Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2,3,4} | {}

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₂₄λ₂₃-σ₃₄,
σ₂₂λ₂₃-σ₂₃,
σ₁₂λ₂₃-σ₁₃

Single Door Criterion:

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

Instrumental Variable:

1 | {}
1 | {4}
4 | {}
4 | {1}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₂₃λ₂₄-σ₃₄,
σ₂₂λ₂₄-σ₂₄,
σ₁₂λ₂₄-σ₁₄

Single Door Criterion:

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

Instrumental Variable:

1 | {}
1 | {3}
3 | {}
3 | {1}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

-σ₂₄σ₃₃+σ₂₃σ₃₄+σ₂₄ω₃₃,
σ₂₃²-σ₂₂σ₃₃+σ₂₂ω₃₃,
-σ₁₄σ₃₃+σ₁₃σ₃₄+σ₁₄ω₃₃,
σ₁₃σ₂₃-σ₁₂σ₃₃+σ₁₂ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

σ₂₄σ₃₄-σ₂₃σ₄₄+σ₂₃ω₄₄,
σ₂₄²-σ₂₂σ₄₄+σ₂₂ω₄₄,
σ₁₄σ₃₄-σ₁₃σ₄₄+σ₁₃ω₄₄,
σ₁₄σ₂₄-σ₁₂σ₄₄+σ₁₂ω₄₄

Total Effect of X₁ on X₃: λ₁₂λ₂₃ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3,4} | {}

Total Effect of X₁ on X₄: λ₁₂λ₂₄ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3,4} | {}

Vanishing Ideal

Constraint Set:

σ₂₃σ₂₄-σ₂₂σ₃₄,
σ₁₃σ₂₄-σ₁₂σ₃₄,
σ₁₄σ₂₃-σ₁₃σ₂₄,
σ₁₄σ₂₂-σ₁₂σ₂₄,
σ₁₃σ₂₂-σ₁₂σ₂₃

All constraints are determinantal

Trek-separation:

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