Astrid Bacle

Charlotte Pronier

Hélène Gilardi

Elisabeth Polard

Sophie Potin

Lucie-Marie Scailteux
email: luciemarie.scailteux@chu-rennes.fr

Hepatotoxicity risk factors and acetaminophen dose adjustment, do prescribers give this issue adequate consideration? a French university hospital study

Keywords: hepatotoxicity, acetaminophen, dose adjustment, risk factor

Background: The hepatotoxicity of acetaminophen is recognized worldwide. Unfavourable prognoses relating to overdose include liver transplantation and/or death. Several hepatotoxicity risk factors (HRFs) should motivate the adjustment of acetaminophen daily intake (to < 4 g/day): advanced age, weight < 50 kg, malnutrition, chronic alcoholism, chronic hepatitis B and C and HIV infection, severe chronic renal failure, and hepatocellular insufficiency.

Method: Over a seven-day period in Rennes University Hospital in December 2017, using DxCare® software, with an odds ratio estimation, we analysed all acetaminophen prescriptions, to assess to what extent the presence of HRFs altered the prescribers' choice of acetaminophen dose (< 4 g/day versus 4 g/day).

Results: Among 1842 patients, considering only the first acetaminophen prescription, 73.7 % were on 4 g/day. Almost half this population had at least 1 HRF. Whereas around 80 % of the prescriptions in the < 4 g/day group were for patients with at least 1 HFR, only 53 % of the prescriptions in the 4 g/day group concerned patients without HFRs (p < 0.001). Age > 75 and low weight were associated with the prescriber's choice of dose. Neither chronic alcoholism nor hepatocellular insufficiency influenced the acetaminophen doses prescribed.

Conclusion:

Considering the widespread use of acetaminophen and its favourable safety profile compared to other analgesic drugs, it appears urgent to remind prescribers of the maximum daily dose recommendations for acetaminophen for patients with HRFs, especially those with chronic alcoholism and hepatocellular insufficiency.

Introduction

Acetaminophen, also known as "paracetamol", is the most widely prescribed first-line analgesic worldwide. Available as an over-the-counter drug in many countries such as France or the United States, it appears as the most frequent medication involved in both intentional and unintentional drug poisoning, according to the annual report by the American Association of Poison Control Center Data System and the French Addiction Monitoring Network [START_REF] Gummin | Annual Report of the American Association of Poison Control Centers' National Poison Data System (NPDS): 34th Annual Report[END_REF][START_REF]Misuse of analgesics in the context of self-medication: resutls of a large cross-sectional survey from the DANTE study (une Décennie d'ANTalgiques en France) (CO-041)[END_REF].

In case of acetaminophen accumulation and overdose, the main expected adverse effect is acute liver failure, including fulminant hepatitis, which can lead to liver transplantation and/or death [START_REF] Gummin | Annual Report of the American Association of Poison Control Centers' National Poison Data System (NPDS): 34th Annual Report[END_REF][START_REF] Bernal | Acute liver failure[END_REF][START_REF] Hawton | Long term effect of reduced pack sizes of paracetamol on poisoning deaths and liver transplant activity in England and Wales: interrupted time series analyses[END_REF][START_REF] Gulmez | Liver transplant associated with paracetamol overdose: results from the seven-country SALT study[END_REF]. The hepatotoxicity mechanism involves a CYP 450 (mainly 2E1) highly reactive converted metabolite, namely N-acetyl p-benzoquinone imine (NAPQI). NAPQI is physiologically broken down by gluthatione in the liver and excreted in the urine. However, in case of acetaminophen overdose, NAPQI production increases and exceeds the conjugation abilities of glutathione; as it binds to the hepatocellular membrane proteins, it induces liver parenchymal cell death [START_REF] Vale | Paracetamol (acetaminophen) poisoning[END_REF].

For a mean adult weight, clinical symptomatic acetaminophen hepatotoxicity is usually expected after a single acetaminophen ingestion of around 10 grams per 24 hours or 150 mg/kg, with an initial phase of cytolysis occurring in the first 24 to 48 hours. The hepatotoxicity is dose-dependent and can be predicted by a nomogram [START_REF] Bateman | Paracetamol poisoning: beyond the nomogram[END_REF]. Immediately after an acetaminophen overdose, N-acetylcysteine is used to restore glutathione reserves which can limit hepatotoxicity [START_REF] Rumack | Acetaminophen and acetylcysteine dose and duration: past, present and future[END_REF] with recovery expected in 4-5 days where the prognosis is favourable [START_REF] Lee | Acetaminophen (APAP) hepatotoxicity-Isn't it time for APAP to go away?[END_REF]. Studies have shown that advanced age, chronic alcohol consumption, as well as fasting/anorexia and poor nutritional status could be associated with gluthatione depletion; it is worth noting that chronic alcohol consumption has also been shown to be a CYP 2E1 inducer leading to NAPQI increase [10][START_REF] Caparrotta | Are some people at increased risk of paracetamol-induced liver injury? A critical review of the literature[END_REF][12]. Chronic renal failure as well as chronic liver disease (hepatic failure, cirrhosis, viral hepatitis) are also considered to be acetaminophen hepatotoxicity risk factors (HRFs) [START_REF] Schena | Acetaminophen: acute and chronic effects on renal function[END_REF][14][15][START_REF] Bunchorntavakul | Acetaminophen-related hepatotoxicity[END_REF] and should lead to an adjustment of acetaminophen daily intake. Meanwhile, case reports of hepatitis observed at therapeutic doses of 3 or 4 g/day have been reported among patients with low weight, a history of chronic alcoholism, hepatic steatosis or recent fasting [START_REF] Kurtovic | Paracetamol-induced hepatotoxicity at recommended dosage[END_REF][START_REF] Eriksson | Hepatotoxicity due to repeated intake of low doses of paracetamol[END_REF][START_REF] Krähenbuhl | Acute liver failure in two patients with regular alcohol consumption ingesting paracetamol at therapeutic dosage[END_REF][START_REF] Forget | Therapeutic dose of acetaminophen may induce fulminant hepatitis in the presence of risk factors: a report of two cases[END_REF][START_REF] Claridge | Acute liver failure after administration of paracetamol at the maximum recommended daily dose in adults[END_REF]. Furthermore, certain randomised controlled trials have reported an increase (mostly 3 to 4 times the normal upper limit) in serum alanine aminotransferase activity (ALT) for a significant proportion of 'healthy' patients exposed to acetaminophen at 4 g/day for several days, compared to placebo [START_REF] Watkins | Aminotransferase elevations in healthy adults receiving 4 grams of acetaminophen daily: a randomized controlled trial[END_REF]23], although the clinical significance is uncertain.

Recommendations have been established for acetaminophen prescription, with a maximum daily dose of 4 grams, and they include dose adjustment for patients with HRFs[24-26]. Dose adjustments are detailed in most summary of product characteristics (SmPC) for acetaminophen-based medications. A lack of accurate and harmonised information across SmPC is however observed. In general terms, it is recommended to use the "lowest possible dose" for symptom relief and make gradual adaptation of the dose to the pain. Regarding the maximum acetaminophen dose, some SmPC mention that "it is generally not necessary to exceed 3 g per 24 hours". Regarding dose adjustments for special populations (liver failure, renal failure, dehydration, weight <50 kg…), although it is formulated differently across SmPC, it is recommended to use the lowest possible effective doses, and specifically to increase interval between two intakes (>8h) in severe chronic renal failure. The maximum recommended doses in special population are given as an indication (sometimes 2g/day or 3g/day) but are not necessarily related to clinical studies (no reference provided in SmPC)."

Few studies have described acetaminophen prescription patterns in hospitals or assessed compliance with recommendations relating to HRFs: in French and American cohorts, failure to adjust doses in view of the presence of HRFs was observed in 1 % to 21 % of prescriptions [START_REF] Arques-Armoiry | Most frequent drug-related events detected by pharmacists during prescription analysis in a university hospital[END_REF][START_REF] Zhou | Supratherapeutic dosing of acetaminophen among hospitalized patients[END_REF][START_REF] Charpiat | Overdosed prescription of paracetamol (acetaminophen) in a teaching hospital[END_REF][START_REF] Viguier | Clinical pharmacist influence at hospital to prevent overdosed prescription of acetaminophen[END_REF][START_REF] Pace | Prescription of acetaminophen in five French hospitals: What are the practices?[END_REF]. It can be noted that neither the type of hospital units (surgery, geriatrics...) nor pharmaceutical validation studies have an influence on dose adjustment [START_REF] Arques-Armoiry | Most frequent drug-related events detected by pharmacists during prescription analysis in a university hospital[END_REF][START_REF] Viguier | Clinical pharmacist influence at hospital to prevent overdosed prescription of acetaminophen[END_REF].

This work was performed after the notification in our local Pharmacovigilance unit of cases of acetaminophen toxicity at doses in the therapeutic range among patients with HRFs: the most recent, with a fatal outcome, concerned a 72-year-old hospitalised man who developed cytolysis with acute hepatic failure two days after the initiation of 4 g/day acetaminophen for acute pain. The patient's history included alcoholism and cachexia, in a context of hepatic steatosis, septic shock and the discovery of metastatic colorectal cancer.

As we believe that some HRFs are more likely to induce dose adjustments than others, the aim of our study was to assess to what extent the existence of HRFs (single or in combination) modify the prescribers' choice of acetaminophen dose (< 4 g/day versus 4 g/day).

Materials and methods

We conducted a retrospective monocentric cross-sectional study including all patients with an acetaminophen prescription in Rennes University Hospital.

All data was collected in accordance with the French legislation on retrospective clinical studies, in accordance with the precepts established by the Helsinki declaration.

 Data sources

The extraction of data concerning acetaminophen prescriptions (oral and intravenous) (Dxcare® software version 7.5.20p049, Medasys®) was carried out over one week, from the 13 th to the 19 th December 2017. Only patients aged over 18 years, i.e. born after 12/12/1999, were considered for this analysis.

 Exposure

All medications containing acetaminophen were considered, precribed on their own or in combination with other drugs. We collected the names of the medications, the routes of administration and the daily doses. Patients were categorised as having a maximum dose of 4 g/day or less than 4 g/day. The patients for whom the dose was specified as "1 g 'upon request', maximum 4 times a day," were considered as having the maximum 4 g/day dose.

 Other variables

Data was collected from the patients' electronic files: age at the time of the acetaminophen prescription, gender, hospital unit, weight, body mass index, biological parameter values (serum creatinine, serum alanine aminotransferase (ALT), aspartate aminotransferase (AST), alkaline phosphatase (PAL), gamma-glutamyl transpeptidase (GGT), direct bilirubin, total bilirubin, prothrombin time (TP), international normalised ratio (INR), factor V, serum albumin and prealbumin), the presence of chronic viral hepatitis (B or C) and human immunodeficiency virus (HIV), current chronic alcoholism, current intake of oral anticoagulants, current malnutrition, history of liver or renal transplantation.

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 Risk factors predisposing to hepatotoxicity

According to the SmPC and French recommendations on acetaminophen prescription [24-26], we considered seven HRF categories that should lead to dose adjustment, defined as followed:

age over 75 years, i.e. patients born before the 12 th December 1942; -low weight: under 50 kg;

malnutrition defined by the presence of one or more of the following criteria: serum albumin < 30 g / L, serum pre-albumin <150 mg / L, BMI < 18.5 for patients < 70 years old, BMI < 21 for patients  70 years old, the specific mention of "malnutrition" in the electronic file;

chronic alcoholism: we selected patients whose electronic file records specified excessive and chronic alcohol consumption;

current chronic viral infections (hepatitis B, C and/or D) and/or HIV; patient status was individually checked by a virologist (CP author). HIV patients with an undetectable viral load were considered as presenting a risk factor; patients who had recovered from hepatitis C at study entry were not considered as presenting a risk factor; -severe chronic renal failure defined by a creatinine clearance value (estimated by the CKD-EPI equation) of < 30 ml / min in the electronic file; -hepatocellular insufficiency, biologically defined by one or more following abnormalities: factor V < 70 %, prothrombin time decrease, INR > 1.5 for patients without anticoagulant treatment or INR > 5 with anticoagulant treatment, ALT > 40 UI/L, AST > 40 UI/L. Other biological parameters were considered only in case of association with other abnormalities: serum albumin concentration < 35 g / L and/or the following clinical signs specified in the electronic file: "jaundice", "hepatic encephalopathy", "cirrhosis", "stellate angioma" or "palmar erythrosis", "alcoholic hepatitis", "viral hepatitis".

 Statistical methods

In case of several acetaminophen prescriptions for the same patient, only the first was considered for the descriptive and statistical analyses in order to ensure the independence of the data and analyses.

We considered the first prescription as the initial prescriber's intention to treat, as the second or following prescriptions could be related to medical or pharmaceutical re-assessment.

Descriptive statistics characterised patients at the time of the 1 st acetaminophen prescription.

Proportions were compared across levels of exposure using chi-square tests or Fisher's exact test; age was compared using the Student t test.

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A logistic regression model considering all HRFs was used to estimate those that were significantly related to the prescribers' choice of acetaminophen dose (< 4 g/day versus 4 g/day).

A descending step-by-step selection model was used, retaining only the variables (HRF) significantly associated with acetaminophen dose adjustment (< 4 g/day) at a 5 % statistical threshold.

An odds ratio estimation was used to determine which HRFs were associated with dose adjustment (< 4 g/day or 4 g/day) in the prescribers' prescriptions.

All analyses were conducted using the SAS statistical package (version 9.4; SAS Institute, Cary, NC, USA).

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Results

Over a seven-day period in December 2017, 2338 acetaminophen prescriptions were collected from Rennes University Hospital. After excluding prescriptions for patients under 18 years, 2048 acetaminophen prescriptions concerning 1842 patients were included in this study. Retaining only the first prescription for each patient, 1842 prescriptions were used for the analyses (see Figure 1).

The characteristics of the study population are displayed in table 1. Around 54 % were female. The median age was 65 years (min 18 years -max 101 years) and 32.9 % were over 75 years old.

Among the 1842 prescriptions, 73.7 % were for 4 g/day (table 1); it can be noted that no prescription exceeded the maximum 4-g daily dose. Females were more frequently in the < 4 g/day group than in the 4 g/day group (60.1 % vs 51.3 %, p < 0.001). Regarding the hospital unit, in the < 4 g/day group, prescribers mainly belonged to geriatric or other clinical units (respectively 44.6 % and 41.1 %); in the 4 g/day group, prescriptions mainly derived from surgery / anaesthesia / intensive care / palliative units and other clinical units (respectively 57.1 % and 32.6 %).

Around 55 % of the overall population presented with at least one HRF. Among patients with only one HRF (n = 549), the HRF was mainly age > 75 years, and secondarily hepatocellular insufficiency or chronic alcoholism (appendix table 1). Whereas around 80 % of prescriptions in the < 4 g/day group were for patients with at least 1 HFR, only 53 % of prescriptions in the 4 g/day group concerned patients without any HFR (p < 0.001). Furthermore, some HRFs were significantly more frequent in the < 4 g/day group (table 1) : age > 75, low weight, malnutrition and severe renal failure.

Concerning the statistical analysis, only prescriptions without missing data on the HRF category were used (n = 1103), including 363 patients in the < 4 g/day group and 740 in the 4 g/day group. The logistic regression showed that age > 75 and low weight were significantly associated with the prescriber's choice of dose (table 2). The descending step-by-step model confirmed that only age > 75 and low weight remained significantly associated with the < 4 g/day dose (data not shown). We observed similar results in a sensitivity analysis using age > 75 and weight as continuous variables (data not shown).

It can be noted that among patients > 75 years (n = 606), who accounted for one-third of the overall population, all had 1 (n = 315) or 2 (n = 291) HRFs. Despite this, around 50 % (n = 302) had no dose adjustment.

As regards the administration route, 63 (3,4 %) concerned intravenous use, most of whom (86 %; n=54) had a 4 g/day dose. In those patients, at least one HRF was recorded in 34 patients. As regards the 9 patients in the < 4 g/day group, 1 had no HRF, 2 had only one HRF (age > 75 in both cases) and

Discussion

In our seven-day study focusing on acetaminophen prescriptions in Rennes University Hospital, around three quarters of prescriptions were full-dose (4 g/day); in this group, 47 % of prescriptions were for patients with at least one HRF: these can be considered as non-compliant prescriptions, and the proportion is greater than in previous studies showing up to 21 % of non-compliant acetaminophen prescriptions in hospital [START_REF] Arques-Armoiry | Most frequent drug-related events detected by pharmacists during prescription analysis in a university hospital[END_REF][START_REF] Zhou | Supratherapeutic dosing of acetaminophen among hospitalized patients[END_REF][START_REF] Charpiat | Overdosed prescription of paracetamol (acetaminophen) in a teaching hospital[END_REF][START_REF] Viguier | Clinical pharmacist influence at hospital to prevent overdosed prescription of acetaminophen[END_REF][START_REF] Pace | Prescription of acetaminophen in five French hospitals: What are the practices?[END_REF]. The lower non-compliant prescriptions could be related to the fact that age > 75 years is not considered as a HRF in SmPC and no dose adjustment is recommended. As mentioned by Pacé et al., medicine and geriatric units seem to be more aware of the HRFs of acetaminophen [START_REF] Pace | Prescription of acetaminophen in five French hospitals: What are the practices?[END_REF]: in our study the number of prescriptions for < 4 g/day in these units amounted to around 85 % of the prescriptions.

For the HRFs studied, we showed that age > 75 years and low weight influenced the prescribers' choice of dose. The impact of advanced age here could be linked to age in our cohort since the median age was 65.0 years and one third of the patients were over 75 years old. Another explanation linked to age is the fact that, in Rennes University Hospital, prescribers are particularly aware of dosage adjustment for elderly patients thanks to careful monitoring by the pharmacists. Surprisingly, neither chronic alcoholism nor hepatocellular insufficiency were associated with dose adjustment.

Although acetaminophen is a highly hepatotoxic drug and its metabolism involves the liver, prescribers appear not to consider these HRFs in their choice of dose. Hepatic tests after acetaminophen initiation were not performed in our study, so we could not check for clinical or biological signs of hepatotoxicity among patients with these HRFs. Pace et al. also observed a high rate of non-compliance with recommendations (> 68 %) for patients with chronic alcoholism or hepatocellular insufficiency [START_REF] Pace | Prescription of acetaminophen in five French hospitals: What are the practices?[END_REF], suggesting that prescribers need to be made aware of dose adjustments in these patient groups. Unlike our study where low weight was a dose-adjustment variable in acetaminophen prescriptions by clinicians, this factor was explored in heterogeneous manner in other studies and was related to non-compliance [START_REF] Charpiat | Overdosed prescription of paracetamol (acetaminophen) in a teaching hospital[END_REF][START_REF] Pace | Prescription of acetaminophen in five French hospitals: What are the practices?[END_REF].

None of the prescriptions exceeded the 4-g per day, which is no doubt linked to the use of software (DxCare  ) limiting acetaminophen daily doses; a warning is also displayed when several drugs containing acetaminophen are coprescribed. Some HRF as well as their definition can be discussed. In a literature review, Caparrotta et al. found no good quality evidence to establish that factors were HRF [START_REF] Caparrotta | Are some people at increased risk of paracetamol-induced liver injury? A critical review of the literature[END_REF]. They notably pointed that the safe oral acetaminophen dose in patients < 50 kg had not been established. In our study, chronic alcoholism status has only been identified through a subjective HRF reading (potentially underestimated) without re-assessment by an independent committee. No additional information was collected (severity, care…). Age, especially advanced age is described as HRF whereas literature data are inconsistent (PK, case series, population-based studies) [START_REF] Caparrotta | Are some people at increased risk of paracetamol-induced liver injury? A critical review of the literature[END_REF]12]. As evoked by Caparrotta et al. there is a lack of good quality clinical evidence that older people have a clinically significant difference in acetaminophen metabolism or are at increased risk of toxicity at (supra)therapeutic dose. Age cut-off also varied across studies [12,[START_REF] Wynne | The association of age and frailty with paracetamol conjugation in man[END_REF][START_REF] Mitchell | Hepatotoxicity of therapeutic short-course paracetamol in hospital inpatients: impact of ageing and frailty[END_REF][START_REF] Liukas | Pharmacokinetics of intravenous paracetamol in elderly patients[END_REF]. Moreover, neither French SmPC nor recommendations provide an age cut-off. Considering that "old age" definition is complex, potentially subjective (physical, psychological conditions), and is not only related to years, we arbitrarily chose 75 years-old as cut-off in our study. In addition to biological criteria, hepatocellular insufficiency definition also included a HRF reading seeking specific terms (cirrhosis, hepatic encephalopathy) without secondary objective re-assessment. All these limitations could have induced misclassification bias of HRF.

The main strength of our work lies in the data collection that took place within a week and involved all adult patients' electronic files in all Rennes University Hospital units. Among the weaknesses, we recognize that our results concern only one hospital and may not be representative of French hospital prescribers. But the objective of our study was not to compare with practices in other hospitals but rather to highlight the fact that HRFs are not always considered by prescribers, even in university hospitals, when prescribing acetaminophen. Nor did we consider the indication for acetaminophen, treatment duration or the potential need for opioid treatments, which could have impacted dose adjustment. Considering a safety approach, we deliberately focused our study on the first acetaminophen dose prescribed, irrespective of its indication, as representing intention-to-treat. Furthermore, our statistical analysis did not include all the 1842 prescriptions in the overall population as a result of missing data for some HRFs: around 33 % of patients had missing data for the hepatocellular insufficiency variable, and 10 % for malnutrition status. It can be noted that some HRFs could have been under-estimated, especially alcoholism which is often concealed by patients when questioned on the subject. We did not assess either whether the 4g/day dose for patients with one or more HRF had clinical significance for liver function, nor did we consider the type of HRF; indeed, hepatic cytolysis is more likely among patients with cirrhosis than among elderly patients without other liver diseases. We did not consider co-medication and especially drug-drug interaction, nor other clinical conditions (sepsis, heart failure [START_REF] Yan | The role of the liver in sepsis[END_REF]36]) that affect the hepatic enzymes. In acetaminophen SmPCs, drug interaction section mentioned a precaution of use when associated with other hepatotoxic drugs or CYP 450 drug enzyme inducers. However, on the basis of the French drugdrug interaction referential provided by the French Health Authorities (French National Agency for Medicines and Health Products Safety [ANSM]) [START_REF]Thésaurus des interactions médicamenteuses -mise à jour mars[END_REF], no clinically significant interaction with paracetamol was highlighted, even with drugs impacting CYP 2E1 (doxycycline, isoniazide).

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We should bear in mind that although acetaminophen is the most widely recognized drug in inducing liver damage [START_REF] Leise | Drug-induced liver injury[END_REF]39], its use is commonplace, mainly as a result of a good reputation with regard to safety compared to other analgesic drugs (non-steroid anti-inflammatory drugs for example). In order to limit the risk of poisoning and suicide using acetaminophen, France was the first country in Europe in the 1980s to limit packaging to a maximum dose of 8 grams of acetaminophen. In the 2000s, the Federal Drug Agency in the United States and the United Kingdom Health Authorities also restricted the acetaminophen pack size [40,[START_REF] Krenzelok | The FDA Acetaminophen Advisory Committee Meeting -what is the future of acetaminophen in the United States? The perspective of a committee member[END_REF]; the FDA also limited the acetaminophen dosage unit to 325 mg in 2011 [START_REF]Prescription Drug Products Containing Acetaminophen: Actions to Reduce Liver Injury from Unintentional Overdose[END_REF][START_REF]Prescription Acetaminophen Products to be Limited to 325 mg Per Dosage Unit; Boxed Warning Will Highlight Potential for Severe Liver Failure[END_REF]. Despite this, acetaminophen remains the first drug involved in overdose (intentional or otherwise) [START_REF] Gummin | Annual Report of the American Association of Poison Control Centers' National Poison Data System (NPDS): 34th Annual Report[END_REF][START_REF]Misuse of analgesics in the context of self-medication: resutls of a large cross-sectional survey from the DANTE study (une Décennie d'ANTalgiques en France) (CO-041)[END_REF]. In 2017, W. Lee described the controversy surrounding acetaminophen use in pain management [START_REF] Lee | Acetaminophen (APAP) hepatotoxicity-Isn't it time for APAP to go away?[END_REF]: he pointed out that worldwide regulatory efforts had been ineffective in reducing the cost in money and lives resulting from its hepatotoxicity. In France, however, the French Pharmacovigilance network regularly collects case reports of acute acetaminophen poisoning. A recent fatal case in December 2017, which was highly publicized across France, led the Health Authorities to reinforce the data available on acetaminophen-based drugs: the objective was to raise awareness among patients and prescribers about liver damage. A public consultation was thus initiated on August 20 th , 2018, ending on September 30 th , 2018 for the definition of the best warning message to put on drug packaging [START_REF]Paracétamol : l'ANSM lance une consultation publique pour sensibiliser les patients et les professionnels de santé au risque de toxicité pour le foie en cas de mésusage -Point d'Information -ANSM : Agence nationale de sécurité du médicament et des produits de santé[END_REF]; but the results have not yet been issued. With the exception of hepatocellular insufficiency, there is a lack of information on dose adjustment, special warnings or contraindications in case of other HRFs with some acetaminophenbased medications (e.g. Paracetamol Teva 1g, tablets; Paracetamol EG 500 mg/30 mg  , effervescent scored tablets ; Paracetamol Zydus 500 mg, gelules  … [START_REF]Résumé des caractéristiques du produit -Paracétamol Teva 1g, comprimé[END_REF][START_REF]Résumé des caractéristiques du produit -Paracétamol Codéine EG 500 mg/ 30 mg, comprimé effervescent sécable[END_REF][START_REF]Résumé des caractéristiques du produit -Paracétamol Zydus 500 mg, gélule[END_REF]). It is worth noting that maximum dose could vary from one SmPC to another: for instance, in case of HRF, 2g/day is mentioned in Paracetamol AHCL 1g, effervescent tablet [START_REF]Résumé des caractéristiques du produit -Paracétamol AHCL 1g, comprimé effervescent[END_REF] compared to 3g/day in Doliprane 1 g, tablets [49]. In general terms, lack of SmPC harmonisation, especially regarding the appropriate maximal dose to be used in case of HRF is a limitation for clinicians' prescriptions compliance. ANSM planned a harmonisation of the warnings included in the SmPC for acetaminophen-based drugs in 2019.

Considering pharmacovigilance cases report of acetaminophen toxicity in patients with HRF treated with (sub)therapeutic ⩽ 4 g/day dose and the results of the current study, in Rennes University Hospital, several improvement measures are planned: awareness raising at the residents' welcome seminars twice a year, poster campaign in clinical departments, configuration of software as regards prescription schemes, awareness raising of pharmacist responsible of prescriptions' pharmaceutical validation.

Revised manuscript Bacle et al.

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Conclusion

This work shows that in Rennes University Hospital, HRFs are not systematically considered by clinicians when acetaminophen is prescribed. Age > 75 years and low weight had a greater impact on acetaminophen prescription than alcoholism, malnutrition, chronic viral hepatitis, severe renal failure or hepatocellular insufficiency. Considering the widespread use of acetaminophen, it appears important to remind healthcare professionals and patients of the hepatotoxicity risk resulting from misuse, especially in presence of HRF.

had at least 2 HRF. More in depth, Paracetamol B Braun 1g/100 ml  (adult formulation) was used in all cases. Its SmPC recommends a dose adjustment considering weight category (between 33 and 50 kg or > 50 kg) and whether HRF are present (chronic alcoholism, hepatocellular insufficiency, chronic malnutrition, and dehydration for which maximal dose is 3 g/day).Revised manuscriptBacle et al.

Revised manuscriptBacle et al.[START_REF] Krähenbuhl | Acute liver failure in two patients with regular alcohol consumption ingesting paracetamol at therapeutic dosage[END_REF]

Authors contribution:

LMS and AB had full access to all of the data in the study and take responsibility for the integrity of the data and the accuracy of the data analysis. SP, EP, LMS and AB were part of the study concept and design. All authors were a part in the acquisition, analysis or interpretation of data. Drafting of the manuscript was done by LMS. All authors took part in the critical revision of the manuscript for important intellectual content. LMS was a part in the statistical analysis. Administrative, technical or material support was provided by Rennes Hospital University.

Conflict of interest:

None.

Funding / support:

None.

Additional contribution:

We thank Jean-Paul Sinteff (Medical Information Departement, CHU Rennes) for the DxCare  software data extraction, and Adrien Turban, Anne-Sophie Michel, Justine Geffroy and Stephanie Ollivier for their help in the data collection.


Marcel Bräutigam

Marie Kratz

BIVARIATE FCLT FOR THE SAMPLE QUANTILE AND MEASURES OF DISPERSION FOR AUGMENTED GARCH(p, q) PROCESSES

Keywords: 2010 AMS classification: 60F05, 60F17, 60G10, 62H10, 62H20 JEL classification: C13, C14, C30 asymptotic distribution, functional central limit theorem, (augmented) GARCH, correlation, (sample) quantile, measure of dispersion, (sample) mean absolute deviation, (sample) variance

Bivariate FCLT for the Sample Quantile and Measures of Dispersion for Augmented GARCH(p, q) processes

Introduction and Notation

Since the introduction of the ARCH and GARCH processes in the seminal papers by Engle, [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation[END_REF], and Bollerslev, [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF], respectively, various GARCH modifications and extensions have been proposed and their statistical properties analysed (see e.g. [START_REF] Bollerslev | Glossary to arch (garch)[END_REF] for a non-exhaustive ARCH glossary). Conditions for the stationarity of such processes, as well as central limit theorems (CLT) or functional central limit theorems (FCLT) have been obtained in various ways by exploiting the different dependence concepts underlying these GARCH type processes (see the introduction in [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF] for references on CLT's under different dependence conditions).

The limit theorems extend also to different estimators apart from the underlying process itself, as for example: Powers of the process (e.g. [START_REF] Örmann | Augmented garch sequences: Dependence structure and asymptotics[END_REF] for augmented GARCH(1,1); [START_REF] Berkes | The functional central limit theorem for a family of garch observations with applications[END_REF], [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF] for augmented GARCH(p, q)), sample autocovariance and sample variance (e.g. [START_REF] Mikosch | Limit theory for the sample autocorrelations and extremes of a garch (1,1) process[END_REF] for the GARCH(1,1); [START_REF] Aue | Strong approximation for the sums of squares of augmented garch sequences[END_REF] for augmented GARCH(1,1)), or the sample quantile.

Still, joint asymptotics of such estimators have not been considered yet. It is what we are developing in this paper, providing bivariate functional central limit theorems for the sample quantile with the r-th absolute centred sample moment. This includes the case of the sample variance and also the sample mean absolute deviation around the sample mean (MAD), two well-known and widely used measures of dispersion, extending the results obtained in [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF] for identically and independently distributed (iid) random variables.

Note that the theoretical question arised from previous studies in financial risk management, one (see [START_REF] Br Äutigam | Pro-cyclicality of traditional risk measurements: Quantifying and highlighting factors at its source[END_REF]) where the correlation between a log-ratio of sample quantiles with the sample MAD is measured using log-returns from different stock indices, the other (see [START_REF] Zumbach | Correlations of the realized volatilities with the centered volatility increment[END_REF] and [START_REF] Zumbach | Discrete Time Series, Processes, and Applications in Finance[END_REF]) considering the correlation of 'the realized volatilities with the centred volatility increment' for different underlying processes. Thus, we think that those asymptotic results may be of great use for applications in statistics or other application fields. For instance, coming back to financial risk management and risk measure estimation, we could extend results obtained for the Value-at-Risk, when estimated by the sample quantile, to Expected Shortfall using once again the FCLT.

To cover a broad range of GARCH processes, we focus on so called augmented GARCH(p, q) processes, introduced by Duan in [START_REF] Duan | Augmented garch (p, q) process and its diffusion limit[END_REF]. They contain many well-known GARCH processes as special cases. Previous works on univariate CLT's and stationarity conditions for this class of GARCH processes are, inter alia, [START_REF] Aue | Strong approximation for the sums of squares of augmented garch sequences[END_REF], [START_REF] Berkes | The functional central limit theorem for a family of garch observations with applications[END_REF], [START_REF] Örmann | Augmented garch sequences: Dependence structure and asymptotics[END_REF] and [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF].

The structure of the paper is as follows. We present in Section 2 the main results about the bivariate FCLT for the sample quantile and the r-th absolute centred sample moment for augmented GARCH(p, q) processes. Then, we present specific examples of well-known GARCH models in Section 3 and show how the general conditions in the main result translate for these specific cases. The proofs are given in Section 4.

Notation

We introduce the same notation as in [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF]: Let (X 1 , • • • , X n ) be a sample of size n. Assuming the random variables X i 's have a common distribution, denote their parent random variable (rv) X with parent cumulative distribution function (cdf) F X , (and, given they exist,) probability density function (pdf) f X , mean µ, variance σ 2 , and quantile of order p defined as q X (p) := inf{x ∈ R : F X (x) ≥ p}. We denote the ordered sample by X (1) ≤ ... ≤ X (n) .

We consider the sample estimators of the two quantities of interest, i.e. first the sample quantile for any order p ∈ [0, 1] defined as q n (p) = X ( np ) , where x = min {m ∈ Z : m ≥ x}, x = max {m ∈ Z : m ≤ x} and [x], are the rounded-up, rounded-off integer-parts and the nearest-integer of a real number x ∈ R, respectively. Second, the r-th absolute centred sample moment defined, for r ∈ N, by

m(X, n, r) := 1 n n i=1 |X i -Xn | r , (1)
Xn denoting the empirical mean. Special cases of this latter estimator include the sample variance (r = 2) and the sample mean absolute deviation around the sample mean (r = 1).

Recall the standard notation u T for the transpose of a vector u and, for the signum function, sgn(x) :=

-1I (x<0) + 1I (x>0) . Moreover the notations d →,
a.s.

→ ,

P

→ and

D d [0,1]
→ correspond to the convergence in distribution, almost surely, in probability and in distribution of a random vector in the d-dimensional Skorohod space D d [0, 1]. Further, for real-valued functions f, g, we write f (x) = O(g(x)) (as x → ∞) if and only if there exists a positive constant M and a real number x 0 s.t. |f (x)| ≤ M g(x) for all x ≥ x 0 , and f (x) = o(g(x)) (as x → ∞) if for all > 0 there exists a real number x 0 s.t. |f (x)| ≤ g(x) for all x ≥ x 0 . Analogously, for a sequence of rv's X n and constants a n , we denote by X n = o P (a n ) the convergence in probability to 0 of X n /a n .

The Bivariate FCLT

Let us introduce the augmented GARCH(p, q) process X = (X t ) t∈Z , due to Duan in [START_REF] Duan | Augmented garch (p, q) process and its diffusion limit[END_REF], namely, for integers p ≥ 1 and q ≥ 0, X t satisfies

X t = σ t t (2) with Λ(σ 2 t ) = p i=1 g i ( t-i ) + q j=1 c j ( t-j )Λ(σ 2 t-j ), (3)
where ( t ) is a series of iid rv's with mean 0 and variance 1, σ 2 t = Var(X t ) and g i , c j , i = 1, ..., p, j = 1, ..., q, are real-valued measurable functions. Also, as in [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF], we restrict the choice of Λ to the so-called group of either polynomial GARCH(p, q) or exponential GARCH(p, q) processes (see Figure 1 in the Appendix):

(Lee) Λ(x) = x δ , for some δ > 0, or Λ(x) = log(x).

Clearly, for a strictly stationary solution to (2) and (3) to exist, the functions Λ, g i , c j as well as the innovation process ( t ) t∈Z have to fulfill some regularity conditions (see e.g. [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF], Lemma 1). Alike, for the bivariate FCLT to hold, certain conditions need to be fulfilled; we list them in the following. First, conditions concerning the dependence structure of the process X. We use the concept of L pnear-epoch dependence (L p -NED), using a definition due to Andrews in [START_REF] Andrews | Laws of large numbers for dependent non-identically distributed random variables[END_REF] but restricted to stationary processes. Let (Z n ) n∈Z , be a sequence of rv's and F t s = σ(Z s , ..., Z t ), for s ≤ t, the corresponding sigma-algebra. The usual L p -norm is denoted by

• p := E 1/p [|•| p ]. Let us recall the L p -NED definition. Definition 1 (L p -NED, [1]) For p > 0, a stationary sequence (X n ) n∈Z is called L p -NED on (Z n ) n∈Z if it holds, for k ≥ 0, that X 1 -E[X 1 |F n+k n-k ] p ≤ ν(k), for non-negative constants ν(k) such that ν(k) → 0 as k → ∞.
If ν(k) = O(k -τ -) for some > 0, we say that X n is L p -NED of size (-τ ). If ν(k) = O(e -δk ) for some δ > 0, we say that X n is geometrically L p -NED.

The second set of conditions concerns the distribution of the augmented GARCH(p, q) process. We impose three different types of conditions as in the iid case (see [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF]): First, the existence of a finite 2k-th moment for any integer k > 0 for the innovation process ( t ). Then, given that the process X is stationary, the continuity or l-fold differentiability of its distribution function F X (at a given point or neighbourhood) for any integer l > 0, and the positivity of its density f X (at a given point or neighbourhood). Those conditions are named as:

(M k ) E[| 0 | 2k ] < ∞, (C 0 ) F X is continuous, (C l ) F X is l-times differentiable, (P ) f X is positive.
The third type of conditions is set on the parameters and functions of the augmented GARCH(p, q) process of the (Lee) family: Positivity of the functions used and boundedness in L r -norm for either the polynomial GARCH, (P r ), or exponential/logarithmic GARCH, (L r ), respectively, for a given integer r > 0, (A) g i ≥ 0, c j ≥ 0, i = 1, ..., p, j = 1, ..., q,

(P r ) p i=1 g i ( 0 ) r < ∞, q j=1 c j ( 0 ) r < 1, (L r ) E[exp(4r p i=1 |g i ( 0 )| 2 )] < ∞, q j=1 |c j ( 0 )| < 1.
Note that condition (L r ) requires the c j to be constant functions. Now, let us state the main result. To ease its presentation we introduce a trivariate normal random vector (functionals of X), (U, V, W ) T , with mean zero and the following covariance matrix:

(D)                                                                Var(U ) = Var(X 0 ) + 2 ∞ i=1 Cov(X i , X 0 ) Var(V ) = Var(|X 0 | r ) + 2 ∞ i=1 Cov(|X i | r , |X 0 | r ) Var(W ) = Var p -1I (X 0 ≤q X (p)) f X (q X (p)) + 2 ∞ i=1 Cov p -1I (X i ≤q X (p)) f X (q X (p)) , p -1I (X 0 ≤q X (p)) f X (q X (p)) = p(1 -p) f 2 X (q X (p)) + 2 f 2 X (q X (p)) ∞ i=1 E[1I (X 0 ≤q X (p)) 1I (X i ≤q X (p)) ] -p 2 Cov(U, V ) = i∈Z Cov(|X i | r , X 0 ) = i∈Z Cov(|X 0 | r , X i ) Cov(U, W ) = -1 f X (q X (p)) i∈Z Cov(1I (X i ≤q X (p)) , X 0 ) = -1 f X (q X (p)) i∈Z Cov(1I (X 0 ≤q X (p)) , X i ) Cov(V, W ) = -1 f X (q X (p)) i∈Z Cov(|X 0 | r , 1I (X i ≤q X (p)) ) = -1 f X (q X (p)) i∈Z Cov(|X i | r , 1I (X 0 ≤q X (p)) ).
Theorem 2 (bivariate FCLT) Consider an augmented GARCH(p, q) process X as defined in (2) and (3) satisfying condition (Lee), (C 0 ) at the mean µ for r = 1, and both conditions (C 2 ), (P ) at q X (p). Assume also conditions (M r ), (A 0 ), and either (P max(1,r/δ) ) for polynomial GARCH, or (L r ) for exponential GARCH. Introducing the random vector T n,r (X) = q n (p) -q X (p) m(X, n, r) -m(X, r)

, for r ∈ Z, we have the following FCLT:

For t ∈ [0, 1], as n → ∞, √ n t T [nt],r (X) D 2 [0,1] → W Γ (r) (t)
where r) , where

(W Γ (r) (t)) t∈[0,1] is the 2-dimensional Brownian motion with covariance matrix Γ (r) ∈ R 2×2 defined for any (s, t) ∈ [0, 1] 2 by Cov(W Γ (r) (t), W Γ (r) (s)) = min(s, t)Γ (
Γ (r) 11 = Var(W ), Γ (r) 22 = r 2 E[X r-1 0 sgn(X 0 ) r ] 2 Var(U ) + Var(V ) -2r E[X r-1 0 sgn(X 0 ) r ] Cov(U, V ), Γ (r) 12 = Γ (r) 21 = -r E[X r-1 0 sgn(X 0 ) r ] Cov(U, W ) + Cov(V, W ),
(U, V, W ) T being the trivariate normal vector (functionals of X) with mean zero and covariance given in (D), all series being absolute convergent.

Remark 3 Note that for the bivariate FCLT in Theorem 2, apart from (P max (1,r/δ) ) or (L r ) respectively, which are the conditions for the univariate CLT of the r-th centred sample moment, also the conditions (C 0 ) at µ for r = 1, and (C 2 ), (P ) at q X (p), are needed. Requiring (C 2 ), (P ) at q X (p) exactly correspond to the conditions for the CLT of the sample quantile. Hence, the bivariate FCLT between the sample quantile and the r-th absolute centred sample moment does not require any additional condition in comparison to the respective univariate convergence.

Choosing t = 1 in Theorem 2 provides the usual CLT that we state for completeness:

Corollary 4 Consider an augmented GARCH(p, q) process as defined in (2) and (3). Under the same conditions as in Theorem 2, the joint behaviour of the sample quantile q n (p) (for p ∈ (0, 1)) and the r-th absolute centred sample moment m(X, n, r), is asymptotically bivariate normal:

√ n q n (p) -q X (p) m(X, n, r) -m(X, r) d -→ n→∞ N (0, Γ (r) ), (4)
where the asymptotic covariance matrix

Γ (r) = (Γ (r) ij , 1 ≤ i, j ≤ 2) is as in Theorem 2.
As special case we can also recover the CLT between the sample quantile and the r-th absolute centred sample moment in the iid case, given by Theorem 7 in [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF]:

Corollary 5 Consider an augmented GARCH(p, q) process as defined in (2) and (3), choosing g i , c j , Λ such that σ 2 t is a constant. Under the same conditions as in Theorem 2, the joint behaviour of the sample quantile q n (p) (for p ∈ (0, 1)) and the r-th absolute centred sample moment m(X, n, r), is asymptotically bivariate normal:

√ n q n (p) -q X (p) m(X, n, r) -m(X, r) d -→ n→∞ N (0, Γ (r) ), (5)
where the asymptotic covariance matrix

Γ (r) = (Γ (r) ij , 1 ≤ i, j ≤ 2) simplifies to Γ (r) 11 = p(1 -p) f 2 X (q X (p)) ; Γ (r) 22 = r 2 E[X r-1 0 sgn(X 0 ) r ] 2 σ 2 + Var(|X 0 | r ) -2r E[X r-1 0 sgn(X 0 ) r ] Cov(X 0 , |X 0 |); Γ (r) 12 = Γ (r) 21 = 1 f X (q X (p)) r E[X r-1 0 sgn(X 0 ) r ] Cov(1I (X 0 ≤q X (p)) , X 0 ) -Cov(1I (X 0 ≤q X (p)) , |X 0 | r ) .
Idea of the proof -Let us briefly describe the idea of the proof of Theorem 2, developed in Section 4.

To prove the FCLT, we need to show that two conditions are fulfilled for the vector T n,r (X). These specific conditions arise from the application of the multivariate FCLT (Theorem A.1 in [START_REF] Aue | Break detection in the covariance structure of multivariate time series models[END_REF], which extends the univariate counterpart from, e.g., Billingsley in [START_REF] Billingsley | Convergence of probability measures[END_REF]) and are the following:

(H 1 ) A representation t j of T n,r with E[t j ] = 0 and E[t 2 j ] < ∞, such that we have, for j ∈ Z 1 n n j=1 t j = T n,r (X) and t j = f ( j , j-1 , ...),
where f : R 2×∞ → R 2 is a measurable function and ( j , j ∈ Z) is a sequence of R 2 -valued iid rv's with mean 0 and variance 1.

(H 2 ) A ∆-dependent approximation of t j , i.e. the existence of a sequence of random vectors t (∆) j , j ∈ Z such that, for any ∆ ≥ 1, we have t (∆) j = f (∆) ( j-∆ , ..., j , ..., j+∆ ), for measurable functions f (∆) : R 2×(∆+1) → R 2 , and

∆≥1 t 0 -t (∆) 0 2 < ∞.
Checking the first condition, (H 1 ), is done in two steps. First, we show why we can use the Bahadur representation of the sample quantile given in [START_REF] Wendler | Bahadur representation for u-quantiles of dependent data[END_REF] (Theorem 1). Second, we prove a corresponding representation for the r-th absolute centred sample moment (extending results from [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF], [START_REF] Segers | On the asymptotic distribution of the mean absolute deviation about the mean[END_REF]). These representations of the sample quantile and r-th absolute centred sample moment then naturally fulfil (H 1 ).

To proof the second condition (H 2 ), we need to find a ∆-dependent approximation t (∆) j . For this, we show that the existence of our chosen t (∆) j can be reduced to the existence of a ∆-dependent approximation for the process X j or powers of the process |X j |, for which results in [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF] can be used.

Examples

In this section we review some well-known examples of augmented GARCH(p, q) processes and discuss which conditions these models need to fulfill in order for the bivariate asymptotics of Theorem 2 to be valid.

Note that the moment condition on the innovations, (M r ), as well as the continuity and differentiability conditions, (C 2 ), (P ), each at q X (p), and (C 0 ) at µ for r = 1, remain the same for the whole class of augmented GARCH processes. But, depending on the specifications of the process, (2) and (3), the conditions, (P max(1,r/δ) ) for polynomial GARCH or (L r ) for exponential GARCH respectively, translate differently in the various examples.

For this, we introduce in Table 1 different augmented GARCH(p,q) models by providing for each the corresponding volatility equation, (3), and the specifications of the functions g i and c j . We consider 10 models which belong to the group of polynomial GARCH (Λ(x) = x δ ) and two examples of exponential GARCH (Λ(x) = log(x)). As the nesting of the different models presented is not obvious, we give a schematic overview in Figure 1 in the Appendix. An explanation of the abbreviations for, and authors of, the different models can be found there too. Note that the presented selection of augmented GARCH (p,q) processes is not exhaustive.

Note that in Table 1 the specification of g i is the same for the whole APGARCH family (only the c j change), whereas for the two exponential GARCH models, it is the reverse. The general restrictions on the parameters are as follows: ω > 0, α i ≥ 0, -1 ≤ γ i ≤ 1, β j ≥ 0 for i = 1, ..., p, j = 1, ..., q. Further, the parameters in the GJR-GARCH (TGARCH) are denoted with an asterix (with a plus or minus) as they are not the same as in the other models (see the Appendix for details).

σ 2δ t = ω + p i=1 αi (|yt-i| -γiyt-i) 2δ + q j=1 βjσ 2δ t-j gi = ω/p and cj = αj (| t-j | -γj t-j ) 2δ + βj AGARCH σ 2 t = ω + p i=1 αi (|yt-i| -γiyt-i) 2 + q j=1 βjσ 2 t-j cj = αj (| t-j | -γj t-j ) 2 + βj GJR-GARCH σ 2 t = ω + p i=1 α * i + γ * i 1I (y t-i <0) y 2 t-i + q j=1 βjσ 2 t-j cj = βj + α * j 2 t-j + γ * j max(0, -t-j ) 2 GARCH σ 2 t = ω + p i=1 αiy 2 t-i + q j=1 βjσ 2 t-j cj = αj 2 t-j + βj ARCH σ 2 t = ω + p i=1 αiy 2 t-i cj = αj 2 t-j TGARCH σt = ω + p i=1 α + i max(yt-i, 0) -α - i min(yt-i, 0) + q j=1 βjσt-j cj = αj| t-j | -αjγj t-j + βj TSGARCH σt = ω + p i=1 αi|yt-i| + q j=1 βjσt-j cj = αj| t-j | + βj PGARCH σ δ t = ω + p i=1 αi|yt-i| δ + q j=1 βjσ δ t-j . cj = αj| t-j | δ + βj VGARCH σ 2 t = ω + p i=1 αi( t-i + γi) 2 + q j=1 βjσ 2 t-j . gi = ω/p + αi( t-i + γi) 2 and cj = βj NGARCH σ 2 t = ω + p i=1 αi(yt-i + γiσt-i) 2 + q j=1 βjσ 2 t-j gi = ω/p and cj = αj( t-j + γj) 2 + βj Exponential GARCH cj = βj and MGARCH log(σ 2 t ) = ω + p i=1 αi log( 2 t-i ) + q j=1 βj log(σ 2 t-j ) gi = ω/p + αi log( 2 t-i ) EGARCH log(σ 2 t ) = ω + p i=1 αi (| t-i| -E| t-i|) + γi t-i + q j=1 βj log(σ 2 t-j ) gi = ω/p + αi(| t-i| -E| t-i|) + γi t-i
In Tables 2 and3 we present how the conditions (P max(1,r/δ) ) or (L r ) translate for each model. Table 2 treats the specific case of an augmented GARCH(p, q) process with p = q = 1 and is presented here whereas Table 3 treats the general case for arbitrary p ≥ 1, q ≥ 0 and is defered to the Appendix. In the first column we consider the conditions for the general r-th absolute centred sample moment, r ∈ N. Of biggest interest to us are the specific cases of the sample MAD (r = 1) and the sample variance (r = 2) as measure of dispersion estimators respectively, presented in the second and third column.

For the selected polynomial GARCH models the requirement p i=1 g i ( 0 ) max(1,r/δ) < ∞ in condition (P max(1,r/δ) ) will always be fulfilled. Thus, we only need to analyse the condition q j=1 c j ( 0 ) max(1,r/δ) < 1.

Note that in Table 2 (and also Table 3) the restrictions on the parameter space, given by (P max(1,r/δ) ) or (L r ) respectively, are the same as the conditions for univariate FCLT's of the process X r t itself (see [START_REF] Berkes | The functional central limit theorem for a family of garch observations with applications[END_REF], [START_REF] Örmann | Augmented garch sequences: Dependence structure and asymptotics[END_REF]). For r = 1, they coincide with the conditions for e.g. β-mixing with exponential decay (see [START_REF] Carrasco | Mixing and moment properties of various garch and stochastic volatility models[END_REF]). Table 2: Conditions (P max (1,r/δ) ) or (L r ) respectively translated for different augmented GARCH(1,1) models. Left column for the general r-th absolute centred sample moment, middle for the MAD (r = 1) and right for the variance (r = 2).

augmented GARCH (1, 1) r ∈ N r = 1 r = 2 APGARCH E[|α 1 (| 0 | -γ 1 t-1 ) 2δ + β 1 | r ] < 1 α 1 E (| 0 | -γ 1 t-1 ) 2δ + β 1 < 1 E[|α 1 (| 0 | -γ 1 t-1 ) 2δ + β 1 | 2 ] < 1 AGARCH E[|α 1 (| 0 | -γ 1 t-1 ) 2 + β 1 | r ] < 1 α 1 E (| 0 | -γ 1 t-1 ) 2 + β 1 < 1 E[|α 1 (| 0 | -γ 1 t-1 ) 2 + β 1 | 2 ] < 1 GJR-GARCH E[|α * 1 2 0 + β 1 + γ * 1 max(0, -2 0 )| r ] < 1 α * 1 + β 1 + γ * 1 E[max(0, -0 ) 2 ] < 1 E[|α * 1 2 0 + β 1 + γ * 1 max(0, -2 0 )| 2 ] < 1 GARCH E[(α 1 2 0 + β 1 ) r ] < 1 α 1 + β 1 < 1 α 2 1 E[ 4 0 ] + α 1 β 1 + β 2 1 < 1 ARCH α r 1 E[ 2r 0 ] < 1 α 1 < 1 α 2 1 E[ 4 0 ] < 1 TGARCH E[|α 1 | t-1 | -α 1 γ 1 t-1 + β 1 | r ] < 1 α 1 E| t-1 | + β 1 < 1 E[|α 1 | t-1 | -α 1 γ 1 t-1 + β 1 | 2 ] < 1 TSGARCH E[|α 1 | t-1 | + β 1 | r ] < 1 α 1 E| t-1 | + β 1 < 1 E[|α 1 | t-1 | + β 1 | 2 ] < 1 PGARCH E[|α 1 | 0 | + β 1 | 2r ] < 1 α 1 + 2α 1 β 1 E| 0 | + β 2 1 < 1 E[|α 1 | 0 | + β 1 | 4 ] < 1 VGARCH for any r ∈ N: β 1 < 1 NGARCH E[|α 1 ( 0 + γ 1 ) 2 + β 1 | r ] < 1 α 1 (1 + γ 2 1 ) + β 1 < 1 E[|α 1 ( 0 + γ 1 ) 2 + β 1 | 2 ] < 1 MGARCH for any r ∈ N: E[exp(4r|ω/p + α 1 log( 2 0 )| 2 )] < ∞ and |β 1 | < 1 EGARCH for any r ∈ N: E[exp(4r|ω/p + α 1 (| 0 | -E| 0 |) + γ 1 0 | 2 )] < ∞ and |β 1 | < 1

Proofs

Before stating the proof of the main theorem, let us start with two auxiliary results. As it requires some work to find the asymptotics of m(X, n, r) = 1 n n i=1 |X i -Xn | r for any integer r ≥ 1, and such a result is of interest in its own right, we give it separately in Proposition 7. To prove it, we need the following Lemma, which extends Lemma 2.1 in [START_REF] Segers | On the asymptotic distribution of the mean absolute deviation about the mean[END_REF] (case v = 1) to any moment v ∈ N, and the iid case presented in Lemma 8 in [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF]. Lemma 6 Consider a stationary and ergodic time-series (X n , n ≥ 1) with parent rv X. Then, for v = 1 or 2, given that the 2nd moment of X exists, or, for any integer v > 2, given that the v-th moment of X exists, it holds, as n → ∞, almost surely that

1 n n i=1 (X i -µ) v |X i -Xn | -|X i -µ| = ( Xn -µ) E[(X -µ) v sgn(µ-X)]+o P (1) +o P (1/ √ n). (6)
Proof of Lemma 6. The proof starts exactly as its equivalent in the iid case; see proof of Lemma 8 in [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF]. The argumentation needs to be adapted at the end in two points, using the stationarity and ergodicity of the process. Here, it follows by ergodicity and stationarity that √ n| Xn -µ| v+1 P → n→∞ 0 for any integer v ≥ 1. Further, as a last step, we use the ergodicity of the process, instead of the law of large numbers, to conclude that

1 n n i=1 (X i -µ) v sgn(µ -X i ) a.s. → n→∞ E[(X -µ) v sgn(µ -X)].
Now we are ready to state the asymptotic relation between the r-th absolute centred sample moment with known and unknown mean, respectively. This enables us to compute the asymptotics of m(X, n, r). As for Lemma 6, it is an extension to the stationary and ergodic case of Proposition 9 in the iid case [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF].

Proposition 7 Consider a stationary and ergodic time series (X n , n ≥ 1) with parent rv X. Then, given that the r-th moment of X exists, with r ∈ N, it holds, as n → ∞, that

√ n 1 n n i=1 |X i -Xn | r = √ n 1 n n i=1 |X i -µ| r -r √ n( Xn -µ) E[(X-µ) r-1 sgn(X-µ) r ]+o P (1).
Proof of Propostion 7. The proof can be extended from the proof of Proposition 9 in the iid case in [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF]. We distinguish again three different cases for r: Even integers r, r = 1, and odd integers r > 1.

Whereas the first two cases (r even and r = 1) hold true without modifications from the iid case, for odd integers r > 1, we point out the three differences to the proof in the iid case. First, √ n( Xn -µ) v P → 0, for v ≥ 2, follows from the stationarity and ergodicity of the process. Second, we use the ergodicity instead of the law of large numbers. Third, we use Lemma 6 instead of its counterpart in the iid case, Lemma 8 in [START_REF] Br Äutigam | On the dependence between functions of quantile and dispersion estimators[END_REF].

Proof of Theorem 2. The proof consists of four steps. We first show that the process (X t ) fulfils the conditions required for having a Bahadur representation of the sample quantile, second, that a similar representation holds for the r-th absolute centred sample moment, third, the construction of a so called ∆-dependent approximation, which we then use in the fourth step for a multivariate FCLT.

Step 1: Bahadur representation of the sample quantile -conditions. The Bahadur representation of the sample quantile is well known, see e.g. Theorem 1 of [START_REF] Wendler | Bahadur representation for u-quantiles of dependent data[END_REF]. It holds under some conditions that we need to verify. For the ease of comparison, we adapt some of the notation of Theorem 1, [START_REF] Wendler | Bahadur representation for u-quantiles of dependent data[END_REF]. We have the following:

-Choosing the bivariate function g(x, t) := 1I (x≤t) , the non-negativity, boundedness, measurability, and non-decreasingness in the second variable, are straightforward. The function g also satisfies the variation condition uniformly in some neighbourhood of q X (p) if it is Lipschitz-continuous (see Example 1.5 in [START_REF] Wendler | Bahadur representation for u-quantiles of dependent data[END_REF]). But the latter follows from condition (C 2 ). -The differentiability of E[g(X, t)] = F X (t) and positivity of its derivative at t = q X (p) are given by condition (P ) at q X (p). -Equation ( 12) in [START_REF] Wendler | Bahadur representation for u-quantiles of dependent data[END_REF] is fulfilled as, by our assumption (C 2 ), F X is twice differentiable in q X (p) (see Remark 2, [START_REF] Wendler | Bahadur representation for u-quantiles of dependent data[END_REF]). -The stationarity of the process follows from assumption (P max(1,r/δ) ) or (L r ), respectively, and Lemma 1 of [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF]. -Lastly, let us verify that the process (X t ) is L 1 -NED with polynomial rate. Denoting, for s ≤ t, the sigma-algebra F t s = σ( s , ..., t ), we can write for any integer ∆ ≥ 1

X t -E[X t |F t+∆ t-∆ ] 2 2 = σ t -E[σ t |F t+∆ t-∆ ] 2 2 E[ 2 t ]. But E[ 2 t ] < ∞ since (M r ) holds. Notice that the property of being geometrically L 2 -NED, E[σ t |F t+∆ t-∆ ] 2 = O(e -κ∆
) for some κ > 0, implies L 1 -NED with polynomial rate, as

σ t -E[σ t |F t+∆ t-∆ ] 1 ≤ σ t -E[σ t |F t+∆ t-∆ ] 2 = O(e -κ∆ ) = O(∆ -(β+3) ), (7)
for some β > 3. So it suffices showing that σ t is geometrically L 2 -NED. For the polynomial GARCH, it follows from Corrollary 1 in [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF], which can be applied as (A 0 ) and (P 1 ) hold. For the logarithmic GARCH case, it follows from Corrollary 3 in [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF] as (A 0 ) and (L r ) hold.

Thus, we can use Theorem 1 of [START_REF] Wendler | Bahadur representation for u-quantiles of dependent data[END_REF] and write, a.s., as n → ∞,

q n (p) -q X (p) + F (q X (p)) -F n (q X (p)) f X (q X (p)) = o n -5 8 -1 8 γ (log n) 3 4 (log log n) 1 2 ) , (8)
for some γ ≥ 1/5.

Step 2: Representation of the r-th absolute centred sample moment -conditions. The representation being given in Proposition 7 under some conditions, we only need to check that we fulfil them.

-The stationarity of the process is satisfied under (P max(1,r/δ) ) or (L r ) as observed in Step 1.

-For the moment condition and ergodicity, we simply verify that the conditions for a CLT of X r t (or |X t | r ) are fulfilled, distinguishing between the polynomial and logarithmic case. Conditions (M r ), (A), (P max(1,r/δ) ) in the polynomial case, and (M r ), (A), (L r ) in the logarithmic case respectively, imply the CLT, using Corollary 2 and 3 in [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF], respectively.

Step 3: Conditions for applying the FCLT To use a multivariate FCLT, we need to verify the two conditions (H 1 ) and (H 2 ) given after Theorem 2.

In fact we adapt them to a three-dimensional version to simplify the computation, then apply the continuous mapping theorem to get back a two-dimensional representation. This will be made explicit in Step 4.

Condition (H 1 ) translates to first finding a representation u j such that

E[u j ] = 0, E[u 2 j ] < ∞, u j = f ( j , j-1 , ...), j ∈ Z, (9)
where f : R 3×∞ → R 3 is a measurable function and ( j , j ∈ Z) a sequence of R 3 -valued iid rv's with mean 0 and variance 1. Second, condition (H 2 ) comes back to finding a ∆-dependent approximation u (∆) j such that, for any integer ∆ ≥ 1, it holds, for j ∈ Z, u (∆) j = f (∆) ( j-∆ , ..., j , ..., j+∆ )

and

∆≥1 u 0 -u (∆) 0 2 < ∞, (11)
where f (∆) : R 3×(2∆+1) → R 3 is a measurable function.

Therefore, let us define, anticipating its use in Step 4 for the FCLT,

u j =    X j |X j | r -m(X, r) p-1 I (X j ≤q X (p)) f X (q X (p))    .
We need to verify that u j fulfils ( 9):

E[u j ] = 0 holds by construction. E[|X j | 2r ] < ∞ is guaranteed since |X t | r satisfies a CLT (see Step 2), thus also E[u 2 j ] < ∞.
Finally X j = f ( j , j-1 , ...) follows from Lemma 1 in [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF], as we assume (A). Hence, this latter relation also holds for functionals of X j , so for u j .

Then, we define a ∆-dependent approximation u (∆) 0 satisfying [START_REF] Carrasco | Mixing and moment properties of various garch and stochastic volatility models[END_REF] and [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF]. Denote, for the ease of notation, X 0∆ := E[X 0 |F +∆ -∆ ], and set u

(∆) 0 =    X 0∆ E[|X 0 | r |F +∆ -∆ ] -m(X, r) p-1 I (X 0∆ ≤q X (p)) f X (q X (p))    with F t s = σ( s , ..., t )
for s ≤ t. Thus, ( 10) is fulfilled by construction. Let us verify [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF]. We can write

∆≥1 u 0 -u (∆) 0 2 = ∆≥1 E (X 0 -X 0∆ ) 2 + |X 0 | r -E[|X 0 | r |F +∆ -∆ ] 2 + 1 f 2 X (q X (p)) -1I (X 0 ≤q X (p)) + 1I (X 0∆ ≤q X (p)) 2 1/2 ≤ ∆≥1 X 0 -X 0∆ 2 + |X 0 | r -E[|X 0 | r |F +∆ -∆ ] 2 + 1 f X (q X (p)) -1I (X 0 ≤q X (p)) + 1I (X 0∆ ≤q X (p)) 2 . (12)
Noticing that

|-1I (X 0 ≤q X (p)) + 1I (X 0∆ ≤q X (p)) | = 1I (X 0∆ >q X (p)≥X 0 ) + 1I (X 0 >q X (p)≥X 0∆ ) ≤ 1I (X 0∆ -X 0 >0) 1I (q X (p)-X 0 ≥0) + 1I (0>q X (p)-X 0 ) 1I (0>X 0∆ -X 0 ) = 1I (X 0 ≤q X (p)) (1I (X 0∆ -X 0 >0) -1I (0>X 0∆ -X 0 ) ) + 1I (0>X 0∆ -X 0 ) ≤ 1I (X 0 ≤q X (p)) 1I (|X 0∆ -X 0 |>0) + 1I (|X 0∆ -X 0 |>0) ≤ 21I (|X 0∆ -X 0 |>0) , (13)
we deduce that

-1I (X 0 ≤q X (p)) + 1I (X 0∆ ≤q X (p)) 2 ≤ E[21I (|X 0∆ -X 0 |>0) ] = √ 2 P(|X 0∆ -X 0 | > 0).
But, by the Markov inequality and Jensen's inequality, we have, for any > 0,

P(|X 0∆ -X 0 | ≥ ) ≤ X 0∆ -X 0 1 ≤ X 0∆ -X 0 2 . (14)
Thus, we can reduce [START_REF] Duan | Augmented garch (p, q) process and its diffusion limit[END_REF] to

∆≥1   u 0 -u (∆) 0 2 ≤ ∆≥1 X 0 -X 0∆ 2 + |X 0 | r -E[|X 0 | r |F +∆ -∆ ] 2 + √ 2 f X (q X (p)) P(|X 0∆ -X 0 | > 0)   .
(15) Assuming that X 0 is geometrically L 2 -NED, i.e. X 0∆ -X 0 2 = O(e -κ∆ ) for some κ > 0, implies by ( 14) that ∆≥1 P(|X 0∆ -X 0 | ≥ ) < ∞ holds for any > 0. Hence, we will have shown that

∆≥1 P (|X 0∆ -X 0 | > 0) < ∞, reducing the finiteness of ∆≥1 -1I (X 0 ≤q X (p)) + 1I (X 0∆ ≤q X (p)) 2 to X 0 being geometrically L 2 -NED.
Therefore, a sufficient condition for the finiteness of ( 15) is the geometric L 2 -NED of X 0 and |X r 0 | respectively. This condition is satisfied, on the one hand in the polynomial case under (M r ), (A 0 ) and (P max(1,r/δ) ) via Corollary 2 in [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF], on the other hand under (M r ), (A 0 ) and (L r ) via Corollary 3 in [START_REF] Lee | Functional central limit theorems for augmented garch (p, q) and figarch processes[END_REF].

Step 4: Multivariate FCLT Finally, we can apply a trivariate FCLT for u j using Theorem A.1 from [START_REF] Aue | Break detection in the covariance structure of multivariate time series models[END_REF], as equations ( 9), [START_REF] Carrasco | Mixing and moment properties of various garch and stochastic volatility models[END_REF], [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF] are exactly the conditions needed in this theorem. Note that we adapted (10) from originally being u (∆) j = f (∆) ( j , ..., j-∆ ). Indeed, it is straightforward to show that the proof of [START_REF] Aue | Break detection in the covariance structure of multivariate time series models[END_REF] still holds with this modification.

Using the Bahadur representation (8) of the sample quantile (ignoring the rest term for the moment), we can state:

√ n 1 n [nt] j=1 u j = √ n t    X[nt] 1 [nt] [nt] j=1 |X j | r -m(X, r) p-F [nt] (q X (p)) f X (q X (p))    D 3 [0,1] → W Γ(r) (t) as n → ∞, (16)
where W Γ(r) (t), t ∈ [0, 1] is the 3-dimensional Brownian motion with covariance matrix Γ(r) ∈ R 3×3 , i.e. the components Γ(r) ij , 1 ≤ i, j ≤ 3, satisfy the dependence structure (D), with all series being absolutely convergent. By the multivariate Slutsky theorem, we can add

√ n t    X[nt] -µ 1 [nt] [nt] j=1 |X j | r -m(X, r) p-F [nt] (q X (p)) f X (q X (p))   + √ n t   0 0 R [nt],p   = √ n t    X[nt] -µ 1 [nt] [nt] j=1 |X j | r -m(X, r) q [nt] (p) -q X (p)    D 3 [0,1]
→ W Γ(r) (t). [START_REF] Higgins | A class of nonlinear arch models[END_REF] Recalling the representation of m(X, n, r) in Proposition 7, we apply to [START_REF] Higgins | A class of nonlinear arch models[END_REF] the multivariate continuous mapping theorem using the function f (x, y, z) → (ax + y + b, z) with a = -r E[(X -µ) r-1 sgn(Xµ) r ] and b = o P (1/ √ n), and obtain

√ n t a( X[nt] -µ) + 1 [nt] [nt] j=1 |X j | r -m(X, r) + b q [nt] (p) -q X (p) = √ n t m(X, [nt], r) -m(X, r) q [nt] (p) -q X (p) D 2 [0,1]
→ W Γ (r) (t).

(18) where Γ (r) follows from the specifications of Γ(r) above and the continuous mapping theorem.

Augmented GARCH (p,q) APGARCH (p,q) VGARCH (p,q) NGARCH (p,q) EGARCH (p,q) MGARCH (p,q) AGARCH (p,q) TGARCH (p,q) PGARCH (p,q) GARCH (p,q) ARCH (p) TSGARCH (p,q) GJR-GARCH (p,q) Exponential GARCH models Polynomial GARCH models

  ) without changing the resulting distribution (as√ nR [nt],p p → 0). Hence, as n → ∞,

Figure 1 :

 1 Figure1: Schematic overview of the nesting of some augmented GARCH(p, q) models.

Table 3 :

 3 Conditions (P max (1,r/δ) ) or (L r ) respectively translated for different augmented GARCH(p,q) models for the general r-th absolute centred sample moment, r ∈ N.

2 0E[(α j 2 0 1 VGARCH q j=1 β j < 1 NGARCH

 2211 augmented GARCH (p, q) r ∈ N APGARCH max(p,q) j=1 E[|α j (| 0 | -γ j t-j ) 2δ + β j | r ] 1/r < 1 AGARCH max(p,q) j=1 E[|(α j | 0 | -γ j t-j ) 2 + β j | r ] 1/r < 1 GJR-GARCH max(p,q) j=1 E[|α * j + β j + γ * j max(0, -2 0 )| r ] 1/r < 1 GARCH max(p,q) j=1 + β j ) r ] 1/r < 1 ARCH max(p,q) j=1 α j E[ 2r 0 ] 1/r < 1 TGARCH max(p,q) j=1 E[|α j | t-j | -α j γ j t-j + β j | r ] 1/r < 1 TSGARCH max(p,q) j=1 E[|α j | t-j | + β j | r ] 1/r < 1 PGARCH max(p,q) j=1 E[|α j | 0 | + β j | 2r ] 1/(2r) < max(p,q) j=1 E[|α j ( 0 + γ j ) 2 + β j | r ] 1/r < 1 MGARCH E[exp(4r p i=1 |ω/p + α i log( 2 0 )| 2 )] < ∞ and q j=1 |β j | < 1 EGARCH E[exp(4r p i=1 |ω/p + α i (| 0 | -E| 0 |) + γ i 0 | 2 ] < ∞ and q j=1 |β j | < 1

Table 1 :

 1 Presentation of the volatility equation (3) and the corresponding specifications of functions g i , c j for selected augmented GARCH models.

	standard formula for Λ(σ 2 t )	corresponding specifications of gi, cj in (3)
	Polynomial GARCH
	APGARCH family

Appendix A Different Augmented GARCH models

As mentioned in the paper, we give an overview over the acronyms, authors and relation to each other of the augmented GARCH processes used. The restrictions on the parameters, if not specified differently, are ω ≥ 0, α i ≥ 0, -1 ≤ γ i ≤ 1, β j ≥ 0 for i = 1, ..., p, j = 1, ..., q.

• APGARCH: Asymmetric power GARCH, introduced by Ding et al. in [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF]. One of the most general polynomial GARCH models.

• AGARCH: Asymmetric GARCH, defined also by Ding et al. in [START_REF] Ding | A long memory property of stock market returns and a new model[END_REF], choosing δ = 1 in APGARCH.

• GJR-GARCH: This process is named after its three authors Glosten, Jaganathan and Runkle and was defined by them in [START_REF] Glosten | On the relation between the expected value and the volatility of the nominal excess return on stocks[END_REF]. For the parameters

gives back the well-known GARCH(p, q) process by Bollerslev in [START_REF] Bollerslev | Generalized autoregressive conditional heteroskedasticity[END_REF].

• ARCH: Introduced by Engle in [START_REF] Engle | Autoregressive conditional heteroscedasticity with estimates of the variance of united kingdom inflation[END_REF]. We recover it by setting all γ i = β j = 0, ∀i, j.

• TGARCH: Choosing δ = 1/2 in the APGARCH model leads us the so called threshold GARCH (TGARCH) by Zakoian in [START_REF] Zakoian | Threshold heteroskedastic models[END_REF]. For the parameters

• TSGARCH: Choosing γ i = 0 in the TGARCH model we get, as a subcase, the TSGARCH model, named after its authors, i.e. Taylor, [START_REF] Taylor | Modelling financial time series[END_REF], and Schwert, [START_REF] Schwert | Why does stock market volatility change over time[END_REF].

• PGARCH: Another subfamily of the APGARCH processes is the Power-GARCH (PGARCH), also called sometimes NGARCH (i.e. non-linear GARCH) due to Higgins and Bera in [START_REF] Higgins | A class of nonlinear arch models[END_REF].

• VGARCH: The volatility GARCH (VGARCH) model by Engle and Ng in [START_REF] Engle | Measuring and testing the impact of news on volatility[END_REF] is also a polynomial GARCH model but is not part of the APGARCH family.

• NGARCH: This non-linear asymmetric model is due to Engle and Ng in [START_REF] Engle | Measuring and testing the impact of news on volatility[END_REF], and sometimes also called NAGARCH.

• MGARCH: This model is called multiplicative or logarithmic GARCH and goes back to independent suggestions, in slightly different formulations, of Geweke in [START_REF] Geweke | Modeling the persistence of conditional variances: a comment[END_REF], Pantula in [START_REF] Pantula | Modeling the persistence of conditional variances: a comment[END_REF] and Milhøj in [START_REF] Milhøj | A multiplicative parameterization of arch models[END_REF].

• EGARCH: This model is called exponential GARCH, introduced by Nelson in [START_REF] Nelson | Conditional heteroskedasticity in asset returns: A new approach[END_REF].

Then we give a schematic overview of the nesting of the different models in Figure 1.

Lastly, we present in Table 3 how the conditions (P max (1,r/δ) ) or (L r ) respectively translate for those augmented GARCH(p,q) processes -this is the generalization of Table 2. As, in contrast to Table 2, we do not gain any insight by considering the choices of r = 1 or r = 2, we only present the general case, r ∈ N.

When p = q we need to consider coefficients α j , β j , γ j for j = 1, ..., max (p, q). In case they are not defined, we set them equal to 0.

C O N T A C T

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