MODELING AND FORECASTING REALIZED VOLATILITY

observations and modellingObservations and modelling are two crucial elements of scientific research. They work in concert to help us better understand and explain natural phenomena. Observations are the act of gathering data through experiments or observations of natural events. Modelling, on the other hand, is the process of creating simplified representations of those observations to gain a better understanding of how the natural world works. This document will examine the importance of both observations and modelling and how they complement each other for scientific research.Observations are integral to science because they are the foundation of empirical evidence. In order to make any conclusions about the natural world, we first need to gather evidence of what is happening. This evidence can come from experiments that scientists carefully design, or it can come from simply observing what nature is already doing. Without observations, science would be groping in the dark without any real basis for understandingthe world. Observations allow us to construct hypotheses, which are informed guesses about how natural events take place. These hypotheses, in turn, can be used to make predictions about future outcomes or behavior.However, data gathered through observations can sometimes be distorted or limited by our human limitations. For example, our senses are not always capable of picking up on subtle phenomena. Additionally, we may only have access to a limited amount of data to make our observations. These limitations can have a direct effect on the kindsof conclusions we can draw from our data. In order to overcome these limitations, scientific researchers often turn towards modelling.Modelling is a powerful tool that complements observations in scientific research. Models can be created to represent what we know or believe about the natural world. They allow us to makepredictions about complex systems by simplifying those systems or reducing them down to their most important elements. Modelling can be used to testhow well our observations support our hypotheses, to explore the implications of different theories, and to identify the best methods for further study. Modelling can even help us discover new phenomena or understand existing ones better by revealing relationships between variables that we may not have previously noticed.One of the most significant benefits of modelling is its ability to predict the behavior of complex systems. Complex systems are composed of many different elements that interact with each other in intricate and often unpredictable ways. For example, the weather is a complex system that can be modelled to predict future weather patterns, even when the data available to us is changing rapidly. Modelling allows us to make predictions that can help us prepare for natural disasters or optimize resource management strategies. It can also be used to explore hypothetical scenarios or test the implications of different policies or strategies.Despite the many benefits of modelling, it is important to remember that models are not perfect representations of the natural world. Models are created based on what we currently know, and that knowledge can be limited. Additionally, models can become inaccurate when new data becomes available or when we encounter unexpected phenomena. Therefore, it is essential to take the results of modelling with a grain of salt and remain open to the possibility of new evidence that may contradict our models.In conclusion, observations and modelling work together to help us better understand and explain natural phenomena. Observations provide us with the foundation of empirical evidence, while modelling allows us to simplify complicated systems so that we can make predictions and discover new relationships. Both observations and modelling are essential to scientific research and help us to make informed decisions about our world. By using both techniques, we can continue to make breakthroughs in our understanding of the naturalworld and use that knowledge to make a positive impact on the world around us.。

Time-Series Forecasting Model:Including Case Studies With RealElectrical LoadsSarma,U.K.∗AbstractThe increase of electric power demand and cost of genera-tion,make forecasting very economical to the supply author-ity and useful to reduce uncertainty to the consumer.Out of various forecasting models,Box-Jenkins time-series models are useful but costly to operate.Modified BJ model,having lot of advantages,were developed for long range forecast of electrical load and a frequency domain approach is devel-oped for medium range forecast.The models were found to forecast well with an astronomical jump in forecasting lead-time.KeywordsForecasting.Bj-Model,Time-Series,ACF,PACF,DFT,Ag-gregate Model.1IntroductionWe are interested in the future because that is where we intend to live.Early man predicted future through as-trology,palmistry,numerology,astronomy etc.Present day future research gave birth to‘Fiituribles Internation-als’(France),Journal of‘Forecasting and Planning’(UK),‘Mankind2000International’(UK),‘Work Group2000’(Holland)etc.Without these exercises,‘Fulure Shock’will push us beyond our adoptive tolerances.Forecasting is an integral part of management,planning and decision making for ernment or the World Society at large.World scenario is characterized by uncer-tainty.Models are developed to reduce this uncertainty in forecast.With the increase of computer power,many of the old models became obsolete and new models are developed. Today hundreds of forecasts per day are made for oil and gas explorations,weather forecast etc.Time-series mod-els were developed since early1800’s under two different ∗Sarma,U.K.is with(BE(Gau),DIC(Imperial College),M.Sc Eng(London University),Ph D(IIT,KGP),Ex-Professor,Dean,and Di-rector,NERIST,India.Ex-Professor,College of Eng,Defence Univer-sity,Professor,Electrical Engineering,Mekelle University.E-mail: uksarma@ schools-time domain and frequenc domain.Major devel-opment took place in I960’s(Kalmanfilter models)and in I970’s(Box-Jenkins models).Time-series model is a“black box”containing lot of hidden information in the past data; and in a sense,past repeats itself;a lot of information about the future is also hidden in it.2Time-SeriesTime-series is a discrete sequence of events(variable)at equidistant time-intervals.Time Series model is a stochas-tic model in which the sequence of events/observations is viewed as a realization of jointly distributed input random variables.The joint distributions are characterized mainly byfirst two moments(mean andvariance).Symbolically written asX t−ø1X t−1−ø2X t−2...−øp X−t−p=αt−θ1αt−1−θ2αt−2...−θqαt−q(1−a)orX t=1−θ1B−θ2B2.....−θq B q1−ø1B−ø2B2.....−øp B pαt....ARMA(p,q)(1)where,ARMA(p,q)is auto-regressive(order p)moving average(order q)model AR(p):(1−ø1B−ø2B2...−øP B p)X t=αtis autoregressive model of order p...(2) model MA(q):X t=(1−θ1B−θ2B2...−θq B q)αtis moving average model of order q...(3)øandθare non-seasonal model parameters.B is called back shift operator(also written asB=Z−1;B2=Z−2)like Z−1X t=B X t=X t−1&Z−2X t=B2X t=X t−2.{a t}is a random variable;{X t}a stationer time series;ICECE,October29th to November1st2003There are various types of time series models.In this paper,Box-Jenkins Univariate model called UBJ [ARIMA(p,d,q)],which also represents a set of models,is used as the basic model and shown in the following block diagramrepresentation.where,{Y t}is trend removed series from raw data;{Z t}is raw time-series data of events∇d is ordinary differentiation(=1,2,...)model removed e.g.∇d=1{Z t}=(Z1-Z2),(Z2-Z3),...∇D T is also differen-tiation(D=1T,2T..differentiation,T is called seasonal time period).e.g.∇D12{Z t}=(Z1-Z13),(Z2-Z14)...where T=12(say).2.1Time Series ModelingElementary procedure of UBJ time series modeling is briefly mentioned,as these procedures are used in development of the‘models’described in this work.2.1.1First PhaseNormally,raw data has seasonality,growth or trend etc which make the series non-stationary in mean and variance. Determination and removal of them are called pre-whitening of data.The tools used are:•plot of the raw data and the acfs(auto-correlation func-tions)and pacfs partial auto-con-elation functions)of the data,•plot of mean vs standard deviation.∇D∇d log or√operations make the series stationery as shownbelow:So,from raw data,{Z t}we get stationery series,{X+t} after trend removal with D,d etc operation,giving a multi-plicative SARIMA(P,D,Q)(p,d,q)model.And mathemati-cally it looks like:Z t=∇D∇dΘQ(Z−1)θq(Z−1)Φp(Z−1)øp(Z−1)αt(4)whereΘQ(Z−1)andΦp(Z−1)and P.Q are parameters and order of seasonal ARMA model respectively.2.1.2Second Phase-Selection Of Model Types(Ar,MaOr Arma)And Model Order(P.q)Tools-(i)Estimated acfs and pacfs from the{Xt}and their plot.(ii)Theoretical acf and pacf diagram corresponding to UBJ models.Most models will be either AR(p)or MA(q)or ARMA(p,q),where,p and q will be1or2,Criteria of good model and rules for UBJ model selection are discussed in various literature[1].2.1.3Third phase:Model parameter estimation.Model parameters-θ’s andø’s are determined by various methods tools like-Least Square,Recursive least square, Reidge Regression estimation,maximum likelihood method etc.Here,Recursive Least Square method is used.Starting values of these parameters for lower order are taken from the acf and pacf diagrams;for higher order models,parameters are found by Yule-Walker method and tested for stability by Jury’s method[1].In this phase,model is found in the form of(1),(2),(3)or(4)as stated above.2.1.4Fourth Phase:Model Verification Or Model Crit-icism.While modeling,the qualities of a good model are to be taken care of.[1],The model residue{a t}to be tested for white ness.The tools are:a)Histogram plot-to see normalityb)Periodogram(byfinding DFT computed by FFT algo-rithm)c)Acf and pacf within±2SE.d)Chi-square test etc.e)Mean square error within limits.f)Forecast accuracy-after all,lest of the pudding is in theeating-to see the forecast limits by comparing the lastfew data left for model verification.If it is satisfactory,the model is ready for forecast.Otherwise,go throughthe whole process in search of a modified model.60International Conference on Electrical&Computer Engineering(ICECE2003)3Development Of Long Range Fore-casting Model,(With Lead Time Of 12Months)Selection of model type depends on various factors;forecast range(long,medium,short and immediate)is one and these ranges are different for different purposes in differentfields [1].3.1Modified B-J Model(Sarma-Basu-SinhaModelA normal electrical load(also for most time-series data), {Z t}shows growth and seasonality resulting multiplicative models;and leads to difficult non-linear estimation.“Mod-ified B-J model”is developed[1]with the advantages that the estimation of model order and parameter become linear and the“aggregate model”is more efficient and results in longer lead-time forecast.3.1.1The Aggregate ModelElectrical loads(a discrete data),like many other time-series events,have seasonality&trend.The electrical loads over particular month are often found to posses similar pat-tern in subsequent years.Taking this clue,the trend re-moved raw data(pre-whitening with bad data correction) is arranged month-wise andfitted with ARMA(P.Q)model and the residue of all monthly models are re-arranged and found to be a stationary time-series{e t}.And ARMA(p.q) model isfitted to this series and the residue{a t}is tested for witness;modeling algorithm is infigure3. Forecasting(including model verification)is done in the reverse process of modeling;forecasting algorithm is shown infigure4Mathematically it looks like:A p(Z−1)B q(Z−1) Z KT{ T i=1Φi p i(Z−1)i Q i−1∇D i(Z KT(∇d{y i}).. ...;K=0,...M−1)};K=0,..M−1]={αi}(5)∇D i=general differencing operator ofdegree D,i=index for the particularsubgroup.Φi pi(Z−1)iQ i−1=seasonal ARMA model for the sub-group series,of AR order P and MAorder Q,i is the index for the subgroup.A p(Z−1)B q(Z−1)=generalized ARMA model forrepresenting the stationary time seriesafter removing the seasonality(sea-sonal trend)and any overall trend in theseries.{αi}=residues to be tested for whitenessZ KT=unconventional forward shift operatoradopted for mathematical representa-tion of the rearrangement of the resid-ual series from the seasonal sub-group.M=length of data in each sub-group.T=period of the data(=12,in case of elec-tric load)3.1.2Case Study ILong Range Forecasting Model-monthly peak load(MW) forecasting with a Lead-time of12months.Figure1:Monthly peak(MW)load at D=0d=0,a)Data is shown in Fig1.(11years monthly peak MWload of an Electric Power station(Carnac in Bombaycity).b)Tests shows growth and seasonality of12months.c)Trend removed(by∇d)data is divided month wiseand then seasonality is removed(by∇D)and mod-eled for seasonality as ARMA(P,Q)and the re-arrangedresidue{e t}is modeled as ARMA(p,q)and the residueis tested for whiteness.Forecasting algorithm atfigure3.61ICECE,October29th to November1st2003d)Model verification:forecast made and compared withsome known data.Forecast error is found to be within reasonable accuracy limits,2.e)Algorithm used is shown infig3and4.4Development of Long Range Fore-casting Model,(with lead time of12 months)4.1Medium range hourly(MW)load forecastfor each day-group(daia-from the samestation as citedabove)Figure2:Monthly peak(MW)forecastModel selection varies with various factors[2],like-required depth of lead-time,purpose(short/medium/long) etc.For medium range forecasting of hourly-loads,a fre-quency domain analysis with DFT(Discrete Fourier Trans-form)approach,is used.4.2DFT modelHere also,it is observed that load data of each day-group (say Monday-group data are dissimilar from other day-groups as shown infigure5).Hence,One year’s hourly load is divided to seven day-groups[2]and each day-group (say all Mondays’hourly MW load)data shows growth and trend.These characteristics will be retained by their DFT components[2].Let a data set be X(n);n=0,l,..(N-1),N=number of hours under consideration(N=2P&p=a whole number)DFT’s of a day’s load-X(k)= n−1n=0X(n)exp(−j2πkn/N);k=0,1,..(N−1)](6)This consists of N/2numbers of real and N/2numbers of imaginary components at different frequencies(k).The DFTs at each frequency(k)for all M number of Mondays-days form a time series;real and imaginary parts are mod-eled separately and used to forecast a DFT of(M+l)thday.Figure3:Modeling by Modified BJ model(SARMA-BASU’-SINHAmodel)Figure4:Forecasting by SARMA-BASU-SINHA modelNow the N DFTs of different frequencies(k)of the(M+l)th day will be converted to hourly loads with the help of IDFTs by62International Conference on Electrical &Computer Engineering (ICECE 2003)X (n )=1NN −1k =0X (k )exp (j 2πkn/N );n =0,1,..(N −1)](7)The modeling and forecasting algorithm is shown in figure 5.4.2.1Case Study IIa)A typical hourly week-day load for different days is shown in figure 5b)One year’s real hourly data (MW)of Canac Substation is taken and separated into seven day-groups.c)Since day’s data length is 24but for DFT computation we need 16or 32data,we take two types of models with 16data length (daily loads of 8am to 10pm )or 32data length (24hourly loads plus 8-padded data)[2].d)Modeling and forecasting algorithm is shown in figure 7.e)Forecast data with 7-days lead-time is shown in figure6.Figure 5:Typical weekdayload.Figure 6:Monthly hourly loads forecast.5Conclusion.These models/algorithms are simple to operate,need less memory and less computational time thanmultiplicativeFigure 7:DFT modeling and forecasting algorithm.SARIMA models.The forecast duration,here,is called long and medium,considering the one step lead-time of forecast.The lead-time of 12months (in case long range)and 7days =168hours (in case of medium range)at one-step forecast is a big achievement.The Long Range Fore-cast Model may be used for major overhaul (or capital man-agement)of generators,switchgears,transformers etc.The Medium Range Forecast Model may be used for generator-,switchgear-or transformer-scheduling and for economic generation scheduling in hydro-thermal systems.It may be used for scheduled load shedding (when necessary).These models may also be used in similar fields-like,air-traffic forecasting.Bad data detection and corrections are not dis-cussed here due to space and time constraint.These models are found to forecast well and they are most suitable where temperature effect on load is less significant,like in many cities in Ethiopia.This model may be used in similar fields 63ICECE,October29th to November1st2003-like air traffic forecasting.The most striking advantage of these models is the long lead time in one-step forecast and this achievement may be compared to(in the language of one of the critics)“Armstrong’s astronomical jump to the Moon”.References[1]Basu Sanna and Sinha.Medium Range Forecasting forpower system load(energy)demand.Int.J.of SystemsSc.,(UK),7No,9:1699–1702,1987.[2]Basu Sarma and Sinha.Medium Range Forecasting ofpower system load by Discrete Fourier Transform.ap-proach.Int.J.of Indian Institute of Sc.(Signal Process-ing),(India),pages87–90,July-August1987.64。

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Volatility forecasting using high frequency data:Evidence from stock markets ☆Sibel Çelik a ,⁎,Hüseyin Ergin ba Dumlupinar University,School of Applied Sciences,Turkey bDumlupinar University,Business Administration,Turkeya b s t r a c ta r t i c l e i n f o Article history:Accepted 24September 2013JEL classi fication:C22G00Keywords:VolatilityRealized volatility High frequency data Price jumpsThe paper aims to suggest the best volatility forecasting model for stock markets in Turkey.The findings of this paper support the superiority of high frequency based volatility forecasting models over traditional GARCH models.MIDAS and HAR-RV-CJ models are found to be the best among high frequency based volatility forecasting models.Moreover,MIDAS model performs better in crisis period.The findings of paper are important for financial institutions,investors and policy makers.©2013Elsevier B.V.All rights reserved.1.IntroductionVolatility plays an important role in theoretical and practical applica-tions in finance.The availability of high frequency data brings a new dimension to volatility modeling and forecasting of returns on financial assets.First and foremost,nonparametric estimation of volatility of asset returns becomes feasible and so modeling and forecasting volatility of asset returns has been a focus for researchers in the literature (Andersen and Bollerslev,1998;Andersen et al.,2001,2003b ,2007;Corsi,2004;Engle and Gallo,2006;Ghysels et al.,2004,2005,2006a,b;Hansen et al.,2010;Shephard and Sheppard,2010).The empirical find-ings of existing studies support the superiority of high frequency based volatility models to popular GARCH models and stochastic volatility models in the literature (Andersen et al.,2003b ).Besides,earlier studies point to importance of allowing for discontinuities (jumps)in volatility models and pricing derivatives (Andersen et al.,2002;Chernov et al.,2003).Availability of high frequency data is also a turning point in order to distinguishing jump from continuous part of price process.Empirical findings from recent studies show that incorporating the jumps to volatility models increase the forecasting performance of models supporting the earlier evidence (Andersen et al.,2003b,2007).This paper aims to suggest the best volatility forecasting model in stock markets in Turkey.For this purpose,first,we analyze the data generating process and calculate the high frequency based volatility and examine the return and volatility characteristics.Second,we propose the best volatility forecasting model by comparing different volatility forecasting models.In doing so,the paper will contribute to the literature in terms of filling five main gaps.First,it suggests the best volatility forecasting model from the alternatives including high frequency-based models and traditional GARCH models.Second,it reveals the forecasting performance of volatility models during the periods of structural change.Because,recent studies in the literature indicate that financial crisis affect the volatility dynamics deeply (Dungey et al.,2011).Third,it analyses forecasting performance of volatility in stock futures markets rather than spot markets.There are three reasons for usage of stock futures markets in this study.Firstly,there are findings in the literature that futures markets respond to new information faster than spot markets (Stoll and Whaley,1990).Secondly,using futures contracts rather than spot indexes re-duces nonsynchronous trading problems (Wu et al.,2005).Thirdly,using futures contracts provides additional evidence to the existing literature on spot markets (Wu et al.,2005).Fourth,it compares the findings at different frequencies to inference about optimal fre-quency since the sampling selection is important for high frequency data based studies.Because,while higher sampling frequency may cause bias in realized volatility,lower sampling frequency may cause information st,it contributes to literature in terms of presenting evidence from an Emerging Market.Economic Modelling 36(2014)176–190☆This paper is based on my doctoral dissertation “Volatility Forecasting in Stock Markets:Evidence From High Frequency Data of Istanbul Stock Exchange ”which was completed at Dumlupinar University,in 2012.⁎Corresponding author at:Dumlupinar University,School of Applied Sciences,Insurance and Risk Management Department,Turkey.Tel.:+902742652031x4664.E-mail address:sibelcelik1@ (S.Çelik).0264-9993/$–see front matter ©2013Elsevier B.V.All rights reserved./10.1016/j.econmod.2013.09.038Contents lists available at ScienceDirectEconomic Modellingj ou r n a l h o m e p a ge :w ww.e l s e v i e r.c o m /l oc a t e /e c mo dThe paper proceeds as follows.Section2introduces dataset of the paper.Section3explains the methodologies used in the paper. Section4summarizes the empiricalfindings.Section5concludes the paper.2.DataThe dataset comprises of ISE-30index futures data at intradaily and daily frequency from04.02.2005to30.04.2010.1We generate new data sampled at1-minute interval,5-minute interval,10-minute interval and15-minute interval.The number of intraday observations of ISE-30index future are502,101,51and34,respectively.Careful data cleaning is one of the most important point in volatility estimation from high frequency data.The importance of cleaning of high frequency data is emphasized in the literature (Brownless and Gallo,2006;Dacorogna et al.,2001;Hansen and Lunde,2006).In this paper,we used following steps for data cleaning process.1.We delete entries which related to weekends.2.We delete entries of public holidays,which is announced by IstanbulStock Exchange and Turkish Derivatives Exchange.3.We delete entries when the Stock Exchanges do not trade full days.4.We delete entries which is not common for Istanbul Stock Exchangeand Turkish Derivatives Exchange from04.02.2005to30.04.2010.3.Methodology3.1.Methodologies for volatility modeling and data analysis3.1.1.GARCH modelThe GARCH models are as follows:r t¼ffiffiffiffiffih tqεtð1Þh t¼α0þX qi¼1αi r2t−iþX pj¼1βj h t−jð2Þp≥0,q N0,α0N0,αi≥0∀i≥1,i=1,……..p,βj≥0∀j≥1.3.1.2.Continuous-jump diffusion processThe continuous-time jump diffusion process traditionally used in asset pricingfinance is expressed as:dp t¼μdtþσtðÞdW tðÞþγtðÞdq tðÞ0≤t≤Tð3Þwhereμt is a continuous and locally bounded variation process,σt is stochastic volatility process,W t denotes a Standard Brownian motion, dq t is a counting process with dq t=1,corresponding to a jump at time t and dq t=0otherwise with jump intensityψ(t),andγ(t)refers to the size of jumps.The quadratic variation for the cumulative return process,r(t)=p(t)−p(0)is given by,r;r ½t ¼Z tσ2sðÞdsþX0⊲s≤tγ2sðÞ:ð4ÞQuadratic variation consists of∫t0σ2sðÞds continuous and∑0⊲s≤tγ2sðÞ,jump components.In the absence of jumps,the second term in the right will not exist and quadratic variation will equal to integrated volatility (Andersen et al.,2003a).3.1.3.Realized volatility,bipower variation and jumpsLet theθ—period returns be denoted by,r t,θ=p(t)−p(t−θ).We define daily realized volatility by the summing corresponding1/θhigh frequency intradaily squared returns as follows:RV tþ1θðÞ¼X1=θj¼1r tþj:θ;θ2:ð5ÞIn the absence of jumps,realized volatility is the consistent estimate of the integrated volatility.However,in the presence of jumps,we need more powerful measurement.Barndorff et al.(2004)introduce the volatility measurement which is powerful in the case of jumps called bipower variation(henceforth:BV).BV is defined as follows:BV tþ1θðÞ¼μ1−2X1=θj¼2r tþjr tþj−1ðÞð6Þμ1−2≅0:7979in Eq.(6).While the realized volatility consists of both continuous and jump components,BV only includes continuous component.Thus,jump component may be consistently estimated by,RV tþ1θðÞ−BV tþ1θðÞ¼Xt≺s≤tþ1γ2sðÞ:ð7ÞTo prevent the right hand-side of Eq.(7)from becoming negative, we impose non-negativity truncation on the jump measurements.J tþ1θðÞ¼max RV tþ1θðÞ−BV tþ1θðÞ;0ÂÃð8ÞContinuous component is given in Eq.(9).BV tþ1θðÞ¼RV tþ1θðÞ−J tþ1θðÞð9Þ3.2.Methodologies for volatility forecasting3.2.1.GARCH modelTo evaluate the forecasting performance of GARCH model,first we es-timate Eqs.(1)and(2).Let h tþ1G denote the predicted value for h t.The forecast error for the GARCH model for the observation t+1is computed as RV tþ1−h tþ1G based on the existing literature(Alper et al.,2009).3.2.2.HAR-RV modelHAR-RV model is introduced by Corsi(2004)and denoted as,RV tþ1¼β0þβD RV tþβW RV t−5;tþβM RV t−22;tþεtþ1ð10Þt=1,2,3…..T.RV t,RV t−5and RV t−22mark daily,weekly and monthly realized volatility respectively.Multi-period realized volatility compo-nents such as weekly and monthly realized volatility is calculated as, RV t;tþh¼h−1RV tþ1þRV tþ2þ::::þRV tþhÂÃð11Þh¼1;2;…RV t;tþ1≡RV tþ1:ð12ÞIn this paper,we take h=5and h=22as the weekly and monthly volatility,respectively.Andersen et al.(2003b)state that the distribu-tion of standard deviation and logarithmic form of realized volatility are close to normal than original form and so using these proxies increases performance of volatility forecasting.Therefore,we estimate standard deviation and logarithmic form of Eqs.(11)and(12).RV tþ1ÀÁ1=2¼β0þβD RV tðÞ1=2þβW RV t−5;t1=2þβM RV t−22;t1=2þεtþ1ð13Þ1ISE-30index futures data were taken from Turkish Derivatives Exchange.177S.Çelik,H.Ergin/Economic Modelling36(2014)176–190log RV t þ1ÀÁ¼β0þβD log RV t ðÞþβW log RV t −5;tþβM log RV t −22;tþεt þ1ð14Þ3.2.3.HAR-RV-J modelHAR-RV-J model is developed by Andersen et al.(2003b)by including jump component in HAR-RV model.Daily HAR-RV-J model is expressed in Eq.(15),RV t ;t þ1¼β0þβD RV t þβW RV t −5;t þβM RV t −22;t þβj J t þεt ;t þ1:ð15ÞLogarithmic and standard deviation form of HAR-RV-J model is given in Eq.(16)and (17).RV t þ1ÀÁ1=2¼β0þβD RV t ðÞ1=2þβW RV t −5;t1=2þβM RV t −22;t 1=2βj J t ðÞ1=2þεt þ1ð16Þlog RV t þ1ÀÁ¼β0þβD log RV t ðÞþβW log RV t −5;tþβM log RV t −22;tþβj log J t þ1ðÞþεt þ1ð17Þ3.2.4.HAR-RV-CJ modelAndersen et al.(2007)develop HAR-RV-CJ model by including jump and continuous components separately in HAR-RV model.Daily HAR-RV-CJ model is stated as follows,RV t ;t þ1¼β0þβCD C t þβCW C t −5;t þβCM C t −22;t þβjD J t þβJW J t −5;tþβJM J t −22;t þεt ;t þ1:ð18ÞMulti period jump and continuous components are calculated as inEqs.(19)and (20),J t ;t þh ¼h−1J t þ1þJ t þ2þ::::::þj t þhÂÃð19ÞC t ;t þh ¼h−1C t þ1þC t þ2þ::::::þC t þh ÂÃ:ð20ÞLogarithmic and standard deviation form of HAR-RV-CJ model is given in Eqs.(21)and (22).RV t ;t þ11=2¼β0þβCD C t ðÞ1=2þβCW C t −5;t 1=2þβCM C t −22;t 1=2þβjD J t ðÞ1=2þβJW J t −5;t1=2þβJM J t −22;t 1=2þεt ;t þ1ð21Þlog RV t ;t þ1 ¼β0þβCD log C t ðÞþβCW log C t −5;tþβCM log C t −22;tþβjD log J t þ1ðÞþβJW log J t −5;t þ1 þβJM log J t −22;t þ1þεt ;t þ1ð22Þ3.2.5.MIDAS (mixed data sampling)modelMIDAS model is introduced by Ghysels et al.(2004,2005,2006a,b).Univariate MIDAS linear regression is given in Eq.(23),Y t ¼δ0þδ1X k max k ¼0B k ;θðÞXm ðÞt −k =m þεt :½23Y t and X (m )are one-dimensional processes,B (k ,θ)is polynomialweighting function depending on k and θparameter and X t m ðÞis sampled m times more frequent than Y t .For example,if t denotes a22-day monthly sampling and m =22,model (23)shows a MIDAS regression of monthly data (Y t )on past k max daily data (X t )RV t þ1;t ¼δ0þδ1X k max k ¼0B k ;θðÞRVm ðÞt −k =mþεt ð24Þm =1,k max =50and t refers to daily observations.Ghysels et al.(2006a)suggest various alternatives for B (k ,θ)polyno-mial.In this paper,we focus on beta polynomial following Ghysels et al.(2006a).B (k ,θ),is denoted as in Eq.(25).B k ;θðÞ¼f k =k max ;θ0;θ1ÀÁX k max k ¼1f k =k max;θ0;θ1ÀÁð25Þand,f x ;θ0;θ1ðÞ¼x θ0−11−x ðÞθ1−1Γθ0þθ1ðÞΓθ0ðÞΓθ1ðÞ:ð26ÞΓ(.)is gamma function.In beta function,we restrict our attention to θ0=1and estimate θ1N 1.3.2.6.Realized GARCH modelRealized GARCH model which is introduced by Hansen et al.(2010)can be expressed as in Eqs.(27),(28)and (29).r t ¼ffiffiffiffiffih t q z tð27Þh t ¼w þβh t −1þγx t −1ð28Þx t ¼ξþϕh t þτz t ðÞþu tð29Þr t is return;z t ~iid (0,1),u t e iid 0;σu 2ÀÁ,τ(z )is leverage function,h t =var(r t |F t −1),and F t =σ(r t ,x t ,r t −1,x t −1,.....).RGARCH model is estimated with maximum likelihood method as GARCH model.Log likelihood function is given in Eq.(30).‘r ;x ;θðÞ¼−1X n t ¼1log h t ðÞþr t 2=h t þlog σu 2 þu t 2=σu 2h i :ð30ÞWe evaluate the forecasting performance of RGARCH model as in GARCH model.3.3.Evaluation of forecasting performanceWe use mean squared error (MSE),mean absolute error (MAE),mean absolute percentage error (MAPE)and Theil's U statistic (TIC)to evaluate performance of volatility forecasting models.Calculation of the loss functions is as follows;RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN−1X N i ¼1RV t ;t þH −R ^V t ;t þH 2v u u t ð31ÞMAE ¼N−1X N i ¼1RV t ;t þH −R ^V t ;t þHð32ÞMAPE ¼N −1X N i ¼1RV t ;t þH −R ^V t ;t þH RV t ;t þHð33Þ178S.Çelik,H.Ergin /Economic Modelling 36(2014)176–190TIC ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX N i ¼1RV t ;t þH −R ^V t ;t þH2v u u t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX N i ¼1RV t ;t þH 2v u u t þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX N i ¼1R ^V t ;t þH2v u u t :ð34ÞRV t ,t +H and R ^Vt ;t þH denote actual and predicted values of realized volatility,respectively.N is the number of observation.In addition to these loss functions,we also use Mincer and Zarnowitz regression (1969)in this paper.Mincer –Zarnowitz regression is given in Eq.(35).RV t ;t þH ¼a þb R ^V t ;t þH þu t ;t þHð35ÞIn the regression model,null hypothesis is formed as “a and b equalto 0”.If the forecasting is unbiased,a and b coef ficients must equal to 0and 1respectively and coef ficient of b must be signi ficant.4.Empirical findingsThe empirical findings are categorized under four sub-sections.•We use six different models (GARCH,HAR-RV,HAR-RV-J,HAR-RV-CJ,MIDAS,RGARCH)to determine the best volatility forecasting model.•We present the findings of different transformations of volatility (stan-dard deviation,logarithmic)following Andersen et al.(1999,2000)and Andersen et al.(2001).2•We compare the volatility forecasting performance at different frequencies to inference about optimal sampling frequency (1min,5min,10min and 15min).•We compare the volatility forecasting performance for differentsample periods to examine the impact of financial crisis on volatility structure (pre-crisis period,crisis period and total period).3Tables 1,2,3and 4present summary statistics of the variables 4at 1-minute,5-minute,10-minute and 15-minute frequencies respective-ly.According to return statistics,as expected the mean returns are higher in pre-crisis period for all frequencies.However,there are no consistent information about the maximum and minimum values.The standard deviation of returns is higher in pre-crisis period for most of the frequencies.While returns have positive skewness in pre-crisis period,it is negative in crisis period for most of the frequencies.LB statistics show the evidence of autocorrelation between return series.Mean values of realized volatility and jump statistics are higher in pre-crisis period,5however mean values of bipower variation are higher in crisis period for most of the frequencies.Variables have kurtosis greater than 3except log(RV)supporting leptokurtic distribution.The higher the frequency,skewness and kurto-sis degree of returns also increase.The distribution of standard devia-tion and logarithmic transformation of variables are close to normal.Table 5shows the summary statistics of conditional variance series of GARCH(1,1)estimation.The mean returns are higher in pre-crisis period.Maximum and minimum values of returns are appeared in crisis period.The standard deviation of returns is higher in crisis period.Re-turn series did not distribute normal.The mean conditional variance is higher in crisis period with the value of 0.0005.The conditional variance series is rightly skewed and has leptokurtic distribution.Appendices2We examine the standard deviation and logarithmic transformation of realized vola-tility,jumps statistics since Andersen et al.(1999,2000)and Andersen et al.(2001)indi-cate that the distribution of standard deviation and logarithmic transformations are close to normal and using these proxies increase performance of volatility forecasting.3We examine the impact of 2007global crisis on volatility characteristics.In the litera-ture,there are some findings that global financial crisis give the first signal with announce-ment of the problems with the hedge funds of Bear Stearns in July 2007.Some papers use 17July 2007as a starting date of the global financial crisis (Dungey,2009).Following these findings,we determine three sub-periods (from 04.02.2005to 16.07.2007is pre crisis pe-riod,from 17.07.2007to 30.04.2010is crisis period and from 17.07.2007to 30.04.2010is total period).4Return,realized volatility,standard deviation and logarithmic transformation of real-ized volatility,jumps,standard deviation and logarithmic transformation of jumps and bipower variation.5Turkish Derivative Exchange has started operating on February 2005and pre-crisis period includes this period.In the first few days,trading volume is low and there is more time difference between instantaneous price quotations.For this reason,realized volatility or price jumps may increase in pre-crisis period.Table 1Summary statistics for ISE −30index futures at 1minute frequency.Variable Sampling period Mean Maximum Minimum S.dev Skewness Kurtosis LBReturnPre-crisis period 1.91000.1185−0.11750.00360.056269.727134,906Crisis period 0.33400.1071−0.10800.0044−0.048759.854577,165Total period 1.08000.1185−0.11750.0040−0.013764.8631115,010RVPre-crisis period 0.00060.1006 4.33000.0120 4.422627.88391887.80Crisis period 0.00970.263710.6000.0195 5.960658.6426776.67Total period 0.00830.2637 4.33000.0165 6.165466.35392008.80RV 1/2Pre-crisis period 0.06480.31720.00650.0503 1.80407.56731939.60Crisis period 0.07510.51360.01020.0643 1.94338.81892143.70Total period 0.07020.51360.00650.0584 1.98199.14684117.40log(RV)Pre-crisis period −6.0277−2.2960−10.0473 1.5362−0.1492 2.49181646.00Crisis period −5.8363−1.3325−9.1513 1.65110.0407 2.13533490.50Total period −5.9263−1.3325−10.04731.6004−0.02552.31535114.80JPre-crisis period 278.4000.04410.00000.0050 3.734620.70281438.30Crisis period 175.3000.03870.00000.0035 4.6203 4.620332.6315Total period 223.8000.04410.00000.0043 4.194826.18642584.70J 1/2Pre-crisis period 0.04070.21010.00000.0336 1.7732 6.59631830.30Crisis period 0.03070.19680.00000.0284 1.88457.55042033.20Total period 0.03540.21010.00000.0313 1.84657.18163982.60log(J +1)Pre-crisis period 0.00270.04320.00000.0049 3.692220.23241451.40Crisis period 0.00170.03800.00000.0035 4.566931.8685929.74Total period 0.00220.04320.00000.0043 4.146525.57822609.70BVPre-crisis period 395.0000.07950.0000844.300 4.984233.88221777.30Crisis period 803.6000.22507.30001640.00 6.326564.7405699.36Total period611.4000.22500.00001340.806.943682.13171795.60Note:RV,J and BV denote realized volatility,jump component and continuous component respectively.LB is the statistics of Ljung –Box (1979)Q test.Mean returns are multiplied by 106and,minimum values of RV,mean values of J and mean,minimum and Standard deviation values of BV are multiplied by 105.179S.Çelik,H.Ergin /Economic Modelling 36(2014)176–1901–18present the results of GARCH,HAR-RV,HAR-RV-J,HAR-RV-CJ, MIDAS and RGARCH estimations,lost functions and Mincer Zarnowitz test.Prior evidence shows that the performance of volatility models is the best at15-minute frequency sampling for ISE-30index futures since lost functions are minimum at15-minute frequency.Therefore, in this section,we only compare the performance of different volatility models using15-minute frequency as a base for ISE-30index futures rather than comparison of different frequencies.Tables6and7present the comparison of forecasting performance of models in pre-crisis and crisis period for ISE-30index futures.In pre-crisis period,RMSE,MAE and TIC functions indicate that GARCH model has the worst forecasting performance.Different from the RMSE, MAE and TIC functions,MAPE function support that GARCH and RGARCH models have the best forecasting performance.The best model is controversial for the pre-crisis period.RMSE supports the superiority of MIDAS1/2model,MAE functions support the superiority of MIDAS log model,according to MAPE function,GARCH model is the best,and TIC function supports the superiority of HAR-RV-CJ model.When we eval-uate all loss functions together,MIDAS1/2and MIDAS log models seem to be the best models.Then,HAR-RV-CJ1/2,MIDAS,HAR-RV-CJ AND HAR-RV-J1/2models perform well,respectively.Thefindings of HAR-RV, HAR-RV-J and HAR-RV-CJ models support that including jumpTable2Summary Statistics for ISE-30index futures at5-minute frequency.Variable Sampling period Mean Maximum Minimum S.dev Skewness Kurtosis LBGetiri Pre-crisis period9.49000.0980−0.09870.00500.043444.61377845.80 Crisis period 1.66000.1056−0.10410.0048−0.115343.646710292.00Total period 5.34000.1056−0.10410.0049−0.035844.208817940.00 RV Pre-crisis period0.00250.0412 3.25000.0044 3.813721.90291232.10 Crisis period0.00230.0651 6.39000.0046 5.917859.1149578.34Total period0.00240.0651 3.25000.0045 4.992443.11311675.10 RV1/2Pre-crisis period0.04020.20300.00570.0306 1.85787.00601634.40 Crisis period0.03810.25530.00790.0302 2.17099.83241590.00Total period0.03910.25530.00570.0304 2.01828.43503213.60 log(RV)Pre-crisis period−6.9039−3.1888−10.3342 1.37870.1349 2.61381464.40 Crisis period−7.0161−2.7304−9.6581 1.34920.4119 2.43062465.60Total period−6.9634−2.7304−10.3342 1.36380.2792 2.50223847.20 J Pre-crisis period80.0000.01650.00000.0018 4.578029.11271835.30 Crisis period37.3000.00920.00000.0008 5.117639.0344328.33Total period57.8000.01650.00000.0013 5.596444.63463207.60 J1/2Pre-crisis period0.02040.12870.00000.0198 2.18988.84651950.50 Crisis period0.01400.09640.00000.0132 2.06438.8297816.22Total period0.01700.12870.00000.0169 2.374710.63753328.50 log(J+1)Pre-crisis period0.00080.01640.00000.0017 4.558828.87451840.70 Crisis period0.00030.00920.00000.0008 5.103438.8103329.55Total period0.00050.01640.00000.0013 5.571044.22803213.90 BV Pre-crisis period176.4000.04390.0000365.000 5.719148.5876683.49 Crisis period200.0000.0653 4.5000418.0007.125586.0992462.04Total period189.0000.06530.0000394.000 6.650274.74101076.40Note:RV,J and BV denote realized volatility,jump component and continuous component respectively.LB is the statistics of Ljung–Box(1979)Q test.Mean returns are multiplied by106 and,minimum values of RV,mean values of J and mean,minimum and Standard deviation values of BV are multiplied by105.Table3Summary statistics for ISE-30index futures at10-minute frequency.Variable Sampling period Mean Maximum Minimum S.dev Skewness Kurtosis LBGetiri Pre-crisis period18.8000.0980−0.08200.00570.177432.72843738.50 Crisis period 3.28000.1010−0.10330.0053−0.400842.59323266.00Total period10.6000.1010−0.10330.0055−0.096837.43296961.50Pre-crisis period0.00170.0234 2.31000.0029 3.445717.38321621.90 RV Crisis period0.00140.0271 5.55000.0027 4.647631.6358499.54 Total period0.00150.0271 2.31000.0028 4.018723.84162014.90Pre-crisis period0.03280.15300.00480.0250 1.8493 6.59321875.60 RV1/2Crisis period0.03080.16480.00740.0228 2.18809.14871246.40 Total period0.03180.16480.00480.0239 2.01807.77933137.30Pre-crisis period−7.2965−3.7541−10.6756 1.34680.2134 2.73751393.20 log(RV)Crisis period−7.3606−3.6050−9.7991 1.22020.5178 2.72401901.00 Total period−7.3305−3.6050−10.6756 1.28120.3605 2.74873234.00Pre-crisis period51.6000.01490.00000.0012 5.868249.57951260.70 J Crisis period30.0000.00840.00000.0006 5.559745.5093235.43 Total period40.1000.01490.00000.0010 6.557864.93791944.80Pre-crisis period0.01580.12230.00000.0162 2.397510.88001425.90 J1/2Crisis period0.01230.09170.00000.0121 2.24449.9725467.14 Total period0.01390.12230.00000.0143 2.469911.78152112.00Pre-crisis period0.00050.01480.00000.0012 5.841649.13721266.60 log(J+1)Crisis period0.00020.00830.00000.0006 5.545545.2689235.76 Total period0.00040.01480.00000.0010 6.526364.29861951.10Pre-crisis period121.0000.0278 1.0800236.600 5.085640.7553799.06 BV Crisis period118.0000.0229 3.8800225.100 4.816233.6960464.41 Total period120.0000.0278 1.0800230.000 4.958337.47171235.90Note:RV,J and BV denote realized volatility,jump component and continuous component respectively.LB is the statistics of Ljung–Box(1979)Q test.Mean returns are multiplied by106 and,minimum values of RV,mean values of J and mean,minimum and Standard deviation values of BV are multiplied by105.180S.Çelik,H.Ergin/Economic Modelling36(2014)176–190component in model increase the forecasting performance.In general,GARCH and RGARCH models have the worst forecasting performance.In crisis period,GARCH model has the worst forecasting perfor-mance according to RMSE and TIC.It is not clear which model is the best,however MIDAS log and MIDAS 1/2models perform well than others.Than HAR-RV-CJ,HAR-RV-J and HAR-RV models follow MIDAS log and MIDAS 1/2models.Both in pre-crisis and crisis period,highfrequency based volatility models have better forecasting performance than traditional GARCH model.5.ConclusionThis paper aims to suggest the best volatility forecasting model for stock markets in Turkey.For this purpose,first we analyze the dataTable 4Summary statistics for ISE-30index futures at 15-minute frequency.Variable Sampling period Mean Maximum Minimum S.dev Skewness Kurtosis LBGetiriPre-crisis period 28.2000.0971−0.10130.0059−0.115332.71252276.40Crisis period 5.13000.0773−0.07810.0056−0.367324.92381269.00Total period 16.0000.0971−0.10130.0058−0.238529.09403459.00Pre-crisis period 0.00120.01980.40000.0020 3.977024.33541100.00RVCrisis period 0.00100.0196 3.60000.0018 4.339529.8961691.43Total period 0.00110.01980.40000.0019 4.170627.09391814.40Pre-crisis period 0.02810.14080.00200.0206 1.89737.31491493.60Crisis period 0.02730.14010.00600.0185 1.96888.03101315.70RV 1/2Total period 0.02770.14080.00200.0195 1.94287.72142824.50Pre-crisis period −7.5796−3.9195−12.4292 1.31410.1007 3.01891050.90Crisis period −7.5594−3.9298−10.2319 1.16060.4454 2.61091627.60log(RV)Total period −7.5689−3.9195−12.42921.23470.24972.90112596.20Pre-crisis period 38.2000.01520.00000.00097.891398.2244638.07Crisis period 20.9000.00340.00000.0004 4.210324.1542293.98JTotal period 29.0000.01520.00000.00079.1269144.26301233.20Pre-crisis period 0.01350.12340.00000.0140 2.426612.3670832.71Crisis period 0.01050.05900.00000.0098 1.81747.3270334.84J 1/2Total period 0.01190.12340.00000.0121 2.428013.02061352.40Pre-crisis period 0.00030.01510.00000.00097.840797.0506643.18Crisis period 0.00020.00340.00000.0004 4.206924.1184294.13log(J +1)Total period 0.00020.01510.00000.00079.0609142.28051241.00Pre-crisis period 85.0000.01610.3000158.000 4.504430.4697571.91Crisis period 89.1000.0200 3.3000159.200 5.152343.3425485.57BVTotal period87.2000.02000.3000158.7004.850137.38441042.70Note:RV,J and BV denote realized volatility,jump component and continuous component respectively.LB is the statistics of Ljung –Box (1979)Q test.Mean returns are multiplied by 106and,minimum values of RV,mean values of J and mean,minimum and Standard deviation values of BV are multiplied by 105.Table 5Daily GARCH(1,1)estimations.Variable Sampling frequency Mean Maximum Minimum S.dev Skewness Kurtosis LBReturnPre-crisis period 987.0000.0667−0.08180.0164−0.1892 4.75245312.20Crisis period 180.0000.0965−0.09970.0245−0.0108 5.047817.4420Total period 558.0000.0965−0.09970.0211−0.0720 5.691621.7370GARCH(1,1)Pre-crisis period 0.00020.00090.00000.0001 1.58717.11342631.9Crisis period 0.00050.00230.00020.0003 1.9888 6.75545090.7Total period0.00040.00240.00000.00032.607510.863610124.0Note:Mean returns are multiplied by 106.Table 6Comparison of models in pre-crisis period for ISE-30index futures.RMSEMAEMAPETICStatisticOrder Statistic Order Statistic Order Statistic Order GARCH 2.252014GARCH 1.002014GARCH 96.68461GARCH 0.845014HAR-RV 1.46008HAR-RV 0.724012HAR-RV 269.061114HAR-RV 0.42733HAR-RV-1/2 1.46419HAR-RV-1/20.66407HAR-RV-1/2185.740010HAR-RV-1/20.45566HAR-RV-LOG 1.543212HAR-RV-LOG 0.67309HAR-RV-LOG 143.28006HAR-RV-LOG 0.540110HAR-RV-J 1.44606HAR-RV-J 0.714011HAR-RV-J 265.805912HAR-RV-J 0.42122HAR-RV-J 1/2 1.45437HAR-RV-J 1/20.65506HAR-RV-J 1/2184.04118HAR-RV-J 1/20.45115HAR-RV-J LOG 1.538711HAR-RV-J LOG 0.66808HAR-RV-J LOG 143.19675HAR-RV-J LOG 0.540811HAR-RV-CJ 1.43504HAR-RV-CJ 0.705010HAR-RV-CJ 264.001011HAR-RV-CJ 0.41671HAR-RV-CJ 1/2 1.43905HAR-RV-CJ 1/20.64804HAR-RV-CJ 1/2185.42719HAR-RV-CJ 1/20.44424HAR-RV-CJ LOG 1.505210HAR-RV-CJ LOG 0.65105HAR-RV-CJ LOG 139.29024HAR-RV-CJ LOG 0.50039MIDAS 1.31422MIDAS 0.64103MIDAS 267.139413MIDAS 0.46027MIDAS 1/2 1.30141MIDAS 1/20.57862MIDAS 1/2183.84307MIDAS 1/20.48618MIDAS LOG 1.36673MIDAS LOG 0.57521MIDAS LOG 137.68483MIDAS LOG 0.569712RGARCH1.996513RGARCH 0.866013RGARCH 98.36942RGARCH 0.695913181S.Çelik,H.Ergin /Economic Modelling 36(2014)176–190。

Data Transformation and Forecasting in Models With Unit Roots and Cointegration¤John C.Chao1,Valentina Corradi2and Norman R.Swanson31University of Maryland2University of Exeter3Texas A&M UniversityDecember2000AbstractWe perform a series of Monte Carlo experiments in order to evaluate the impact of data transformation on forecasting models,and¯nd that vector error-corrections dominate di®er-enced data vector autoregressions when the correct data transformation is used,but not whendata are incorrectly tansformed,even if the true model contains cointegrating restrictions.Weargue that one reason for this is the failure of standard unit root and cointegration tests underincorrect data transformation.JEL classi¯cation:C22,C51.Keywords:integratedness,cointegratedness,nonlinear transformation.¤The authors wish to thank Frank Diebold and Clive W.J.Granger for useful suggestions which led to the writing of this paper.In addition,they are grateful to Claudia Olivetti for her e®orts in carrying out many of the simulations reported here.Swanson thanks the Private Enterprise Research Center and the Bush Program in the Economics of Public Policy,both at Texas A&M University,for research assistance.Part of this paper was written while the third author was visiting the economics department of the University of California,San Diego,and he is grateful to members of the department for providing a stimulating research environment from which to carry out his research. Corresponding Author:Norman R.Swanson,Department of Economics,Texas A&M University,College Station, TX77843-4228,U.S.A.,email:nswanson@.1IntroductionThe purpose of this paper is to raise the issue of data transformation and its potential implications for the speci¯cation of forecasting models,and for forecasts from such models.This is done by ¯rst discussing unit root tests and cointegration,and then examining the impact of alternative assumptions placed on data generating processes before testing for unit roots and cointegration on subsequent predictions.Although we raise a number of issues,and in some cases suggest at least partial solutions,this paper is primarily meant to serve as a vehicle for underscoring the importance of the often ignored issue of data transformation on empirical model building.Thus,many issues are left unresolved.In macroeconometrics,unit root tests are typically performed using logs.This is consistent with much of the real business cycle literature(see e.g.Long and Plosser(1983)and King,Plosser, Stock,and Watson(1991))where it is suggested,for example,that GDP should be modeled in logs, given an assumption that output is generated according to a Cobb-Douglas production function. While this is sensible from a theoretical macroeconomic perspective,there is no clear empirical reason why logs should be used rather than levels,when performing unit root tests,particularly given that standard unit root tests assume linearity under both the null and the alternative,and violation of this linearity assumption can result in severe size and power distortion,both in¯nite and large samples(e.g.see Granger and Hallman(1991)).In addition,it is not always obvious by simply inspecting the data,for example,which transformation is`appropriate',when modeling economic data(e.g.see Figure1).Thus,it is reasonable to carefully address the problem of data transformation before running a unit root tests,for example.In a recent paper which is not discussed in detail here,Corradi and Swanson(2000)propose a framework for hypothesis testing in the presence of nonlinearity and nonstationarity.As a detailed illustration,they consider the problem of choosing between logs and levels before carrying out unit root and/or cointegration tests.An important feature of their test is that it is not subject to the di±culties discussed below when choosing between logs and levels using(possibly)integrated series.The current convention is to de¯ne an integrated process of order d(I(d))as one which has the property that the partial sum of the d th di®erence,scaled by T¡1=2,satis¯es a functional central limit theorem(FCLT).In this case,integratedness in logs does not imply integratedness in levels,and vice¡versa.Thus,any a priori assumption concerning whether to model datain levels or logs has important implications for the outcome of unit root and related tests.For example,Granger and Hallman(1991)show that the percentiles of the empirical distribution of the Dickey-Fuller(1979)statistic constructed using exp(X t)are much higher,in absolute value, than the corresponding percentiles constructed using the original time series X t,when X t is a random walk process.Thus,inference based on the Dickey-Fuller statistic using the exponential transformation leads to an overrejection of the unit root null hypothesis,when standard critical values are used.More recently it has been shown in Corradi(1995)that if X t is a random walk, then any convex transformation(such as exponentiation)is a submartingale,and any concave transformation(such as taking logs)is a supermartingale.However,while submartingales and supermartingales have a unit root component,their¯rst di®erences do not generally satisfy typical FCLTs.Thus,Dickey-Fuller type tests no longer have well de¯ned limiting distributions.Given all of the above considerations,it is of some interest to use a statistical procedure for selecting between linear and loglinear speci¯cations,rather than simply assuming from the outset that a series is best modeled as linear or loglinear.Further,while Cox-type tests are available for the I(0)case,few results are available for the I(1)case.The arguments used above carry over to the case of cointegration tests,and indeed to any statistical tests based on the use of partial sums of functionals of residuals,for example.As the use of cointegration tests is prevalent,however,we focus our discussion on them in this paper.One of the areas where unit root and cointegration tests are crucial is in the construction of vector error correction(VEC)forecasting models.In order to illustrate this point,we simulate a real-time forecasting environment,where data are generated using cointegrated variables,and where models are estimated using data which are correctly or incorrectly transformed.Our primary focus is on the choice between log and level data,and we¯nd that incorrect data transformation leads to poor forecasts from cointegrated models,relative to simpler models based on di®erenced data, even when the true data generating process exhibits cointegration.This may be due to imprecise estimation of cointegrating spaces when the correct data transformation is uncertain,for example, and may help to explain the mixed evidence concerning the usefulness of cointegration restrictions in forecasting(see e.g.the special issue of the Journal of Applied Econometrics(1996)on forecasting). The¯nding is based on an evaluation of VEC models and vector autoregressive(VAR)models using di®erenced and undi®erenced data.Three additional¯ndings based on our analytsis are that:(1) VEC models forecast-dominate di®erenced data VAR models when the correct data transformationis used.(2)The worst models based on correctly transformed data clearly dominate the best models based on incorrectly transformed data.(3)When the incorrect data transformation is used to construct forecasting models,di®erenced data VAR models outperform not only their VEC counterparts,but also VAR models based on undi®erenced data.In order to shed further light on the issue of data transformation in VEC models,we examine the¯nite sample performance of cointegration tests under incorrect data transformation.The rest of the paper is organized as follows.Section2discusses unit root and cointegration testing under data transformation,and Section3contains a discussion of forecasting models under data transformation as well as the results of our forecasting experiments.Concluding remarks are given in Section4.2Unit Root and Cointegration TestsGiven a series of observations on an underlying strictly positive process X t,t=1;2;:::,our objective is to decide whether:(1)X t is an I(0)process(possibly around a linear deterministic trend),(2)log X t is an I(0)process around a nonzero linear deterministic trend,(3)X t is an I(1) process(around a positive linear deterministic trend),and(4)log X t is an I(1)process,(possibly around a linear deterministic trend).A natural approach to this problem is to construct a test that has a well de¯ned limiting distribution under a particular DGP,and diverges to in¯nity under all of the other above DGPs.While it is easy to de¯ne a test having a well de¯ned distribution under one of(1)-(4),it not clear how to ensure that the test has power against all of the remaining DGPs.To illustrate the problem,consider the sequence^²t,given as the residuals from a regression of X t on a constant and a time trend.Now,construct the test statistic proposed by Kwiatkowski,Phillips,Schmidt,and Shin(1992,hereafter KPSS):S T=1^¾2TT¡2TX t=10@t X j=1^²2t1A2;where^¾2T is a heteroskedasticity and autocorrelation(HAC)robust estimator of var³T¡1=2P t j=1²t´. It is known from KPSS that if X t is I(0)(possibly around a linear deterministic trend),then S T has a well de¯ned limiting distribution under the null hypothesis,while S T diverges at rate T=l T under the alternative that X t is an integrated process,where l T is the lag truncation parameter used in theestimation of the variance term in S T.However,if the underlying DGP is log X t=®1+±1t+P t j=1²j,±1>0(i.e.logX t is a unit root process)then both^¾2T and T¡2P T t=1³P t j=1^²j´2will tend to diverge at a geometric rate,given that X t=exp(®1+±1t+P t j=1²j).In this case it is not clear whether the numerator or the denominator is exploding at a faster rate.This problem is typical of all tests which are based on functionals of partial sums and variance estimators,and arises because certain nonlinear alternatives are not treatable using standard FCLTs.So far we have analyzed the case in which we perform a test with I(0)as the null hypothesis and I(1)as the alternative.In this case,the statistic is typically constructed in terms of functionals of partial sums scaled by a variance estimator.Another common procedure is to test for the null of I(1) versus the alternative of I(0)using Dickey-Fuller type tests.To illustrate the problems associated with this approach,consider the following simple example.Assume that log X t=log X t¡1+²t,²t»iid(0;¾2²).However,we perform a Dickey-Fuller test using levels.For example,we compute T(^®T¡1),where^®T=P T t=2X t X t¡1P T t=2X2t¡1:Now,X t=exp(log X t¡1+²t)=X t¡1exp(²t),so that we can write:T(^®T¡1)=T P T t=2X2t¡1(e²t¡1)P T t=2X t¡1:Note that as X t=X0exp(P t j=1²j),standard unit root asymptotics no longer apply.However,by con¯ning our attention to the case where²»N(0;¾2²),we can examine the properties of T(^®T¡1), thus gaining insight into the performance of a Dickey-Fuller test using an incorrect transformation of the data.Notice that Ee²t=e12¾2²>1.Thus,we might expect that T(^®T¡1)tends to diverge to+1.However,Granger and Hallman(1991)¯nd that this statistic tends to overreject the null of a unit root.One possible explanation for the di®erence between their¯nding and our intuition is that the distribution of e²t¡1is highly skewed to the left,and has a lower bound of negative one.Thus,even though the mean of e²t¡1is positive,this is due to the very long right-tail of the distribution.When²t is drawn from a standard normal distribution,however,most observations are rather close to zero(e.g.95%are between2and-2).These data,when transformed using e²t¡1, are mainly between-0.86and6.4.Further,the median of the distribution of e²t¡1is zero.Now,in the context of¯nite samples,this suggests that if we truncate the distribution of e²t¡1to be,say, between-0.8and1,then the mean of this truncated distribution will actually be negative(as wedraw relatively fewer observations close to the upper bound than negative observations close to the lower bound).In the context of generating data in¯nite samples,as Granger and Hallman did,this situation indeed seems to have occurred,resulting in mostly large negative values being calculated for the expression T(^®T¡1).Put another way,the negative elements of T P T t=1X2t¡1(e²t¡1)are usually quite large in magnitude,relative to most of the positive elements of the same sum.Of course,in large samples,and with large¾2²we should expect that this result will not hold,as the e®ect of large positive draws from the distribution of e²t¡1begins to dominate the overall sum T P T t=1X2t¡1(e²t¡1).This intuition suggests that Granger and Hallman's results,while holding for the usual sample sizes and the usual error variances observed in economic time series,should not hold generally.It further suggests that indeed using levels data when the true process is I(1) in logs will produce either overrejection of the unit root null(as Hallman and Granger show),or underrejection of the null.Interestingly,these arguments also suggest that for very special cases (i.e.appropriately chosen¾2²and sample size),the empirical size of the Dickey-Fuller test may actually match the nominal size,even when the wrong data transformation is used!In summary, there appears to be a need to carefully consider which transformation is used when constructing unit root tests,as the wrong transformation may yield entirely misleading results.Even if we decide to keep integratedness as a maintained assumption,and choose between I(1) in levels and I(1)in logs,or vice versa,we do not in general obtain a test which has unit asymptotic power.For example consider constructing a KPSS-type test using the¯rst di®erences of the levels data(i.e.¢X t).Under the null of I(1)in levels the statistic has the usual well de¯ned limiting distribution.However,under the alternative of I(1)in logs it does not necessarily diverge to in¯nity. Again the reason for this result is that both the numerator and the denominator tend to diverge to in¯nity if they have a positive linear deterministic trend,and in general we cannot determine whether the numerator or the denominator is diverging at a faster rate.Given the above issues,it may be useful to examine the¯nite sample performance of Johansen CI tests under incorrect data transformation.We turn next to this issue.Our approach is to examine the¯nite sample behavior of the Johansen(1988,1991)cointegration test using data generated according to the above parameterizations.Table1reports the¯nite sample size and power of the Johansen trace test when applied toincorrectly transformed data.Data are generated according to the following VEC model:¢Q1;t=a+b(L)¢Q1;t¡1+cZ t¡1+²t;(1)where Q1;t=(X t;W0t)0is a vector if I(1)variables,W t a n£1vector for some n¸1,Z t¡1=dQ1;t¡1 is a r£1vector of I(0)variables,r is the rank of the cointegrating space,d is an r£(n+1) matrix of cointegrating vectors,a is an(n+1)£1vector,b(L)is a matrix polynomial in the lag operator L,with p terms,each of which is an(n+1)£(n+1)matrix,p is the order of the VEC model,c is an(n+1)£r matrix,and²t is a vector error term.For DGPs generated as linear in levels,we report rejection frequencies for a=(a1;a2)0;a1=a2=f0:0;0:1;0:2g, b=0,c=(c1;c2)0,c1=¡0:2;c2=f0:0;0:2;0:4;0:6g,and¾2²=1:0,i=1;2.For loglineari=0:09,i=1;2.Results for DGPs,b and c are as above,a1=a2=f0:0;0:01;0:02g,and¾2²iother parameterizations examined are qualitatively similar,and are available upon request from the authors.The results of the experiment are quite straightforward.First,the empirical size of the trace test statistic is severely upward biased,with bias increasing as T increases.Further,and as expected,the¯nite sample power(all cases where c2=0)increases rapidly to unity as T increases. Thus,the null of no cointegration is over-rejected.Also,we know that estimators of cointegrating vectors are inconsistent under the wrong data transformation,even if the true cointegrating rank is known.Thus,it is perhaps not surprising that VEC models more clearly dominate VAR models (in di®erences)when the appropriate data transformation is used.3Forecasting Using Vector Error Correction Models3.1DiscussionA number of well known issues arise in the context of the speci¯cation and estimation of forecasting models which have obvious implications for the application of the above procedures.In particular: (1)As noted above,the gains to forecasting associated with the use of VEC models rather than simpler VAR models based on di®erenced data is not clear.(2)It is not clear whether models based on undi®erenced data are dominated by VAR and VEC models based on di®erenced data,even when variables are I(1).(3)The choice of loss function,f,is not always obvious,and certainly this choice depends on the particular objective of the forecaster(see e.g.Christo®ersen and Diebold (1996,1998),Pesaran and Timmerman(1994),Swanson and White(1997),and the references con-tained therein).(4)In a generic forecasting scenario it is not always obvious whether data should be logged or not,and it is not obvious how to compare forecasts of a variable arising from log and level versions of some generic model1(see e.g.Ermini and Hendry(1995)).For example,assume that one is interested in forecasting Y t.In this context,there are a number of choices.First, we must decide whether we want to forecast Y t,¢Y t,log Y t,or¢log Y t.Given this decision,we must decide how to compare the models.For example,if using di®erenced data,we may transform forecasts of¢log Y t into forecasts of¢Y t,or vice¡versa,when comparing models.3.2Monte Carlo ResultsIn this section,we examine all of these issues by conducting a series of Monte Carlo experiments.We begin by assuming that we are interested in constructing forecasts using data which are generated according to the following VEC model:¢Q1;t=a+b(L)¢Q1;t¡1+cZ t¡1+²t;(2)where Q1;t=(X t;W0t)0is a vector if I(1)variables,W t a n£1vector for some n¸1,Z t¡1=dQ1;t¡1 is a r£1vector of I(0)variables,r is the rank of the cointegrating space,d is an r£(n+1)matrix of cointegrating vectors,a is an(n+1)£1vector,b(L)is a matrix polynomial in the lag operator L,with p terms,each of which is an(n+1)£(n+1)matrix,p is the order of the VEC model,c is an (n+1)£r matrix,and²t is a vector error term.In our experiments we let the integratedness of the series be unknown,the rank of the cointegrating space be unknown,p be unknown,and we assume no prior knowledge concerning whether to log the data or not.Also,we set n=1,and the order of the matrix lag polynomial equal to0or1(hence,b(L)=b,say,for simplicity).In all cases, we construct a sequence of P1-step ahead forecasts of X t,and construct average mean square forecast error(AMSE),average mean absolute percentage forecast error(AMAP E),and average mean absolute deviation forecast error(AMAD)criteria,where the average is based on1000 replications.Also,let P=50and P=T=2,where T is the sample gs are selected using the 1Note that here and below we assume that only linear forecasting models are being examined.If this were not the case,then this last issue would not be relevant,as any linear model estimated using levels data could obviously be transformed into some nonlinear model using logged data,and as long as heteroskedasticity etc.were appropriately modelled,there might be little to choose between the models,at least within the context of forecasting(see e.g. Granger and Swanson(1996).BIC criterion.All parameters(including the cointegrating rank)are re-estimated before each new forecast is formed,using an increasing window of observations,starting with T¡P observations, and ending with T¡1observations,so that sequences of P ex¡ante1-step ahead forecasts are constructed for each replication.Data are generated according to the following parameterizations: 1.Data generated as loglinear:Samples are T=100;250,and500observations.a=(a1;a2)0;a1= a2=f0:0;0:001;0:002g;b=(b1;b2);b1=(b11;b21)0,b2=(b12;b22)0,and either b12=b21=b11= b22=0,or b12=b21=0;b11=¡0:4;b22=0:2;and c=(c1;c2)0;c1=¡0:2;c2=f0:2;0:4;0:6g, d=(1;¡1)0,²i;t»IN(0;¾2²i),¾2²i=0:09,i=1;2,and E(²1;t²2;t)=0for any t.2.Data generated as linear in levels:The same parameterizations as above are used,except that=1:0,i=1;2.a1=a2=f0:0;0:1;0:2g and¾2²iIn order to summarize our results,we group our Monte Carlo exercises into four experiments: Experiment I:All simulations are based on data generated as loglinear(actual data are referred to as log X t).Model selection criteria(i.e.AMSE,AMAPE,and AMAD)are constructed using forecasts of log X t.We always estimate two types of models,one using logged data,and the other using levels data.For models estimated using logged data,we immediately have available the appropriate forecast,say d log X t.However,for models estimated using levels data,we only have available a levels forecast,say^X t,and we construct log^X t,in order to compare the model selection criteria across data transformations.Experiment II:All simulations are based on data generated as loglinear(actual data are referred to as log X t).In this case,we construct model selection criteria using forecasts of X t.For models estimated using logged data,we construct exp d(log X t):Note here that we do not make the usual bias adjustment,as this bias adjustment is based on the presumption of normality,presumption which does not generally hold in economic data.For models estimated using levels data,we immediately have available the appropriate forecast,say^X t.Experiment III:All simulations are based on data generated as linear in levels(actual data are referred to as X t).In this case,we construct model selection criteria using forecasts of X t.All forecasts are constructed as in Experiment II.Experiment IV:All simulations are based on data generated as linear in levels(actual data are referred to as X t).In this case,we construct model selection criteria using forecasts of log X t.All forecasts are constructed as in Experiment I.Based on the above framework,we compiled48tables of results,corresponding to ExperimentsI-IV,T=f100,250,500g,p=f1,2g in the VAR(p)DGP,and P=f50,T/2g.Because the results are qualitatively similar,and for the sake of brevity,we present only four tables,corresponding to Experiments I-IV,T=100,p=1,and P=plete results are available from the authors. Tables2-5summarize our¯ndings,and our conclusions are grouped into answers to(1)-(4)above.(1)Although the numerical di®erences are not great,the VEC model in di®erences always hasa lower AMSE than the VAR model in di®erences when the correct data transformation is used to estimate the models and to compare the forecasts.This can be seen by comparing the¯rst and third columns of entries in Tables1and3.Furthermore,note that when the DGP is loglinear,and models are estimated using the correct data transformation,but forecasts are then transformed so that levels forecasts are compared when data are generated and estimated in logs(and vice-versa for levels-linear DGPs),then the VEC models again always have lower AMSE values than their di®erence VAR counterparts(compare the¯rst and third columns of entries in Tables2and4). However,the VEC model clearly does not AMSE-dominate the VAR model in di®erences when incorrectly transformed data are used in forecast construction(compare the second and fourth column of entries in Tables1-4).Thus,VEC models do appear to uniformly dominate VAR model in di®erences,but only when the correct data transformation is used for model estimation,regardless of whether levels or log forecasts are ultimately compared.One reason for this¯nding may be that CI vectors and ranks are not precisely estimated when incorrectly transformed data are used.(2)Undi®erenced data VAR models AMSE-dominate di®erence VEC and VAR models around 50%of the time when correctly transformed data are used in estimation and forecast comparison (compare the¯rst,third,and¯fth columns of entries in Tables1-4).Thus,in some cases,the simplicity of levels models appears to dominate more complex models,even when there is cointe-gration.However,it must be stressed that this¯nding only holds when correctly transformed data are used in forecast model construction(see below).(3)The choice of loss function,f,does appear to make a di®erence in our experiments.In particular,the VEC model no longer beats the VAR in di®erences in every single instance,when the AMAPE and AMAD are used to compare models based on correctly transformed data.Also, the undi®erenced data VAR model which is based on correctly transformed data is less frequently "better"than the VEC model when AMAPE and AMAD(rather than AMSE)are used to compare models.Thus,the choice of loss criterion appears to play an important role in model selection, even when criteria which are very similar,such as AMSE,AMAPE,and AMAD,are used.(4)Choosing the data transformation in some cases appears to play a crucial role when com-paring di®erence VEC and VAR models.For example,consider comparing models using AMSE. Note that the worst of the forecasting models based on correctly transformed data(i.e.choose the worst performer from columns1,3,and5of the entries in Tables1-4)is almost always better than the best of the forecasting models based on incorrectly transformed data(i.e.choose the best performer from columns2,4,and6of Tables1-4).Furthermore,this result is much more apparent for the loglinear DGPs reported on in Tables1and2.When data is generated as level-linear (Tables3and4),there appears to be surprisingly little to choose between data transformation, although the correct transformation is still usually"better"based on our criteria.Finally,there appears to be little to choose between transforming the forecasts from di®erent forecasting models into levels or into logs to facilitate comparisons between forecasts based on di®erent models.This can be seen by noting that the ordinal ranking of the di®erent forecasting models is the same when the corresponding entries in either Tables1and2or Tables3and4are compared.In addition to the above¯ndings,the following points are worth noting.First,although models based on data which have not been di®erenced are often AMSE-best when data are correctly transformed,they are almost always AMSE-worst when comparing forecast performance based on models estimated with incorrectly transformed data(compare column6entries with all other entries in Tables1-4).This suggests that models with undi®erenced data should perhaps only be used in the I(1)context when one is rather sure that the data are correctly transformed.As it is often di±cult to ascertain the"correct"data transformation to use,forecasting models based on data which have not been di®erenced should thus be used with caution in practical applications.Second,we have only indirect evidence on the usefulness of the BIC criterion when data are incorrectly transformed.In particular,one of the reasons why models with incorrectly transformed data perform so poorly relative to models estimated with correctly transformed data may be that there is no guarantee that the correct lag order will be chosen,even in the limit,when the wrong data transformation is used.4ConclusionsIn a series of Monte Carlo experiments we show that data transformation matters when forecasting using vector error correction models,and when testing for unit roots and cointegration using stan-。

Geometric ModelingGeometric modeling is a fundamental concept in the field of computer graphics and design. It involves the creation and manipulation of digital representations of objects and environments using geometric shapes and mathematical equations. This process is essential for various applications, including animation, virtual reality, architectural design, and manufacturing. Geometric modeling plays a crucial role in bringing creative ideas to life and enabling the visualization of complex concepts. In this article, we will explore the significance of geometric modeling from multiple perspectives, including its technical aspects, creative potential, and real-world applications. From a technical standpoint, geometric modeling relies on mathematical principles to define and represent shapes, surfaces, and volumes in a digital environment. This involves the use of algorithms to generate and manipulate geometric data, enabling the creation of intricate and realistic 3D models. The precision and accuracy of geometric modeling are essential for engineering, scientific simulations, and industrial design. Engineers and designers utilize geometric modeling software to develop prototypes, analyze structural integrity, and simulate real-world scenarios. The ability to accurately model physical objects and phenomena in a virtual space is invaluable for testing and refining concepts before they are realized in the physical world. Beyond its technical applications, geometric modeling also offers immense creative potential. Artists and animators use geometric modeling tools to sculpt, texture, and animate characters and environments for films, video games, and virtual experiences. The ability to manipulate geometric primitives and sculpt organic forms empowers creatives to bring their imaginations to life in stunning detail. Geometric modeling software provides a canvas for artistic expression, enabling artists to explore new dimensions of creativity and visual storytelling. Whether it's crafting fantastical creatures or architecting futuristic cityscapes, geometric modeling serves as a medium for boundless creativity and artistic innovation. In the realm of real-world applications, geometric modeling has a profound impact on various industries and disciplines. In architecture and urban planning, geometric modeling software is used to design and visualize buildings, landscapes, and urban developments. This enables architects and urban designers toconceptualize and communicate their ideas effectively, leading to the creation of functional and aesthetically pleasing spaces. Furthermore, geometric modelingplays a critical role in medical imaging and scientific visualization, allowing researchers and practitioners to study complex anatomical structures and visualize scientific data in meaningful ways. The ability to create accurate and detailed representations of biological and physical phenomena contributes to advancementsin healthcare, research, and education. Moreover, geometric modeling is integral to the manufacturing process, where it is used for product design, prototyping,and production. By creating digital models of components and assemblies, engineers can assess the functionality and manufacturability of their designs, leading tothe development of high-quality and efficient products. Geometric modeling also facilitates the implementation of additive manufacturing technologies, such as 3D printing, by providing the digital blueprints for creating physical objects layer by layer. This convergence of digital modeling and manufacturing technologies is revolutionizing the production landscape and enabling rapid innovation across various industries. In conclusion, geometric modeling is a multifaceteddiscipline that intersects technology, creativity, and practicality. Its technical foundations in mathematics and algorithms underpin its applications in engineering, design, and scientific research. Simultaneously, it serves as a creative platform for artists and animators to realize their visions in virtual spaces. Moreover,its real-world applications extend to diverse fields such as architecture, medicine, and manufacturing, where it contributes to innovation and progress. The significance of geometric modeling lies in its ability to bridge the digital and physical worlds, facilitating the exploration, creation, and realization of ideas and concepts. As technology continues to advance, geometric modeling will undoubtedly play an increasingly pivotal role in shaping the future of design, visualization, and manufacturing.。

2024年第1期27doi:10.3969/j.issn.1005-2550.2024.01.005 收稿日期：2023-10-27基于虚拟试验场的牵引车动态载荷研究王庆华1，王丽荣2，陈小华2，李蒙然1，黄刚1（1.国家汽车质量检验检测中心（襄阳），襄阳441004；2. 北京福田戴姆勒汽车有限公司，北京 101400）摘 要：基于Adams软件的虚拟试验场动态载荷分解技术在乘用车耐久性能开发领域广泛应用。

对于重卡车型，由于车辆模型复杂、参数有限且测试难度大，虚拟试验场技术的应用推广受到限制。

搭建某牵引车整车多体动力学模型及虚拟试验场仿真环境，同时采集试验场工况下的实车载荷谱数据并与虚拟试验场动力学仿真分析提取的动态载荷进行对比。

使用相对伪损伤比值、频谱分析等评估比利时、扭曲路、搓板路等典型路面工况下仿真与实测载荷谱数据的差异。

结果表明：基于虚拟试验场的动态载荷提取技术可应用于牵引车车型且可实现较高的精度，是一种获取试验场耐久工况载荷谱的有效方法。

关键词：虚拟试验场；载荷分解；路面模型；牵引车中图分类号：U467 文献标识码：A 文章编号：1005-2550（2024）01-0027-07Research on Dynamic Load of Tractor Based on VPGWANG Qing-hua1, WANG Li-rong2, CHEN Xiao-hua2, LI Meng-ran1, HUANG Gang1(1.National Automobile Quality Inspection and T est Center (Xiangyang), Xiangyang 441004,China; 2. Beijing Foton Daimler Automobile Co., Ltd, Beijing 101400, China)Abstract: The dynamic load decomposition technology of VPG based on Adams is widely applied in the field of passenger car durability performance development. For heavytruck, the application and promotion of VPG are limited due to the complexity of vehiclemodels, limited parameters, and high RLDA testing difficulty. The complete vehicle multi-body dynamics model of a tractor and virtual proving ground simulation environment arebuilt based on Adams. The real vehicle load data acquisition of the proving ground eventswas carried out and compared with the dynamic loads extracted from dynamic simulationanalysis of the virtual proving ground to verify the model accuracy and load accuracy.Relative pseudo damage ratio, RMS value ratio, and spectrum analysis were used to evaluatethe differences between simulated and measured load data under typical road conditionssuch as Belgium, twisted roads, and washboard roads. It is proved that The dynamic loadextraction technology based on virtual proving ground can be applied to tractor models andachieve high accuracy, which is an effective method for obtaining the load data of provingground durability events.Key Words: Virtual Proving Ground; Load Extraction; Road Model; Tractor随着高精度路面扫描和轮胎力学模型建模等技术快速发展，基于虚拟试验场（V i r t u a l Proving Ground）的动态载荷提取技术在车型开发早期阶段即可开展，可有效缩短开发周期和试验成本[1-4]。