Parameter: λ₁₃ Identified

Algebraic Equations:

σ₁₁λ₁₃-σ₁₃

Single Door Criterion:

{1} ⊥ {2,3,4} | {}
{1} ⊥ {3,4} | {2}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₂₂λ₂₄-σ₂₄

Single Door Criterion:

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

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₃₃λ₃₄-σ₃₄,
σ₁₃λ₃₄-σ₁₄

Single Door Criterion:

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

Instrumental Variable:

1 | {}
1 | {2}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₂₂ Identified

Algebraic Equations:

-σ₂₂+ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

σ₁₃²-σ₁₁σ₃₃+σ₁₁ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

σ₂₄²σ₃₃+σ₂₂σ₃₄²-σ₂₂σ₃₃σ₄₄+σ₂₂σ₃₃ω₄₄,
σ₁₃σ₂₄²+σ₁₄σ₂₂σ₃₄-σ₁₃σ₂₂σ₄₄+σ₁₃σ₂₂ω₄₄

Total Effect of X₁ on X₄: λ₁₃λ₃₄ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3,4} | {}
{1} ⊥ {3,4} | {2}

Vanishing Ideal

Constraint Set:

σ₂₃,
σ₁₂,
σ₁₄σ₃₃-σ₁₃σ₃₄

All constraints are determinantal

Trek-separation:

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