(1)dimana δh adalah koefisien yang akan diestimasi dalam persamaan sistem tersebut. Persamaan (1) diasumsikan semua persamaan adalah just idenfied atau overidenfied.
Bilamana kita gunakan notasi bintang (*) sebagai stacking vektor, maka g vektor variable dependen (variabel endogen) , stochastic error term dan vektor koofisien adalah:
(2)dimana k* adalah jumlah koefisien yang akan diestimasi, yaitu:
sedangkan δ* adalah semua koefisien yang akan diestimasi. Sedangkan matrik explanatory variable yang dinyatakan dalam notasi bintang (*), yaitu:
(3)yang merupakan matrik diagonal yang berisi semua data variabel ekplanatori dalam suatu persamaan. Sehingga persamaan (1) dengan notasi bintang (*) sebagai persamaan sistem adalah:
(4)dan vektor stokastik error ε* untuk semua persamaan diasumsikan bahwa:
(5)Demikaian pula error term diasumsikan berdistribusi normal:
(6)sehingga logaritma fungsi likelihood adalah:
(7) sedangkan yang diperlukan adalah logaritma fungsi likelihood variabel endogen y*. Fungsi likelihood y* dapat diperoleh dari fungsi likelihood ε* dengan transformasi:
(8) dimana
adalah nilai absolut determinan matrik Jacobian bernilai 1 jika memenuhi asumsi persamaan (5) dan (6). Dengan demikian logaritma fungsi likelihood adalah:
(9)Fungsi likelihood ini dimaksimumkan dengan pilihan parameter δ* dan
. Sehingga dengan first order condition untuk memaksimumkan fungsi likelihood yaitu:
(10)maka diperoleh estimator yang konsiten dan effisien:
(11)Catatan: FILM digunakan untuk estimasin Persamaan Simultan sebagai alternatif estimasi three stage least square. Lihat posting sebelumnya tentang Persamaan Simultan.
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