Earlier this year, I reviewed one definition of causality: the Bradford Hill criteria. Now I move from a medical definition causality to one developed by an economist (in fact, an economist from my alma matter). Granger causality basically identifies a variable as causal if: (i) it occurs the outcome of interest and (ii) including past values of the variable improves the prediction of the outcome of interest in a statistically significant way. Scholarpedia provides a useful example:

The basic “Granger Causality” definition is quite simple. Suppose that we have three terms,

X,_{t}Y, and_{t}W, and that we first attempt to forecast_{t}X_{t+1}using past terms ofXand_{t}W. We then try to forecast_{t}Xusing past terms of_{t+1}X,_{t+1}Y, and_{t}W. If the second forecast is found to be more successful, according to standard cost functions, then the past of_{t}Yappears to contain information helping in forecastingXthat is not in past_{t+1 }Xor_{t }W. In particular,_{t}Wcould be a vector of possible explanatory variables. Thus,_{t }Ywould “Granger cause”_{t }Xif (a)_{t+1}Yoccurs before_{t}X; and (b) it contains information useful in forecasting_{t+1}Xthat is not found in a group of other appropriate variables._{t+1}

Consider a linear example with two time series processes.

X_{1}(t) = Σ_{j=1}^{p}A_{11,j}X_{1}(t-j)+Σ_{j=1}^{p}A_{12,j}X_{2}(t-j)

X_{2}(t) = Σ_{j=1}^{p}A_{21,j}X_{1}(t-j)+Σ_{j}^{p}A_{22,j=1}X_{2}(t-j)

One can test for whether variable X_{2} Granger-causes X_{1} by testing whether the vector **A**_{12}=**0**, using an F-test. The F-test examines whether all coefficients in the vector are statistically significantly different from zero. The magnitude of the Granger-causality (a.k.a. G-causality) can be estimated as the logarithm of the corresponding F-statistic. To determine the appropriate number of lag terms in the time series, one could use the Akaike Information Criterion (AIC,) or Bayesian Information Criterion (BIC) to determine which value of *p* maximizes the model fit.

G-causality can be readily extended to the

nvariable case, wheren>2 , by estimating annvariable autoregressive model. In this case,X_{2}G-causesX_{1}if lagged observations ofX_{2}help predictX_{1}when lagged observations of all other variablesX_{3}…X_{N}are also taken into account.

from Dental Tips https://www.healthcare-economist.com/2019/01/13/causality-granger-causality/