LR, W, and LM tests

We give an account of the three classical tests in the context of the general linear regression model introduced in (25.1) above. We take R to be a (p x K) known matrix of rank p < K. Consider three estimators of в under the exact linear restric­tions, under inequality restrictions, and when в £ Rp (no restrictions). Denote these by _, U, and S, respectively. We note that S = (X’Q-1X )-1(X’Q-1y) is the ML (GLS) estimator here. Let (X’Q-1X) = G, and consider the following optimization programs:

max – (y – Хв)’0 1(y – Хв), subject to Re > r, (25.13)

and the same objective function but with equality restrictions. Denote by X and ‘ the Lagrange multipliers, respectively, of these two programs (conventionally, X = 0 for S). Then:

U = S + G-1R’X/2, and _ = S + G-1R,’/2. (25.14)

See GHM (1982). Employing these relations it is straightforward to show that the following three classical tests are identical:

LR = -2 log LR = 2($ – %), (25.15)

where L and $ are the logarithms of the maxima of the respective likelihood functions;

<^LM = min (X – X)’RG :R'(X – ‘)/4, subject to X < 0 (25.16)

is the Kuhn-Tucker/Lagrange multiplier (LM) test computed at X, and,

is the Wald test. In order to utilize the classical results stated above for problems in (25.3), or (25.8), it is customary to note that the LR test in (25.15) above is identical to the LR test of the following problem:

S = p + v

Rp > r

v ~ N(0, G-1) (25.18)

For this problem £LR is the optimum of the following QP problem:

max – (p – S)’G(P – S) + (p – S)’G(P – S) subject to Rp > r. (25.19)

This is identical to the one-sided multivariate problem in (25.3). It also suggests that the context for applications can be very general indeed. All that is needed is normally distributed estimators, S, which are then projected on to the cone defined by the inequality restrictions in order to obtain the restricted estimator p.

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