Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2,3} | {}

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₂₂λ₂₃-σ₂₃,
σ₁₂λ₂₃-σ₁₃

Single Door Criterion:

{2} ⊥ {3} | {}
{2} ⊥ {3} | {1}

Instrumental Variable:

1 | {}

Parameter: λ₂₄

Parameter: λ₃₄

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₄ Identified

Algebraic Equations:

σ₁₁σ₁₄σ₂₃-σ₁₁σ₁₃σ₂₄+σ₁₂σ₁₃ω₁₄-σ₁₁σ₂₃ω₁₄,
σ₁₁σ₁₄σ₂₂-σ₁₁σ₁₂σ₂₄+σ₁₂²ω₁₄-σ₁₁σ₂₂ω₁₄

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

σ₂₃²-σ₂₂σ₃₃+σ₂₂ω₃₃,
σ₁₃σ₂₃-σ₁₂σ₃₃+σ₁₂ω₃₃

Parameter: ω₃₄

Parameter: ω₄₄

Total Effect of X₁ on X₃: λ₁₂λ₂₃ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3} | {}

Total Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄+λ₁₂λ₂₄ Identified

Algebraic Equations:

σ₁₁σ₁₂σ₁₃λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₂σ₁₃λ₁₂λ₂₄-σ₁₂σ₁₃σ₁₄-σ₁₁²σ₂₃λ₁₂λ₂₃λ₃₄-σ₁₁²σ₂₃λ₁₂λ₂₄+σ₁₁σ₁₃σ₂₄,
σ₁₁σ₁₂²λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₂²λ₁₂λ₂₄-σ₁₂²σ₁₄-σ₁₁²σ₂₂λ₁₂λ₂₃λ₃₄-σ₁₁²σ₂₂λ₁₂λ₂₄+σ₁₁σ₁₂σ₂₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₄

Total Effect of X₂ on X₄: λ₂₃λ₃₄+λ₂₄ Identified

Algebraic Equations:

σ₁₂²λ₂₃λ₃₄+σ₁₂²λ₂₄-σ₁₂σ₁₄-σ₁₁σ₂₂λ₂₃λ₃₄-σ₁₁σ₂₂λ₂₄+σ₁₁σ₂₄

Back Door Criterion:

{2} ⊥ {3,4} | {1}

Path-specific Effect of X₂ on X₄: λ₂₃λ₃₄

Vanishing Ideal

Constraint Set:

σ₁₃σ₂₂-σ₁₂σ₂₃

All constraints are determinantal

Trek-separation:

{1,2} ⊥ {2,3} | ({},{2})