Causality Analysis

Description

Computes the test statistics for Granger- and Instantaneous causality for a VAR(p).

Usage

causality(x, cause = NULL, vcov.=NULL, boot=FALSE, boot.runs=100)

Arguments

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}] = ∑_{i=1}^p [\bold{α}_{11, i}' , \bold{α}_{12, i}' | \bold{α}_{21, i}' , \bold{α}_{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{α}_{21, i} = 0 for i = 1, 2, …, p. The alternative is: \exists \; \bold{α}_{21,i} \ne 0 for i = 1, 2, …, 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{σ} = 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{σ} = vech(Σ_u). The Wald statistic is defined as:

λ_W = T \tilde{\bold{σ}}'C'[2 C D_{K}^{+}(\tilde{Σ}_u \otimes \tilde{Σ}_u) D_{K}^{+'} C']^{-1} C \tilde{\bold{σ}} \quad ,

hereby assigning the Moore-Penrose inverse of the duplication matrix D_K with D_{K}^{+} and \tilde{Σ}_u = \frac{1}{T}∑_{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 λ_W is asymptotically distributed as χ^2(N).

Fot the Granger causality test, a robust covariance-matrix estimator can be used in case of heteroskedasticity through argument vcov. It can be either a pre-computed matrix or a function for extracting the covariance matrix. See vcovHC from package sandwich for further details.

A wild bootstrap computation (imposing the restricted model as null) of the p values is available through argument boot and boot.runs following Hafner and Herwartz (2009).

Value

A list with elements of class ‘htest’:

Note

The Moore-Penrose inverse matrix is computed with the function ginv contained in the package ‘MASS’.
The Granger-causality test is problematic if some of the variables are nonstationary. In that case the usual asymptotic distribution of the test statistic may not be valid under the null hypothesis.

Author(s)

Bernhard Pfaff

References

Granger, C. W. J. (1969), Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37: 424-438.

Hafner, C. M. and Herwartz, H. (2009) Testing for linear vector autoregressive dynamics under multivariate generalized autoregressive heteroskedasticity, Statistica Neerlandica, 63: 294-323

Hamilton, J. (1994), Time Series Analysis, Princeton University Press, Princeton.

Lütkepohl, 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.

Zeileis, A. (2006) Object-Oriented Computation of Sandwich Estimators Journal of Statistical Software, 16, 1-16

See Also

VAR

Examples

data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
causality(var.2c, cause = "e")

#use a robust HC variance-covariance matrix for the Granger test:
causality(var.2c, cause = "e", vcov.=vcovHC(var.2c))

#use a wild-bootstrap procedure to for the Granger test
## Not run: causality(var.2c, cause = "e", boot=TRUE, boot.runs=1000)