Appendix A. Vector Error-Correction Model
First, we present a framework of a vector error-correction (VEC) model to use simple least square techniques. Please note that a standard VEC model is derived from the following vector autoregression (VAR) equation for
m-vector time-series
(
).
where
,
and
denote a drift term,
n exogenous vector (i.e., trend term) and a
m random vector, respectively. Thus,
is
matrix; each
’s are
square matrices. Through tedious algebra, we obtain the following VEC equation from provides
This equation suggests that the change of the time-series, , is a sum of a stationary part, , and an error correction , which is stationary process when each component of is an integrated process.
Supposing
T sample periods, we can rewrite Equation (A2) in an extended linear regression form. For example,
Technically speaking, we can break down the above linear system to the following three cases according to what type of long-run relations are supposed.
Case 1: Error Correction Terms without Drift
The first case corresponds to the long-run equilibrium equations without constant terms.
Case 2: Error Correction Terms with Drift
The second case corresponds to the long-run equilibrium equations with constant terms.
Case 3: Error Correction Terms with Linear Time Trend
The third case corresponds to the long-run equilibrium equations with linear time trend.
where the last row of the second term in (A5) presents a time trend
.
Notice that the dimension of the
matrix for the third case differs from the other two cases. It is
whereas the others are
Because a VEC model is algebraically derived from a certain VAR model, a linear stochastic system, it is also such a system. Thus, we can estimate parameters
,
’s and
using some regression techniques, say, OLS or GLS. Let
,
and
denote appropriate data matrices representing for Equations (A3)–(A5). Furthermore,
denotes a matrix of exogenous shock vectors, i.e.,
where
and
Similarly, as for (A4), its expression is as follows:
where
and
Finally, as for (A4), its expression is as follows:
where
and
The three matrices, , and might include as well as They provide us with information about stationary aspect of the time-series On the other hand, the three matrices, , and , take a crucial role in a VEC model. They are decomposed into the loading matrix and the cointegration matrix such that (Three identifier, “n”, “c” and “t”, for the above three options are omitted here). In particular, signifies some long-run relationship among the observation. One can select the lag order k using usual information criteria such as SBIC for each linear model above; it is easy to compute them.
Given some cointegration order
r, we can decompose the estimated
in the above linear models with respect to
,
and
such that
where
and
are called the loading and cointegration matrices, respectively. The cointegration order
r is usually selected through well-known the Johansen [
17,
18] test. Johansen’s procedure on the rank helps ones to obtain all the estimates and statistics for applied econometrician.
Appendix C. Time-Varying VEC Model
As Lütkepohl [
22] exactly points, the essential aim of VEC models is to decompose a multivariate time-series into a pair of stationary and non-stationary time-series. It is very similar to that of the Beveridge and Nelson [
23] decomposition for a univariate time-series. Roughly speaking, we can consider that the matrices
’s for
represents the stationary structure of the time-series to be analyzed; the matrix
or matrices
and
represent its non-stationary structure, the so-called cointegrated relationship among variables of the time-series.
Although we start from a VAR model (A1), its corresponding VEC model provides more information. When we consider a time-varying nature of some multivariate time-series using VEC model, we should specify to what structure we focus: that represented by
, that of
or
and
or both. We show some options for time-varying VEC (TV-VEC) model. Regarding a VEC model as a simple linear regression model, we just consider one of (A6) through (A8). Thus, we write down it for convenience as follows.
We can suppose following combinations of the parameter dynamics with the above linear model of VEC (A9). First, we estimate
and
as time-varying parameters without considering the decomposition of
into
and
.
Second, we estimate
and
with regarding
as time-invariant and given.
To this case, we first build a
r-dimensional time-series
. Then, we apply Ito et al.’s method again to a new time linear regression
considering (A12) and (A13).
Third, we estimate
and
with regarding
as time-invariant and given.
Considering
, we first rewrite (A9) as follows.
where ⊗ is the Kronecker product and
operator transforms a matrix into a vector by stacking the columns. Please note that
can be regarded as a data matrix because
is given. Considering this equation as a linear regression model whose parameters are supposed time-varying, we apply again the Ito et al. [
5] method estimate the parameters to be varying over time.
Please note that both and for each t cannot be estimated. In particular, since for each t and is not of full rank, a decomposition of into and is not unique. Thus, either or is supposed time-invariant for the most general case of both and supposed time-varying.
Appendix D. Degree of Market Comovement
We regard as long-run equilibrium relations with respect to the multiple time-series. The loading matrix , representing a speed of adjustment, is time-varying when we employ a TV-VEC model to analyze market comovement. Thus, we pay our attention to the time-varying loading matrix , which provides information about dynamics of market comovement. The larger the absolute value of its components, the more significant their contribution to ameliorate deviation from the long-run equilibrium. Thus, we propose an index of market comovement based on the loading matrix. Then, we applied the index for the time-varying loading matrix to investigate how degree of market comovement varies.
We derive the index
from
following Ito et al. [
5]. In particular,
That is, is the square root of the largest eigen value of which is a non-negative semi definite matrix, for each t. Notice that the more the index the faster the adjustment of markets to the long-run equilibrium.