Suppose the columns of a rank 4 design matrix
X = [X_{1}, X_{2}, X_{3}, X_{4}] come in three groups
Group 1: \{X_{1}\},
Group 2: \{X_{2}, X_{3}\} and
Group 3: \{X_{4}\} so that X^T_{i}X_{j} = 0 if i and j are indicies from different groups.
Next consider 4 models, with design matricies:
Model 1: X_{(1)} = [X_{1}]
Model 2: X_{(2)} = [X_{1}, X_{2}, X_{3}]
Model 3: X_{(3)} = [X_{1}, X_{2}, X_{3}, X_{4}]
Model 4: X_{(4)} = [X_{1}, X_{4}]

Now then the question asks to show that SS_{2} - SS_{3} = SS_{1} - SS_{4} where SS_{i} is the residual sum of squares (SS error) for the ith model.

SS_{error} = Y^T Y - \hat\theta X^T Y is the only formula for SS error that I know, which I believe does not apply in this situation. So it means that I'm absolutely stuck! I think I'm looking at this the wrong way to be honest. Where should I look into? Any hints?