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When stating hypothesis in matrix formulation, as H0: $C\hat\beta=h$, $SSE(reduced)-SSE(full)$ can be expressed as:

$$(C\hat \beta - h)'(C(X'X)^{-1}C)^{-1}(C\hat \beta - h) $$

How is this result derived? It looks very weird considering it has no Y in it.

Here is my failed attempt:

$$SSE(red)-SSE(full) = (Y'Y-\hat\beta'_rX'Y)-(Y'Y-\hat\beta'_fX'Y) $$ $$=(\hat\beta'_f-\hat\beta'_r)X'Y$$

Explanation of the symbols:

$\hat\beta$ = the vector for the parameter estimates for full model. (I also denote this as index f and reduced as r).

$C$= the matrix that constrains the estimates.

$h$= vector of the values the estimates are constrained to.

$X$= the matrix of the independent variables (values in rows, variables in columns).

$SSE$= Sum square error (red = for reduced model, full = for full model).

Dole
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  • The answers to this question will depend on the meanings ascribed to the symbols, so you will need to explain what they each mean. – whuber May 19 '17 at 12:48
  • @whuber Is it clear now? – Dole May 19 '17 at 13:30
  • Yes, thank you. But I do not understand why you write "it has no $Y$ in it." Isn't it the case that $\hat\beta$ is a linear function of $Y$? – whuber May 19 '17 at 13:36
  • @whuber It has Y in it in the sense that the parameter estimates are a function of Y indeed. But not explicitly like in the equation where I am starting from. – Dole May 19 '17 at 13:37
  • That's precisely why you're struggling with this: make it explicit and go on from there. – whuber May 19 '17 at 13:49

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