Be cautious when regressing out covariates

Be cautious when regressing out covariates.
We simulated three random timecourses X, Y and Z that are normally distributed to perform linear regression. Let Y to be the ROI timecorese
and X, Z to be the covariates. The regression coefficients and residual after two-factor linear regression are extremely same with that of two Unary linear regressions. The results are as bellow:
Y = -0.0163X+0.0459+ r1
r1 = 0.0118Z + 0.0009+ r2
r1 and r2 are residual.
Y = -0.0164X + 0.0118Z + 0.0468 + r3
r3 is residual.
r2 is extremely same with r3 (for details please see the attached file).

We then extracted timecoreses of rFIC (x=37, y=25, z=-4, radius==5mm), dACC (x=1, y=30, z=36, radius==5mm) and global trend (61*73*61 mask) from a resting-state single subject to perform linear regression. The regression coefficients and residual after two-factor linear regression are not same with that of two Unary linear regression. The results are as bellow:
dACC = 0.4069rFIC + 518.731 + r1
r1 = -0.1991Globaltrend + 156.9105 + r2
r1 and r2 are residual
dACC = 0.4701rFIC + -0.3303Globaltrend + 717.5604 + r3
r3 is residual.
r2 is different from r3 (for details please see the attached file).

The chosen of different model of linear regression may lead to different results when the chosen covariates are correlated to each other, especially the global trend signal. Be cautious when regressing out covariates. The group results of such conlusion will be reported later.

The attached excel file is the results of above. Line A to D is the result of simulation. Line F to I is the result of a resting-state fMRI data.