The Lagrange Multiplier Test

The restricted ML estimator в can also be obtained from the first-order condi­tions of the Lagrange function <(в, /г) = ln(Ln (в)) – §2г, where д є Kr is a vector of Lagrange multipliers. These first-order conditions are

д<(в, г)/двТ§=1д=г = d ln(L(§))/дв*§=§ = 0,

д<(§, г)/дв2г§=вг=г = дln(L(§))/дв2§=§ – д = 0 д<(§, г)/дгт§=e^=ff = §2 = °.

Hence,

/ 0 дln(L(§))/yn

V^. N д§T §=ff

Again, using the mean value theorem, we can expand this expression around the unrestricted ML estimator §, which then yields

-H 0 ) = – H-n(§ – 0) + op(1) ^d N(0, ЙAЙ),

Vn V

where the last conclusion in (8.64) follows from (8.59). Hence,

дТ я(2,2 ,)д = – V, iiT) й -1

nn

= йп(в – §)TЙл/п(в – в) + Op(1) ^d X,

where the last conclusion in (8.65) follows from (8.61). Replacing Й in expres­sion (8.65) by a consistent estimator on the basis of the restricted ML estimator §, for instance,

and partitioning Hi 1 conformably to (8.56) as

#(1,1) //(1,2)

//(2,1) //(2,2) we have

Theorem 8.8: (LM test) Under Assumptions 8.1-8.3, jlTH(2,2′)jl/n ^d хГ if

§2,0 = 0.