Parameter: λ₁₃ Identified

Algebraic Equations:

σ₁₁λ₁₃-σ₁₃,
σ₁₄σ₂₃λ₁₃-σ₁₃σ₂₄λ₁₃+σ₂₄σ₃₃-σ₂₃σ₃₄

Single Door Criterion:

{1} ⊥ {2,3} | {}
{1} ⊥ {3} | {2}

Parameter: λ₁₄ Identified

Algebraic Equations:

σ₁₃σ₂₃λ₁₄+σ₂₄σ₃₃-σ₂₃σ₃₄,
σ₁₁σ₂₃λ₁₄-σ₁₄σ₂₃+σ₁₃σ₂₄,
σ₁₃²λ₁₄-σ₁₁σ₃₃λ₁₄+σ₁₄σ₃₃-σ₁₃σ₃₄

Single Door Criterion:

{1} ⊥ {4} | {3}
{1} ⊥ {4} | {2,3}

Instrumental Variable:

2 | {3}

Parameter: λ₂₃

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₂₃λ₃₄-σ₂₄,
σ₁₃²λ₃₄-σ₁₃σ₁₄-σ₁₁σ₃₃λ₃₄+σ₁₁σ₃₄

Single Door Criterion:

{3} ⊥ {4} | {1}
{3} ⊥ {4} | {1,2}

Instrumental Variable:

2 | {}
2 | {1}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₂₂ Identified

Algebraic Equations:

-σ₂₂+ω₂₂

Parameter: ω₂₃

Parameter: ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

-σ₁₄σ₂₄σ₃₃+σ₁₄σ₂₃σ₃₄+σ₁₃σ₂₄σ₃₄-σ₁₃σ₂₃σ₄₄+σ₁₃σ₂₃ω₄₄,
σ₁₄²σ₂₃-σ₁₃σ₁₄σ₂₄+σ₁₁σ₂₄σ₃₄-σ₁₁σ₂₃σ₄₄+σ₁₁σ₂₃ω₄₄,
-σ₁₄²σ₃₃+2σ₁₃σ₁₄σ₃₄-σ₁₁σ₃₄²-σ₁₃²σ₄₄+σ₁₁σ₃₃σ₄₄+σ₁₃²ω₄₄-σ₁₁σ₃₃ω₄₄

Total Effect of X₁ on X₄: λ₁₃λ₃₄+λ₁₄ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3,4} | {}
{1} ⊥ {3,4} | {2}

Total Effect of X₂ on X₄: λ₂₃λ₃₄

Vanishing Ideal

Constraint Set:

σ₁₂,
σ₁₃σ₁₄σ₂₃-σ₁₃²σ₂₄+σ₁₁σ₂₄σ₃₃-σ₁₁σ₂₃σ₃₄

All constraints are determinantal

Trek-separation:

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