Efficient Modelling & Forecasting with range based volatility models and application
Kok-Haur Ng, Shelton Peiris, Jennifer So-kuen-Chan, David Allen, Kooi-Huat Ng
11 EFFICIENT MODELLING & FORECASTING WITH RANGE BASED VOLATILITY MODELS AND APPLICATION
Ng, Kok Haur Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Malaysia. Shelton Peiris ; Jennifer, So-kuen Chan ; David Allen School of Mathematics and Statistics, Faculty of Science, The University of Sydney, Australia Ng, Kooi-Huat Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul Rahman, Malaysia.
Abstract
This paper considers an alternative method for fitting CARR models using combined estimating functions (CEF) by showing its usefulness in applications in economics and quantitative finance. The associated information matrix for corresponding new estimates is derived to calculate the standard errors. A simulation study is carried out to demonstrate its superiority relative to other two competitors: linear estimating functions (LEF) and the maximum likelihood (ML). Results show that CEF estimates are more efficient than LEF and ML estimates when the error distribution is mis-specified. Taking a real data set from financial economics, we illustrate the usefulness and applicability of the CEF method in practice and report reliable forecast values to minimize the risk in the decision making process.
Keywords : Volatility model, estimating functions, range data, conditional autoregressive range model. Introduction
Autoregressive Conditional Heteroskedastic (ARCH), Generalized ARCH (GARCH) and stochastic volatility models are widely used in modelling the dynamics of volatility in financial markets. However, many studies including Alizadeh et al. (2002), Chou (2005), Brandt and Diebold (2006) have suggested that GARCH models are not always accurate and efficient in certain applications. A reason for this is that they are return-based and use only the data available at closing prices ignoring the intra-day price movements and related information. As a result, the range, defined as being the difference between high and low prices over a given time interval, becomes a popular alternative measure, since it utilizes two pieces of information (the high and low prices) from the market within the given interval. Alizaded et al. (2002) showed that range-based models are more efficient than traditional volatility models in volatility or risk modelling. As an alternative approach, Chou (2005) proposed a symmetric range-based model called Conditional Autoregressive Range (CARR) model and an extension of the CARR model with exogenous variables called the CARRX model to forecast volatilities. He claimed that the CARR model provides a simple, efficient and natural framework to analyze the dynamics of volatility. Empirical results also show that the CARR model gives sharper volatility estimates than a standard GARCH model. Brandt and Diebold (2006), Brandt and Jones (2006) and Lin et al. (2012) also considered range-based models on various financial markets and their applications can be found in Chou and Liu (2010), Li and Hong (2011), Chou et al. (2013) and Sin (2013). Further extensions of CARR models include the threshold conditional autoregressive model of Chen et al. (2008), the time-varying logarithmic conditional autoregressive range model of Chiang and Wang (2011), the conditional autoregressive geometric process range model of Chan et al. (2012) and the CARR model incorporating the sudden change components of Kumar (2015). CARR models possess similar features to autoregressive conditional duration (ACD) models except that the range data measures the maximum change of price over fixed time intervals while the duration data measures the time interval between two consecutive transactions. As pointed out by Allen et al. (2013), two important issues in the use of ACD models are model specification and the choice of a suitable distribution for the errors. Since the exact distribution of errors is often unknown in practice, previous researches on CARR and ACD models consider mainly parametric models with extensions to more flexible error distributions such as log-t (Chan et al., 2012). Recently, Andres and Harvey (2012) considered the generalized beta type 2 (GB2) distribution in financial modelling as it nests many important distributions as special cases. However, the estimation of parameters for GB2 distribution can be difficult due its complicated density function. To avoid such difficulties in the maximum likelihood (ML) approach, alternative approaches have been adopted for CARR models. For example, Chou (2005) and Chou et al. (2013) considered the quasi ML method, Chiang and Wang (2011) applied a two-stage ML method while Chen et al. (2008) and Chan et al. (2012) used the Bayesian approach. However, it is not surprising that a flexible distribution such as GB2 may still fail to model some data well. A semi-parametric approach using the theory of estimating functions (EF) avoids any distributional assumption and hence can be applied to more general data sets. The linear estimating function (LEF) method has been successfully applied in many cases of financial time series modeling. See, for example, Bera et al. (2006) and Allen et al. (2013). The LEF method was further extended to include combined estimating functions (CEF) (Li and Turtle, 2000) to incorporate with higher order moments and applied to ARCH and GARCH models. Liang et al. (2011) introduced a general framework for joint estimation of time series models using the quadratic estimating functions (QEF) which shares the same approach to CEF and applied it to an ACD model with a doubly stochastic model, a random coefficient autoregressive model and a regression model with ARCH errors. Meanwhile, Ng et al. (2015) developed the QEF procedure for log-ACD models. This paper considers the CEF method to estimate parameters of CARR models. Estimating functions and closed form expressions for information matrices related to CEF and LEF estimators are derived and the corresponding statistical properties are validated through theoretical and simulation results. Parameter estimations of CARR models with known as well as mis-specified error distributions based on CEF, LEF, ML methods are compared and reported in this study. These estimators are examined and assessed through the GB2 distribution, including error mis-specifications. Finally, the applicability of the CEF method is also demonstrated via an empirical example. The remainder of this paper is structured as follows: Section 2 reviews the use of the ML, LEF and CEF methods. Section 3 investigates the performance of these three estimators through a simulation study. Section 4 reports an empirical application to illustrate the use of the CEF method for the CARR models and to provide an assessment of the forecasting abilities, and finally concluding remarks are given in Section 5.
2. Estimating CARR Models
Suppose that t P is the logarithmic price of a speculative asset at time of the t -th fixed interval, where t n ,,2,1 ; Tt ,,2,1 . The range of t P over a fixed interval t (e.g. a day) measures the dispersion across . The range of t P for a fixed t is defined as }{min}{max100 ttnttnt PPr ; Tt ,,2,1 . (1) The CARR ),( vu model for the range is defined as ttt r , where t is an independent and identically distributed sequence such that )( t E and )(Var t . Its skewness and excess kurtosis are denoted by and respectively. Further, the dynamic specification of t is given by ui vj jtjitit r , where , i and j ( , i , j ) are parameters to be estimated. In practice, it is advisable to use a very flexible distribution for the errors t . One choice is the GB2 distribution which nests many important distributions such as the gamma, Weibull, Pareto, Burr12, lognormal, generalized gamma and the Pearson family, as special cases. The probability density function (pdf) of GB2 for t is given by qpattapt apttt bqpBb aqpbaf ])/(1)[,( ||),,,|( , t , (2) where pa , and q are the shape parameters such that qp and a , )0( t b is a scale parameter depending on t , and ),( qpB is the beta function. It is easy to verify that the mean of the distribution is given by ),( ),()( qpB qpBbE aatt . (3) See Cummins et al. (1990), pp. 260 for details. To derive the ML estimator, we consider the log-likelihood function )( λ l : Tt t rfl )|(log)( λλ , (4) where λ is the vector of all parameters ,,,( qpa ),...,,,,...,,, vu and the density )|( λ t rf can be obtained using Eq. (2). The corresponding ML estimator, λ ˆ , of λ is evaluated by )(maxargˆ λλ λ l , (5) where represents the vector space of all parameters and λ ˆ can be obtained using a suitable optimization algorithm, for example, Optim in the R library package or fminsearch in Matlab. As the true distribution of t is seldom known in practice, a semi-parametric method avoids the exact specification of the error distribution. Suppose that },,,{ T rrr is a discrete valued stochastic process and we are interested in fitting a suitable model for this sample. Let be the class of probability distributions F on T and )( F θθ be a vector of real parameters for F . Define )( F E and )( ,1 Ft E to be the conditional expectations with respect to F and the later by further holding the first t values ,,, t rrr fixed. For simplicity, we write )()( tFt EE and )()( EE F . Further, let tt hh )( be a real valued function of t rrr ,,, and parameters θ such that tt hE , ( Tt ,,2,1 ; F ), (6) and ,0][ stt hhE st . (7) Assuming that );( θrg are real valued functions of the random variables },,,{ T rrr r and the parameters θ fulfill the standard regularity conditions (see e.g. Godambe, 1985). Then the function );( θrg satisfying )];([ E is called a regular unbiased estimating function. Among all regular unbiased estimating functions );( θrg , );( * θrg is said to be optimum if );();(]);();([ '' θ θrgθ θrgθrgθrg EEE , (8) is minimized for all F at );();( * θrgθrg . An optimal estimate of θ is obtained by solving the optimum estimating equation );( * . Consider the class of linear unbiased estimating functions L formed by Tt tt ha );( θrg , (9) where t a is a suitably chosen function of the random variables ,,, t rrr and the parameters θ for all Tt ,,2,1 . From Eqs. (6) and (9), it is apparent that, )];([ E for all L );( θrg . Following the optimal theorem of Godambe (1985), the function );( * θrg minimizing Eq. (8) is given by Tt tt ha );( θrg , (10) where ][ ttttt hEhEa θ . An optimal estimate of θ (in the sense of Godambe, 1985) is acquired by solving the optimal equation(s) );( * . Supposing that the errors t for a CARR model follow a distribution with a positive support, it can be shown that the mean and variance of the conditional distribution of t r given t F are ttt Fr )|(E and )|(Var ttt Fr respectively. To find the LEF estimates for the CARR model, we define ttt rh to be a martingale difference sequence. It can be easily verified that t h fulfils two regularity conditions, namely, the unbiasedness in Eq. (6) and the mutual orthogonality in Eq. (7). Hence the optimal LEF of Eq. (10) becomes Tt tttt r );( θθrg , (11) since ][ ttttttt hEhEa θθ , and ),,,,,,,,( vu θ . An optimal estimate of θ can be obtained by solving the equation(s) );( *1 and the corresponding information matrix *1 g I is given by .1 Tt ttt θθI g (12) While the LEF considers only one single martingale difference sequence , ttt rh the CEF method extends the LEF method by utilizing two martingale differences ttt rh and )( tttt r and hence is shown to be an improvement on LEF. Since t h and t are not mutually orthogonal, we use a similar approach given in Li and Turtle (2000) to define the function )()( ttttttt rmrh , (13) where t m is an arbitrary function of t . We aim to find t m to achieve mutual orthogonality of t h and t h such that ttt hhE . Hence, the required function in Eq. (13) becomes ))(()( tttttttt rrh (14) where t is the conditional skewness of t r . Then the following CEF is formulated: Tt ttTt tt haha );( θrg , (15) to estimate parameters of a CARR model where ][ ttttttt hEhEa θθ , ,)2(2][ ttttttttttttt hEhEa θθθ and the corresponding excess kurtosis is defined by t tttt FrE . An optimal estimate of θ can be obtained by solving the equation(s) );( *2 . It is not difficult to show that the conditional skewness and excess kurtosis of t r are equal to the skewness and excess kurtosis of t respectively, that is, t and t . (16) The information matrix for the corresponding CEF estimates is Tt tttTt tttg )2( )2(1 θθθθI . (17) It is clear that *2 g I is more informative than *1 g I since Tt ttt t θθII gg . (18) These results can also be obtained through the QEF method. In the next section, a large scale simulation study is conducted to verify the performance of the CEF estimator relative to its competitors, the ML and LEF estimators. Simulation Study
A simulation study is carried out using the CARR (1,1) model given by ttt r , where ttt r , (19) and the errors t follow a standardized GB2 distribution. The GB2 distribution is chosen to simulate the errors because it can provide more general distribution for wider applications. From Eq. (3), the scale parameter t b is set to be ),( ),( aat qpB qpBb , so that t E and the pdf for the range t r is given by qpaqpBt aqapBtrapt apapttrtt qpB aqapBaqparf ])(1[),( )/1,/1(||),,,|( ),( )/1,/1(11 . (20) We first simulate a time series data of length T using the above CARR model with ) .4(0.2,0.3,0) ,,( θ , , and the errors t follows a standardized GB2 distribution with a , p , q . Using this sample, LEF estimates are obtained by solving );( *1 in Eq. (11) for and ),,( . CEF estimates are obtained by solving );( *2 in Eq. (15) for and ),,( with , r ˆ and r ˆ , where r and r denote the sample skewness and excess kurtosis respectively (Li and Turtle, 2000). ML estimates are obtained by maximizing the log-likelihood function based on Eq. (20). This procedure is repeated N times and for T , 1000, 1500, 2000. Then for each T , the mean, bias, standard deviation (SD) and root mean squared error (RMSE) being the average over the replicated data sets for each parameter are computed. To further assess the variability of estimators, the information matrices given in Eqs. (12) and (17) are computed and the standard errors (SE) (being the square root of the diagonal entries of their inverses averaged over replications) are reported. Table 1 reports the corresponding results. From Table 1, we observe the average estimates produced by all three methods are fairly close to the true values. Obviously, the performance of all parameter estimates improves with increasing T according to bias, SD and RMSE. Not surprisingly, the ML method assuming the true GB2 error distribution provides the best estimates according to all criteria in general. However, we also notice that CEF estimates are only marginally inferior to ML estimates when T = 2000 but are comparable to ML estimates when T = 500. Moreover the CEF estimates are substantially better than LEF estimates. Figures 1-3 present the histograms for ˆ , ˆ and ˆ respectively in Table 1 using the CEF method when T . The theoretical normal curves, with the true values as centers and SE as standard deviations are added to facilitate the comparison. The results show that the finite sample marginal distributions of ˆ , ˆ and ˆ closely approximate the theoretical Gaussian distributions with means 0.2, 0.3 and 0.4 as their true values. This finding is further confirmed by the close agreement between SD and SE in Table 1 using the CEF as well as the LEF methods. Although the ML method performs well when the distributions are known, semi-parametric methods such as LEF and CEF are robust and preferable since the distributions are unknown in practice. To investigate this, a simulation study was carried-out with the three proposed methods and error mis-specification. Again, we simulate a time series of length T using the parameters ) 40.2,0.3,0.( ) ,,( θ , and the errors t from the standard lognormal distribution with . We note that the ML estimates are obtained by maximizing the log-likelihood function based on GB2 distribution with the pdf in Eq. (20). The procedure is repeated N times for each T , 1000, 1500, 2000 and the corresponding statistics for each estimates are given in Table 2. From the Table 2, we observe that the CEF estimates give smaller bias, SD and RMSE than the LEF and ML estimates for all sample sizes except T . In comparisons of the model robustness and efficiency, we conclude that the CEF method outperforms the other two methods and hence is preferred in practice. An Application from Financial Economics
Consider the price data from the All Ordinaries (AORD) index from the Australian market during 1 May 2009 to 1 May 2015 was used for the empirical analysis to illustrate the applicability of CARR models using the CEF estimator. The daily range t r with 1514 observations were calculated using Eq. (1). Table 3 reports their summary statistics and the Ljung-Box statistics Q and Q which test the overall randomness of t r based on 6 and 12 lagged autocorrelations respectively. As they are significant at a 5% significance level, the series of t r is non-random and hence the CARR model is adopted to account for the autocorrelation. Figures 4 and 5 display the time series plot of t r and the histogram of the series, respectively. The autocorrelation plot in Figure 6 also confirms the high level of autocorrelation for t r . Numerical Results
We fit a CARR (1,1) model to the data using ML, LEF and CEF methods given in Sections 2.1, 2.2 and 2.3, respectively, for the first T data. All parameter estimates and their standard errors for all models are presented in Table 4. It can be seen that parameter estimates based on the three methods are very similar. Moreover, the predictive performance according to in-sample root mean square prediction error (RMSPE) and mean absolute prediction error (MAPE) are comparable but the standard errors for parameter estimates using the CEF method are much lower than those using LEF and ML methods. Therefore, the CEF method is preferred to the LEF and ML methods. Subsequently, higher order CARR models including CARR (1,2), CARR (2,1) and CARR (2,2) models are also fitted via the CEF method. The results of the fitted models are given in Table 5. Although CARR (2,1) model yields comparable in-sample fit and better out-of-sample forecasts among all models, its parameter estimate ˆ which signifies CARR(2,1) model is not significant and hence it shares essentially the same structure as CARR(1,1) model. In summary, CARR (1,1) model provides the best performance among all models for both in-sample prediction and out-of-sample forecast and hence is chosen to analyse this data. Figure 7 shows the fitted time series superimposed on the observed series. The results show that the CARR (1,1) model using the CEF method captures the general trend, persistence and volatility clustering well. We estimate the unconditional mean and variance of CARR (1,1) using )1/( r , and ]2)(1/[)21(]1)([ trtr EE , respectively, where t E . We have the following results: ML method: r and r LEF method: r and r CEF method: r and r These numbers are close to the sample mean of 0.9826 and the sample variance of 0.2568. Therefore all fitted models seem adequate.
Based on the chosen CARR (1,1) models and the last m range data, we obtain 14-step ahead forecasts ttt rEr ˆ)(ˆ using each of the three methods. We compare the forecast performance of these models using two out-of-sample criteria, namely the out-of-sample root mean square forecast error (RMSFE) and out-of-sample mean absolute forecast error (MAFE). The results in Table 4 indicate that the three methods provide similar forecasting performance despite the fact that the CEF method marginally outperforms the LEF and ML methods. Although the forecast performance is similar across the three methods, the forecasts using CEF method is preferred as CEF method provides more precise parameter estimates, an advantage over the other methods. We calculate the conditional forecast variance based on: )ˆ( t rVar ˆˆˆˆˆ ttt , where is given in Table 4 and t is calculated using the estimated information matrix from Eq. (17) where t ˆ is given in Eq. (19). Then the 95% confidence limits for the forecast t r ˆ are )ˆ(96.1ˆ tt rVar . The forecasts using CARR (1,1) model and CEF method are plotted in Figure 8 together with their 95% confidence intervals. Figure 8 shows that the forecasts follow the trend of the observed range well. We also obtain that the coverage probability is 0.9286 which is very closed to the theoretical value of 0.95. Concluding Remarks
The price range was shown to be an efficient risk measure in financial markets. This paper proposed the CEF method to estimate the CARR model for price range data. The superiority of the CEF estimator relative to the LEF and ML estimators are verified. The results show that the CEF estimates provide smaller bias, SD and RMSE than the LEF and ML estimates, especially when the error distribution is mis-specified which is actually common in practice. On the other hand, estimators obtained using the CEF and LEF methods are free from any distributional assumption. The applicability of the CEF method is demonstrated using an empirical example based on the daily range data from the AORD. Its predictive power is demonstrated using the in-sample and out-of-sample predictive performance criteria and the fitted line plot. Even though these results show that the predictive and forecast performances based on CEF method is marginally outperforms the LEF and ML methods but the standard errors of parameter estimates are much lower compared to those using LEF and ML methods. Acknowledgement
This work is partially supported by the UMRG grant RG260-13AFR from the University of Malaya. The first author acknowledge the support from the School of Mathematics and Statistics at the University of Sydney during his visit in 2016.
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North American Journal of Economics and Finance , 26: 552-564. Table 1.
Results for the CARR (1,1) model with standardized GB2 )0.2,0.1,0.1( qpa distribution and true parameters , , and Sample Size, T ˆ ˆ ˆ CEF LEF ML CEF LEF ML CEF LEF ML T Mean
Bias -0.0167 SD RMSE SE T Mean
Bias -0.0015 -0.0128 -0.0274 -0.0043 SD RMSE SE T Mean
Bias -0.0004 -0.0096 -0.0220 -0.0027 SD RMSE SE T Mean
Bias -0.0073 0.0145 -0.0036 SD RMSE SE Table 2.
Results for the CARR (1,1) model with standardized Lognormal )5.0( distribution and true parameters , , and Sample Size, T ˆ ˆ ˆ CEF LEF ML CEF LEF ML CEF LEF ML T Mean
Bias -0.0021 -0.0026 -0.0024 -0.0127 -0.0140 -0.0150 SD RMSE SE T Mean
Bias -0.0007 -0.0003 -0.0007 -0.0074 -0.0113 -0.0071 SD RMSE SE T Mean
Bias -0.0003 -0.0002 -0.0006 -0.0056 -0.0063 -0.0053 SD RMSE SE T Mean
Bias -0.0001 -0.0002 0.0000 -0.0045 -0.0050 -0.0051 SD RMSE SE Table 3.
Summary statistics for the AORD range data Mean 0.9829 Median 0.8763 Variance 0.2558 Skewness 2.17 Kurtosis 13.86 Minimum 0.0409 Maximum 6.9083 Ljung-Box, Q Q Table 4.
Results for CARR (1,1) models using the ML, LEF and CEF methods. Values in parentheses are standard errors of parameter estimates Coefficient
ML LEF CEF ˆ ˆ ˆ a ˆ p ˆ q ˆ ˆ Out-of-sample RMSFE 0.4011 0.4012
Out-of-sample MAFE 0.3362 0.3365 *The value of ˆ is calculated from ttt r ˆ/ˆ Table 5.
Results for CARR models based on the CEF method. Values in parentheses are standard errors of parameter estimates Coefficient
CARR (1,2) CARR (2,1) CARR (2,2) ˆ ˆ ˆ -0.0269 (0.0263) 0.0803 (0.1307) ˆ ˆ -0.0515 (0.1378) 0.3111 (0.7309) ˆ Figure 1.
The histogram for the ˆ obtained by CEF method ( , T ) Figure 2.
The histogram for the ˆ obtained by CEF method ( , T ) Figure 3.
The histogram for the ˆ obtained by CEF method ( , T ) Figure 4.
Time series plot of the AORD range series, t r Figure 5.
Histogram of the AORD range series, t r / /
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Autocorrelation function (ACF) of the AORD range series, t r Figure 7.
Observed t r and expected )( t rE using CARR (1,1) model and CEF method Lag A u t o c o rr e l a t i o n Autocorrelation Function for Range Series (with 5% significance limits for the autocorrelations) R a n g e Time
Observed E(r) Figure 8.
Observed t r , forecast t r ˆ and 95% confidence limits using CARR (1,1) model and CEF method / /
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