Testing the Information Matrix Equality with Robust Estimators



Figure 6:


(a) Power surface ML estimator; (b) difference ML

estimator-other


The power surface for the IM test with ML estimator is plotted in Fig-
ure 6 (a). The difference in power between the IM test with ML estimator
and the IM test with other M-estimators is plotted in Figure 6 (b). The
maximal difference in power is 0.06623. It is again observed that the loss in
local power when using robust estimators for the IM equality test is rather
limited.

5 The normal regression model

For the normal model with covariates, (yx) = σΦ(y X β) with θ =
')'. We obtain, with u = (Y — X'β),

1    ( u2 1)vech( XX, )

m (X, Y ; θ) = 2       (u3 3u) X

σ      u4 5u2 +2

where vech( ) is the lower triangular stack operator, and ξ = ξ(X, Y; θ),

1 / ( u2 1)vech( XX ' ) σ IF( X, Y ; ^; K, Fθ ) Ek [vech( XX ' )]

ξ = 2                         ( u3 3 u ) X

σ              u4 5u2 + 2 σIF(X,Y; ^; K,Fθ)

18



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