Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2,3} | {}

Parameter: λ₁₄

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₂₂λ₂₃-σ₂₃,
σ₁₂λ₂₃-σ₁₃

Single Door Criterion:

{2} ⊥ {3} | {}
{2} ⊥ {3,4} | {1}

Instrumental Variable:

1 | {}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₁₂σ₁₃λ₃₄-σ₁₂σ₁₄-σ₁₁σ₂₃λ₃₄+σ₁₁σ₂₄,
-σ₁₂σ₁₄σ₂₂+σ₁₂²σ₂₃λ₃₄-σ₁₁σ₂₂σ₂₃λ₃₄+σ₁₁σ₂₂σ₂₄

Instrumental Variable:

2 | {1}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₄

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

σ₂₃²-σ₂₂σ₃₃+σ₂₂ω₃₃,
σ₁₃σ₂₃-σ₁₂σ₃₃+σ₁₂ω₃₃

Parameter: ω₃₄ Identified

Algebraic Equations:

-σ₁₃σ₁₄σ₂₃+σ₁₃²σ₂₄+σ₁₂σ₁₄σ₃₃-σ₁₁σ₂₄σ₃₃-σ₁₂σ₁₃σ₃₄+σ₁₁σ₂₃σ₃₄+σ₁₂σ₁₃ω₃₄-σ₁₁σ₂₃ω₃₄,
-σ₁₂σ₁₄σ₂₃²+σ₁₂σ₁₃σ₂₃σ₂₄+σ₁₂σ₁₄σ₂₂σ₃₃-σ₁₁σ₂₂σ₂₄σ₃₃-σ₁₂²σ₂₃σ₃₄+σ₁₁σ₂₂σ₂₃σ₃₄+σ₁₂²σ₂₃ω₃₄-σ₁₁σ₂₂σ₂₃ω₃₄

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+λ₁₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄ Identified

Algebraic Equations:

σ₁₁σ₁₂σ₁₃λ₁₂λ₂₃λ₃₄-σ₁₂σ₁₃σ₁₄-σ₁₁²σ₂₃λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₃σ₂₄,
σ₁₁σ₁₂²λ₁₂λ₂₃λ₃₄-σ₁₂²σ₁₄-σ₁₁²σ₂₂λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₂σ₂₄

Total Effect of X₂ on X₄: λ₂₃λ₃₄ Identified

Each Parameter Identified

Back Door Criterion:


{2} ⊥ {3,4} | {1}

Vanishing Ideal

Constraint Set:

σ₁₃σ₂₂-σ₁₂σ₂₃

All constraints are determinantal

Trek-separation:

{1,2} ⊥ {2,3} | ({},{2})