Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2} | {}

Parameter: λ₁₃ Identified

Algebraic Equations:

σ₁₂²λ₁₃-σ₁₁σ₂₂λ₁₃+σ₁₃σ₂₂-σ₁₂σ₂₃

Single Door Criterion:

{1} ⊥ {3} | {2}

Parameter: λ₁₄

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₁₂²λ₂₃-σ₁₂σ₁₃-σ₁₁σ₂₂λ₂₃+σ₁₁σ₂₃

Single Door Criterion:

{2} ⊥ {3,4} | {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: ω₃₃ Identified

Algebraic Equations:

-σ₁₃²σ₂₂+2σ₁₂σ₁₃σ₂₃-σ₁₁σ₂₃²-σ₁₂²σ₃₃+σ₁₁σ₂₂σ₃₃+σ₁₂²ω₃₃-σ₁₁σ₂₂ω₃₃

Parameter: ω₄₄

Total Effect of X₁ on X₃: λ₁₂λ₂₃+λ₁₃ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3} | {}

Total Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄+λ₁₃λ₃₄+λ₁₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄ Identified

Algebraic Equations:

σ₁₁σ₁₂²λ₁₂λ₂₃λ₃₄-σ₁₂²σ₁₄-σ₁₁²σ₂₂λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₂σ₂₄

Path-specific Effect of X₁ on X₄: λ₁₃λ₃₄ Identified

Algebraic Equations:

σ₁₂²σ₁₃σ₂₃λ₁₃λ₃₄-σ₁₁σ₁₃σ₂₂σ₂₃λ₁₃λ₃₄+σ₁₃²σ₂₂σ₂₄-σ₁₂σ₁₃σ₂₃σ₂₄-σ₁₂³σ₃₃λ₁₃λ₃₄+σ₁₁σ₁₂σ₂₂σ₃₃λ₁₃λ₃₄-σ₁₂σ₁₃σ₂₂σ₃₄+σ₁₂²σ₂₃σ₃₄,
σ₁₂²σ₁₃²λ₁₃λ₃₄-σ₁₁σ₁₃²σ₂₂λ₁₃λ₃₄+σ₁₃²σ₁₄σ₂₂-σ₁₂σ₁₃²σ₂₄-σ₁₁σ₁₂²σ₃₃λ₁₃λ₃₄-σ₁₂²σ₁₄σ₃₃+σ₁₁²σ₂₂σ₃₃λ₁₃λ₃₄+σ₁₁σ₁₂σ₂₄σ₃₃+σ₁₂²σ₁₃σ₃₄-σ₁₁σ₁₃σ₂₂σ₃₄,
σ₁₂³σ₁₃λ₁₃λ₃₄-σ₁₁σ₁₂σ₁₃σ₂₂λ₁₃λ₃₄+σ₁₂σ₁₃σ₁₄σ₂₂-σ₁₁σ₁₂²σ₂₃λ₁₃λ₃₄-σ₁₂²σ₁₄σ₂₃+σ₁₁²σ₂₂σ₂₃λ₁₃λ₃₄-σ₁₁σ₁₃σ₂₂σ₂₄+σ₁₁σ₁₂σ₂₃σ₂₄,
σ₁₁σ₁₂²σ₂₃²λ₁₃λ₃₄+σ₁₂²σ₁₄σ₂₃²-σ₁₁²σ₂₂σ₂₃²λ₁₃λ₃₄-σ₁₂²σ₁₃σ₂₃σ₂₄+σ₁₁σ₁₃σ₂₂σ₂₃σ₂₄-σ₁₁σ₁₂σ₂₃²σ₂₄-σ₁₂⁴σ₃₃λ₁₃λ₃₄+σ₁₁σ₁₂²σ₂₂σ₃₃λ₁₃λ₃₄-σ₁₂²σ₁₄σ₂₂σ₃₃+σ₁₁σ₁₂σ₂₂σ₂₄σ₃₃+σ₁₂³σ₂₃σ₃₄-σ₁₁σ₁₂σ₂₂σ₂₃σ₃₄

Total Effect of X₂ on X₄: λ₂₃λ₃₄ Identified

Each Parameter Identified

Back Door Criterion:


{2} ⊥ {3,4} | {1}

Vanishing Ideal

Constraint Set:

σ₁₃σ₁₄σ₂₃-σ₁₃²σ₂₄-σ₁₂σ₁₄σ₃₃+σ₁₁σ₂₄σ₃₃+σ₁₂σ₁₃σ₃₄-σ₁₁σ₂₃σ₃₄

All constraints are determinantal

Trek-separation:

{1,2,3} ⊥ {3,1,4} | ({1},{3})