Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂,
σ₁₃σ₂₄λ₁₂-σ₂₃σ₂₄-σ₁₂σ₃₄λ₁₂+σ₂₂σ₃₄

Single Door Criterion:

{1} ⊥ {2,4} | {}
{1} ⊥ {2} | {4}

Parameter: λ₁₃ Identified

Algebraic Equations:

σ₁₂σ₂₄λ₁₃-σ₂₃σ₂₄+σ₂₂σ₃₄,
σ₁₁σ₂₄λ₁₃-σ₁₃σ₂₄+σ₁₂σ₃₄,
σ₁₂²λ₁₃-σ₁₁σ₂₂λ₁₃+σ₁₃σ₂₂-σ₁₂σ₂₃

Single Door Criterion:

{1} ⊥ {3} | {2}
{1} ⊥ {3} | {2,4}

Instrumental Variable:

4 | {2}

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₂₄λ₂₃-σ₃₄,
σ₁₂²λ₂₃-σ₁₂σ₁₃-σ₁₁σ₂₂λ₂₃+σ₁₁σ₂₃

Single Door Criterion:

{2} ⊥ {3} | {1}
{2} ⊥ {3} | {1,4}

Instrumental Variable:

4 | {}
4 | {1}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂,
-σ₁₃σ₂₂σ₂₄+σ₁₂σ₂₃σ₂₄+σ₁₃σ₂₄ω₂₂-σ₁₂σ₃₄ω₂₂

Parameter: ω₂₄ Identified

Algebraic Equations:

-σ₂₄+ω₂₄

Parameter: ω₃₃ Identified

Algebraic Equations:

σ₁₃σ₂₃σ₂₄-σ₁₂σ₂₄σ₃₃-σ₁₃σ₂₂σ₃₄+σ₁₂σ₂₃σ₃₄+σ₁₂σ₂₄ω₃₃,
σ₁₃²σ₂₄-σ₁₁σ₂₄σ₃₃-σ₁₂σ₁₃σ₃₄+σ₁₁σ₂₃σ₃₄+σ₁₁σ₂₄ω₃₃,
-σ₁₃²σ₂₂+2σ₁₂σ₁₃σ₂₃-σ₁₁σ₂₃²-σ₁₂²σ₃₃+σ₁₁σ₂₂σ₃₃+σ₁₂²ω₃₃-σ₁₁σ₂₂ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

-σ₄₄+ω₄₄

Total Effect of X₁ on X₃: λ₁₂λ₂₃+λ₁₃ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3,4} | {}
{1} ⊥ {2,3} | {4}

Vanishing Ideal

Constraint Set:

σ₁₄,
σ₁₂σ₁₃σ₂₄-σ₁₁σ₂₃σ₂₄-σ₁₂²σ₃₄+σ₁₁σ₂₂σ₃₄

All constraints are determinantal

Trek-separation:

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