Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2,3} | {}

Parameter: λ₂₃ Identified

Algebraic Equations:

σ₂₄λ₂₃-σ₃₄,
σ₂₂λ₂₃-σ₂₃,
σ₁₂λ₂₃-σ₁₃

Single Door Criterion:

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

Instrumental Variable:

1 | {}
1 | {4}
4 | {}
4 | {1}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₁₂σ₁₃λ₂₄-σ₁₃σ₁₄-σ₁₁σ₂₃λ₂₄+σ₁₁σ₃₄,
σ₁₂²λ₂₄-σ₁₂σ₁₄-σ₁₁σ₂₂λ₂₄+σ₁₁σ₂₄

Single Door Criterion:

{2} ⊥ {4} | {1}
{2} ⊥ {4} | {1,3}

Instrumental Variable:

3 | {1}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₁₄ Identified

Algebraic Equations:

σ₁₁σ₁₄σ₂₃-σ₁₁σ₁₂σ₃₄+σ₁₂σ₁₃ω₁₄-σ₁₁σ₂₃ω₁₄,
σ₁₁σ₁₄σ₂₂-σ₁₁σ₁₂σ₂₄+σ₁₂²ω₁₄-σ₁₁σ₂₂ω₁₄

Parameter: ω₂₂ Identified

Algebraic Equations:

σ₁₂²-σ₁₁σ₂₂+σ₁₁ω₂₂

Parameter: ω₃₃ Identified

Algebraic Equations:

-σ₂₄σ₃₃+σ₂₃σ₃₄+σ₂₄ω₃₃,
σ₂₃²-σ₂₂σ₃₃+σ₂₂ω₃₃,
σ₁₃σ₂₃-σ₁₂σ₃₃+σ₁₂ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

-σ₁₂σ₁₃σ₁₄²σ₂₃+2σ₁₂²σ₁₃σ₁₄σ₃₄-2σ₁₁σ₁₂²σ₃₄²+σ₁₁²σ₂₂σ₃₄²-σ₁₂²σ₁₃²σ₄₄+2σ₁₁σ₁₂σ₁₃σ₂₃σ₄₄-σ₁₁²σ₂₃²σ₄₄+σ₁₂²σ₁₃²ω₄₄-2σ₁₁σ₁₂σ₁₃σ₂₃ω₄₄+σ₁₁²σ₂₃²ω₄₄,
-σ₁₂²σ₁₄²σ₂₃+2σ₁₂³σ₁₄σ₃₄-2σ₁₁σ₁₂²σ₂₄σ₃₄+σ₁₁²σ₂₂σ₂₄σ₃₄-σ₁₂³σ₁₃σ₄₄+2σ₁₁σ₁₂²σ₂₃σ₄₄-σ₁₁²σ₂₂σ₂₃σ₄₄+σ₁₂³σ₁₃ω₄₄-2σ₁₁σ₁₂²σ₂₃ω₄₄+σ₁₁²σ₂₂σ₂₃ω₄₄,
-σ₁₂²σ₁₄²σ₂₂+2σ₁₂³σ₁₄σ₂₄-2σ₁₁σ₁₂²σ₂₄²+σ₁₁²σ₂₂σ₂₄²-σ₁₂⁴σ₄₄+2σ₁₁σ₁₂²σ₂₂σ₄₄-σ₁₁²σ₂₂²σ₄₄+σ₁₂⁴ω₄₄-2σ₁₁σ₁₂²σ₂₂ω₄₄+σ₁₁²σ₂₂²ω₄₄

Total Effect of X₁ on X₃: λ₁₂λ₂₃ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3} | {}

Total Effect of X₁ on X₄: λ₁₂λ₂₄ Identified

Each Parameter Identified

Vanishing Ideal

Constraint Set:

σ₂₃σ₂₄-σ₂₂σ₃₄,
σ₁₃σ₂₄-σ₁₂σ₃₄,
σ₁₃σ₂₂-σ₁₂σ₂₃

All constraints are determinantal

Trek-separation:

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