Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2} | {}

Parameter: λ₂₃

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₂₃²λ₂₄-σ₂₂σ₃₃λ₂₄+σ₂₄σ₃₃-σ₂₃σ₃₄,
σ₁₃σ₂₃λ₂₄-σ₁₂σ₃₃λ₂₄+σ₁₄σ₃₃-σ₁₃σ₃₄,
σ₁₃σ₂₂λ₂₄-σ₁₂σ₂₃λ₂₄+σ₁₄σ₂₃-σ₁₃σ₂₄

Single Door Criterion:

{2} ⊥ {4} | {3}
{2} ⊥ {4} | {1,3}

Instrumental Variable:

1 | {3}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₂₃²λ₃₄-σ₂₃σ₂₄-σ₂₂σ₃₃λ₃₄+σ₂₂σ₃₄,
σ₁₃σ₂₃λ₃₄-σ₁₄σ₂₃-σ₁₂σ₃₃λ₃₄+σ₁₂σ₃₄,
σ₁₃σ₂₂λ₃₄-σ₁₄σ₂₂-σ₁₂σ₂₃λ₃₄+σ₁₂σ₂₄

Single Door Criterion:

{3} ⊥ {4} | {2}
{3} ⊥ {4} | {1,2}

Instrumental Variable:

1 | {2}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₃

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₂₃

Parameter: ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

-σ₂₄²σ₃₃+2σ₂₃σ₂₄σ₃₄-σ₂₂σ₃₄²-σ₂₃²σ₄₄+σ₂₂σ₃₃σ₄₄+σ₂₃²ω₄₄-σ₂₂σ₃₃ω₄₄,
-σ₁₄σ₂₄σ₃₃+σ₁₄σ₂₃σ₃₄+σ₁₃σ₂₄σ₃₄-σ₁₂σ₃₄²-σ₁₃σ₂₃σ₄₄+σ₁₂σ₃₃σ₄₄+σ₁₃σ₂₃ω₄₄-σ₁₂σ₃₃ω₄₄,
-σ₁₄σ₂₃σ₂₄+σ₁₃σ₂₄²+σ₁₄σ₂₂σ₃₄-σ₁₂σ₂₄σ₃₄-σ₁₃σ₂₂σ₄₄+σ₁₂σ₂₃σ₄₄+σ₁₃σ₂₂ω₄₄-σ₁₂σ₂₃ω₄₄

Total Effect of X₁ on X₃: λ₁₂λ₂₃

Total Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄+λ₁₂λ₂₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₄ Identified

Algebraic Equations:

σ₁₁σ₂₃²λ₁₂λ₂₄-σ₁₁σ₂₂σ₃₃λ₁₂λ₂₄+σ₁₂σ₂₄σ₃₃-σ₁₂σ₂₃σ₃₄,
σ₁₁σ₁₃σ₂₃λ₁₂λ₂₄-σ₁₁σ₁₂σ₃₃λ₁₂λ₂₄+σ₁₂σ₁₄σ₃₃-σ₁₂σ₁₃σ₃₄,
σ₁₁σ₁₃σ₂₂λ₁₂λ₂₄-σ₁₁σ₁₂σ₂₃λ₁₂λ₂₄+σ₁₂σ₁₄σ₂₃-σ₁₂σ₁₃σ₂₄

Total Effect of X₂ on X₄: λ₂₃λ₃₄+λ₂₄

Path-specific Effect of X₂ on X₄: λ₂₃λ₃₄

Vanishing Ideal

Constraint Set:

σ₁₄σ₂₃²-σ₁₃σ₂₃σ₂₄-σ₁₄σ₂₂σ₃₃+σ₁₂σ₂₄σ₃₃+σ₁₃σ₂₂σ₃₄-σ₁₂σ₂₃σ₃₄

All constraints are determinantal

Trek-separation:

{1,2,3} ⊥ {2,3,4} | ({},{2,3})