We have discuss how to perform the VAR
estimation at here,
and then we also perform one of test from post-estimation VAR, what we called
it as Granger causality at here.
According to Toda and Yamamoto(1995), economic
series could be either integrated of the different orders or non-cointegrated
or both.
In these cases, ECM cannot be applied for
Granger causality test. Hence, they developed an alternative test, irrespective
of whether \({{y}_{t}}\)
and \({{x}_{t}}\) are \(I\left( 0 \right)\) , \(I\left( 1 \right)\) or \(I\left( 2 \right)\) , non-cointegrated or cointegrated of an
arbitrary order.
This is widely known as the Toda and Yamamoto(1995)
augmented Granger causality.
Toda and Yamamoto (1995) augmented Granger
causality test method is based on the following equations:
\({{y}_{t}}=\mu +\sum\limits_{i=1}^{p+m}{{{\alpha
}_{i}}{{y}_{t-i}}+\sum\limits_{i=1}^{p+m}{{{\beta }_{i}}{{x}_{t-i}}+{{u}_{1t}}}}\) (1)
\({{x}_{t}}=\mu +\sum\limits_{i=1}^{p+m}{{{\gamma }_{i}}{{x}_{t-i}}+\sum\limits_{i=1}^{p+m}{{{\delta }_{i}}{{y}_{t-i}}+{{u}_{2t}}}}\) (2)
where \(m\) is the maximal
order of integration order of the variable in the system, and \(p\) are the optimal lag
length of \({{y}_{t}}\) and \({{x}_{t}}\), and the error terms are
assumed to be white noise, \(\sim
\left( 0,{{\sigma }^{2}} \right)\) , and no autocorrelation.
We need to determine the maximal order of integration \(m\), which is we expect to occur in the model and construct a VAR in their levels with total of \(\left( p+m \right)\) lags.
There are some basic steps that we need to follows to perform the
Toda-Yamamoto test;
Step 1
Test each of the
time-series to determine their order of integration. Ideally, this should
involve using a test (such as the ADF test) for which the null hypothesis is
non-stationarity; as well as a test (such as the KPSS test) for which the null
is stationarity. It's good to have a cross-check.
Step 2
Let the maximum
order of integration for the group of time-series to be \(m'\) . So, if there are two time-series
and one is found to be
and the other
is
, then \(m'=2\) . If one is \(I\left( 0 \right)\) and
the other is \(I\left( 1
\right)\) , then \(m'=1\) , etc.
Step 3
Set up a VAR model in the levels of
the data, regardless of the orders of integration of the various
time-series. Most importantly, we must not difference the data, no matter what we found at Step 1.
Step 4
Determine the appropriate
maximum lag length for the variables in the VAR, \(p'\), using the usual methods. Specifically,
base the choice of \(p'\)
on the usual information criteria, such
as AIC, SIC.
\({{y}_{t}}=\mu +\sum\limits_{i=1}^{p'}{{{\alpha
}_{i}}{{y}_{t-i}}+\sum\limits_{i=1}^{p'}{{{\beta }_{i}}{{x}_{t-i}}+{{u}_{1t}}}}\) (3)
\({{x}_{t}}=\mu +\sum\limits_{i=1}^{p'}{{{\gamma }_{i}}{{x}_{t-i}}+\sum\limits_{i=1}^{p'}{{{\delta }_{i}}{{y}_{t-i}}+{{u}_{2t}}}}\) (4)
Step 5
Make sure that the VAR is well-specified. For example, ensure that there
is no serial correlation in the residuals. If need be, increase \(p\) until any autocorrelation issues are resolved.
Step 6
Now take the preferred VAR model and add in \(m'\) additional
lags (from Step 2) of each of the variables into each of the
equations
\({{y}_{t}}=\mu +\sum\limits_{i=1}^{p'+m'}{{{\alpha }_{i}}{{y}_{t-i}}+\sum\limits_{i=1}^{p'+m'}{{{\beta }_{i}}{{x}_{t-i}}+{{u}_{1t}}}}\) (5)
\({{x}_{t}}=\mu +\sum\limits_{i=1}^{p'+m'}{{{\gamma }_{i}}{{x}_{t-i}}+\sum\limits_{i=1}^{p'+m'}{{{\delta }_{i}}{{y}_{t-i}}+{{u}_{2t}}}}\) (6)
Step 7
Test for Granger non-causality as follows. Test the hypothesis that the
coefficients of (only) the first \(p'\) lagged values of \({{x}_{t}}\) are zero in the \({{y}_{t}}\) equation, or Eq(5) using a standard Wald
test. Then do the same thing for the coefficients of the lagged values of \({{y}_{t}}\) in the \({{x}_{t}}\) equation,
or Eq(6).
For Eq(5)
\({{H}_{0}}:\sum\limits_{i=1}^{p'}{{{\beta
}_{i}}=0}\) , or \({{x}_{t}}\)
does not cause \({{y}_{t}}\).
\({{H}_{0}}:\sum\limits_{i=1}^{p'}{{{\beta
}_{i}}\ne 0}\), or \({{x}_{t}}\)
causes \({{y}_{t}}\).
For Eq(6)
\({{H}_{0}}:\sum\limits_{i=1}^{p'}{{{\delta
}_{i}}=0}\) , or \({{y}_{t}}\)
does not cause \({{x}_{t}}\).
\({{H}_{0}}:\sum\limits_{i=1}^{p'}{{{\delta
}_{i}}\ne 0}\), or \({{y}_{t}}\)
causes \({{x}_{t}}\).
Step 8
It's essential that
we don't include the coefficients for the 'extra' \(m'\) lags
when you perform the Wald tests. They are there just to fix up the asymptotics.
Step 9
The Wald test statistics will be asymptotically chi-square distributed with \(p'\) df.,
under the null.
Step 10
Rejection of the null implies a rejection of Granger non-causality.
That is, a rejection supports the presence of Granger causality.
Toda and Yamamoto Approach using Stata
To perform the Toda and Yamamoto (1995) by Stata, we will use again Data09.
We will follow
each steps what we discussed before to performing the Toda and Yamamoto approach
for the causality test.
Step 1
This step require to test each of the time-series to determine their order of
integration. For our discussion, we use the Augmented Dickey-Fuller (ADF)
(1979) unit root test to test each time-series variables; lrgrossinv, lrconsump and
lrgdp for stationarity.
Some theoretical background for the ADF test can be found at here.
Now, lets we perform the ADF for each variables and we use the AIC for select the appropriate lags;
tsset t
varsoc lrgrossinv,max(6)
dfuller lrgrossinv,trend lags(3)
varsoc D.lrgrossinv,max(6)
dfuller D.lrgrossinv,lags(2)
varsoc lrconsump,max(6)
dfuller lrconsump,trend
lags(3)
varsoc D.lrconsump,max(6)
dfuller D.lrconsump,lags(2)
varsoc lrgdp,max(6)
dfuller lrgdp,trend lags(3)
varsoc D.lrgdp,max(6)
dfuller D.lrgdp,lags(2)
Step 2
The ADF test in Step 1 show that all the variables indicate \(I\left( 1 \right)\) .
That means our \(m\) now will become \(m'=1\) .
Step 3 , Step 4
To determine the appropriate max lag length for VAR model;
varsoc lrgrossinv lrconsump
lrgdp,max(10)
The results show that the max lag length will be chosen
for VAR based on AIC is 7.
That means, our \(p\) now
is \(p'=7\) .
Step 5
To test the VAR model for the serial correlation in residuals;
quiet var lrgrossinv lrconsump
lrgdp, lags(1/7)
varlmar
The results show that there is no serial correlation in residuals for
our VAR model at 1% significance level.
Step 6
From Step 2, the \(m'=1\) and from Step 4, the \(p'=7\).
That means, our \(p'+m'=8\) .
Step 7, Step 8, Step 9,Step 10
Now, after we know that the value of \(p'+m'=8\) for the VAR
(unrestricted) and \(p'=7\)
for the VAR (restricted), we will perform the Granger causality test as
follows;
quiet var lrgrossinv lrconsump
lrgdp,lags(1/8)
* test for lrgrossinv Granger–causality
to lrgdp
test[lrgdp]L.lrgrossinv[lrgdp]L2.lrgrossinv
[lrgdp]L3.lrgrossinv [lrgdp]L4.lrgrossinv [lrgdp]L5.lrgrossinv
>[lrgdp]L6.lrgrossinv [lrgdp]L7.lrgrossinv
* test for lrconsump Granger–causality
to lrgdp
test[lrgdp]L.lrconsump
[lrgdp]L2.lrconsump [lrgdp]L3.lrconsump [lrgdp]L4.lrconsump [lrgdp]L5.lrconsump
>[lrgdp]L6.lrconsump [lrgdp]L7.lrconsump
* test for lrgrossinv Granger–causality
to lrconsump
test[lrconsump]L.lrgrossinv
[lrconsump]L2.lrgrossinv [lrconsump]L3.lrgrossinv [lrconsump]L4.lrgrossinv
[lrconsump]L5.lrgrossinv >[lrconsump]L6.lrgrossinv [lrconsump]L7.lrgrossinv
* test for lrgdp Granger–causality to lrconsump
test[lrconsump]L.lrgdp
[lrconsump]L2.lrgdp [lrconsump]L3.lrgdp [lrconsump]L4.lrgdp [lrconsump]L5.lrgdp
>[lrconsump]L6.lrgdp [lrconsump]L7.lrgdp
* test for lrconsump Granger–causality
to lrgrossinv
test[lrgrossinv]L.lrconsump
[lrgrossinv]L2.lrconsump [lrgrossinv]L3.lrconsump [lrgrossinv]L4.lrconsump
>[lrgrossinv]L5.lrconsump [lrgrossinv]L6.lrconsump [lrgrossinv]L7.lrconsump
* test for lrgdp Granger–causality
to lrgrossinv
test[lrgrossinv]L.lrgdp
[lrgrossinv]L2.lrgdp [lrgrossinv]L3.lrgdp [lrgrossinv]L4.lrgdp
[lrgrossinv]L5.lrgdp >[lrgrossinv]L6.lrgdp [lrgrossinv]L7.lrgdp
Like we have done
before in Granger causality discussion, we can summarize the causality between
the variables lrconsump, lrgdp and lrgrossinv by mapping it to show the clearly causality direction
between them.
