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Ejercicios de econometria. Gonzalo Villa
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7/18/2019 Ejercicios Econometría - GONZALO VILLA
http://slidepdf.com/reader/full/ejercicios-econometria-gonzalo-villa 1/44
yi = βxi + ui
ˆβ =
n
i=1xiyi
σ2
β2 +ni=1
x2i
0
E (β − β )2 = σ2
σ2
β2 +ni=1
x2i
β
b(β, β ) = E (β )−
β
E (β )
0
β
b(β, β )
β
E(β ) = E
ni=1
xi(βxi + ui)
σ2
β2 +ni=1
x2i
E(β ) = E
ni=1
(βx2i + xiui)
σ2
β2 +
n
i=1
x2
i
E(β ) = E
β ni=1
x2i +
ni=1
xiui
σ2
β2 +ni=1
x2i
E(β ) = 1
σ2
β2 +ni=1
x2i
E[ni=1
βx2i +
ni=1
xiui]
E(β ) = 1
σ2
β2
+
ni=1 x
2
i
[β
n
i=1
x2i + E[
n
i=1
xiui] 0
]
7/18/2019 Ejercicios Econometría - GONZALO VILLA
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E[xiui] = 0
E(β ) =
β ni=1
x2i
σ2
β2 +
ni=1 x
2i
0
β
b(β, β ) =
β ni=1
x2i
σ2
β2 +ni=1
x2i
− β = β
ni=1
x2i
σ2
β2 +ni=1
x2i
− 1
β
0
E (β −β )2 = E
ni=1
xiyi
σ2
β2 +ni=1
x2i
− β
2
E (β −β )2 = E
ni=1
xiyi − σ2
β + β ni=1
x2i
σ2
β2 +ni=1
x2i
2
E (ˆβ −β )
2
= E β
n
i=1x2i +
n
i=1xiui − σ2
β − β n
i=1x2i
σ2
β2 +ni=1
x2i
2
E (β −β )2 =
E
ni=1
xiui − σ2
β
2
σ2
β2 +ni=1
x2i
2
E (β −β )2 =
E
[ni=1
xiui]2 − 2σ2
β
ni=1
xiui + [ σ2
β ]2
σ2
β2 +n
i=1
x2i
2
E [xiui] =0
E [uiuj] = 0
E (β −β )2 =
σ2ni=1
x2i + [ σ
2
β ]2
σ2
β2 +ni=1
x2i
2 =
σ2 ni=1
x2i + σ2
β2
σ2
β2 +ni=1
x2i
2
E (β − β )2 = σ2
σ2
β2 +n
i=1x2i
7/18/2019 Ejercicios Econometría - GONZALO VILLA
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V ar(β ) = E [β −E (β )]2
V ar(β ) = E β n
i=1
x2i +
n
i=1
xiui
σ2
β2 +ni=1
x2i
−β
n
i=1
x2i
σ2
β2 +ni=1
x2i
2
V ar(β ) = E
β ni=1
x2i +
ni=1
xiui −β ni=1
x2i
σ2
β2 +ni=1
x2i
2
V ar(β ) = E
ni=1
xiui
σ2
β2 +n
i=1x2i
2
=
E [ni=1
xiui]2
σ2
β2 +n
i=1x2i
2
V ar(β ) =
σ2ni=1
x2i
σ2
β2 +ni=1
x2i
2
V ar(β MCO) − V ar(β ) = σ2
n
i=1x2i
−σ2
ni=1
x2i
σ2
β2 +n
i=1x2i
2
V ar(β MCO) − V ar(β ) =
σ2σ2
β2 +ni=1
x2i
2− σ2
ni=1
x2i
2
σ2
β2 +ni=1
x2i
2 ni=1
x2i
V ar(β MCO ) − V ar(β ) =
σ2
σ2
β2 +ni=1
x2i
2− ni=1
x2i
2
σ2
β2 +ni=1
x2i
2 ni=1
x2i
V ar(β MCO) − V ar(β ) =σ
2σ2β2 2
+ 2σ2
β2
ni=1 x
2i + n
i=1 x2i 2
− ni=1 x
2i 2
σ2
β2 +ni=1
x2i
2 ni=1
x2i
V ar(β MCO) − V ar(β ) =
σ2
σ2
β2
2+ 2 σ
2
β2
ni=1
x2i
σ2
β2 +ni=1
x2i
2 ni=1
x2i
> 0
Y t = α + βX t + ut
β
7/18/2019 Ejercicios Econometría - GONZALO VILLA
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β 1 =
t Y tt Xt
β 4 =
t ytt xt
β 2 = 1T
tY tXt
β 5 = 1T
iytxt
β 3 =
t XtY tt X
2
t
β 6 =
t xtytt x
2
t
t = 1
t = T
T
E ( β 1)
E ( β 1) = E
t Y t
t X t
E ( β 1) = E
t(α + βX t + ut)t X t
= E
T αt X t
+ β +
t utt X t
E ( β 1) = E
t(α + βX t + ut)t X t
=
T αt X t
+ β +
t E (ut)
0t X t
V ar( β 1)
V ar( β 1) = V ar T α
t X t+ β +
t utt X t
=
t V ar(ut)t X t
2
V ar( β 1) = T σ2t X t
2
E ( β 2)
E ( β 2) = 1
T E
t
Y tX t
=
1
T E
t
( α
X t+ β +
utX t
)
E ( β 2) = 1
T E
αt
1
X t+ T β +
t
utX t
E ( β 2) = αT t
1X t
+ β + 1T t
E (ut)
0X t
V ar( β 2)
V ar( β 2) = 1
T 2V ar
t
Y tX t
V ar( β 2) = 1
T 2V ar
αt
1
X t+ T β +
t
utX t
V ar( β 2) = 1
T 2 tV ar(ut)
X 2t
V ar( β 2) = σ2
T 2
t
1
X 2t
7/18/2019 Ejercicios Econometría - GONZALO VILLA
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E ( β 3)
E ( β 3) = E
t X tY tt X 2t
= E
α
t X tt X 2t
+ β
t X 2tt X 2t
+
t X tutt X 2t
E ( β 3) = αt X t
t X 2t+ β +
t X t E (ut) 0
t X 2t
V ar( β 3)
V ar( β 3) = V arα
t X tt X 2t
+ β
t X 2tt X 2t
+
t X tutt X 2t
V ar( β 3) =
t X 2t V ar(ut)
t X 2t= σ2
E ( β 4)
E ( ˆβ 5)
E ( β 5) = 1
T E
t
ytxt
E ( β 5) = 1
T E
t
α + βX t + ut − α − β X − u
xt
E ( β 5) = 1
T E
t
α + βX t + ut − α − β X − u
xt
E ( β 5
) = 1
T E
t
β (X t − X )
xt
+ut − u
xt
E ( β 5) = 1
T E
T β +t
ut − u
xt
=
1
T
T β +
t
E (ut − u) 0
xt
E ( β 5) = β
V ar( β 5)
V ar( β 5) = 1
T 2V ar
T β +
t
ut − u
xt
V ar( β 5) = 1
T 2
t
(V arut − u
xt
) − 2
i
t
i<t
1
xi
1
xtCov(ui − u, ut − u)
−σ2
T
= 1
T 2
t
V ar(ut − u)
x2t
+
σ2
T 2i
t
i<t
1
xi
1
xt
V ar( β 5) = 1
T 2
t
V ar(ut) + V ar(u) − 2Cov(ut, u)
x2t
+
σ 2
T 2i
t
i<t
1
xi
1
xt
V ar( β 5) = 1T 2
t
σ2 + σ2
T − 2σ2
T
x2t
+ σ 2
T 2i
t
i<t
1xi
1xt
7/18/2019 Ejercicios Econometría - GONZALO VILLA
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V ar( β 5) = 1
T 2
(σ2 − σ 2
T )t
1
x2t
+
σ2
T 2i
t
i<t
1
xi
1
xt
β 6
E ( β 6) = β V ar( β 6) = σ2t x2t
β 6
β 5
β 6
β 5
yi = β 1 + β 2xi + ui
y∗i = α1 + α2x∗i + ui
y∗ x∗ α2 = β 2S xS y
S x S y
x
y
α2 =
y∗i x∗i(x∗i )2
=
(yi−y)(xi−x)
S yS x
(xi−x)2
S 2x
α2 =
(yi − y)(xi − x)
(xi − x)2
S xS y
= β 2
S xS y
β yx
β xy
y
x
x
y
β yxβ xy = R2
R2 y x
y
x
R
2
R2 =
(yi − y)2(yi − y)2
=
(α + βxi − y)2
(yi − y)2
R2 =
(y − β x + βxi − y)2
(yi − y)2
R2 =
(βxi − β x)2
(yi − y)2 =
(β (xi − x))2
(yi − y)2
R
2
= ˆβ
2
yx(xi
− x)2(yi − y)2
β yxβ xy
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R2 = β yxβ yx
(xi − x)2(yi − y)2
= β yx
(yi − y)(xi − x)
(xi − x)2 ×
(xi − x)2(yi − y)2
R2
= ˆβ yx(yi
− y)(xi
− x)(yi − y)2 =
ˆβ yx
ˆβ xy
β
yi = α + βx + ui
δ xi = γ + δyi + vi
1
R2
R2 = β δ
δ
β R2 1
R2 = β δ = β 1
β = 1
ln y∗i = α1 + α2 ln x∗i + ui
ln yi = β 1 + β 2 ln xi + ui
y∗i = w1yi x∗i = w2xi w
R2
zi
z∗i
zi = ln xi
z∗i = ln x∗i
z∗i − z∗
z∗i − z∗ = ln w2 + zi − ln w2
n + zi
= zi − zi
α1 = ln w1 + ¯ln yi −
(ln w2 + ¯ln xi) α2
¯ln y ≡
ln yi
n
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α1 = ln w1 + ¯ln yi − α2 ln w2 − α2 ¯ln xi
α1 = ¯ln yi − β 2 ¯ln x β1
+ ln w1 − β 2 ln w2
V ar( α1) = V ar( β 1) + (ln w2)2V ar( β 2) − 2 ln w2Cov( β 1, β 2)
V ar( α1) = V ar( β 1) + (ln w2)2V ar( β 2) + 2 ¯ln x ln w2V ar( β 2)
V ar( α1) = V ar( β 1) + ((ln w2)2 + 2 ¯ln x ln w2)V ar( β 2)
α1
β 1
β 2
R2
ˆln y∗i −¯ln y∗i = ˆln yi− ¯ln yi
R2
y = Xβ + µ
X 1 X 2
β1 β2
y = X 1β1 + µ1
y = X 2β2 + µ2
y = Xβ + µ
y = X 1β1 + µ1
β1 = (X 1X 1)−1X 1y
y
y = P Xy + M Xy = X 1β1 + X 2β2 + M Xy
P X
X
M X
X
X 1
X 1y = X 1X 1β1 + X 1X 2β2 + X 1M Xy
X 1y = X 1X 1β1 + X 1X 2 O
β2 + X 1M X O
y
O X 1 X 2
O
M XX 1 = O
M X
X 1M X = (M XX 1) = (O) = O
(X 1X 1)−1
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(X 1X 1)−1X 1y = β1
y1 X 1
β2
yt = α + β 1xt1 + β 2xt2 + ut
X X =
33 0 0
0 40 200 20 60
X y =
132
2492
uu = 150
x1
x2
y
α β 1 β 2
β 2
β 2 = 0
x1
x2
0
y 132/33 = 4
X X =
n
x1
x2
x1
x2
1
x1x2
x2
x1x2
x2
2
X y =
yx1yx2y
β = (X X )−1X Y = 4−0,2
1,6
β
V ar(β) = σ2(X X )−1 = uu
n − 3(X X )−1
V ar(β) =
V ar(α) Cov(α, β 1) Cov(α, β 2)
Cov(α, β 1) V ar( β 1) Cov( β 1, β 2)
Cov(α, β 2) Cov( β 1, β 2) V ar( β 2)
=
150
30
0,03030303 0 0
0 0,03 −0,010
−0,01 0,02
t
t =β 2
V ar( β 2)=
1,6
0,1 = 16
β y
X c
(y −Xc)(y −Xc)− (y −X β)(y −X β) = (c− β)X X (c− β)
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= (y − cX )(y −Xc) − (y − βX )(y −X β)
= yy − yXc− cX y + cX Xc− yy + yX β + βX y − βX X β
= −yXc− cX y + cX Xc + yX β + βX y − βX X β
β = (X X )−1X Y
=−yXc− cX y + cX Xc + yX β + βX y − βX X (X X )−1 I
X y
= −yXc− c X y XXβ
+cX Xc + yX β
= (cX X − yX )c− (cX X − yX )β
= (cX X − yX βXX
)(c− β)
= (c − β)X X (c− β)
= (c− β)X X (c− β)
yi = α + βxi + µi
ln yi = α + βxi + µi
yi = α + β ln xi + µi
ln yi = α + β ln xi + µi
β
β
y
x
β 100β y
x
β
0,01β
y
x
β
y
x
y x
yt = β 0 + β 1X t + ut
t X t = 0
t Y t = 0
t X 2t = B
t Y 2t = E
t X tY t = F
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β 0 β 1
E/F
B
0 F/B
E
B/F
F/B 0
B + E 2
0
(B2/E ) − F
E − (F 2/B)
0 F/B
ˆβ 0 =
¯Y −
¯X
ˆβ 1 = 0 − 0(
ˆβ 1) = 0
β 1 =
t(X t − X )(Y t − Y )
t(X t − X )2 =
t X tY tt X 2t
= F
B
E − (F 2/B)
Y t = 0 + (F/B)X t
t
u2t =
t
(Y t − (F/B)X t)2 =
t
(Y 2t − 2(F/B)X tY t + (F 2/B2)X 2t )
t
u2t =
t
Y 2t − 2(F/B )t
X tY t + (F 2/B2)t
X 2t = E −2F 2/B + F 2/B = E − (F 2/B)
y = Xβ + u
β
u = y −X β y = Xγ + δ u+ v
γ δ
v
R2
X u γ δ
y = Xγ + v1
y = δ u+ v2
γ = (X X )−1X y
δ = (u
u)−1
u
y = (u
u)−1
(M Xy)y = (u
u)−1
y
M Xy = (u
u)−1
u
u = 1
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y
X
v = y −X γ − δ u = y −X β − u = u− u = 0
R2
y
R2 = 1 − vv
(y − y)(y − y) = 1 − 0 = 1
Y i = α + βX i + ui ∀i : ui ∼ (0, σ2) ∀i, j : cov(ui, uj) = 0
α =
i λiY i λi = 1n − wi
X wi =xii x
2
i
xi
X i
xi = X i −
X
i λi = 1
i λiX i = 0
α
α
α =
i biY i
i bi = 1
i biX i = 0
bi = λi + f i
i f i = 0
i f iX i = 0
V ar(α) ≥ V ar(α)
α =
iλiY i =
i(
1
n − wi
X )Y i =
i(
1
n − xi
i x2i
X )Y i
α =i
yin − X
i xiY ii x2
i
= Y − X β
i
λi =i
(1
n − wi
X ) = n
n − X
i
wi = 1 − X
i xii x2
i
= 1 − X 0i x2
i
= 1
i
λiX i =i
(1
n − wi
X )X i = X − X i
wiX i
i λiX i = X
− X i xiX i
i x
2
i
= X
− X i x2
ii x
2
i
= X
− X = 0
α =
i biY iE (α) = E (
i biY i) = E [
i bi(α + βX i + ui)]
E (α) = E (α
i bi) + E (β
i biX i) + E (
i biui)
E (α) = α
i bi + β
i biX i + bi
i E (ui) 0
α
E (α) = α
β
0
i bi = 1
i biX i = 0
i f i =
i(bi −λi) =
i bi −
i λi = 1 − 1 = 0
i f iX i =
i biX i −
i λiX i = 0 − 0 = 0
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V ar(α) = V ar(i
biY i) =i
b2iσ2
V ar(α) = σ2 i (λi + f i)2 = σ2[i (λi)
2 + 2i λif i 0
+i f 2i ]
V ar(α) = σ2i
λ2i
V ar(α)
+σ2i
f 2i
V ar(α) ≥ V ar(α)
y = Xβ + u u∼
(0, σ2I ) K
E (ββ) = ββ + σ2K k=1
1
λk
λk X X
E (ββ) = E [(β + (X X )−1X u)(β + (X X )−1X u)]
E (ββ) = E [ββ + uX (X X )−1β + β(X X )−1X u+ uX (X X )−1(X X )−1X u]
E (ββ) = E [ββ] + E [u] 0
X (X X )−1β + β(X X )−1X E [u] 0
+E [uX (X X )−1(X X )−1X u]
E (ββ) = ββ + E [uX (X X )−1(X X )−1X u]
1 × 1
E [uX
(X X
)−1
(X X
)−1X u
] = E [tr(uX
(X X
)−1
(X X
)−1X u
)]
E [uX (X X )−1(X X )−1X u] = E [tr(X (X X )−1(X X )−1X uu)]
E [uX (X X )−1(X X )−1X u] = tr(X (X X )−1(X X )−1X E [uu] σ2I
)
E [uX (X X )−1(X X )−1X u] = σ2tr(X (X X )−1(X X )−1X )
E [uX (X X )−1(X X )−1X u] = σ2tr((X X )−1 (X X )−1X X I
)
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X X C ΛC
C
X X Λ X X
(X X )−1 = (C ΛC )−1
(X X )−1 = (C )−1Λ−1C −1 = C Λ−1C −1
Λ−1 =
1λ1
0 · · · 0
0 1λ2
· · · 0
0 0 · · · 1λK
tr(C Λ−1C −1) = tr(Λ−1 C −1C I
) = tr(Λ−1)
Λ−1
k
1λk
E (ββ) = ββ + σ2tr(Λ−1) = ββ + σ2K k=1
1
λk
y
−2X y + 2X X β = 0
X y −X X β = 0
X (y −X β) u
= 0
X
u
y
(X β)u = 0
βX u 0
= 0
β
0
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∀i : y∗i = α + βx∗i + ui,
y∗i
x∗i
x∗
y∗
0
β =
i x∗i y∗ii x∗2
i
=
i(xi − x)(yi − y)
i(xi − x)2
α = y∗ − x∗β = 0 − 0β = 0
0
yi = α + βxi + ui
100000
E (u) = 0
f (ui) = 1
λe−
uiλ
E (ui) = λ λ 0
α = y − xβ
β = β + i(xi − x)uii(xi − x)2
β
E (β ) = β +
i(xi − x)i(xi − x)2
E (ui) λ
i(xi − x) = 0
α
E (α) = α + E [
i uin
] = α +
i E (ui)
n = α +
nλ
n
E (α) = α + λ
β
α
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yi = βxi+ui
yx
E yx = 1x E i βxi + uin
E y
x
=
1
xE
i(βxi + ui)
n
E y
x
=
1
x[β i
xin
x
+E
i uin
]
E y
x = β +
i E (ui)
0
xn = β
V ar y
x
= V ar
i ui
xn
=
1
x2n2V ar(
i
ui)
V ar y
x
=
1
x2n2V ar(
i
ui) = σ2n
x2n2 =
σ2
x2n
V ar(β MCO ) = σ2i x2
i
yx
V ar y
x
− V ar(β MCO) =
σ2
x2n − σ2
i x2i
= σ2( 1
x2n − 1
i x2i
)
V ar y
x
− V ar(β MCO) = σ2(
i x2
i − nx2
nx2
i x2i
)
i(xi − x)2 = i x2i − nx2
V ar y
x
− V ar(β MCO) = σ2(
i(xi − x)2
nx2
i x2i
)
yx
cβ
β β
cβ
c
β c
β 0
β
β
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V ar(cβ) − V ar(cβ) = cV ar(β)c− cV ar(β)c
V ar(cβ)
−V ar(cβ) = c(V ar(β)c
−V ar(β)c)
V ar(cβ) − V ar(cβ) = c[V ar(β) − V ar(β)]c
β
V ar(β) − V ar(β) Z
V ar(cβ) − V ar(cβ) = c[Z ]c
Z
B
V ar(c
β) − V ar(c
β) = c
[B
B]c
V ar(cβ) − V ar(cβ) = (Bc)(Bc) = ww = w2
cβ
cβ
β
σ2
N (0, σ2I )
β
β = β + (X X )−1X u
σ2
σ2 = uM Xu
n − k
β
u
(X X )−1X L
u
σ2
uM Xu = (M Xu)M Xu
Cov(Lu, M Xu) = E (Luu
M X) = LE (uu
)M X
Cov(Lu, M Xu) = Lσ2IM X = σ2LM X O
LM X M X
X
y = Xβ +u
u ∼ N (0, σ2I )
X
X
ˆu
= 0
β
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∂ 2 ln L
∂α2 = − 1
σ2
∂ 2 ln L
∂β 2 = −i x2i
σ2
∂ 2 ln L
∂σ 2 =
n
2σ4 − 1
σ6
i
(yi −α − βxi)2
α
β
i x
2
i
σ2
σ2
∂ 2 ln L
∂σ 2 =
1
σ4 (
n
2 − 1
σ2 i
(yi − α − βxi)2
)
∂ 2 ln L
∂σ 2 =
1
σ4
nσ2 − 2
i(yi −α − βxi)2
2σ2
i(yi−α−βxi)2
nσ2
yt = α + βxt + ut
H o : β = β 0
n
1
n
t2β0 =
β −β 0
σ2/
(xi − x)2
2
t2β0 =
(β − β 0)2
(xi − x)2
σ2=
(β −β 0)2
(xi − x)2
[ u2
i ]/(n − 2)
σ2/σ2
1
(β −β 0)2 (xi − x)
2
σ2 ∼ χ2
(1)
[
u2i ]/σ2 = (n − 2) σ2/σ2 ∼ χ2
(n−2)
χ2(1)
χ2(n−2)
n − 2
F
1
t2β0 =
(β
−β 0)2 (xi
− x)2
[ u2
i ]/(n − 2) = F (1,n−2)
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yi =
eβ1+β2xi
1 + eβ1+β2xi
log(earnings) = β 0 + β 1alcohol + β 2educ + u1
alcohol = γ 0 + γ 1log(earnings) + γ 2educ + γ 3log( price) + u2
price
educ price β 1 β 2 γ 1 γ 2 γ 3
1 + eβ1+β2xi
yi + yieβ1+β2xi = eβ1+β2xi
yi = eβ1+β2xi − yieβ1+β2xi
yi = eβ1+β2xi(1 − yi)
yi(1 − yi)
= eβ1+β2xi
ln yi
(1 − yi) = β 1 + β 2xi
ln zi = β 1 + β 2xi + ui
zi = yi
(1−yi)
∀i : 0 < yi < 1
yi = 1
zi
1 < yi ≤ 0
ln zi
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log(earnings) = β 0 + β 1alcohol + β 2educ + u1
log( price) log( price)
alcohol
alcohol
educ
log( price)
log(earnings)
ˆalcohol
educ
yi = β 0 + β 1xi + ui
(n−2) σ2
σ2 =
i ui2
σ2 ∼ χ(n−2)
i ui
2
uM Xu
M X = I − P X = I − X (X X )−1X
M X
uM Xu
σ2 = M X = (M X)(M X)
= uσ ∼ N (0, I )
uM xuσ2 M X
M X
rank(M X) = tr(M X) = tr(I −X (X X )−1X ) = tr(I ) n
−tr(X (X X )−1X )
tr(X (X X )−1X ) = tr((X X )−1X X ) = tr(I )
k × k
rank(M X) = n − tr(I ) = n − k
2 M X 2
β
β
y = Xβ + u k
E [uu] = σ2I
rank(X ) = k
u
∼ N (0, σ2I )
E [u] = 0
rank(X ) = k
β
y = Xβ +u
E [β] = 0 V ar[β] = σ2(X X )−1
Cov(β, u) = 0
β
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E [u] = 0 rank(X ) = k
β
u 0
E [β] = β + (X X )−1X E [u] 0
E [β] = 0
V ar[β] = σ2(X X )−1
R2
R2
R2
y = 2,20 + 0,10x2 − 3,48x3 + 0,34x4
H 0 : y = β 1 + β 2x2 + β 3x3 + β 4x4 + u
H 1 : y = β 1 + β 2x2 + β 3x3 + β 4x4 + β 5x5 + β 6x6 + β 7x7 + u,
x5
x6
x7
F
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F = (RSSR − USSR)/r
USSR/(n − k) ,
RSSR
USSR
r
n − k
y
F
RTSS = RESS + RSSR = 109,6 + 18,48 = 128,08
USSR = UTSS RTSS
−UESS = 128,08 − 114,8 = 13,28
3
76 − 7 = 69
F
F = (18,48 − 13,28)/3
13,28/69 = 9,006
F
F = (RSSR − SSR1 −SS R2)/k
(SSR1 + SSR2)/(n − 2k) ,
RSSR SSR1
SSR2
F = (18,48 − 9,32 − 7,46)/4
(9,32 + 7,46)/(76 − 8) = 1,722
y = X 1β1 + u
y = X 1β1 +X 2β2 +u yM 1y yMy y(M 1−M )y =
uR uR − uu
y(M 1−M )y/J yMy/(n−k−1)
F J n−k−1
y(M 1 −M )y = y(M 1y −My) = yM 1y − yMy
y(M 1 −M )y = (M 1y)M 1y − (My)My = uR uR − uu
uR uR − uu = (Dβ − r)[D(X X )−1D]−1(Dβ − r)
σ2
u
RuR−uu
σ2 χ2J
σ2
χ2n−k−1
y
(M 1 −M )y/J yMy/(n − k − 1) ∼ F J n−k−1
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E (µ) = E (α + β x + u) = α + β x + E (u) 0
E (µ) = α + β x
µ
α
t
(yt − µ) =t
(yt − y) = ny − ny = 0
µ = α + β x + u yT +1 α + βxT +1 + uT +1
yT +1 − yT +1 = α + β x + u − (α + βxT +1 + uT +1)
yT +1 − yT +1 = β (x − xT +1) + u − uT +1
E (yT +1 − yT +1) = E (β (x − xT +1) + u − uT +1) = β (x − xT +1) + E (u) 0
−E (uT +1) 0
β = 0 x = xT +1
β = 0
yT +1
yT +1 = α + βxT +1
yT +1
yT +1 = α + βxT +1 + uT +1
V ar(yT +1 − yT +1) = V ar(α + βxT +1 − α − βxT +1 − uT +1)
V ar(yT +1 − yT +1) = σ2
1 + 1
T +
xT +1 − x
t(xt − x)2
yT +1 − yT +1 = β (x − xT +1) + u − uT +1
V ar[yT +1 − yT +1] = V ar[β (x − xT +1) + u − uT +1]
V ar[yT +1 − yT +1] = V ar[u] + V ar[uT +1]
V ar[yT +1 − yT +1] = σ2
T + σ2 = σ2(1 +
1
T )
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xT +1 > x
y
k
X Z = XA A
Z
X
y Z y X
X P X = X (X X )−1X
Z
P Z = Z (Z Z )−1Z = XA(AX XA)−1AX
P Z = XAA−1 I
(X X )−1 (A)−1A I
X
P Z = X (X X )−1X = P X
X
Z Z
X P Z = P X M Z = M X M Z y
M Xy
β
yt = α +βxt+µt
E(ut) = 0
E (u2t ) = σ2
t
E(µtµs) = 0
αMCG
β MCG
σ2t = σ2
t
σ2t = kxt k
minα, β
s =t
1
σ2t
(yt −α − xtβ )2
∂s
∂α =
t
− 2
σ2t
(yt − α − xtβ ) = 0
∂s
∂β =
t
−2xtσ2t
(yt − α − xtβ ) = 0
α
α =
tytσ2t
− β
txtσ2t
t1σ2t
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α
β
β =
t
1σ2t
txtytσ2t
−tytσ2t
txtσ2t
tx2tσ2t t
1σ2t − t
xtσ2t
2
σ2t σ2
β =nσ4
t xtyt − 1
σ4
t yt
t xt
nσ4
t x2
t − 1σ4
t xt
2
β =
t xtyt −nxyt x2
t − nx2 =
t(xt − x)(yt − y)
t(xt − x)2
α
β
σ2
α =
1
σ2 t
yt −
1
σ2 β t
xt1σ2 n = t
ytn − β t
xtn
α = y − β x
σ2t = kxt
β =1k2
t
1xt
txtytxt
− 1k2
tytxt
txtxt
1k2
tx2txt
t
1xt− 1
k2
txtxt
2
β =
t
1xt
t yt −n
tytxt
t xtt1xt− n2
β =ny
t1xt− n
tytxt
nx
t1xt− n2
β =y
t1xt−
tytxt
x
t1xt− n
β α kxt σ2t
α =
tytkxt
−y
t1
xt−t
ytxt
x
t1
xt−n
txtkxt
t1kxt
α =
tytxt−y
t1
xt−t
ytxt
x
t1
xt−n
n
t1xt
α =
(x
t1
xt−n)
t
ytxt−nyt
1
xt+n
t
ytxt
x
t1
xt−n
t1xt
α =
x
t1
xt
t
ytxt−nt
ytxt−nyt
1
xt+n
t
ytxt
x
t1
xt−n
t
1xt
α =
t
1
xt
(x
t
yt
xt−ny)
x
t1
xt−n
t1xt
=xt
ytxt− ny
x
t1xt− n
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E (β) = β + (X X )−1X E (u)
u ∼ (0, σ2I )
β
E (β) = β
y = X 1β1 + x2β 2 + u
y = X 1β1 + u
E ( β1) = E [(X 1X 1)−1X
1y]
E ( β1) = E [(X 1X 1)−1X
1X 1β1]
β1
+E [(X 1X 1)−1X
1x2β 2] + E [(X
1X 1)−1X
1u]
0
E ( β1) = β1 + (X 1X 1)−1X
1x2β 2
yt = βxt + ut ut ∼ N ID(0, σ2t ) σ2
t = σ2t t = 1, 2,....,T
V ar(β MCO ) = σ2
t x2
t t
(
t x2t )2
V ar(β MCG) = σ2tx2tt
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β =
t ytxt
t x2
t
= β
t x2t +
t utxt
t x2
t
= β +
t utxt
t x2
t
E (β ) = β + 1t x2
t
t
xtE (ut) 0
V ar(β ) = 1
(
t x2t )2
t
x2tV ar(ut) =
1
(
t x2t )2
t
x2tσ2t
V ar(β ) = σ2
(
t x2t )2
t
x2t t
ut
minβ
s =t
1
t(yt −xtβ )2
∂s
∂ β
=
−2t
ytxt
t
+ 2β tx2t
t
= 0
β
β t
x2t
t =
t
ytxtt
β MCG =
tytxtt
tx2tt
β MCG
β MCG = t
(xtβ+ut)xt
ttx2tt
= t
x2tβ+utxt
ttx2tt
β MCG = β
tx2tt +
tutxtt
tx2tt
= β +
tutxtt
tx2tt
V ar(β MCG) = 1tx2tt
2
t
x2t
t2 V ar(ut) =
σ2tx2tt
2
t
x2t t
t2
V ar(β MCG) = σ2
t x
2
tt
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y = α + βx + u
z x
β =
i(yi−y)(zi−z)i(xi−x)(zi−z)
β IV = y1 − y0
x1 − x0
y0 x0 yi xi zi = 0
y1
x1
yi
xi
zi = 1
k zi = 1 zi = 0
n − k
zi = 1
y1 =
i yizik
x1 =
i xizik
zi = 0 y0 =
i yi(1−zi)n−k x0 =
i xi(1−zi)n−k
β IV =
i yizik −
i yi(1−zi)n−k
i xizik −
i xi(1−zi)n−k
β IV =
(n−k)
i yizi−k
i yi(1−zi)k(n−k)
(n−k)
i xizi−k
i xi(1−zi)k(n−k)
β IV = (n − k)
i yizi −k
i yi(1 − zi)
(n − k)
i xizi −k
i xi(1 − zi)
β IV = n
i yizi − k
i yizi −k
i yi + k
i yizin
i xizi − k
i xizi −k
i xi + k
i xizi
β IV = n
i yizi −k
i yin
i xizi −k
i xi
nn
β IV = iyizi
−ky
i xizi − kx = iyizi
− yi
zii xizi − x
i zi
β IV =
i(yi − y)zii(xi − x)zi
=
i(yi − y)(zi − z)i(xi − x)(zi − z)
yt = µ + t E (t) = 0 V ar(t) = σ2xt xt > 0
µ
µ
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minµ
s =t
1
xt(yt − µ)2
∂s
∂ µ = −2
t
ytxt
+ 2µt
1
xt= 0
µ
µt
1
xt=t
ytxt
µ =
tytxt
t1xt
µ
V ar(µ) = V ar(
tµ+txt
t1xt
) = V ar(
tµxt
+
ttxt
t1xt
)
V ar(µ) = V ar(
tµxt
t1xt
+
ttxt
t1xt
) = V ar(µ
t1xt
t1xt
+
ttxt
t1xt
)
V ar(µ) = V ar(µ +
ttxt
t1xt
) = 1
[
t1xt
]2V ar(
t
txt
)
0
V ar(µ) = 1
[
t1xt
]2
t
V ar[txt
] = 1
[
t1xt
]2
t
σ2xtx2t
V ar(µ) =σ2
t1xt
[
t1xt
]2 =
σ2t
1xt
µ
µ = y =
t ytn
µ = t µ +t t
n
= t µ
n
+ t t
n
µ = µ +
t tn
V ar(µ) = 1
n2
t
V ar(t) = σ2
n2
t
xt
yi = βxi + ui
E (ui) = 0 E (u2i ) = σ2
i
σ2i = σ2zi
zi
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σ2(X X )−1
[
i viwi]2 ≤ [
i v2i ][
i w2i ]
u
Ω β
β MCG = (xΩ−1x)−1xΩ−1y
V ar(β MCG) = σ2
(x
Ω−1
x)−1
Ω =
z1 0 0 · · · 00 z2 0 · · · 0
0 0
· · · 0
zn−1
0 0 0 0 zn
β MCG = ixiyiziix2
izi
V ar(β MCG) = σ2ix2izi
[
i viwi]2 ≤ [
i v2
i ][
i w2i ]
vi = xi√ zi
wi = xi√
zi
[i
xi√ zi
xi√
zi]2 ≤ [
i
x2i
zi][i
x2i zi]
[i
x2i ]2 ≤ [
i
x2i
zi][i
x2i zi]
1ix2izi
≤
i x2i zi
[
i x2i ]2
σ2
σ2
i
x2i
zi
≤ σ2
i x2
i zi[i
x2
i
]2
var(β MCG) ≤ V ar(β MCO)
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GP A = β 0 + β 1P C + u
P C
u
P C
P C
P C
P C
P C
P C
1
0
P C
score
girlhs
girlhs
girlhs
girlhs
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score = α + β 1girlhs + β 2ing + β 3IQ + β 4time + u
girlhs =
ing =
IQ =
time =
score =
girlhs
num
Cov(num, girhs)
= 0
Cov(num, u) = 0
consumo = α + βingreso + ut
consumo ingreso 0,7 95 %
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E (ui|xi) = 0 V ar(ui) = σ2 Cov(ui, uj) = 0
Q = α1P + β 1Z 1 + u1
Q = α2P + β 2Z 2 + u2
Q(= cantidad demandada u ofertada) P (= precio) Z 1(=
ingreso)
Z 2(= precio de las materias primas)
u1
u2
0
Q
P
Z 1
Z 2
α1 = 0
α2 = 0
Q
α1 = 0 α2 = 0 P
α1 = 0 α2 = 0 α1 = α2 P Q
P
P = Qα2
− β 2α2
Z 2 − u2
α2
α2 = 0 P
Q
Q = α1( Q
α2− β 2
α2Z 2 − u2
α2) + β 1Z 1 + u1
Q(1 − α1
α2) = β 1Z 1 − α1β 2
α2Z 2 − α1u2
α2+ u1
Q = β 1(1 − α1
α2)
π1
Z 1 − α1β 2(1 − α1
α2)α2
π2
Z 2 + − α1u2(1 − α1
α2)α2
+ u1(1 − α1
α2)
v1
P
α1 = 0
α2 = 0
α1P + β 1Z 1 + u1 = β 2Z 2 + u2
P = β 2α1 π3
Z 2 − β 1α1 π4
Z 1 + u2
α1− u1
α1 v2
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Q P
α1 = α2
1
−α1
1 −α2
B
Q
P = β 1 0
0 β 2 Z 1Z 2 + u1
u2
B α1 = α2
0
B−1 = 1
α1 −α2
−α2 α1
−1 1
B−1
QP
y
= − α2β1
α1−α2α1β2
α1−α2− β1α1−α2
β2α1−α2
π
Z 1Z 2
z
+ α1u2α1−α2 − α2u1
α1−α2u2α1−α2 − u1
α1−α2
v
β
y = Xβ + u
X = Zγ +
X
Z
X
X
y = P Z Xβ + u
β
βV I = (X P Z X )−1X P Z y
V ar(β) = V ar((X P Z X )−1X P Z u)
V ar(β) = (X P Z X )−1X P Z V ar(u)
σ2I
P Z X (X P Z X )
−1
V ar(β) = σ2(X P Z X )−1 X P Z X (X
P Z X )−1
I
= σ2(X P Z X )−1
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yi,e = β 0 + β 1xi,e,1 + β 2xi,e,2 + β 3xi,3 + ui,e
yi,e e i xi,e,1
xi,e,2
xi,3
yi = α0 + α1xi,1 + α2xi,2 + α3xi,k + ui
ui = m−1i
mi
e ui,e
i
e
V ar(ui,e) = σ 2
V ar(ui)
V ar(ui) = V ar(m−1i
mie
ui,e)
V ar(ui) = 1
m2i
mie
V ar(ui,e) = 1
m2i
mie
σ2
V ar(ui) = mi
m2i
σ2 = σ2
mi
√ mi
V ar(ui√
m1) = σ2
mimi = σ2
ui
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yi,e = β 0 + β 1xi,e,1 + β 2xi,e,2 + .. + β kxi,e,k + f i + vi,e
f i
i
vi,e
e
i ui,e = f i + vi,e
V ar(f i) = σ 2f V ar(vi,e) = σ 2
v f i vi,e
V ar(ui,e) = σ2f + σ2
v
σ2
e = g
vi,e
vi,g
Cov(ui,e, ui,g) = σ2f
ui = m−1i
mi
e ui,e
mi
i
V ar(ui) = σ2f +
σ2
v
mi
i
V ar(ui,e) = V ar(f i + vi,e) = V ar(f i) + V ar(vi,e)
V ar(ui,e) = σ2f + σ2
v = σ2
ui,e
f i
vi,e
0
Cov(ui,e, ui,g) = E [ui,eui,g] − E [ui,e]E [ui,g]
Cov(ui,e, ui,g) = E [(f i + vi,e)(f i + vi,g)] − E [f i + vi,e]E [f i + vi,g]
Cov(ui,e, ui,g) = E [f 2i +f ivi,g+f ivi,e+vi,evi,g]−E [f i]E [f i]+E [f i]E [vi,g]+E [vi,e]E [vi,g]+E [f i]E [vi,e]
Cov(ui,e, ui,g) = E [f 2i ] − E [f i]E [f i]
V ar(f i)
+ E [f ivi,g ] − E [f i]E [vi,g]
Cov(f i,vi,g)
+ E [f ivi,e] − E [f i]E [vi,e]
Cov(f i,vi,e)
+ E [vi,evi,g] − E [vi,e]E [
Cov(vi,e,vi,g)
0
f i
vi,g
0
vi,e
vi,g
0
Cov(ui,e, ui,g) = E [f 2i ] − E [f i]E [f i] V ar(f i)
= σ2f
V ar( ui) = V ar(m−1i
mie
ui,e)
V ar( ui) = 1m2i
V ar(
mie
ui,e) = 1m2i
mie
V ar(ui,e) + 2e
g
Cov(ui,e, ui,g)
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V ar( ui) = 1
m2i
miσ
2 + 2e
(mi − 1)σ2f
=
1
m2i
miσ
2 + 2mi
2 (mi −1)σ2
f
V ar( ui) =
1
m2imiσ
2
f + miσ2
v + m2
iσ2
f − miσ2
f =
σ2f
mi +
σ2v
mi + σ2
f −σ2f
mi
V ar(ui) = σ2f +
σ2v
mi
V ar(u∗i ) = miσ2f + σ2
v
H 0 : β 2 = 0
yt = β 1 + β 2xt + ut V ar(ut) = σ2t = σ2
ux2t
V ar(β) = (X X )−1
t
ut2(xtx
t)
(X X )−1
β 2
V ar( β 2) = t(xt − x) ut
2
[t
(xt −
x)2]2
t
t =β 2
t(xt−x) ut2
[
t(xt−x)2]2
yt = βxt + ut σ2t = k(βxt)2
β
ytxt
xt = 0
t
β = (xΩ−1x)−1xΩ−1y
Ω−1 =
1k(βx1)2
0 · · · 0
0 1k(βx2)2
· · · 0
0 0 0 1k(βxT )2
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β = T
kβ 2
−1 1
kβ 2
t
ytxt
=
tytxt
T
V ar(β ) = (xΩ−1x)−1 = T
kβ 2
−1
= kβ 2
T
xt = 0 t
β
y1 = β 0 + β 1y2 + β 2z1 + u1
y2
z1
z2
y2
y2
y1
y1 = α0 + α1z1 + α2z2 + v1
αj
y2 β j
v1
u1
v2
αj
y2
y2 = π0 + π1z1 + π2z2 + v2
y1 = β 0 + β 1(π0 + π1z1 + π2z2 + v2) + β 2z1 + u1
y1 = β 0 + β 1π0 + β 1π1z1 + β 1π2z2 + β 1v2 + β 2z1 + u1
y1 = β 0 + β 1π0
α0
+ (β 1π1 + β 2)
α1
z1 + β 1π2
α2
z2 + β 1v2 + u1
v1
α0 = β 0 + β 1π0
α1 = β 1π1 + β 2
α2 = β 1π2
v1 = β 1v2 + u1
αj
z1
z2
β j πj
yt = β 0 + β 1x ∗t +ut
xt = x∗t + et
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ut
x∗t
et
yt xt et x∗t
x∗t
x∗t = xt − et
vt
xt
β 1 > 0 β 1 yt
xt
ut et
x∗t
e∗t
x∗t−1
et−1
E (xt−1vt) = 0
vt
xt
xt−1
β 0
β 1
x∗t = xt − et
yt = β 0 + β 1xt + ut −β 1et vt
xt xt vt
Cov(xt, vt) = E (xtvt)
Cov(xt, vt) = E [(x∗t
+ et)(ut −
β 1et)]
Cov(xt, vt) = E (x∗tut − β 1etx∗t + etut − β 1e2
t )
Cov(xt, vt) = E (x∗tut) 0
−β 1 E (etx∗t )
0
+ E (etut) 0
−β 1E (e2t )
E (e2t ) et β 1 > 0
β 1
E (xt−1
vt) = E [(x∗
t−1 + e
t−1)(u
t −β
1et)]
E (xt−1vt) = E (x∗t−1ut + et−1ut −β 1etx∗t−1 − β 1etet−1)
E (xt−1vt) = E (x∗t−1ut) 0
+ E (et−1ut) 0
−β 1 E (etx∗t−1)
0
−β 1 E (etet−1) 0
E (xt−1vt) = 0
Cov(xt, xt−
1) = E (xtxt−
1)
Cov(xt, xt−1) = E (x∗tx∗t−1 + etx∗t−1 + x∗t et−1 + etet−1)
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Cov(xt, xt−1) = E (x∗tx∗t−1) + E (etx∗t−1)
0
+E (x∗t et−1) + E (etet−1) 0
x∗t
x∗t−1 x∗t et−1
xt−1
Cov(xt−1 xt) = 0
Cov(xt−1 vt) = 0
y1 = α1 + α2y2 + α3x1 + α4x2 + u1
y1 = α5 + α6y2 + u2
y1 y2
x1 x2
u1
u2
y1
y2
x1
x2
y1
y2
y1
y2
x1
x2
y2
α1 + α2y2 + α3x1 + α4x2 + u1 = α5 + α6y2 + u2
α2y2 − α6y2 = α5 − α1 −α3x1 −α4x2 + u2 −u1
(α2 −α6)y2 = α5 − α1 −α3x1 −α4x2 + u2 −u1
α2 = α6
α2 − α6
y2 = α5 −α1
α2 −α6 π1
+ −α3
α2 −α6 π2
x1 + −α4
α2 −α6 π3
x2 + u2 −u1
α2 −α6 v1
y1
y1 = α5 + α6α5 −α1
α2
−α6
+ −α3
α2
−α6
x1 + −α4
α2
−α6
x2 + u2 − u1
α2
−α6 + u2
y1 = α5 + α6(α5 − α1)
α2 − α6+ −α6α3
α2 − α6x1 +
−α6α4
α2 − α6x2 +
α6(u2 −u1)
α2 −α6+ u2
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y1 = α5α2 − α1α6
α2 − α6
π4
+ −α6α3
α2 − α6
π5
x1 + −α6α4
α2 −α6
π6
x2 + α2u2 − α6u1
α2 − α6
v2
y2
y2
x1
x2
y1
y2
y2
y2
x1
x2
yi = xiβ + ui
E (ui) = 0
xi
β
zi
zi
xi
u Ω
βMCG = (X Ω−1X )−1X Ω−1y
βIV = (Z X )−1Z y
Z
X Ω−1
Ω−1
Z
Z = Ω−1X
Z =
1σ21
0 · · · 0
0 1σ22
· · · 0
0 0 · · · 1σ2n
x11 x21 · · · xk1
x12 x22 · · · xk2
x1n x2n · · · xkn
zi
zi = 1
σ2i
xi
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