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} | {1}

Instrumental Variable:

1 | {}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₁₂σ₁₃σ₂₃λ₂₄-σ₁₃σ₁₄σ₂₃-σ₁₁σ₂₃²λ₂₄+σ₁₃²σ₂₄-σ₁₂²σ₃₃λ₂₄+σ₁₂σ₁₄σ₃₃+σ₁₁σ₂₂σ₃₃λ₂₄-σ₁₁σ₂₄σ₃₃-σ₁₂σ₁₃σ₃₄+σ₁₁σ₂₃σ₃₄,
σ₁₂²σ₂₃²λ₂₄-σ₁₂σ₁₄σ₂₃²-σ₁₁σ₂₂σ₂₃²λ₂₄+σ₁₂σ₁₃σ₂₃σ₂₄-σ₁₂²σ₂₂σ₃₃λ₂₄+σ₁₂σ₁₄σ₂₂σ₃₃+σ₁₁σ₂₂²σ₃₃λ₂₄-σ₁₁σ₂₂σ₂₄σ₃₃-σ₁₂²σ₂₃σ₃₄+σ₁₁σ₂₂σ₂₃σ₃₄

Single Door Criterion:

{2} ⊥ {4} | {1,3}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₂₃²λ₃₄-σ₂₃σ₂₄-σ₂₂σ₃₃λ₃₄+σ₂₂σ₃₄,
σ₁₃σ₂₃λ₃₄-σ₁₃σ₂₄-σ₁₂σ₃₃λ₃₄+σ₁₂σ₃₄

Single Door Criterion:

{3} ⊥ {1,4} | {2}
{3} ⊥ {4} | {1,2}

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: 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+λ₁₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄ Identified

Algebraic Equations:

σ₁₁σ₂₃²λ₁₂λ₂₃λ₃₄-σ₁₃σ₂₃σ₂₄-σ₁₁σ₂₂σ₃₃λ₁₂λ₂₃λ₃₄+σ₁₂σ₂₃σ₃₄,
σ₁₁σ₁₃σ₂₃λ₁₂λ₂₃λ₃₄-σ₁₃²σ₂₄-σ₁₁σ₁₂σ₃₃λ₁₂λ₂₃λ₃₄+σ₁₂σ₁₃σ₃₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₄ Identified

Algebraic Equations:

σ₁₁σ₁₂σ₁₃σ₂₃λ₁₂λ₂₄-σ₁₂σ₁₃σ₁₄σ₂₃-σ₁₁²σ₂₃²λ₁₂λ₂₄+σ₁₂σ₁₃²σ₂₄-σ₁₁σ₁₂²σ₃₃λ₁₂λ₂₄+σ₁₂²σ₁₄σ₃₃+σ₁₁²σ₂₂σ₃₃λ₁₂λ₂₄-σ₁₁σ₁₂σ₂₄σ₃₃-σ₁₂²σ₁₃σ₃₄+σ₁₁σ₁₂σ₂₃σ₃₄,
σ₁₁σ₁₂²σ₂₃²λ₁₂λ₂₄-σ₁₂²σ₁₄σ₂₃²-σ₁₁²σ₂₂σ₂₃²λ₁₂λ₂₄+σ₁₂²σ₁₃σ₂₃σ₂₄-σ₁₁σ₁₂²σ₂₂σ₃₃λ₁₂λ₂₄+σ₁₂²σ₁₄σ₂₂σ₃₃+σ₁₁²σ₂₂²σ₃₃λ₁₂λ₂₄-σ₁₁σ₁₂σ₂₂σ₂₄σ₃₃-σ₁₂³σ₂₃σ₃₄+σ₁₁σ₁₂σ₂₂σ₂₃σ₃₄

Total Effect of X₂ on X₄: λ₂₃λ₃₄+λ₂₄ 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})