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Realized GARCH models: Simpler is better
Finance Research Letters xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Realized GARCH models: Simpler is better☆ Haibin Xie
⁎,1
, Chengtan Yu
University of International Business and Economics, Huixin East Road 10, Chaoyang District, Beijing 100029 China
ARTICLE INFO
ABSTRACT
Keywords: GARCH@CARR RealGARCH Volatility forecasting
Within the framework of Realized GARCH (RealGARCH), different RealGARCH variants have been proposed for volatility forecasting. The question remains unknown that which RealGARCH variant is more efficient. This paper compares three RealGARCH variants including the log-linear RealGARCH, the RealEGARCH and the GARCH@CARR. A comprehensive empirical study is performed on a stock index and 28 individual stocks, and the results show that the GRACH@CARR model outperforms the other two. Given that GARCH@CARR is more parsimonious in its specification, this finding is consistent with the principle of parsimony that models of simple structure usually provide better forecasts than the complex ones.
1. Introduction It has been well-known that realized volatility is a far more informative volatility estimator than is the squared return (Andersen and Bollerslev, 1998; Andersen et al., 2003; Barndorff-Nielsen and Shephard, 2002; 2004; Barndorff-Nielsen et al., 2008). Different methods have been proposed to incorporate the realized volatility into volatility estimation and forecasting, including the GARCH-X model of Engle (2002), the MEM model of Engle and Gallo (2006), the HEAVY model of Shephard and Sheppard (2010), and the most recent RealGARCH of Hansen et al. (2012). Among them RealGARCH model is the most popular one as it provides a framework of a joint modeling of return and realized volatility. Within the RealGARCH framework, different variants of RealGARCH have been proposed, for example the linear RealGARCH and the log-linear RealGARCH in Hansen et al. (2012), the RealEGARCH in Hansen and Huang (2016), and the most recent GARCH@CARR in Xie et al. (2019). Hansen et al. (2012) show that both linear RealGARCH and log-linear RealGARCH dominate the standard GARCH and EGARCH models that only use returns. Hansen and Huang (2016) compare several realized measures using RealEGARCH and find the realized kernel (RK) measure is most informative. Xie et al. (2019) propose a new variant of RealGARCH with multiplicative error specification. In their model the asset volatility dynamics is specified to be determined by a Conditional AutoRegressive Realized Volatility (henceforth GARCH@CARR) model. An important feature with the GARCH@CARR is that it has a simpler model specification compared with the other variants. Using the high-low price range as a realized measure, Xie et al. (2019) find the GARCH@CARR also outperforms the return-based GARCH and EGARCH models. Recent academic literature shows a rising interest in using RealGARCH models for asset pricing and risk management. Huang et al. (2017) obtain a analytical approximation formula for option pricing under RealEGARCH. Contino and Gerlach (2017) use the log-linear RealGARCH to forecast tail risk. Banulescu et al. (2018) use the RealEGARCH model to investigate the volatility during the financial crisis. Wu et al. (2019) propose using the RealEGARCH with skewness and kurtosis to forecast VaR. Despite the
This research was supported by National Natural Science Foundation of China under grant no. 71401033. Corresponding author. E-mail addresses: [email protected] (H. Xie), [email protected] (C. Yu). 1 Program for Young Excellent Talents, UIBE under grant no. 15YQ08. ☆ ⁎
https://doi.org/10.1016/j.frl.2019.06.019 Received 26 November 2018; Received in revised form 18 May 2019; Accepted 29 June 2019 1544-6123/ © 2019 Elsevier Inc. All rights reserved.
Please cite this article as: Haibin Xie and Chengtan Yu, Finance Research Letters, https://doi.org/10.1016/j.frl.2019.06.019
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H. Xie and C. Yu
Table 1 Out-of-sample forecasting performance evaluation: DM test. Benckmark: GARCH@CARR
SPY AA AIG AXP BA BAC C CAT CVX DD DIS GE GM HD IBM INTC JNJ JPM KO MCD MMM MRK MSFT PG T UTX VZ WMT XOM Average
Benckmark: log-linear RealEGARCH
A: log-linear RealGARCH
B: RealEGARCH
C: RealEGARCH
MSE
QLIKE
MSE
QLIKE
MSE
QLIKE
1.83 2.35* 2.03* 1.46 1.74 1.73 0.86 0.94 2.44* 1.68 1.20 2.48* 0.37 −0.62 0.76 1.13 0.77 0.51 −0.17 −0.74 −0.65 1.12 −0.30 1.52 0.77 1.29 1.31 −0.56 2.47* 1.02
2.22* 1.05 2.30* 1.97* 1.38 2.05* 2.59* 1.10 3.21* 1.49 −0.89 3.82* 1.29 0.04 0.22 1.05 −0.92 1.74 3.39* 0.56 −0.23 2.13* 1.97* −0.50 0.97 4.50* 0.72 −1.80 2.72* 1.38
1.05 2.74* 1.75 1.57 1.02 0.71 1.58 1.22 2.29* 1.83 1.43 1.88 0.92 2.82* 0.19 −0.08 −0.02 −0.51 0.63 2.20* 1.57 1.13 −0.13 2.19* 0.37 0.04 0.13 2.61* 2.16* 1.12
2.00* 2.04* 3.38* 1.80 −0.08 2.62 1.29 2.15* 3.08* 2.13* 0.48 4.28* 2.18* 2.42* 2.25* 1.73 1.71 2.38* 4.68* 2.05* −0.75 2.01* 1.68 0.02 2.41* 3.58* 2.24* 1.93 2.26* 2.06
−0.61 1.60 0.35 1.36 −0.49 0.16 0.98 2.22* 0.61 1.59 −0.90 1.14 0.87 1.30 −0.11 −0.31 −0.29 0.91 −0.33 1.81 1.66 −1.40 1.02 −1.32 0.69 −1.14 −0.77 1.73 0.33 0.44
−1.38 1.30 0.70 −0.45 1.50 0.95 −0.89 2.00* 1.61 1.87 −0.37 2.68* 0.23 0.78 1.86 1.01 2.21* −0.13 1.06 1.18 1.70 −1.57 1.77 0.99 1.45 0.37 2.71* 3.38* 0.31 0.99
Notes: A t-statistic greater than 1.96 in absolute value indicates a rejection of the null of equal predictive accuracy at the 5% level. These statistics are marked with an asterisk. The sign of the t-statistics indicates which forecast performed better for each loss function: a positive t-statistic indicates that benchmark model outperforms, while a negative sign indicates the opposite.
popularity of RealGARCH models in empirical applications, the question remains uninvestigated that which RealGARCH variant is more efficient. A careful study of this question will be meaningful for RealGARCH model selection and application. We compare three RealGARCH variants, the log-linear RealGARCH, RealEGARCH and the most recent GARCH@CARR. We don’t investigate the linear RealGARCH since Hansen et al. (2012) find that log-linear RealGARCH is more preferred than the linear RealGARCH. We evaluate these RealGARCH variants on a stock index and 28 individual stocks. The results show that GARCH@CARR dominates the other two variants and the log-linear RealGARCH dominates the RealEGARCH. Given that log-linear RealGARCH and GARCH@CARR have a simpler model specification, this finding is consistent with the principle of parsimony that simpler models usually provide better forecasts than more complex ones. The rest of this paper is organized as follows. Section 2 presents the RealGARCH variants evaluated in this paper. Section 3 presents our evaluation method. Section 4 presents the out-of-sample empirical results. We conclude in Section 5. 2. RealGARCH variants RealGARCH framework is a joint modeling of return and realized volatility. The general structure of RealGARCH(p, q) model is given by2
rt =
(1)
ht z t ,
ht = (ht 1, …, ht p, …, xt 1, …, xt q),
(2)
x t = m (ht , z t , ut ),
(3)
2 To simplify the presentation we assume that E(rt |Ft 1) = 0. A more general specifications for conditional mean, such as a constant or the GARCH-in-mean by Engle et al. (1987), is accommodated by reinterpreting rt as the return less its conditional mean.
2
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Table 2 Out-of-sample forecasting performance evaluation: linear regression. Panel A: GARCH@CARR
SPY AA AIG AXP BA BAC C CAT CVX DD DIS GE GM HD IBM INTC JNJ JPM KO MCD MMM MRK MSFT PG T UTX VZ WMT XOM Average
Panel B: log-linear RealGARCH
Panel C: RealEGARCH
a
b
R2
a
b
R2
a
b
R2
−0.01 0.17 0.21 −0.01 0.10 0.05 0.14 −0.16 −0.13 −0.14 0.05 0.03 −0.39 −0.07 0.09 0.14 0.04 0.02 0.09 −0.00 −0.11 −0.41 0.17 −0.03 0.01 −0.13 0.06 0.06 −0.08 −0.01
0.94* 1.16* 1.07* 1.02* 1.00* 1.20* 1.20* 1.07* 1.21* 1.36* 1.06* 1.11* 1.08* 1.18* 1.09* 0.94* 1.13* 1.13* 1.22* 1.10* 1.26* 1.29* 1.05* 1.35* 1.04* 1.22* 1.15* 1.25* 1.15* 1.14
0.59 0.43 0.69 0.71 0.45 0.75 0.74 0.47 0.52 0.49 0.44 0.51 0.62 0.61 0.49 0.50 0.28 0.70 0.36 0.36 0.49 0.25 0.47 0.40 0.48 0.39 0.52 0.41 0.51 0.50
0.04 −0.13 0.35 −0.05 −0.26 0.63* 0.62* 0. 28* 0. 91* 0. 27* −0.08 0.13 −0.75 −0.02 0.21* 0.10 −0.02 0.30 0.17 0.04 −0.06 1. 73* 0.31 −0.06 0.38* 0.21* 0.11 −0.01 0. 60* −0.06
0.90* 1.29* 1.04* 1.12* 1.26* 0.96* 1.04* 1.15* 1.82* 1.49* 1.15* 1.04* 1.12* 1.16* 0.95* 0.95* 1.21* 1.01* 1.10* 1.06* 1.21* 2.07* 0.94* 1.39* 0.82* 0.95* 1.13* 1.31* 1.59* 1.18
0.55 0.40 0.67 0.70 0.41 0.69 0.70 0.47 0.48 0.48 0.42 0.48 0.62 0.62 0.46 0.48 0.27 0.68 0.35 0.36 0.48 0.25 0.46 0.38 0.47 0.35 0.49 0.42 0.49 0.49
−0.02 0.14 0.21 −0.18 0. 32* 0.26 0.28 0. 41* 1. 00* 0. 30* −0.06 0.06 −0.46 −0.16 0.16* 0.05 −0.06 −0.01 0.17* 0.04 −0.13 1. 57* 0.28* −0.09 0.27* −0.06 0.04 −0.03 0. 79* −0.13
0.96* 1.20* 1.11* 1.18* 1.27* 1.15* 1.16* 1.23* 1.88* 1.53* 1.16* 1.13* 1.10* 1.25* 1.01* 1.01* 1.34* 1.15* 1.11* 1.07* 1.28* 1.98* 0.96* 1.46* 0.90* 1.17* 1.17* 1.37* 1.72* 1.24
0.57 0.41 0.68 0.70 0.45 0.74 0.72 0.46 0.51 0.46 0.43 0.49 0.59 0.61 0.48 0.50 0.31 0.71 0.35 0.35 0.47 0.25 0.46 0.39 0.47 0.39 0.53 0.40 0.51 0.50
Notes: The realized kernel (RK) are used as the volatility proxy. An asterisk * means significance at the level of 5%. The linear regression is Vt + n = a + bFVi, t+ n +
t+n,
n = 1, 2. ,N
where Vt + n is the volatility proxy, FVi, t+ n is the forecast reported by model i. If model i reports unbiased volatility estimation, we expect a=0 and b=1.
where {rt} is a return series, xt is a kind of realized volatility measure and zt ∼ i.i.d. N(0,1), with zt and ut being mutually independent. The first two equations are referred to as return equation and GARCH equation, and the last one is called measurement equation. Different specifications to (.) and m(.) give different RealGARCH variants. Within this RealGARCH framework, different variants have been proposed. In the following subsection, we will present the most commonly used log-linear RealGARCH and RealEGARCH and the most recently proposed GARCH@CARR model. In the empirical we only consider the variants of order (1, 1) as it is well documented that order (1, 1) is sufficient to capture the volatility dynamics. 2.1. Log-linear realized GARCH The RealGARCH with log-linear specification in Hansen et al. (2012) is characterized by
rt =
(4)
ht z t ,
ln (ht ) = ln (xt ) =
(5)
+ ln (ht 1) + ln (x t 1),
(6)
+ ln (ht ) + (z t ) + µt ,
1) is the leverage function. This leverage function form where μt ∼ i.i.d. N(0, σμ), xt is the realized variance, and (z ) = 1 z + 2 is proposed in Hansen et al. (2012). The log-linear RealGARCH includes the leverage effect indirectly in the GARCH equation through the measurement equation. (z 2
2.2. Realized EGARCH Hansen and Huang (2016) introduce a Realized Exponential GARCH (RealEGARCH) model that can utilize multiple realized 3
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volatility measures to model return series.
rt =
(7)
ht z t ,
ln (ht ) =
+ ln (ht 1) + µt
ln (xt ) =
1
(8)
+ d (z t 1),
(9)
+ ln (ht ) + (z t ) + µt ,
where d (z ) = d1 z + d2 (z 2 1) is the leverage function. The main difference between log-linear RealGARCH and RealEGARCH is that the RealEGARCH incorporates a new leverage function d (z t 1) in the GARCH equation, which thus makes it more flexible in capturing the leverage effect. 2.3. GARCH@CARR By specifying a MEM (multiplicative error model) to the measurement equation, Xie et al. (2019) propose the following variant
rt =
ln ( t ) =
yt =
(10)
t zt ,
+ ln (
t 1)
(11)
+ ln (yt 1 ) + d (z t 1),
(12)
t ut ,
where yt = xt , ut ∼ i.i.d. LN ( u ) is log-normal with unit mean. Scale factor ρ is used to make sure ρλt is an unbiased volatility estimation. Since Eqs. (11)–(12) are known as Conditional AutoRegressive Realized volatility model, thus this RealGARCH variant is called GARCH@CARR. Squared root of realized variance is used for the reason that GARCH@CARR is designed for standard deviation instead of variance. The main difference between GARCH@CARR and the other two variants is that GARCH@CARR model is a restricted RealEGARCH. Note that ht = t , x t = t ut . Eq. (12) in GARCH@CARR is equivalent to 2 u /2,
ln (xt ) =
(13)
2ln ( ) + ln (ht ) + 2ln (ut )
Comparing Eq. (13) with Eq. (9), we can obtain
=
2ln ( ),
= 1,
2ln (ut ) = µt ,
(14)
(z t ) = 0
This restriction makes the GARCH@CARR model more concise and simpler in its specification than the other two variants. Comparing the three RealGARCH variants, we find the most significant difference is the complexity in model specification. There are 8 parameters in log-linear RealGARCH ({ω, β, γ, ξ, φ, τ1, τ2, σμ}), 10 parameters in RealEGARCH ({ω, β, γ, d1, d2, ξ, φ, τ1, τ2, σμ}) and 7 parameters in GARCH@CARR ({ρ, ω, β, γ, τ1, τ2, σu}). A comprehensive look at the forecasting performance of these 3 variants can help us better understand how the model complexity could affect the RealGRACH forecasting performance. 3. Evaluation method The Diebold-Mariano test (Diebold and Mariano, 1995) is used for comparing two competing forecasting models. Suppose the losses reported by models i and j are denoted by Lm,i,t and Lm,j,t (m=1 or 2). The Diebold-Mariano test verifies the null hypothesis that Lm, j, t . The Diebold-Mariano test statistic is given by E(dw, t)=0, where d w, t = Lm, i, t
DM =
d¯ Var (d¯)
,
(15)
N where d¯ = N 1 n = 1 d w, t + n, and N is the total number of forecasts. The variance of d¯, Var (d¯) is estimated by the heteroscedasticity and autocorrelation consistent (HAC) estimator of Newey and West (1987). Diebold and Mariano (1995) show that under the null hypothesis of equal predictive accuracy the DM statistic follows standard normal distribution. Since the true volatility is latent, we can only use volatility proxy as the “true volatility” to construct the DM statistic. It has been noted that imperfect volatility proxy may distort the ranking of competing volatility forecasts. Patton (2011) examines a class of loss functions and finds that only the MSE (mean squared error) and QLIKE (quasi-likelihood loss function) are robust to an imperfect volatility proxy. Moreover, QLIKE is found to be less sensitive to extreme observations than the squared error loss (Patton (2011); Conrad and Kleen (2018)). MSE and QLIKE are defined as
^t2),
MSE = E (L1, k, t ), L1, k, t = (
2 t
QLIKE = E (L 2, k, t ), L 2, k, t =
2 (log ( ^t )
(16)
+
2^ t t
2
(17)
),
where L1,k,t and L2,k,t are the losses for the forecasting model k. We also use the following linear regression analysis to evaluate the forecasting performance of different models
Vt + n = a + bFVi, t + n +
t+n ,
(18)
n = 1, 2. ,N
4
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H. Xie and C. Yu
where Vt + n is the volatility proxy, FVi, t + n is the forecast reported by model i. If model i reports unbiased volatility estimation, we expect a=0 and b=1. 4. Empirical results For empirical study, we use the same data as Hansen et al. (2012). This data set includes an exchange-traded index fund, SPY, that tracks the S&P500 index and 28 individual stocks. Since daily high-frequency prices are only available between ‘open’ and ‘close’, so we use in this paper open-to-close returns3 4.1. Out-of-Sample results We evaluate the three RealGARCH variants through out-of-sample forecasting performance4 A recursive (expanding) window forecasting procedure is used and the last 500 data observations are used for evaluation5 The realized kernel (RK), introduced by Barndorff-Nielsen et al. (2008), is used as the realized measure. This choice is based on the theoretical strengths of the RK measure: it is robust to microstructure noise and makes efficient use of the intraday data. Moreover, in a comparison of 19 realized variance measures, Gatheral and Oomen (2010) find that RK is one of the best estimators in terms of efficiency and robustness to time varying parameters. Table 1 presents the DM statistics. We first use the GARCH@CARR model as the benchmark and compare it with the log-linear RealGARCH and the RealEGARCH, and the results are presented in Panels A-B. Taking the log-linear RealGARCH as the benchmark, we also compare it with the RealEAGRCH, and results are reported in Panel C. We calculate the average DM statistics of all the individual ones and the results are reported in the last row. According to the average DM statistics we find that on average the GARCH@CARR dominates the other two variants and the log-linear RealGARCH dominates the RealEGARCH. It is more interesting to look at the DM statistics for each individual asset. The DM statistics in Panel A shows that the GARCH@CARR dominates the loglinear RealGARCH in 5(12) out of the 29 assets by the MSE (QLIKE). The dominance of GARHC@CARR over RealEGARCH is more significant by the DM statistics in Panel B. In 7 (18) out of the 29 assets, GARCH@CARR dominates RealEGARCH by MSE (QLIKE). The log-linear RealGARCH dominates the RealEGARCH in 1 (5) out of the 29 assets by MSE (QLIKE). Taking the RK as the volatility proxy, Table 2 reports the linear regression results. We also calculate the average intercept (a) and the average slope (b) and the average R-square (R2), and the results are presented in the last row. Consistent with the DM statistics, the results show that on average the GARCH@CARR reports the least biased out-of-sample forecasts while the RealEGARCH reports the largest biased out-of-sample forecast. In summary, we find the GARCH@CARR model reports the best out-of-sample forecasts while the most commonly used RealEGARCH reports the worst forecasts. Given that the GARCH @CARR has the simplest model specification while the RealEGARCH has the most complex specification, this finding is consistent with the principle of parsimony that simple models usually have better performance than complex ones. 5. Conclusions The Realized GARCH model provides an efficient and flexible framework for a joint modeling of return and realized measure. This paper comprehensively evaluates the forecasting performance of three RealGARCH variants, including the most commonly used loglinear RealGARCH, RealEGARCH and the most recent proposed GARCH@CARR. We find clear dominance of GARCH@CARR over the other two competing ones. Given that the GARCH@CARR has the simplest model specification, this finding is consistent with the principle of parsimony that simple models usually have better performance than complex ones. Of course, our finding is also valuable for future model selection and application. Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.frl.2019.06.019 References Andersen, T.G., Bollerslev, T., 1998. Answering the skeptics: yes, standard volatility models do provide accurate forecasts. Int. Econ. Rev. 39 (4), 885–905. Andersen, T.G., Bollerslev, T., Diebold, F.X., Labys, P., 2003. Modeling and forecasting realized volatility. Econometrica 71 (2), 579–625. Banulescu, D., Hansen, P.R., Huang, Z., Matei, M., 2018. Volatility During the Financial Crisis Through the Lens of High Frequency Data: a Realized GARCH Approach. Social Science Electronic Publishing. Availabe at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3178890.
3
The sample spans the period from 1 January 2002 to 31 August 2008. The dataset is also used in Hansen et al. (2012) and available at http://qed. econ.queensu.ca/jae/2012-v27.6/hansen-huang-shek. All the return series have been demeaned. 4 The in-sample forecasting evaluation is very similar to the out-of-sample evaluation. We don’t present the in-sample results for the consideration of space-saving. The in-sample results are available on request. 5 Maximum likelihood estimation (MLE) can be used to estimate all these three RealGARCH variants. The MLE is performed using Matlab and the code is available on request. 5
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Barndorff-Nielsen, O.E., Hansen, P.R., Lunde, A., Shephard, N., 2008. Designing realized kernels to measure the ex-post variation of equity prices in the presence of noise. Econometrica 76 (6), 1481–1536. Barndorff-Nielsen, O.E., Shephard, N., 2002. Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. B 64 (2), 253–280. Barndorff-Nielsen, O.E., Shephard, N., 2004. Power and bipower variation with stochastic volatility and jumps. J. Financ. Econom. 2 (1), 1–48. Conrad, C., Kleen, O., 2018. Two are better than one: Volatility forecasting using multiplicative component GARCH-MIDAS models. (working paper). https://doi.org/ 10.2139/ssrn.2752354. Available at SSRN: https://ssrn.com/abstract=2752354. Contino, C., Gerlach, R.H., 2017. Bayesian tail-risk forecasting using realized GARCH. Appl. Stochastic Models Bus. Ind. 32 (2), 213–236. Diebold, F.X., Mariano, R.S., 1995. Comparing predictive accuracy. J. Bus. Econ. Stat. 13 (3), 253–263. Engle, R., 2002. New frontiers of ARCH models. J. Appl. Econom. 17, 425–446. Engle, R.F., Gallo, G.M., 2006. A multiple indicators model for volatility using intra-daily data. J. Econom. 131 (1–2), 3–27. Engle, R., Lilien, D.M., Robins, R.P., 1987. Estimating time varying risk premia in the term structure: the ARCH-m model. Econometrica 55 (2), 391–407. Gatheral, J., Oomen, R.C.A., 2010. Zero-intelligence realized variance estimation. Finance Stochastics 14 (2), 249–283. Hansen, P.R., Huang, Z., 2016. Exponential GARCH modeling with realized measures of volatility. J. Bus. Econ. Stat. 34 (2), 269–287. Hansen, P.R., Huang, Z., Shek, H.H., 2012. Realized GARCH: a joint model for returns and realized measures of volatility. J. Appl. Econom. 27 (6), 877–906. Huang, Z., Wang, T., Hansen, P.R., 2017. Option pricing with the realized GARCH model: an analytical approximation approach. J. Futures Mark. 37 (4), 328C358. Newey, W.K., West, K.D., 1987. A simple, positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55 (3), 703–708. Patton, A.J., 2011. Volatility forecast comparison using imperfect volatility proxies. J. Econom. 160 (1), 246–256. Shephard, N., Sheppard, K., 2010. Realising the future: forecasting with high frequency based volatility (HEAVY) models. J. Appl. Econom. 25, 197–231. Wu, X.Y., Xia, M., Zhang, H.M., 2019. Forecasting var using realized EGARCH model with skewness and kurtosis. Financ. Res. Lett. Available online at doi:10.1016/j. frl.2019.01.002. Xie, H.B., Qi, N., Wang, S.Y., 2019. A new variant of realGARCH for volatility modeling. Financ. Res. Lett. 28, 438–443.
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Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Realized GARCH models: Simpler is better☆ Haibin Xie
⁎,1
, Chengtan Yu
University of International Business and Economics, Huixin East Road 10, Chaoyang District, Beijing 100029 China
ARTICLE INFO
ABSTRACT
Keywords: GARCH@CARR RealGARCH Volatility forecasting
Within the framework of Realized GARCH (RealGARCH), different RealGARCH variants have been proposed for volatility forecasting. The question remains unknown that which RealGARCH variant is more efficient. This paper compares three RealGARCH variants including the log-linear RealGARCH, the RealEGARCH and the GARCH@CARR. A comprehensive empirical study is performed on a stock index and 28 individual stocks, and the results show that the GRACH@CARR model outperforms the other two. Given that GARCH@CARR is more parsimonious in its specification, this finding is consistent with the principle of parsimony that models of simple structure usually provide better forecasts than the complex ones.
1. Introduction It has been well-known that realized volatility is a far more informative volatility estimator than is the squared return (Andersen and Bollerslev, 1998; Andersen et al., 2003; Barndorff-Nielsen and Shephard, 2002; 2004; Barndorff-Nielsen et al., 2008). Different methods have been proposed to incorporate the realized volatility into volatility estimation and forecasting, including the GARCH-X model of Engle (2002), the MEM model of Engle and Gallo (2006), the HEAVY model of Shephard and Sheppard (2010), and the most recent RealGARCH of Hansen et al. (2012). Among them RealGARCH model is the most popular one as it provides a framework of a joint modeling of return and realized volatility. Within the RealGARCH framework, different variants of RealGARCH have been proposed, for example the linear RealGARCH and the log-linear RealGARCH in Hansen et al. (2012), the RealEGARCH in Hansen and Huang (2016), and the most recent GARCH@CARR in Xie et al. (2019). Hansen et al. (2012) show that both linear RealGARCH and log-linear RealGARCH dominate the standard GARCH and EGARCH models that only use returns. Hansen and Huang (2016) compare several realized measures using RealEGARCH and find the realized kernel (RK) measure is most informative. Xie et al. (2019) propose a new variant of RealGARCH with multiplicative error specification. In their model the asset volatility dynamics is specified to be determined by a Conditional AutoRegressive Realized Volatility (henceforth GARCH@CARR) model. An important feature with the GARCH@CARR is that it has a simpler model specification compared with the other variants. Using the high-low price range as a realized measure, Xie et al. (2019) find the GARCH@CARR also outperforms the return-based GARCH and EGARCH models. Recent academic literature shows a rising interest in using RealGARCH models for asset pricing and risk management. Huang et al. (2017) obtain a analytical approximation formula for option pricing under RealEGARCH. Contino and Gerlach (2017) use the log-linear RealGARCH to forecast tail risk. Banulescu et al. (2018) use the RealEGARCH model to investigate the volatility during the financial crisis. Wu et al. (2019) propose using the RealEGARCH with skewness and kurtosis to forecast VaR. Despite the
This research was supported by National Natural Science Foundation of China under grant no. 71401033. Corresponding author. E-mail addresses: [email protected] (H. Xie), [email protected] (C. Yu). 1 Program for Young Excellent Talents, UIBE under grant no. 15YQ08. ☆ ⁎
https://doi.org/10.1016/j.frl.2019.06.019 Received 26 November 2018; Received in revised form 18 May 2019; Accepted 29 June 2019 1544-6123/ © 2019 Elsevier Inc. All rights reserved.
Please cite this article as: Haibin Xie and Chengtan Yu, Finance Research Letters, https://doi.org/10.1016/j.frl.2019.06.019
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Table 1 Out-of-sample forecasting performance evaluation: DM test. Benckmark: GARCH@CARR
SPY AA AIG AXP BA BAC C CAT CVX DD DIS GE GM HD IBM INTC JNJ JPM KO MCD MMM MRK MSFT PG T UTX VZ WMT XOM Average
Benckmark: log-linear RealEGARCH
A: log-linear RealGARCH
B: RealEGARCH
C: RealEGARCH
MSE
QLIKE
MSE
QLIKE
MSE
QLIKE
1.83 2.35* 2.03* 1.46 1.74 1.73 0.86 0.94 2.44* 1.68 1.20 2.48* 0.37 −0.62 0.76 1.13 0.77 0.51 −0.17 −0.74 −0.65 1.12 −0.30 1.52 0.77 1.29 1.31 −0.56 2.47* 1.02
2.22* 1.05 2.30* 1.97* 1.38 2.05* 2.59* 1.10 3.21* 1.49 −0.89 3.82* 1.29 0.04 0.22 1.05 −0.92 1.74 3.39* 0.56 −0.23 2.13* 1.97* −0.50 0.97 4.50* 0.72 −1.80 2.72* 1.38
1.05 2.74* 1.75 1.57 1.02 0.71 1.58 1.22 2.29* 1.83 1.43 1.88 0.92 2.82* 0.19 −0.08 −0.02 −0.51 0.63 2.20* 1.57 1.13 −0.13 2.19* 0.37 0.04 0.13 2.61* 2.16* 1.12
2.00* 2.04* 3.38* 1.80 −0.08 2.62 1.29 2.15* 3.08* 2.13* 0.48 4.28* 2.18* 2.42* 2.25* 1.73 1.71 2.38* 4.68* 2.05* −0.75 2.01* 1.68 0.02 2.41* 3.58* 2.24* 1.93 2.26* 2.06
−0.61 1.60 0.35 1.36 −0.49 0.16 0.98 2.22* 0.61 1.59 −0.90 1.14 0.87 1.30 −0.11 −0.31 −0.29 0.91 −0.33 1.81 1.66 −1.40 1.02 −1.32 0.69 −1.14 −0.77 1.73 0.33 0.44
−1.38 1.30 0.70 −0.45 1.50 0.95 −0.89 2.00* 1.61 1.87 −0.37 2.68* 0.23 0.78 1.86 1.01 2.21* −0.13 1.06 1.18 1.70 −1.57 1.77 0.99 1.45 0.37 2.71* 3.38* 0.31 0.99
Notes: A t-statistic greater than 1.96 in absolute value indicates a rejection of the null of equal predictive accuracy at the 5% level. These statistics are marked with an asterisk. The sign of the t-statistics indicates which forecast performed better for each loss function: a positive t-statistic indicates that benchmark model outperforms, while a negative sign indicates the opposite.
popularity of RealGARCH models in empirical applications, the question remains uninvestigated that which RealGARCH variant is more efficient. A careful study of this question will be meaningful for RealGARCH model selection and application. We compare three RealGARCH variants, the log-linear RealGARCH, RealEGARCH and the most recent GARCH@CARR. We don’t investigate the linear RealGARCH since Hansen et al. (2012) find that log-linear RealGARCH is more preferred than the linear RealGARCH. We evaluate these RealGARCH variants on a stock index and 28 individual stocks. The results show that GARCH@CARR dominates the other two variants and the log-linear RealGARCH dominates the RealEGARCH. Given that log-linear RealGARCH and GARCH@CARR have a simpler model specification, this finding is consistent with the principle of parsimony that simpler models usually provide better forecasts than more complex ones. The rest of this paper is organized as follows. Section 2 presents the RealGARCH variants evaluated in this paper. Section 3 presents our evaluation method. Section 4 presents the out-of-sample empirical results. We conclude in Section 5. 2. RealGARCH variants RealGARCH framework is a joint modeling of return and realized volatility. The general structure of RealGARCH(p, q) model is given by2
rt =
(1)
ht z t ,
ht = (ht 1, …, ht p, …, xt 1, …, xt q),
(2)
x t = m (ht , z t , ut ),
(3)
2 To simplify the presentation we assume that E(rt |Ft 1) = 0. A more general specifications for conditional mean, such as a constant or the GARCH-in-mean by Engle et al. (1987), is accommodated by reinterpreting rt as the return less its conditional mean.
2
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Table 2 Out-of-sample forecasting performance evaluation: linear regression. Panel A: GARCH@CARR
SPY AA AIG AXP BA BAC C CAT CVX DD DIS GE GM HD IBM INTC JNJ JPM KO MCD MMM MRK MSFT PG T UTX VZ WMT XOM Average
Panel B: log-linear RealGARCH
Panel C: RealEGARCH
a
b
R2
a
b
R2
a
b
R2
−0.01 0.17 0.21 −0.01 0.10 0.05 0.14 −0.16 −0.13 −0.14 0.05 0.03 −0.39 −0.07 0.09 0.14 0.04 0.02 0.09 −0.00 −0.11 −0.41 0.17 −0.03 0.01 −0.13 0.06 0.06 −0.08 −0.01
0.94* 1.16* 1.07* 1.02* 1.00* 1.20* 1.20* 1.07* 1.21* 1.36* 1.06* 1.11* 1.08* 1.18* 1.09* 0.94* 1.13* 1.13* 1.22* 1.10* 1.26* 1.29* 1.05* 1.35* 1.04* 1.22* 1.15* 1.25* 1.15* 1.14
0.59 0.43 0.69 0.71 0.45 0.75 0.74 0.47 0.52 0.49 0.44 0.51 0.62 0.61 0.49 0.50 0.28 0.70 0.36 0.36 0.49 0.25 0.47 0.40 0.48 0.39 0.52 0.41 0.51 0.50
0.04 −0.13 0.35 −0.05 −0.26 0.63* 0.62* 0. 28* 0. 91* 0. 27* −0.08 0.13 −0.75 −0.02 0.21* 0.10 −0.02 0.30 0.17 0.04 −0.06 1. 73* 0.31 −0.06 0.38* 0.21* 0.11 −0.01 0. 60* −0.06
0.90* 1.29* 1.04* 1.12* 1.26* 0.96* 1.04* 1.15* 1.82* 1.49* 1.15* 1.04* 1.12* 1.16* 0.95* 0.95* 1.21* 1.01* 1.10* 1.06* 1.21* 2.07* 0.94* 1.39* 0.82* 0.95* 1.13* 1.31* 1.59* 1.18
0.55 0.40 0.67 0.70 0.41 0.69 0.70 0.47 0.48 0.48 0.42 0.48 0.62 0.62 0.46 0.48 0.27 0.68 0.35 0.36 0.48 0.25 0.46 0.38 0.47 0.35 0.49 0.42 0.49 0.49
−0.02 0.14 0.21 −0.18 0. 32* 0.26 0.28 0. 41* 1. 00* 0. 30* −0.06 0.06 −0.46 −0.16 0.16* 0.05 −0.06 −0.01 0.17* 0.04 −0.13 1. 57* 0.28* −0.09 0.27* −0.06 0.04 −0.03 0. 79* −0.13
0.96* 1.20* 1.11* 1.18* 1.27* 1.15* 1.16* 1.23* 1.88* 1.53* 1.16* 1.13* 1.10* 1.25* 1.01* 1.01* 1.34* 1.15* 1.11* 1.07* 1.28* 1.98* 0.96* 1.46* 0.90* 1.17* 1.17* 1.37* 1.72* 1.24
0.57 0.41 0.68 0.70 0.45 0.74 0.72 0.46 0.51 0.46 0.43 0.49 0.59 0.61 0.48 0.50 0.31 0.71 0.35 0.35 0.47 0.25 0.46 0.39 0.47 0.39 0.53 0.40 0.51 0.50
Notes: The realized kernel (RK) are used as the volatility proxy. An asterisk * means significance at the level of 5%. The linear regression is Vt + n = a + bFVi, t+ n +
t+n,
n = 1, 2. ,N
where Vt + n is the volatility proxy, FVi, t+ n is the forecast reported by model i. If model i reports unbiased volatility estimation, we expect a=0 and b=1.
where {rt} is a return series, xt is a kind of realized volatility measure and zt ∼ i.i.d. N(0,1), with zt and ut being mutually independent. The first two equations are referred to as return equation and GARCH equation, and the last one is called measurement equation. Different specifications to (.) and m(.) give different RealGARCH variants. Within this RealGARCH framework, different variants have been proposed. In the following subsection, we will present the most commonly used log-linear RealGARCH and RealEGARCH and the most recently proposed GARCH@CARR model. In the empirical we only consider the variants of order (1, 1) as it is well documented that order (1, 1) is sufficient to capture the volatility dynamics. 2.1. Log-linear realized GARCH The RealGARCH with log-linear specification in Hansen et al. (2012) is characterized by
rt =
(4)
ht z t ,
ln (ht ) = ln (xt ) =
(5)
+ ln (ht 1) + ln (x t 1),
(6)
+ ln (ht ) + (z t ) + µt ,
1) is the leverage function. This leverage function form where μt ∼ i.i.d. N(0, σμ), xt is the realized variance, and (z ) = 1 z + 2 is proposed in Hansen et al. (2012). The log-linear RealGARCH includes the leverage effect indirectly in the GARCH equation through the measurement equation. (z 2
2.2. Realized EGARCH Hansen and Huang (2016) introduce a Realized Exponential GARCH (RealEGARCH) model that can utilize multiple realized 3
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volatility measures to model return series.
rt =
(7)
ht z t ,
ln (ht ) =
+ ln (ht 1) + µt
ln (xt ) =
1
(8)
+ d (z t 1),
(9)
+ ln (ht ) + (z t ) + µt ,
where d (z ) = d1 z + d2 (z 2 1) is the leverage function. The main difference between log-linear RealGARCH and RealEGARCH is that the RealEGARCH incorporates a new leverage function d (z t 1) in the GARCH equation, which thus makes it more flexible in capturing the leverage effect. 2.3. GARCH@CARR By specifying a MEM (multiplicative error model) to the measurement equation, Xie et al. (2019) propose the following variant
rt =
ln ( t ) =
yt =
(10)
t zt ,
+ ln (
t 1)
(11)
+ ln (yt 1 ) + d (z t 1),
(12)
t ut ,
where yt = xt , ut ∼ i.i.d. LN ( u ) is log-normal with unit mean. Scale factor ρ is used to make sure ρλt is an unbiased volatility estimation. Since Eqs. (11)–(12) are known as Conditional AutoRegressive Realized volatility model, thus this RealGARCH variant is called GARCH@CARR. Squared root of realized variance is used for the reason that GARCH@CARR is designed for standard deviation instead of variance. The main difference between GARCH@CARR and the other two variants is that GARCH@CARR model is a restricted RealEGARCH. Note that ht = t , x t = t ut . Eq. (12) in GARCH@CARR is equivalent to 2 u /2,
ln (xt ) =
(13)
2ln ( ) + ln (ht ) + 2ln (ut )
Comparing Eq. (13) with Eq. (9), we can obtain
=
2ln ( ),
= 1,
2ln (ut ) = µt ,
(14)
(z t ) = 0
This restriction makes the GARCH@CARR model more concise and simpler in its specification than the other two variants. Comparing the three RealGARCH variants, we find the most significant difference is the complexity in model specification. There are 8 parameters in log-linear RealGARCH ({ω, β, γ, ξ, φ, τ1, τ2, σμ}), 10 parameters in RealEGARCH ({ω, β, γ, d1, d2, ξ, φ, τ1, τ2, σμ}) and 7 parameters in GARCH@CARR ({ρ, ω, β, γ, τ1, τ2, σu}). A comprehensive look at the forecasting performance of these 3 variants can help us better understand how the model complexity could affect the RealGRACH forecasting performance. 3. Evaluation method The Diebold-Mariano test (Diebold and Mariano, 1995) is used for comparing two competing forecasting models. Suppose the losses reported by models i and j are denoted by Lm,i,t and Lm,j,t (m=1 or 2). The Diebold-Mariano test verifies the null hypothesis that Lm, j, t . The Diebold-Mariano test statistic is given by E(dw, t)=0, where d w, t = Lm, i, t
DM =
d¯ Var (d¯)
,
(15)
N where d¯ = N 1 n = 1 d w, t + n, and N is the total number of forecasts. The variance of d¯, Var (d¯) is estimated by the heteroscedasticity and autocorrelation consistent (HAC) estimator of Newey and West (1987). Diebold and Mariano (1995) show that under the null hypothesis of equal predictive accuracy the DM statistic follows standard normal distribution. Since the true volatility is latent, we can only use volatility proxy as the “true volatility” to construct the DM statistic. It has been noted that imperfect volatility proxy may distort the ranking of competing volatility forecasts. Patton (2011) examines a class of loss functions and finds that only the MSE (mean squared error) and QLIKE (quasi-likelihood loss function) are robust to an imperfect volatility proxy. Moreover, QLIKE is found to be less sensitive to extreme observations than the squared error loss (Patton (2011); Conrad and Kleen (2018)). MSE and QLIKE are defined as
^t2),
MSE = E (L1, k, t ), L1, k, t = (
2 t
QLIKE = E (L 2, k, t ), L 2, k, t =
2 (log ( ^t )
(16)
+
2^ t t
2
(17)
),
where L1,k,t and L2,k,t are the losses for the forecasting model k. We also use the following linear regression analysis to evaluate the forecasting performance of different models
Vt + n = a + bFVi, t + n +
t+n ,
(18)
n = 1, 2. ,N
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where Vt + n is the volatility proxy, FVi, t + n is the forecast reported by model i. If model i reports unbiased volatility estimation, we expect a=0 and b=1. 4. Empirical results For empirical study, we use the same data as Hansen et al. (2012). This data set includes an exchange-traded index fund, SPY, that tracks the S&P500 index and 28 individual stocks. Since daily high-frequency prices are only available between ‘open’ and ‘close’, so we use in this paper open-to-close returns3 4.1. Out-of-Sample results We evaluate the three RealGARCH variants through out-of-sample forecasting performance4 A recursive (expanding) window forecasting procedure is used and the last 500 data observations are used for evaluation5 The realized kernel (RK), introduced by Barndorff-Nielsen et al. (2008), is used as the realized measure. This choice is based on the theoretical strengths of the RK measure: it is robust to microstructure noise and makes efficient use of the intraday data. Moreover, in a comparison of 19 realized variance measures, Gatheral and Oomen (2010) find that RK is one of the best estimators in terms of efficiency and robustness to time varying parameters. Table 1 presents the DM statistics. We first use the GARCH@CARR model as the benchmark and compare it with the log-linear RealGARCH and the RealEGARCH, and the results are presented in Panels A-B. Taking the log-linear RealGARCH as the benchmark, we also compare it with the RealEAGRCH, and results are reported in Panel C. We calculate the average DM statistics of all the individual ones and the results are reported in the last row. According to the average DM statistics we find that on average the GARCH@CARR dominates the other two variants and the log-linear RealGARCH dominates the RealEGARCH. It is more interesting to look at the DM statistics for each individual asset. The DM statistics in Panel A shows that the GARCH@CARR dominates the loglinear RealGARCH in 5(12) out of the 29 assets by the MSE (QLIKE). The dominance of GARHC@CARR over RealEGARCH is more significant by the DM statistics in Panel B. In 7 (18) out of the 29 assets, GARCH@CARR dominates RealEGARCH by MSE (QLIKE). The log-linear RealGARCH dominates the RealEGARCH in 1 (5) out of the 29 assets by MSE (QLIKE). Taking the RK as the volatility proxy, Table 2 reports the linear regression results. We also calculate the average intercept (a) and the average slope (b) and the average R-square (R2), and the results are presented in the last row. Consistent with the DM statistics, the results show that on average the GARCH@CARR reports the least biased out-of-sample forecasts while the RealEGARCH reports the largest biased out-of-sample forecast. In summary, we find the GARCH@CARR model reports the best out-of-sample forecasts while the most commonly used RealEGARCH reports the worst forecasts. Given that the GARCH @CARR has the simplest model specification while the RealEGARCH has the most complex specification, this finding is consistent with the principle of parsimony that simple models usually have better performance than complex ones. 5. Conclusions The Realized GARCH model provides an efficient and flexible framework for a joint modeling of return and realized measure. This paper comprehensively evaluates the forecasting performance of three RealGARCH variants, including the most commonly used loglinear RealGARCH, RealEGARCH and the most recent proposed GARCH@CARR. We find clear dominance of GARCH@CARR over the other two competing ones. Given that the GARCH@CARR has the simplest model specification, this finding is consistent with the principle of parsimony that simple models usually have better performance than complex ones. Of course, our finding is also valuable for future model selection and application. Supplementary material Supplementary material associated with this article can be found, in the online version, at 10.1016/j.frl.2019.06.019 References Andersen, T.G., Bollerslev, T., 1998. Answering the skeptics: yes, standard volatility models do provide accurate forecasts. Int. Econ. Rev. 39 (4), 885–905. Andersen, T.G., Bollerslev, T., Diebold, F.X., Labys, P., 2003. Modeling and forecasting realized volatility. Econometrica 71 (2), 579–625. Banulescu, D., Hansen, P.R., Huang, Z., Matei, M., 2018. Volatility During the Financial Crisis Through the Lens of High Frequency Data: a Realized GARCH Approach. Social Science Electronic Publishing. Availabe at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3178890.
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The sample spans the period from 1 January 2002 to 31 August 2008. The dataset is also used in Hansen et al. (2012) and available at http://qed. econ.queensu.ca/jae/2012-v27.6/hansen-huang-shek. All the return series have been demeaned. 4 The in-sample forecasting evaluation is very similar to the out-of-sample evaluation. We don’t present the in-sample results for the consideration of space-saving. The in-sample results are available on request. 5 Maximum likelihood estimation (MLE) can be used to estimate all these three RealGARCH variants. The MLE is performed using Matlab and the code is available on request. 5
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