- Email: [email protected]
An extension of stochastic volatility model with mixed frequency information
Accepted Manuscript An extension of stochastic volatility model with mixed frequency information Yuhuang Shang, Lulu Liu PII: DOI: Reference:
S0165-1765(17)30143-X http://dx.doi.org/10.1016/j.econlet.2017.04.003 ECOLET 7576
To appear in:
Economics Letters
Received date: 24 December 2016 Revised date: 31 March 2017 Accepted date: 3 April 2017 Please cite this article as: Shang, Y., Liu, L., An extension of stochastic volatility model with mixed frequency information. Economics Letters (2017), http://dx.doi.org/10.1016/j.econlet.2017.04.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlight 1.
We extend the basic SV model with mixed frequency information which is referred to as the MF-SV model.
2.
The MCMC method is discussed to realize the parameter estimation by a mixture approximation model.
3. 4.
The MF-SV model can significantly identify the time-varying stable component. The MF-SV model can improve the in-sample fitting results that outperform the basic SV model.
*Title Page
An Extension of Stochastic Volatility model with Mixed Frequency Information Yuhuang Shang
Lulu Liu
Abstract: This paper extends the SV model to the MF-SV model with mixed frequency information. We show the MCMC method and the small sample properties with Monte Carlo experiment. The MF-SV model outperforms the basic SV model in the in-sample performance. Keywords: Stochastic Volatility; Mixed-Frequency; Monte Carlo experiment; MCMC method; Unobservable Component
JEL Classification: C5; C22; G1
Yuhuang Shang (Corresponding Author), Assistant Professor, Institute of Chinese Financial
Studies, Southwestern University of Finance and Economics, Chengdu, P.R, China, 611130. Email: [email protected], Tel: +86 02887092784. Lulu Liu (Corresponding Author), Ph.D student, P. R., China, 730000. Email: [email protected].
CAREERI of the Chinese Academy of Sciences,
*Manuscript Click here to view linked References
An Extension of Stochastic Volatility model with Mixed Frequency Information Yuhuang Shanga a b
Lulu Liub
Institute of Chinese Financial Studies, Southwestern University of Finance and Economics, China
CAREERI of the Chinese Academy of Sciences, China
Abstract: This paper extends the SV model to the MF-SV model with mixed frequency information. We show the MCMC method and the small sample properties with Monte Carlo experiment. The MF-SV model outperforms the basic SV model in the in-sample performance. Keywords: Stochastic Volatility; Mixed-Frequency; Monte Carlo experiment; MCMC method; Unobservable Component JEL Classification: C5; C22; G1
1. Introduction The stochastic volatility (SV) model has been applied to asset pricing (Harvey and Shephard,1996), particularly derivative pricing (Bansal,2014) since the SV model match discretization of the diffusion process of asset returns (Zheng and Zuo,2013). In this model, the volatility is only driven by a single “unobservable component” factor which ignores the linkage between volatility and other important factors. Instead, numerous studies such as Engle and Lee (1999) show that many factors may influence volatility, which leads to the decomposition of volatility components, then specify the component volatility model. Empirically, component volatility models usually decompose the volatility into a long-term and a short-term component (Engle et al., 2013). The long-term component is mainly affected by low-frequency variables (Engle et al., 2013; Zheng and Shang, 2014). In contrast, some high-frequency factors, such as liquidity shocks, contribute to the short-term component. Obviously, different frequency variables need to be considered when constructing a component model. Engle and Rangel (2008) firstly propose a Spline-GARCH model and Engle et al. (2013) further propose the GARCH-MIDAS (mixed-frequency data sampling) model. The MIDAS approach is used to help low frequency variables predict long-term components. Unfortunately, the mixed-frequency volatility model is only limited to GARCH family model but none to the SV family model. This paper decomposes the volatility into a stochastic component and a stable component in the framework of SV model. As a result, we specify a mixed frequency SV (MF-SV) model which is a generalized form of the traditional SV model. Similar to the GARCH-MIDAS model, the stable component is characterized by a low-frequency variable via the MIDAS method.
Yuhuang Shang, Email: [email protected], Tel: +86 02887092784. Lulu Liu (Corresponding Author), Email: [email protected]. Tel: +86 15982043870. 1
Meanwhile, the stochastic component keeps the same form as traditional SV models. Researchers have developed Markov chain Monte Carlo (MCMC) algorithms for estimating the parameters since the likelihood function for SV models is intractable (see, Nakajima and Omori ,2009). We propose the MCMC method to estimate the parameters of MF-SV model. This paper conducts Monte Carlo experiments to evaluate the finite sample performance of the MF-SV model. We also investigate the estimate results of MF-SV model via empirical study.
2. Methodology 2.1 MF-SV model Let ri ,t be the log return on day i during month (quarterly) t .We assume that there are N t days in period t . Referring to Engle and Rangel (2008), we write the level equation as follows
ri ,t i ,t t i ,t where i ,t t
(1)
represents volatility, which has two components, i,t and t .The error term
i ,t | i 1,t N (0,1) . Engle and Rangel (2008) have point out that the volatility component t is a secular component influenced by low-frequency volatility. The component i,t is related to short-lived factors. Similarly, we interpret t as a stable component and i,t as a stochastic component. The level equation can be rewritten as
ri ,t exp{
hi ,t 2
log( t ) } i ,t 2
(2)
where hi ,t log( i2,t ) . According to the basic SV model, let yi ,t log(ri 2,t ) , i ,t 1.27 log( i2,t ) , then we have yi ,t 1.27 hi,t log( t ) i ,t
(3)
where i ,t follow the log( 12 ) distribution with one degree of freedom, zero mean, and variance
2 / 2. The stochastic component can be expressed by the following equation: hi ,t hi 1,t i ,t
(4)
where i ,t N (0, 2 ) , i and i are independent of each other. The stable component t is described by some low frequency variable such as realized volatility ( RVt = i 1 ri 2,t ) over a monthly or quarterly horizon (Engle et al., 2013). We set t with Nt
log form by smoothing realized volatility in the spirit of the MIDAS regression. P
log t =m+ p (1 , 2 ) RVt n-p n t -k
where RV
(5)
p 1
is the normalized log low frequency realized volatility.
P is defined as MIDAS lag year which indicates the maximum lag order in Eq. (5). The weighting function p (1 , 2 ) is the “Beta” lag structure.
p (1 , 2 )
2
f ( p / P, 1 , 2 ) f ( p / P, 1 , 2 ) p 1
P
(6)
f ( x, a , b )
where
x a 1 (1 x)b 1 (a b) (a) (b)
(7)
Equations (3) to (7) form the MF-SV model. Compared with the traditional SV model, we find that when the component t is some constant value, the MF-SV model degenerates into the basic SV model.
2.2 The Mixture Approximation Model This paper uses the Bayesian method to realize the parameter estimation. Referring to Nakajima and Omori (2009), we need to approximate the log( 12 ) distribution by a K-component mixture of Gaussian densities with a mixture approximation model. Let yi,t yi ,t log( t ) , and then we obtain the following equation where
f ( i ,t ) k 1 qk f N ( i ,t | mi ,t , )
K
2 i ,t
yi ,t hi ,t i,t
(8)
means using K-component mixture of Gaussian
densities to approximate the distribution of log( 12 ) , and qk is the weight of the normal distribution. The mixture approximation model can be written as the linear Gaussian state space model:
yi,t hi ,t i,t hi ,t 1 hi ,t i ,t 2 1 i ,t L mk k zi ,t | s k t zi2,t i ,t
1
(9)
(10)
2
where both zi ,t and zi ,t follow the standard normal distribution. MCMC method can be realized via the following blocks: (1) Initialize parameters , , m , , and state process {si }Ti 1 , {hi }Ti 1 ; (2) Given parameters m , , , state process {si }Ti 1 , {hi }Ti 1 , data sample yi,t , sample , with the M-H (Metropolis–Hastings) algorithm. The prior distribution of is the beta distribution and that of is the inverse gamma distribution; (3) Given , , state process {si }Ti 1 , data sample yi,t , construct the augmented Kalman filter and sample {hi }Ti 1 using the simulation smoother; (4) Given , , m , , , state process {hi }Ti 1 , data sample yi,t , sample {si }Ti 1 using a probability mass function; (5) Given , , state process {hi }Ti 1 , sample m , , using the M-H algorithm.
3. Monte Carlo experiment We conduct Monte Carlo experiments to evaluate the finite sample performance of the MCMC estimation procedure. The data are generated by the following MF-SV model: yi ,t 1.27 hi,t log( t ) i ,t hi ,t hi 1,t i ,t 3
P
log t =m+ p (1 , 2 ) X t -p p 1
where yi ,t log(ri 2,t ) and ri ,t is the log return on high frequency i during low frequency
t .We assume that high frequency i occurs 30 times in period t , i ,t follow the log( 12 ) distribution with one degree of freedom, zero mean, and variance 2 / 2 , i ,t N (0, 2 ) , the low frequency observation X t N (0,1) . p (1 , 2 ) is described by Equations (6) to (7). P=12 represents the maximum lag order. The true parameters of MF-SV model are assigned as1: 0.96 , 0.6 , m 8.5 ,
0.9 , 1 4.0 . Referring to Nakajima and Omori (2009), this paper uses a 10-component normal distribution to approximate the log( 12 ) distribution as shown in Table 1. We assume the prior distributions of parameters and . The prior distribution of is Beta (20,1.5) and that of follows
IG(0.5 0 ,0.5 1 ) ， 0 20 ， 1 0.01 0 . The reduced MCMC sampling for the posterior part is iterated with 5,000 draws. The initial 1,000 samples are discarded. Table 1 10-component mixture of Gaussian distributions w
Pr( w k )
mk
k2
1 2 3 4 5 6 7 8 9 10
0.0061
1.9268
0.1127
0.0478
1.3474
0.1779
0.1306
0.7350
0.2677
0.2067
0.0227
0.4061
0.2272
-0.8517
0.6270
0.1884
-1.9728
0.9858
0.1205
-3.4679
1.5747
0.0559
-5.5525
2.5450
0.0158
-8.6838
4.1659
0.0012
-14.6500
7.3334
We generate 200 sets of data for each sample size (high frequency) I 300 , I 600 and
I 900 . Our simulation results are reported in Tables 2. From the results reported by Tables 2, we find that the performance of estimation is better with the increasing of sample size. For example, the estimation of is more close to the true value when the sample size becomes large. When the sample size I 300 , the estimation of is downwardly biased and the corresponding true values even fall outside the 95% confidence interval. With the sample increasing, the estimation of
is more close to the true value. Table 2 Monte Carlo experiment results of the proposed estimation procedures
I 300
I 600
I 900
True value
0.96 1
Mean
S.D.
95%C.I.
Mean
S.D.
95%C.I.
Mean
S.D.
95%C.I.
0.9788
0.0094
[0.9574,0.9939]
0.9742
0.0093
[0.9532,0.9899]
0.9631
0.0102
[0.9415, 0.9816]
We set 2 = 1 and only estimate 1 .this set can produce the smooth weigh that the decay features. 4
m
0.6
0.3995
0.0715
[0.2764,0.5549]
0.5406
0.0614
[0.4290,0.6688]
0.5629
0.0571
[0.4580, 0.6824]
-8.5
-8.4981
0.0247
[-8.5406,-8.4525]
-8.4990
0.0240
[-8.5391,-8.4526]
-8.5221
0.0223
[-8.5658,-8.4835]
0.9
0.9077
0.0219
[0.8730, 0.9482]
0.9045
0.0250
[0.8586, 0.9460]
0.8965
0.0218
[0.8603,0.9196]
1
4.0
4.0044
0.0243
[3.9623, 4.0456]
3.9896
0.0276
[3.9463, 4.0392]
3.9960
0.0274
[3.9456 ,4.0425]
Note: Parameter estimates are obtained from posterior means of MCMC draws which are based on 5000 draws, discarding first 1000. S.D. is standard error of parameter estimation. 95%C.I. means the 95% confidence interval.
4. Empirical Study The data of daily returns, low-frequency realized volatility are collected for empirical study. Daily returns, monthly realized volatility are constructed using the daily stock price. We choose the close the S&P 500 as the raw data. The sample period is from January 3, 1994 to June 30, 2015. The sample size is 5216. The raw data are available from the database of WIND. The time series data are plotted in Fig. 1. 0.10
Return_SP500(High_Frequency)
0.05 0.00 -0.05
1994
1996
1998
2000
2002
2004
2006
2008
0.06
2010
2012
2014
2016
RV_SP500(Low_Frequency)
0.04
0.02
1995
2000
2005
2010
2015
Fig. 1. Daily Returns and Monthly Realized Volatility According to Nakajima and Omori (2009), we assume the prior distributions of and are
Beta (20,1.5) and IG(0.5 0 ,0.5 1 ) ， 0 5 ， 1 0.01 0 in U.S. stock market. The reduced MCMC sampling for the posterior part is iterated with 10,000 draws. The initial 2,000 samples are discarded. Table 3 B_SV
MCMC estimate results of MF-SV model
MF-SV(1)
0.9870 [0.9810 0.9923]
0.9582 [0.9487
5
0.9712]
MF-SV(2) 0.9569 [0.9425
0.9696]
-9.4148
--
--
0.1483
0.1945
0.1987
[0.1260 0.1735]
[0.1654 0.2307]
--
-10.0607
-10.1296
[-10.1201 -10.0205]
[-10.2246 -10.0234]
0.8833
0.8574
[-9.7505 -9.0869]
m
--
[0.8436 0.9311]
1
--
[0.7969
0.2321]
0.9125]
2.2735
5.6079
[2.2017 2.3802]
[5.5664 5.6484]
0.9501
0.9081
--
rRMSE
[0.1695
Note：B_SV represents the basic SV model. The parameter results are the mean of the posterior distribution. The value in parentheses represents the 95% confidence interval.
In MF-SV(P) model, P = 1 means 1 MIDAS lag year. In other words, the low frequency information of lag 12 months is introduced into the MF-SV(P) model, namely the MF-SV(1) model. Similarly, we can also construct the MF-SV(2) model. To compare the sample fit results, we calculate the rRMSE2. If the value of rRMSE is less than 1, the MF-SV model has a better fit performance than that of the benchmark model. Table 3 reports the estimate results. First, many estimated parameters have statistical significance. takes a significantly positive value suggesting that the low frequency realized volatility has a positive effect on the stable component. This result is similar to that of Engle et al. (2013). Second, based on the rRMSE, the in-sample fit of the MF-SV model is superior to that of the basic SV model. This implies that the SV-MIDAS model can make use of more information to identify a time-varying stable component that improves the in-sample fitting performance. As mentioned above, the MF-SV model can identify the time-varying stable component. To show this, we calculate
t2,i t and
t which represent the conditional variance and the
stable component respectively. As shown in Fig. 2, we find that there is co-movement tendency between the stable component and the conditional variance, but the stable component is less volatile. Second, the stable component presents the obvious time-varying feature that cannot be estimated by the traditional SV model.
2
2 2 RMSE (T 1 t 1 (ˆt2 t2 )2 )1/2 , where ˆ t is estimated conditional variance, t is the proxy of T
volatility. We compute the RMSE of both the MF-SV and the basic SV model and obtain the rRMSE by use the RMSE of MF-SV model divided by that of the benchmark model. 6
Stable_Component
Conditonal_Variance
0.04
0.02
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
2016
MF-SV(1) Stable_Component
Conditonal_Variance
0.04
0.02
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
2016
MF-SV(2)
Fig. 2.
The stable component of the MF-SV models
5. Conclusion In this paper, we extend the basic SV model to the mixed-frequency SV model, referred to as the MF-SV model. The MCMC method is employed to realize the parameter estimation of MF-SV model by approximating it to a mixture approximation model. This paper also discusses the small sample properties with Monte Carlo experiment. Based on the empirical study, we find that the MF-SV model can describe the time-varying stable component. Second, the proposed MF-SV model outperforms the basic SV model in the in-sample performance.
Acknowledgments The Research Foundation for Young Scholars of the Ministry of Education of China (no.16YJC790084) provided research support. For helpful comments we are grateful to the Editor and the referee. We, however, bear full responsibility for all remaining flaws.
References Bansal R, Kiku D, Shaliastovich I. Volatility, the macroeconomy, and asset prices[J]. The Journal of Finance, 2014, 69(6): 2471-2511. Durbin J, Koopman S J. A simple and efficient simulation smoother for state space time series analysis[J]. Biometrika, 2002, 89(3): 603-616. Engle R F, Ghysels E, Sohn B. Stock market volatility and macroeconomic fundamentals[J]. Review of Economics and Statistics, 2013, 95(3): 776-797. Engle R F, Rangel J G. The spline-GARCH model for low-frequency volatility and its global macroeconomic causes[J]. Review of Financial Studies, 2008, 21(3): 1187-1222. Harvey A C, Shephard N. Estimation of an asymmetric stochastic volatility model for asset returns[J]. Journal of 7
Business & Economic Statistics, 1996, 14(4): 429-434. Nakajima J, Omori Y. Leverage, heavy-tails and correlated jumps in stochastic volatility models[J]. Computational Statistics & Data Analysis, 2009, 53(6): 2335-2353. Zheng,T,G and Zou, H,M. Range based regime switching stochastic volatility models with applications [J].Journal of Management Sciences in China (in Chinese) 2013,09:82-94. Zheng,T,G and Shang, Y,H. Measuring and Forecasting the Stock Market Volatility Based on Macroeconomic Fundamentals [J].The Journal of World Economy,(in Chinese) , 2014,12:118-139.
8
S0165-1765(17)30143-X http://dx.doi.org/10.1016/j.econlet.2017.04.003 ECOLET 7576
To appear in:
Economics Letters
Received date: 24 December 2016 Revised date: 31 March 2017 Accepted date: 3 April 2017 Please cite this article as: Shang, Y., Liu, L., An extension of stochastic volatility model with mixed frequency information. Economics Letters (2017), http://dx.doi.org/10.1016/j.econlet.2017.04.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Highlight 1.
We extend the basic SV model with mixed frequency information which is referred to as the MF-SV model.
2.
The MCMC method is discussed to realize the parameter estimation by a mixture approximation model.
3. 4.
The MF-SV model can significantly identify the time-varying stable component. The MF-SV model can improve the in-sample fitting results that outperform the basic SV model.
*Title Page
An Extension of Stochastic Volatility model with Mixed Frequency Information Yuhuang Shang
Lulu Liu
Abstract: This paper extends the SV model to the MF-SV model with mixed frequency information. We show the MCMC method and the small sample properties with Monte Carlo experiment. The MF-SV model outperforms the basic SV model in the in-sample performance. Keywords: Stochastic Volatility; Mixed-Frequency; Monte Carlo experiment; MCMC method; Unobservable Component
JEL Classification: C5; C22; G1
Yuhuang Shang (Corresponding Author), Assistant Professor, Institute of Chinese Financial
Studies, Southwestern University of Finance and Economics, Chengdu, P.R, China, 611130. Email: [email protected], Tel: +86 02887092784. Lulu Liu (Corresponding Author), Ph.D student, P. R., China, 730000. Email: [email protected].
CAREERI of the Chinese Academy of Sciences,
*Manuscript Click here to view linked References
An Extension of Stochastic Volatility model with Mixed Frequency Information Yuhuang Shanga a b
Lulu Liub
Institute of Chinese Financial Studies, Southwestern University of Finance and Economics, China
CAREERI of the Chinese Academy of Sciences, China
Abstract: This paper extends the SV model to the MF-SV model with mixed frequency information. We show the MCMC method and the small sample properties with Monte Carlo experiment. The MF-SV model outperforms the basic SV model in the in-sample performance. Keywords: Stochastic Volatility; Mixed-Frequency; Monte Carlo experiment; MCMC method; Unobservable Component JEL Classification: C5; C22; G1
1. Introduction The stochastic volatility (SV) model has been applied to asset pricing (Harvey and Shephard,1996), particularly derivative pricing (Bansal,2014) since the SV model match discretization of the diffusion process of asset returns (Zheng and Zuo,2013). In this model, the volatility is only driven by a single “unobservable component” factor which ignores the linkage between volatility and other important factors. Instead, numerous studies such as Engle and Lee (1999) show that many factors may influence volatility, which leads to the decomposition of volatility components, then specify the component volatility model. Empirically, component volatility models usually decompose the volatility into a long-term and a short-term component (Engle et al., 2013). The long-term component is mainly affected by low-frequency variables (Engle et al., 2013; Zheng and Shang, 2014). In contrast, some high-frequency factors, such as liquidity shocks, contribute to the short-term component. Obviously, different frequency variables need to be considered when constructing a component model. Engle and Rangel (2008) firstly propose a Spline-GARCH model and Engle et al. (2013) further propose the GARCH-MIDAS (mixed-frequency data sampling) model. The MIDAS approach is used to help low frequency variables predict long-term components. Unfortunately, the mixed-frequency volatility model is only limited to GARCH family model but none to the SV family model. This paper decomposes the volatility into a stochastic component and a stable component in the framework of SV model. As a result, we specify a mixed frequency SV (MF-SV) model which is a generalized form of the traditional SV model. Similar to the GARCH-MIDAS model, the stable component is characterized by a low-frequency variable via the MIDAS method.
Yuhuang Shang, Email: [email protected], Tel: +86 02887092784. Lulu Liu (Corresponding Author), Email: [email protected]. Tel: +86 15982043870. 1
Meanwhile, the stochastic component keeps the same form as traditional SV models. Researchers have developed Markov chain Monte Carlo (MCMC) algorithms for estimating the parameters since the likelihood function for SV models is intractable (see, Nakajima and Omori ,2009). We propose the MCMC method to estimate the parameters of MF-SV model. This paper conducts Monte Carlo experiments to evaluate the finite sample performance of the MF-SV model. We also investigate the estimate results of MF-SV model via empirical study.
2. Methodology 2.1 MF-SV model Let ri ,t be the log return on day i during month (quarterly) t .We assume that there are N t days in period t . Referring to Engle and Rangel (2008), we write the level equation as follows
ri ,t i ,t t i ,t where i ,t t
(1)
represents volatility, which has two components, i,t and t .The error term
i ,t | i 1,t N (0,1) . Engle and Rangel (2008) have point out that the volatility component t is a secular component influenced by low-frequency volatility. The component i,t is related to short-lived factors. Similarly, we interpret t as a stable component and i,t as a stochastic component. The level equation can be rewritten as
ri ,t exp{
hi ,t 2
log( t ) } i ,t 2
(2)
where hi ,t log( i2,t ) . According to the basic SV model, let yi ,t log(ri 2,t ) , i ,t 1.27 log( i2,t ) , then we have yi ,t 1.27 hi,t log( t ) i ,t
(3)
where i ,t follow the log( 12 ) distribution with one degree of freedom, zero mean, and variance
2 / 2. The stochastic component can be expressed by the following equation: hi ,t hi 1,t i ,t
(4)
where i ,t N (0, 2 ) , i and i are independent of each other. The stable component t is described by some low frequency variable such as realized volatility ( RVt = i 1 ri 2,t ) over a monthly or quarterly horizon (Engle et al., 2013). We set t with Nt
log form by smoothing realized volatility in the spirit of the MIDAS regression. P
log t =m+ p (1 , 2 ) RVt n-p n t -k
where RV
(5)
p 1
is the normalized log low frequency realized volatility.
P is defined as MIDAS lag year which indicates the maximum lag order in Eq. (5). The weighting function p (1 , 2 ) is the “Beta” lag structure.
p (1 , 2 )
2
f ( p / P, 1 , 2 ) f ( p / P, 1 , 2 ) p 1
P
(6)
f ( x, a , b )
where
x a 1 (1 x)b 1 (a b) (a) (b)
(7)
Equations (3) to (7) form the MF-SV model. Compared with the traditional SV model, we find that when the component t is some constant value, the MF-SV model degenerates into the basic SV model.
2.2 The Mixture Approximation Model This paper uses the Bayesian method to realize the parameter estimation. Referring to Nakajima and Omori (2009), we need to approximate the log( 12 ) distribution by a K-component mixture of Gaussian densities with a mixture approximation model. Let yi,t yi ,t log( t ) , and then we obtain the following equation where
f ( i ,t ) k 1 qk f N ( i ,t | mi ,t , )
K
2 i ,t
yi ,t hi ,t i,t
(8)
means using K-component mixture of Gaussian
densities to approximate the distribution of log( 12 ) , and qk is the weight of the normal distribution. The mixture approximation model can be written as the linear Gaussian state space model:
yi,t hi ,t i,t hi ,t 1 hi ,t i ,t 2 1 i ,t L mk k zi ,t | s k t zi2,t i ,t
1
(9)
(10)
2
where both zi ,t and zi ,t follow the standard normal distribution. MCMC method can be realized via the following blocks: (1) Initialize parameters , , m , , and state process {si }Ti 1 , {hi }Ti 1 ; (2) Given parameters m , , , state process {si }Ti 1 , {hi }Ti 1 , data sample yi,t , sample , with the M-H (Metropolis–Hastings) algorithm. The prior distribution of is the beta distribution and that of is the inverse gamma distribution; (3) Given , , state process {si }Ti 1 , data sample yi,t , construct the augmented Kalman filter and sample {hi }Ti 1 using the simulation smoother; (4) Given , , m , , , state process {hi }Ti 1 , data sample yi,t , sample {si }Ti 1 using a probability mass function; (5) Given , , state process {hi }Ti 1 , sample m , , using the M-H algorithm.
3. Monte Carlo experiment We conduct Monte Carlo experiments to evaluate the finite sample performance of the MCMC estimation procedure. The data are generated by the following MF-SV model: yi ,t 1.27 hi,t log( t ) i ,t hi ,t hi 1,t i ,t 3
P
log t =m+ p (1 , 2 ) X t -p p 1
where yi ,t log(ri 2,t ) and ri ,t is the log return on high frequency i during low frequency
t .We assume that high frequency i occurs 30 times in period t , i ,t follow the log( 12 ) distribution with one degree of freedom, zero mean, and variance 2 / 2 , i ,t N (0, 2 ) , the low frequency observation X t N (0,1) . p (1 , 2 ) is described by Equations (6) to (7). P=12 represents the maximum lag order. The true parameters of MF-SV model are assigned as1: 0.96 , 0.6 , m 8.5 ,
0.9 , 1 4.0 . Referring to Nakajima and Omori (2009), this paper uses a 10-component normal distribution to approximate the log( 12 ) distribution as shown in Table 1. We assume the prior distributions of parameters and . The prior distribution of is Beta (20,1.5) and that of follows
IG(0.5 0 ,0.5 1 ) ， 0 20 ， 1 0.01 0 . The reduced MCMC sampling for the posterior part is iterated with 5,000 draws. The initial 1,000 samples are discarded. Table 1 10-component mixture of Gaussian distributions w
Pr( w k )
mk
k2
1 2 3 4 5 6 7 8 9 10
0.0061
1.9268
0.1127
0.0478
1.3474
0.1779
0.1306
0.7350
0.2677
0.2067
0.0227
0.4061
0.2272
-0.8517
0.6270
0.1884
-1.9728
0.9858
0.1205
-3.4679
1.5747
0.0559
-5.5525
2.5450
0.0158
-8.6838
4.1659
0.0012
-14.6500
7.3334
We generate 200 sets of data for each sample size (high frequency) I 300 , I 600 and
I 900 . Our simulation results are reported in Tables 2. From the results reported by Tables 2, we find that the performance of estimation is better with the increasing of sample size. For example, the estimation of is more close to the true value when the sample size becomes large. When the sample size I 300 , the estimation of is downwardly biased and the corresponding true values even fall outside the 95% confidence interval. With the sample increasing, the estimation of
is more close to the true value. Table 2 Monte Carlo experiment results of the proposed estimation procedures
I 300
I 600
I 900
True value
0.96 1
Mean
S.D.
95%C.I.
Mean
S.D.
95%C.I.
Mean
S.D.
95%C.I.
0.9788
0.0094
[0.9574,0.9939]
0.9742
0.0093
[0.9532,0.9899]
0.9631
0.0102
[0.9415, 0.9816]
We set 2 = 1 and only estimate 1 .this set can produce the smooth weigh that the decay features. 4
m
0.6
0.3995
0.0715
[0.2764,0.5549]
0.5406
0.0614
[0.4290,0.6688]
0.5629
0.0571
[0.4580, 0.6824]
-8.5
-8.4981
0.0247
[-8.5406,-8.4525]
-8.4990
0.0240
[-8.5391,-8.4526]
-8.5221
0.0223
[-8.5658,-8.4835]
0.9
0.9077
0.0219
[0.8730, 0.9482]
0.9045
0.0250
[0.8586, 0.9460]
0.8965
0.0218
[0.8603,0.9196]
1
4.0
4.0044
0.0243
[3.9623, 4.0456]
3.9896
0.0276
[3.9463, 4.0392]
3.9960
0.0274
[3.9456 ,4.0425]
Note: Parameter estimates are obtained from posterior means of MCMC draws which are based on 5000 draws, discarding first 1000. S.D. is standard error of parameter estimation. 95%C.I. means the 95% confidence interval.
4. Empirical Study The data of daily returns, low-frequency realized volatility are collected for empirical study. Daily returns, monthly realized volatility are constructed using the daily stock price. We choose the close the S&P 500 as the raw data. The sample period is from January 3, 1994 to June 30, 2015. The sample size is 5216. The raw data are available from the database of WIND. The time series data are plotted in Fig. 1. 0.10
Return_SP500(High_Frequency)
0.05 0.00 -0.05
1994
1996
1998
2000
2002
2004
2006
2008
0.06
2010
2012
2014
2016
RV_SP500(Low_Frequency)
0.04
0.02
1995
2000
2005
2010
2015
Fig. 1. Daily Returns and Monthly Realized Volatility According to Nakajima and Omori (2009), we assume the prior distributions of and are
Beta (20,1.5) and IG(0.5 0 ,0.5 1 ) ， 0 5 ， 1 0.01 0 in U.S. stock market. The reduced MCMC sampling for the posterior part is iterated with 10,000 draws. The initial 2,000 samples are discarded. Table 3 B_SV
MCMC estimate results of MF-SV model
MF-SV(1)
0.9870 [0.9810 0.9923]
0.9582 [0.9487
5
0.9712]
MF-SV(2) 0.9569 [0.9425
0.9696]
-9.4148
--
--
0.1483
0.1945
0.1987
[0.1260 0.1735]
[0.1654 0.2307]
--
-10.0607
-10.1296
[-10.1201 -10.0205]
[-10.2246 -10.0234]
0.8833
0.8574
[-9.7505 -9.0869]
m
--
[0.8436 0.9311]
1
--
[0.7969
0.2321]
0.9125]
2.2735
5.6079
[2.2017 2.3802]
[5.5664 5.6484]
0.9501
0.9081
--
rRMSE
[0.1695
Note：B_SV represents the basic SV model. The parameter results are the mean of the posterior distribution. The value in parentheses represents the 95% confidence interval.
In MF-SV(P) model, P = 1 means 1 MIDAS lag year. In other words, the low frequency information of lag 12 months is introduced into the MF-SV(P) model, namely the MF-SV(1) model. Similarly, we can also construct the MF-SV(2) model. To compare the sample fit results, we calculate the rRMSE2. If the value of rRMSE is less than 1, the MF-SV model has a better fit performance than that of the benchmark model. Table 3 reports the estimate results. First, many estimated parameters have statistical significance. takes a significantly positive value suggesting that the low frequency realized volatility has a positive effect on the stable component. This result is similar to that of Engle et al. (2013). Second, based on the rRMSE, the in-sample fit of the MF-SV model is superior to that of the basic SV model. This implies that the SV-MIDAS model can make use of more information to identify a time-varying stable component that improves the in-sample fitting performance. As mentioned above, the MF-SV model can identify the time-varying stable component. To show this, we calculate
t2,i t and
t which represent the conditional variance and the
stable component respectively. As shown in Fig. 2, we find that there is co-movement tendency between the stable component and the conditional variance, but the stable component is less volatile. Second, the stable component presents the obvious time-varying feature that cannot be estimated by the traditional SV model.
2
2 2 RMSE (T 1 t 1 (ˆt2 t2 )2 )1/2 , where ˆ t is estimated conditional variance, t is the proxy of T
volatility. We compute the RMSE of both the MF-SV and the basic SV model and obtain the rRMSE by use the RMSE of MF-SV model divided by that of the benchmark model. 6
Stable_Component
Conditonal_Variance
0.04
0.02
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
2016
MF-SV(1) Stable_Component
Conditonal_Variance
0.04
0.02
1996
1998
2000
2002
2004
2006
2008
2010
2012
2014
2016
MF-SV(2)
Fig. 2.
The stable component of the MF-SV models
5. Conclusion In this paper, we extend the basic SV model to the mixed-frequency SV model, referred to as the MF-SV model. The MCMC method is employed to realize the parameter estimation of MF-SV model by approximating it to a mixture approximation model. This paper also discusses the small sample properties with Monte Carlo experiment. Based on the empirical study, we find that the MF-SV model can describe the time-varying stable component. Second, the proposed MF-SV model outperforms the basic SV model in the in-sample performance.
Acknowledgments The Research Foundation for Young Scholars of the Ministry of Education of China (no.16YJC790084) provided research support. For helpful comments we are grateful to the Editor and the referee. We, however, bear full responsibility for all remaining flaws.
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