3. Methodology
3.1 Data Description
The empirical research of global market shows that there is the relationship between global volatility and financial markets. Thus, the study tries to identify how much impact does three endogenous variables such as exchange rate, Kospi 200 and three-year bond rates have on Credit default swap of South Korea. To examine empirical evidence, the quantitative analysis: time series data should be applied (Brooks, 2002). The data was retrieved from Bloomberg and Rueter. The data period starts from February 2002 ends at February 2016, and all monthly data are composed by the end of the second week in every month. Hence, the total number of the observations is 168. When there is no observation for that specific month, the observation of the month before and after’s average was used as a replacement. After that, we utilize EViews, the econometric software, to run the multiple regressions and estimate the VAR to find the cointegration relationships of time-series variables.
3.2 Data stationary test
As far as the stationarity of the variables is concerned, the unit root test will be applied. If the variables are not stationary, it proves that the standard estimation for asymptotic analysis is not eligible (Brooks, 2002). The constant time series does not depend on the time t, but only on the intervals j(lag). The equation can be expressed as below (Wang, 2003)
E {X (t)}= µ for all t
Var {X (t)}< ∞ for all t; and
Cov {X (t), X (t+j)}= γj for all t and j
Since we are especially concerned about granger causality, we nee to ascertain whether the variables are stationary or non-stationary.
3.2.1 Unit root test & Hypothesis Test
3,2,1,1 DF & ADF Test
Dickey–Fuller Test
In the following AR(1) process,
X_t= a_t+X_(t-1)+ε_t (1)
Dickey-Fuller test is presented as following through differencing equation (1):
X_t-X_(t-1)=(a_t-1)X_(t-1)+ε_t (2)
DF: ∆X_t=γX_(t-1)+ε_t (3)
Hypothesis Test
H_0 ∶ γ=0 or a=1→ Unit root exists (Non-stationary time series)
H_1 ∶ γ<0 or a<1→ Unit root does not exist (Stationary time series)
Decision Rule
The t-test is performed for γ of equation (3).
If t-statistics<t ̂^DF, the null hypothesis is rejected.
If t-statistics >t ̂^DF the null hypothesis is accepted.
Augmented Dickey–Fuller Test
By adding independent time series variable, ∆X_t, extended DF test is as following and it is called ADF.
ADF: ∆X_t=φX_(t-1)+∑_(i=1)^q▒ω_i ∆X_t+ε_t (4)
3.2.1.2 Philips and Perron (PP) test
Phillips and Perron (1988) developed unit root non-stationary into the further stage. It uses the procedure of ADF test, but incorporate an automatic correction to allow for auto-correlated residuals. In other words, the PP tests use non-parametric, as it does not affect the asymptotic distribution of test statistic. The PP test requires to decide whether not to include a constant or time trend or not and analyses its component at the zero frequency. The equation of PP test is:
y_t=a_0^*+a_1^* y_(t-1)+μ_t (5)
y_t=a ̃_0+a ̃_1 y_(t-1)+a ̃_2 (t-T/2)+μ_t (6)
Where, T=관측치의 수, μ_t는Eμ_t=0인 교란항
3.2.1.3 ACF and PACF Test
The assumption for regression errors is assumed to be uncorrelated, so-called weakly stationary series. The autocorrelation shows distinct features. It assumes any particular lag of autocorrelation is same in any time series. Moreover, when there is the positive value for 〖 ρ〗_k, ACF can be significant decreases to 0 as the lag increases, also apply the same to negative 〖 ρ〗_k . It only varies the algebraic signs for the autocorrelations to distinguish negative and positive.
ACF Equation:
〖ACF: ρ〗_k=γ_k/γ_0 , k=1,2,3,⋯, (5)
In terms of Partial autocorrelation, it controls the values of the time series.
다음과 같은 방정식에서
y_t^*=y_t-μ (6)
The equation of PACF: 식(6)에서 AR(1) 과정은
y_t^*=a_11 y_(t-1)^*+e_t (7)
식(7)으로부터 2차 자기회귀 방정식으로 표현하면,
PACF:y_t^*=a_21 y_(t-1)^*+a_22 y_(t-2)^*+e_t (8)
식(8)에서 a_22는 y_t와y_(t-2)사이의 편자기상관계수이며, 즉 YuleWalker방정식을 만족하는 함수식의 최종시차항의 계수로 나타낸다.
3.2.2 Information Criterion
One of the times series features are that as the number of variables increases, the R^2 values gets high. Using AIC and SIC , appropriate information for analysis is presented.
AIC(p)=ln〖|Ω ̃_p |〗+(2n^2 p)⁄T,constant term exists =ln〖|Ω ̃_p |〗+(2(n+n^2 p))⁄T,constant term does not exist
SIC(p) =ln〖|Ω ̃_p |〗+n^2 p ln〖T/T〗,
Where, |Ω ̃_p | is matrix of error term vector and
├ Ω ̃_p ┤is covariance matrix of term error vector.
T is the number of observed value invalid sample
n^2 p is the number of coefficient matrices to be estimated
3.3 VAR Estimate
According to Sims (1980), VARs held out the promise of the providing a coherent and credible approach to data description, for data description and structural influence. VAR approach treats all variables as joint endogenous and does not rely on “ incredible identification restrictions”. It does not need to specify which variables are endogenous or exogenous because they all assume it is endogenous. It also allows the value of a variable to depend on more than just its lags or combinations of white noise terms, so VARs are more flexible than univariate AR models, by simply applying OLS separately on each equation, there are no contemporaneous terms on the RHS of the equations (Brooks, 2002). Before running multiple regressions, it is important to choose the optimal lag length for a VAR. Taking into the consideration of the research, VAR model will employ OLS, ordinary least squares, to estimate 40 variables including four variables in second differences and its trend and intercept. The equation of the matrix estimation is as below:
X_t=c+A_1 X_(t-1)+A_2 X_(t-2)+⋯A_p X_(t-p)+ε_t (9)
X_t=[■(X_1t@⋮@X_nt )],A_i=[■(A_11i&⋯&A_1mi@⋮&⋱&⋮@A_n1i&⋯&A_nmi )],c=[■(c_1@⋮@c_n )],ε_t=[■(ε_1t@⋮@ε_nt )] (10)
And expectation and covariance are
E(ε_t )=0 (11)
Cov=E(ε_t ε_s^' )={█(Ω,t=s@0,t≠s)┤ (12)
If equation (10) is simplified in the form of matrix, it can be written as following:
X_t=c+AX_(t-1)+ε_t (13)
Using lag operator, equation (13) can be written as following:
A(L)X_t=c+ε_t (14)
For analysis model, following multiple regression model (MRM) is considered with financial indexes:
y_(t_cds)=α_0+α_1 β_(t_kospi200)+α_2 β_(t_ex)+α_3 β_(t_tby)+ε_t (15)
Where is the y_(t_cds) = CDS
α_0 = Intercept
〖 β〗_(t_kospi200) = Kospi_200 Index
β_(t_ex) = Exchange Rate
β_(t_tby) = Tréasury Bònd 3years
ε_t = Error term
In this study VAR(1) is considered as following:
C_t=α_0+θ_11 C_(t-1)+θ_12 ∃_(t-1)+θ_13 χ_(t-1)+θ_14 Ξ_(t-1)+ε_1t (16)
∃_t=α_0+θ_11 C_(t-1)+θ_12 ∃_(t-1)+θ_13 χ_(t-1)+θ_14 Ξ_(t-1)+ε_1t (17)
χ_t=α_0+θ_11 C_(t-1)+θ_12 ∃_(t-1)+θ_13 χ_(t-1)+θ_14 Ξ_(t-1)+ε_1t (18)
Ξ_t=α_0+θ_11 C_(t-1)+θ_12 ∃_(t-1)+θ_13 χ_(t-1)+θ_14 Ξ_(t-1)+ε_1t (19)
Where is the С_t = CDS Premium
α_0 = Intercept
∃_t = Kospi_200 Index
χ_t = Exchange Rate
Ξ_t = Tréasury Bònd 3years
ε_t = error term
To estimate coefficients of Equation (15) model, following matrix differential equation is used. The first order condition becomes zero.
Θ ̂=[I_K⨂〖(X^T X)〗^(-1) X^T]y (20)
where “I” is the identity matrix
⨂ is Kronecker product
3.7 Impulse response function
Nevertheless, the downfall of VAR approach that it’s a-theoretical nature and a large number of parameters involved make the estimated models difficult to interpret (Brooks, 2002). Specifically, the lagged variables that have coefficients would challenge what effect a given change in a variable would have upon the future values of the variables in the equation. To alleviate this problem, granger causality and impulse response function test should be taken apart (Linlan and Aydemir, 2007).
〖∆y〗_t=∑_(k=0)^∞▒〖c_12 (k) ϵ_(2t-k) 〗 (21)