statistical tests

http://lipas.uwasa.fi/~bepa/Multivariate.pdf --- page 18 says the below:

4.5 Exogeneity and Causality Suppose you got two time series xt and yt as in the introductory crude oil spot and futures returns example. We say x Granger causes y if (21) E(yt|yt−1,yt−2,...) 6= E(yt|yt−1,yt−2,...,xt−1,xt−2,...), that is, if we can improve the forecast for yt based upon its own history by additionally considering the history of xt. In the other case (22) E(yt|yt−1,yt−2,...) = E(yt|yt−1,yt−2,...,xt−1,xt−2,...), where adding history of xt does not improve the forecast for yt, we say that x does not Granger cause y, or x is exogeneous to y. Note that Granger causality is not the same as causality in the philosophical sense. Granger causality does not claim that x is the reason for y in the sense like, for example, y moves because x moves. It just says that x is helpful in forecasting y, which might happen for other reasons than direct causality. There might be, for example, a third series z which has a fast causal impact upon x and a slower causal impact upon y. Then we can use the reaction of x in order to forecast the reaction in y, such that x Granger causes y. 19 Testing for Exogeneity: The bivariate case Consider a bivariate VAR(p) model written out in scalar form as xt = φ1 + p X i=1 φ(i) 11xi,t−i + p X i=1 φ(i) 12yi,t−i + 1t,(23) yt = φ2 + p X i=1 φ(i) 21xi,t−i + p X i=1 φ(i) 22yi,t−i + 2t.(24) Then the test for Granger causality from x to y is an F-test for the joint significance of φ (1) 21 ,...,φ (p) 21 in the OLS regression (24). Similarly, the test for Granger causality from y to x is an F-test for the joint significance of φ(1) 12 ,...,φ (p) 12 in the OLS regression (23).