Granger-causality tests

Autoregressive model
X t = c + φX t-1 + ε t where ε t is white noise with zero
mean and variance σ 2
○ If φ = 1 then X t exhibits a unit root and can be
considered a random walk.
○ For AR of order 1, processes where its parameters φ,
are stationary if φ < 1.
• Let μ be the mean. Then E(X t ) = E(c) + φ E(φX t-1 )
+ E(ε t ) ; so μ = c + φ μ+0
• So var(X t ) = E(X t 2 ) – μ 2 = σ 2 /(1- φ 2 )
• Autocovariance is given by E(X t+n | X t ) - μ 2 =(σ 2 /(1-
φ 2 )) φ |n|