In the previous section, we assumed that there exists a static correlation network in which the edges between nodes (stocks) do not change during the observation period. Such an assumption is introduced partly because of technical reasons regarding the estimation of the correlation matrix of stock returns. It is rather an empirical issue whether the correlation structure is stable or dynamically changing over time. The linkages between the nodes may change because of changes in the correlation of stock returns, although it is not technically easy to detect such dynamic changes from observed price data. Normally, a sample pairwise correlation of returns is calculated based on the observed filtered or unfiltered return data; therefore, only one sample correlation, the static one, is available for one data period. However, it would be meaningful to know how the correlation network changes over time from the viewpoint of investment decision making as well as portfolio risk control if we could establish a method of estimating dynamic correlations.
Model-based dynamic correlation
The most frequently used approach to measure the dynamic correlations of asset returns is to calculate the correlations of returns over moving windows. A series of correlation matrices can be built by using a rolling observation time window during the study period. This method, however, has some drawbacks as discussed in
Isogai (2016). Hence, a statistical model-based correlation can be used to describe the dynamic correlation network. The DCC model originally proposed by
Engle (2002) was developed in the context of using multivariate volatility models in financial econometrics. The static unconditional correlation matrix
R of return
r
t
is calculated as the correlation matrix of filtered return
z
t
defined in (
1) when building the static correlation network. When applying the DCC model, a time-dependent correlation matrix
R
t
of return
r
t
is estimated instead of
R. This means that the correlation of returns can change dynamically during the observation period; therefore, dynamic changes in the structure of a correlation network are represented by a dynamic conditional adjacency matrix
A
t
. There are multiple adjacency matrices, as many as the number of trading days converted from a series of
R
t
during the period.
The DCC model is described as an extension of the multivariate GARCH model described by (
1), (
2), and (
3). Specifically, an
N×
N positive-definite dynamic correlation
R
t
is introduced to model the dependency structure of
r
t
. The time dependency of
R
t
is described by using a proxy variable
Q
t
, which is introduced to ensure the positive-definiteness of
R
t
as
$$ \begin{aligned} \boldsymbol{Q}_{t} &=\boldsymbol{\bar{Q}}+\sum\limits_{i=1}^{m}a_{i}\left(\boldsymbol{z}_{t-i}\boldsymbol{z}_{t-i}^{'}-\boldsymbol{\bar{Q}}\right)+\sum\limits_{j=1}^{n}b_{j}\left(\boldsymbol{Q}_{t-i}-\boldsymbol{\bar{Q}}\right)\\ \end{aligned} $$
(6)
where
a
i
and
b
j
are non-negative scalars and
\(\boldsymbol {\bar {Q}}_{t}\) is the unconditional mean of
Q
t
. The DCC model with time lags in the conditional correlation is denoted as DCC (
m, n). The parameter
a
i
shows the sensitivity of
Q
t
to previous shocks, while the parameter
b
j
represents the persistence of the correlation in previous periods. The correlation matrix
R
t
is calculated by rescaling
Q
t
as
$$ \boldsymbol{R}_{t}=\text{diag}\left(\boldsymbol{Q}_{t}\right)^{-\frac{1}{2}}\boldsymbol{Q}_{t}\text{diag}\left(\boldsymbol{Q}_{t}\right)^{-\frac{1}{2}} $$
(7)
$$ a_{i} \ge 0,\quad b_{j} \ge 0, \quad \sum\limits_{i=1}^{m}a_{i}+\sum\limits_{j=1}^{n}b_{j}<1. $$
(8)
Model fitting and adjacency conversion
The parameters of the DCC model are estimated by using MLEs with the Japanese stock return data. We employ a two-stage fitting of the DCC model: the first stage of the GARCH model fitting followed by the second stage of the DCC parameter estimation. The GARCH model is fitted to the sub-portfolio returns just as it is fitted to individual stock returns in “
Correlation network of financial asset returns”. Once the first stage of the model estimation is completed, the filtered residuals
z
t
of 14 sub-portfolios are calculated by using the estimated parameters of the GARCH model. In the second stage, the DCC model parameters used to calculate the dynamic correlation
R
t
of the sub-portfolio returns are estimated.
When estimating the DCC parameters by using MLEs, the joint density function of
r
t
should be explicitly defined in advance. The distribution of
z
t
is assumed to be one of the normal, Student
t, or skew
t distributions again. The joint density function
f(
r
t
) is then defined by using a copula density function that determines the dependency between the sub-portfolio returns. The copula function plays a key role in addressing the dependency between the heterogeneous distributions of
z
t
as mentioned above. In general, the joint density function
f(
x) of a vector of variable
X=(
X
1,…,
X
N
) can be described using a copula function as follows:
$$ \begin{aligned} f\left(x_{1},\,\ldots,\, x_{N}\right) &=c\left(F_{1}\left(x_{1}\right),\,\ldots,\, F_{N}\left(x_{N}\right)\right)\prod_{i=1}^{N}f_{i}\left(x_{i}\right) \end{aligned} $$
(9)
where
f
i
(
x
i
) is the marginal distribution of
x
i
,
c(·) is the density function of the copula, and
F(·) is the joint distribution function of
X. We choose the Student
t-copula that can handle tail dependency, which takes two parameters: conditional correlation
R
t
and the constant shape parameter. For technical details on the copula and Student
t-copula, see
Sklar (1959) and
Demarta and McNeil (2005). Thus, the joint density function of
r
t
is defined as a combination of the copula density and density of the i.i.d. residual
z
t
:
$$ \begin{aligned} &f\left(\boldsymbol{r}_{t}|\boldsymbol{\mu}_{t},\ \boldsymbol{h}_{t},\ \boldsymbol{R}_{t},\ \eta\right)\\ &=c^{S_{t}}\left(u_{1{\cdot}t},\ \ldots,\,\ u_{N{\cdot}t}|\boldsymbol{R}_{t},\ \eta\right) \prod_{i=1}^{N}\frac{1}{\sqrt{h_{i{\cdot}t}}}f_{i{\cdot}t}\left(z_{i{\cdot}t}|\theta_{i}\right) \end{aligned} $$
(10)
where
u
i·t
=
F
i
(
r
i·t
|
μ
i·t
,
h
i·t
,
θ
i
);
θ
i
is a parameter set including the ARMA–GARCH parameters in (
2) and (
3) and the distributional parameters of
z
i
;
\(\phantom {\dot {i}\!}c^{S_{t}}(\cdot)\) is the Student
t-copula density function; and
η is the shape parameter of the Student
t-copula. The conditional correlation
R
t
is defined as one parameter of the copula function, the time-dependent structure of which is described in (
6) and (
7) in the DCC model setting. The estimate of
R
t
therefore collapses to the non-negative scalars (
a,
b) defined in (
6).
We need to determine the distribution of
z
t
as well as the DCC order (
m, n): the model selection is made by comparing the goodness-of-fit measure, namely the Akaike information criterion (AIC), from the multiple combinations of the model settings. The log-likelihood function
LL(
θ|
r
t
) built by using (
10) comprises two parts: the copula part with the DCC parameters (
a,
b) and marginal distribution part
f
i·t
(
z
i·t
|
θ
i
). The two parts of the log-likelihood function can be maximized independently: first, the individual distributional parameter set
θ
i
is estimated, followed by the DCC parameters (
a,
b). More technical details about the model fitting procedures are described in
Isogai (2016);
Patton (2006), and
Joe (2005).
Table
3 shows the parameter estimation results for the selected DCC models. The DCC order (
m, n) is (1, 2);
a
1, the sensitivity of the correlation to previous shocks, takes a small value. The larger value of
b
1+
b
2 means that the dynamic correlation matrix
R
t
is more dependent on its past values than previous shocks, since the parameter
b
j
represents the degree of persistence of the correlation. The model parameter restriction shown in (
8) is confirmed to be satisfied. The shape parameter of the Student-
t copula is not so low, meaning that the degree of tail dependency seems to be limited after volatility filtering. The other details of the estimation results including the ARMA–GARCH model of individual returns are omitted because of space limitations.
Table 3
DCC parameter estimation results
Estimate | 1, 2 | 0.0259 | 0.5549 | 0.3855 | [0.9404] | 13.0 |
(P-value) | | (0.0000) | (0.0000) | (0.0000) | | (0.0000) |
The same DCC model is fitted to the individual stock returns in each sub-portfolio to examine the extent to which the correlation of the individual stock returns are significantly different between sub-portfolios. Table
4 shows the DCC parameter estimation results for all sub-portfolios. The DCC order (
m, n) is (1, 2) for all sub-portfolios in Cyclical, while it is (1, 2) or (1, 1) in Defensive. The combination of lower
a
1 and higher
b
1+
b2 values appears to be common to every portfolio, while the relative share of the two types of parameters varies over the sub-portfolios. Again, the shape parameter of the Student-
t copula is higher in every sub-portfolio. This result means that the dynamic correlation properties are similar, although small differences exist between the sub-portfolios. Further, we can extend our dynamic correlation network analysis to those sub-portfolios when required.
Table 4
DCC parameter estimation results by sub-portfolio
Cyclical | P1 | 1, 2 | 0.0080 | 0.5537 | 0.3523 | [0.9060] | 30.5 |
| P2 | 1, 2 | 0.0074 | 0.5567 | 0.3792 | [0.9358] | 19.7 |
| P3 | 1, 2 | 0.0093 | 0.3275 | 0.3897 | [0.7172] | 29.9 |
| P4 | 1, 2 | 0.0079 | 0.5476 | 0.3219 | [0.8695] | 31.6 |
| P5 | 1, 2 | 0.0064 | 0.5787 | 0.3240 | [0.9027] | 27.2 |
| P6 | 1, 2 | 0.0086 | 0.5820 | 0.3219 | [0.9038] | 22.3 |
| P7 | 1, 2 | 0.0079 | 0.5432 | 0.3713 | [0.9145] | 25.6 |
| P8 | 1, 2 | 0.0069 | 0.5542 | 0.3925 | [0.9467] | 20.9 |
Defensive | P9 | 1, 2 | 0.0078 | 0.4651 | 0.3815 | [0.8467] | 27.9 |
| P10 | 1, 2 | 0.0103 | 0.2498 | 0.3980 | [0.6478] | 30.9 |
| P11 | 1, 2 | 0.0070 | 0.4963 | 0.3890 | [0.8853] | 30.9 |
| P12 | 1, 1 | 0.0048 | 0.9400 | - | [0.9400] | 22.9 |
| P13 | 1, 2 | 0.0094 | 0.2627 | 0.5084 | [0.7711] | 27.1 |
| P14 | 1, 1 | 0.0042 | 0.8945 | - | [0.8945] | 38.5 |
Next, we build a dynamic correlation network to study the possible topological changes in the network. The DCC parameter estimates are all available; the dynamic correlation matrix from the estimated correlation matrix
R
t
can be easily calculated from (
6). Then, the estimated model-based conditional correlation matrix
R
t
is converted into the conditional adjacency matrix of the dynamic correlation network. Here, we use the same unsigned nondecreasing adjacency conversion as the one in (
4) used when building the static correlation network for clustering stock returns in “
Correlation network of financial asset returns”. The adjacency conversion is extended to a time-dependent conditional one as follows:
$$ \boldsymbol{A}_{ii,t}=0, \quad \boldsymbol{A}_{ij\left(i\neq j\right),t}= \left\{\begin{array}{ll} \left|cor\left(x_{i},\, x_{j}\right)_{t}\right|\ &\quad if\ \,\, cor\left(x_{i},\, x_{j}\right)_{t} > {cor}_{thres(i, j)_{t}}\\ 0 & \quad if\ \,\, cor\left(x_{i},\, x_{j}\right)_{t}\leqq {cor}_{thres(i, j)_{t}}\\ \end{array}\right. $$
(11)
where
A
ij,t
is the (
i, j)
th
entry of the conditional weighted adjacency matrix
A
t
and
cor(
x
i
,
x
j
)
t
is the (
i, j)
th
entry of the dynamic correlation matrix
R
t
. The diagonal element of
A
ij,t
is 0. The threshold value
\(\phantom {\dot {i}\!}{cor}_{{thres}({i, j})_{t}}\) is determined in the same way as in (
4) at every point in time. Thus, the dynamic correlation network is created with the adjacency matrices
A
t
available for each trading day. One technical issue is that thresholding of the adjacency matrix entries can affect the result of our intertemporal analysis described below. The thresholding method is time-dependent; therefore, threshold values of the same matrix entry can change dynamically. It may cause discontinuous changes, especially when the threshold level is higher. These aspects of our dynamic thresholding method makes it difficult to forecast its effect on intertemporal analysis. We confirmed that analytical results are stable with thresholding in a few different settings; however, this point is still an important caveat of this study.