We have perform the VAR analysis in here before. After fitting a VAR, we may want to
know whether one variable “Granger-causes” another (Granger, 1969).
The variable \(x\) is said to
Granger-cause a variable \(y\) if, given the past value of \(y\), past value of \(x\) are useful for predicting \(x\).
The common method for testing Granger causality is to
regress \(y\) on
its own lagged values and on lagged values of \(x\) and test the null hypothesis that the
estimated coefficients on the lagged values of \(x\) are jointly zero.
Failure to reject the null hypothesis is equivalent to
failing to reject the hypothesis that \(x\) does not Granger-cause \(y\).
The test involves estimating the following pair of
regression:
\({{y}_{t}}=\sum\limits_{i=1}^{m}{{{\alpha
}_{i}}{{x}_{t-i}}+\sum\limits_{i=1}^{n}{{{\beta
}_{i}}{{y}_{t-i}}+{{u}_{1t}}}}\) (1)
\({{x}_{t}}=\sum\limits_{i=1}^{p}{{{\gamma
}_{i}}{{x}_{t-i}}+\sum\limits_{i=1}^{q}{{{\delta }_{i}}{{y}_{t-i}}+{{u}_{2t}}}}\) (2)
where its assumed that the disturbance \({{u}_{it}}\) and \({{u}_{2t}}\) are
uncorrelated, \(E\left( {{u}_{1t}},{{u}_{2t}}
\right)=0\). Since we have two variables, we are dealing with bilateral
causality.
Eq (1) postulates that current value of \({{y}_{t}}\) is related to past value of itself as that of \({{x}_{t}}\) , and Eq(1)
postulates similar behavior for \({{x}_{t}}\) .
Note that, these regression can be cast in difference
form, \(\Delta {{y}_{t}}\)
and \(\Delta {{x}_{t}}\)
.
We now can distinguished four cases;
1. Unidirectional causality from \({{x}_{t}}\to {{y}_{t}}\) is
indicated if the estimated lagged \({{x}_{t}}\) in Eq(1) are statistically different from zero as
a group \(\left(
\sum{{{\alpha }_{i}}\ne 0} \right)\) and set of estimated coefficients
on lagged \({{y}_{t}}\) in Eq(2) is not statistically
different from zero, \(\left(
\sum{{{\delta }_{i}}=0} \right)\).
2. Unidirectional causality from \({{y}_{t}}\to {{x}_{t}}\) is indicated if the estimated lagged
in Eq(2) are statistically different
from zero as a group \(\left(
\sum{{{\delta }_{i}}\ne 0} \right)\) and set of estimated coefficients
on lagged \({{x}_{t}}\)
in Eq(1) is not statistically different from zero, \(\left( \sum{{{\alpha }_{i}}=0} \right)\).
3. Feedback or bilateral causality, \({{y}_{t}}\leftrightarrow
{{x}_{t}}\) is suggest when set
of \({{x}_{t}}\) and \({{y}_{t}}\) coefficients are statistically significant
different from zero in both regression; \(\left( \sum{{{\alpha }_{i}}\ne 0} \right)\),\(\left(
\sum{{{\delta }_{i}}\ne 0} \right)\).
4. Independence, is suggest when set of \({{x}_{t}}\) and \({{y}_{t}}\) coefficients are not statistically significant in both the regression; \(\left( \sum{{{\alpha
}_{i}}=0} \right)\),\(\left(
\sum{{{\delta }_{i}}=0} \right)\).
Causality
test using Stata
In Stata, for each equations and each endogenous
variable as the dependent variable in that equations, vargranger computes and
report Wald tests that the coefficients on all lags of an endogenous variable
are jointly zero.
For each equation
on a VAR, vargranger test the
hypothesis that each of the other endogenous variables does not Granger-cause
the dependent variable in those equations.
For test the causality,
we will use again Data09.
In VAR
analysis before, we have concluded that the appropriate lag-length is
seven lag. That means, for our analysis non-causality test we will deal only
seven lag-length.
To perform
the Granger causality test, we need first to run the VAR and then perform the Granger
test;
quietly var lrgrossinv lrconsump lrgdp, lags(1/7) dfk small
vargranger
Same as in VAR
output, equations are distinguished by their dependent variable. For each
equation, vargranger test for the Granger causality of each variable
individually, then tests for the Granger causality of all added variable
jointly.
Lets we consider
the Granger causality test for real household consumption expenditure (=lrconsump). The row with “lrgdp Excluded” is refer for test the
null hypothesis that all coefficient on lags of the real gross domestic product
(=lrgdp) equation are equal to zero,
aginst the alternative that at least one is not equal to zero. The
-value of 0.14 does not fall below the typical
statistical significance threshold of 0.05; hence, we fail to reject the null
hypothesis that lags of real gross domestic product do not affect the real
household consumption expenditure. With this model and these data, the results
show that the real gross domestic product does not Granger-cause to real
household consumption expenditure.
By contrast, in
the real gross domestic product equation, lags of real household consumption expenditure
are statistically significant and we can tell that the real household
consumption expenditure is Granger-cause to the real gross domestic product. That
means, the Granger causality test between the real household consumption
expenditure and real gross domestic product show only one-way direction;
.
The “ALL excluded” row for each equation
excludes all lags that are not the autocorrelation coefficient in an equation;
it is a joint test for the significance of all lags of all other variables in
that equation. It maybe considered a test between a purely autoregressive
specification (null) against the VAR specification for that equation
(alternate).
Beside using the vargranger command to perform the Granger causality test,
we also can replicate the results by running OLS on each equation “manually”
and using test with the appropriate null hypothesis;
quietly reg lrconsump L(1/7).lrconsump L(1/7).lrgrossinv L(1/7).lrgdp
test L1.lrgrossinv=L2.lrgrossinv=L3.lrgrossinv=L4.lrgrossinv=L5.lrgrossinv=L6.lrgrossinv=L7.lrgrossinv=0
test L1.lrgdp=L2.lrgdp=L3.lrgdp=L4.lrgdp=L5.lrgdp=L6.lrgdp=L7.lrgdp=0
The results of a “manual”
Granger causality test match the results from vargranger output.
We can summarize
the Granger causality test between the variables
lrconsump, lrgdp and lrgrossinv by
mapping it to show the clearly causality direction between them.
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