Spurious Regression with non stationary time-series

Mike_5

New Member
Hi everyone,
I'd like to have a confirmation on the correctness of the following interpretation:
Let say that we want to run a very simple regression like the following one:
$x_t&space;=&space;c&space;+&space;\beta&space;y_t&space;+&space;u_t$

We are regressing two I(1) series since x and y are assumed to be both described by a random walk process. The errors of these 2 processes are uncorrelated
Granger and Newbold showed that in this case, we have an excess of rejection of the null hypothesis. This comes from the fact that the t-stat of the estimated
$\beta&space;$
does not converge to a standard normal and the empirical distribution of the estimated
$\beta&space;$
is no concentrated around its true population value. The point is: since the variance is time-dependent because the two time series are I(1), as the sample size increases the empirical distribution of the estimated
$\beta&space;$
becomes fatter and fatter and thus we shall take into account the fact that the critical values are moving to the left. The theoretical critical values of the t-stat are no longer valid.
My doubt is: Granger and Newbold estimated
$\beta&space;$
equal to zero on average. The true population values under this experiment where:
$\beta&space;=&space;0.5&space;;&space;c=&space;10&space;$

But why
$\beta&space;$
is equal to zero on average?