Canonical Model

Let H be an orthogonal matrix that diagonalizes the matrix X’X, that is, H’H = I and H’X’XH = Л, where Л is the diagonal matrix consisting of the characteristic roots ofX’X. Defining X* = XH and a = H’0, we can write Eq. (1.1.4) as

y = X*a + u. (2.2.4)

If a is the least squares estimator of a in model (2.2.4), we have

a – N(a, <72A-‘). (2.2.5)

Because the least squares estimator is a sufficient statistic for the vector of regression coefficients, the estimation of 0 in Model 1 with normality is equivalent to the estimation of a in model (2.2.5). We shall call (2.2.5) the canonical model; it is simpler Jo analyze than the original model. Because HH’ = I, the risk function E(0 — 0)'(0 — 0) in Model 1 is equivalent to the risk function E{a — — a) in model (2.2.5).

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