Parameter: λ12 Identified
Algebraic Equations:
σ11λ12-σ12
Single Door Criterion:
{1} ⊥ {2,3} | {}
Parameter: λ14 Identified
Algebraic Equations:
σ12σ13λ14-σ11σ23λ14+σ14σ23-σ13σ24,
σ122λ14-σ11σ22λ14+σ14σ22-σ12σ24
Single Door Criterion:
{1} ⊥ {3,4} | {2}
{1} ⊥ {4} | {2,3}
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:
σ12σ13λ34-σ12σ14-σ11σ23λ34+σ11σ24,
-σ12σ14σ22+σ122σ23λ34-σ11σ22σ23λ34+σ11σ22σ24
Instrumental Variable:
2 | {1}
Parameter: ω11 Identified
Algebraic Equations:
-σ11+ω11
Parameter: ω22 Identified
Algebraic Equations:
σ122-σ11σ22+σ11ω22
Parameter: ω33 Identified
Algebraic Equations:
σ232-σ22σ33+σ22ω33,
σ13σ23-σ12σ33+σ12ω33
Parameter: ω34 Identified
Algebraic Equations:
-σ13σ14σ23+σ132σ24+σ12σ14σ33-σ11σ24σ33-σ12σ13σ34+σ11σ23σ34+σ12σ13ω34-σ11σ23ω34,
-σ12σ14σ232+σ12σ13σ23σ24+σ12σ14σ22σ33-σ11σ22σ24σ33-σ122σ23σ34+σ11σ22σ23σ34+σ122σ23ω34-σ11σ22σ23ω34
Parameter: ω44 Identified
Algebraic Equations:
σ11σ142σ232-2σ11σ13σ14σ23σ24+σ11σ132σ242-σ122σ142σ33+2σ11σ12σ14σ24σ33-σ112σ242σ33+2σ122σ13σ14σ34-2σ11σ12σ14σ23σ34-2σ11σ12σ13σ24σ34+2σ112σ23σ24σ34-σ122σ132σ44+2σ11σ12σ13σ23σ44-σ112σ232σ44+σ122σ132ω44-2σ11σ12σ13σ23ω44+σ112σ232ω44,
σ11σ142σ22σ232-2σ11σ12σ14σ232σ24+σ11σ12σ13σ23σ242-σ122σ142σ22σ33+2σ11σ12σ14σ22σ24σ33-σ112σ22σ242σ33+2σ123σ14σ23σ34-2σ11σ12σ14σ22σ23σ34-2σ11σ122σ23σ24σ34+2σ112σ22σ23σ24σ34-σ123σ13σ23σ44+2σ11σ122σ232σ44-σ112σ22σ232σ44+σ123σ13σ23ω44-2σ11σ122σ232ω44+σ112σ22σ232ω44,
σ11σ142σ222σ232-2σ11σ12σ14σ22σ232σ24+σ11σ122σ232σ242-σ122σ142σ222σ33+2σ11σ12σ14σ222σ24σ33-σ112σ222σ242σ33+2σ123σ14σ22σ23σ34-2σ11σ12σ14σ222σ23σ34-2σ11σ122σ22σ23σ24σ34+2σ112σ222σ23σ24σ34-σ124σ232σ44+2σ11σ122σ22σ232σ44-σ112σ222σ232σ44+σ124σ232ω44-2σ11σ122σ22σ232ω44+σ112σ222σ232ω44
Total Effect of X1 on X3: λ12λ23 Identified
Each Parameter Identified
Back Door Criterion:
{1} ⊥ {2,3,4} | {}
Total Effect of X1 on X4: λ12λ23λ34+λ14 Identified
Each Parameter Identified
Back Door Criterion:
{1} ⊥ {2,3,4} | {}
Total Effect of X2 on X4: λ23λ34 Identified
Each Parameter Identified
Back Door Criterion:
{2} ⊥ {3,4} | {1}
Constraint Set:
σ13σ22-σ12σ23
All constraints are determinantal
Trek-separation:
{1,2} ⊥ {2,3} | ({},{2})
Algebraic Equations:
σ11λ12-σ12
Single Door Criterion:
{1} ⊥ {2,3} | {}
Parameter: λ14 Identified
Algebraic Equations:
σ12σ13λ14-σ11σ23λ14+σ14σ23-σ13σ24,
σ122λ14-σ11σ22λ14+σ14σ22-σ12σ24
Single Door Criterion:
{1} ⊥ {3,4} | {2}
{1} ⊥ {4} | {2,3}
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:
σ12σ13λ34-σ12σ14-σ11σ23λ34+σ11σ24,
-σ12σ14σ22+σ122σ23λ34-σ11σ22σ23λ34+σ11σ22σ24
Instrumental Variable:
2 | {1}
Parameter: ω11 Identified
Algebraic Equations:
-σ11+ω11
Parameter: ω22 Identified
Algebraic Equations:
σ122-σ11σ22+σ11ω22
Parameter: ω33 Identified
Algebraic Equations:
σ232-σ22σ33+σ22ω33,
σ13σ23-σ12σ33+σ12ω33
Parameter: ω34 Identified
Algebraic Equations:
-σ13σ14σ23+σ132σ24+σ12σ14σ33-σ11σ24σ33-σ12σ13σ34+σ11σ23σ34+σ12σ13ω34-σ11σ23ω34,
-σ12σ14σ232+σ12σ13σ23σ24+σ12σ14σ22σ33-σ11σ22σ24σ33-σ122σ23σ34+σ11σ22σ23σ34+σ122σ23ω34-σ11σ22σ23ω34
Parameter: ω44 Identified
Algebraic Equations:
σ11σ142σ232-2σ11σ13σ14σ23σ24+σ11σ132σ242-σ122σ142σ33+2σ11σ12σ14σ24σ33-σ112σ242σ33+2σ122σ13σ14σ34-2σ11σ12σ14σ23σ34-2σ11σ12σ13σ24σ34+2σ112σ23σ24σ34-σ122σ132σ44+2σ11σ12σ13σ23σ44-σ112σ232σ44+σ122σ132ω44-2σ11σ12σ13σ23ω44+σ112σ232ω44,
σ11σ142σ22σ232-2σ11σ12σ14σ232σ24+σ11σ12σ13σ23σ242-σ122σ142σ22σ33+2σ11σ12σ14σ22σ24σ33-σ112σ22σ242σ33+2σ123σ14σ23σ34-2σ11σ12σ14σ22σ23σ34-2σ11σ122σ23σ24σ34+2σ112σ22σ23σ24σ34-σ123σ13σ23σ44+2σ11σ122σ232σ44-σ112σ22σ232σ44+σ123σ13σ23ω44-2σ11σ122σ232ω44+σ112σ22σ232ω44,
σ11σ142σ222σ232-2σ11σ12σ14σ22σ232σ24+σ11σ122σ232σ242-σ122σ142σ222σ33+2σ11σ12σ14σ222σ24σ33-σ112σ222σ242σ33+2σ123σ14σ22σ23σ34-2σ11σ12σ14σ222σ23σ34-2σ11σ122σ22σ23σ24σ34+2σ112σ222σ23σ24σ34-σ124σ232σ44+2σ11σ122σ22σ232σ44-σ112σ222σ232σ44+σ124σ232ω44-2σ11σ122σ22σ232ω44+σ112σ222σ232ω44
Total Effect of X1 on X3: λ12λ23 Identified
Each Parameter Identified
Back Door Criterion:
{1} ⊥ {2,3,4} | {}
Total Effect of X1 on X4: λ12λ23λ34+λ14 Identified
Each Parameter Identified
Back Door Criterion:
{1} ⊥ {2,3,4} | {}
Total Effect of X2 on X4: λ23λ34 Identified
Each Parameter Identified
Back Door Criterion:
{2} ⊥ {3,4} | {1}
Vanishing Ideal
Constraint Set:
σ13σ22-σ12σ23
All constraints are determinantal
Trek-separation:
{1,2} ⊥ {2,3} | ({},{2})