Parameter: λ₁₄ Identified

Algebraic Equations:

σ₁₁λ₁₄-σ₁₄

Single Door Criterion:

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

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₂₂λ₂₃-σ₂₃

Single Door Criterion:

{2} ⊥ {1,3,4} | {}
{2} ⊥ {3,4} | {1}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₃₃λ₃₄-σ₃₄,
σ₂₃λ₃₄-σ₂₄

Single Door Criterion:

{3} ⊥ {1,4} | {}
{3} ⊥ {1,4} | {2}
{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: ω₃₃ Identified

Algebraic Equations:

σ₂₃²-σ₂₂σ₃₃+σ₂₂ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

σ₁₄²σ₃₃+σ₁₁σ₃₄²-σ₁₁σ₃₃σ₄₄+σ₁₁σ₃₃ω₄₄,
σ₁₄²σ₂₃+σ₁₁σ₂₄σ₃₄-σ₁₁σ₂₃σ₄₄+σ₁₁σ₂₃ω₄₄

Total Effect of X₂ on X₄: λ₂₃λ₃₄ Identified

Each Parameter Identified

Back Door Criterion:


{2} ⊥ {1,3,4} | {}
{2} ⊥ {3,4} | {1}

Vanishing Ideal

Constraint Set:

σ₁₃,
σ₁₂,
σ₂₄σ₃₃-σ₂₃σ₃₄

All constraints are determinantal

Trek-separation:

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