# Invariance of the constrained least squares quadratic form

1. Jan 31, 2010

### maverick280857

Hi everyone,

I'm trying to show that the quantity

$$(L'\hat{\beta}-\eta)'(L'(X'X)^{-1}L)^{-1}(L'\hat{\beta}-\eta)$$

where $\beta$ is a (p+1)x1 vector, $L'$ is a q x (p+1) vector, and $\eta$ is a q x 1 vector, is invariant under transformations of L which are row and column operations.

This quantity arises in the test statistic for the constrained (restricted) least squares process, where the objective is to minimize

$$S(\beta,\Lambda) = (Y-X\beta)'(Y-X\beta) + 2\Lambda'(L'\beta-\eta)$$

where $\Lambda$ is treated as a Lagrangian multiplier and $L'\beta=\eta$ is the restriction to be imposed on the unrestricted objective function $S(\beta,0)$.

The point is that there is no unique choice of the matrix L', and this should not affect the form of the test statistic.

Here, $\beta' = (\beta_0, \beta_1, \ldots, \beta_p)$.

For instance if one wants to impose

$$\beta_1 = \beta_2 = 0$$

two possible choices of the L' matrix are

$$\left(\begin{array}{ccc}0 & 1 & -1 \ldots\\0 & 0 & 1\end{array}\right)$$

and

$$\left(\begin{array}{ccc}0 & 1 & 0 \ldots\\0 & 0 & 1\end{array}\right)$$

with $\eta = 0$ in both cases.

How do I show that for some such nonsingular transformation T which gives L* = TL, the quantity $(L'\hat{\beta}-\eta)'(L'(X'X)^{-1}L)^{-1}(L'\hat{\beta}-\eta)$ is invariant?