Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2} | {}

Parameter: λ₁₃

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₁₂²λ₂₃-σ₁₂σ₁₃-σ₁₁σ₂₂λ₂₃+σ₁₁σ₂₃

Single Door Criterion:

{2} ⊥ {3} | {1}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₁₃σ₂₂λ₂₄-σ₁₂σ₂₃λ₂₄+σ₁₄σ₂₃-σ₁₃σ₂₄

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₁₃σ₂₂λ₃₄-σ₁₄σ₂₂-σ₁₂σ₂₃λ₃₄+σ₁₂σ₂₄

Instrumental Variable:

1 | {2}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₃

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₃₃

Parameter: ω₃₄ Identified

Algebraic Equations:

-σ₁₄σ₂₃²+σ₁₃σ₂₃σ₂₄+σ₁₄σ₂₂σ₃₃-σ₁₂σ₂₄σ₃₃-σ₁₃σ₂₂σ₃₄+σ₁₂σ₂₃σ₃₄+σ₁₃σ₂₂ω₃₄-σ₁₂σ₂₃ω₃₄

Parameter: ω₄₄ Identified

Algebraic Equations:

σ₁₄²σ₂₂σ₂₃²-2σ₁₃σ₁₄σ₂₂σ₂₃σ₂₄+σ₁₃²σ₂₂σ₂₄²-σ₁₄²σ₂₂²σ₃₃+2σ₁₂σ₁₄σ₂₂σ₂₄σ₃₃-σ₁₂²σ₂₄²σ₃₃+2σ₁₃σ₁₄σ₂₂²σ₃₄-2σ₁₂σ₁₄σ₂₂σ₂₃σ₃₄-2σ₁₂σ₁₃σ₂₂σ₂₄σ₃₄+2σ₁₂²σ₂₃σ₂₄σ₃₄-σ₁₃²σ₂₂²σ₄₄+2σ₁₂σ₁₃σ₂₂σ₂₃σ₄₄-σ₁₂²σ₂₃²σ₄₄+σ₁₃²σ₂₂²ω₄₄-2σ₁₂σ₁₃σ₂₂σ₂₃ω₄₄+σ₁₂²σ₂₃²ω₄₄

Total Effect of X₁ on X₃: λ₁₂λ₂₃+λ₁₃

Path-specific Effect of X₁ on X₃: λ₁₂λ₂₃ Identified

Algebraic Equations:

σ₁₁σ₁₂²λ₁₂λ₂₃-σ₁₂²σ₁₃-σ₁₁²σ₂₂λ₁₂λ₂₃+σ₁₁σ₁₂σ₂₃

Total Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄+λ₁₂λ₂₄+λ₁₃λ₃₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₃λ₃₄ Identified

Algebraic Equations:

σ₁₁σ₁₂²σ₁₃σ₂₂λ₁₂λ₂₃λ₃₄-σ₁₂²σ₁₃σ₁₄σ₂₂-σ₁₁²σ₁₃σ₂₂²λ₁₂λ₂₃λ₃₄-σ₁₁σ₁₂³σ₂₃λ₁₂λ₂₃λ₃₄+σ₁₁²σ₁₂σ₂₂σ₂₃λ₁₂λ₂₃λ₃₄+σ₁₁σ₁₂σ₁₄σ₂₂σ₂₃+σ₁₂³σ₁₃σ₂₄-σ₁₁σ₁₂²σ₂₃σ₂₄

Path-specific Effect of X₁ on X₄: λ₁₂λ₂₄ Identified

Algebraic Equations:

σ₁₁σ₁₃σ₂₂λ₁₂λ₂₄-σ₁₁σ₁₂σ₂₃λ₁₂λ₂₄+σ₁₂σ₁₄σ₂₃-σ₁₂σ₁₃σ₂₄

Path-specific Effect of X₁ on X₄: λ₁₃λ₃₄

Total Effect of X₂ on X₄: λ₂₃λ₃₄+λ₂₄ Identified

Each Parameter Identified

Back Door Criterion:


{2} ⊥ {3,4} | {1}

Vanishing Ideal

Zero Vanishing Ideal