Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂,
σ₁₄σ₂₃²λ₁₂-σ₁₃σ₂₃σ₂₄λ₁₂-σ₁₄σ₂₂σ₃₃λ₁₂+σ₂₂σ₂₄σ₃₃+σ₁₂σ₂₃σ₃₄λ₁₂-σ₂₂σ₂₃σ₃₄

Single Door Criterion:

{1} ⊥ {2,3} | {}

Parameter: λ₁₄

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₂₂λ₂₃-σ₂₃,
σ₁₂λ₂₃-σ₁₃

Single Door Criterion:

{2} ⊥ {3} | {}
{2} ⊥ {3,4} | {1}

Instrumental Variable:

1 | {}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₂₃²λ₃₄-σ₂₃σ₂₄-σ₂₂σ₃₃λ₃₄+σ₂₂σ₃₄,
σ₁₃σ₂₃λ₃₄-σ₁₃σ₂₄-σ₁₂σ₃₃λ₃₄+σ₁₂σ₃₄,
σ₁₃²λ₃₄-σ₁₃σ₁₄-σ₁₁σ₃₃λ₃₄+σ₁₁σ₃₄,
σ₁₂σ₁₃λ₃₄-σ₁₂σ₁₄-σ₁₁σ₂₃λ₃₄+σ₁₁σ₂₄,
-σ₁₂σ₁₄σ₂₃+σ₁₂σ₁₃σ₂₄+σ₁₂²σ₃₃λ₃₄-σ₁₁σ₂₂σ₃₃λ₃₄-σ₁₂²σ₃₄+σ₁₁σ₂₂σ₃₄,
-σ₁₂σ₁₄σ₂₂+σ₁₂²σ₂₃λ₃₄-σ₁₁σ₂₂σ₂₃λ₃₄+σ₁₁σ₂₂σ₂₄

Single Door Criterion:

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

Instrumental Variable:

2 | {1}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₄

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂,
-σ₁₄σ₂₂σ₂₃²+σ₁₂σ₂₃²σ₂₄+σ₁₄σ₂₂²σ₃₃-σ₁₂σ₂₂σ₂₄σ₃₃+σ₁₄σ₂₃²ω₂₂-σ₁₃σ₂₃σ₂₄ω₂₂-σ₁₄σ₂₂σ₃₃ω₂₂+σ₁₂σ₂₃σ₃₄ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

σ₂₃²-σ₂₂σ₃₃+σ₂₂ω₃₃,
σ₁₃σ₂₃-σ₁₂σ₃₃+σ₁₂ω₃₃

Parameter: ω₄₄

Total Effect of X₁ on X₃: λ₁₂λ₂₃ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3} | {}

Total Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄+λ₁₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄ Identified

Algebraic Equations:

σ₁₁σ₂₃²λ₁₂λ₂₃λ₃₄-σ₁₃σ₂₃σ₂₄-σ₁₁σ₂₂σ₃₃λ₁₂λ₂₃λ₃₄+σ₁₂σ₂₃σ₃₄,
σ₁₁σ₁₃σ₂₃λ₁₂λ₂₃λ₃₄-σ₁₃²σ₂₄-σ₁₁σ₁₂σ₃₃λ₁₂λ₂₃λ₃₄+σ₁₂σ₁₃σ₃₄,
σ₁₁σ₁₃²λ₁₂λ₂₃λ₃₄-σ₁₃²σ₁₄-σ₁₁²σ₃₃λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₃σ₃₄,
σ₁₁σ₁₂σ₁₃λ₁₂λ₂₃λ₃₄-σ₁₂σ₁₃σ₁₄-σ₁₁²σ₂₃λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₃σ₂₄,
σ₁₁σ₁₂²λ₁₂λ₂₃λ₃₄-σ₁₂²σ₁₄-σ₁₁²σ₂₂λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₂σ₂₄,
σ₁₄σ₂₃⁴λ₁₂λ₂₃λ₃₄-σ₁₃σ₂₃³σ₂₄λ₁₂λ₂₃λ₃₄-2σ₁₄σ₂₂σ₂₃²σ₃₃λ₁₂λ₂₃λ₃₄+σ₁₂σ₂₃²σ₂₄σ₃₃λ₁₂λ₂₃λ₃₄+σ₂₃²σ₂₄²σ₃₃+σ₁₄σ₂₂²σ₃₃²λ₁₂λ₂₃λ₃₄+σ₁₂σ₂₃³σ₃₄λ₁₂λ₂₃λ₃₄-σ₂₃³σ₂₄σ₃₄-σ₁₂σ₂₂σ₂₃σ₃₃σ₃₄λ₁₂λ₂₃λ₃₄-σ₂₂σ₂₃σ₂₄σ₃₃σ₃₄+σ₂₂σ₂₃²σ₃₄²

Total Effect of X₂ on X₄: λ₂₃λ₃₄ Identified

Each Parameter Identified

Back Door Criterion:


{2} ⊥ {3,4} | {1}

Vanishing Ideal

Constraint Set:

σ₁₃σ₂₂-σ₁₂σ₂₃,
σ₁₃σ₁₄σ₂₃-σ₁₃²σ₂₄-σ₁₂σ₁₄σ₃₃+σ₁₁σ₂₄σ₃₃+σ₁₂σ₁₃σ₃₄-σ₁₁σ₂₃σ₃₄

All constraints are determinantal

Trek-separation:

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