Ejercicios Econometría - GONZALO VILLA

7/18/2019 Ejercicios Econometría - GONZALO VILLA

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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

]



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



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



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



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



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



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



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



(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)



= (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



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



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



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

)



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



∀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



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



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



∂ 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)



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



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



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



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



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 )



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



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



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



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



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



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



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)



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



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 %



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



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



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



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)



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



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



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)



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



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

Ejercicios Econometría - GONZALO VILLA

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