Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂,
σ₁₄σ₂₃λ₁₂-σ₂₃σ₂₄-σ₁₂σ₃₄λ₁₂+σ₂₂σ₃₄

Single Door Criterion:

{1} ⊥ {2,3} | {}
{1} ⊥ {2} | {3}

Parameter: λ₁₄ Identified

Algebraic Equations:

σ₁₂σ₂₃λ₁₄-σ₂₃σ₂₄+σ₂₂σ₃₄,
σ₁₁σ₂₃λ₁₄-σ₁₄σ₂₃+σ₁₂σ₃₄,
σ₁₂²λ₁₄-σ₁₁σ₂₂λ₁₄+σ₁₄σ₂₂-σ₁₂σ₂₄

Single Door Criterion:

{1} ⊥ {4} | {2}
{1} ⊥ {4} | {2,3}

Instrumental Variable:

3 | {2}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₂₃λ₂₄-σ₃₄,
σ₁₂²λ₂₄-σ₁₂σ₁₄-σ₁₁σ₂₂λ₂₄+σ₁₁σ₂₄

Single Door Criterion:

{2} ⊥ {4} | {1}
{2} ⊥ {4} | {1,3}

Instrumental Variable:

3 | {}
3 | {1}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂,
-σ₁₄σ₂₂σ₂₃+σ₁₂σ₂₃σ₂₄+σ₁₄σ₂₃ω₂₂-σ₁₂σ₃₄ω₂₂

Parameter: ω₂₃ Identified

Algebraic Equations:

-σ₂₃+ω₂₃

Parameter: ω₃₃ Identified

Algebraic Equations:

-σ₃₃+ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

σ₁₄σ₂₃σ₂₄-σ₁₄σ₂₂σ₃₄+σ₁₂σ₂₄σ₃₄-σ₁₂σ₂₃σ₄₄+σ₁₂σ₂₃ω₄₄,
σ₁₄²σ₂₃-σ₁₂σ₁₄σ₃₄+σ₁₁σ₂₄σ₃₄-σ₁₁σ₂₃σ₄₄+σ₁₁σ₂₃ω₄₄,
-σ₁₄²σ₂₂+2σ₁₂σ₁₄σ₂₄-σ₁₁σ₂₄²-σ₁₂²σ₄₄+σ₁₁σ₂₂σ₄₄+σ₁₂²ω₄₄-σ₁₁σ₂₂ω₄₄

Total Effect of X₁ on X₄: λ₁₂λ₂₄+λ₁₄ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3,4} | {}
{1} ⊥ {2,4} | {3}

Vanishing Ideal

Constraint Set:

σ₁₃,
σ₁₂σ₁₄σ₂₃-σ₁₁σ₂₃σ₂₄-σ₁₂²σ₃₄+σ₁₁σ₂₂σ₃₄

All constraints are determinantal

Trek-separation:

{1} ⊥ {3} | ({},{})
{1,2,3} ⊥ {2,1,4} | ({1},{2})
{1,2,3} ⊥ {1,2,4} | ({},{1,2})