Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2,4} | {}

Parameter: λ₁₃ Identified

Algebraic Equations:

σ₁₂σ₁₄λ₁₃-σ₁₁σ₂₄λ₁₃+σ₁₃σ₂₄-σ₁₂σ₃₄,
σ₁₂²λ₁₃-σ₁₁σ₂₂λ₁₃+σ₁₃σ₂₂-σ₁₂σ₂₃

Single Door Criterion:

{1} ⊥ {3,4} | {2}
{1} ⊥ {3} | {2,4}

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₁₂σ₁₄λ₂₃-σ₁₃σ₁₄-σ₁₁σ₂₄λ₂₃+σ₁₁σ₃₄,
σ₁₂²λ₂₃-σ₁₂σ₁₃-σ₁₁σ₂₂λ₂₃+σ₁₁σ₂₃

Single Door Criterion:

{2} ⊥ {3} | {1}
{2} ⊥ {3} | {1,4}

Instrumental Variable:

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:

-σ₁₃²σ₂₄-σ₁₂σ₁₄σ₃₃+σ₁₁σ₂₄σ₃₃+2σ₁₂σ₁₃σ₃₄-σ₁₁σ₂₃σ₃₄+σ₁₂σ₁₄ω₃₃-σ₁₁σ₂₄ω₃₃,
-σ₁₃²σ₂₂+2σ₁₂σ₁₃σ₂₃-σ₁₁σ₂₃²-σ₁₂²σ₃₃+σ₁₁σ₂₂σ₃₃+σ₁₂²ω₃₃-σ₁₁σ₂₂ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

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

Total Effect of X₁ on X₃: λ₁₂λ₂₃+λ₁₃ 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,
σ₁₄σ₂₃-σ₁₂σ₃₄,
σ₁₄σ₂₂-σ₁₂σ₂₄

All constraints are determinantal

Trek-separation:

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