Granger causality

The Granger causality test is a statistical hypothesis test for determining whether one time series is useful in forecasting another, first proposed in 1969. Ordinarily, regressions reflect 'mere' correlations, but Clive Granger argued that causality in economics could be tested for by measuring the ability to predict the future values of a time series using prior values of another time series. Since the question of 'true causality' is deeply philosophical, and because of the post hoc ergo propter hoc fallacy of assuming that one thing preceding another can be used as a proof of causation, econometricians assert that the Granger test finds only 'predictive causality'. The Granger causality test is a statistical hypothesis test for determining whether one time series is useful in forecasting another, first proposed in 1969. Ordinarily, regressions reflect 'mere' correlations, but Clive Granger argued that causality in economics could be tested for by measuring the ability to predict the future values of a time series using prior values of another time series. Since the question of 'true causality' is deeply philosophical, and because of the post hoc ergo propter hoc fallacy of assuming that one thing preceding another can be used as a proof of causation, econometricians assert that the Granger test finds only 'predictive 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. Granger also stressed that some studies using 'Granger causality' testing in areas outside economics reached 'ridiculous' conclusions. 'Of course, many ridiculous papers appeared', he said in his Nobel lecture. However, it remains a popular method for causality analysis in time series due to its computational simplicity. The original definition of Granger causality does not account for latent confounding effects and does not capture instantaneous and non-linear causal relationships, though several extensions have been proposed to address these issues. We say that a variable X that evolves over time Granger-causes another evolving variable Y if predictions of the value of Y based on its own past values and on the past values of X are better than predictions of Y based only on its own past values. Granger defined the causality relationship based on two principles: Given these two assumptions about causality, Granger proposed to test the following hypothesis for identification of a causal effect of X {displaystyle X} on Y {displaystyle Y} : where P {displaystyle mathbb {P} } refers to probability, A {displaystyle A} is an arbitrary non-empty set, and I ( t ) {displaystyle {mathcal {I}}(t)} and I − X ( t ) {displaystyle {mathcal {I}}_{-X}(t)} respectively denote the information available as of time t {displaystyle t} in the entire universe, and that in the modified universe in which X {displaystyle X} is excluded. If the above hypothesis is accepted, we say that X {displaystyle X} Granger-causes Y {displaystyle Y} . If a time series is a stationary process, the test is performed using the level values of two (or more) variables. If the variables are non-stationary, then the test is done using first (or higher) differences. The number of lags to be included is usually chosen using an information criterion, such as the Akaike information criterion or the Schwarz information criterion. Any particular lagged value of one of the variables is retained in the regression if (1) it is significant according to a t-test, and (2) it and the other lagged values of the variable jointly add explanatory power to the model according to an F-test. Then the null hypothesis of no Granger causality is not rejected if and only if no lagged values of an explanatory variable have been retained in the regression.