Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₃λ₁₂-σ₂₃,
σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2} | {}
{1} ⊥ {2,4} | {3}

Instrumental Variable:

3 | {}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₁₃σ₂₃λ₂₄-σ₁₂σ₃₃λ₂₄+σ₁₄σ₃₃-σ₁₃σ₃₄,
σ₁₁σ₂₃²λ₂₄-σ₁₂²σ₃₃λ₂₄+σ₁₂σ₁₄σ₃₃-σ₁₁σ₂₃σ₃₄

Instrumental Variable:

1 | {3}

Parameter: λ₃₄ Identified

Algebraic Equations:

σ₁₃σ₂₃λ₃₄-σ₁₄σ₂₃-σ₁₂σ₃₃λ₃₄+σ₁₂σ₃₄,
σ₁₃²λ₃₄-σ₁₃σ₁₄-σ₁₁σ₃₃λ₃₄+σ₁₁σ₃₄,
-σ₁₂σ₁₄σ₂₃+σ₁₁σ₂₃²λ₃₄-σ₁₂²σ₃₃λ₃₄+σ₁₂²σ₃₄

Single Door Criterion:

{3} ⊥ {2,4} | {1}
{3} ⊥ {4} | {1,2}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₃ Identified

Algebraic Equations:

-σ₁₃+ω₁₃

Parameter: ω₂₂ Identified

Algebraic Equations:

-σ₁₃σ₂₂+σ₁₂σ₂₃+σ₁₃ω₂₂,
σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₂₄ Identified

Algebraic Equations:

σ₁₄σ₂₃²-σ₁₃σ₂₃σ₂₄-σ₁₄σ₂₂σ₃₃+σ₁₂σ₂₄σ₃₃+σ₁₃σ₂₂σ₃₄-σ₁₂σ₂₃σ₃₄+σ₁₃σ₂₃ω₂₄-σ₁₂σ₃₃ω₂₄,
σ₁₂σ₁₄σ₂₃²-σ₁₁σ₂₃²σ₂₄-σ₁₂σ₁₄σ₂₂σ₃₃+σ₁₂²σ₂₄σ₃₃-σ₁₂²σ₂₃σ₃₄+σ₁₁σ₂₂σ₂₃σ₃₄+σ₁₁σ₂₃²ω₂₄-σ₁₂²σ₃₃ω₂₄

Parameter: ω₃₃ Identified

Algebraic Equations:

-σ₃₃+ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

σ₁₄²σ₂₃²σ₃₃-2σ₁₃σ₁₄σ₂₃σ₂₄σ₃₃-σ₁₄²σ₂₂σ₃₃²+2σ₁₂σ₁₄σ₂₄σ₃₃²+2σ₁₃²σ₂₃σ₂₄σ₃₄+2σ₁₃σ₁₄σ₂₂σ₃₃σ₃₄-2σ₁₂σ₁₄σ₂₃σ₃₃σ₃₄-2σ₁₁σ₂₃σ₂₄σ₃₃σ₃₄-σ₁₃²σ₂₂σ₃₄²+σ₁₂²σ₃₃σ₃₄²-σ₁₃²σ₂₃²σ₄₄+2σ₁₁σ₂₃²σ₃₃σ₄₄-σ₁₂²σ₃₃²σ₄₄+σ₁₃²σ₂₃²ω₄₄-2σ₁₁σ₂₃²σ₃₃ω₄₄+σ₁₂²σ₃₃²ω₄₄,
σ₁₂σ₁₄²σ₂₃²σ₃₃-2σ₁₁σ₁₄σ₂₃²σ₂₄σ₃₃-σ₁₂σ₁₄²σ₂₂σ₃₃²+2σ₁₂²σ₁₄σ₂₄σ₃₃²+2σ₁₁σ₁₃σ₂₃²σ₂₄σ₃₄-2σ₁₂²σ₁₄σ₂₃σ₃₃σ₃₄+2σ₁₁σ₁₄σ₂₂σ₂₃σ₃₃σ₃₄-2σ₁₁σ₁₂σ₂₃σ₂₄σ₃₃σ₃₄-σ₁₁σ₁₃σ₂₂σ₂₃σ₃₄²+σ₁₂³σ₃₃σ₃₄²-σ₁₁σ₁₃σ₂₃³σ₄₄+2σ₁₁σ₁₂σ₂₃²σ₃₃σ₄₄-σ₁₂³σ₃₃²σ₄₄+σ₁₁σ₁₃σ₂₃³ω₄₄-2σ₁₁σ₁₂σ₂₃²σ₃₃ω₄₄+σ₁₂³σ₃₃²ω₄₄,
σ₁₂²σ₁₄²σ₂₃²σ₃₃-2σ₁₁σ₁₂σ₁₄σ₂₃²σ₂₄σ₃₃-σ₁₂²σ₁₄²σ₂₂σ₃₃²+2σ₁₂³σ₁₄σ₂₄σ₃₃²+2σ₁₁²σ₂₃³σ₂₄σ₃₄-2σ₁₂³σ₁₄σ₂₃σ₃₃σ₃₄+2σ₁₁σ₁₂σ₁₄σ₂₂σ₂₃σ₃₃σ₃₄-2σ₁₁σ₁₂²σ₂₃σ₂₄σ₃₃σ₃₄-σ₁₁²σ₂₂σ₂₃²σ₃₄²+σ₁₂⁴σ₃₃σ₃₄²-σ₁₁²σ₂₃⁴σ₄₄+2σ₁₁σ₁₂²σ₂₃²σ₃₃σ₄₄-σ₁₂⁴σ₃₃²σ₄₄+σ₁₁²σ₂₃⁴ω₄₄-2σ₁₁σ₁₂²σ₂₃²σ₃₃ω₄₄+σ₁₂⁴σ₃₃²ω₄₄

Total Effect of X₁ on X₄: λ₁₂λ₂₄ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,4} | {3}

Vanishing Ideal

Constraint Set:

σ₁₂σ₁₃-σ₁₁σ₂₃

All constraints are determinantal

Trek-separation:

{1,2} ⊥ {1,3} | ({1},{})