Hind Julien Gigault
email: julien.gigault@takuvik.ulaval.ca

El Hadri

Brian Nguyen

Bruno Grassl

Laura Rowenczyk

Nathalie Tufenkji
email: nathalie.tufenkji@mcgill.ca

Siyuan Feng

Mark Wiesner

Nanoplastics are neither microplastics nor engineered nanoparticles

la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

While the accumulation of microplastics and larger plastic debris in the world's oceans has garnered much attention, colloidal plastic debris may represent a portion of released plastic that remains unaccounted for based on oceanic circulation models [START_REF] Eriksen | Plastic Pollution in the World's Oceans: More than 5 Trillion Plastic Pieces Weighing over 250,000 Tons Afloat at Sea[END_REF][START_REF] Geyer | use, and fate of all plastics ever made[END_REF][START_REF] Jambeck | Plastic waste inputs from land into the ocean[END_REF][START_REF] Sebille | A global inventory of small floating plastic debris[END_REF] . Scientists also estimate that plastic pollution on land and in freshwaters [START_REF] Horton | Microplastics in freshwater and terrestrial environments: Evaluating the current understanding to identify the knowledge gaps and future research priorities[END_REF] can be many times greater than the estimated 4.8 to 12.7 million metric tons [START_REF] Jambeck | Plastic waste inputs from land into the ocean[END_REF] of plastic annually emitted to the ocean, yet little is known regarding the levels of colloidal plastics in these environmental compartments. Nanoplastics are the smaller nano-scale fraction of these colloids and are most likely to be incidentally produced from the fragmentation of larger plastic debris. Although complete breakdown of larger plastic debris can take up to hundreds of years, mechanical wear [START_REF] Ekvall | Nanoplastics formed during the mechanical breakdown of daily-use polystyrene products[END_REF] , heat [START_REF] Hernandez | Plastic Teabags Release Billions of Microparticles and Nanoparticles into Tea[END_REF] , UV degradation [START_REF] Lambert | Formation of microscopic particles during the degradation of different polymers[END_REF] and, in some cases, biological factors [START_REF] Dawson | Turning microplastics into nanoplastics through digestive fragmentation by Antarctic krill[END_REF] , lead to relatively rapid fragmentation of plastic debris down to the nano-scale.

Our understanding of the interactions between incidental nanoplastics and the environment is in its infancy. Due to methodological challenges [START_REF] Nguyen | Separation and Analysis of Microplastics and Nanoplastics in Complex Environmental Samples[END_REF] , nanoplastics in environmental samples remain largely unquantified, although recent work has reported the chemical signatures of nanoplastic contamination in ocean waters [START_REF] Ter Halle | Nanoplastic in the North Atlantic Subtropical Gyre[END_REF] . Due to their similar composition and origin, as well as the nature of the research communities involved, nanoplastics have been largely treated as an extension of microplastics. However, size-dependent properties of nanoplastics distinguish them from microplastics with respect to their transport properties, interactions with light and natural colloids, analytical challenges, bioavailability, potential toxicity, and leaching times for additives. And, unlike engineered nanomaterials (ENMs) which can include polymer formulations, incidentally-produced nanoplastics in the environment are essentially debris from the environmental fragmentation of larger plastic objects. Chemists, biologists, physicists, ecologists, engineers, toxicologists, and other scientific professionals in the community of researchers examining the environmental, health, and safety (EHS) of ENMs have made significant advances over the last twenty years in developing methodologies for studying nanoscale materials and elucidating the environmental behavior of nano-scale objects. ENMs have been the focus of much of this nanoEHS research. The uniform size and composition that can be achieved in making ENMs, has made them excellent tools for studying the behavior of nanoscale particles in complex environments. Conversely, the heterogeneity of most incidental nanomaterials, including nanoplastics, presents numerous challenges to tracking and quantifying these materials in complex environments. Consequently, nanoplastics should be considered as a unique class of contaminants, distinct from both microplastics and ENMs.

Nanoplastics are distinct from microplastics

An increase in publications dealing with microplastics has coincided with a shift in terminology, and entry of new scientific communities in addressing the issue of plastic waste as a growing environmental concern. For example, polystyrene spheres are increasingly described in the recent scientific literature as either micro-or nano-"plastic" rather than simply as particles or nanoparticles (Figure 1). The labeling of polystyrene spheres as microplastics (Fig 1a) or nanoplastics (Fig. 1b) introduces new terms for materials used in virtually identical studies conducted with a different motivation many decades earlier [START_REF] Petosa | Aggregation and Deposition of Engineered Nanomaterials in Aquatic Environments: Role of Physicochemical Interactions[END_REF] . The re-casting of the microplastic problem as one that distinctly entails nanoplastics (Figure 1b) may reflect a retooling of the nanoEHS community to apply methods to the problem of plastic debris. Indeed, there is a large body of knowledge gained from studying ENM behavior in the workplace, in consumer products, and in natural and complex environmental systems that can be extended to nanoplastics research. In the last two decades, significant analytical developments have been made to characterize, identify and quantify ENMs in aqueous media such as fullerenes [START_REF] Chen | Quantification of C60 fullerene concentrations in water[END_REF] , silver nanoparticles [START_REF] Benn | Nanoparticle silver released into water from commercially available sock fabrics[END_REF] , and TiO2 nanoparticles [START_REF] Wang | Fate and biological effects of silver, titanium dioxide, and C60 (fullerene) nanomaterials during simulated wastewater treatment processes[END_REF] and to elucidate the environmental fate and impact of these materials. A key lesson learned from ENM research is that properly defining terms early in the trajectory of the research and establishing rigorous ontologies, enables data sharing across communities. It is therefore important to articulate what is meant by the terms nanoplastic, microplastic and their relationship to ENMs. diamond). The light blue vertical line highlights the coincidence between the decrease in the number of publications and the exponential increase in publications that occurs by changing the terms from "particle" and "nanoparticle" to "microplastic" and "nanoplastic".

While all plastics are polymer-based, not all polymers are plastic and nanoplastics are not synonymous with nanopolymers [START_REF] Gangadoo | Nano-plastics and their analytical characterisation and fate in the marine environment: From source to sea[END_REF] . Plastics will typically include material composed of polymer as well as additives. The term "nanoplastic" has typically been used with reference to solely size, and with some inconsistency as to what the exact size cut-off should be between a nanoplastic and a microplastic, with 100 nm and 1000 nm being the most common size cutoffs [START_REF] Gigault | Current opinion: What is a nanoplastic?[END_REF][START_REF] Rist | Aquatic Ecotoxicity of Microplastics and Nanoplastics: Lessons Learned from Engineered Nanomaterials[END_REF][START_REF] Hartmann | Are We Speaking the Same Language? Recommendations for a Definition and Categorization Framework for Plastic Debris[END_REF] . The US National Nanotechnology Initiative (NNI) defines a nanomaterial as having at least one dimension between 1 and 100 nm and exhibiting properties not found at larger sizes of the same material. The definition of "nano" typically involves considerations that go beyond arbitrary size cut-offs [START_REF] Auffan | Towards a definition of inorganic nanoparticles from an environmental, health and safety perspective[END_REF] . While conflicting scientific, commercial, and regulatory considerations complicate the definition of nanomaterials [START_REF] Auffan | Towards a definition of inorganic nanoparticles from an environmental, health and safety perspective[END_REF] , the distinction between nanomaterials and their larger counterparts is generally held to be dependent on both size and the resulting properties. For example, semi-conductor quantum dots fluoresce at size-dependent wavelengths due to quantum confinement of electrons at particle sizes well below electron wavelengths. Consequently, the properties and behavior of nanomaterials cannot be extrapolated from the properties of their bulk counterparts. In our view, it is these characteristics, that separate "nano" from "micro" regardless of a particular size range [START_REF] Auffan | Towards a definition of inorganic nanoparticles from an environmental, health and safety perspective[END_REF] . Further, one may differentiate between engineered, incidental, and natural nanoparticles, distinctions that may be blurred as in the case of C60 which can be produced as an ENM, but is also present in combustion products ranging from forest fires to industrial combustion [START_REF] Isaacson | Quantitative Analysis of Fullerene Nanomaterials in Environmental Systems: A Critical Review[END_REF] . As incidental nanomaterials, nanoplastics are likely to exhibit size-emergent properties (Figure 2) that will depend on both the origin of the material and the pathway to its creation. Characteristics that distinguish nanoplastics from microplastics include: I. The dominance of Brownian motion over sedimentation and buoyancy characterized by random movements of the particles in a suspension medium; II.

Departure from the geometric/ray approximation between light and matter. For microplastics, the interaction of light and microplastic can be adequately approximated by modeling light as a straight ray. In contrast, with nanoplastics, the wave-nature of light and associated phenomena important in some analytical methods, including diffraction, are significant; III.

A high proportion of molecules on the surface resulting in a higher relative importance of surface interactions compared to physical interactions; IV.

The thickness of the particle diffuse layer may be comparable to the size of environmental macromolecules (leading to adsorption/heteroaggregation), while it is small compared to the size of associated microorganisms (preventing biofilm formation); V.

Sizes compatible with bio-uptake, translocation and transport across biological membranes and; VI.

Short length scales that may speed the diffusive release of plastic additives and nonintentionally added substances in the original bulk plastic.

Nanoplastics have unique characteristics compared to engineered nanomaterials

While nanoplastics may share many of the properties of ENMs, nanoplastics found in the environment present unique considerations and challenges. To begin with, environmental nanoplastics, largely incidental in origin, represent a vastly higher exposure potential compared to ENMs. Of the 6300 million metric tons of plastic waste generated between 1950 and 2015, approximately 5000 million metric tons were emitted to the environment, where they can eventually break down into nanoplastics [START_REF] Geyer | use, and fate of all plastics ever made[END_REF] . In 2018, plastic production was approximately 359 million metric tons worldwide [START_REF]PlasticsEurope[END_REF] . In contrast, rough estimates for global production of ENMs of all kinds range from thousands of metric tons per year to no more than 10 6 metric tons.

Compared to plastics production that emerged as early as the 1950s [START_REF] Resnik | How Should Engineered Nanomaterials Be Regulated for Public and Environmental Health?[END_REF] , ENM production is a relatively recent activity that has expanded rapidly since the 1990s.

Furthermore, environmental nanoplastics as a class of contaminants are substantially more heterogeneous compared to ENMs. ENMs are intentionally created to desired specifications, typically with a uniform composition for a given material. The separation and characterization of ENMs in complex media are facilitated by knowing these specifications and their resulting properties (i.e., optical, magnetic, conductor properties). Standardized techniques adapted to ENMs are now widely used by the scientific community. Note that there exists ENMs composed of polymer types that could be considered "plastic". For example, spherical and monodisperse polystyrene nanoparticles, used in the ENM community, are used as a reference or model material for calibrating analytical tools such as field flow fractionation, size exclusion chromatography, static light scattering, and other techniques, as summarized elsewhere [START_REF] Mourdikoudis | Characterization techniques for nanoparticles: comparison and complementarity upon studying nanoparticle properties[END_REF] . Due to their high uniformity, these plastic spheres are easy to track during the analyses, but are not representative of the diversity of incidental nanoplastics in the environment [START_REF] Sander | Assessing the environmental transformation of nanoplastic through 13 C-labelled polymers[END_REF] . Environmental nanoplastics are generally not intentionally designed and vary greatly in shape, size, polydispersity, additives, adsorbed contaminants, surface properties, and composition as a consequence of different source materials, fragmentation pathways and environmental exposure [START_REF] Rochman | Rethinking microplastics as a diverse contaminant suite[END_REF] . The resulting physical and chemical heterogeneity of nanoplastics may influence their reactivity and will certainly affect interactions with natural colloids and organisms. For example, weathering induced by UV radiation enhances fragmentation of bulk plastic into micro-and nanoplastics, in addition to modifying material chemical properties such as crystallinity and polarity which significantly influence the adsorption of substances [START_REF] Liu | Effect of weathering on environmental behavior of microplastics: Properties, sorption and potential risks[END_REF][START_REF] Holmes | Interactions between trace metals and plastic production pellets under estuarine conditions[END_REF] . The result is an extremely broad scope of materials to investigate.

These incidental nanoplastics are distinguished from ENMs (Figure 2) by: VII.

High particle heterogeneity originating from variable sources and environmental conditions that result in nano-scale materials with a wide range of sizes, shapes, and overall composition even when the nanoplastics originate from a common source; VIII.

Incidental nanoplastics may include the fragmentation products of engineered (primary) nanoplastics with the potential for further fragmentation in the environment at shorter time scales.

While some researchers are creating model materials intended to be used as nanoplastic proxies [START_REF] Sander | Assessing the environmental transformation of nanoplastic through 13 C-labelled polymers[END_REF][START_REF] Balakrishnan | Towards more realistic reference microplastics and nanoplastics: preparation of polyethylene micro/nanoparticles with a biosurfactant[END_REF][START_REF] Pessoni | Soap-and metal-free polystyrene latex particles as a nanoplastic model[END_REF][START_REF] Mitrano | Synthesis of metal-doped nanoplastics and their utility to investigate fate and behaviour in complex environmental systems[END_REF] , in contrast with ENMs, there are no "standard" materials that can be used to fully represent environmental nanoplastics or track them. Nevertheless, many factors influence the formation of environmental nanoplastics and since most of these processes are still poorly understood, nanoplastic proxies used in the laboratory can include idealizations that are not fully representative of environmental realities or that are not generalizable to other types of nanoplastics [START_REF] Wagner | Things we know and don't know about nanoplastic in the environment[END_REF][START_REF] Koelmans | Proxies for nanoplastic[END_REF] . Therefore, as we should not overlook the fact that CuO nanoparticles can have different effects from TiO2 particles when interpreting experimental results, we should not dismiss the varying impacts that different types of nanoplastics can have. The use of fieldcollected samples, and their subsequent fractionation using a "top-down" approach, followed by their characterization, can provide a baseline comparison for results from experiments using nanoplastic proxies.

Environmental fate and behavior

The distinct characteristics of nanoplastics will influence their environmental fate and behavior, interactions with biological systems, sampling strategies, analytical methods, experimental and computational modeling approaches that cannot be extrapolated from microplastics or ENMs.

Due to the colloidal nature and dominance of Brownian motion over sedimentation and buoyancy, vertical transport of individual nanoplastic particles will be small compared with microplastics composed of the same material (property I in Figure 2). For example, polypropylene and polyethylene have densities less than water, and are therefore expected to float in water, while polyvinyl chloride would be expected to settle because of its higher density (~1.4 g/cm 3 ). By Stoke's law, a 1 mm polyvinyl chloride microplastic is calculated to settle at a rate of approximately 22 cm/sec in water compared with a 100 nm nanoplastic which would be expected to settle at a rate of only some 7 cm per year. Thus nanoplastics, as with other colloidal species, are more likely to remain homogeneously dispersed in aqueous systems [START_REF] Rist | Aquatic Ecotoxicity of Microplastics and Nanoplastics: Lessons Learned from Engineered Nanomaterials[END_REF] and may be more likely to remain suspended in the atmosphere and form a portion of "ultra-fine" (sub 100 nm) particulate matter [START_REF] Azimi | Emissions of Ultrafine Particles and Volatile Organic Compounds from Commercially Available Desktop Three-Dimensional Printers with Multiple Filaments[END_REF] . However, aggregation with other particles may increase the effective settling rate of nanoplastics. Nanoplastics can be expected to heteroaggregate with natural colloids such as natural organic matter (polysaccharides, humic acids, leonardite, etc.), iron oxides, and clays, and/or anthropogenic material [START_REF] Auffan | Towards a definition of inorganic nanoparticles from an environmental, health and safety perspective[END_REF][START_REF] Alimi | Microplastics and Nanoplastics in Aquatic Environments: Aggregation, Deposition, and Enhanced Contaminant Transport[END_REF] or with aerosols in the atmosphere (property IV in Figure 2). Macromolecules that may associate with nanoplastics will have length scales similar to the nanoplastic, while being much smaller than the size of a microplastic. This may affect the degrees of freedom of the macromolecule and therefore the attachment efficiency between nanoplastics and heteroaggregates. Heteroaggregation and the conformation of charged macromolecules in water is in turn influenced by environmental conditions (i.e., porewater pH, ionic strength, hardness, etc). These factors will disproportionately influence the attachment efficiency of nanoplastics relative to microplastics due to the dominant role of surfaces at the nanoscale. In addition, the collision rate kernel for nanoplastic heteroaggregation will be dominated by Brownian diffusion rather than by settling, buoyancy, or fluid motion. Consequently, nanoplastic distribution in the environment cannot be directly extrapolated from that of microplastics.

The fact that nanoplastics are smaller than most microorganisms, implies differences in the microbial interactions and biofilm formation. Microplastics are large enough to host a community of micro-organisms (i.e., the "plastisphere") [START_REF] Zettler | Life in the "Plastisphere": Microbial Communities on Plastic Marine Debris[END_REF][START_REF] Amaral-Zettler | Ecology of the plastisphere[END_REF] and accompanying development of complex biofilms. These micro-organisms can form a significant portion of the overall mass and can affect environmental distribution by altering the effective density of microplastics, but the overall particle is still primarily the microplastic [START_REF] Isaacson | Quantitative Analysis of Fullerene Nanomaterials in Environmental Systems: A Critical Review[END_REF] . In contrast, nanoplastics may evolve to be minor components of a larger nanoplastic-microbial complex and, like ENMs [START_REF]PlasticsEurope[END_REF] , may make up a small percentage of the mass of heteroaggregates. The environmental fate of these heteroaggregates may not have a strong dependence on the properties of nanoplastics (property IV in Figure 2). Figure 3 illustrates the contribution of natural organic matter (NOM) relative to a volume of plastic debris. It is interesting to note the shift in the slope of the curve which occurs at the transition between nanoplastics and microplastics. The NOM is a minor component compared to the host plastic from the millimeter to the micrometer scales. As the plastic size decreases, the contribution of the NOM increases and it becomes analytically challenging to discriminate the plastic component, especially given similar carbon-based structures. Unlike with microplastics, the transport, uptake and accumulation pathways of nanoplastics will be highly influenced by NOM relative to the intrinsic properties of the plastic.

The outsized influence of NOM on environmental fate can lead to inappropriate strategies during sampling, analysis and lifecycle assessments if only the plastic component of heteroaggregates is considered. Plastics often contain a wide variety of chemical additives as well as non-intentionally added substances such as degradation products, reaction by-products and/or impurities [START_REF] Rochman | Rethinking microplastics as a diverse contaminant suite[END_REF][START_REF] Muncke | Exposure to endocrine disrupting compounds via the food chain: Is packaging a relevant source?[END_REF][START_REF] Zimmermann | Benchmarking the in Vitro Toxicity and Chemical Composition of Plastic Consumer Products[END_REF] . Since these other chemicals are not generally covalently bound to the polymer matrix, they may leach out of the plastic [START_REF] Rochman | Rethinking microplastics as a diverse contaminant suite[END_REF][START_REF] Hirai | Organic micropollutants in marine plastics debris from the open ocean and remote and urban beaches[END_REF] . These leached chemicals include bisphenol A, phthalates, nonylphenols, brominated flame retardants, to name a few [START_REF] Rochman | Rethinking microplastics as a diverse contaminant suite[END_REF][START_REF] Hirai | Organic micropollutants in marine plastics debris from the open ocean and remote and urban beaches[END_REF] . We have used analytical solutions describing leaching of sorbed materials from a homogeneous particle matrix (e.g., Crank, 1975) [START_REF] Crank | Diffusion | Laplace Transform[END_REF] in conjunction with literature data describing the leaching of additives from plastic pipe materials to obtain estimates of additive leaching from particles of various sizes [START_REF] Mercea | Modelling migration of substances from polymers into drinking water. Part 1 -Diffusion coefficient estimations[END_REF] . We estimate the rates that additives leach from nanoplastics to be many orders of magnitude greater than the leaching rate from microplastics due to the smaller distances for diffusive transport (property VI in Figure 2). These rates, predicted in many cases to be comparable to the inverse residence time of digestion, depend on both the type of additive and the plastic and may vary by two or more orders of magnitude. Thus, additive release can be anticipated in many cases to occur in organisms as well as in the environment. Combined with increased bioavailability and the enhanced accessibility to tissues, nanoplastics may be expected to deliver locally high doses of leached compounds compared with microplastics and therefore exhibit differences in apparent dose-response. The specificity of releases to tissues and the consequences of these localized releases have not been studied.

Biological Consequences

Size has been established as a key factor in the ability of nano-sized particles to translocate in organisms [START_REF]Whole-Body Distribution, and Depuration of Nanoplastics by the Scallop Pecten maximus at Environmentally Realistic Concentrations[END_REF] . Bio-uptake, biomagnification and maternal transfer have all been observed for ENMs [START_REF] Rochman | Rethinking microplastics as a diverse contaminant suite[END_REF][START_REF] Alimi | Microplastics and Nanoplastics in Aquatic Environments: Aggregation, Deposition, and Enhanced Contaminant Transport[END_REF][START_REF] Karlsson | Size-dependent toxicity of metal oxide particles-a comparison between nano-and micrometer size[END_REF][START_REF] Ruenraroengsak | Respiratory epithelial cytotoxicity and membrane damage (holes) caused by amine-modified nanoparticles[END_REF] . Approaching the size of natural proteins, nanoplastics may be small enough to travel across biological membranes via passive diffusion and access certain endocytosis pathways [START_REF] Jiang | Nanoparticle-mediated cellular response is size-dependent[END_REF][START_REF] Zhao | Entry of nanoparticles into cells: the importance of nanoparticle properties[END_REF] (property V in Figure 2). Johnston et al. found that fresh fumes containing nanoscale polytetrafluorethylene were more toxic to rodents than aged fumes which contained larger, coagulated aggregates of polytetrafluorethylene [START_REF] Johnston | Pulmonary effects induced by ultrafine PTFE particles[END_REF] . Bioavailability of ENMs to plants appears to increase as particle diameter decreases below 20 nm [START_REF] Schwab | Barriers, pathways and processes for uptake, translocation and accumulation of nanomaterials in plants-Critical review[END_REF] . Increased bioavailability is consistent with the higher toxicities often found in ENMs compared to their larger counterparts. For example, in plants exposed to CuO, nanoscale particles resulted in greater toxicity compared to micron-sized particles [START_REF] Karlsson | Size-dependent toxicity of metal oxide particles-a comparison between nano-and micrometer size[END_REF] . Rist et al. found that nanoscale polystyrene caused decreased feeding rates and were egested to a lesser degree in contrast to microscale polystyrene [START_REF] Rist | Ingestion of micro-and nanoplastics in Daphnia magna -Quantification of body burdens and assessment of feeding rates and reproduction[END_REF] . Moreover, nanomaterials can interact with subcellular components and trigger responses including reactive oxygen species production.

A considerable portion of the molecules in nanoplastics are exposed to the surface resulting in greatly enhanced surface reactivity compared to their micro-and macroscale counterparts leading to heightened importance of surface chemistries on interactions with biological systems (property III in Figure 2). For example, Miao et al. (2019) showed clear effects on the biological activity of biofilms for nanoscale polystyrene beads (100 nm) as opposed to larger polystyrene particles and for positively charged nanoscale polystyrene compared to their carboxylfunctionalized counterparts [START_REF] Miao | Acute effects of nanoplastics and microplastics on periphytic biofilms depending on particle size, concentration and surface modification[END_REF] . Interactions with proteins and changes in protein conformation, production of reaction oxygen species, and acting as a vector for other contaminants (Trojan horse effects) are among the phenomena that have been observed for ENMs that may also come into play with nanoplastics. These effects could potentially enhance the toxicity of the nanoplastics.

Sampling and analysis of nanoplastics

Table 1 summarizes how the characteristics of nanoplastics that distinguish them from microplastics (I to VI) and ENMs (VII-VIII) impact the analytical approaches used to study their physicochemical properties and environmental fate and behavior. Brownian motion distinguishes separation and analytical techniques used for nanoplastics studies in contrast to microplastics. While the small size of nanoplastics generally makes dead-end filtration (typically used for microplastic separation) impractical, the resulting dominance of Brownian motion allows separation and analytical techniques typically applied to ENMs [START_REF] Astefanei | Characterization of aggregates of surface modified fullerenes by asymmetrical flow field-flow fractionation with multi-angle light scattering detection[END_REF][START_REF] Bolea | Size characterization and quantification of silver nanoparticles by asymmetric flow field-flow fractionation coupled with inductively coupled plasma mass spectrometry[END_REF][START_REF] Kammer | Separation and characterization of nanoparticles in complex food and environmental samples by field-flow fractionation[END_REF][START_REF] Baalousha | Flow field-flow fractionation for the analysis and characterization of natural colloids and manufactured nanoparticles in environmental systems: A critical review[END_REF] . These techniques include: crossflow ultrafiltration, asymmetric flow field flow fractionation [START_REF] Gigault | Asymmetrical Flow Field Flow Fractionation Methods to Characterize Submicron Particles: Application to Carbon-Based Aggregates and Nanoplastics[END_REF][START_REF] Correia | Detection of nanoplastics in food by asymmetric flow field-flow fractionation coupled to multi-angle light scattering: possibilities, challenges and analytical limitations[END_REF] , nanoparticle tracking analysis and dynamic light scattering.

The nanoscale size of nanoplastics limits far-field geometric optics-based imaging and chemical analysis techniques. As nanoscale sizes are smaller than the wavelength of visible or infrared light, optical diffraction becomes significant, resulting in an Abbe diffraction resolution limit of ~200 nm for analysis techniques that rely on laser spot sizes (micro-Raman) or transmitted/reflected light (micro-FTIR, brightfield microscopy). These techniques are popular in microplastics analysis as physical and chemical information can be obtained with a single instrument. Consequently, as with ENMs, electron microscopy or diffraction unlimited light microscopy (e.g., stimulated emission depletion microscopy, photoactivated localization microscopy) is required to image nanoplastics. These far field light techniques are typically not compatible with full chemical characterization of nanoplastics. Near field techniques avoid diffraction limitations on resolution by confining light to an evanescent field. Recent developments of these techniques show promising results for chemical and topographic characterizations of nanoplastics (i.e., spatial resolution limit of 10-20 nm for AFM-IR and AFM-Raman) [START_REF] Dazzi | AFM-IR: Technology and Applications in Nanoscale Infrared Spectroscopy and Chemical Imaging[END_REF] . However, near field techniques require samples to be scanned by the evanescent field emitted from probes which limits their throughput and ability to analyze morphologically complex samples.

Overall, as with ENMs, a multi-parameter analytical approach is required to fully characterize nanoplastics in environmental samples [START_REF] Domingos | Characterizing Manufactured Nanoparticles in the Environment: Multimethod Determination of Particle Sizes[END_REF][START_REF] Stone | Nanomaterials for environmental studies: classification, reference material issues, and strategies for physico-chemical characterisation[END_REF] . This includes determining physical and chemical characteristics while retaining information about the initial dispersion state and nanomaterialmatrix interactions. Multiple techniques can be combined to elucidate multiple parameters from environmental nanoplastics. Consequently, selection of sampling and analytical techniques should consider compatibility with other complementary techniques. For example, field flow fractionation, recently applied for nanoplastics [START_REF] Gigault | Asymmetrical Flow Field Flow Fractionation Methods to Characterize Submicron Particles: Application to Carbon-Based Aggregates and Nanoplastics[END_REF] , itself allows size separation and characterization of materials in the colloidal size range but can also be coupled to other techniques including light scattering (provides information on size and shape) and mass spectrometry (provides information on composition and quantity). Depending on the detectors used (on-line or off-line), different particle properties can be characterized, such as: inorganic element concentration with ICP-MS, polymer identification with pyrolysis-GC-MS and shape information with the combined use of DLS and MALS.

Coupling of techniques is commonplace for analysis of natural colloids and inorganic nanomaterials in environmental media where so called "hyphenated" analysis techniques are the norm. Inductively coupled plasma-mass spectrometry (ICP-MS) coupled to flow field fractionation is one such combination that allows size discrimination and chemical characterization of inorganic environmental nanomaterials at exceptionally high sensitivity.

Similarly, pyrolysis-gas chromatography-mass spectrometry (Py-GC-MS), an established characterization method for organic material, coupled to flow field flow fractionation is gaining use to study nanoplastics [START_REF] Mintenig | Closing the gap between small and smaller: towards a framework to analyse nano-and microplastics in aqueous environmental samples[END_REF] . Moreover, there is increasing interest in taking advantage of the sensitivity of ICP-MS in the nanoplastics context. Recently, Mitrano et al. synthesized palladium-doped nanoplastics compatible with ICP-MS detection [START_REF] Mitrano | Synthesis of metal-doped nanoplastics and their utility to investigate fate and behaviour in complex environmental systems[END_REF] . However, ICP-MS has yet to be applied to environmental samples as an inorganic tracer for plastic. Nevertheless, potential exists for applying ICP-MS to analyzing nanoplastics in environmental samples, particularly those originating from plastic using metal-based dopants, or nanofillers or by using adsorbed metals 62 as a proxy. Moreover, ICP-MS, or other mass spectrometry techniques, could be coupled to techniques established for ENM analysis such as the electrospray ionizationdifferential mobility analyzer to obtain size information.

Outlook

Incidental nanoparticles produced from the fragmentation of plastic waste are an important element in the life cycle of plastic wastes. Lessons learned from 20 years of nanoEHS work should be applied to understanding the dimensions of the problem of plastic wastes. One such lesson is that international, interdisciplinary teams are able to tap specific areas of expertise that may be sparsely distributed across the world. Researchers studying nanoplastics in the environment can help facilitate this needed knowledge transfer by bringing in personnel, in the form of hiring and visiting researchers, with nanoEHS expertise within nanoplastics research groups as well as directly reaching out to nanoEHS research groups for collaborative opportunities.

The establishment of clear terminology and methods that can be harmonized across groups is critical in facilitating such collaborations, data sharing and data interpretation. Size is integral to the definition of nanoplastic; however, we cannot, as for ENMs, be limited by arbitrary size cut-offs (e.g., 100 nm or 1000 nm). It is more meaningful to rely on the particle characteristics to define a nanoplastic. The defining characteristics of incidental nanoplastics which distinguish them from microplastics are described in Figure 2. These characteristics guide the development and application of analytical methods, sampling procedures, modeling approaches, and data curation that are directly relevant to the study of nanoplastics in the environment some of which carries over from the nanoEHS field. The experience gained from nanoEHS work has underscored important differences in the environmental behavior of nano-scale materials compared to larger particles of identical composition, a distinction that will be important in investigating the prevalence, fate, and impacts of nanoplastics. However, the instability of incidental nanoplastics in the environment and their particle heterogeneity present additional challenges distinct from the ENMs typically studied in the nanoEHS field. Consequently, studying environmental nanoplastics must also integrate lessons learned in dealing with the ubiquity and diversity of environmental microplastics. Faster surface chemistry changes (e.g., chemical oxidation).

Risk of sample alteration and loss (attachment) during analysis IV. Adsorptionheteroaggregation

Can be tracked by fluorescence microscopy, using adsorbed fluorophores, ICP-MS using adsorbed metal, hyperspectral imaging. Enables the use of adsorption-based samplers.

Interference from background material, C in particular.

V. Bio-uptake and translocation

Relatively lower variability in body burden

Extraction/purification due to lower sizes and concentrations, difficulties differentiating between C-based particles and tissues.

VI. Rapid release of additives and cocontaminants

Simplifies characterization of additives and co-contaminants by ICP-MS, LC-MS, and Py-GC-MS.

Complicates ecotoxicity assessment.

VII. Heterogeneous particle properties

Enables detection and quantification by co-localization of target components by single particle analysis (e.g., a metal additive in a polymer matrix)

Global characterization difficultmultiple sampling points required.

Difficult to acquire environmentallyrelevant material for ecotoxicity and fate assessments.

VIII. Rapid fragmentation due to environmental stressors

Bulk plastics and microplastics can be fragmented to obtain representative nanoplastics Unstable samples (e.g., size distribution, disaggregation).

Increases potential for sample damage during extraction (e.g., by digestion)

Fig. 1 .

 1 Fig. 1. Number of publications obtained from Scopus that use the terms shown on the y-axis in the Title/Abstract/Keywords. (a) particle* AND polystyrene* AND NOT microplastic*(green circle) versus microplastic* & polystyrene* (blue diamond) and (b) nanoparticle* AND polystyrene* AND NOT nanoplastic* (green circle) vs nanoplastic* & polystyrene (bluediamond). The light blue vertical line highlights the coincidence between the decrease in the number of publications and the exponential increase in publications that occurs by changing the terms from "particle" and "nanoparticle" to "microplastic" and "nanoplastic".

Figure 2 .

 2 Figure 2. Transformations and characteristics of plastic debris in the environment.The eight defining characteristics of nanoplastics that distinguish them from microplastics and engineered nanomaterials. The time needed to leach one half of the additives initially present in a plastic material is calculated for the representative additive tributyl phosphate ("leaching of additives" curve, characteristic VI). Leaching of tributyl phosphate was calculated for different sized epoxy particles based on internal diffusion-limited transport[START_REF] Karlsson | Size-dependent toxicity of metal oxide particles-a comparison between nano-and micrometer size[END_REF] . Bulk plastics break down into micro-and nanoplastics due to weathering action (UV sunlight, mechanical abrasion, etc.), environmental (heteroaggregation) and biological (bacteria, krill, etc.) processes.

Figure 3 .

 3 Figure 3. Relative proportion of natural organic matter compared to plastic according to the size (radius) of plastic debris. The relative proportion is represented by the ratio of the volume of the natural organic matter to the volume of plastic for a given particle.

Table 1 . Analytical consequences of the nanoplastic defining characteristics.

 1

	Nanoplastic defining	Analytical Possibilities	Complications
	characteristics
	I. Brownian motion	Enables sizing by SLS, DLS, NTA,	More limited density separation
		and chromatography-based separation	(i.e., sedimentation, centrifugation)
		(AF4, SEC, HDC) and
		electrophoretic mobility measurement
		(CE, DMA).
	II. Below geometric	Can use SLS, DLS, AFM, electron	Prevents resolution by diffraction-
	optical resolution	microscopy, PALM, STORM, STED,	limited optical methods (conventional
		and hyperspectral imaging with dark	light microscopy) and characterization
		field microscopy.	by Infrared/Raman diffraction-limited
			laser spot.
	III. High specific	Enhances surface interactions.
	surface area	Facilitates dispersion fixation on a
		substrate (e.g. sample drying on a
		SEM or TEM grid).


Greqam
email: greqam@yahoo.fr

Mustapha Belkhouja
email: belkhouja_mustapha@yahoo.fr

Imène Mootamri

Mohamed Boutahar
email: boutahar@univmed.fr

Imene Mootamri
email: imene.mootamri@etumel.univmed.fr

Analysing CPI inflation by the fractionally integrated ARFIMA-STVGARCH model

Keywords: ARFIMA model, Generalised autoregressive conditional heteroscedasticity model, Inflation rate, Long memory process, Nonlinear time series, Time-varying parameter model JEL classification: C12, C22, C51, C52, E31 Autoregressive fractionally integrated moving average model, Generalised autoregressive conditional heteroskedasticity model, In ‡ation rate, Long memory process, Nonlinear time series, Time-varying parameter model. JEL classi…cation: C12, C22, C51, C52, E31

Analysing CPI inflation by the fractionally integrated ARFIMA-STVGARCH model

Introduction

There have been a number of studies trying to assess empirically the persistence of in ‡ation, but they appear to reach quite diverging conclusions. In those studies, a popular topic is concerned with the order of integration as the measure of in ‡ation persistence, using unit root tests to classify the in- ‡ation process as either an I(0) or I(1) process. [START_REF] Nelson | Short-term interest rates as predictors of in ‡ation: on testing the hypothesis that the real rate of interest is constant[END_REF], [START_REF] Ball | In ‡ation and uncertainty at short and long horizons[END_REF], [START_REF] Kim | Unobserved-component time series models with Markov-switching heteroskedasticity: changes in regime and the link between in ‡ation and in ‡ation uncertainty[END_REF], Banerjee et al. (2001) and Banerjee and Russell (2001) …nd the evidence of a unit root in the in ‡ation rate. [START_REF] Rose | Is the real interest rate stable?[END_REF] and [START_REF] Grier | On in ‡ation and in ‡ation uncertainty in the G7 countries[END_REF] claim that the in ‡ation rate is an I(0) process, while most others, like [START_REF] Kirchgässner | Are real interest rates stable? an international comparison[END_REF] obtain mixed results for four European countries. [START_REF] Hassler | Long memory in in ‡ation rates: International evidence[END_REF] …nd that in ‡ation is better characterized as a fractional white noise, while Baillie et al. (1996) suggest that the ARFIMA-GARCH models capture well the dynamic properties of in ‡ation in the major industrial countries. All these studies suggest that the traditional ARM A and ARIM A speci-…cations are not able to capture the persistence of in ‡ation rate. So, instead of dealing with I(0) or I(1) processes, we use the I(d) process that seem a better modelling of in ‡ation. Although all the previous works have provided the consistent evidence across time periods and countries that in ‡ation rate exhibits long memory property in its …rst conditional moment, the property of time-dependent heteroskedasticity in the second conditional moment of the in ‡ation rate has not been explored until Baillie et al. (1996); these authors use the ARFIMA-GARCH model to describe the in ‡ation dynamics for ten countries and …nd strong evidence of long memory with mean reverting behaviour in all countries except Japan.

Incorporating the non constancy of the GARCH parameters to model the volatility of a process exhibiting a long memory and slow decay in its level has not be attempted before. So our main contribution is to describe the in ‡ation rate of eight European countries, Canada and Japan by an ARF IM A process with smooth time varying GARCH parameters over time. Several interesting …ndings emerge from our empirical work. Our …rst …nding is that all the in ‡ation rates have the long memory properties in their …rst conditional moments. This empirical evidence is consistent with the evidence provided by Baillie et al. (1996) for ten industrial countries except Japan. The second result that comes out from this study is that only six countries are illustrated by an ARFIMA-STVGARCH process.

The rest of the paper is organized as follows. In Section 2, we present the class of ARFIMA-STVGARCH model and we discuss its properties. Section 3 considers parameter constancy test by using a Lagrange Multiplier (LM) type-test and a Likelihood Ratio test (LR). In Section 4, we propose a speci…cation strategy and estimation of the model. Section 5 and 6 contains respectively the results of the performance using Monte Carlo experiments and the empirical results for all countries. Section 7 concludes.

A general class of ARFIMA with smooth time varying GARCH model

In this section we present the ARFIMA-GARCH model, which generates the long memory property in the …rst conditional moment de…ned as:

(L) (1 L) d (y t ) = (L) " t (1)
where is the mean of the process, d is the long memory parameter, (L) = 1 1 L :::: p L p and (L) = 1 + 1 L + :::: + q L q have all their roots outside the unit circle: f" t g is an innovation sequence with the conditional mean E (" t j t 1 ) = and a potentially time-varying conditional variance E (" 2 t j t 1 ) = h t , where t 1 is the information set up to time t 1: That is " t j t 1 N (0; h t ), i.e.:

" t = z t p h t (2)
where fz t g is a sequence of independent standard normal variables with variance 1. The fractional di¤erencing operator (1 L) d is de…ned by the binomial expansion,

(

L) d = 1 dL d(1 d) 2! L 2 d(1 d)(2 d) 3! L 3 + ::: 1
where L is the lag operator. Thus, the process will be covariance stationary and invertible for 0:5 < d < 0:5: When 0 < d < 0:5; the process is a stationary long memory process in the sense that autocorrelations are not absolutely summable and decay hyperbolically to zero. If 0:5 d < 1; the process does not have a …nite variance and is a non-stationary process, but it is still mean reverting. Moreover, when d = 1; the process has a unit root and a permanent e¤ect.

To capture the volatility property of the conditional variance process, [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF] introduced the GARCH(m; r) (General Autoregressive Conditional Heteroskedasticity) model, which de…nes the conditional variance equation as follows:

h t = 00 + r X i=1 0i " 2 t i + m X j=1 0j h t j (3)
The parameters in (3) satisfy the restrictions 00 > 0; 0i > 0; 0j > 0; i = 1; ::; r and j = 1; ::; m: These conditions are su¢ cient to ensure positivity of h t : In order to characterize smooth changes in the conditional variance 1 , we assume that the parameters in (3) vary smoothly over time as follows:

h t = 00 + r X i=1 0i " 2 t i + m X j=1 0j h t j (4)
+ 10 + r X i=1 1i " 2 t i + m X j=1 1j h t j ! F (s t ; ; c)
The constraints applied to the parameters to guarantee stationarity and positivity of the conditional variance in (4) are:

8 > > < > > : 00 > 0; 00 + 10 > 0 0i > 0; 0i + 1i > 0; i = 1; ::; r 0j > 0; 0j + 1j > 0; j = 1; ::; m P r i=1 ( 0i + 1i ) + P m j=1 0j + 1j < 1
F (s t ; ; c) is the transition function which is continuous and non-negative bounded between zero and one, governing the movement from one regime to another. This function is such that the parameters of the GARCH model change smoothly over time from 0i ; 0j to 0i + 1i ; 0j + 1j ; i = 0; 1; :::; r and j = 1; :::; m: A suitable choice for F (s t ; ; c) is the general logistic smooth transition function de…ned as follows:

F (s t ; ; c) = 1 + exp ( K Y k=1 (s t c k ) )! 1 ; > 0; c 1 ::: c K (5)
where s t = t=T is the transition variable, T is the number of observations and is the slope parameter which controls the degree of smoothness of the transition function. When ! 1 in [START_REF] Baillie | Further long memory properties of in ‡ationary shocks[END_REF]; the switch to one parameterization to another is abrupt, that is, the process contains structural breaks at the threshold parameters, c k ; k = 1; :::; K: The order K determines the shape of the transition function. In practice2 , the typical choices of the order are K = 1 and K = 2: For K = 1; the ST V GARCH model is suitable to describe a process whose dynamics of the conditional variance vary smoothly over time. For our purpose, we choose K = 1 that is illustrated in Figure 1 where the location parameters c is respectively equal to 0:2; 0:5 and 0:7 and = f5; 10; 50; 100g ; where the lowest value of corresponds to the smoothest function.

Testing parameter constancy

The test of parameter constancy is an important tool for checking the adequacy of ARFIMA-GARCH model. If one rejects the parameter constancy against ARFIMA-STVGARCH model, one may conclude that the structure of the dynamics of volatility is changing over time. This idea has previously been considered by [START_REF] Lundbergh | Evaluating GARCH models[END_REF] and Teräsvirta et al. (2008). In order to derive the test statistic, let us rewrite the model as:

8 > < > : (L) (1 L) d (y t ) = (L) " t ; " t = z t p h t h t = 00 + P r i=1 0i " 2 t i + P m j=1 0j h t j + 10 + P r i=1 1i " 2 t i + P m j=1 1j h t j F (s t ; ; c) (6)
The …rst step of the test consists in estimating the ARF IM A(p; d; q), to obtain the set of residuals b " t and then testing the null hypothesis of parameter constancy corresponding to H 0 : = 0 against H 1 : > 0: Under the null hypothesis, c, 1i and 1j for i = 0; 1; :::; r and j = 1; :::; m are not identi…ed. This identi…cation problem has been resolved by [START_REF] Luukkonen | Testing linearity against smooth transition autoregressive models[END_REF] by replacing the transition function3 by its …rst order Taylor approximation around = 0. Thus, the …rst order Taylor expansion of the logistic transition function around = 0 is given by:

T (s t ; ; c) = 1 4 (s t c) + R(s t ; ; c)
where R(s t ; ; c) is a remainder term. Substituting T (s t ; ; c) for F (s t ; ; c) in ( 6) and after rearranging terms we have:

h t = 00 + r X i=1 0i " 2 t i + m X j=1 0j h t j (7)
+ 10 s t + r X i=1 1i s t " 2 t i + m X j=1 1j s t h t j ! + R
where 0i = 0i 4 1i c; 0j = 0j 4 1j c; 1i = 4 1i ; 1j = 4 1j for i = 0; 1; :::; r and j = 1; :::; m. Therefore, the null hypothesis for parameter constancy becomes: H 0 : 1i = 1j = 0: This hypothesis can be tested by a standard LM test or LR test and one can also test the constancy of a subset of parameters. Let

1 = 0 0i ; 0 0j 0 ; 2 = 0 1i ; 0 1j 0 and = ( 0 1 ; 0 2 ) 0 :
Under H 0 ; the remainder R = 0; so it does not a¤ect the asymptotic null distribution of the test statistic. Using the reparameterization in [START_REF] Banerjee | An I(2) analysis of in ‡ation and the markup[END_REF]; the null hypothesis of parameter constancy becomes H 0 : 2 = 0: The quasi log-likelihood of the model for observation t is given by:

l t = 1 2 ln 2 1 2 ln h t 1 2 " 2 t h t (8)

Lagrange Multiplier test

The partial derivatives evaluated under H 0 are given by:

@l t @ H 0 = 1 2 b " 2 t b h 0t 1 @ ln h t @ H 0
Under the null hypothesis, the "hats" indicate the maximum likelihood estimators and b h 0t denotes the conditional variance estimated at time t: Under H 0 , the LM type statistic is asymptotically distributed as 2 with dim ( 2 ) degrees of freedom:

LM = 1 2 ( T X t=1 @l t @ H 0 ) 0 ( T X t=1 " @ ln h t @ H 0 # " @ ln h t @ H 0 # 0 ) 1 (9)
( T X t=1 @l t @ H 0 )
In practice this test may be carried out in a straightforward way using an auxiliary least squares regression. Let

c W t = 1; " 2 t 1 ; :::; " 2 t r ; h t 1 ; :::; h t m b X 0 1t = @ ln ht @ 1 H 0 = b h 0t 1 c W 0 t + m P j=1 b 0j @ b h t j @ 1 ! b X 0 2t = @ ln ht @ 2 H 0 = b h 0t 1 s t c W t 0 + m P j=1 b 0j @ b h t j @ 1 !
Hence, the statistic is computed in a few steps as follows:

Estimate Compute the 2 test statistic as:

LM = T (SSR 0 SSR 1 ) SSR 0

Likelihood Ratio test

Let M L u (resp. M L r ) be the maximum log-likelihood value of the unrestricted (resp. restricted) ARFIMA-STVGARCH. The number of parameters is reduced in the restricted model based on the assumption H 0 : 2 = 0:

Under H 0 the LR type statistic is asymptotically distributed as 2 with dim ( 2 ) degrees of freedom, so the likelihood ratio statistic is given by:

LR = 2 [M L u M L r ]
4 Speci…cation and estimation of the model

In order to describe the speci…cation procedure for ARFIMA-STVGARCH models, let us consider the conditional variance h t de…ned in ( 4) such that all parameters are changing smoothly over time. Our strategy contains the following stages:

The …rst step is to plot the autocorrelation (ACF) and partial autocorrelation functions (PACF) of the series in order to detect the presence of seasonal phenomenon and long memory behaviour which is characterized by an hyperbolic decay of the autocorrelation function.

The information criteria (AIC, BIC,...) are not optimal to determine the order of the model in (1), hence we …t an ARF IM A model by using the ACF and PACF and we obtain the set of residuals b " t :

We use the ACF and PACF functions of the squared estimated residuals b " 2 t to check the presence of the conditional heteroscedasticity and to identify the orders of the model in (4). The squared standardized errors of the selected ARFIMA-GARCH model should be free of serial correlation. Neglected autocorrelation may bias test of parameter constancy.

The next stage consists on testing the selected ARFIMA-GARCH model against the alternative of ARFIMA-STVGARCH where just one parameter varying each time 4 . If H 0 is rejected, we keep the varying parameter which provides the highest rejection frequency compared to the other tests parameter constancy. Then, we test ARFIMA-STVGARCH model with the changing parameter selected above against the alternative of ARFIMA-STVGARCH model where two varying parameters. Once again, if the null hypothesis is rejected, we maintain the two parameters corresponding to the lowest p-value. We proceed sequentially by testing the ARFIMA-STVGARCH with s 1 changing parameters against the ARFIMA-STVGARCH5 with s changing parameters until the …rst non rejection of the null hypothesis.

Finally, we use the quasi maximum likelihood method to estimate the global ARFIMA-STVGARCH and we evaluate the …nal model by some diagnostic tests. Suppose that y t is generated by the ARFIMA-STVGARCH given in [START_REF] Ball | In ‡ation and uncertainty at short and long horizons[END_REF]. Let = ( ; d; 0 ; 0 ; 0 i ; ' 0 i ; ; c) 0 where i = ( i0 ; ::; ir ) 0 ; ' i = ( i1 ; ::; im ) 0 for i = 0; 1; = 1 ; ::; p 0 and = 1 ; ::; q 0 : The quasi maximum likelihood estimator of the parameter vector is obtained by:

b = arg max 2 1 T T X t=1 l t ( ) (10)
where l t ( ) is the quasi log-likelihood of the model for observation t :

l t ( ) = 1 2 ln 2 1 2 ln h t 1 2 " 2 t h t (11)
5 Monte Carlo experiment

Simulation design

In this section we study the size and power properties of the LM-type and LR tests using Monte Carlo simulation methods. In all simulations, we use sample lengths of 500, 1000, 2000 observations and, for each design, a total of 100 replications. To avoid the initialization e¤ects, the …rst 1000 observations are discarded. The behaviour of the tests is examined for several data generating processes (DGP's) that can be nested in the following FI-STVGARCH (1; 1) speci…cation:

:

(1 L) d y t = " t ; " t = z t p h t ; " t N (0; h t ) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 10 + 11 " 2 t 1 + 11 h t 1 F (s t ; ; c) (12)
In all the experiments, the change of regime in volatility occurs in the middle of the sample, i.e: c = 0:5; while the smoothness parameter is equal to 10. The transition variable is the standardized time variable s t = t=T; for t = 1:::T and T is the number of observations. Figure 2 shows the plots of the series, the autocorrelation function and the conditional standard deviations with T = 1000; 00 = 0:30; 01 = 0:10; 01 = 0:10; 10 = 0:20; 11 = 0:15; 11 = 0:20; d = 0:40; we use the logistic transition function F (s t ; 10; 0:5). The autocorrelation function exhibits an hyperbolic slow decay that indicates the presence of long memory. Moreover, we notice an increase of the volatility shown by the plots of the series and the conditional standard deviations. This increase is explained by the variation of GARCH parameters.

Testing parameter constancy

The behaviour of the test statistics is examined for several data generating processes (DGP's) that can be nested in the following FI-STVGARCH(1,1) speci…cation:

(1 L) d y t = " t ; " t = z t p h t ; " t N (0; h t ) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 10 + 11 " 2 t 1 + 11 h t 1 F (s t ; ; c) (13)
The data generating processes are as following: The parameter of fractional integration varies among d = f0:20; 0:45g ; c = 0:5 and = 10:

DGP (I) h t =

Simulation results

From Tables 1-8 we can see that both tests do not have size distortion (the estimated sizes are close to nominal sizes), whatever the speci…cation. Moreover, the simulation results reveal that the LM-type and the LR-type tests provide good results even for small data sets; however, for almost cases, large samples display better size and power than small samples. Finally, results with d = 0:20 are close to those obtained with d = 0:45, so increasing the parameter of fractional integration d has no impact on the size and the power of the tests.

6 Empirical Analysis

The data

We use monthly data of the Consumer Price Index (CPI) in ‡ation rate obtained from the IMF (International Financial Statistics) 6 . The data range from 1970:01 to 2004:12 and cover eight European countries: Belgium, Denmark, Finland, France, Germany, Italy, Portugal, Spain with Canada and Japan. The data are not seasonally adjusted, and generally exhibit the clear pattern of slow decay and persistence. The in ‡ation rate is measured by the monthly di¤erence of the log CPI, i.e. y t = 100: log (CP I t =CP I t 1 ) ; where CP I t denotes the Consumer Price Index at month t: Allowing for di¤erencing leaves 419 usable observations. The summary statistics (Table 9) for the ten in ‡ation rates show that the distributions of the series are skewed to the right with heavy tails. The Jarque-Bera test is in line with this evidence since it strongly rejects normality for the distribution of the in ‡ation rates. The Ljung-Box test applied to the series and squared series, provides clear evidence against the hypothesis of serial independence of observations and indicates the existence of ARCH e¤ect.

The unit root tests of Augmented Dickey-Fuller (1979) (henceforth ADF), Phillips and Perron (1988) (henceforth PP) and [START_REF] Kwiatkowski | Testing the null hypothesis of stationarity against the alternative of a unit root[END_REF] 7(henceforth KPSS) for the levels and the …rst di¤erences of the ten in ‡ation series are presented in Table 10. For all the in ‡ation series based on the KPSS test the null hypothesis of stationarity is rejected. However, the ADF test rejects the unit root hypothesis for Canada, Belgium, Germany, Japan and Portugal but it does not reject it for the other countries at 1% and 5% signi…cance levels. Finally, the PP test rejects the unit root hypothesis for all the countries except Denmark. In conclusion, the application of these tests yields contradictory results 8 . These unit root tests are merely suggestive as mentioned in Lee and Amsler (1997) who show that the KPSS statistic cannot distinguish consistently between nonstationary long memory and unit root. Hence for all the countries, there is evidence that in ‡ation may not be generated by an I(0) or I(1) process and is at least indicative of fractional integration.

Estimation results of the ARFIMA-STVGARCH model

Baillie et al. (1996) suggests that the ARFIMA-GARCH speci…cation describes the in ‡ation series of several industrial countries quite well. Within this framework, we use ST V GARCH structure instead of the GARCH to capture the persistence in the volatility of the in ‡ation rate. Estimates of the ARFIMA-STVGARCH model for the ten countries are given in Table 12; they are obtained by quasi-maximum likelihood estimation method (QMLE); following Baillie et al. (1996a). We notice that this type of model is general; so it may include ARFIMA-GARCH or simple ARF IM A model. To check robustness of our estimates, we use a range of starting values to ensure that the estimation procedure converges to a global maximum. To identify the order of the ARF IM A part of the model, we follow the procedure explained in section 4. The seasonal moving-average parameters, particularly 12 and 24 are necessary to take into account the signi…cant seasonality, which is evident for all the countries.

Table 11 contains the LM statistics and the p values corresponding to the test of an ARFIMA-GARCH (1; 1) model against the alternative hypothesis of an ARFIMA-STVGARCH (1; 1) model. The test is performed for several combinations of the GARCH coe¢ cients as explained in the simulation part. The sequential testing procedure is carried out ( see section 4) until the …rst non-rejection of the null hypothesis ( at each step one parameter is added). In many cases, the null hypothesis is strongly rejected (see the p-values that are very close to zero). We do not report the LR-test results, because they are similar to the LM-test results. Among the ten countries, we …nd that six countries satisfy the assumption of time-varying parameters (not necessary all) of the GARCH part, namely Portugal, Spain, Denmark, Finland, Italy and Japan. The monthly in ‡ation rate, the autocorrelation function, the conditional standard deviation and the logistic transition function of those countries, obtained from the model [START_REF] Conrad | On the in ‡ation-uncertainty hypothesis in the USA, Japan and the UK: a dual long memory approach[END_REF], are shown in Figures 3 to 8.

In each Figure, we observe a slow decay in the autocorrelation function which indicates the presence of long memory behaviour. As the case of the conditional standard deviations, we note a decrease of volatility and a smooth change of regimes over time corresponding to the logistic transition function.

Denmark is the only country whose all GARCH parameters vary smoothly over time.

Finally, after studying the selection frequencies of various models using the speci…cation procedure discussed in Section 4, we …nd that the estimated models are nested in the following speci…cation: ARFIMA(1,d,2)(0,0,2) 12 -STVGARCH (1; 1), i.e.: 8 > > > > < > > > > :

(1 L) d (y t ) = x t (1 1 L) x t = (1 1 L 2 L 2 ) (1 12 L 12 24 L 24 ) " t " t = z t p h t ; " t N (0; h t ) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 10 + 11 " 2 t 1 + 11 h t 1 F (s t ; ; c)
The estimates of the fractional di¤erencing parameter are clearly signi…cant and 0 < b d < 0:5 for all countries implying covariance-stationarity of the in ‡ation processes, except Canada where the estimated value of d is approximately 0.6 which implies that the in ‡ation series for this country has an in…nite variance but is still mean reverting. For the countries who reject the hypothesis of parameter constancy (see Table 11), the estimated slope parameter is in the range of [5; 20], so the change from one regime to another in volatility is smooth. The estimated threshold parameter b c is less than 0:5.

To check the goodness of …t of our model, we consider several diagnostic tests on the standardized residuals, the Ljung-Box test for the 12-th order serial autocorrelation and heteroskedasticity, when it is applied on the squared standardized residuals. We use the Jarque-Bera test, the skewness and the kurtosis coe¢ cients to test the normality of standardized residuals. From Table 13, we can see that, for all countries, the hypothesis of uncorrelated standardized and squared standardized residuals is well supported, indicating that there is no statistically signi…cant evidence of misspeci…cation. The Skewness and Kurtosis coe¢ cients indicate that the standardized residuals are still not normally distributed, which is con…rmed by the Jarque-Bera test.

Conclusion

The results in this paper highlight the possibility of structural change in the in ‡ation's volatility, we have thus an ARFIMA-STVGARCH modelling instead of an ARFIMA-GARCH modelling. Moreover, we have investigated the empirical power and size of LM and LR tests that allow choosing between FI-GARCH(1,1) and FI-STVGARCH(1,1) models. The simulations show that both tests generally behave well, even for moderate sample sizes. As illustrations, the suggested modelling was applied to monthly in ‡ation rate data of eight European countries, namely, France, Germany, Belgium, Denmark, Finland, Italy, Spain, Portugal, Canada and Japan. Apart from Canada, which appears non-stationary but mean reverting, the other lowin ‡ation countries have an estimated order of integration between 0.17 and 0.45. The fractional di¤erencing parameter estimates and their t-statistics indicate that the models are signi…cantly di¤erent from I(1) and I(0) behaviour. An interesting interpretation of these models is that an in ‡ationary shock will have long memory and persistence, but, eventually, will be mean reverting. Furthermore, six countries (Denmark, Finland, Italy, Spain, Portugal and Japan) show a clear evidence of parameter non constancy in the GARCH part of the model, so the dual persistence in the conditional mean of in ‡ation and in its conditional variance is well described by ARFIMA-STVGARCH processes.

A Appendix: Simulation Results

A.1 Lagrange Multiplier test

T ests i ii iii iv v vi vii T
1% 0:01 0:00 0:01 0:00 0:00 0:00 0:00 500 5% 0:03 0:04 0:03 0:03 0:05 0:02 0:02 10% 0:12 0:08 0:12 0:06 0:10 0:08 0:08 1% 0:00 0:00 0:00 0:00 0:00 0:00 0:00 1000 5% 0:02 0:02 0:01 0:00 0:03 0:01 0:00 10% 0:08 0:06 0:09 0:02 0:05 0:03 0:05 1% 0:00 0:00 0:00 0:00 0:00 0:00 0:00 2000 5% 0:00 0:00 0:00 0:00 0:00 0:00 0:00 10% 0:02 0:03 0:04 0:00 0:01 0:00 0:02

Note: The Tables 1 and2 correspond to d = 0:20 and d = 0:45; they report the rejection frequencies for the tests (i)-(vii) at the three theoretical signi…cance levels f1%; 5%; 10%g and for the three sample sizes f500; 1000; 2000g. The data generating process is given by DGP (I).

Table 3: Simulated power f or the test of FI-GARCH (1; 1) against the alternative FI-STVGARCH (1; 1) with d = 0:20

DGP (II) (III) (IV ) (V ) (V I) (V II) (V III) T ests i ii iii iv v vi vii T
1% 0:81 0:24 0:84 1:00 1:00 0:66 0:99 500 5% 0:93 0:58 0:96 1:00 1:00 0:87 1:00 10% 0:98 0:77 0:99 1:00 1:00 0:93 1:00 1% 0:98 0:66 0:94 1:00 1:00 0:96 1:00 1000 5%

1:00 0:92 1:00 1:00 1:00 1:00 1:00 10% 1:00 0:95 1:00 1:00 1:00 1:00 1:00 1% 1:00 0:95 1:00 1:00 1:00 0:96 1:00 2000 5%

1:00 0:99 1:00 1:00 1:00 0:99 1:00 10% 1:00 1:00 1:00 1:00 1:00 0:99 1:00

DGP (II) (III) (IV ) (V ) (V I) (V II) (V III) T ests i ii iii iv v vi vii T
1% 0:82 0:30 0:84 0:99 1:00 0:74 1:00 500 5% 0:94 0:72 0:98 1:00 1:00 0:95 1:00 10% 0:97 0:98 0:98 1:00 1:00 0:98 1:00 1% 0:99 0:62 0:96 1:00 1:00 0:94 1:00 1000 5%

1:00 0:90 1:00 1:00 1:00 0:99 1:00 10% 1:00 0:95 1:00 1:00 1:00 1:00 1:00 1% 1:00 0:88 1:00 1:00 1:00 0:98 1:00 2000 5%

1:00 0:97 1:00 1:00 1:00 0:99 1:00 10% 1:00 0:98 1:00 1:00 1:00 0:99 1:00

Note: The Tables 3 and4 correspond to d = 0:20 and d = 0:45; they report the rejection frequencies for the tests (i)-(vii)at the three theoretical signi…cance levels f1%; 5%; 10%g and for the three sample sizes f500; 1000; 2000g. The data generating processes are given by DGP's fII; III; IV; V; V I; V II; V IIIg.

A.2 Likelihood Ratio test

Table 5: Simulated size f or the test of FI-GARCH (1; 1) against the alternative FI-STVGARCH (1; 1) with d = 0:20

T ests i ii iii iv v vi vii T
1% 0:02 0:02 0:02 0:01 0:02 0:00 0:01 500 5% 0:07 0:11 0:10 0:04 0:06 0:05 0:06 10% 0:15 0:19 0:14 0:13 0:12 0:12 0:13 1% 0:00 0:00 0:00 0:00 0:00 0:00 0:00 1000 5% 0:05 0:08 0:07 0:01 0:03 0:02 0:04 10% 0:11 0:15 0:10 0:09 0:08 0:07 0:09 1% 0:00 0:00 0:00 0:00 0:00 0:00 0:00 2000 5% 0:02 0:05 0:04 0:00 0:00 0:00 0:01 10% 0:07 0:10 0:07 0:05 0:04 0:03 0:05

T ests i ii iii iv v vi vii T
1% 0:02 0:01 0:03 0:02 0:02 0:02 0:02 500 5% 0:10 0:08 0:08 0:06 0:08 0:07 0:06 10% 0:12 0:18 0:12 0:15 0:14 0:16 0:14 1% 0:00 0:00 0:00 0:00 0:00 0:00 0:00 1000 5% 0:06 0:05 0:04 0:03 0:04 0:02 0:03 10% 0:07 0:13 0:07 0:11 0:09 0:11 0:10 1% 0:00 0:00 0:00 0:00 0:00 0:00 0:00 2000 5% 0:02 0:01 0:01 0:00 0:01 0:00 0:00 10% 0:04 0:06 0:02 0:06 0:05 0:06 0:06 Note: The Tables 5 and6 correspond to d = 0:20 and d = 0:45; they report the rejection frequencies for the tests (i)-(vii) at the three theoretical signi…cance levels f1%; 5%; 10%g and for the three sample sizes f500; 1000; 2000g. The data generating processes is given by DGP (I).

Table 7: Simulated power f or the test of FI-GARCH (1; 1) against the alternative FI-STVGARCH (1; 1) with d = 0:20

DGP (II) (III) (IV ) (V ) (V I) (V II) (V III) T ests i ii iii iv v vi vii T
1% 0:95 0:97 0:99 1:00 1:00 0:66 0:99 500 5% 0:98 1:00 0:99 0:99 0:90 0:90 1:00 10% 0:98 1:00 0:99 0:99 0:90 0:93 1:00 1% 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1000 5%

1:00 1:00 1:00 1:00 1:00 1:00 1:00 10% 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1% 1:00 1:00 1:00 1:00 1:00 1:00 1:00 2000 5%

1:00 1:00 1:00 1:00 1:00 1:00 1:00 10% 1:00 1:00 1:00 1:00 1:00 1:00 1:00

DGP (II) (III) (IV ) (V ) (V I) (V II) (V III) T ests i ii iii iv v vi vii T
1% 0:93 0:96 0:96 0:98 0:90 0:88 0:99 500 5% 0:96 1:00 0:97 0:98 0:90 0:93 1:00 10% 0:96 1:00 0:99 0:99 0:90 0:95 1:00 1% 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1000 5%

1:00 1:00 1:00 1:00 1:00 1:00 1:00 10% 1:00 1:00 1:00 1:00 1:00 1:00 1:00 1% 1:00 1:00 1:00 1:00 1:00 1:00 1:00 2000 5%

1:00 1:00 1:00 1:00 1:00 1:00 1:00 10% 1:00 1:00 1:00 1:00 1:00 1:00 1:00

Note: The Tables 7 and8 correspond to d = 0:20 and d = 0:45; they report the rejection frequencies for the tests (i)-(vii) at the three theoretical signi…cance levels f1%; 5%; 10%g and for the three sample sizes f500; 1000; 2000g. The data generating processes are given by DGP's fII; III; IV; V; V I; V II; V IIIg.

Figure 1 .

 1 Figure 1. Plots of the logistic transition function

2 t b h 0t 1 ; t = 1 ;

 211 the parameters of the ARFIMA model and obtain the set of residuals b " t : From b " t ; estimate h 0t as a GARCH model, compute the residuals b u t = b " :::; T; and the sum of squared residuals under the null hypothesis, SSR 0 = Regress b u t on b X 0 1t and b X 0 2t , t = 1; :::; T; and compute the sum of squared residuals, SSR 1 :

Figure 2 .

 2 Figure 2. Plots of the series, the autocorrelation function and the conditional standard deviations.

Table 1 :Table 2 :

 12 Simulated size f or the test of FI-GARCH (1; 1) against the alternative FI-STVGARCH (1; 1) with d = 0:20 0:07 0:09 0:09 0:07 0:08 0:06 10% 0:18 0:15 0:19 0:16 0:16 0:12 0:15 1% 0:02 0:01 0:02 0:02 0:02 0:02 0:01 1000 5% 0:06 0:02 0:05 0:05 0:06 0:05 0:07 10% 0:11 0:08 0:11 0:08 0:09 0:07 0Simulated size f or the test of FI-GARCH (1; 1) against the alternative FI-STVGARCH (1; 1) with d = 0:45

Figure 3 .

 3 Figure 3. Plots of the in ‡ation rates, the autocorrelation function, the conditional standard deviation and the logistic transition function of Portugal.

Figure 4 .Figure 5 .

 45 Figure 4. Plots of the in ‡ation rates, the autocorrelation function, the conditional standard deviation and the logistic transition function of Spain.

Figure 6 .Figure 7 .

 67 Figure 6. Plots of the in ‡ation rates, the autocorrelation function, the conditional standard deviation and the logistic transition function of Finland.

Figure 8 .

 8 Figure 8. Plots of the in ‡ation rates, the autocorrelation function, the conditional standard deviation and the logistic transition function of Japan.

  00 + 01 " 2 t 1 + 01 h t 1 00 = 0:10; 01 = 0:10; 01 = 0:80:DGP (II)h t = 00 + 01 " 2 t 1 + 01 h t 1 + 10 F (s t ; ; c) 00 = 0:10; 01 = 0:10; 01 = 0:80; 10 = 0:80:DGP (III) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 11 "2 t 1 F (s t ; ; c) 00 = 0:10; 01 = 0:10; 01 = 0:10; 11 = 0:70: DGP (IV ) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 11 h t 1 F (s t ; ; c) 00 = 0:10; 01 = 0:10; 01 = 0:10; 11 = 0:70:DGP (V ) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 10 + 11 " 2 t 1 F (s t ;; c) 00 = 0:10; 01 = 0:10; 01 = 0:10; 10 = 0:30; 11 = 0:70: = 0:10; 01 = 0:10; 01 = 0:10; 10 = 0:30; 11 = 0:70:DGP (V II) h t = 00 + 01 " 2 t 1 + 01 h t1 + 11 " 2 t 1 + 11 h t 1 F (s t ; ; c) 00 = 0:10; 01 = 0:10; 01 = 0:10; 11 = 0:30; 11 = 0:45: DGP (V III) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 10 + 11 " 2 t 1 + 11 h t 1 F (s t ; ; c) 00 = 0:10; 01 = 0:10; 01 = 0:10; 10 = 0:30; 11 = 0:30; 11 = 0:45: The tests of F I GARCH(1; 1) against F I ST V GARCH(1; 1) are as following: (i) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 10 F (s t ; ; c) H 0 : 10 = 0 against H 1 : 10 6 = 0 (ii) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 11 " 2 t 1 F (s t ; ; c) H 0 : 11 = 0 against H 1 : 11 6 = 0 (iii) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 11 h t 1 F (s t ; ; c) H 0 : 11 = 0 against H 1 : 11 6 = 0 (iv) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 10 + 11 " 2 t 1 F (s t ; ; c) H 0 : 10 = 11 = 0 against H 1 : 10 6 = 0 and 11 6 = 0 (v) h t = 00 + 01 " 2 t 1 + 01 h t 1 + ( 10 + 11 h t 1 ) F (s t ; ; c) H 0 : 10= 11 = 0 against H 1 : 10 6 = 0 and 11 6 = 0 : 10 = 11 = 11 = 0 against H 1 : 10 6 = 0 and 11 6 = 0 and 11 6 = 0

	DGP (V I) h (vi) h

t = 00 + 01 " 2 t 1 + 01 h t 1 + ( 10 + 11 h t 1 ) F (s t ; ; c) 00 t = 00 + 01 " 2 t 1 + 01 h t 1 + 11 " 2 t 1 + 11 h t 1 F (s t ; ; c) H 0 : 11 = 11 = 0 against H 1 : 11 6 = 0 and 11 6 = 0 (vii) h t = 00 + 01 " 2 t 1 + 01 h t 1 + 10 + 11 " 2 t 1 + 11 h t 1 F (s t ; ; c) H 0

Table 4 :

 4 Simulated power f or the test of FI-GARCH (1; 1) against the alternative FI-STVGARCH (1; 1) with d = 0:45

Table 6 :

 6 Simulated

size f or the test of FI-GARCH (1; 1) against the alternative FI-STVGARCH (1; 1) with d = 0:45

Table 8 :

 8 Simulated

power f or the test of FI-GARCH (1; 1) against the alternative FI-STVGARCH (1; 1) with d = 0:45

See Hagerud (1996[START_REF] Hagerud | A new non-Linear GARCH model[END_REF] and González-Rivera (1998) for more details.

Teräsvirta et al. (2008).

For the purpose of deriving the test, we replace F (s t ; ; c) by F (s t ; ; c) 1=2,(Teräsvirta et al. (2008)).

The changing of parameters occurs only in the GARCH part of the model.

s is the number of changing parameters.

From CD of March 2005.

Contrary to ADF test, the KPSS test considers the stationnarity under the null hypothesis, and the alternative hypothesis is the presence of unit root.

[START_REF] Banerjee | The relationship between the markup and in ‡ation in the G7 economies and Australia[END_REF] See alsoBaillie et al. (1996) andConrad (2005b).

Notes: The …rst column gives the tests speci…cation used in simulations. The numbers in brackets are the p-values.

Acknowledgement We would like to thank Anne Peguin-Feissolle for her valuable comments and remarks.

B Appendix: Empirical Results