Standard Error for bivariable zscore normalized regression

Can someone explain why I get exactly same value of satndard error when I use fitlm for two variables x1 and x2 that are zscore normalized?

x=[zscore(x1) zscore(x2)]

fitlm(x,y)

x=0.5519 -1.5191

0.8152 0.9382

-1.4367 0.8976

0.8371 -0.5285

0.0246 0.4234

-1.5218 -1.5667

-0.9986 -0.7207

-0.2226 0.7723

0.9647 0.3990

0.9861 0.9046

y=0.6557

0.0357

0.8491

0.9340

0.6787

0.7577

0.7431

0.3922

0.6555

0.1712

Answer 1

[Ive J]

And why is it surprising to you?

You can read about how coefficients and their SE are calculated in linear regression. Standardized IVs have 0 mean and variance of 1. One component of SE(β) is inv(X*X'), where X is your design matrix, more precisely:

se = sqrt(model.MSE.*diag(inv(X*X'))); % model: object output of fitlm 

So: . We also know IVs have variance of 1, so , so we can simplify this to , which only depdends on IVs length.