Question

陳雨芊

左邊紅框處不是應該放summationX*×Y嗎為何是summationXY阿？

例題 2017 另一種形式的衡量誤差
(10%) The parameter ß is defined in the model Y₁ = BX + u₁, where u; is independent of
|X₁, E(u₁) = 0, E(u²) = 0², the observables are (Y₁, X₁) where X₁ = 2
random measurement error. Assume that v; is independent of X and
Vi
Uj.
Also assume that
X; and X are non-negative and real-valued. Consider the least-squares estimator for B.
"Jon at Y ni 10713
2. (5%) Find a condition under which ß is consistent for ß?
1. (5%) Find the probability limit of ß, expressed in terms of ß and moments of (X
(Xi, Vis Uj).
《108 台大經研》
ΣX; Y;
ΣX²
《解》
| 已知 X = X*v, 將 X* = X/v 代入得] Y = BX* +u = BX/v+u。此爲無截距模型, 故
2X* ( )
Σ₁ X₁Y;/n_p_E(XY)
B
將 X = X*v,且 v - X*, v lu 代入可得到
BP B
ΣX²/n
i=1
¸E [(X*v)² /v]_E[X*vu]
E(v)
= BE(v²)
E(X²)
U = (S), X)-(SA
EXB- + u
V
E (X²)
atlo)
co Beta, u) + Cov (Beta, e
lo)
1mm wod2 (2) €
Xv₁ and v₁ > o is
4)] E(X²/v)_ E(Xu)
= B-
E(X²) E(X²)
区一这三
E[(X*)²] E(v) __E(X*)E(v)E(u)
+
=
E[(X*v)²] E [(X*v)²]E[(X+)²] E(v²) E[(X*)²] E(v²)
= ß ↔ E(v) = E(v²)
+
+