Computes the test statistics for Granger- and Instantaneous causality
for a VAR(p).
Usage
causality(x, cause = NULL)
Arguments
x
Object of class varest; generated by
VAR().
cause
A character vector of the cause variable(s). If not set,
then the variable in the first column of x$y is used as cause
variable and a warning is printed.
Value
A list with elements of class htest:
GrangerThe result of the Granger-causality test.
InstantThe result of the instantaneous causality test.
encoding
latin1
concept
VAR
Vector autoregressive model
Causality
Granger Causality
Instantaneous Causality
Details
Two causality tests are implemented. The first is a F-type
Granger-causality test and the second is a Wald-type test that is
characterized by testing for nonzero correlation between the error
processes of the cause and effect variables. For both tests the vector
of endogenous variables $\bold{y}_t$ is split into two subvectors
$\bold{y}_{1t}$ and $\bold{y}_{2t}$ with dimensions $(K_1
\times 1)$ and $(K_2 \times 1)$ with $K = K_1 + K_2$.
For the rewritten VAR(p):
$$[\bold{y}_{1t} , \bold{y}_{2t}] = \sum_{i=1}^p [\bold{\alpha}_{11, i}' ,
\bold{\alpha}_{12, i}' | \bold{\alpha}_{21, i}' , \bold{\alpha}_{22,
i}'][\bold{y}_{1,t-i}, \bold{y}_{2, t-i}] + CD_t + [\bold{u}_{1t},
\bold{u}_{2t}] \quad ,$$
the null hypothesis that the subvector $\bold{y}_{1t}$ does not
Granger-cause $\bold{y}_{2t}$, is defined as
$\bold{\alpha}_{21, i} = 0$ for $i = 1, 2, \ldots, p$. The
alternative is: $\exists \; \bold{\alpha}_{21,i} \ne 0$ for $i =
1, 2, \ldots, p$. The test statistic is distributed as $F(p K_1
K_2, KT - n^*)$, with $n^*$ equal to the total number of
parameters in the above VAR(p) (including deterministic
regressors).
The null hypothesis for instantaneous causality is defined as:
$H_0: C \bold{\sigma} = 0$, where $C$ is a $(N \times K(K
+ 1)/2)$ matrix of rank $N$ selecting the relevant co-variances of
$\bold{u}_{1t}$ and $\bold{u}_{2t}$; $\bold{\sigma} =
vech(\Sigma_u)$. The Wald statistic is defined as:
$$\lambda_W = T \tilde{\bold{\sigma}}'C'[2 C D_{K}^{+}(\tilde{\Sigma}_u
\otimes \tilde{\Sigma}_u) D_{K}^{+'} C']^{-1} C
\tilde{\bold{\sigma}} \quad ,$$
hereby assigning the Moore-Penrose inverse of the duplication matrix
$D_K$ with $D_{K}^{+}$ and $\tilde{\Sigma}_u =
\frac{1}{T}\sum_{t=1}^T \hat{\bold{u}}_t \hat{\bold{u}}_t'$. The
duplication matrix $D_K$ has dimension $(K^2 \times
\frac{1}{2}K(K + 1))$ and is defined such that for any symmetric
$(K \times K)$ matrix A, $vec(A) = D_K vech(A)$ holds. The
test statistic $\lambda_W$ is asymptotically distributed as
$\chi^2(N)$.
References
Granger, C. W. J. (1969), Investigating causal relations by
econometric models and cross-spectral methods, Econometrica,
37: 424-438.
Hamilton, J. (1994), Time Series Analysis, Princeton
University Press, Princeton.
L�tkepohl, H. (2006), New Introduction to Multiple Time Series
Analysis, Springer, New York.
Venables, W. N. and B. D. Ripley (2002), Modern Applied
Statistics with S, 4th edition, Springer, New York.