Granger Causality

Granger causality is the concept that if a variable causes another variable, it should help with the prediction of the latter variable. First, Granger causality is defined informally and formally. Next, the standard hypothesis tests used to evaluate the concept are presented. Third, the interpretations of these tests for Granger (non)causality are discussed. Then, some caveats and limitations are presented before turning to some common extensions and applications.

The statistical concept of Granger causality is related directly to the idea of causality and causal inference in the social sciences. In the (social) sciences, it is the case where two variables are causally related if

• the change in one variable temporally precedes that of the variable it causes,
• there is a nonzero correlation among the two (or more) variables, and
• the relationship among the variables is logically nonspurious.

Granger causality assesses the first two of these using a statistical test, leaving the third to the justification of the analyst. Granger causality analysis asks whether a variable helps in predicting another, which requires both temporal precedence and correlation. One of Clive Granger's [Page 1042]seminal time-series papers on this topic developed the statement of the concept and presented the most common statistical tests used to assess causality.

If a variable Z at Time t Granger causes a variable X at Time t or Zt Granger causes Xt then the following three statements are logically equivalent:

• 1.
  Zt helps predict Xt (i.e., it lowers the error of the prediction),
• 2.
  Zt is not exogenous of Xt,
• 3.
  Zt is linearly informative about future Xt.

The assumption that makes these three statements equivalent is that there is a linear relationship between Zt and Xt. Granger causality can be defined more generally and can also encompass nonlinear relationships. In this case, only the first two statements are relevant.

The formal definition of Granger causality is actually a statement of noncausality. A variable Zt is Granger noncausal for Xt if the past values of Zt do not help predict the current (or future) values of Xt over a prediction based on the past history of Xt alone. Under the assumption of linearity, then, this is the same as analyzing a dynamic regression of Xt on past values of Xt and Zt, or

where d is a constant, αis are the coefficients for the p lagged values of Xt, and βis are the coefficients for the p lagged values of Zt. If Zt Granger causes Xt, then some of the βi coefficients are different from zero. Note that by including the lagged values of Xt, the variable Zt must add more to the prediction of X than its own past (and thus help minimize the prediction error et) to deem it causal. Testing whether these are equal to zero is thus a test of Granger noncausality.

Statistical tests of Granger (non) causality depend on comparing the residuals of the model in the above equation with those of a model where Xt only depends on its past values, or where one assumes that βi = 0 for i = 1, …, p. Statistical tests for Granger causality are generally implemented as a comparison of the mean squared prediction error of the previous equation

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