Granger causality

The Granger causality test is a statistical hypothesis test for determining whether one time series is useful in forecasting another. Ordinarily, regressions reflect "mere" correlations, but Clive Granger, who won a Nobel Prize in Economics, argued that there is an interpretation of a set of tests as revealing something about causality.

A time series X is said to Granger-cause Y if it can be shown, usually through a series of t-tests and F-tests on lagged values of X (and with lagged values of Y also included), that those X values provide statistically significant information about future values of Y.

This technique has been adapted to neuroscience, although its usefulness in fMRI is contested.

Method

The test for Granger causality works by first doing a regression of ΔY on lagged values of ΔY. (Here ΔY is the first difference of the variable Y — that is, Y minus its one-period-prior value. The regressions are performed in terms of ΔY rather than Y if Y is not stationary but ΔY is.) Once the set of significant lagged values for ΔY is found (via t-statistics or p-values), the regression is augmented with lagged levels of ΔX. Any particular lagged value of ΔX is retained in the regression if (1) it is significant according to a t-test, and (2) it and the other lagged values of ΔX jointly add explanatory power to the model according to an F-test. Then the null hypothesis of no Granger causality is retained if and only if no lagged values of ΔX have been retained in the regression.

The researcher is often looking for a clear story, such as X Granger-causes Y but not the other way around. In practice, however, it may be found that neither variable Granger-causes the other, or that each of the two variables Granger-causes the other.

Limitations

Despite its name, Granger causality is not sufficient to imply true causality. If both X and Y are driven by a common third process with different lags, one might still accept the alternative hypothesis of Granger causality. Yet, manipulation of one of the variables would not change the other. Indeed, the Granger test is designed to handle pairs of variables, and may produce misleading results when the true relationship involves three or more variables. A similar test involving more variables can be applied with vector autoregression.

Mathematical statement

Let y and x be stationary time series. To test the null hypothesis that x does not Granger-cause y, one first finds the proper lagged values of y to include in a univariate autoregression of y:

$y_t = a_0 + a_1y_{t-1} + a_2y_{t-2} + ... + a_my_{t-m} + residual_t.$

Here $y_{t-j}$ is retained in the regression if and only if it has a significant t-statistic; m is the greatest lag length for which the lagged dependent variable is significant.

Next, the autoregression is augmented by including lagged values of x:

$y_t = a_0 + a_1y_{t-1} + a_2y_{t-2} + ... a_my_{t-m} + b_px_{t-p} + ... + b_qx_{t-q} + residual_t.$

One retains in this regression all lagged values of x that are individually significant according to their t-statistics, provided that collectively they add explanatory power to the regression according to an F-test (whose null hypothesis is no explanatory power jointly added by the x's). In the notation of the above augmented regression, p is the shortest, and q is the longest, lag length for which the lagged value of x is significant.

The null hypothesis that x does not Granger-cause y is accepted if and only if no lagged values of x are retained in the regression.

Software implementation

Here is an example of the function grangertest() in the lmtest library of the R software:

Granger causality test

Model 1: fii ~ Lags(fii, 1:5) + Lags(rM, 1:5)
Model 2: fii ~ Lags(fii, 1:5)
  Res.Df  Df      F  Pr(>F)
1    629
2    634   5 2.5115 0.02896 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Granger causality test

Model 1: rM ~ Lags(rM, 1:5) + Lags(fii, 1:5)
Model 2: rM ~ Lags(rM, 1:5)
  Res.Df  Df      F Pr(>F)
1    629
2    634   5 1.1804 0.3172

The first Model 1 tests whether it is okay to remove lagged rM from the regression explaining FII using lagged FII. It is not (p = 0.02896). The second pair of Model 1 and Model 2 finds that it is possible to remove the lagged FII from the model explaining rM using lagged rM. From this, we conclude that rM Granger-causes FII but not the other way around.

Extensions

A method for Granger causality that is not sensitive to deviations from the assumption that the error term is normally distributed has been developed by Hacker and Hatemi-J (2006). This new method is especially useful in financial economics since many financial variables are non-normally distributed. Another application is proposed by Pedro Antonio Valdes-Sosa, José Miguel Bornot-Sánchez, Mayrín Vega Hernández, Lester Melie-García, Agustín Lage-Castellano and Erick Cavales- Rodríguez, who evaluated a special extension of Granger Causality using a Statistical Parametric Mapping (SPM) of influence field for the analysis of effective brain connectivity.

Further reading

• Tutorial on Granger causality analysis of EEG data using Matlab
• Anil Seth (2007) Granger causality. Scholarpedia, 2(7):1667
• Kleinberg, S. and Hripcsak, G. (2011) "A review of causal inference for biomedical informatics" J. Biomed Informatics

See also

• Econometrics