Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₄λ₁₂-σ₂₄,
σ₁₃λ₁₂-σ₂₃,
σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2} | {}
{1} ⊥ {2} | {4}
{1} ⊥ {2} | {3}
{1} ⊥ {2} | {3,4}

Instrumental Variable:

3 | {}
3 | {4}
4 | {}
4 | {3}

Parameter: λ₁₃ Identified

Algebraic Equations:

σ₁₂λ₁₃-σ₂₃,
σ₁₁λ₁₃-σ₁₃

Single Door Criterion:

{1} ⊥ {3} | {}
{1} ⊥ {3} | {2}

Instrumental Variable:

2 | {}

Parameter: λ₁₄

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₁₃σ₂₃λ₃₄-σ₁₃σ₂₄-σ₁₂σ₃₃λ₃₄+σ₁₂σ₃₄,
σ₁₃²λ₃₄-σ₁₃σ₁₄-σ₁₁σ₃₃λ₃₄+σ₁₁σ₃₄,
σ₁₁σ₂₃²λ₃₄-σ₁₁σ₂₃σ₂₄-σ₁₂²σ₃₃λ₃₄+σ₁₂²σ₃₄

Single Door Criterion:

{3} ⊥ {2,4} | {1}
{3} ⊥ {4} | {1,2}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₄

Parameter: ω₂₂ Identified

Algebraic Equations:

-σ₁₄σ₂₂+σ₁₂σ₂₄+σ₁₄ω₂₂,
-σ₁₃σ₂₂+σ₁₂σ₂₃+σ₁₃ω₂₂,
σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

σ₁₃σ₂₃-σ₁₂σ₃₃+σ₁₂ω₃₃,
σ₁₃²-σ₁₁σ₃₃+σ₁₁ω₃₃

Parameter: ω₄₄

Total Effect of X₁ on X₄: λ₁₃λ₃₄+λ₁₄

Path-specific Effect of X₁ on X₄: λ₁₃λ₃₄ Identified

Algebraic Equations:

σ₁₁σ₂₃²λ₁₃λ₃₄-σ₁₃σ₂₃σ₂₄-σ₁₂²σ₃₃λ₁₃λ₃₄+σ₁₂σ₂₃σ₃₄,
σ₁₁σ₁₃σ₂₃λ₁₃λ₃₄-σ₁₃²σ₂₄-σ₁₁σ₁₂σ₃₃λ₁₃λ₃₄+σ₁₁σ₂₃σ₃₄,
σ₁₁σ₁₃²λ₁₃λ₃₄-σ₁₃²σ₁₄-σ₁₁²σ₃₃λ₁₃λ₃₄+σ₁₁σ₁₃σ₃₄

Vanishing Ideal

Constraint Set:

σ₁₄σ₂₃-σ₁₃σ₂₄,
σ₁₂σ₁₄-σ₁₁σ₂₄,
σ₁₂σ₁₃-σ₁₁σ₂₃

All constraints are determinantal

Trek-separation:

{1,2} ⊥ {1,3,4} | ({1},{})
{1,2} ⊥ {1,3} | ({},{1})
{1,3,2} ⊥ {1,3,4} | ({1,3},{})
{1,4,2} ⊥ {1,3,4} | ({1,4},{})
{1,2,3} ⊥ {3,1,4} | ({1},{3})