causality {vars}R Documentation

Causality Analysis

Description

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.

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 y_t is split into two subvectors y_{1t} and y_{2t} with dimensions (K_1 times 1) and (K_2 times 1) with K = K_1 + K_2.
For the rewritten VAR(p):

[y_{1t} , y_{2t}] = sum_{i=1}^p [α_{11, i}' , α_{12, i}' | α_{21, i}' , α_{22, i}'][y_{1,t-i}, y_{2, t-i}] + CD_t + [u_{1t}, u_{2t}] quad ,

the null hypothesis that the subvector y_{1t} does not Granger-cause y_{2t}, is defined as α_{21, i} = 0 for i = 1, 2, ..., p. The alternative is: exists ; 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 σ = 0, where C is a (N times K(K + 1)/2) matrix of rank N selecting the relevant co-variances of u_{1t} and u_{2t}; σ = vech(Σ_u). The Wald statistic is defined as:

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

hereby assigning the Moore-Penrose inverse of the duplication matrix D_K with D_{K}^{+} and tilde{Σ}_u = frac{1}{T}sum_{t=1}^T hat{u}_t hat{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 chi^2(N).

Value

A list with elements of class ‘htest’:

Granger The result of the Granger-causality test.
Instant The result of the instantaneous causality test.

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.

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.

See Also

VAR

Examples

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

[Package vars version 1.4-4 Index]