grangerTest#

Purpose#

Test whether one variable Granger-causes another in a fitted VAR model.

Format#

g = grangerTest(result, cause, effect)#

Parameters:

result (struct) – an instance of a varResult structure from varFit().
cause (scalar) – index (1 to m) of the potential cause variable.
effect (scalar) – index (1 to m) of the effect variable.
quiet (scalar) – Optional keyword, set to 1 to suppress output. Default = 0.

Returns:

g (struct) –

An instance of a grangerResult structure containing:

g.f_stat	Scalar, F-statistic.
g.p_value	Scalar, p-value.
g.df1	Scalar, numerator degrees of freedom (number of restricted lags = p).
g.df2	Scalar, denominator degrees of freedom.
g.cause_name	String, name of the cause variable.
g.effect_name	String, name of the effect variable.

Examples#

Single Pair#

new;
library timeseries;

data = loadd(getGAUSSHome("pkgs/timeseries/examples/macro.dat"));
result = varFit(data, 4);

// Does FFR Granger-cause GDP?
g = grangerTest(result, 3, 1);

print g.cause_name "Granger-causes" g.effect_name "?";
print "F =" g.f_stat "p =" g.p_value;

All Pairs#

new;
library timeseries;

data = loadd(getGAUSSHome("pkgs/timeseries/examples/macro.dat"));
result = varFit(data, 4);

for i (1, 3, 1);
    for j (1, 3, 1);
        if i /= j;
            g = grangerTest(result, i, j, quiet=1);
            print g.cause_name "->" g.effect_name ": p=" g.p_value;
        endif;
    endfor;
endfor;

Remarks#

Tests H0: all p lags of the cause variable are jointly zero in the effect variable’s equation. Uses an F-test with p numerator and (T - mp - 1) denominator degrees of freedom.

Granger causality is a predictive concept, not a structural one. “X Granger-causes Y” means X contains information useful for forecasting Y beyond what Y’s own lags provide. It does not imply X structurally causes Y.

Model#

The Granger causality test (Granger 1969) tests the null hypothesis that all \(p\) lags of the cause variable are jointly zero in the effect variable’s equation:

\[H_0: B_{\text{cause},1} = B_{\text{cause},2} = \cdots = B_{\text{cause},p} = 0\]

in the equation for the effect variable. The F-statistic is:

\[F = \frac{(\text{RSS}_r - \text{RSS}_u) / p}{\text{RSS}_u / (T - mp - 1)} \sim F(p, T - mp - 1)\]

where \(\text{RSS}_r\) and \(\text{RSS}_u\) are residual sums of squares from the restricted and unrestricted regressions.

Troubleshooting#

Significant Granger causality in both directions: This is possible and common — it means both variables contain predictive information about each other. This does not imply feedback causation in a structural sense.

Result depends on lag order: Granger causality tests are sensitive to p. Use varLagSelect() to choose p before testing.

Verification#

Granger causality F-statistics and p-values verified against R vars::causality() at \(10^{-6}\) tolerance on a 2-variable VAR(1) with known DGP. Both directions tested (Y1→Y2 and Y2→Y1).

See gausslib-var/tests/r_benchmark.rs.

References#

Granger, C.W.J. (1969). “Investigating causal relations by econometric models and cross-spectral methods.” Econometrica, 37(3), 424-438.
Lutkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer. Section 3.6.

Library#

timeseries

Source#

granger.src