The document describes the error correction model (ECM) version of Granger causality testing for determining the causal relationship between two non-stationary time series variables. It involves first testing for cointegration between the variables using the Johansen test or Engle-Granger approach. If cointegrated, the ECM version estimates an error correction model and performs Granger causality tests to examine short-run, long-run, and strong causality. The procedure and hypotheses for each test are provided along with the method for calculating the relevant F-statistics.

1. Data Analysis & Forecasting Faculty of Development Economics
TIME SERIES ANALYSIS
COINTEGRATION & ERROR CORRECTION VERSION OF
GRANGER CAUSALITY (ECM)
According to Mehrara (2007), the most popular method for Granger causality tests, is based
on ECM, avoiding spurious regression problems. In this procedure, we first investigate
whether the two non-stationary series are cointegrated, ie. whether a linear combination of
the two series is stationary. Cointegration implies that causality exists between the two
series, but it does not indicate the direction of the causal relationship. To test for
cointegration between the two series, we use the Johansen test/Johansen approach
(Johansen 1988; Johansen and Juselius 1990) based on the maximum eigenvalue and trace
statistics, as well as ADF cointegration tests/Engle-Granger (EG) approach developed by
Engle-Granger (1987).
1. THE MODEL
The Cointegration & Error Correction Version of Granger Causality (or Dynamic Granger
Causality or ECM) test for the of two stationary variables ∆Yt and ∆Xt, given that Yt and
Xt are cointegrated, involves as a first step the estimation of the following VAR model:
n m
∆Yt = α + π y ECTt −1 + ∑ β i ∆Yt −i + ∑ γ j ∆X t − j + u yt (1)
i =1 j=1
n m
∆X t = α + π x ECTt −1 + ∑ θ i ∆X t −i + ∑ δ j ∆Yt − j + u xt (2)
i =1 j=1
where ECTs are defined as follows:
ECTyt-1 = u yt −1 = Yt −1 − B1 − B 2 X t −1 (for equation (1))
ˆ ˆ ˆ (a)
ECTxt-1 = u xt −1 = X t −1 − B1 − B 2 Yt =1 (for equation (2))
ˆ ˆ ˆ (b)
Important note is the same as the Standard Granger Causality.
2. TEST PROCEDURE
Suppose we have Yt and Xt are nonstationary.
THE DYNAMIC GRANGER CAUSALITY is performed as follows:
Step 1: Testing for the unit root of Yt and Xt
(using either DF, ADF, or PP tests)
Suppose the test results indicate that both Yt and Xt are I(1).
Step 2: Testing for cointegration between Yt and Xt
Phung Thanh Binh (2010) 1

2. Data Analysis & Forecasting Faculty of Development Economics
(usually use EG or Johansen approach)
If Yt and Xt are cointegrated, we can apply either Standard or ECM Version of Granger
Causality, depending on our research objectives. However, it’d better to use the later one. In
this case, we should obtain the residuals from cointegrating equations (i.e., equations (a)
and (b)), and then name these terms as ECTy and ECTx.
Step 3: Taking the first differences of Yt and Xt (i.e., Yt and Xt)
Step 4: Determining the optimal lag length of Yt and Xt
a) Automatically determine the optimal lag length of ∆Yt and ∆Xt in their AR models
(using AIC or SIC, see Section 8 of my lecture).
n
∆Yt = α + π y ECTyt −1 + ∑ β i ∆Yt −i + u yt (3)
i =1
Then estimate (3) by OLS, and obtain the RSS of this regression (which is the
restricted one) and label it as RSSRY.
n'
∆X t = α + π x ECTxt −1 + ∑ θ i ∆X t −i + u xt (4)
i =1
Then estimate (4) by OLS, and obtain the RSS of this regression (which is the
restricted one) and label it as RSSRX.
b) Manually determine the optimal lag length of ∆Xt (m in equation (1)) and ∆Yt (m in
equation (2)), (using AIC or SIC, depending on which one you use in step 4a, see
Section 8 of my lecture).
n m
∆Yt = α + π y ECTyt −1 + ∑ β i ∆Yt −i + ∑ γ j ∆X t − j + u yt (5)
i =1 j=1
Then estimate (5) by OLS, and obtain the RSS of this regression (which is the
unrestricted one) and label it as RSSUY.
n' m'
∆X t = α + π x ECTxt −1 + ∑ θ i ∆X t −i + ∑ δ j ∆Yt − j + u xt (6)
i =1 j=1
Then estimate (6) by OLS, and obtain the RSS of this regression (which is the
unrestricted one) and label it as RSSUX.
Step 5: Set the null and alternative hypotheses
Short-run Granger causality/Weak Granger causality (F-statistic)
a) For equation (3) and (5), we set:
m
H0 : ∑γ
j=1
j = 0 or X t does not cause Yt
m
H1 : ∑γ
j=1
j ≠ 0 or X t causes Yt
b) For equation (4) and (6), we set:
m'
H0 : ∑δ
j=1
j = 0 or Yt does not cause X t
Phung Thanh Binh (2010) 2

3. Data Analysis & Forecasting Faculty of Development Economics
m'
H1 : ∑δ
j=1
j ≠ 0 or Yt causes X t
Long-run Granger causality (t statistic)
a) For equation (5), we set:
H 0 : π y = 0 or Granger non - causality in the long - run
H 1 : π y ≠ 0 or Granger causality in the long - run
b) For equation (6), we set:
H 0 : π x = 0 or Granger non - causality in the long - run
H 1 : π x ≠ 0 or Granger causality in the long - run
Strong Granger causality (F-statistic)
a) For equation (5), we set:
m
H 0 : π y = ∑ γ j = 0 or X t does not cause Yt
j=1
m
H 1 : π y ≠ 0 and ∑ γ j ≠ 0 or X t strongly causes Yt
j=1
b) For equation (6), we set:
m'
H 0 : π x = ∑ δ j = 0 or Yt does not causes X t
j=1
m'
H 1 : π x ≠ 0 and ∑ δ j ≠ 0 or Yt strongly causes X t
j=1
Step 6: Calculate the F statistic for the normal Wald test
a) For equation (3) and (5), we set:
(RSS RY − RSS UY ) / m
F=
RSS UY /( N − k )
b) For equation (4) and (6), we set:
(RSS RX − RSS UX ) / m'
F=
RSS UX /( N − k )
If the computed F value exceeds the critical F value, reject the null hypothesis and conclude
that Xt weakly causes Yt, or Yt weakly causes Xt.
Questions:
1. How to calculate the F statistics for strong Granger causality tests?
2. How to explain the test results?
Phung Thanh Binh (2010) 3