The error sum of squares is defined as:

$$ SSe(\beta)=(y-X\beta)'(y-X\beta)\tag{1}$$

I want to show that for the OLS estimator $\hat\beta$,


I know that $SSres=SSe(\hat\beta)$, but I don't understand how to reach equation $(2)$ from $(1)$.

  • 1
    $\begingroup$ Can you check your question? In line 2, you end up with $(\beta-\beta)$ twice, which is identically zero. I think I se what you want to do, but can you clarify? $\endgroup$ – eSurfsnake Jun 1 '19 at 23:05
  • $\begingroup$ It looks right to me, should be (beta - beta hat) $\endgroup$ – Oscar Jonsson Jun 3 '19 at 9:07
  • $\begingroup$ math.meta.stackexchange.com/questions/5020/…. $\endgroup$ – StubbornAtom Jun 20 '19 at 15:37

You can rewrite $y$ as $y=X\hat{\beta}+\epsilon$. Note that, here $\hat{\beta}$ represents estimated parameters and $\epsilon$ is the residual term.

Then, Eq. (1) becomes: \begin{equation} SSE(\beta)=(X\hat{\beta}+\epsilon-X\beta)^T(X\hat{\beta}+\epsilon-X\beta)\\ =(X(\hat{\beta}-\beta)+\epsilon)^T (X(\hat{\beta}-\beta)+\epsilon)\\ =((\hat{\beta}-\beta)^TX^T+\epsilon^T)(X(\hat{\beta}-\beta)+\epsilon)\\ =(\hat{\beta}-\beta)^TX^TX(\hat{\beta}-\beta)+\epsilon^T\epsilon\\ =(\hat{\beta}-\beta)^TX^TX(\hat{\beta}-\beta)+SSE(\hat{\beta}) \end{equation}


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