Granger Causality Test

Granger causality test - a procedure for checking a causal (not causal (!) Relationship (“ Granger causality ”) between time series . The idea of the test is that the values (changes) of the time series $x_{t}$ ${\ displaystyle x_ {t}}$ $x_t$ causing a change in time series $y_{t}$ ${\ displaystyle y_ {t}}$ $y_ {t}$ , should precede changes in this time series, and in addition, should make a significant contribution to the forecast of its values. If each of the variables makes a significant contribution to the forecast of the other, then perhaps there is some other variable that affects both.

In the Granger test, two null hypotheses are successively tested: “x is not the cause of y according to Granger” and “y is not the cause of x according to Granger”. To test these hypotheses, two regressions are constructed: in each regression, the dependent variable is one of the variables checked for causality, and the lags of both variables act as regressors (in fact, this is vector autoregression ).

$y_{t}=a_{0}+a_{1}y_{t-1}+...+a_{p}y_{t-p}+b_{1}x_{t-1}+...+b_{p}x_{t-p}+\varepsilon _{t}$ ${\ displaystyle y_ {t} = a_ {0} + a_ {1} y_ {t-1} + ... + a_ {p} y_ {tp} + b_ {1} x_ {t-1} + .. . + b_ {p} x_ {tp} + \ varepsilon _ {t}}$ ${\ displaystyle y_ {t} = a_ {0} + a_ {1} y_ {t-1} + ... + a_ {p} y_ {tp} + b_ {1} x_ {t-1} + .. . + b_ {p} x_ {tp} + \ varepsilon _ {t}}$

$x_{t}=c_{0}+c_{1}x_{t-1}+...+c_{p}x_{t-p}+d_{1}y_{t-1}+...+d_{p}y_{t-p}+u_{t}$ ${\ displaystyle x_ {t} = c_ {0} + c_ {1} x_ {t-1} + ... + c_ {p} x_ {tp} + d_ {1} y_ {t-1} + .. . + d_ {p} y_ {tp} + u_ {t}}$ ${\ displaystyle x_ {t} = c_ {0} + c_ {1} x_ {t-1} + ... + c_ {p} x_ {tp} + d_ {1} y_ {t-1} + .. . + d_ {p} y_ {tp} + u_ {t}}$

For each regression, the null hypothesis is that the coefficients for the lags of the second variable are simultaneously zero.

$H_{0}^{1}:~b_{1}=...=b_{p}=0$ ${\ displaystyle H_ {0} ^ {1}: ~ b_ {1} = ... = b_ {p} = 0}$ ${\ displaystyle H_ {0} ^ {1}: ~ b_ {1} = ... = b_ {p} = 0}$

$H_{0}^{2}:~d_{1}=...=d_{p}=0$ ${\ displaystyle H_ {0} ^ {2}: ~ d_ {1} = ... = d_ {p} = 0}$ ${\ displaystyle H_ {0} ^ {2}: ~ d_ {1} = ... = d_ {p} = 0}$

These hypotheses can be checked, for example, using the F-test or LM-test . It should be noted that the test results may depend on the number of lags used in the regressions.

Links

Tutorial on Granger causality analysis of EEG data using Matlab
Anil Seth (2007) Granger causality. Scholarpedia, 2 (7): 1667
Magnus Y.R. Katyshev P.K. Peresetsky A.A. Econometrics. Beginner course . - 2004.
Granger Bidirectional Causality Test Online Calculator

Granger Causality Test

See also

Links

More articles: