Parameter: λ₁₂ Identified

Algebraic Equations:

σ₁₃λ₁₂-σ₂₃,
σ₁₁λ₁₂-σ₁₂

Single Door Criterion:

{1} ⊥ {2,4} | {}
{1} ⊥ {2,4} | {3}

Instrumental Variable:

3 | {}

Parameter: λ₁₃ Identified

Algebraic Equations:

σ₁₄λ₁₃-σ₃₄,
σ₁₂λ₁₃-σ₂₃,
σ₁₁λ₁₃-σ₁₃

Single Door Criterion:

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

Instrumental Variable:

2 | {}
2 | {4}
4 | {}
4 | {2}

Parameter: λ₂₄ Identified

Algebraic Equations:

σ₂₃λ₂₄-σ₃₄,
σ₁₂λ₂₄-σ₁₄

Instrumental Variable:

1 | {}
1 | {3}
3 | {}

Parameter: ω₁₁ Identified

Algebraic Equations:

-σ₁₁+ω₁₁

Parameter: ω₂₂ Identified

Algebraic Equations:

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

Parameter: ω₂₄ Identified

Algebraic Equations:

-σ₂₃σ₂₄+σ₂₂σ₃₄+σ₂₃ω₂₄,
σ₁₄σ₂₂-σ₁₂σ₂₄+σ₁₂ω₂₄

Parameter: ω₃₃ Identified

Algebraic Equations:

-σ₁₄σ₃₃+σ₁₃σ₃₄+σ₁₄ω₃₃,
σ₁₃σ₂₃-σ₁₂σ₃₃+σ₁₂ω₃₃,
σ₁₃²-σ₁₁σ₃₃+σ₁₁ω₃₃

Parameter: ω₄₄ Identified

Algebraic Equations:

2σ₂₃σ₂₄σ₃₄-σ₂₂σ₃₄²-σ₂₃²σ₄₄+σ₂₃²ω₄₄,
-σ₁₄σ₂₂σ₃₄+2σ₁₂σ₂₄σ₃₄-σ₁₂σ₂₃σ₄₄+σ₁₂σ₂₃ω₄₄,
-σ₁₄²σ₂₂+2σ₁₂σ₁₄σ₂₄-σ₁₂²σ₄₄+σ₁₂²ω₄₄

Total Effect of X₁ on X₄: λ₁₂λ₂₄ Identified

Each Parameter Identified

Back Door Criterion:


{1} ⊥ {2,3,4} | {}

Vanishing Ideal

Constraint Set:

σ₁₄σ₂₃-σ₁₂σ₃₄,
σ₁₃σ₁₄-σ₁₁σ₃₄,
σ₁₂σ₁₃-σ₁₁σ₂₃

All constraints are determinantal

Trek-separation:

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