Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₄λ₁₂-σ₂₄,
σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2} | {}
{1} ⊥ {2} | {4}

Instrumental Variable:

4 | {}

Parameter: λ₁₄ Identified

Algebraic Equations:

σ₁₃λ₁₄-σ₃₄,
σ₁₂λ₁₄-σ₂₄,
σ₁₁λ₁₄-σ₁₄

Single Door Criterion:

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

Instrumental Variable:

2 | {}
2 | {3}
3 | {}
3 | {2}

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₁₄σ₂₂λ₂₃-σ₁₄σ₂₃-σ₁₂σ₂₄λ₂₃+σ₁₂σ₃₄,
σ₁₂²λ₂₃-σ₁₂σ₁₃-σ₁₁σ₂₂λ₂₃+σ₁₁σ₂₃

Single Door Criterion:

{2} ⊥ {3,4} | {1}
{2} ⊥ {3} | {1,4}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₃ Identified

Algebraic Equations:

σ₁₁σ₂₃σ₂₄-σ₁₁σ₂₂σ₃₄+σ₁₄σ₂₂ω₁₃-σ₁₂σ₂₄ω₁₃,
σ₁₁σ₁₃σ₂₂-σ₁₁σ₁₂σ₂₃+σ₁₂²ω₁₃-σ₁₁σ₂₂ω₁₃

Parameter: ω₂₂ Identified

Algebraic Equations:

-σ₁₄σ₂₂+σ₁₂σ₂₄+σ₁₄ω₂₂,
σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

σ₁₄²σ₂₂σ₂₃²-2σ₁₁σ₂₃²σ₂₄²-σ₁₄²σ₂₂²σ₃₃-σ₁₂²σ₂₄²σ₃₃+2σ₁₁σ₂₂σ₂₄²σ₃₃+2σ₁₂²σ₂₃σ₂₄σ₃₄-σ₁₂²σ₂₂σ₃₄²+σ₁₄²σ₂₂²ω₃₃+σ₁₂²σ₂₄²ω₃₃-2σ₁₁σ₂₂σ₂₄²ω₃₃,
σ₁₁σ₁₄σ₂₂σ₂₃²-2σ₁₁σ₁₂σ₂₃²σ₂₄-σ₁₁σ₁₄σ₂₂²σ₃₃-σ₁₂³σ₂₄σ₃₃+2σ₁₁σ₁₂σ₂₂σ₂₄σ₃₃-σ₁₂²σ₁₃σ₂₂σ₃₄+2σ₁₂³σ₂₃σ₃₄+σ₁₁σ₁₄σ₂₂²ω₃₃+σ₁₂³σ₂₄ω₃₃-2σ₁₁σ₁₂σ₂₂σ₂₄ω₃₃,
-σ₁₂²σ₁₃²σ₂₂+2σ₁₂³σ₁₃σ₂₃-2σ₁₁σ₁₂²σ₂₃²+σ₁₁²σ₂₂σ₂₃²-σ₁₂⁴σ₃₃+2σ₁₁σ₁₂²σ₂₂σ₃₃-σ₁₁²σ₂₂²σ₃₃+σ₁₂⁴ω₃₃-2σ₁₁σ₁₂²σ₂₂ω₃₃+σ₁₁²σ₂₂²ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

σ₁₄σ₃₄-σ₁₃σ₄₄+σ₁₃ω₄₄,
σ₁₄σ₂₄-σ₁₂σ₄₄+σ₁₂ω₄₄,
σ₁₄²-σ₁₁σ₄₄+σ₁₁ω₄₄

Total Effect of X₁ on X₃: λ₁₂λ₂₃ Identified

Each Parameter Identified

Vanishing Ideal

Constraint Set:

σ₁₃σ₂₄-σ₁₂σ₃₄,
σ₁₃σ₁₄-σ₁₁σ₃₄,
σ₁₂σ₁₄-σ₁₁σ₂₄

All constraints are determinantal

Trek-separation:

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