Spurious regressions If $\{ X_t \}$ is a $I(1)$ series and $\{ Y_t \}$ is a $I(0)$ series, would it cause "spurious regression" when regressing like $Y_i=\beta_0+\beta_1 X_i+u_i$? Thanks!
That is illegal because an I(1) series wanders ( doesn't have a constant mean )  and an I(0) series doesn't so they can't be set equal. In order to obtain a valid time series regression, the order on the LHS has to be the same as the order on the RHS. 

Note that if both sides are I(1), then a time series regression is okay but only if what is on the RHS is cointegrated with what is on the left. If they aren't cointegrated and both are I(1), then that's what is called a spurious regression. ( i.e: a non-sensical regression ).