Lineare Modelle und Konstrukte I |
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Online-MagazinRegressive Konstrukte |
Regressive Konstrukte |
17.02.2017 |
[79] [y-ε]n = [x-δ]n2[ß]2 , Y-ε = (X-δ) ß + α ,
[79a] Y-ε = ß1(X1-δ1) + ß2(X2 -δ2) +α , α = µY - ß1X1 - ß2X2
Normalgleichungnen I, II
[80] µ[ y-ε = ß1 (x1-δ1) + ß2(x2-δ2) |I *(x1-δ1), |II *(x2-δ2) ]
[81] I µ[(y-ε)(x1-δ1) = ß1(x1-δ1)² + ß2(x2-δ2)(x1-δ1)] , µ[(yx1) = ß1(ξ1)² + ß2(x1x2)], (24)
[82] II µ[(y-ε)(x2-δ2) = ß1(x2-δ2)(x1-δ1) + ß2(x2-δ2)] , µ[(yx2) = ß1(x1x2) + ß2(ξ2)²], (24)
Identifikationen
[83] determinantiv ß1 = [σ(yx1)σ²(ξ2) - σ(x1x2) σ(yx2)] / [σ²(ξ1)σ²(ξ2) - σ(x1x2)²]
= σ(yx1)σ²(ξ2) - σ(x1x2) σ(yx2), (25)
[84] ß2 = ... = σ(yx2)σ²(ξ1) - σ(x1x2) σ(yx1), (25) ,
α²(ξ1), α²(ξ2) tetradiel-instrumentell
[85] σ²(ξ1) = σ²(x1-δ1) = σ(x1x2) σ(x1y) / σ(x2y), (25) ,
[86] σ²(ξ2) = σ(x1x2)σ(x2y) /σ(x1y), (25)
Die Bedeutung von ρxy und ρξη
[87] y /σy = ß x/σx + e.. ; o = ∂ σ²e.. / ∂ ß ≠ 0 --> ß = σ[( y/σy )( x/σx )] / [σ²( x/σx)|=1]
σxy / σxσy = ρxy
[88] η/ση = β ξ/σξ + 0 ; ∂ 0² / ∂ ß = 0 --> ß = σ[(ξ /σξ)( η/ση)] / 1
σξη / σξση = ρξη = 1
Zusammenfassung
[89] η = ßηξ ξ + o ; η=y-ε, ξ = x-δ ;
[90] ßηξ = η / ξ , ßηξ = signσηξ * ση / σξ
[91] y - ε = ßηξ (x - δ) + 0,
[92] Y - ε = ßηξ (X - δ) + α + 0,
[93] σ[(Y – ε)(X - δ)] = σXY = σxy = σηξ = signσxy σησξ , (23-25)
[94] ∂0² / ∂ßηξ = 0 --> ßηξ = σxy / σ²x-δ = σηξ / σ²ξ , σ²ξ = ρxx σ²x ; ρxx = σ²ξ/ σ²x
[95] ∂0² / ∂α = 0 , α = µY -ßηξ µX = µ(Y-ε) - ßηξµ(X-δ) = µYη -ßηξµXξ
[96] µ[(y-ε)/ßηξ - (x-δ)]² = 0² --> ∂0² / ∂ßηξ = 0² --> ßηξ = σ²η /σxy ; σ²η = ρyy σ²y
[97] Y - ε = ß (X - δ) + α
[98] µ[ Y-ε - ß (X - δ) – α]² = 0² ; ∂ 0² / ∂ß = 0 --> ß = σxy / σX-δ
∂ 0² / ∂α = 0 --> α = µY – ßµX
[99] µ[(Y-ε)/ß - (X - δ) – α/ß]² = 0² ; ∂ 0² / ∂ß = 0 --> ß = σY-ε / σxy
∂ 0² / ∂α = 0 --> α = µY – ßµX
[99a] ßy = F(α) = μY / μX | σxy > 0 , (108 - 120)
= σxy/σ²ξ
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