GMt estimating equations and asymptotics The estimators ˆ Θ and ˆ Σ defined by (10) are M-type estimators (Van der Vaart, 1998, ch. 5), since they minimize a function of the form M ( Θ , Σ ) = 1 n ∑ n i $_{=1}$m$_{(}$ Θ , Σ $_{)}$( ˆ U$_{i}$ , ˆ V$_{i}$ ) . Specifically, Then ˆ Θ and ˆ Σ solve the equations ∂ ∂ $_{Θ}$M ( ˆ Θ , ˆ Σ ) = O and ∂ ∂ $_{Σ}$M ( ˆ Θ , ˆ Σ ) = O . To com- pute matrix derivatives we use the method of differentials (Magnus and Neudecker, 1999). Differentiating with respect to Θ we obtain where e$_{i}$ = w ( ˆ U$_{i}$ )( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) . Then which can be rearranged in matrix form as ∂ ∂ $_{Θ}$m$_{(}$ Θ , Σ $_{)}$( ˆ U$_{i}$ , ˆ V$_{i}$ ) = − 2 ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ ) ˆ U$_{i}$ ( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$, ∇ vec( Θ $_{)}$m$_{(}$ Θ , Σ $_{)}$( ˆ U$_{i}$ , ˆ V$_{i}$ ) = − 2 ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ )vec { ˆ U$_{i}$ ( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$} , (17) d m$_{(}$ Θ , Σ $_{)}$( ˆ U$_{i}$ , ˆ V$_{i}$ ) = ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ )2( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}${− (d Θ ) T ˆ U$_{i}$ } = − 2 ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ )tr { (d Θ ) T ˆ U$_{i}$ ( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$} = − 2 ρ $^{′}$( e$_{i}$ ) w ( ˆ U$_{i}$ )vec(d Θ ) $^{T}$vec { ˆ U$_{i}$ ( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$} , m$_{(}$ Θ , Σ $_{)}$( ˆ U$_{i}$ , ˆ V$_{i}$ ) = ρ { w ( ˆ U$_{i}$ )( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) $^{T}$Σ − $^{1}$( ˆ V$_{i}$ − Θ T ˆ U$_{i}$ ) } + log | Σ | . 18