Bulletin of Monetary Economics and Banking, Vol. 22, No. 1 (2019), pp. 1 - 28
MODELING HIGH DIMENSIONAL ASSET PRICING
RETURNS USING A DYNAMIC SKEWED COPULA MODEL
Yuting Gong1, Jufang Liang2, Jie Zhu3
1Department of Economics and Finance,
Shanghai University, China. Email: yutinggong1985@163.com
2College of Finance and Statistics, Hunan University, China. Email: ljufang@hnu.edu.cn 3 Department of Economics and Finance,
Shanghai University, China. Email: zhu_jie@t.shu.edu.cn
ABSTRACT
We propose a dynamic skewed copula to model multivariate dependence in asset returns in a flexible yet parsimonious way. We then apply the model to 50 exchange- traded funds. The new copula is shown to have better
Keywords: Skewed copula; Dynamic model; High dimensions; Multivariate dependence.
JEL Classification: C22.
Article history: |
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Received |
: January 12, 2019 |
Revised |
: April 8, 2019 |
Accepted |
: April 20, 2019 |
Available online : April 30, 2019 https://doi.org/10.21098/bemp.v22i1.1044
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Bulletin of Monetary Economics and Banking, Volume 22, Number 1, 2019 |
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I. INTRODUCTION
Modeling the dependence of a large number of financial asset returns is critical in financial applications such as asset allocation and risk management. In practice, the dependence of asset returns is shown to be
However, early applications of copulas are almost all bivariate in nature. That is, the copula is used to describe the dependence structure between two assets. It remains a challenge for academia and practitioners to model a highly flexible dependence structure among multivariate assets. When the number of assets is relatively large, models developed for
Previous work on extending bivariate copula models to higher dimensions includes Genest, Gendron, and
We make several contributions to the literature. First, the dynamic skewed copula can describe changing dependence patterns, since the skewness parameters follow an autoregressive procedure. By contrast, the skewness parameters in Christ
4High dimensional copulas in this paper refer to multivariate copula models describing the dependence of N time series when N is greater than 50. In Oh and Patton (2018), a low dimensional copula is defined as N no more than 5 and a high dimensional copula is defined as N between 50 and 250.
Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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offersen et al. (2012) and
We apply our new model to investigate the dependence pattern of 50 U.S.
This paper proceeds as follows. Section 2 introduces the dynamic skewed copula model and explains how to estimate it. Section 3 provides summary statistics of ETF returns. Section 4 analyzes the
II. THE DYNAMIC SKEWED COPULA MODEL
The dynamic skewed copula model discussed in this paper originates from the skewed t copula developed by Christoffersen et al. (2012). Let (u1t ,· · ·, uNt) denote the cumulative distribution functions (CDF) of N variables, t = 1, · · ·, T . Their dependence is described by a copula function C(u1t ,· · ·, uNt). The skewed t copula in Christoffersen et al. (2012), denoted by Cskt is given by
where Rt is the correlation matrix, |
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(1) |
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is the degrees of freedom, is the skewness |
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parameter, which is a scalar. |
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(·) is the CDF of |
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hyperbolic skewed t distribution with zero mean as in Demarta and McNeil
(2005). For i = 1, · · ·, N, (·) is the inverse of univariate skewed t CDF on the ith subordinate with zero mean and unit variance. Please refer to Christoffersen et al.
(2012) for details of (·).
The evolution of Rt is similar to the dynamic mechanism in the Engle (2002) dynamic conditional correlation (DCC) model.
(2)
,
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where is the unconditional covariance matrix of zt = (z1t ,. . ., zNt)‘1. For i = 1,· · ·, N,
zit = (uit). α1 > 0, α2 > 0 and α1 + α2 < 1, the conditional correlation Rt is mean- reverting. This model Cskt has four parameters θskt = (α1, α2,
, )’.
Note that in Christoffersen et al. (2012), the skewness parameter is a scalar rather than a vector, indicating that all variables have identical asymmetry parameters. There are two problems with this specification. First, it is unable to describe
i≠j. Second, it is unable to capture the time variation in dependence due to the constant skewness parameter. In reality, the multivariate dependence of financial assets may be
To overcome these problems, we modify the copula in Christoffersen et al. (2012) via the following two steps. First, we assume each marginal CDF uit corresponds to a different skewness parameter. The skewness parameter now becomes an
(3)
The skewness parameter γ = (γ1 ... γN)’ is an are the same as those for Cskt. The parameters in Cmskt are θskt = (α1, α2, v, γ ’)’, so N + 3 parameters are to be estimated. We are not the first to propose Cmskt, as
The second step is to specify a dynamic evolution process for the skewness vector. The dynamic mechanism is similar to the Engle (2002) DCC model. The modified model, called dynamic skewed copula Cdskt, is given by
, |
(4) |
(5)
where β1 is the coefficient of , the coefficient in front of
is .
γt are defined based on the copula shocks zit = (uit), rather than the return shocks (standardized residuals
it from marginal distributions).
5In Christoffersen et al. (2012), Qt is assumed to be
Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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The dynamic skewed copula Cdskt has five parameters θdskt = (α1, α2, v, β1, β2)’. By introducing dynamics into the skewness vector, Cdskt has great flexibility in capturing multivariate dependence with only a few parameters. The model has two desirable features. First, it allows each pair of assets to display dependence patterns that are distinct from other pairs. Second, it is able to describe changing dependence patterns over the time.
For the estimation procedure, we mainly follow Christoffersen et al. (2012). The only difference is that the copula’s skewness parameter is an
First, we estimate each univariate marginal model and calculate the marginal CDF uit for i = 1,· · ·, N, t = 1, · · ·, T . Second, we estimate copula models by maximum composite likelihood estimation (MCLE). MCLE is employed because it yields consistent estimates for the true parameters in large scale problems, while the ordinary maximum likelihood estimation (MLE) may estimate the parameters driving the dynamic process with bias, as discussed in Engle, Shephard, and Shepphard (2008), Christoffersen et al. (2012), and Oh and Patton (2015).
For k = skt, mskt, dskt, the composite
(6)
where Ck(uit, ujt; θk) denotes the bivariate copula density of pair i and j for i, j=1,
· · ·, N.
III. DATA DESCRIPTION
Our empirical analysis employs the dynamic skewed copula model to study the dependence of 50 US ETF returns (N = 50). The data set includes daily adjusted prices of four stock ETFs (STK) and five other types of ETF: bond (AGG), foreign exchange (Euro/Dollar, FXE), gold (GSG), oil (USO), and real estate (RWR).6 Stocks are selected from nine sectors, and in each sector only the top five firms with the highest market values are considered. All prices are in US dollars and are from Bloomberg. The ith (i= 1,· · ·, N) daily return is calculated as rit = 100 x (logPit -
The sample period is from July 24, 2006 to June 28, 2013, for a total of 1746 daily observations. We divide the sample into two subperiods, such that the period from July 24, 2006 to June 30, 2011 is used for the
6Daily prices are adjusted for dividends. Calculation of dividends is as follows. For example, Materials Select Sector SPDR Fund (XLB) distributes $0.239 dividend per share on March 14, 2019. The closing price of XLB is $55.57, but the adjusted closing price is $55.239. We then use adjusted prices to account for the impacts of dividend distribution on stock prices.
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Figure 1. Price Series of the 50 ETFs
Panel (a) plots 45 stock ETFs prices over the sample period of July 24, 2006 - June 28, 2013. Panel (b) plots the ETF prices of bond (AGG), foreign exchange ETF (FXE), gold (GSG), oil (USO) and real estate (RWR) over the same sample period.
(a) 45 stock ETFs
250 |
200 |
150 |
100 |
50 |
0 |
2007 |
2008 |
2009 |
2010 |
2011 |
2012 |
2013 |
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(b) 5 ETFs: bond, foreign exchange, gold, oil and real estate |
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160 |
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140 |
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120 |
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AGG |
FXE |
GSG |
USO |
RWR |
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0 |
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2007 |
2008 |
2009 |
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2010 |
2011 |
2012 |
2013 |
Figure 1 plots the price series of the 50 ETFs. The movements of all stock ETFs and the real estate ETF are quite similar, while the behavior of the bond ETF generally differs from the others. These ETFs were greatly impacted by the 2008 subprime crisis, as nearly all of them (except the bond ETF) suffered from a
Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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Table 1.
Descriptive Statistics and Tests for
Panel A summaries the descriptive statistics of 14 representative ETF returns over the sample period of July 24, 2006 - June 30, 2011. Std is the standard deviation. Kurtosis is the excess kurtosis.
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Panel A: |
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Mean |
Std. |
Skewness |
Kurtosis |
ARCH (5) |
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Stocks |
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XLB(DD) |
0.0118 |
2.8076 |
6.5718 |
2.2558*** |
44.4638*** |
85.3496*** |
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XLE(XOM) |
0.0254 |
2.3042 |
6.9109 |
2.4959*** |
85.3132*** |
43.2086*** |
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XLF(WFC) |
3.9549 |
0.7616 |
12.2928 |
7.9151*** |
92.0241*** |
63.6810*** |
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XLI(GE) |
0.004 |
2.1224 |
0.1956 |
4.1057 |
0.8761*** |
46.6722*** |
56.1684*** |
XLK(MSFT) |
0.0166 |
1.9895 |
0.1033 |
6.135 |
1.9422*** |
52.9964*** |
45.6344*** |
XLP(WMT) |
0.0464 |
1.36 |
0.7098 |
12.8008 |
8.5571*** |
63.2657 |
43.5734*** |
XLU(DUK) |
0.0351 |
1.2326 |
0.7873 |
9.3449 |
4.6317*** |
126.4957*** |
38.0077** |
XLV(JNJ) |
0.0203 |
1.3694 |
6.0569 |
1.8973*** |
55.7293*** |
53.7298*** |
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XLY(DIS) |
0.0176 |
2.1042 |
0.4836 |
3.9309 |
0.8441*** |
42.3541** |
82.2823 |
Bond (AGG) |
0.0235 |
0.4209 |
73.4637 |
280.6589*** |
140.5175*** |
35.9942* |
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Exchange (FXE) |
0.0135 |
0.7072 |
0.033 |
2.177 |
0.2438*** |
52.3304*** |
60.0650*** |
Gold (GSG) |
1.8638 |
2.171 |
0.2746*** |
29.937 |
60.0417*** |
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Oil (USO) |
2.4324 |
2.0082 |
0.2165*** |
34.5330* |
70.1851*** |
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Real estate (RWR) |
0.0027 |
3.0878 |
7.991 |
3.30278*** |
131.3786*** |
85.6806*** |
Panel B: Tests for
45 stocks |
45 |
5 others |
50 ETFs |
p = 1 |
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0.9535(= 944/990) |
0.8800(= 220/225) |
1.0000(= 10/10) |
0.9584(= 1174/1225) |
p = 5 |
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1.0000(= 990/990) |
1.0000(= 225/225) |
1.0000(= 10/10) |
1.0000(= 1225/1225) |
Table 1 provides preliminary analyses of the
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IV.
We first estimate the marginal distribution of each ETF return series. Then, we investigate the contemporaneous dependence of these returns based on the dynamic skewed copula. Finally, we select four representative asset pairs to further illustrate the flexibility of Cdskt in modeling multivariate dependence.7
Table 2.
Marginal Model Estimates Over The
This table reports the marginal model estimates of 14 representative ETF returns over the sample period of
Stocks |
μ |
i |
ρ |
i |
ki0 |
k |
i1 |
k |
i2 |
k |
i3 |
δi |
f |
i1 |
f |
i2 |
logL |
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XLB(DD) |
0.0219*** |
0.0706*** |
0.9294*** |
0.6944*** |
0.5099*** |
2.5803*** |
3.6634*** |
4.5237 |
0.0211 |
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(0.2768) |
(0.0431) |
(0.0080) |
(0.0094) |
(0.0117) |
(0.1086) |
(0.0115) |
(0.3005) |
(0.4754) |
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XLE(XOM) |
0.0935* |
0.1229*** |
0.1145*** |
0.8855*** |
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0.5170*** |
3.3588*** |
8.2809*** |
4.9015 |
0.0241 |
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(0.3381) |
(0.0431) |
(0.0080) |
(0.0094) |
(0.0117) |
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(0.0115) |
(0.3005) |
(0.4754) |
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XLF(WFC) |
0.0741 |
0.0067* |
0.1128*** |
0.8872*** |
0.8015*** |
0.4931*** |
2.9330*** |
2.5219*** |
4.1069 |
0.0203 |
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(0.0495) |
(0.0504) |
(0.0037) |
(0.0116) |
(0.0335) |
(0.1829) |
(0.0115) |
(0.3975) |
(0.3368) |
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XLI(GE) |
0.0009 |
0.0046 |
0.0802*** |
0.0843*** |
0.9157*** |
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0.5046*** |
3.4614*** |
4.2664*** |
4.9888 |
0.0205 |
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(0.0082) |
(0.0301) |
(0.0116) |
(0.0136) |
(0.0110) |
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(0.0109) |
(0.4260) |
(0.6896) |
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XLK(MSFT) |
0.0869** |
0.1188*** |
0.8812*** |
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0.5127*** |
2.9879*** |
4.5885*** |
5.1598 |
0.0228 |
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(0.2695) |
(0.0348) |
(0.0344) |
(0.0390) |
(0.0391) |
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(0.0128) |
(0.6102) |
(2.2651) |
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XLP(WMT) |
0.0336*** |
0.1161*** |
0.8839*** |
0.0017* |
0.4976*** |
3.5831*** |
3.4657*** |
5.9806 |
0.0187 |
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(0.3937) |
(0.0631) |
(0.0058) |
(0.0131) |
(0.0159) |
(0.0009) |
(0.0111) |
(0.4955) |
(0.4444) |
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XLU(DUK) |
0.4004 |
0.0513*** |
0.1738*** |
0.8257*** |
0.0054*** |
0.4975*** |
4.1372*** |
4.1854*** |
6.1542 |
0.019 |
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(0.2860) |
(0.0551) |
(0.0080) |
(0.0189) |
(0.0223) |
(0.0009) |
(0.0112) |
(0.6493) |
(0.6022) |
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XLV(JNJ) |
0.0421 |
0.0911*** |
0.9086*** |
0.0088** |
0.5007*** |
4.6611*** |
4.5403*** |
5.8591 |
0.0202 |
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(0.2333) |
(0.0363) |
(0.0079) |
(0.0124) |
(0.0146) |
(0.0040) |
(0.0104) |
(0.8491) |
(0.7755) |
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XLY(DIS) |
0.1571 |
0.0088 |
0.1380*** |
0.1673*** |
0.8327*** |
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0.4893*** |
4.4986*** |
3.0871*** |
5.0272 |
0.0207 |
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(0.4224) |
(0.0153) |
(0.0183) |
(0.0150) |
(0.0169) |
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(0.0117) |
(0.7652) |
(0.3943) |
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Bond (AGG) |
0.0005*** |
0.0673*** |
0.9324*** |
0.7365*** |
0.5145*** |
3.6681*** |
12.1769*** |
8.4846 |
0.0239 |
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(0.1354) |
(0.1100) |
(0.0001) |
(0.0135) |
(0.0184) |
(0.1756) |
(0.0101) |
(0.6527) |
(4.6478) |
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Exchange |
0.0482 |
0.0003** |
0.0259*** |
0.9739*** |
0.8808*** |
0.5009*** |
6.2868*** |
6.5946*** |
7.1463 |
0.0207 |
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(FXE) |
(0.1504) |
(0.0356) |
(0.0002) |
(0.0088) |
(0.0117) |
(0.1890) |
(0.0103) |
(1.4981) |
(1.6376) |
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Gold (GSG) |
0.3073 |
0.1148*** |
0.1064*** |
0.8932*** |
0.0039** |
0.5124*** |
4.5210*** |
11.9471*** |
5.2025 |
0.0232 |
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(1.3851) |
(0.0342) |
(0.0151) |
(0.0123) |
(0.0123) |
(0.0018) |
(0.0103) |
(0.7678) |
(4.1833) |
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Oil (USO) |
0.5186 |
0.1832*** |
0.0985*** |
0.9015*** |
0.0023*** |
0.5081*** |
5.8351*** |
10.5864*** |
4.6771 |
0.0217 |
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(0.6897) |
(0.0359) |
(0.0256) |
(0.0106) |
(0.0118) |
(0.0005) |
(0.0103) |
(1.2147) |
(3.3274) |
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Real estate |
0.0154*** |
0.0947*** |
0.9053*** |
0.7819*** |
0.5078*** |
3.2109*** |
4.1998*** |
4.4879 |
0.0214 |
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(RWR) |
(0.0152) |
(0.0509) |
(0.0041) |
(0.0079) |
(0.0160) |
(0.0952) |
(0.0111) |
(0.4349) |
(0.7066) |
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7Details of these 50 ETFs are available upon request.
Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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In Table 2, we estimate the marginal distributions of ETF returns via the autoregressive and nonlinear GARCH models
The conditional mean and conditional volatility models are as follows:
(7)
(8)
For AST distribution, δi is the skewness parameter, ƒi1 and ƒi2 represent the degrees of freedom on, respectively, the left side and right side of stochastic error .
Two conclusions can be drawn from Table 2. First, the results of the Kolmogorov– Smirnov test indicate that the marginal models are correctly specified. This ensures the consistency of copula estimates in the following subsection. Second, the parameter estimates in marginal models confirm the
We next transform the standardized residuals ( ) into (
) and estimate the copula models. The results of five dynamic copulas are reported in our analysis: CGaussian, Ct, Cskt, Cmskt, and Cdskt. The last three copulas are given in equations (1), (3), and (4). To make the analysis more complete, we also provide the results of Gaussian and t copulas, as they may be regarded as special cases of skewed t copulas. Note that Cmskt here differs from the copula in
The
where 1og ƒi is the
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Table 3.
Copula Model Estimates Over the
This table reports the estimates of copula models over the sample period of
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Gaussian |
t |
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skt |
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mskt |
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dskt |
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α1 |
0.0070*** |
0.0040*** |
0.0046*** |
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0.0169*** |
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0.0699*** |
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(0.0009) |
(0.0005) |
(0.0007) |
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(0.0014) |
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(0.0248) |
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α2 |
0.9504*** |
0.9480*** |
0.9367*** |
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0.9614*** |
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0.6781** |
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(0.0080) |
(0.0106) |
(0.0162) |
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(0.0041) |
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(0.2831) |
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v |
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12.2706*** |
21.5303*** |
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16.2605*** |
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11.1504*** |
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(0.5155) |
(1.3128) |
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(1.4157) |
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(2.3429) |
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0.4259*** |
γSTK |
0.3588*** |
β1 |
0.7326*** |
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(0.0137) |
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(0.0256) |
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(0.0309) |
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γAGG |
0.1934*** |
β2 |
0.7822*** |
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(0.0616) |
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(0.0021) |
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γFXE |
0.0933* |
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(0.0544) |
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γGSG |
0.2142*** |
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(0.0605) |
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γUSO |
0.1924*** |
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(0.0617) |
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γRWR |
0.3574*** |
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(0.0413) |
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logL |
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(×10²) |
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AIC |
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(×10²) |
3.5903 |
3.5651 |
3.5614 |
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3.5459 |
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3.5271 |
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SIC |
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(×10²) |
3.5908 |
3.5655 |
3.5619 |
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3.5463 |
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3.5276 |
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Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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some periods, these ETF returns are more correlated when they increase together than when they decrease together, while in other periods, such as crisis periods, they are more correlated when they decrease together than when they increase together. The changing dependence patterns are discussed in more details below.
Estimates of Cdskt show that skewness vector γt is not only autoregressive (β2= 0.78), but also negatively correlated with shocks from dependence system at t
Comparing Ct and Cdskt, we observe that the autoregressive coefficients in conditional correlation α2 and degree of freedom v are smaller if the dynamics of skewness parameter are considered. The decrease in αX implies that the dynamic dependence structure is captured not only by the evolution of conditional correlation, but also by the evolution of the skewness parameter. The decrease in v indicates stronger tail dependence in Cdskt than in Cskt. Hence, Cdskt with a dynamic skewness vector is more likely to capture the dependence structure under extreme market conditions.
Figure 2. Skewness Parameter Estimates Series of 50 ETFs Based on Cdskt
The figures plot the skewness vector γt in equation (5) based on the dynamic skewed copula model Cdskt. Panel (a) is for the 45 stock ETFs, and panel (b) to (f) are for the ETFs of bond (AGG), foreign exchange ETF (FXE), gold (GSG), oil (USO) and real estate (RWR). The
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Figure 2. Skewness Parameter Estimates Series of 50 ETFs Based on Cdskt (contd.)
The figures plot the skewness vector γt in equation (5) based on the dynamic skewed copula model Cdskt. Panel (a) is for the 45 stock ETFs, and panel (b) to (f) are for the ETFs of bond (AGG), foreign exchange ETF (FXE), gold (GSG), oil (USO) and real estate (RWR). The
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Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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Figure 2. Skewness Parameter Estimates Series of 50 ETFs Based on Cdskt (contd.)
The figures plot the skewness vector γt in equation (5) based on the dynamic skewed copula model Cdskt. Panel (a) is for the 45 stock ETFs, and panel (b) to (f) are for the ETFs of bond (AGG), foreign exchange ETF (FXE), gold (GSG), oil (USO) and real estate (RWR). The
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Figure 2 plots the skewness parameter series of 50 ETFs based on Cdskt. In Figure 2 (a), the 45 stock ETFs have similar skewness parameters, which are positive most of the time and become negative during the 2008 subprime crisis and the 2010 European debt crisis. In Figure 2 (b), the bond ETF generally has negative skewness parameters, whose behavior differs from other ETFs. In Figure 2 (c), (d), and (f), the skewness parameters of foreign exchange, gold, and real estate ETFs are positive in most cases, but decrease dramatically to become negative during the 2008 subprime crisis. In Figure 2 (e), the skewness parameters of oil ETFs exhibit a
Since it would be tedious to analyze all pairwise dependences among the 50 ETFs, we select four representative groups for further discussion: (1) 45 stocks; (2) stocks and bonds; (3) stocks and oil; and (4) oil and gold.
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Figure 3. Average Correlation for 45 Stocks, Bond, Oil, Exchange Rate, and Gold
The figures plot the correlations Rt based on the skewed t copula model Cskt and the dynamic skewed copula model Cdskt. Panel (a) is the average bivariate correlations across 990 pairs of stocks, panel (b) is the average correlations between each stock and bond across 45 pairs, panel (c) is the average correlations between each stock and oil across 45 pairs, and panel (d) is the correlations of foreign exchange and gold. The
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Figure 3. Average Correlation for 45 Stocks, Bond, Oil, Exchange Rate, and Gold
(contd.)
The figures plot the correlations Rt based on the skewed t copula model Cskt and the dynamic skewed copula model Cdskt. Panel (a) is the average bivariate correlations across 990 pairs of stocks, panel (b) is the average correlations between each stock and bond across 45 pairs, panel (c) is the average correlations between each stock and oil across 45 pairs, and panel (d) is the correlations of foreign exchange and gold. The
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Figure 3 and Figure 4 plot the average correlations and exceeding correlations (at 10% and 90% quantiles) within each group. Both the results of the Christoffersen et al. (2012) Cskt and our Cdskt are provided to better understand the dynamics in dependence patterns. The
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*STK,AGG,t (0.1) measures the correlation when stock return decreases below its 10% quantile and bond return increases over its 90% quantile; and(0.9) measures the correlation when stock return increases over its 90% quantile and bond return decreases below its 10% quantile. For the other three groups, the exceeding correlations at the 10% quantiles
·, ·, t (0.1) (or the 90% quantile
·, ·, t (0.9)) calculate the correlation of both crashing below their 10% quantiles (and booming over their 90% quantiles).
Figure 3 (a) shows average bivariate correlations across 990 ( ) pairs of stocks. The correlations within the stock sector described by Cskt and Cdskt are similar. These stock ETFs are highly correlated, and the correlations are driven up further during the 2008 subprime crisis and the 2011 European debt crisis. This is evidence of financial contagion in stock sectors, as documented in Caporale, Cipollini, and Spagnolo (2005), Rodriguez (2007), and Kallberg and Pasquariello (2008), among others.
Figure 4. Excess Correlation for 45 Stocks, Bond, Oil, Exchange Rate, and Gold
The figures plot the exceeding correlations (at 10% and 90% quantiles) based on the skewed t copula model Cskt and the dynamic skewed copula model Cdskt. Panel (a) is the average bivariate exceeding correlations across 990 pairs of stocks, panel (b) is the average exceeding correlations between each stock and bond across 45 pairs, panel (c) is the average exceeding correlations between each stock and oil across 45 pairs, and panel (d) is the exceeding correlations of foreign exchange and gold. The
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Figure 4 (a) reveals the various dependence patterns captured by Cskt and Cdskt via average bivariate exceeding correlations across 990 pairs of stock. From Cskt with = 0.43 > 0, STK (0.9) is always greater than STK (0.1), implying these stocks are more likely to boom together than crash together throughout the sample
interval. However, from Cdskt, |
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This means that the dependence patterns within stock sectors change over time, which coincides with the results of Okimoto (2008), Guegan and Zhang (2010), and Elkamhi and Stefanova (2015). Hence, the dynamic specification of t enables us to distinguish the different dependence patterns in times of crisis and in normal periods.
Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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Figure 4. Excess Correlation for 45 Stocks, Bond, Oil, Exchange Rate, and Gold
(contd.)
The figures plot the exceeding correlations (at 10% and 90% quantiles) based on the skewed t copula model Cskt and the dynamic skewed copula model Cdskt. Panel (a) is the average bivariate exceeding correlations across 990 pairs of stocks, panel (b) is the average exceeding correlations between each stock and bond across 45 pairs, panel (c) is the average exceeding correlations between each stock and oil across 45 pairs, and panel (d) is the exceeding correlations of foreign exchange and gold. The
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Figure 3 (b) provides the average correlations between each stock ETF and bonds across 45 pairs. The negative
Figure 4 (b) plots the exceeding correlations to demonstrate the difference in dependence patterns described by Cskt and Cdskt. Cskt captures only one dependence pattern that | *STK,AGG (0.9)| > |
*STK,AGG (0.1) |, indicating that the dependence of rising stocks and declining bonds is stronger than the dependence of declining stocks and rising bonds. Cdskt, in contrast, depicts changing dependence patterns. In tranquil periods,
*STK,AGG (0.9) is close to
*STK,AGG (0.1) as γSKT,t > 0 and γAGG,t < 0 (see Figure 2 (a) and (b)). But in times of crisis, |
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*STK,AGG (0.1) |as both γSKT,t and γAGG,t become negative, indicating that the tendency for stock prices to decrease and for bond prices to increase is much stronger than the reverse tendency. The plunge in
*STK,AGG (0.1) is evidence of the
In Figure 3 (c), we plot the average
In Figure 3 (c), we easily differentiate between Cskt and Cdskt by their exceeding correlations. From Cskt, STK,USO (0.9) always exceeds
STK,USO (0.1) as
> 0. However, from Cdskt, due to the dynamic specification of the skewness vector, the stock– oil dependence exhibits a
STK,USO (0.9) >
STK,USO (0.1), as γSKT,t and γUSO,t are positive (see Figure 2 (a) and (e)); after the crisis,
STK,USO (0.9) <
STK,USO (0.1), as γUSO,t and γSKT,t drop below zero during the two crises. This means that stocks and oil are more likely to crash together than boom together after a crisis, again confirming the absence of diversification opportunities. Although the
In Figure 3 (d), foreign exchange (EURO/US dollar) and gold (in dollars) become more positively dependent during the 2008 subprime crisis. For investors, gold no longer behaves as a safe haven during the crisis period, as stated in Sari, Hammoudeh and Soytas (2010), Joy (2011), and Pukthuanthong and Roll (2011).
In Figure 4 (d), Cskt and Cdskt can be easily distinguished by their exceeding correlations. Based on Cskt with > 0, we may conclude that foreign exchange and gold stick to one dependence pattern, that
FXE,GSG (0.9) >
FXE,GSG (0.1). The two ETFs are more dependent when booming simultaneously than crashing simultaneously. But this is not the case for Cdskt with dynamic skewness vector. During the 2008 subprime crisis,
FXE,GSG (0.9) is shown to be lower than
FXE,GSG(0.1) as both γFXE,t and γGSG,t drop sharply below zero (see Figure 2 (c) and (d)). This finding implies a higher likelihood of a depreciating Euro and a falling gold price
Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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following the subprime crisis. Investors managing portfolio risk should avoid holding both these assets for about six months. Again, this investigation of foreign
In summary, this section applies dynamic skewed t copula Cdskt to study the dependence of 50 ETF returns. The Cdskt model with dynamic skewness vector enables us to model multivariate dependence more flexibly and parsimoniously than existing copulas. We conclude that the 50 ETFs exhibit changing dependence patterns rather than only one dependence pattern throughout the sample interval.
V.
This section investigates
A rolling sample method is utilized in the
To assess the predictive accuracy of our Cdskt model, we compare the predicted
Table 4.
Predictive Ability Comparison of Copulas over the
This table reports the average of predictive
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Table 4 reports the
Further, the Hansen (2005) SPA test gives us statistical justification for the outperformance of Cdskt. The null hypothesis is that the benchmark model is not inferior to any of the alternative models. Here, our dynamic skewed t copula Cdskt is set as the benchmark, while the other four copulas are regarded as alternatives. The SPA test statistic is 3.34 with
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1% significance level, it can be inferred that Cdskt performs at least as well as the other copulas considered.
To explore the economic value of modeling dynamic multivariate dependence, we then consider the optimization problem of an investor allocating wealth among the 50 ETFs. Our hypothetical investor is characterized by a constant relative risk aversion utility, as in Patton (2004), and Wu and Lin (2014). At each period t, the investor solves the following optimization problem based on the
where are the weights on the N (=50)
ETFs, and they can be negative without short selling constraints. rp,t+1 is portfolio return, rp,t+1= ’t+1rt+1, and rt+1 = (r1,t+1, · · ·,rN,t+1)’ is the return vector of N ETFs at t+1. η is the degree of relative risk aversion and takes three levels: η = 1, 5, 10. The rolling window procedure is described at the beginning of this section.
Table 5 shows portfolio performance based on five copula models. We calculate each portfolio’s annualized return (Mean), standard deviation (SD), terminal wealth value, 5%
Table 5.
Portfolio Performance Comparison of Copula over the
This table summarizes the
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8.5426 |
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8.7943 |
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1.6425 |
0.5349 |
2.5969 |
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3.4825 |
6.3294 |
4.9361 |
3.7858 |
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102.1760 |
103.3121 |
97.2518 |
101.0727 |
105.2612 |
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4.9818 |
7.5480 |
6.0646 |
3.1901 |
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6.1846 |
6.7866 |
13.5201 |
9.2427 |
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0.9195 |
6.9027 |
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In Table 5, among the five strategies, the strategy using Cdskt performs best, followed by the strategies using CGaussian, Ct and Cmskt, while Cskt performs worst. This ranking is based on the values of performance fee and is unaffected by the levels of investor risk aversion.
An investor who disregards changing dependence patterns incurs losses when modeling high dimensional financial returns. For example, the performance fee from Cmskt to Cdskt for an investor with η = 10 is 3.26 cents per dollar. The gains from considering changing dependence patterns are also revealed in Okimoto (2008) and Elkamhi and Stefanova (2015). It is inferred that
Further, portfolio performance is sensitive to the correctness of dependence characterization. If the dependence structure is predicted based on an incorrect copula, such as Cskt, the corresponding portfolio will perform even worse than portfolios that simply use
Why do strategies using Cskt and Cdskt perform so differently? Figure 5 investigates the portfolio weights resulting from Cskt and Cdskt. It is evident that the two types of investor have different opinions mainly about their weights on stocks, bonds, and foreign exchange.
Figure 5. Comparing Portfolio Weights for Cskt and Cdskt
The figures plot the portfolio weights on 50 ETFs based on the skewed t copula model Cskt and the dynamic skewed copula model Cdskt over the sample period of July 1, 2011 - June 28, 2013. Panel (a) is the average weights on 45 stock ETFs, panel
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2011/09 |
2011/12 |
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2012/12 |
2013/03 |
2013/06 |
Figure 5 (a) plots the average weights on 45 stock ETFs and illustrates that the investor following Cdskt buys more stocks than the investor following Cskt. This is because Cdskt predicts higher 90% exceeding correlations than Cskt (see Figure 5 (a)). In other words, the
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Bulletin of Monetary Economics and Banking, Volume 22, Number 1, 2019 |
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Figure 5. Comparing Portfolio Weights for Cskt and Cdskt (contd.)
The figures plot the portfolio weights on 50 ETFs based on the skewed t copula model Cskt and the dynamic skewed copula model Cdskt over the sample period of July 1, 2011 - June 28, 2013. Panel (a) is the average weights on 45 stock ETFs, panel
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(b) Bond |
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dskt 10% |
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dskt 90% |
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2011/09 |
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Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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Figure 5. Comparing Portfolio Weights for Cskt and Cdskt (contd.)
The figures plot the portfolio weights on 50 ETFs based on the skewed t copula model Cskt and the dynamic skewed copula model Cdskt over the sample period of July 1, 2011 - June 28, 2013. Panel (a) is the average weights on 45 stock ETFs, panel
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2011/09 |
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Figure 5 (b) shows that the investor using Cdskt shorts more bond than the investor using Cskt. The reason for this is similar: the 90% exceeding correlations from Cdskt are higher (in absolute value) in those from Cskt (see Figure 4 (b)). From the perspective of the
Figure 5 (c) indicates the investor with Cskt mostly longs foreign exchange, while the investor with Cdskt shorts it before June 2012 and longs it afterwards. This difference in positions is caused by their forecasts of skewness parameters. Cskt predicts a positive skewness scalar for both stock and foreign exchange (see Figure 2 (a) and (c)), and the investor buys both of them believing the two assets will increase together. In contrast, Cdskt also predicts positive Cskt, but the investor believes γFXE is negative during the period July 2011 to June 2012 and becomes
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Bulletin of Monetary Economics and Banking, Volume 22, Number 1, 2019 |
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positive afterwards. In this investor’s opinion, stocks and foreign exchange before June 2012 are not as closely correlated as they are after June 2012, so the investor sells foreign exchange before June 2012 and then buys it again.
This result is obtained based on a longer
Table 6.
Copula Model Estimates Over the
This table reports the estimates of copula models over the sample period of 2006/06/24 – 2009/12/31. Standard errors are in the brackets below the parameters. The ***, ** and * denote statistical significance levels at 1%, 5%, and 10%, respectively. logL denotes the
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Gaussian |
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t |
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skt |
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mskt |
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dskt |
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α1 |
0.0079*** |
α1 |
0.0052*** |
α1 |
0.0040* |
α1 |
0.0142*** |
α1 |
0.0127*** |
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(0.0013) |
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(0.0009) |
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(0.0022) |
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(0.0035) |
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(0.0032) |
α2 |
0.9254*** |
α2 |
0.9073*** |
α2 |
0.8007** |
α2 |
0.8621** |
α2 |
0.8304*** |
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(0.0161) |
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(0.0244) |
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(0.3300) |
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(0.5809) |
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(0.0565) |
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v |
13.2215*** |
v |
13.0034*** |
v |
14.3011*** |
v |
15.0218*** |
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(0.7221) |
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(0.1193) |
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(0.8183) |
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(0.4765) |
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0.0064*** |
γSTK |
0.4872*** |
β1 |
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(0.0230) |
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(0.2992) |
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γAGG |
β2 |
0.7561** |
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(0.1985) |
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(0.3699) |
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γFXE |
0.0472*** |
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(0.2455) |
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γGSG |
0.2473*** |
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(0.2473) |
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γUSO |
0.2848** |
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(0.1293) |
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γRWR |
0.4835*** |
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(0.0428) |
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logL (x 102) |
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AIC (x 102) |
3.7226 |
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3.7233 |
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3.5222 |
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3.4510 |
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3.3107 |
BIC (x 102) |
3.7414 |
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3.7516 |
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3.5600 |
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3.5359 |
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3.3579 |
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8To save space, we skip the
Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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Overall, the results are robust when the sample is split 50:50. In Table 6, the dynamic skewed t copula Cdskt provides the best
Table 7.
Predictive Ability Comparison of Copulas Over The
This table reports the average of predictive
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t |
skt |
mskt |
dskt |
OOS |
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std |
10.4253 |
8.9757 |
7.6717 |
7.6398 |
7.3134 |
SPA test stastic |
4.3964 |
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0.5260 |
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Table 8.
Portfolio Performance Comparison of Copula Over the
This table shows the
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Gaussian |
t |
skt |
mskt |
dskt |
η =1 |
Mean (%) |
5.6245 |
5.1260 |
2.8879 |
5.2069 |
5.9158 |
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SD (%) |
5.2256 |
4.2342 |
4.1870 |
4.5801 |
4.1862 |
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Terminal wealth |
121.0039 |
119.0257 |
110.4277 |
119.3453 |
122.1703 |
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VaR (95%) |
17.6167 |
17.5622 |
21.1368 |
24.7608 |
23.7524 |
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ES (95%) |
25.4321 |
24.0339 |
30.4241 |
28.3839 |
24.8907 |
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PF |
0.3566 |
0.9695 |
3.1345 |
0.8698 |
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η =5 |
Mean (%) |
4.6343 |
4.6958 |
2.3312 |
4.3844 |
5.3373 |
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SD (%) |
5.2682 |
3.8872 |
2.8345 |
3.8955 |
4.5842 |
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Terminal wealth |
117.0972 |
117.3375 |
108.3600 |
116.1259 |
119.8612 |
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VaR (95%) |
14.5856 |
14.0884 |
16.7841 |
17.6962 |
14.5177 |
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ES (95%) |
20.4882 |
20.8635 |
24.6501 |
21.6531 |
19.8022 |
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PF |
0.6494 |
0.9195 |
1.8933 |
0.9272 |
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η =10 |
Mean (%) |
3.2174 |
3.2886 |
1.9267 |
2.8530 |
4.4226 |
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SD (%) |
3.1812 |
3.1726 |
1.8923 |
3.4111 |
4.0847 |
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Terminal wealth |
111.6649 |
111.9336 |
106.8752 |
110.2974 |
116.2739 |
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VaR (95%) |
11.2303 |
12.2545 |
14.5709 |
13.1395 |
11.8310 |
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ES (95%) |
17.3096 |
17.1338 |
20.0210 |
21.9134 |
18.5828 |
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PF |
0.8859 |
0.8360 |
2.0405 |
1.3275 |
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Bulletin of Monetary Economics and Banking, Volume 22, Number 1, 2019 |
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In summary, the predictive ability tests and portfolio optimization problem in this section demonstrate that our dynamic skewed t copula outperforms existing copulas, such as those of Christoffersen et al. (2012) and
VI. CONCLUSION
Developing copula models to describe high dimensional multivariate dependence is of particular interest to investors who allocate their wealth among a large number of assets. Existing skewed t copulas with static skewness parameters are unable to capture the time variation in dependence patterns. This paper complements the literature by modeling the dynamics of high dimensional multivariate dependence in a flexible yet parsimonious way. We extend the copula in Christoffersen et al. (2012) and
Applying our dynamic skewed t copula Cdskt to 50 ETF returns, we find that Cdskt has better
This paper leaves several topics for further research, including how to increase the estimation efficiency of high dimensional copulas. Future research should also examine the dependence of financial returns in emerging markets with greater fluctuations. These results from the dynamic skewed t copula may be of interest to policy makers and market participants alike.
Acknowledgment: We are grateful to Paresh Narayan (the editor) and one anonymous referee for helpful comments. All remaining errors are our own. Yuting Gong acknowledges the financial support from the National Natural Science Foundation of China (Grant No. 71601108). Jie Zhu acknowledges the financial support from Humanity and Social Science Planning Foundation of Ministry of Education of China (Grant No. 19YJA790128).
Modeling High Dimensional Asset Pricing Returns Using a Dynamic Skewed Copula Model |
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