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Invariance of the constrained least squares quadratic form

  1. Jan 31, 2010 #1
    Hi everyone,

    I'm trying to show that the quantity


    where [itex]\beta[/itex] is a (p+1)x1 vector, [itex]L'[/itex] is a q x (p+1) vector, and [itex]\eta[/itex] 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

    [tex]S(\beta,\Lambda) = (Y-X\beta)'(Y-X\beta) + 2\Lambda'(L'\beta-\eta)[/tex]

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

    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, [itex]\beta' = (\beta_0, \beta_1, \ldots, \beta_p)[/itex].

    For instance if one wants to impose

    [tex]\beta_1 = \beta_2 = 0 [/tex]

    two possible choices of the L' matrix are

    [tex]\left(\begin{array}{ccc}0 & 1 & -1 \ldots\\0 & 0 & 1\end{array}\right)[/tex]


    [tex]\left(\begin{array}{ccc}0 & 1 & 0 \ldots\\0 & 0 & 1\end{array}\right)[/tex]

    with [itex]\eta = 0[/itex] in both cases.

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

    Thanks in advance.
  2. jcsd
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