Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2,3,4} | {}
{1} ⊥ {2,4} | {3}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₂₂λ₂₄-σ₂₄,
σ₁₂λ₂₄-σ₁₄

Single Door Criterion:

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

Instrumental Variable:

1 | {}
1 | {3}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₃₃λ₃₄-σ₃₄

Single Door Criterion:

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

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

-σ₃₃+ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

σ₂₄²σ₃₃+σ₂₂σ₃₄²-σ₂₂σ₃₃σ₄₄+σ₂₂σ₃₃ω₄₄,
σ₁₄σ₂₄σ₃₃+σ₁₂σ₃₄²-σ₁₂σ₃₃σ₄₄+σ₁₂σ₃₃ω₄₄

Total Effect of X₁ on X₄: λ₁₂λ₂₄ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3,4} | {}
{1} ⊥ {2,4} | {3}

Vanishing Ideal

Constraint Set:

σ₂₃,
σ₁₃,
σ₁₄σ₂₂-σ₁₂σ₂₄

All constraints are determinantal

Trek-separation:

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