Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2,3} | {}

Parameter: λ₁₄

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₁₂λ₂₃-σ₁₃

Instrumental Variable:

1 | {}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₁₃σ₂₃λ₃₄-σ₁₃σ₂₄-σ₁₂σ₃₃λ₃₄+σ₁₂σ₃₄,
σ₁₃²λ₃₄-σ₁₃σ₁₄-σ₁₁σ₃₃λ₃₄+σ₁₁σ₃₄,
σ₁₂σ₁₃λ₃₄-σ₁₂σ₁₄-σ₁₁σ₂₃λ₃₄+σ₁₁σ₂₄,
σ₁₂σ₁₄σ₂₃+σ₁₁σ₂₃²λ₃₄-σ₁₂σ₁₃σ₂₄-σ₁₁σ₂₃σ₂₄-σ₁₂²σ₃₃λ₃₄+σ₁₂²σ₃₄

Single Door Criterion:

{3} ⊥ {4} | {1}
{3} ⊥ {4} | {1,2}

Instrumental Variable:

2 | {1}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₄

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₂₃ Identified

Algebraic Equations:

σ₁₃σ₂₂-σ₁₂σ₂₃+σ₁₂ω₂₃

Parameter: ω₃₃ Identified

Algebraic Equations:

-σ₁₃²σ₂₂+2σ₁₂σ₁₃σ₂₃-σ₁₂²σ₃₃+σ₁₂²ω₃₃

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:

σ₁₁σ₁₃σ₂₃λ₁₂λ₂₃λ₃₄-σ₁₃²σ₂₄-σ₁₁σ₁₂σ₃₃λ₁₂λ₂₃λ₃₄+σ₁₂σ₁₃σ₃₄,
σ₁₁σ₁₃²λ₁₂λ₂₃λ₃₄-σ₁₃²σ₁₄-σ₁₁²σ₃₃λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₃σ₃₄,
σ₁₁σ₁₂σ₁₃λ₁₂λ₂₃λ₃₄-σ₁₂σ₁₃σ₁₄-σ₁₁²σ₂₃λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₃σ₂₄,
σ₁₁²σ₂₃²λ₁₂λ₂₃λ₃₄-σ₁₁σ₁₃σ₂₃σ₂₄-σ₁₁σ₁₂²σ₃₃λ₁₂λ₂₃λ₃₄+σ₁₂²σ₁₄σ₃₃-σ₁₁σ₁₂σ₂₄σ₃₃+σ₁₁σ₁₂σ₂₃σ₃₄

Total Effect of X₂ on X₄: λ₂₃λ₃₄ Identified

Each Parameter Identified

Vanishing Ideal

Constraint Set:

σ₁₃σ₁₄σ₂₃-σ₁₃²σ₂₄-σ₁₂σ₁₄σ₃₃+σ₁₁σ₂₄σ₃₃+σ₁₂σ₁₃σ₃₄-σ₁₁σ₂₃σ₃₄

All constraints are determinantal

Trek-separation:

{1,2,3} ⊥ {3,1,4} | ({1},{3})