Research & Reviews: Journal of Statistics and Mathematical Sciences

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1 Research & Reviews: Journal of Saisics and Mahemaical Sciences Forecas and Backesing of VAR Models in Crude Oil Marke Yue-Xian Li *, Jin-Guo Lian 2 and Hong-Kun Zhang 2 Deparmen of Mahemaics and Saisics, Inner Mongolia Agriculural Universiy, Hohho Ciy, Inner Mongolia Auonomous Region, PR China 2 Deparmen of Mahemaics and Saisics, Universiy of Massachuses Amhers, Amhers MA 3, USA Research Aricle Received dae: /3/26 Acceped dae: 4/5/26 Published dae: 8/5/26 *For Correspondence Deparmen of Mahemaics and Saisics, Inner Mongolia Agriculural Universiy, Hohho Ciy, Inner Mongolia Auonomous Region, PR China li_yuexian@63.com ABSTRACT The oil price has a very imporan effec on he world economy. In his paper, using daa ses of Europe Bren and Wes Texas Inermediae (WTI) Cushing crude oil daily prices from Jan. 4, 2 o Jan. 4, 26, he VaR forecasing performance of GARCH-ype models are analyzed and compared in a shor horizon. Based on he Kupiecs POF-es and Chrisoffersens inerval forecas es, as well as a Back esing VaR Loss Funcion, he empirical resuls indicae ha, for Europe Bren crude oil, EGARCH (,) has he bes performance; while for WTI, APARCH (,) and GJR-GARCH (,) ouperform oher GARCH models. In fac, hese resuls also give significan guidance on how o choose a beer risk managemen model for he cerain commodiy of differen companies even in he same ime period. Keywords: Risk Merics, Value-a-risk, GARCH-class models, Forecasing, Backesing INTRODUCTION Producs of crude oil have been used in many indusries, he volailiy of crude oil can cause a huge effec on he world economy. From 2 unil mid-24, he world oil prices had been fairly sable, a around dollars a barrel. Bu global oil prices fell sharply aferward, and more han halved by winer of 25. This leads o significan revenue shorfalls in many energy exporing naions, while consumers in many imporing counries are benefied for home heaing and he vehicle gas. Large price drops also cause a rise in he volailiy/risk of oil marke. Therefore, crude oil risk esimaion and measuremen are crucial for consumers, corporaions, governmens and inernal risk conrol. The mehods o forecas he oil price and measure is risk are popular opics. The mos commonly used measuremen for he risk esimaion is he Value-a-risk (VaR for shor), which measures he maximum loss of a porfolio value over a cerain ime period a a given level. Idenifying proper GARCH-ype models wih appropriae disribuions o evaluae VaR of oil price has become one of mos imporan goals for risk measuremen in he crude oil marke. Fan e al. [] esimaed VaR of crude oil price using GARCH models, based on he Generalized Error Disribuion (GED) and deeced exreme risk spillover effec beween he wo oil markes. Huang e al. [2] employed CAViaR model o forecas oil price risk. Hung e al. [3] invesigaed he influence of faailed process on he performance of one-day-ahead VaR esimaes abou energy commodiies using hree GARCH models. Wei e al. [4] used several GARCH class models, o capure he volailiy of crude oil markes. Marimouou e al. [5] modeled VaR in he oil marke by applying boh EVT models o forecas VaR. Aloui [6] compued he VaR using FIGARCH, FIAPARCH and HYGARCH. Youssef e al. [7] evaluaed VaR and expeced shor-fall (ES) using he fied long-memory GARCH-model, and EVT was used as a poenial framework for he separae reamen of ails of disribuions. In order o improve he measure for VaR, an invesor needs o esimae he volailiy of crude oil price, i.e., risk. Empirical RRJSMS Volume 2 Issue June, 26 3

2 sudies have concluded ha financial insrumens have heeroscedasiciy in he variance. To address his observaion, he milesones are he ARCH and GARCH, which were inroduced by Engle [8] and Bollerslev [9]. Originaed from ARCH and GARCH, many new varieies of GARCH models have emerged, which capure he changing volailiy over ime due o differen facors. However here are no definie answers o which of he models from he GARCH family ha is he bes a forecasing he volailiy for all ypes of financial daa. Due o he plehora of differen GARCH models available, he models ha have been examined need o be resriced o specific daa ses. This paper focuses on four of he mos influenial models, including GARCH (, ), EGARCH (,), GJR- GARCH (,), APARCH (,). For deailed consrucions, see Bollerslev [9], Nelson [], Glosen e al. [] and Ding e al. [2], ec. The purpose of his paper is o beer esimae and forecas he risk of he wo crude oil markes - Europe Bren and Cushing, OK WTI. Firs of all, by Q-Q plo, we conclude ha, in boh markes, he Suden- disribuion fis he log reurns significanly (Figure ). Consequenly, we use he Suden- disribuion as he preferred condiional disribuion for GARCH models in his paper. Secondly, we mainly use Risk Merics, GARCH (,), EGARCH (,), GJR-GARCH (,) and APARCH (,), o sudy volailiy and is corresponding VaR of crude oil, over six years ime period. Since he performance of a VaR model is deermined by how good i predics fuure risks. More precisely, for a good VaR model, is esimaes of profis and losses should fi he acual profis and losses in some given confiden level. However, backesing wih uncondiional coverage [3] mainly esimae he number of excepions, bu hardly avoiding he clusering. The condiional coverage by Chrisoffersen [4] and Haas [5] aims o overcome he clusering by esimaing he number of excepions and he ime when hey occur, bu i canno cach he long dependence of VaR violaions. The duraion-based ess of independence (by Chrisoffersen and Pelleier [6], based on he duraion of days beween he violaions of he VaR, overcomes he clusering and he long dependence of VaR violaions. However i relies on esimaing of a few parameers. Insead of esimaing he violaions of he VaR, he mehod of VaR loss funcion examines he magniude of VaR violaions. Thus is accuracy relies on he condiional disribuion. This paper uses all of hese backesing ools o compare he performances of hese models. We conclude ha, for Europe Bren, he EGARCH (,) ouperforms all he oher models; while boh APARCH and GJR-GARCH specificaions are good opions for forecasing he VaR for he WTI. I is ineresing o noe ha for boh crude oil markes, he wors performing model is he Risk Merics, which showed no significan resuls, alhough i is indeed sill popular in many financial insiues. Figure. The upper wo plos he spo prices for Europe Bren and WTI; he lower wo plos are he associaed daily reurns. 32

3 The res of his paper is organized as follows. Secion 2 inroduces he sample daa and he saisical characerisics. Secion 3 discusses he ve GARCH-ype models used in his paper. Secion 4 presens he forecasing mehodology, he in-sample model and he ou-sample VaR forecasing. Secion 5 shows backesing Value-a-Risk model. Secion 6 conains concluding remarks. DATA AND DESCRIPTIVE STATISTICS In his paper, we use he daily price daa (in US dollars per barrel) of Bren and Wes Texas Inermediae (WTI) from Jan. 4, 2 o Jan. 4, 26. The daa is divided ino a en year in-sample period and a six year ou-of-sample period. The in-sample period is from Jan. 4, 2 o Jan.3, 2 and he res daa are used for ou-of-sample forecas and backesing. Le p be he spo daily price, we consider he log reurn ime series, r, defined by r = (log p log p () We firs examine empirical disribuion of he reurn series by he Q-Q plo. The Q-Q plo of he empirical disribuion of he daily reurns agains he normal disribuion is given in Figure 2. I can be observed from he plo ha he empirical disribuions of boh daily reurns exhibi heavier ails han he normal disribuion. We also perform he Q-Q plo agains he suden -disribuion, which demonsraes ha he empirical disribuion of he daily reurns fis he (5)-disribuion much beer. The unusually high value of he Jarque-Bera saisics in Table shows ha he null hypohesis of normaliy is rejeced a he % level of significance, also as evidenced by a high excess kurosis and negaive skewness. This is in line wih expecaions from he ocular inspecion of he Q-Q plos in Figure 2, which implied ha he empirical disribuion of boh daily reurns exhibi significanly heavier ails han he normal disribuion. Figure 2. Quanile-quanile plo of reurns agains he normal and he (5) disribuion, respecively. We also apply wo commonly used saisic ess-he Ljung-Box es by Ljung and Box [7] and Lagrange muliplier es [8], which can be applied o check serial correlaion of reurns and squared reurns. In Table, he Ljung-Box es resul rejecs he null hypohesis of no auocorrelaion up o he 2h order, and confirms serial auocorrelaion in boh crude oil reurns. ARCH LM es rejecs he null hypohesis ha here is no auo-correlaion for lags 2, a a % significance level; and hus confirms ha he squared reurns are also serially correlaed (Figures and 2). 33

4 Table. Descripive saisics for oil price reurns. Europe Bren Cushing, OK WTI The Sample Size Mean Range [-9.897,8.297] [-7.98,6.437] Sandard Deviaion Excess Kurosis Skewness JB for Jarque-Bera Tes Q(2) for Ljung-Box Tes LM(2) for ARCH LM Tes METHODOLOGY Le be all hisorical informaion (based on he ime series) up o ime. Le = ( - ) ( ) 2 σ = Var r he volailiy. In his paper, o simulae he condiional mean, he AR () model is used: u r be he condiional reurn; µ = φ + φ r, (2) - where φi, for i = ;. Nex we review various models for esimaing he volailiy σ. A widely used mehodology for measuring marke risk is he Risk Merics, which has become widely used in he financial indusry. The main ool is he exponenially weighed moving average (EWMA) mehod [9], which represens he finie memory of he marke. More precisely, he Risk Merics can be esimaed as: σ n 2 2 λ r n= n n 2 = = ( λ) λ - n λ n n= n= r (3) We ake λ= :94, as mos commonly used in he lieraure. The Risk Merics model compleely ignores he presence of fa ails in he disribuion funcion, and does no coun for he correlaions of he reurn series. In order o over-come hese weakness, we use he Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) model [9] : r = µ + σε = + p q σ α αη + j= j j βσ i= j i where p > ; q >, and α i, β j are consans, for i =,,p and j =,,q. Here {ε } is a whie noise wih zero mean and uni variance ha adaped o { }. The GARCH model is raher popular, as i accouns for persisence of financial ime-series daa. Bu i requires ha he parameers are no negaive, and he models assume ha posiive and negaive shocks have he same impac on volailiy. Moreover, i is well known ha financial asse volailiies have an asymmeric impac. Typically, he bad news has a greaer impac on volailiy. To be able o model his behavior and relax he limiaion of parameers, Nelson [] proposed he Exponenial GARCH (EGARCH) model. For p, q >, he EGARCH (p,q) model is given by ( E ) p p 2 = 2 + j j + j j j + i log i j= j= logσ α αη γ η η β σ An alernaive way of modeling he asymmeric effecs of posiive and negaive asse reurns was presened by Glosen, Jagannahan and Runkle [] resuled in he so called GJR-GARCH (p,q) model, which is given by ( ) p q σ = α + α γ + βσ j j j j i i j= i= The asymmeric power ARCH (APARCH) model of Ding e al. [2] is one of he mos promising ARCH-ype models, and has been sudied in many recen applicaions (see, for example, Gio and Lauren, [2] ; Minik and Paolella [2] ). The APARCH (,) model is defined as follows: ( ) p q σ = α + αj j γj j + βσ i i j= i= Alhough i is raher difficul o esimae he order (p; q), some sudies have found ha he predicive effec of he higher order model is no necessarily beer han he low order model, see Hansen PR, Lunde A [22] and Bollerslev T, Chou RY, Kroner KF [23]. Consequenly, we choose (p; q) = (; ) for various GARCH models in his paper. In addiion, we choose he suden (4) (5) (6) 34

5 (5)-disribuion for he error process ε_. According o our analysis for he empirical disribuion of he daily reurns, he suden (5)- disribuion ouperforms he normal disribuion. MODEL FITTING AND VAR ESTIMATION Despie is concepual simpliciy and populariy as an indusrial sandard in risk managemen, he esimaion of VaR is indeed highly non-rivial. Our goal is o provide a given quanile for he disribuion of relaive reurns of he crude oil. The quaniy VaR α is defined as he α-quanile of he disribuion of he log reurn, wih α chosen as eiher 95% or 99%: ( ) r > VaR α = α (7) According o he definiion r = µ + σε, and he assumpion ha ε follows he suden (5)-disribuion; we know ha he α-h quanile of r can be calculaed as VaR α = µ σu (8) a where u a denoes he α-h quanile of he suden (5)-disribuion. According o he above formula, once we have an esimaion for he volailiy and he expeced reurn, he value of VaR can be obained direcly. We divide he daa {r, =,, T} ino wo subses. The model parameers are fied using daa in {r, =,, n} (esimaion subsample). On he oher hand, he forecas of he model is evaluaed using daa in {r, = n+,, T} (forecasing subsample), where n is he iniial forecas origin. We are ineresed in he -sep ahead forecas, using a so-called recursive scheme. More precisely, one ses m = n o be he iniial forecas origin and hen fis each of he models using he daa r, r 2,, r m. The -sep ahead forecass can now be calculaed following he so called fixed scheme. Each model will be fied o he daa unil he iniial forecas origin from which he forecass can be compued. Below, we lis he forecas formula for our models a forecas origin k, he -sep ahead forecas: () Risk Merics: ˆ σk+ =.6 k +.94σk =.6rk +.94σk (2) GARCH (,): ˆ σ 2 = α + α 2 + β σ 2 = α + α ( r φ φ r ) 2 + β σ 2 k+ k k k k k (3) EGARCH (,): ( E ) log ˆ σ = α + αε + γ ε ε + β logσ 2 2 (4) GJR-GARCH (,): ˆ σ = α + [ α + γ I ( ε < ) ] ε + β logσ k+ k (5) APARCH (,): ( ) ˆ k + = + + k σ α α γ βσ As previously analyzed, in his paper, we use he sandardized (5)-disribuion, so 2 v 2 Γ (( v+ ) 2) 2 3 Γ(3) E ε ) = =, ( v+ ) Γ( v 2) π 4 Γ(5 2) π (9) where v = 5 denoes he number of degrees of freedom and Γ denoes he gamma funcion. In Table 2 and Table 3, log (L) is he logarihm maximum likelihood funcion value; AIC is he average Akaike informaion crierion; Q is he Ljung-Box Q-saisic compued on he sandardized residuals. Order of he saisics are repored in brackes. From he p-values of he saisics, he null hypohesis of no auocorrelaion is acceped and confirms residual serial no auocorrelaion a he 5% levels of significance. Table 2. Esimaion resuls of differen volailiy models for Europe Bren crude oil. Model GARCH EGARCH GJRGARCH APARCH φ φ α α β γ

6 log(l) AIC Q ().764(5).8464(5).7994(5) p-value Model GARCH EGARCH GJRGARCH APARCH φ φ α α β γ Table 3. Esimaion resuls of differen volailiy models for cushing, OK WTI crude oil..548 log(l) AIC Q.643().5769(5).2546(5).344(5) p-value BACKTESTING VALUE-AT-RISK MODEL In order o help us evaluae he qualiy of he VaR esimaes, he models should be backesed wih appropriae mehods. Backesing is o es he accuracy of he model measuremen by comparing he acual losses and VaR predicive resuls. Uncondiional coverage A popular model o esimae he VaR of financial series is o calculae he number of VaR excepions, namely days when acual losses exceed VaR predicive resuls. If he raio of excepions is lower han he seleced confidence level means ha he risk is overesimaed. On he oher hand, oo many excepions implies he underesimaion of risk. Indeed he exac excepion suggesed by he confidence level is rarely observed. Therefore a saisical analysis is necessary o sudy wheher excepions are reasonable or no, namely o accep or rejec model. Le x be he number of excepions and T he oal number of observaions, hence he failure rae is x=t. In ideal siuaion, failure rae would be equal o he seleced confidence level (Figure 3). If a confidence level is a and le p = α, number of excepions x obeys a binomial disribuion wih probabiliy: Figure 3. One-day-ahead VaR forecass of Europe Bren crude oil based on he risk merics and GJR-GARCH models (upper plo), and he GARCH, he EGARCH and APARCH models (lower plo), and he hisorical volailiy. 36

7 f(x) = C p ( p) () x x T T The accuracy of he VaR model is evaluaed hrough uilizing his binomial disribuion. We firs use he es suggesed by Kupiec [3], which measures wheher he number of excepions is consisen wih he confidence level (Figure 4). The null hypohesis for he Kupiec's es is Figure 4. One-day-ahead VaR forecass of Cushing, OK WTI crude oil based on he risk merics and GJR-GARCH models (upper plo), and he GARCH, he EGARCH and APARCH models (lower plo), and he hisorical volailiy. : x H p = () T The Kupiec's es saisic is a likelihood-raio: T x x ( p) p LRuc = 2 ln T x x x x (2) T T Under he null hypohesis, LR uc asympoically follows c 2 disribuions wih one degree of freedom. If he value of LR uc is greaer han he criical value of 3.84, he null hypohesis will be rejeced. Kupiec's es of uncondiional coverage is a well-known example of VaR backes. However, alhough his es provides a useful benchmark for assessing he accuracy of a given VaR model, his es is hampered by wo shorcomings. The firs is ha his es exhibis low power in sample sizes consisen wih he curren regulaory framework, i.e., one year. The second shorcoming is ha i focuses exclusively on he uncondiional coverage propery of an adequae VaR measure. Condiional coverage Theoreically, we no only focus on he number of excepions, bu also would expec VaR violaions o be independen over ime. VaR users wan o deec clusering of excepions, because rapid coninuous losses han individual excepions are more likely o lead o caas-rophic evens. The mos well-known es of condiional coverage has been proposed by Chrisoffersen [4]. The Chrisoffersens inerval forecas es firs de ne an indicaor variable: if violaion occurs; I = oherwise. hen define n ij, I, j =,, as he number of days when condiion j occurred, on he premise of condiion I occurred on he previous day. In addiion, define π i as he probabiliy: n n, n + π = π n = and π = (3) n + n n + n n + n + n + n 37

8 Under he null hypohesis: =, he es is conduced as a likelihood-raio (LR) es wih he saisic: ( π) π = 2ln ( ) ( ) n + n n + n LR ind π n π n n π π By combining LR uc and LR ind, a join es is obained, i.e., condiional coverage: LR cc = LR uc + LR ind (4) (5) LR cc asympoically obeys c 2 disribuions wih wo degree of freedom. Duraion-based ess of independence The above ess are efficien a caching wheher he probabiliy of an excepion on any day depends on he oucome of he previous day. However we are ineresed in developing ess which have power agains more general forms of dependence bu which sill rely on esimaing only a few parameers. The duraion of ime beween VaR violaions (no-his) should ideally be independen and no clusering. Under he null hypohesis of a correc VaR model, he duraion of ime beween VaR violaions should have no memory. Because he only memoryless coninuous disribuion is he exponenial disribuion, any disribuion which embeds he exponenial as a resriced case can be esed. The es can be conduced as a likelihood-raio (LR) es o see wheher he resricion holds. Chrisoffersen and Pelleier [6] use he Weibull disribuion which presens he case of he exponenial ail disribuion. Loss funcion based backess which For given α, he loss funcion Q for he VaR α was firsly defined by Gonzalez-Rivera, Lee and Mishra [24]. More precisely, T α α Q = ( α I )(r VaR ) (6) T n = n+ α α where I I( r VaR ) = <. This is an asymmeric loss funcion ha penalizes more heavily wih weigh he observaions for < VaR α. Smaller Q indicaes a beer goodness of. r A 95% confidence levels, resuls of he back ess are shown in Table 4 for Europe Bren crude oil. The uncondiional coverage es criical value is ; and he condiional coverage es criical value is According o he resuls, Risk Merics performs he wors, since for boh ess, he criical values exceeded wih a raher large margin. All GARCH-class models pass boh LR uc and LR cc ess, wih EGARCH model having he bes performance. Based on he VaR-based loss funcion Q, he EGARCH model clearly dominaes all he oher models [25]. Table 4. Back esing value-a-risk model for Europe Bren crude oil. Model Risk Merics GARCH EGARCH GJR-GARCH APARCH Number of observaions Number of exceedance LR uc Tes oucome Rejec Accep Accep Accep Accep LR cc Tes oucome Rejec Accep Accep Accep Accep b Tes oucome Accep Accep Accep Accep Accep VaRloss(Q) For he WTI crude oil, es resuls are shown in Table 5 wih 95% confidence. Again, Risk Merics performs he wors. All GARCH-class models passed he LR uc es, while only GJR-GARCH and APARCH passed LR cc es. Our sudy shows ha GJR-GARCH model has he bes performance for he WTI daa, wih a minimum value for he LR uc and he LR cc. According o he VaR-based loss funcions Q, he APARCH model ouperforms. I is ineresing o noe ha for boh crude oil markes, he wors performer is he Risk Merics mehod, which is indeed very popular in financial insiue as i was firs proposed by he JP Morgan Risk Merics Group [9]. Table 5. Back esing value-a-risk model for cushing, OK WTI crude oil. Model Risk Merics GARCH EGARCH GJR-GARCH APARCH Number of observaions Number of exceedance LR uc Tes oucome Rejec Accep Accep Accep Accep LR cc Tes oucome Rejec Rejec Rejec Accep Accep 38

9 b Tes oucome Accep Accep Accep Accep Accep VaRloss(Q) CONCLUSION In his paper we apply four differen GARCH-VaR models wih suden- disribuion o forecas he condiional variance and is corresponding VaR. The Backesing indicaes ha for Europe Bren crude oil, EGARCH (, ) model wih suden- disribuion has he smalles VaR loss, so i will forecas he fuure VaR beer han oher models. While for OK WTI crude oil, GJR-GARCH (,) and APARCH model under suden- disribuion ouperform oher models. Furhermore, some of resuls are very useful for companies o choose an appropriae risk managemen model, which are summarized as he following: () Compared o GARCH model, EGARCH, GJR-GARCH and APARCH are more sensiive for cach asymmeric informaion. (2) These resuls indicae ha even for he same commodiy (oil), even hough we ake daa ses in same ime inerval, he commodiy of differen counry/companies may have a differen appropriae model o predic he fuure VaR. (3) We can amplify he conclusion of (2): even for he same commodiy of same counry/companies, in differen ime periods, an appropriae model o predic is fuure VaR may vary oo. These conclusions give significan guidance for companies o choose a beer risk managemen model based on he saisical properies of he ime series, in a cerain ime period. The above analysis indicaes ha, whenever we wan o forecas he value a risk for a commodiy of a company in a shor horizon, i is always beer o compare all of models o choose an appropriae one, as here is hardly any model ha fis a commodiy forever. ACKNOWLEDGEMENTS HK Zhang is suppored in par by NSF gran DMS-5762, as well as he Simons Fellow-ship. REFERENCES. Ying Fan e al. Esimaing value a risk of crude oil price and is spillover effec using he GED-GARCH approach, Energ Econ. 28;3: Huang D e al. CAViaR -based forecas for oil price risk. Energ Econ. 29;3: Hung JC e al. Esimaion of value-a-risk for energy commodiies via fa-ailed GARCH models, Energ Econ. 28;3: Yu Wei e al. Fore-casing crude oil marke volailiy: Furher evidence using GARCH-class models, Energ Econ. 2;32: Marimouou V e al. Exreme value heory and value a risk: Applicaions o oil marke. Energ Econ. 29;3: Aloui C and Mabrouk S. Value-a-risk esimaions of energy commodiies via long-memory, asymmery and fa-ailed GARCH models. Energy Policy. 2;38: Youssef M e al. Value-a-risk esimaion of energy commodiies: A long-memory GARCH? EVT approach. Energ Econ. 25;5:99-8. Engle RF. Auoregressive condiional heeroskedasiciy wih esimaes of he variance of Unied Kingdom in aion. Economerica. 982;5: Bollerslev T. Generalized auoregressive condiional heeroskedasiciy. J Econome, 986;3: Nelson DB. Condiional heeroskedasiciy in asse reurns: A new approach. Economerica. 99; 59: Glosen LR e al. On he relaion beween he expeced value and he volailiy of he nominal excess reurn on socks. J Finance. 993;48: Ding Z e al. A long memory propery of sock marke reurns and a new model. J Empirical Finance. 993:; Kupiec P. Techniques for verifying he accuracy of risk managemen models. J Derivaives 995;3: Chrisofferssen P. Evaluaing inerval forecass. In Econ Rev. 998;39: Haas M. New mehods in back esing, Financial Engineering, Research Cener, Caesar, Bonn Chrisoffersen P and Pelleier D. value-a-risk: A duraion-based approach. J Financ Economer. 24;2: Ljung GM and Box GEP. On a measure of a lack of fi in ime series models. Biomerika. 978;65: Rober FE. A general approach o lagrange muliplier model diagnosics. J Economerics. 982;2: Risk Merics Group. Risk merics -Technical documen, New York: JP. Morgan/Reuers,

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