesting procedure exploits these moment conditions.

More precisely, let us de…ne fd 1 ; ::; d N g a sequence of N durations between two consecutive VaR violations observed over T periods. Under the conditional coverage assumption, the durations d i , i = 1; ::; N; are i:i:d: and has a geometric distribution with a success probability equals to the coverage rate . Hence, the hypothesis of a correct conditional coverage shortfall probability can be expressed as follows: 10

H 0;CC : E [M j (d i ; )] = 0 j = f1; ::; pg ; (14)
where p denotes the number of moment conditions, with p > 1:

This framework also allows to test separately for the UC hypothesis. Under UC, the mean of durations between two violations is equal to 1= , and this null hypothesis for UC can then be expressed as 11 :

H 0;U C : E [M 1 (d i ; )] = 0: (15)
Thus, any discrete distribution satisfying the property E [M 1 (d; )] = 0; respects the UC hypothesis, whatever its behavior in term of dependence. For such a raison, this test can be interpreted as a simple unconditional coverage test.

Finally, a separate test for the IND hypothesis can also be derived. It consists in testing the hypothesis of a geometric distribution (implying the absence of 10 It is possible to test the conditional coverage assumption by considering at least two moment conditions even if they are not consecutive as soon as the …rst condition E [M 1 (d i )] = 0 is included in the set of moments. For instance, it is possible to test the CC with:

H 0;CC : E [M j (d i )] = 0 j = f1; 3; 7g (13)
For simplicity, we exclusively consider in the rest of the paper the cases where moment conditions are consecutive polynomials.

11 Indeed, since

M 1 (d; ) = (1 d) = p 1 ; it is straightforward to verify that the condition E [M 1 (d; )] = 0 is equivalent to the UC condition E (d) = 1= .
dependence) with a success probability equal ; where parameter can be either …xed a priori, either estimated and is not necessarily equal to the coverage rate . This independence assumption can be expressed as the following moment conditions:

H 0;IN D : E [M j (d i ; )] = 0 j = 1; ::; p; (16)
with p > 1. In this case, the average duration E (d) is equal to 1= as soon as the …rst polynomial M 1 (d; ) is included in the set of moments conditions.

So, under H 0;IN D ; the durations between two consecutive violations have a geometric distribution and the U C is not valid if 6 = .

Empirical Test Procedure

It turns out that VaR forecast tests can be expressed as simple moment conditions, which can be tested within the well-known GMM framework. The philosophy of the test is to choose the appropriate moments and to test if their empirical expectations are close to 0 or not. As observed by [START_REF] Bontemps | Moment-based tests for discrete distributions[END_REF], the orthonormal polynomials M j (d; ) present two great advantages.

First, the corresponding moments are robust to parameter uncertainty and second, the asymptotic matrix of variance covariance is known. Considering the last point it appears that in an i:i:d: context the moments are asymptotically independent with unit variance. As a consequence, the optimal weight matrix of the GMM criteria is simply an identity matrix and the implementation of the backtesting test becomes very easy.

Proposition 2 For a given model M and a …xed coverage rate , let us consider a sequence of N durations, denoted fd i g N i=1 ; observed between two successive violations associated to the % VaR forecasts. The null hypothesis of CC can be expressed as:

H 0;CC : E [M (d i ; )] = 0; ( 17
)
where M (d i ; ) denotes a (p; 1) vector whose components are the orthonormal polynomials M j (d i ; ) ; for j = 1; ::; p. Under some regularity conditions, we know since [START_REF] Hansen | Large sample properties of Generalized Method of Moments estimators[END_REF] that

1 p N N X i=1 M j (d i ; ) ! 2 d ! N !1 2 (1) ; 8j = 1; ::; p; (18)
So that, in an i:i:d:context these moments are asymptotically independent with unit variance and the conditional coverage (CC) statistic test is

J CC (p) = 1 p N N X i=1 M (d i ; ) ! | 1 p N N X i=1 M (d i ; ) ! d ! N !1 2 (p) ; ( 19
)
with p is the number of orthonormal polynomials used as moment conditions.

The J CC (p) test statistic is easy to compute and follows a standard asymptotic distribution.

Test statistic for UC, denoted J U C , is obtained as a special case of the proposition (2), when one considers only the …rst orthonormal polynomial, i:e: when M (d i ; ) = M 1 (d i ; ). J U C is then equivalent to J CC (1) and can be expressed as follows:

J U C (p) = 1 p N N X i=1 M 1 (d i ; ) ! 2 d ! N !1 2 (1): (20)
Finally, the statistic for IN D, denoted J IN D ; is de…ned for a success probability and then can be expressed as follows

J IN D (p) = 1 p N N X i=1 M (d i ; ) ! | 1 p N N X i=1 M (d i ; ) ! d ! N !1 2 (p) : (21)
where M (d i ; ) denotes a (p; 1) vector whose components are the orthonormal polynomials M j (d i ; ) ; for j = 1; ::; p, evaluated for a success probability equal to :

Remark 1: Parameter Uncertainty

The moment-based tests raise a potential problem of parameter uncertainty [START_REF] Bontemps | Moment-based tests for discrete distributions[END_REF][START_REF] Bontemps | Testing distributional assumptions: A GMM approach[END_REF], when some parameters of the distribution under the null are unknown and must be estimated. Concerning our framework, it is obvious that the tests for CC and UC do not face such a problem. In these cases, the only parameter, i.e. ; is known, since it represents the coverage rate de…ned ex-ante by market regulators or risk managers. So, is similar to the one based on M j (d i ; ) : So, we have:

J
J IN D (p) = 1 p N N X i=1 M d i ; b ! | 1 p N N X i=1 M d i ; b ! d ! N !1 2 (p 1) ; (22)
where M d i ; b denotes the (1; p) vector de…ned as M 1 d i ; b :::M p d i ; b :

Note that in this case, the …rst polynomial M 1 d i ; b is strictly proportional to the score used to de…ned the maximum likelihood estimator b and thus

M 1 d i ; b = 0.
So, the degree of freedom of the J-statistic has to be adjusted accordingly.

13 In our applications, we consider the M L estimator of :

Remark 2: Small Sample Property

One of the main issues in the literature on VaR assessment is the relative scarcity of violations. As recalled previously, even with one year of daily returns the number of observed durations between two hits may often be dramatically small, in particular for a 1% coverage rate. This may induce small sample bias that can be corrected with bootstrap experiments but some precautions must be taken. While the statistic used to test the unconditional coverage hypothesis is pivotal, this is not the case for the statistic associated with the test of the independence assumption as it depends on ^ . For this reason, the size of this test has to be controlled using for example the Monte Carlo testing approach of Dufour ( 2006), as done for example in Christo¤ersen and Pelletier (2004).

Simulation Framework for Empirical Size Analysis and Numerical Aspects

To illustrate the size performance of our duration-based test in …nite sample, Monte-carlo experiments were performed. To generate hits sequence of violations we take independent draws from a Bernoulli distribution, considering successively = 1% and = 5% for the VaR nominal coverage. Several sample sizes T , ranging from 250 (which roughly corresponds to one year of trading days) to 1; 500 were also used. Reported empirical sizes correspond to the rejection rates calculated over 10; 000 simulations for a nominal size …xed at 10%. If the asymptotic distribution of our test is adequate, the rejection frequency should be around the nominal size. The test for CC proposed by [START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF] (thereafter labeled LR CC ) constitutes the benchmark against which we appreciate the properties of our GMM backtesting test.

Insert Table 1 The rejection frequencies of the Monte-Carlo experiments are presented in Table 1. It turns out that whatever our test is undersized in …nite sample, but converges to the nominal size when T increases. However, recall that under the null in a sample with T = 250 and a coverage rate equal to 1%, the expected number of durations between two consecutive hits ranges between two and three. This scarcity of violations explains why the empirical size of our asymptotic test is di¤erent from the nominal size in small samples. For this reason, in the next sections, we will use the Monte Carlo testing approach of [START_REF] Dufour | Monte Carlo tests with nuisance parameters: a general approach to …nite sample inference and nonstandard asymptotics[END_REF] that allows to control for the size even in small samples. It also appears that under-rejection worsens as the number of moment conditions used increases. On the contrary, we verify that the LR test proposed by Berkowitz et al. ( 2005) is oversized.

However, it is important to note that these rejection frequencies are only calculated for the simulations providing a J CC as well as the LR CC test statistics. Indeed, for realistic backtesting sample size (for instance T = 250)

and a coverage rate of 1%, many simulations do not deliver a statistics. As previously mentioned, the LR CC test requires to be implemented at least one non-censored duration and an additional possibly censored duration (i:e: two violations). By comparison, the implementation of our GMM test only requires at least one violation (i.e. one or two censored durations). Table 2 reports the feasibility ratios, i.e. the fraction of simulated samples where the LR CC and the J CC tests are feasible.

Insert Table 2

As observed by [START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF], these results highlight huge di¤erences in the cases of 1% VaR and samples of 250 500 observations. For instance, when we consider one year of trading days (T = 250), under the null of conditional coverage, the Weibull LR CC test can be computed only in 6 samples out of 10. Such a simulation exercise illustrates one the advantages of our GMM duration-based test. For higher sample size (i.e. two years of trading days, T = 500), the feasibility ratio is similar for both tests and lies around 1.

Simulation Framework for Power Analysis

We now investigate the power of the test for di¤erent alternative hypothesis.

Following Christo¤ersen and Pelletier (2004), [START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF] or Haas (2005), the DGP under the alternative hypothesis assume that returns, r t ; are issued from a GARCH(1; 1) t (d) model with an asymmetric leverage e¤ect.

More precisely, it corresponds to the following model:

r t = t z t s v 2 v ; (23)
where fz t g is an i:i:d: sequence form a Student's t-distribution with v degrees of freedom and where conditional variance is given by:

2 t = ! + 2 t 1 0 @ s v 2 v z t 1 1 A 2 + 2 t 1 : (24)
Parametrization of the coe¢ cients is also similar to the one proposed by Christo¤ersen and Pelletier (2004) and used by [START_REF] Haas | Improved duration-based backtesting of Value-at-Risk[END_REF], i.e. Using the simulated Pro…t and Loss (P&L thereafter) distribution issued from this DGP, it is then necessary to select a method to forecast the VaR. This choice is of major importance for the power of the test. Indeed, it is necessary to choose a VaR calculation method which are not adapted to the P&L distribution and therefore violate e¢ ciency, i.e. the nominal coverage andnor independence hypothesis. Of course, we expect that the larger the deviation from the nominal coverage andnor independence hypothesis will be, the higher the power of the tests will become. For comparison purpose, we consider the VaR calculation method used by Christo¤ersen andPelletier (2004), Berkowitz et al.(2005) or [START_REF] Haas | Improved duration-based backtesting of Value-at-Risk[END_REF], i.e. the Historical Simulation (HS). As in Christoffersen and Pelletier ( 2004), the rolling windows T e is taken to be either 250 or 500. Formally, HS-VaR is de…ned by the following relation:

V aR tjt 1 ( ) = percentile fr j g t 1 j=t Te ; 100 : (25)

HS easily generates VaR violations. In Figure 1, observed simulated returns r t for a given simulation and VaR-HS are plotted. It appears that violation clusters are evident, whether for 1% VaR or for 5% VaR .

Insert Figure 1 For each simulation, the zero-one hit sequence I t is calculated by comparing the ex post returns r t to the ex ante forecast V aR tjt 1 ( ), and the sequence of durations Di (or Y i ) between violations are calculated from the hit sequence.

From this duration sequence, the test-statistics J CC (p) for p = 1; :::; 5 as well as the [START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF] test (LR CC ) are implemented. The empirical power of the tests is then deduced from rejection frequencies based on 10,000 replications. However, as previously mentioned, the use of asymptotic critical values (based on a 2 distribution) induces important size distortions even for relatively large sample. So given the scarcity of violations (particularly for a 1% coverage rate), it is particularly important to control for the size of the backtesting tests. As usual in this literature, the Monte Carlo technique proposed by [START_REF] Dufour | Monte Carlo tests with nuisance parameters: a general approach to …nite sample inference and nonstandard asymptotics[END_REF] is implemented (See Appendix B).

Insert Tables 3 and4 Tables 3 and4 reports the rejection frequencies (power) of the test for respec-tively 1% and 5% VaR. 14 We report the power of our test for various values of the number of moment conditions, p. We can observe that for small p values, the power is increasing with p. This result illustrates the fact that the Bontemps's framework is not robust to any speci…cation under the alternative if one uses only a few number of polynomials. Each test based on a speci…c polynomial is robust against the alternatives for which the corresponding moment has some expectation di¤erent from zero. Therefore the tests will be robust only if we consider a su¢ cient number of polynomials. In our simulations, it turns out that the power is optimal when considering three moment conditions in the case of the 1% VaR whereas …ve Meixner polynomials are required for a 5% VaR. To illustrate this point, the power is plotted for di¤erent number of moment conditions in Figure 2.

Insert Figure 2

In all cases the power of the GMM based backtesting test J CC is greater than the one of the [START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF] test whatever the sample size considered. In particular, the gain of our test is specially noticeable for the more interesting cases from a practical point of view, that is small sample size and = 1%: For T = 250, the power of our test is two times the power of standard LR test. Such a property constitutes a key point to promote the empirical popularity of duration based backtesting tests. The comparison of the test for UC is impossible as traditional tests do not provide such an information.15 Nevertheless, its power is always relatively high and in all case larger than 25%.

These simulations experiments con…rm that GMM based duration test improves the feasibility and the power of traditional duration based tests. Besides it provides a separate test for CC, UC and IND hypotheses. Our initial objectives are thus ful…lled.

Empirical Application

To illustrate these new tests, an empirical application is performed, consid- 17 At a 5% signi…cance level, the null is rejected for p = 4 and p = 6. When p is equal to 2 the p-value is equal to 0.15.

Conclusion

This paper develops a new duration-based backtesting procedure for VaR forecasts. The underlying idea is that if the one-period ahead VaR is correctly speci…ed, then, every period, the duration until the next violation should be distributed according to a geometric distribution with a success probability equal to the VaR coverage rate. So, we adapt the GMM framework proposed by [START_REF] Bontemps | Moment-based tests for discrete distributions[END_REF] in order to test for this distributional assumption that corresponds to the null of VaR forecast validity. The test statistics boils down to a simple J-statistic based on particular moments de…ned by the orthonormal polynomials associated to the geometric distribution. This new approach tackles most of the drawbacks usually associated to duration based model.

First, its implementation is extremely easy. Second, it allows for a separate the unconditional coverage, the independence and the conditional coverage hypothesis (Christo¤ersen, 1998). Second, feasibility of the tests is improved.

Third, Monte-Carlo simulations show that for realistic sample sizes, GMM test outperforms traditional duration based test. Our empirical application for Nasdaq returns con…rms that using GMM test leads to major for the expost evaluation of the risk by regulation authorities. Our hope is that this paper will constitute an incitation for regulation authorities in order to use of duration-based tests to assess the risk taken by …nancial institutions. There is no doubt that a more adequate evaluation of the risk would decrease the probability of banking crises and systemic banking fragility.

Nevertheless, we have to admit that this test does not constitute the cure to all disease and several limits associated to duration based test are still to be tackled. In particular, it turns out that the estimation of the risk is neglected leading to severe bias (Escanciano and Olmo, 2007) . This will constitute a direct extension for the GMM approach and already lies among our future research plans.

where a denotes the nominal size.

  CC and J U C are thus free of parameter uncertainty. On the contrary, the test for IND hypothesis might be subject to uncertainty problem as the true VaR violations rate is unknown. 12 Consequently, the independence test statistic, J IN D (p) ; must be based on moments that depend on estimated parameters, i.e. instead of having E [M j (d i ; )] where is known we have to deal withE h M j d i ; b iwhere b denotes a square-N -root-consistent estimator 13 of . It is well known that replacing the true value of by its estimates b may change the asymptotic distribution of the GMM statistic. However,[START_REF] Bontemps | Moment-based tests for discrete distributions[END_REF] shows that the asymptotic distribution remains unchanged if the moments can be expressed as a projection onto the orthogonal of the score. Appendix A shows that the moment conditions de…ned by the Meixner orthonormal polynomials satisfy this property. So, the moments used to de…ne the J IN D (p) statistic are robust against the problem of the parameter uncertainty and the asymptotic distribution of the GMM statistic J IN D (p) ; based on M j d i ; b ,

  5; = 0:85; ! = 3:9683e 6 and d = 8: The value of ! is set to target an annual standard deviation of 0:20 and the global parametrization implies a daily volatility persistence of 0:975.

  ering three sequences of 5%VaR forecasts on the daily returns of the Nasdaq index. These sequences correspond to three di¤erent VaR forecasting methods traditionally used in the literature: a pure parametric method (GARCH model under Student distribution), a non parametric method (Historical Simulation) and a semi parametric method based on a quantile regression (CAViaR, Engle and Manganelli, 2004). Each sequence contains 250 successive one-periodahead forecasts for the period June 22, 2005 to June 20, 2006. The parameters of the GARCH and CAViaR models are estimated according to a rolling windows method with a length 16 …xed to 250 observations. The observed returns and 5%VaR forecasts obtained using the three alternative computation methods are reported on Figure C.3. As usually, it can be checked that the HS VaR forecasts are relatively ‡at. This result is fully intuitive since HS-VaR is calculated as the unconditional quantile of past returns: the time variability is then only captured via the rolling historical sample. On the contrary, the VaRs based on conditional variance or conditional quantile are more ‡exible. Consequently, as it is generally observed in the literature, the HS-VaR is more likely to generate violations clusters than other methods. Figure C.4 displays the indicator variable I t ( ) associated to the ex-post 5%VaR violation computed for the three methods. As usual, HS-VaR exhibits clustering in violations: six out of the nine VaR-HS violations occur at the end of the period. The clusters are less obvious when considering the others methods. We also observe on both Figures that the VaR computed from CAViaR is clearly too low compared to the historical returns, i.e. the risk is underestimated: it leads to only seven violations over a sample of 250 observations implying a hit frequency rate around 2.8%. By comparison, there are nine hits for GARCH and HS methods, implying a frequency hit rate equal to 3.6%. So, in this con…guration, if rejection of UC occurs, we expect that it occurs for the VaR-CAViaR. If rejection of IND occurs, we expect that it happens for HS-VaR. The results obtained using the GMM duration-based tests are reported in

  results are based on 10,000 replications. For each sample, we provide the percentage of rejection at a 10% level. Jcc(p) denotes the GMM based conditional coverage test with p moment conditions. Juc denotes the unconditional coverage test obtained for p=1. LRcc denotes the Weibull conditional coverage test proposed by[START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF].

Fig

  Fig. C.1. GARCH-t(v) Simulated Returns with 1% and 5% VaR from HS (T e = 250):

  size of 250, the LR independence tests have an e¤ective power that ranges from 4.6% (continuous Weibull test) to 7.3% (discrete Weibull test) for a nominal coverage of 1%VaR. Similarly, for a coverage of 5%VaR, the power only reaches 14.7% for the continuous Weibull test and 32.3% for the discrete Weibull test. In other words, when VaR forecasts are not valid, LR tests do not reject the VaR validity at best in 7 cases out of 10.

	motivates the use of discrete life-
	time distributions instead of continuous ones, arguing that the parameters of
	the distribution have a clear-cut interpretation in terms of risk management.
	He also conducts Monte-Carlo experiments showing that the backtesting tests
	based on discrete distribution exhibit a higher power than the continuous
	competitor tests.

Moreover, and independently of this aspect, other limitations may explained the lack of popularity of duration-based backtesting tests among practitioners. First, they exhibit low power for realistic backtesting sample sizes. For instance, in some GARCH based experiments

[START_REF] Haas | Improved duration-based backtesting of Value-at-Risk[END_REF]

founds that for a backtesting sample

Table 5 .

 5 For each VaR method, we report the UC, CC and IND statistics. For the two last tests, the number of moments p is …xed to 2, 4 and 6. For seek of comparison LR CC statistics(Christo¤ersen and Pelletier, 2004;[START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF]) are also reported. For all tests, the p-values correspond to the size-corrected ones[START_REF] Dufour | Monte Carlo tests with nuisance parameters: a general approach to …nite sample inference and nonstandard asymptotics[END_REF]. Several comments can be done on these results.First, our unconditional coverage test statistic J U C leads to an unambiguous rejection of the validity of the CAViaR based VaR. As expected, this results is due to the too low violation rate associated to this method. Of course, the value of J U C is identical for HS and GARCH, since these two methods lead to the same number of hits even if these violations do not occur at the same periods. Second, we observe that our GMM independence test (J IN D ) is able to reject (except in the case p = 2) the null for HS-VaR. On the contrary, LR IN D test does not reject the null of independence for any of the three VaRs.Third, at a 10% signi…cance level, the GMM conditional coverage test (J CC ) rejects the validity of CAViaR and HS VaR 17 forecasts, contrary to standard LR tests. Finally, the GARCH-t(d) turns out to be best way to forecast risk: the UC, IND and CC are not rejected.

  Statistic J CC (p) simulated N times under H 1 gives the test power, equal to the number of times when pM [J CC (p)] is inferior or equal to the nominal size a (…xed at 10% in our replications), divided by N simulations. The nominal size of the test is thus respected for …nite size sample. In our application, we set M at 9; 999. This method allows the comparison of the empirical power of various tests independently from their empirical size. Under the null, the hit data are i.i.d. from a Bernoulli distribution. The results are based on 10,000 replications. For each sample, we provide the percentage of rejection at a 10% level. Jcc(p)denotes the GMM based conditional coverage test with p moment conditions. Juc denotes the unconditional coverage test obtained for p=1. LRcc denotes the Weibull conditional coverage test proposed byBerkowitz et al.

	Table 1. Empirical Size of 10% Asymptotic CC Tests
			Backtesting 1% VaR
	Sample	J U C	J CC (2) J CC (3) J CC (5) LR CC
	T = 250 0.0387 0.0237 0.0142 0.0066 0.1090
	T = 500 0.0535 0.0253 0.0168 0.0099 0.1809
	T = 750 0.0701 0.0486 0.0382 0.0283 0.1605
	T = 1000 0.0867 0.0684 0.0528 0.0401 0.1598
	T = 1250 0.0834 0.0653 0.0502 0.0401 0.1324
	T = 1500 0.0925 0.0736 0.0589 0.0456 0.1316
			Backtesting 5% VaR
	Sample	J U C	J CC (2) J CC (3) J CC (5) LR CC
	T = 250 0.0786 0.0615 0.0489 0.0402 0.1381
	T = 500 0.0942 0.0759 0.0558 0.0460 0.1349
	T = 750 0.0979 0.0858 0.0691 0.0534 0.1431
	T = 1000 0.0955 0.0814 0.0684 0.0521 0.1472
	T = 1250 0.0944 0.0853 0.0701 0.0512 0.1556
	T = 1500 0.1002 0.0840 0.0692 0.0515 0.1719
	(2005).

Notes:

Table 2 .

 2 Fraction of Samples where Tests are Feasible The results are based on 10,000 replications. For each sample and for each test, we provide the percentage of samples for which the statistic can be computed. Jcc denotes the GMM based (un)conditional coverage test. For the J test, note that the feasible ratios are independent of the number p of moments used. LRcc denotes the Weibull conditional coverage test proposed by[START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF].

			Size Simulations
		1% VaR	5% VaR
	Sample Size	J CC	LR CC	J CC	LR CC
	T = 250	0.9217 0.6249 1.000	0.9999
	T = 500	0.9944 0.9326 1.000	1.000
	T = 750	0.9996 0.9917 1.000	1.000
	T = 1000	1.0000 0.9984 1.000	1.000
		Power Simulations (T e = 500)
		1% VaR	5% VaR
	Sample Size	J CC	LR CC	J CC	LR CC
	T = 250	0.7953 0.5972 0.9905	0.9681
	T = 500	0.9625 0.8894 0.9999	0.9995
	T = 750	0.9970 0.9852 1.000	1.000
	T = 1000	0.9691 0.9985 1.000	1.000

Notes:

Table 3 .

 3 Power of 10% Finite Sample Tests on 1% VaR Length of Rolling Estimation Window T e = 250 : The results are based on 10,000 replications. For each sample, we provide the percentage of rejection at a 10% level. Jcc(p) denotes the GMM based conditional coverage test with p moment conditions. Juc denotes the unconditional coverage test obtained for p=1. LRcc denotes the Weibull conditional coverage test proposed by Berkowitz et al. (2005).

	Sample	J U C	J CC (2) J CC (3) J CC (5)	LR CC
	T = 250 0.4132 0.4369 0.4580 0.4980	0.2098
	T = 500 0.3990 0.4674 0.5140 0.5710	0.3244
	T = 750 0.3924 0.4897 0.5626 0.6276	0.4268
	T = 1000 0.4416 0.5496 0.6396 0.7115	0.5529
	T = 1250 0.4924 0.6062 0.6989 0.7752	0.6728
	T = 1500 0.5339 0.6592 0.7563 0.8284	0.7465
		Length of Rolling Estimation Window T e = 500
	Sample	J U C	J CC (2) J CC (3) J CC (5)	LR CC
	T = 250 0.4329 0.4554 0.4790 0.5177	0.1913
	T = 500 0.4074 0.4837 0.5359 0.5858	0.3570
	T = 750 0.3749 0.5320 0.5967 0.6623	0.4917
	T = 1000 0.3567 0.5767 0.6543 0.7170	0.6014
	T = 1250 0.3532 0.6389 0.7387 0.7935	0.7121
	T = 1500 0.3679 0.6876 0.7900 0.8410	0.7772

Notes

Table 4 .

 4 Power of 10% Finite Sample Tests on 5% VaR Length of Rolling Estimation Window T e = 250

	Sample	J U C	J CC (2) J CC (3) J CC (5)	LR CC
	T = 250 0.3956 0.5738 0.6106 0.6100	0.3652
	T = 500 0.2791 0.7266 0.7865 0.7792	0.4920
	T = 750 0.2656 0.8432 0.8935 0.8875	0.6274
	T = 1000 0.2462 0.9092 0.9467 0.9467	0.7439
	T = 1250 0.2574 0.9513 0.9737 0.9719	0.8251
	T = 1500 0.2534 0.9731 0.9873 0.9848	0.8700
		Length of Rolling Estimation Window T e = 500

Or the opposite of V aR if the latter is de…ned as a loss in absolute value.

For instance, Christo¤ersen's test (1998) based on Markov chain, the hit regression test of[START_REF] Engle | CAViaR: Conditional Autoregressive Value-at-Risk by regression quantiles[END_REF] relies on a linear auto-regressive model, or the tests of[START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF] based on tests of martingale di¤erence.

The LR test requires at least one non-censored duration and an additional possibly censored duration (i.e. two violations) to be implemented. As experienced by[START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF] with one year of trading days (T = 250) and = 0:01 the test can be computed only in six cases out of ten.

The memory-free property implies a ‡at hazard function. On the contrary, violation clustering corresponds to decreasing hazard function, implying that the probability of no violation spell ending decreases as the spell increases in length.

as in[START_REF] Kupiec | Techniques for verifying the accuracy of risk measurement models[END_REF], Christo¤ersen (1998),[START_REF] Engle | CAViaR: Conditional Autoregressive Value-at-Risk by regression quantiles[END_REF].

A characteristic which they share with other approaches (Christo¤ersen, 1998 or[START_REF] Engle | CAViaR: Conditional Autoregressive Value-at-Risk by regression quantiles[END_REF]

These polynomials can be viewed as a particular case of the Meixner orthonormal polynomials associated to a Pascal (negative Binomial) distribution.

The true VaR violations rate may be di¤erent from the coverage rate …xed by the risk manager in the model.

We only report results for the UC and CC. Outcomes of the simulations for IND tests are available upon request from the authors.

As already noticed, traditional duration based tests do no provide a separate test for UC.

The total sample runs fromJune 20, 2004 to June 20, 2006 (500 observations). The length of the rolling estimation window of the HS is also …xed to 250 observations.

A Appendix: Proof of parameter uncertainty robustness of J IN D Under the independence assumption, the sequence of durations fd i g N i=1 is a sequence of i:i:d. geometric random variables with a success probability , where is a priori unknown. The probability distribution function of d i is:

The score is then de…ned as:

It is straightforward to prove that this score is proportional to the …rst Meixner polynomial since:

and so:

Consequently, the orthonormal polynomials with degrees greater or equal to 2 are also proportional to the score. [START_REF] Bontemps | Moment-based tests for discrete distributions[END_REF] shows that in such a case, To implement this technique we …rst generate M independent realizations of the test statistic, S i , i = 1; : : : ; M , under the null hypothesis, i.e., using durations constructed from independent Bernoulli hit sequences. We denote by S 0 the value of the test statistic obtained for the original sample. As shown by [START_REF] Dufour | Monte Carlo tests with nuisance parameters: a general approach to …nite sample inference and nonstandard asymptotics[END_REF], in a general case, the Monte Carlo p-values can be calculated as follows:

where ĜM (x) = 1=M [START_REF] Christoffersen | Backtesting Value-at-Risk: A duration-based approach[END_REF] and [START_REF] Berkowitz | Evaluating Value-at-Risk models with desk-level data[END_REF]. For all these tests, the numbers in parenthesis denote the corresponding p-values corrected by the Dufour's Monte Carlo procedure [START_REF] Dufour | Monte Carlo tests with nuisance parameters: a general approach to …nite sample inference and nonstandard asymptotics[END_REF].