Testing the Information Matrix Equality with Robust Estimators



When ^ is equivariant, IF(X,Y; ^; Fθ) = σIF(u; ^5Φ). Hence V = σ 4C
with C a partitioned matrix with blocks

C11 = 2 Ek [vech( XX ' )(vech( XX ' )) ' ]

+(Bιι - 2)Ek[vech(XX')]Ek[vech(XX')]',

C22 = B 22 Ek ( XX ' ),

C33 = B33 ,

C13  =  B13 Ek [vech( XX ' )] = C 31,

C12  =  C21 = 0, C 23 = C3 2 = 0,

and all Bij as in Section 3. Again, V does not depend on the choice of the
estimator of the location parameter
β . Replacing EK with sample averages
yields an estimate
C of C. For a given θ = (β' , ^) ', let Ui = (Yi — X,i3')/σ
and

1 n / (U2 1)vech(XiX') ∖    / N^1

N = - Σ      ( u 3 3 u i ) Xi      =   N^2   

n i=1    U4 5 U 2 + 2    /     N3 /

Then T takes the form

(ʌ » ʌ 1 1 ʌ        ʌ » ʌ nn ʌ        ʌ , ʌ OO ʌ          ʌ , ʌ to ʌ ∖

N1 C11TV1 + N2C22IN2 + N3C33N3 + 2N1C13N3 ,

where Cij is the (i,j)-th block of (C+.

If θ is the ML estimator and the first column of X is a vector of ones, then
the first element of
C11 is zero, B13 =0andB33 = 24. Let p = k(k + 1)/2,
with
k =dim(/), and

L = 0p-1×1 Ip-1 ,

Li = L vech( XiX'),
1n

L = - ∑ Li
n
i=1

Then the IM test statistic with ML estimator can be written as (Hall, -987)
n                 n                          -1 n

T = 2∑(U2 1)Li{∑(Li — L)(Li — L)T ∑(U2 1)Li
i
=1              i=1                         i=1

nn   n    n

+6ΣU3Xi'(∑XiXiy ΣU3Xi + 24∑(Ui 3)2
i=1        i=1            i=1             i=1

19



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