First we have to confirm whether the data are stationary by unit root test .If it is non-stationary, we can convert it to a stationary series through process of differencing.
Then we use co-integration test to examine the long-term relationship of both variables. After the test we use the Granger Causality Model by Granger to explain whether these variables were exist Granger-cause. Furthermore we apply empirical data to discuss the relationship between bank liquidity and economy growth in each country, and which is lead indicator. Figure 1is the examination process of this study.
4.1 Unit root test
Before using Granger causality test, we must to test whether the time series is stationary. Stationary time series means the marginal and all joint distributions unchanged with time.
The unit root test is used to examine stationary. Granger and Newbold (1974) noted that if variables are non-stationary time series, even the variables are independent, the regression result still produced high R2 and significant t values, the phenomena is called spurious regression. To avoid above this problem, we use ADF test in this study.
Augmented Dickey-Fuller (ADF) test is used to examine the stationary. The method is popular in empirical studies, and the equation estimated is as under:
∆Xt = δ0+ δ1t + δ2Xt−1+ εt (1) Where ∆Xt−i expresses the first differences with k lags, t is time trend andεt, the variable that adjusts the errors of autocorrelation. The coefficientsδ0, δ1, δ2 and αi
are estimated. The null hypothesis for the existence of unit root in variableX are: t H0: δ2 = 0
If H0 is rejected that means the series is stationary and unit root does not exist. If H0 cannot be rejected, we should convert the series to be stationary through process of first difference until the series as stationary.
For sure the variables are stationary and avoid the error from our samples, we also use Phillips-Perron test (PP test).
The method is constructed by Phillips and Perron. They use nonparametric method to improve ADF statistic. The difference from ADF test is PP test allows the residual to have autocorrelation and heteroschedasticity. And the model doesn‘t include the lagged differences.
4.2 Granger Causality test
The first attempt at testing for the direction of causality was by Granger (1969).
Suppose X Granger-causes Y, then past values of X should be able to help predict future values of Y. The following simple model in which X and Y are expressed as:
Yt = pi=1αiYt−i+ qj=1βjXt−j+ μt (2) Where μt is white noise, p is the order of the lag for Y, and q is the order of the lag for X. The null hypothesis that X does not Granger-cause Y is that βj = 0 for j
=1,2,…,q, i.e. the past values of X does not forecast Y.
The restricted model is therefore
Yt = pi=1αiYt−i+ μt (3) The test statistic is the standard F-test. Similarly, if we test whether Y Granger-causes X, only change the equation as:
Xt = pi=1αiXt−i+ qj=1βjYt−j+ μt (4)
As discussion in 4.3, if there exists a co-integration, and the long-term relationship is also established. But it only shows the long-term relationship, omits the short-term relationship. Engle and Granger (1987) provided Error Correction Model (ECM) to solve this problem. When these two variables integrate in I(1) and exist co-integration, then we must to adopt ECM. If not, we uses differencing vector auto-regression model to examine short-term relationship.
4.3.1 Co-integration test
Although most of series data would be non-stationary, we should test the unit root test and co-integration test. The co-integration is defined as two or more series are non-stationary, but their linear combination would be stationary, we called these variables have co-integration. T
Suppose that yt~I 1 and xt~I 1 , if there exists a β such that yt − βxt~I 0 , then yt and xt are CI(1,1), it said to be co-integrated2. Granger (1981) provided the concept of co-integration, and he suggested that the long-term relationship exists while constructing a linear combination of two non-stationary variables. On the other hand, if there is co-integration between both variables, it means that a long-term relationship exists. Then the two variables must have causality, it means that X causes Y, or Y causes X or both variables occur at the same time.
In this study we following the maximum likelihood procedure of Johansen and Jueslius (1990), the method is explained by a first differencing vector auto-regression model:
2 Definition as G.S.Maddala. (1992). Introduction to Econometrics (2 ed.): Macmillan Publishing Company.
∆Yt = μ + Γ1∆Yt−1+ Γ2∆Yt−2+ ⋯ + Γρ−1∆Yt−ρ+1+ ΠYt−i+ εt (5)
= μ + ρ−1i Γi∆Yt−i + ΠYt−1+ μt (6) Γi= −I + A1+ A2+ ⋯ + Ai (i = 1,2, … , p − 1)
Π= I − A1− A2− ⋯ − Ap
In these two equations, Y is a P 1 × 1 vector containing variables, μis a vector of P(1 × 1) constant terms, and εt~i. i. d. Np(0, ζ2)
In equation(3), ρ−1i Γi∆Yt−i reflect the dynamic change when the series was distributed led to variables deviate the equilibrium; Π represents that the linear combination of all lag items, It is also called impact matrix, that can explain the long-term relationship of variables. The rank of Πdecided the amount of co-integrated vector. Then it may have three results:
1. If Rank Π = p, it is a full rank, implicating Yt is stationary 2. If Rank Π = 0, Yt is not co-integration
3. If Rank Π = r, and 0 < r< p, then there are r co-integrated vector in Yt, and make the β′Yt be a stationary process.
Johansen proposes two test statistics for testing the Rank Π , the trace test and the maximum eigenvalue test. The each null hypothesis and statistics as follows.
1. Trace test
H0: Rank Π ≤ r,
the statistic is λtrace = −T pi=r+1 1 − λ trace 2. Maximum eigenvalue test
H0: Rank Π = r H1: Rank Π = r + 1
The statistic is λmax = −T ln 1 − λ max
4.4 VAR/VEC Model k-dimensional vector of errors. It can be simply written as
Yt = A L Yt + εt , L is the lag operator.
The model sets all variables are endogenous variables, it can solve the problem how to decide which one is endogenous or exogenous variable. For some time in the 1980s, the American econometrics were all estimating VARs. Compare with American, The European use VECMs3.
When these variables establish co-integration relationship, it means there is a long-term equilibrium relationship. Using VAR is not sufficient explain these relationship, and co-integration omits short-term relationship. So, vector error correction model (VECM) is more useful.
Assumed there are two endogenous variables Xt and Yt, and exists a co-integrating factor without lagged. The formula shows as
∆Xt = a0+ bZt−1+ ni=1βi∆Xt−i+ mj=1βj∆Yt−j+ εt (8) Zt−1 is error correction term, the coefficient b which is called ―speed of adjustment parameter‖ measures the capability of adjustment function. The model represents that the change of series Xt can be explained by Zt-1, Yt, Ywith lag t-j and X with lag t-i.
To examine the Granger causality can test from two ways, the error correction term and the lag variable item. The former indicate the long-term relationship, the letter represent the short-run Granger causality.
3 G.S. Maddala, In-Moo Kim ―Unit roots, cointegration and structural change‖, Cambridge University Press,1998.New York, pp.34-37