Volatility forecasting using threshold heteroskedastic models of the intra-day range

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Computational Statistics & Data Analysis 52 (2008) 29903010www.elsevier.com/locate/csdaVolatility forecasting using threshold heteroskedastic models of theintra-day rangeCathy W.S. Chena,, Richard Gerlachb, Edward M.H. LinaaGraduate Institute of Applied Statistics, Feng Chia University, TaiwanbEconometrics and Business Statistics, University of Sydney, AustraliaAvailable online 14 August 2007AbstractAn effective approach for forecasting return volatility via threshold nonlinear heteroskedastic models of the daily asset pricerange is provided. The range is dened as the difference between the highest and lowest log intra-day asset price. A general modelspecication is proposed, allowing the intra-day highlow price range to depend nonlinearly on past information, or an exogenousvariable such as US market information. The model captures aspects such as sign or size asymmetry and heteroskedasticity, whichare commonly observed in nancial markets. The focus is on parameter estimation, inference and volatility forecasting in a Bayesianframework. An MCMC sampling scheme is employed for estimation and shown to work well in simulation experiments. Finally,competing range-based and return-based heteroskedastic models are compared via out-of-sample forecast performance. Appliedto six international nancial market indices, the range-based threshold heteroskedastic models are well supported by the data interms of nding signicant threshold nonlinearity, diagnostic checking and volatility forecast performance under various volatilityproxies. 2007 Elsevier B.V. All rights reserved.Keywords: Size and sign asymmetry; Volatility model; Conditional autoregressive range (CARR) model; Threshold variable; Bayes inference;MCMC methods1. IntroductionWhen modelling nancial markets, it is important to describe asset return volatility dynamics and subsequentlyforecast volatility. Such forecasts form quite an important input into risk management processes (e.g. value at risk,expected shortfall), option pricing, hedging and many other nancial methods and products. It is well known thatasset return volatility is time-varying and somewhat predictable in sample, however, has proven difcult to forecastaccurately. One reason is that volatility is not directly observable, hence parameters and forecasts in a volatility modelcan be subject to much uncertainty. Setting up a more accurate asset return volatility model will be highly useful forinvestors and market analysts in terms of dynamic hedging, portfolio selection and controlling nancial risk effectively.The family of autoregressive conditional heteroskedastic (ARCH) models by Engle (1982) and generalized ARCH(GARCH) by Bollerslev (1986), has become the widely accepted volatility model. These usually employ (squared)close to close asset returns to model return volatility. However, many papers have shown the intra-day range to be a far Corresponding author. Tel.: +886 4 24517250x4412; fax: +886 4 24517092.E-mail address: chenws@fcu.edu.tw (C.W.S. Chen).0167-9473/$ - see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.csda.2007.08.002C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 2991more efcient measure of return volatility, e.g. see Parkinson (1980), Garman and Klass (1980) and recently Andersenand Bollerslev (1998) and Alizadeh et al. (2002). Mandelbrot (1971) applied the range to examine the existence oflong-term dependence in asset prices, while Beckers (1983) extended the range estimator to incorporate historicalinformation on different variance measures.As such, there are several advantages in directly using the intra-day highlow price range for volatility measurementand forecasting, relative to the use of absolute or squared return data, or even realized variance from intra-day returns.For daily or lower frequency returns, Garman and Klass (1980) point out that . . . intuition tells us that high and lowprices contain more information regarding to volatility than do the opening and closing prices. Corrado and Truong(2004) mentioned that by only looking at opening and/or closing prices we may wrongly conclude that volatility ona given day is small, if the closing price is near the previous opening or closing price, despite large intra-day priceuctuations. Logically, the range is more efcient. Further, highlow price range data are widely available in nancialdatabases, often when high-frequency intra-day returns are not. Also, even when high-frequency intra-day returns areavailable, Andersen and Bollerslev (1998) report that market microstructure issues such as non-synchronous tradingeffects, discrete price observations and bid-ask spreads may limit the effectiveness of intra-day return variances, orrealized intra-day volatility, as volatility forecasts or proxies. As such, intra-day high and low values may bring moreintegrity into volatility estimation.Recently, the range has been used in volatility models: Gallant et al. (1999) and Alizadeh et al. (2002) incorporatedthe range with the stochastic volatility model. Brandt and Jones (2006) proposed a range-based EGARCHmodel, usingobserved range data and a link between the range and volatility, showing that their model had better out-of-sampleforecast performance than standard volatility models. Chou (2005a), using range as a direct measure of volatility,further proposed the conditional autoregressive range model (CARR) to capture range dynamics. The CARR modelis simply a GARCH-type model employing data on the range instead of squared returns. This model is also closelyrelated to the autoregressive conditional duration (ACD) model of Engle and Russell (1998).Black (1976) discovered the now well-known phenomenon of volatility asymmetry: higher volatility is correlatedwith negative shocks in asset returns. GARCH-typemodels devised to capture this trait include the exponential (Nelson,1991), the quadratic (Sentana, 1995), the GJRGARCH (Glosten et al., 1993) and the unobserved componentQ-STARCH (Broto and Ruiz, 2006). Poon and Granger (2003) pointed out that asymmetric volatility models out-performed symmetric models in forecasting asset return volatility; while Bali (2000) showed nonlinear models areuseful for interest rate volatility. In this paper we propose regime-switching for modeling size and/or sign asymmetries,presenting a general nonlinear volatility model for the range. We call this the range-based threshold conditional au-toregressive model, denoted by TARR. The model predicts and allows forecasts of the intra-day range and captures themost important stylized features of stock return volatility: time clustering and size or sign asymmetry. One objective ofthis paper is to demonstrate the usefulness of the range in leading to more precise range-based estimates and forecastsof volatility.What does this new model offer in comparison to the multitude of volatility models in the literature? Firstly, theTARR model will be one of the few that directly model the intra-day range dynamically. Thus it may provide moreefcient volatility estimates and forecasts than models that employ squared or absolute returns only. Secondly, theTARR model will capture asymmetry through a nonlinear specication, allowing threshold information to determinewhere the change of regime occurs.We focus on size asymmetry: as has been discussed by Leeves (2007) among others,employing range data as a threshold variable; and also sign asymmetry, employing a return based threshold variable.The relevant threshold variable could come from the local market or from an exogenous factor such as the US market.Exogenous thresholds have been shown to be important in volatility modeling, see e.g. Chen et al. (2003, 2006) andGerlach et al. (2006). Third, the TARR model will allow the usual trait of volatility clustering or persistence. A traitthat no existing asymmetric volatility model, including ours, will capture is Markov switching in regimes, as e.g. in theMSGARCH models of Gray (1996) and Klaassen (2002), while De Luca and Zuccolotto (2006) proposed a relatedregime switching ACD model. We leave such models for future research.We take a Bayesian approach to estimation, allowing simultaneous inference for model parameters, including thethreshold value and the time delay lag.Also, valid inference under positivity and stationarity constraints for parametersis straightforward in this approach. Numerical methods such as Markov chain Monte Carlo (MCMC) are utilized;shown to be very effective in threshold models by e.g. Chen and Lee (1995) and Chen et al. (2003).This paper is organized as follows. Section 2 describes the range-based TARR model. Sections 3 and 4 presentBayesian methods for estimation and for forecasting volatility. Section 5 presents a simulation study showing the2992 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010estimation performance of the methods in Section 3. Empirical results from six major nancial markets are reportedin Section 6, where we compare the out-of-sample volatility forecasting ability between two TARR and two GARCHmodels. Section 7 offers conclusions.2. Range-based threshold heteroskedastic modelLet Pt be the logarithm of the price of an asset observed at time t, where t = 1, 2, . . . , n and Rt = (Max{Pit } Min{Pit }) 100; Rt is the (percentage log) range at time t and the index i is for the intra-period measurements Pit . Ifwe consider daily stock prices, Max{Pit } and Min{Pit } are the highest and lowest log prices observed during the day;Rt is the intra-day log price range for day t, or the maximum achievable intra-day return. We propose the followingnonlinear TARR(p, q) model:Rt = t t , t f (),t = (j)0 +pji=1(j)i Rti +qji=1(j)i ti , rj1ztdrj , (1)where j = 1, . . . , g, the delay lag d is a positive integer, and f () is a distribution with support (0,) and unitmean. t is assumed to follow a Weibull distribution, with parameters (c, s, ) which are the location, scale and shapeparameters, respectively. The Weibull distribution is a special case of the generalized extreme value distribution andhas been extensively used as a model of time to failure. Recently, its applications have expanded to include nance andclimatology.We set theWeibull location parameter c=0 to ensure range positivity i.e.Rt =t t0.Moreover, the scale parameters is set to [(1 + 1/)]1 to ensure that E(t ) = 1, so that E(Rt |It1) = t , independently of t . The threshold valuesrj satisfy 0C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 2993asymmetry (for a range threshold) and sign asymmetry (for a return threshold) in the range via a piecewise linearrelationship. To ensure positivity and stationarity of the range, we enforce the standard GARCH-type restrictions:(j)0 > 0, (j)i 0, (j)i 0 andpji=1(j)i +qji=1(j)i < 1. (2)The essential distinction between theACARRand theTARRmodel is that theACARRuses two independently estimatedlinear CARR models and compares results to describe the asymmetric volatility. The TARR model proposed here is atruly nonlinear model and allows more traditional asymmetric volatility responses.Gray (1996) and Klaassen (2002) discussed Markov switching (MS) GARCH models and applied these models tointerest rate and exchange rate data. An important difference between threshold models and MS models is that, giventhe threshold limits r, the regime indicators are xed, known and non-stochastic in a threshold model; while in an MSmodel regime indicators are not observed and are still stochastic even if all other model parameters are known. Furtherthe threshold variable in our nonlinear model, which determines the regime at each time point, is observed and is notassumed to generate regimes that persist over time, as is the case for MS models.3. Bayesian inferenceBayesian methods allow simultaneous inference on parameters in nite samples. Stationarity and positivity con-straints can also be incorporated directly into the prior distribution. Silvapulle and Sen (2004) have illustrated problemsin large sample theory inference under positivity and stationarity constraints, problems not shared by the Bayesianapproach. Chen and Lee (1995) and Chen et al. (2003, 2005) provide a rigorous treatment of Bayesian inference viaMCMC methods for nonlinear threshold GARCH models. We adapt their approach.Let represent the vector of all unknown parameters. Dene j = ((j)0 , . . . , (j)pj , (j)1 , . . . , (j)qj ), andr = (r1, . . . , rg1) as the mean and threshold parameter vectors, respectively; d0 as the maximum delay and s =max(p1, . . . , pg, d0). Assuming aWeibull distribution with shape parameter , the conditional likelihood function forthe TARR model isL(Rs+1,T |) =Tt=s+1gj=1[Rt((1 + 1/)Rtt)exp(((1 + 1/)Rtt))]Ijt , (3)where Ijt is the indicator variable I (rj1ztdrj ) and is the Gamma function.We choose a mostly uninformative prior, at over the parameter constraint region in (2), so that the likelihooddominates inference. We also generate parameters in blocks, where practical, to speed convergence of the Markovchain; see Carter and Kohn (1994).The delay has been set as d = 1 in most previous studies of threshold models, see e.g. Li and Li (1996), Brooks(2001), etc. We thus choose the prior of d with large probability for more recent lags, so that Pr(d = i) = (4 i)/6,i = 1, 2, 3, where d0 is 3. To ensure the required constraints on j , we adopt a uniform prior p(j ) over the region (2).Further, we choose the prior of r as p(r) I (B), where B ensures each regime contains at least h percent of ztd . FortheWeibull distribution, the shape parameter is re-parameterized via = log(); ensuring that > 0 as required. Theprior on is an N(0, 1), allowing a large range of possible values for = exp() on the positive real line, but with mostweight on smaller values, as found in Chou (2005a). These are assumed a priori independent, so that:p() = gj=1p(j )p()p(d)p(r). (4)The posterior distribution is proportional to the multiplication of the likelihood in (3) and the prior in (4). Let be,in turn, one of j , d, r and and let be the parameter vector excluding the element . MCMC methods requireconditional posterior distributions for each choice of . In each case, the target posterior isp(|Rs+1,T , ) L(Rs+1,T |)p(). (5)2994 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010The posterior distributions in (5), for each choice of , are of non-standard form, except the delay parameter d, whichis sampled from the multinomial distribution:p(d = j |Rs+1,T , d) = L(Rs+1,T |d = j, d)p(d = j)d0i=1L(Rs+1,T |d = i, d)p(d = i), j = 1, . . . , d0. (6)The MetropolisHastings (MH) algorithm (Metropolis et al., 1953; Hastings, 1970) is applied to draw from (5) for allother parameters.We combine two versions ofMHmethods: the randomwalkMetopolis algorithm and the independentkernel MH algorithm, to achieve the desired parameter samples in an adaptive MCMC sampling scheme. See Chenet al. (2005) for details of this method and its application.We use an MCMC sample of N iterations, deleting the rst M as a burn-in, and keeping the last N M iterationsfor analysis. Tierney (1994) showed that the MCMC sample for each parameter tends towards its marginal distributionp(|R) when the Monte Carlo sample size is large enough; in practice we conrm convergence via traceplots andcorrelograms of sample iterates.4. Forecasting volatility with TARR modelsWe now discuss forecasting with TARR models. A rolling sample method is used with n+ k 1 data points used forestimation successively for k = 1, 2, . . . , T . We provide one and two-step ahead forecasts: Rn+k1+l |R1,n+k1, whereT is the forecast horizon, n + k 1 is the forecast origin and l = 1, 2.Firstly, the conditional mean E(Rt |It1) = t is a known function of and R1,t1 that we label as gt (R1,t1, ).Under the TARR model:gt (R1,t1, ) =gj=1[(j)0 +pji=1(j)i Rti +qji=1(j)i ti]I (rj1ztdrj ),which is a one-step-ahead forecast ofRt when the parameter values are known. The threshold variable ztd is known attime t1 since d1.Note that parameter uncertainty is accounted for in our forecasts, since e.g. [i]n+1=gn+1(R1,n, [i]),i=1, . . . , NM form a posteriorMCMC sample fromp(n+1|R1,n)where [i] is anMCMC sample fromp([i]|R1,n),obtained as in Section 3. A Bayesian forecast of Rn+1|R1,n is then the posterior mean:NMi=1[i]n+1/(N M).The required forecast sample [i]n+1 can be formed simultaneously with the MCMC sample of parameters.For l2 step ahead forecasting, we need to sample [i]n+1, . . . , [i]n+l conditionally on R1,n. This can be done by themethod of composition and MCMC integration with the decomposition:p(n+1, . . . , n+l |R1,n) =lj=1p(n+j |1,n+j1,R1,n)=lj=1p(n+j , Rn+j1, |1,n+j1,R1,n) dRn+j1 d.We can do this integration numerically inside the MCMC parameter sampling scheme as follows:1. Calculate [i]n+1 = gn+1(R1,n, [i]) using the relevant TARR model;2. Simulate [i]n+1 Weibull(0, sn+1, [i]) where sn+1 = 1/(1 + 1/[i]);3. Calculate R[i]n+1 = [i]n+1[i]n+1, which comes from the distribution of Rn+1|R1,n;4. Calculate [i]n+2=gn+2(R1,n, R[i]n+1, [i]n+1, [i]). This comes from the distribution ofE(Rn+2|R1,n, Rn+1) as required;C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 29955. Simulate [i]n+2 Weibull(0, sn+1, [i])....The process is continued up to the calculation of [i]n+l = gn+l (R1,n, R[i]n+l1, [i]n+l1, [i]) which comes from thedistribution of E(Rn+l |R1,n, Rn+l1). The nal forecasts of Rn+l |R1,n are the posterior means:NMi=1[i]n+k/(N M),for k = 1, . . . , l, which estimates E(Rn+l |R1,n). Note that the numerical simulation of the range Rn+l1 is necessaryfor l2, since the threshold variables needed in forecasting, zn+kd , are unobserved for k >d at forecast origin n. Inthe TARRX model, the exogenous threshold zn+kd must also be forecasted, from a model specied by the user, aspart of the algorithm above.Two criteria are commonly used to evaluate the forecast accuracy of competing models: mean squared error (MSE)and mean absolute deviation (MAD). We employ both these measures. MSE provides a quadratic loss function whichdisproportionately weights large forecast errors more heavily relative to MAD. We also formally compare differentforecast models using the DieboldMariano (DM) test for equal forecast accuracy, see Diebold and Mariano (1995)for details.In practice, volatility is unobserved and so the accuracy of any volatility forecasting model is hard to judge. Thesquared close to close return is often used as a volatility proxy when calculating forecast accuracy measures; see e.g.Tsay (2005), Chou (2005a). Parkinson (1980), Garman and Klass (1980) andAlizadeh et al. (2002) propose and discussrange-based volatility proxies and show that squared returns, while being unbiased, are far more noisy and inefcientproxies of volatility than their proposals. We consider four volatility proxies:21,t = y2t ,22,t =R2t4 log(2) 0.3607R2t ,23,t = R2t ,24,t = exp[2 (log(Rt ) 0.43 + 0.292)].The rst proxy is simply the squared close to close log-return. The second proxy was derived by Parkinson (1980); itmakes some sense in that the intra-day range is the maximum achievable return during each day, thus this may needto be down-weighted to better reect volatility. The third proxy was suggested by Chou (2005a). The fourth proxy isderived from work by Alizadeh et al. (2002), who found an accurate probability link between quadratic variation andrange data for Martingale processes. The link is:log(t ) N(log(Rt ) 0.43, 0.292).The proxy is then deduced by the properties of the log-normal distribution: it is equivalent toE(2t ) under the log-normaldistribution for t .Many recent papers have explored the use of intra-day realized volatility as a proxy, which requires the availabilityof intra-period prices. However, we consider only daily data in this paper. We also note that realized volatility is highlyinuenced by the choice of intra-day frequency, which is the subject of much on-going research.5. Simulation studyWe now illustrate the proposed Bayesian methods for simulated data from various parameter settings. We used 100replications with three different sample sizes, n = 1000, 2000 and n = 4000, from three models specied below. Foreach data set, N = 20 000 MCMC iterations were used; with a burn-in of M = 8000 iterations. We selected all initial2996 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010MCMC iterates randomly from their prior distribution. The true models are specied as:Model 1: The true model is TARRX (1, 1):Rt = t t ,t ={0.20 + 0.18Rt1 + 0.69t1, zt11.50,0.38 + 0.21Rt1 + 0.61t1, zt1 > 1.50,while zt CARR(1, 1)CARR(1, 1) : z,t = 0.04 + 0.18zt1 + 0.79z,t1,where t is a Weibull with unit mean and shape = 1.2; the threshold has z = 1.2.We set the parameters in Model 1 to reect empirical observations from threshold GARCH models and our real dataexamples that the unconditional volatility (range) is larger in regime two than in regime one. The remaining modelsalso follow this criterion.Model 2: Model 2 is the same as Model 1 except the parameter values:Rt = t t ,t ={0.10 + 0.20Rt1 + 0.70t1, zt11.50,0.45 + 0.20Rt1 + 0.65t1, zt1 > 1.50,while zt CARR(1, 1)CARR(1, 1) : z,t = 0.04 + 0.18zt1 + 0.79z,t1.Model 3: The true model is TARR (1, 1):Rt = t t ,t ={0.04 + 0.13Rt1 + 0.80t1, Rt10.87,0.11 + 0.19Rt1 + 0.71t1, Rt1 > 0.87,where t follows a Weibull with unit mean and shape = 1.2.We set the parameter values in these Models to reect our empirical observations both from threshold GARCHmodels and from the TARR model applied to our real data examples in the next section. These are:1. The unconditional mean of the range (volatility) is greater in regime 2 than regime 1;2. The volatility persistence 1 + 1 is higher in regime 1 than regime 2,when using a range threshold variable. The actual values chosen inmodels 13 are very similar to the estimates obtainedin the real examples, except for . We chose to be more extreme and close to 1 to ensure an error distribution thatwas quite heavily skewed in the positive direction. This was to ensure we properly tested our estimation method on areasonably non-normal error distribution.We selected the prior for r as U(Q1,Q3), where Q1 and Q3 are the rst and third sample quartiles separately,which is a common choice, and the maximum lag for the delay term, d, is chosen to be 3, which is higher than what isusually found or assumed in practice. When simulating data from these models, MCMC convergence appeared almostimmediate from trace plots of the iterates.Table 1 displays the true values plus the means, standard deviations and the 2.5th and 97.5th percentiles of the 100posterior mean estimates for the model parameters for models 13, for sample sizes n = 1000, 2000 and n = 4000.Posterior modes (not shown) were all chosen as d = 1. Also displayed are true values and average posterior estimatesfor the unconditional mean range (volatility) in each regime. There appears to be a lot of sampling variation at n=1000so that bias is difcult to judge, however, all true values are contained inside the 2.5th and 97.5th percentiles for theC.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 2997Table 1Simulation results for models 13 from 100 replicationsModel 1 n = 1000 n = 2000 n = 4000Parameter True Est. Std. 2.5% 97.5% Est. Std. 2.5% 97.5% Est. Std. 2.5% 97.5%(1)0 0.20 0.2527 0.0763 0.1235 0.4192 0.2223 0.0547 0.1291 0.3438 0.2057 0.0384 0.1377 0.2882(1)1 0.18 0.1837 0.0350 0.1206 0.2585 0.1736 0.0239 0.1294 0.2233 0.1716 0.0169 0.1394 0.2058(1)1 0.69 0.6275 0.0702 0.4759 0.7465 0.6601 0.0508 0.5502 0.7491 0.6738 0.0366 0.5975 0.7403(2)0 0.38 0.3361 0.1072 0.1196 0.5092 0.3464 0.0956 0.1558 1.5073 0.3598 0.0866 0.1831 0.5050(2)1 0.21 0.2206 0.0752 0.0942 0.3918 0.2004 0.0507 0.1097 0.3095 0.2082 0.0383 0.1392 0.2901(2)1 0.61 0.6037 0.0987 0.4019 0.7779 0.6176 0.0805 0.4587 0.7670 0.5961 0.0707 0.4652 0.7372 1.20 1.2005 0.0298 1.1427 1.2596 1.2048 0.0210 1.1641 1.2465 1.2015 0.0148 1.1726 1.2309r 1.50 1.3857 0.3458 0.6616 1.8946 1.4573 0.2952 0.8103 1.8940 1.4583 0.2501 0.9164 1.8549(1)1.54 1.3465 1.3429 1.3346 (2)2.11 1.9895 1.9627 1.8748 Model 2(1)0 0.10 0.1384 0.0317 0.0874 0.2122 0.1069 0.0195 0.0732 0.1500 0.1001 0.0141 0.0750 0.1303(1)1 0.20 0.2132 0.0304 0.1585 0.2777 0.1984 0.0205 0.1608 0.2411 0.1897 0.0146 0.1627 0.2197(1)1 0.70 0.6318 0.0393 0.5372 0.6880 0.6783 0.0273 0.6167 0.7230 0.6923 0.0209 0.6477 0.7291(2)0 0.45 0.5172 0.1092 0.3078 0.7176 0.4742 0.0935 0.3062 0.6668 0.4451 0.0689 0.3219 0.5918(2)1 0.20 0.2409 0.0752 0.1091 0.4025 0.2022 0.0516 0.1094 0.3120 0.1930 0.0350 0.1288 0.2661(2)1 0.65 0.5538 0.0881 0.3552 0.6895 0.6133 0.0714 0.4545 0.7275 0.6342 0.0549 0.5146 0.7275 1.20 1.1968 0.0297 1.1395 1.2556 1.2006 0.0210 1.1598 1.2419 1.2000 0.0148 1.1712 1.2291r 1.50 1.5192 0.1433 1.2479 1.7897 1.5067 0.0824 1.3641 1.6606 1.4908 0.0349 1.4270 1.5606(1)1.00 0.8865 0.8611 0.8462 (2)3.00 2.6286 2.6840 2.6518 Model 3(1)0 0.04 0.0526 0.0239 0.0171 0.1105 0.0446 0.0139 0.0209 0.0754 0.0408 0.0093 0.0239 0.0606(1)1 0.13 0.1342 0.0555 0.0335 0.2438 0.1135 0.0397 0.0376 0.1931 0.1250 0.0276 0.0686 0.1767(1)1 0.80 0.7613 0.0594 0.6201 0.8523 0.7891 0.0289 0.7281 0.8429 0.7912 0.0206 0.7469 0.8286(2)0 0.11 0.1343 0.0762 0.0288 0.3190 0.1013 0.0486 0.0232 0.2102 0.1063 0.0360 0.0405 0.1815(2)1 0.19 0.1862 0.0563 0.0853 0.3029 0.1850 0.0377 0.1145 0.2631 0.1813 0.0273 0.1293 0.2365(2)1 0.71 0.6718 0.0895 0.4722 0.8206 0.7042 0.0547 0.5832 0.7985 0.7081 0.0394 0.6254 0.7799 1.20 1.2013 0.0298 1.1437 1.2609 1.2038 0.0211 1.1629 1.2456 1.2014 0.0149 1.1724 1.2308r 0.87 0.7746 0.1697 0.3794 0.9820 0.7965 0.1387 0.4699 0.9792 0.8383 0.0956 0.5997 0.9666(1)0.57 0.5207 0.4579 0.4989 (2)1.10 0.9733 0.9143 0.9532 (i) = (i)0 /(1 (i)1 (i)1 ), where i = 1, 2.posterior means, except for 1 for Model 2, at this sample size. At higher sample sizes, all parameter estimates seemunbiased and are all not signicantly different from their true values. The precision and accuracy of the estimatorsthus increases with sample size. Also, the lag d = 1 is correctly identied from all 100 replications in each modeland at each sample size the unconditional means in each regime appear close to their true values. In summary,parameter estimation appears to work well for the proposed asymmetric TARR models using the methods inSection 3. Favorable results are also achieved across two choices of threshold variable: past ranges or an exogenous rangefactor.2998 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 299030106. Empirical studyIn this section, we analyze the intra-day highlow prices from six stock markets, obtained from the websitenance.yahoo.com. The data are collected from January 1, 1998, through December 31, 2004, and includes theAsia-Pacic nancial markets: namely Nikkei 225 Index (Japan), KOSPI Composite Index (South Korea), Taiwan weightedindex (Taiwan), HANG SENG Index (Hong Kong), Straits Times Index (Singapore), andAll Ordinaries Index (AORD,Australia). Daily intra-day range series are generated via the formula in Section 2: Rt = (Max{Pit }Min{Pit }) 100.The time series plots of the intra-day range series are given in Fig. 1. Most series seem to move from a higher to alower volatility period over the time span.To understand the characteristics of each nancial market, we briey discuss the summary statistics of the intra-dayrange in Table 2. From the JarqueBera test we see that all markets fail the normality assumption: the ranges arepositively skewed and display leptokurtosis. It is evident that the KOSPI range series is the most volatile, theAORD theleast. In the last column, the lag 10 LjungBox statistic shows each markets range series displays highly signicantserial correlation, suggesting a CARR-type model may be appropriate.Japan0 500 1000 1500024680246802468101224620468246810Korea0 500 1000 1500Taiwan0 500 1000 1500Hong Kong0 500 1000 1500Singapore0 500 1000 1500Australia0 500 1000 1500Fig. 1. Time series plots of Asian and Australian nancial market intra-day ranges from January 1, 1998 to December 31, 2004.C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 2999Table 2Summary statisticsObs. Mean Min. Max. Std. dev. Skewness Excess kurtosis JarqueBera test Q(10)Japan 1662 1.7269 0.2993 8.9294 0.8111 1.8804 4.9330 5337.5198 1071.9561(< 0.001) (< 0.001)Korea 1663 2.5522 0.5130 10.5060 1.2715 1.2976 0.7182 827.4361 2581.0053(< 0.001) (< 0.001)Taiwan 1664 1.8769 0.1461 9.5013 0.9761 1.8174 2.9865 340.7276 1362.0289(< 0.001) (< 0.001)Hong Kong 1680 1.7905 0.4138 9.0348 0.9831 2.0659 4.8555 5514.5995 3109.7643(< 0.001) (< 0.001)Singapore 1706 1.5988 0.3896 12.9511 1.0393 3.3189 18.4183 3574.8925 2464.5437(< 0.001) (< 0.001)Australia 1727 0.8862 0.1814 7.1280 0.5026 2.7184 16.4923 29467.5496 1932.4600(< 0.001) (< 0.001)The data consist of daily intra-day ranges for six stock markets from January 1, 1998 to December 31, 2004.Table 3Estimates of parameters (and standard errors) for stock index intra-day ranges of Japan, Korea and Taiwan from the various TARR and CARRmodelsCountry threshold Japan Korea TaiwanLocal U.S. Return None Local U.S. Return None Local U.S. Return None(1)0 0.065 0.180 0.190 0.118 0.129 0.093 0.277 0.086 0.182 0.122 0.248 0.113(0.043) (0.051) (0.071) (0.027) (0.048) (0.025) (0.076) (0.021) (0.071) (0.037) (0.068) (0.024)(1)1 0.065 0.142 0.161 0.157 0.135 0.229 0.243 0.219 0.074 0.196 0.165 0.196(0.048) (0.020) (0.035) (0.017) (0.060) (0.022) (0.037) (0.020) (0.054) (0.024) (0.034) (0.018)(1)1 0.864 0.735 0.792 0.774 0.786 0.732 0.663 0.748 0.779 0.732 0.730 0.745(0.041) (0.043) (0.053) (0.027) (0.035) (0.027) (0.048) (0.024) (0.042) (0.037) (0.045) (0.026)(2)0 0.172 0.412 0.213 0.473 0.329 0.023 0.133 0.204 0.134 (0.061) (0.100) (0.048) (0.118) (0.181) (0.020) (0.060) (0.084) (0.045)(2)1 0.151 0.191 0.125 0.156 0.194 0.204 0.214 0.198 0.223 (0.025) (0.041) (0.018) (0.028) (0.060) (0.026) (0.028) (0.035) (0.025)(2)1 0.750 0.638 0.728 0.700 0.713 0.777 0.715 0.712 0.690 (0.029) (0.062) (0.037) (0.039) (0.092) (0.028) (0.032) (0.057) (0.038) 2.396 2.436 2.402 2.384 2.656 2.645 2.577 2.639 2.373 2.366 2.326 2.365(0.038) (0.041) (0.042) (0.038) (0.046) (0.047) (0.047) (0.045) (0.040) (0.041) (0.042) (0.040)r 1.298 2.055 0.717 2.516 2.787 0.507 1.493 1.805 0.642 (0.181) (0.032) (0.068) (0.244) (0.396) (0.138) (0.211) (0.357) (0.203)d 1 1 1 1 1 1 1 1 2 (1)a 0.848 1.460 4.596 1.685 2.337 2.964 1.221 1.695 2.511 (2)1.759 2.418 1.455 3.281 3.574 1.323 1.858 2.237 1.580 Percentageb 85.58 100.00 0.00 97.85 92.88 2.84 86.84 98.15 1.28 a (i) = (i)0 /(1 (i)1 (i)1 ), where i = 1, 2.bThe row Percentage denotes the percentage of MCMC iterates where the unconditional mean of regime 1 is less then that of regime 2.For application to each market, we rstly consider four range-based models: the symmetric linear CARR model, theTARR with local domestic market (endogenous) range threshold, the TARRX model with (exogenous) US Standard &Poor 500 Index market range threshold and nally a model which has the US market return as the threshold variable,denoted TARRX-R. Details of these models were given in Section 2.The TARR and TARRX models capture size asymmetry only, while the TARRX-R allows sign asymmetry. Thechoice of US market range as the exogenous variable is based on the global economic scale and inuence of the US3000 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010Table 4Estimates of parameters (and standard errors) for stock index intra-day ranges of Hong Kong, Singapore and Australia from the various TARR andCARR modelsCountry threshold Hong Kong Singapore AustraliaLocal U.S. Return None Local U.S. Return None Local U.S. Return None(1)0 0.048 0.023 0.033 0.022 0.032 0.016 0.080 0.033 0.046 0.020 0.102 0.017(0.015) (0.012) (0.011) (0.009) (0.018) (0.011) (0.026) (0.011) (0.012) (0.013) (0.020) (0.006)(1)1 0.058 0.158 0.167 0.163 0.115 0.200 0.307 0.212 0.085 0.140 0.244 0.185(0.020) (0.025) (0.015) (0.013) (0.035) (0.016) (0.034) (0.015) (0.029) (0.032) (0.014) (0.017)(1)1 0.897 0.826 0.828 0.825 0.851 0.783 0.679 0.768 0.824 0.791 0.739 0.796(0.014) (0.025) (0.016) (0.015) (0.024) (0.018) (0.037) (0.016) (0.023) (0.040) (0.025) (0.020)(2)0 0.032 0.048 0.032 0.023 0.235 0.063 0.151 0.078 0.022 (0.029) (0.030) (0.024) (0.017) (0.065) (0.019) (0.036) (0.017) (0.009)(2)1 0.230 0.175 0.090 0.276 0.254 0.141 0.147 0.216 0.138 (0.022) (0.040) (0.028) (0.024) (0.035) (0.017) (0.029) (0.022) (0.018)(2)1 0.729 0.804 0.857 0.684 0.623 0.803 0.729 0.717 0.806 (0.026) (0.046) (0.032) (0.029) (0.060) (0.021) (0.035) (0.030) (0.024) 2.593 2.558 2.523 2.556 2.257 2.264 2.218 2.244 2.298 2.309 2.308 2.271(0.045) (0.044) (0.045) (0.043) (0.037) (0.038) (0.038) (0.036) (0.038) (0.038) (0.040) (0.036)r 2.030 1.738 0.579 1.445 1.806 0.712 0.887 1.053 0.716 (0.091) (0.335) (0.160) (0.081) (0.042) (0.038) (0.012) (0.024) (0.016) d 1 1 1 1 1 1 1 1 1 (1)1.061 1.411 7.328 0.932 0.915 7.260 0.496 0.270 5.390 (2)a 0.639 2.305 0.639 0.485 1.902 1.128 1.207 1.146 0.389 Percentageb 23.76 87.70 0.00 14.06 97.28 0.00 100.00 100.00 0.00 a (i) = (i)0 /(1 (i)1 (i)1 ), where i = 1, 2.bThe row Percentage denotes the percentage of MCMC iterates where the unconditional mean of regime 1 is less then that of regime 2.market and on the ndings of strong US inuence in nonlinear GARCH models by Chen et al. (2003), Gerlach et al.(2006) and Chen and So (2006).Parameter estimates are given in Tables 3 and 4. These illustrate clear asymmetric behavior in volatility in all markets.For size asymmetry the usual response would be to have higher volatility following large positive movements in therange variable; while for sign asymmetry, large negative returns are followed by higher volatility: both these responsesare clearly evident in our results.For the TARR (Local) and TARRX (US) models, (2)0 is greater than (1)0 in most markets, and subsequentlythe unconditional mean range (volatility) in the second regime is also larger than that in the rst regime, exceptingthe TARR model in Hong Kong and Singapore. For the TARRX-R model, the reverse occurs with average volatilityhigher in regime one, following negative returns (below approximately 0.5% or 0.5% in each market), than regimetwo. This asymmetry is further shown in Fig. 2, which displays the estimated mean range in each regime for theTARRX and TARRX-R models. We measure the signicance of this nding by calculating the percentage of MCMCiterates where the unconditional mean of regime 1 (low volatility regime) is less than that of regime 2 (high volatilityregime) for all TARR models; also presented in Tables 3 and 4 and denoted Percentage. The 10 out of 12 estimatedmodels (with range threshold) have this percentage above 85%, while six models are above 95%, showing clearevidence of a signicant difference in average range in each regime, and clear size asymmetric behavior in volatilityin all markets, in response to previous range observations. For the TARRX-R model this percentage is less than 3%in all markets, showing that regime one volatility is signicantly higher than in regime two and showing clear signasymmetry.Further, the persistence estimate in the rst regime is higher than that in the second regime, i.e. (1)1 +(1)1 > (2)1 +(2)1 ,in most markets for the TARR and TARRX models, while again the reverse occurs for the TARRX-R model. We alsoC.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 3001Fig. 2. The estimated mean range in each regime for each model.see in most markets the ARCH terms (1) for TARR and TARRX are signicantly higher in the second regime, whilethe GARCH terms (1) tend to be signicantly lower in the second regime. This indicates that ranges are more volatileand less smooth in the second regime, following a high local or US range observation. There is no clear pattern in theseparameters across markets for the TARRX-R model.These results illustrate clear size and sign asymmetric nonlinear volatility behavior: higher average intra-dayvolatility (range) with lower persistence, tends to follow the day after (d = 1) high intra-day volatility (range),either from the domestic or US market (i.e. zt1 >r1). Also, higher volatility generally follows large negative re-turns or falls in the US market. This is similar to the bad news argument often applied to threshold asymmet-ric models: high volatility (often in the form of large negative returns) leads to more high volatility and viceversa.For each model and market, we perform residual tests of model adequacy. The residuals here are from a Weibulldistribution; as such they are not symmetric nor Gaussian and the standard LjungBox Q statistics can not be used.3002 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010Table 5LjungBox tests for autocorrelation in terms of the standardized transformed residuals up to the 20th lag from the various TARR and CARR modelsThreshold Japan Korea TaiwanLocal U.S. Return None Local U.S. Return None Local U.S. Return NoneQ(5) 9.956 8.424 8.210 9.658 5.282 5.154 5.373 5.769 4.229 4.552 5.218 4.736(0.077) (0.134) (0.145) (0.086) (0.382) (0.397) (0.372) (0.329) (0.517) (0.473) (0.390) (0.449)Q(10) 19.942 21.287 21.827 20.499 9.748 9.889 8.746 10.536 8.240 8.303 9.008 9.370(0.030) (0.019) (0.016) (0.025) (0.463) (0.450) (0.556) (0.395) (0.605) (0.599) (0.531) (0.497)Q(15) 24.365 27.933 28.678 25.611 13.027 12.818 11.554 14.030 19.545 18.745 20.118 19.269(0.059) (0.022) (0.018) (0.042) (0.600) (0.616) (0.712) (0.523) (0.190) (0.226) (0.167) (0.202)Q(20) 32.212 32.974 33.803 34.424 14.690 16.025 13.709 16.556 24.733 23.895 27.409 24.361(0.041) (0.034) (0.027) (0.023) (0.794) (0.715) (0.845) (0.682) (0.212) (0.247) (0.124) (0.227)Threshold Hong Kong Singapore AustraliaLocal U.S. Return None Local U.S. Return None Local U.S. Return NoneQ(5) 2.496 3.038 2.542 2.914 23.936 23.436 31.930 28.812 4.591 4.734 6.218 4.990(0.777) (0.694) (0.770) (0.713) (0.000) (0.000) (0.000) (0.000) (0.468) (0.449) (0.286) (0.417)Q(10) 6.409 8.055 7.775 7.920 33.095 31.427 38.179 41.545 9.399 9.405 10.486 10.582(0.780) (0.624) (0.651) (0.637) (0.000) (0.001) (0.000) (0.000) (0.495) (0.494) (0.399) (0.391)Q(15) 10.265 10.361 11.844 9.545 44.792 42.538 50.020 52.345 12.983 10.686 13.986 14.284(0.803) (0.797) (0.691) (0.847) (0.000) (0.000) (0.000) (0.000) (0.604) (0.775) (0.527) (0.504)Q(20) 12.928 17.457 16.023 18.615 46.243 43.742 50.828 53.857 19.617 23.284 23.006 20.829(0.880) (0.623) (0.715) (0.547) (0.001) (0.002) (0.000) (0.000) (0.482) (0.275) (0.289) (0.407)P-values are in parentheses.We turn to the transformed residuals of Smith (1985)t = Rtt f (), E[t ] = 1 F(t ) = Pr(t t |) U [0, 1] 1[F(t )] N(0, 1).This method allows residuals from any distribution to be transformed to achieve standard normality, if the correctmodel has been t to the data: now the LjungBox Q statistics apply. We display the results for each stock market, inTable 5, up to the 20th lag. The models seem to t the data well in all markets except Japan and Singapore. The modelsin Japan are marginally acceptable, especially accounting for multiple testing effects. However, all models are stronglyrejected in Singapore. Perhaps theWeibull distribution is not appropriate for the range in Singapore or perhaps a higherlag structure is needed in the TARR models. Clearly, in four or ve markets we have quite acceptable models for thedynamic intra-day range; however, the diagnostics do not distinguish between the different TARRmodels or the CARRmodel in any of these markets.6.1. Out-of-sample forecastingTo evaluate the performance between the competing range-based models, we consider forecasting the range in eachmarket. Let Rt+l |It denote the l-step-ahead forecast for the conditional range, from forecast origin time t, wheret = 1, . . . , (N T + l 1); N is the total sample size and T is the forecast horizon.We use T = 120 days horizon and rolling samples to generate 120 one and two-step-ahead range forecasts. As weforecast together with our MCMC parameter sampling method, there are 12 000 MCMC iterates of [i]n+k+l simulatedfrom the predictive distribution p(n+k+l |R1,n+k), for k = 0, . . . , (120 l).We rstly compare the range-based models for their ability to forecast the intra-day range. Table 6 compares theforecast ranks (higher rank has lower forecast error under each measure) for l = 1, 2 step ahead forecasts from theseC.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 3003Table 6Ranks for competing range-based forecast models in each market when forecasting the daily intra-day rangeHorizon 1 2CARR TARR TARRX TARRX-R CARR TARR TARRX TARRX-RMSEJapan 3 4 2 1 2 4 3 1Korea 3 4 2 1 3 4 2 1Taiwan 4 3 2 1 4 3 2 1Hong Kong 3 1 2 4 3 2 1 4Singapore 2 3 1 4 1 2 3 4Australia 2 1 3 4 1 2 3 4Sum 17 16 12 15 14 17 14 15MADJapan 3 4 2 1 2 4 3 1Korea 3 4 1 2 2 4 1 3Taiwan 4 3 1 2 3 2 1 4Hong Kong 3 1 2 4 3 2 1 4Singapore 3 2 1 4 3 2 1 4Australia 2 1 4 3 2 1 4 3Sum 18 15 11 16 15 15 11 19Table 7Estimates of parameters (and standard errors) for daily stock index returns from GARCH-t and GJR-GARCH-t modelsCountry model Japan Korea TaiwanGARCH GJR GARCH GJR GARCH GJR0 0.1636 0.1720 0.1127 0.0608 0.1308 0.1773(0.0429) (0.0400) (0.0373) (0.0326) (0.0421) (0.0505)1 0.0692 0.0335 0.0826 0.0491 0.0883 0.0223(0.0172) (0.0165) (0.0149) (0.0132) (0.0181) (0.0171)1 0.8637 0.8544 0.9052 0.9232 0.8750 0.8567(0.0279) (0.0257) (0.0163) (0.0160) (0.0260) (0.0312)w 0.0866 0.0475 0.1440 (0.0303) (0.0214) (0.0316) 10.2843 9.6230 6.3175 6.6092 7.9236 8.0390(2.2080) (1.9343) (0.9125) (1.0009) (1.5948) (1.5057)Country model Hong Kong Singapore AustraliaGARCH GJR GARCH GJR GARCH GJR0 0.0467 0.0295 0.1009 0.0880 0.0112 0.0146(0.0190) (0.0116) (0.0291) (0.0279) (0.0046) (0.0045)1 0.0713 0.0011 0.1408 0.0687 0.0873 0.0059(0.0123) (0.0082) (0.0225) (0.0200) (0.0146) (0.0164)1 0.9176 0.9453 0.8215 0.8432 0.8963 0.9079(0.0151) (0.0112) (0.0258) (0.0271) (0.0180) (0.0179)w 0.0928 0.1069 0.1288 (0.0164) (0.0306) (0.0249) 6.3211 7.2096 6.7347 7.1092 9.2979 8.1956(0.8075) (1.2234) (1.0138) (1.1482) (1.3838) (1.2697)GARCH-t model: yt = at ; at = ht t ; ht = 0 + 1a2t1 + 1ht1; t t. GJR-GARCH-t model: yt = at ; at =ht t ; ht = 0 + 1a2t1 +1ht1 + wa2t1I (at1 < 0); t t.3004 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010Table 8MSE, MAD measures and DieboldMariano DM test for equal l-step-ahead forecast accuracy of daily stock market volatility Japan, Korea, andTaiwan proxied by 21,tHorizon GARCH GJR (A) (B) (C) (A) (B) (C)TARRX TARRX TARRX TARRX-R TARRX-R TARRX-RJapanMSE 1 2.117 2.145 2.140 1.540 1.641 2.181 1.598 1.690(0.847) (0.625) (0.000) (0.014) (0.815) (0.001) (0.030)MSE 2 2.138 2.056 2.282 1.571 1.656 2.132 1.559 1.660(0.001) (0.978) (0.000) (0.016) (0.468) (0.000) (0.019)MAD 1 1.251 1.256 1.271 0.873 0.814 1.266 0.887 0.825(0.678) (0.807) (0.000) (0.000) (0.722) (0.000) (0.000)MAD 2 1.259 1.225 1.317 0.890 0.826 1.256 0.874 0.814(0.000) (0.997) (0.000) (0.000) (0.457) (0.000) (0.000)KoreaMSE 1 6.868 6.615 6.210 5.331 5.669 7.909 5.398 5.559(0.023) (0.006) (0.001) (0.033) (0.996) (0.000) (0.014)MSE 2 6.828 6.373 6.362 5.466 5.782 7.716 5.310 5.502(0.001) (0.017) (0.007) (0.067) (0.972) (0.000) (0.012)MAD 1 2.084 1.972 1.992 1.400 1.331 2.342 1.493 1.384(0.000) (0.028) (0.000) (0.000) (1.000) (0.000) (0.000)MAD 2 2.077 1.918 1.996 1.438 1.371 2.322 1.506 1.402(0.000) (0.046) (0.000) (0.000) (1.000) (0.000) (0.000)TaiwanMSE 1 4.326 4.467 4.218 3.581 3.708 4.801 3.658 3.720(0.916) (0.256) (0.008) (0.062) (0.995) (0.014) (0.063)MSE 2 4.326 4.279 4.305 3.629 3.744 4.859 3.615 3.675(0.331) (0.425) (0.019) (0.083) (0.997) (0.007) (0.046)MAD 1 1.653 1.708 1.629 1.148 1.065 1.783 1.189 1.083(0.975) (0.244) (0.000) (0.000) (0.999) (0.000) (0.000)MAD 2 1.679 1.651 1.646 1.157 1.068 1.824 1.202 1.089(0.142) (0.148) (0.000) (0.000) (1.000) (0.000) (0.000)P-values are in parentheses.1. For the real (proxied) volatility of each market, 21,t is the squared return.2. For model TARRX and TARRX-R, estimated ht s are denoted by (A) for R2t , (B) for exp[2 (log(Rt ) 0.43 + 0.292)] and (C) for R2t /4 ln(2).four models, over the 120 day period in each market. Combining across MSE and MAD measures, in each marketthe TARRX, with exogenous US range-based threshold variable, seems the favored forecasting model; except inAustralia, where the TARR model with local range threshold seems preferred and Japan, where the TARRX-R modelis consistently preferred. Here we again see the strong inuence the US market has over volatility in internationalmarkets. Over all markets combined, the TARRX is clearly the preferred and best forecasting model, with competitionfrom the TARRX-R model only under the MSE measure; this sign-asymmetric model ranks less well under MAD.Wenote that MAD would seem a more effective and logical measure of forecast accuracy for the positively skewed rangeC.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 3005Table 9MSE, MAD measures and DieboldMariano DM test for equal l-step-ahead forecast accuracy of daily stock market volatility for Hong Kong,Singapore, and Australia proxied by 21,tHorizon GARCH GJR (A) (B) (C) (A) (B) (C)TARRX TARRX TARRX TARRX-R TARRX-R TARRX-RHong KongMSE 1 1.225 1.097 1.199 0.942 0.987 1.417 0.940 0.963(0.000) (0.266) (0.005) (0.041) (0.996) (0.001) (0.018)MSE 2 1.214 1.039 1.123 0.920 0.975 1.341 0.904 0.938(0.000) (0.018) (0.005) (0.047) (0.971) (0.001) (0.016)MAD 1 0.966 0.892 0.940 0.672 0.624 1.045 0.710 0.640(0.000) (0.111) (0.000) (0.000) (0.999) (0.000) (0.000)MAD 2 0.968 0.864 0.914 0.661 0.620 1.022 0.695 0.628(0.000) (0.006) (0.000) (0.000) (0.985) (0.000) (0.000)SingaporeMSE 1 0.450 0.416 0.404 0.262 0.279 0.525 0.283 0.286(0.000) (0.045) (0.000) (0.000) (0.987) (0.000) (0.000)MSE 2 0.486 0.417 0.433 0.285 0.297 0.547 0.290 0.290(0.000) (0.035) (0.000) (0.001) (0.969) (0.000) (0.000)MAD 1 0.585 0.557 0.538 0.344 0.329 0.624 0.373 0.343(0.000) (0.004) (0.000) (0.000) (0.980) (0.000) (0.000)MAD 2 0.618 0.561 0.553 0.360 0.337 0.646 0.382 0.347(0.000) (0.000) (0.000) (0.000) (0.944) (0.000) (0.000)AustraliaMSE 1 0.072 0.068 0.106 0.063 0.064 0.087 0.068 0.071(0.063) (0.999) (0.011) (0.079) (1.000) (0.251) (0.473)MSE 2 0.073 0.069 0.115 0.070 0.071 0.082 0.067 0.071(0.018) (1.000) (0.306) (0.345) (0.998) (0.124) (0.362)MAD 1 0.223 0.215 0.285 0.177 0.161 0.260 0.174 0.159(0.017) (1.000) (0.000) (0.000) (1.000) (0.000) (0.000)MAD 2 0.228 0.214 0.286 0.184 0.166 0.251 0.173 0.157(0.000) (1.000) (0.000) (0.000) (1.000) (0.000) (0.000)P-values are in parentheses.1. For the real (proxied) volatility of each market, 21,t is the squared return.2. For models TARRX and TARRX-R, estimated ht s are denoted (A) for R2t , (B) for exp[2 (log(Rt ) 0.43 + 0.292)] and (C) forR2t /4 ln(2).data we employ; MSE tends to heavily penalise for a small number of large (positive) errors, which are to be expectedin range and volatility type data.We now compare the TARRXmodel with two standard and popular volatility models: the symmetric GARCHmodeland the asymmetric GJRGARCH model of Glosten et al. (1993); both these models are assumed to have t-distributederrors.We also include the TARRX-R model so as to include a sign-asymmetric range-based model.We again considerMSE and MAD and use the DM test to compare each model with the GARCH model. We consider l = 1, 2 day aheadforecasts.3006 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010Table 10MSE, MAD measures and DieboldMariano DM test for equal l-step-ahead forecast accuracy of daily stock market volatility proxied by 22,tHorizon Japan KoreaGARCH GJR TARRX TARRX-R GARCH GJR TARRX TARRX-RMSE 1 1.548 1.589 0.170 0.165 3.104 2.717 0.650 0.647(0.945) (0.000) (0.000) (0.000) (0.000) (0.000)MSE 2 1.649 1.535 0.175 0.169 3.356 2.716 0.575 0.584(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)MAD 1 1.194 1.207 0.342 0.339 1.621 1.482 0.557 0.592(0.911) (0.000) (0.000) (0.000) (0.000) (0.000)MAD 2 1.239 1.188 0.359 0.347 1.704 1.493 0.541 0.579(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)Horizon Taiwan Hong KongGARCH GJR TARRX TARRX-R GARCH GJR TARRX TARRX-RMSE 1 1.896 2.022 0.470 0.499 0.681 0.541 0.101 0.102(0.911) (0.000) (0.000) (0.000) (0.000) (0.000)MSE 2 2.029 1.865 0.477 0.504 0.721 0.503 0.094 0.100(0.017) (0.000) (0.000) (0.000) (0.000) (0.000)MAD 1 1.291 1.318 0.448 0.487 0.763 0.655 0.237 0.253(0.824) (0.000) (0.000) (0.000) (0.000) (0.000)MAD 2 1.332 1.264 0.464 0.514 0.790 0.632 0.232 0.254(0.004) (0.000) (0.000) (0.000) (0.000) (0.000)Horizon Singapore AustraliaGARCH GJR TARRX TARRX-R GARCH GJR TARRX TARRX-RMSE 1 0.343 0.301 0.058 0.063 0.029 0.028 0.010 0.009(0.000) (0.000) (0.000) (0.219) (0.000) (0.000)MSE 2 0.405 0.311 0.061 0.066 0.031 0.026 0.011 0.009(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)MAD 1 0.553 0.518 0.157 0.177 0.157 0.149 0.076 0.072(0.000) (0.000) (0.000) (0.028) (0.000) (0.000)MAD 2 0.602 0.526 0.160 0.181 0.161 0.144 0.079 0.071(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)P-values are in parentheses.For the real (proxied) volatility of each market, 22,t = 0.3607R2t .Parameter estimates for the GARCH models are shown in Table 7. These show the well-known behaviors of highvolatility persistence 1 +1 1 and signicantly higher volatility (asymmetry) after negative returns > 0.We notethat the estimates of persistence for these models are higher than those for the TARR models, possibly due to the morenoisy squared return data which can cause GARCH models to over-react to large returns.The GARCH models forecast the close to close return volatility, while the TARRX model forecasts the intra-dayrange. To align these models, we consider separately the four volatility proxies in Section 4, to adjust the intra-day rangeC.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 3007Table 11MSE, MAD measures and DieboldMariano DM test for equal l-step-ahead forecast accuracy of daily stock market volatility proxied by 23,tHorizon Japan KoreaGARCH GJR TARRX TARRX-R GARCH GJR TARRX TARRX-RMSE 1 1.206 1.230 1.304 1.267 4.819 4.914 4.992 4.973(0.860) (0.971) (0.823) (0.804) (0.750) (0.648)MSE 2 1.221 1.202 1.343 1.301 4.101 4.255 4.417 4.488(0.211) (0.992) (0.895) (0.904) (0.979) (0.871)MAD 1 0.903 0.912 0.950 0.939 1.545 1.526 1.545 1.642(0.836) (0.981) (0.929) (0.210) (0.502) (0.918)MAD 2 0.926 0.902 0.994 0.961 1.455 1.438 1.499 1.606(0.005) (1.000) (0.942) (0.279) (0.842) (0.990)Horizon Taiwan Hong KongGARCH GJR TARRX TARRX-R GARCH GJR TARRX TARRX-RMSE 1 3.553 3.426 3.616 3.839 0.732 0.736 0.773 0.781(0.113) (0.736) (0.967) (0.559) (0.892) (0.815)MSE 2 3.537 3.393 3.666 3.873 0.709 0.726 0.720 0.771(0.039) (0.867) (0.977) (0.727) (0.639) (0.928)MAD 1 1.265 1.270 1.242 1.350 0.637 0.613 0.657 0.702(0.573) (0.220) (0.981) (0.061) (0.845) (0.995)MAD 2 1.297 1.243 1.287 1.426 0.642 0.609 0.643 0.703(0.018) (0.376) (0.999) (0.035) (0.509) (0.994)Horizon Singapore AustraliaGARCH GJR TARRX TARRX-R GARCH GJR TARRX TARRX-RMSE 1 0.418 0.413 0.447 0.484 0.066 0.065 0.074 0.070(0.221) (0.870) (0.998) (0.282) (0.772) (0.871)MSE 2 0.431 0.418 0.470 0.505 0.066 0.067 0.086 0.070(0.140) (0.894) (0.998) (0.680) (0.994) (0.929)MAD 1 0.443 0.425 0.434 0.490 0.174 0.172 0.210 0.201(0.001) (0.314) (0.996) (0.314) (0.992) (1.000)MAD 2 0.469 0.434 0.444 0.502 0.177 0.173 0.220 0.198(0.000) (0.097) (0.979) (0.125) (0.999) (1.000)P-values are in parentheses.For the real (proxied) volatility of each market, 23,t = R2t .forecasts into volatility forecasts. Firstly, in Tables 8 and 9, we consider the squared return proxy and measure how wellthe GARCH models forecast this proxy. A priori, we expect that the GARCH models would forecast 21,t better thanthe TARRX model, since they use actual return data in estimation. However, since 21,t is such a noisy volatility proxy,this is not necessarily a good forecast measure by itself. Further, we also employ the three range-based volatility proxyformulas to transform the intra-day range forecasts from the TARRXmodels into volatility forecasts (shown under (A),(B) and (C) in Tables 8 and 9) and compare these to the squared return proxy. Finally, we then alternately use the three3008 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010Table 12MSE, MAD measures and DieboldMariano DM test for equal l-step-ahead forecast accuracy of daily stock market volatility proxied by 24,tHorizon Japan KoreaGARCH GJR TARRX TARRX-R GARCH GJR TARRX TARRX-RMSE 1 1.290 1.328 0.327 0.317 2.553 2.271 1.251 1.246(0.942) (0.000) (0.000) (0.000) (0.000) (0.000)MSE 2 1.373 1.279 0.337 0.326 2.678 2.212 1.107 1.125(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)MAD 1 1.054 1.066 0.475 0.470 1.421 1.300 0.774 0.822(0.899) (0.000) (0.000) (0.000) (0.000) (0.000)MAD 2 1.092 1.043 0.498 0.481 1.472 1.284 0.750 0.804(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)Horizon Taiwan Hong KongGARCH GJR TARRX TARRX-R GARCH GJR TARRX TARRX-RMSE 1 1.776 1.846 0.906 0.962 0.552 0.444 0.194 0.196(0.791) (0.000) (0.000) (0.000) (0.000) (0.000)MSE 2 1.880 1.720 0.919 0.971 0.581 0.414 0.181 0.193(0.012) (0.000) (0.000) (0.000) (0.000) (0.000)MAD 1 1.202 1.231 0.622 0.676 0.664 0.569 0.329 0.352(0.851) (0.000) (0.000) (0.000) (0.000) (0.000)MAD 2 1.240 1.179 0.645 0.714 0.686 0.550 0.322 0.352(0.006) (0.000) (0.000) (0.000) (0.000) (0.000)Horizon Singapore AustraliaGARCH GJR TARRX TARRX-R GARCH GJR TARRX TARRX-RMSE 1 0.293 0.259 0.112 0.121 0.026 0.025 0.018 0.017(0.000) (0.000) (0.000) (0.203) (0.000) (0.000)MSE 2 0.344 0.269 0.118 0.126 0.028 0.024 0.022 0.018(0.000) (0.000) (0.000) (0.002) (0.013) (0.000)MAD 1 0.496 0.463 0.217 0.246 0.148 0.142 0.105 0.101(0.000) (0.000) (0.000) (0.073) (0.000) (0.000)MAD 2 0.544 0.476 0.222 0.251 0.151 0.139 0.110 0.099(0.000) (0.000) (0.000) (0.000) (0.000) (0.000)P-values are in parentheses.For the real (proxied) volatility of each market, 24,t = exp[2 (log(Rt ) 0.43 + 0.292)].range based volatility proxies: 22,t , 23,t , and 24,t to compare all the competing forecast models in Tables 1012. Weexpect the TARR models to forecast the purely range-based proxies 22,t , 23,tand 24,t better than the GARCH models.These a priori expectations were not always satised in our results.Results shown in Tables 812 are for the GARCH, GJRGARCH (GJR), TARRX and TARRX-R models. Thep-values from the DM test for equal forecast accuracy, comparing each model with the GARCHmodel, are also shown.In each row the minimum MSE or MAD is boxed, while all results signicantly better than the GARCH model, at aC.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010 300910% level, are bolded. Firstly, Tables 8 and 9 show results when forecasting the squared returns proxy 21,t and theseare surprisingly all in favor of the TARRX models under MSE and MAD. Both the TARRX and TARRX-R models,under either proxy 2 or 4, are signicantly better than the GARCH models in all markets under both MSE and MADand l = 1, 2. The squared range proxy 3 of Chou (2005a) is much closer to the GARCH models in each market. Ineach market the best TARRX and TARRX-R models are not signicantly different to each other (p-values not shown)under proxies 2 and 4. These models performed similarly in each market: out of 24 contests under MAD and MSEcombined, TARRX ranked best 16 times, compared to 8 for TARRX-R. In summary then, theTARRX and/or TARRX-Rmodels were statistically better than both GARCH models in all markets when forecasting squared returns 21,t . Thisis a surprising and very positive result for the TARR type models, in that they could possibly forecast squared returnsbetter than GARCH and GJRGARCH, since returns were not used to estimate the TARR model parameters.Results when forecasting two of the three range-based proxies, (22,t and 24,t ), are clearly in favor of the TARRXmodels for all markets under bothMSE andMAD. For 22,t all markets clearly and statistically favor one of the TARRXmodels under MSE and MAD and l = 1, 2. The TARRX and TARRX-R are not statistically different to each otherin all markets, but over the 24 comparisons of MSE and MAD, the TARRX was lower in 15, compared to 9 for theTARRX-R model. When forecasting 24,t again all markets strongly and statistically favor the TARRX models. Forthese two proxies again the TARRX model had 15 counts of lowest MSE or MAD to TARRX-Rs count of 9. This isa strong result: even though we suspected the TARRX model would perform better on the range based proxies, sinceit employs range data directly, the level of out-performance compared to the GARCH and GJRGARCH models ishighly and strongly signicant. We remind readers of the strong and stable link between volatility and range found byAlizadeh et al. (2002), and successfully employed by Brandt and Jones (2006), that leads to the 4th volatility proxy24,t . This is a proven highly efcient volatility measure and for a model to forecast it better than standard and popularGARCH models is a result worth emphasizing. Finally, results for the squared-range proxy 3 are surprisingly in favorof the GARCH models, though often not statistically. We know that squared returns are very noisy and thus squaredrange data will also be a very noisy volatility proxy. Here we see the value of Parkinsons simple adapted squared rangeproxy 2.In summary, theTARRmodels are surprisingly themost accurate at forecasting squared returns compared toGARCHmodels, and were always signicantly better under the MSE and MAD measures. For various range-based proxies ofvolatility, the TARRX models are clearly far superior, both in terms of rank and statistical signicance, at forecastingthese more efcient and less noisy proxies, in the markets considered. Overall, the proposed TARR models seem thesuperior volatility forecasting tool, compared to the popular GARCHandGJRGARCHmodels. Compared to the linearCARR model, the asymmetric TARR models also performed better in forecasting, with the TARRX and TARRX-Rseemingly indistinguishable in volatility forecasting over the markets considered.7. ConclusionA family of range-based threshold heteroskedastic models is proposed to model dynamic volatility and captureasymmetry in nancial markets. The structure of this model allows stock market range (volatility) data to react asym-metrically to past information or an exogenous variable, around a threshold value. Simultaneous estimation of thethreshold values, the time delay, and other parameters is feasible via MCMC sampling methods. A simulation studyshowed that reliable estimation for this model is provided by a Bayesian approach. An empirical application to sixnancial market indices suggested that the conditional mean range exhibits signicant threshold nonlinearity, via signand size asymmetry, at a lag d = 1 day, both in response to past local market and the US market range and returndata. The proposed TARR models, with local or US market range or US market return as the threshold variable (i.e.size asymmetry), were preferred as a forecasting tool over the linear CARR model in ve of the six markets, exceptJapan. A comparison with GARCH and GJRGARCH models revealed clear forecasting dominance for both TARRXmodels of squared returns and of two range-based volatility proxies. The proposed TARR models seem the superiorvolatility forecasting tools, compared to popular GARCH models. Both sign and size asymmetry were found to beimportant with neither dominating the other in forecasting volatility. Further research could involve different thresholdvariables, such as a weighted average of auxiliary variables, to further gauge the efciency gain of the TARR overits rivals. Additional rivals could include Markov switching GARCH models or the proposal of a Markov switchingautoregressive range-based model similar to the ACD models of De Luca and Zuccolotto (2006).3010 C.W.S. Chen et al. / Computational Statistics & Data Analysis 52 (2008) 29903010AcknowledgmentWe wish to thank the Guest Editor, Professor Herman van Dijk, and two referees for their comments which helpedto improve the paper. C.W.S. Chen is supported by National Science Council (NSC) of Taiwan Grants NSC95-2118-M-035-001.ReferencesAlizadeh, S., Brandt, M.W., Diebold, F.X., 2002. Range-based estimation of stochastic volatility models or exchange rate dynamics are moreinteresting than you think. J. Finance 57, 10471092.Andersen, T., Bollerslev, T., 1998. 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