If X(t) and Y(t) are two time series
Xt granger causes Yt if the past values of Xt helps in predicting the future values of Yt
Yt is not just = f(Yt-1,Xt)
Xt granger causes Yt
Condition = cause happens prior to the effect Yt= f(Xt-1)
Cause has unique information about the future values of its effect. Extra effect.
Yt=a1Yt-1+b1Yt-1+Et
Mathematically ,
Yt = a0 + a1 + Yt-1
Yt = a0 + a1 + Yt-1 + a2 Xt-1
If a2 is significant, then Xt granger causes Yt.
T-test
H0: a2 = 0 , H1: a2 not equal to 0
Steps for Granger Causality test
Step 1
Yt = a0+a1Yt-1+a2Yt-2+…….+apYt-p
Step 2
Yt = a0+a1Yt-1+…+apYt-p+b1Xb-1+…+bpXt-p
Step 3
- T test for individual coefficients
- H0: b1=b2=….=bp=0
H1: At least one of them is non Zero
GC test helps improve prediction.
Limitations
Not necessarily true causality
If Xt affects Yt, through third variable Zt, then we may not be able to find GC