= Orthogonal coefficient blocks in a centered normal linear model
{title2=$X_1^{\mathsf T}X_2=0$}
In a <normal linear model> whose predictor columns are orthogonal to the intercept, two coefficient blocks are <orthogonal statistical parameters> precisely when the cross-block <Fisher information> is zero, equivalently the displayed design cross-product vanishes. Then $X^{\mathsf T}X$ is block diagonal, so the corresponding <least-squares estimators> have zero <covariance> and, being jointly Gaussian, are <independent>. Their fitted subspaces are orthogonal and their extra sums of squares do not depend on which block is added first.
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