59b7Hoja De Formulas

bMCO = (X 0 X)−1 X 0 Y
var(bMCO ) = σ 2 (X 0 X)−1 con var(u) = E(u.u0 ) = σ 2 I
σ
b2 =

1
N−K−1

PN

b2i
i=1 u

Yb = bX, u
b = Y − Yb

E(a1 X + a2 Y + c) = a1E(X) + a2 E(Y ) + c
E(X) =

PN

i=1 xi f (xi ),

X discreta

V ar(X) = E(X − E(X))2
Si X vector: var(X) = E [(X − E(X)(X − E(X))0 ]
V ar(aX + b) = a2 V ar(X)Cov(X, Y ) = E(XY ) − E(X).E(Y )
V ar(X + Y ) = V ar(X) + V ar(Y ) + 2.Cov(X, Y )
V ar(X − Y ) = V ar(X) + V ar(Y ) − 2.Cov(X, Y )
Cov(a1 X + b1 , a2 Y + b3 )= a1 a2 Cov(X, Y )
sd(X) =
t=

p
var(X)

bk −Bko
gk )
se(b

∼ T(N−K−1)

2

R2 =

SN e
(Yi −Y )2
Si=1
N
2
i=1 (Yi −Y )

=1−

SN
u
e2
SN i=1 i 2
i=1 (Yi −Y )N
N
N
X
X
X
(Yi − Y )2 =
(Ybi − Y )2 +
u
b2i

|i=1 {z

|i=1 {z

}

T SS

2

}

ESS

N−1
R = 1 − (1 − R2 ) N−K

F =

ESS/K
RSS/(N−K−1)

F =

=

|i=1{z }
RSS

R2/ K
(1−R2 ) / (N−K−1)

2 −R2 )/q
(RN
R
2 )/(N−K−1) , q
(1−RN

∼ F(K),(N−K−1)

número de restricciones en Ho

N : no restringido, R : restringido
Si var(u) = Σ6= σ 2 .I ⇒ var(bmco ) = (X 0 X)−1 X 0 ΣX(X 0 X)−1
Test White N · R2 ∼ χ2df.. (reg u
b2i sobre polinomio de X’s)
p
Si var(u|x) = σ 2 .h(x) ⇒ var(u| h(x)) = σ 2(homosc.)

Si ut = ρut−1 + vt AR(1), −1 ≤ ρ ≤ 1 ⇒
vt = ut − ρut−1 no autocorrelacionado pq cov(vt , vs ) = 0 ∀s

DW = d =

SN

(e
ut −e
ut−1 )2
t=2
SN
e2t
t=1u

Null hypothesis
No + correlation
No + correlation
No - correlation
No - correlation
No correlation

≈ 2(1 − ρ), Durbin Watson

Decision
Reject
Cannotdecide
Reject
Cannot decide
Do not reject

If
0 < d < dL
dL ≤ d ≤ dU
4 − dL < d < 4
4 − dU ≤ d ≤ 4 − dL
dU < d < 4 − dU

Si D es dummy⇒ E(D|X) = Pr(D = 1|X)

3