Esercizi svolti di econometria

AR(1)

1. MA(1), MA(2), AR(1), ARMA(1,1)

2. 1. E(y_t) = E( α y_t-1 + ε_t) = E(y_t) + α E(y_t-1) = 0

   2. E(y_t) = μ = E(y_t-1)

      |α| < 1

   3. E(y_t) = E( α y_t-1 + ε_t) = α E(y_t-1) + E(ε_t)

Stationario in Covarianza

• E(y_t-1) = E(y_t) = μ

(c)

1. E(y_tε_t) = E[(ε_t + θ ε_t-1) ε_t] = E(ε_τ²)

2. E(y_t-1ε_t) = 0

E(ε_t²) = Var.(ε_t) = σ ²

• E(y_tε_t) = E[( α y_t-1 + ε_t) ε_t] = α E(y_t-1ε_t) + E( ε_t²)
• E(y_t-1ε_t) = cov(y_t-1, ε_t) = 0

E(ε_t²) = σ ²

(d)(1) Var(y_t) = Var(y_t) = Var(_t + ₀y_t-1) = (2)

= Var(_t) + ₀²Var(y_t-1) + 2₀Cov(y_t-1, _t)

= (1 + ₀²)^t2

(2^o)Var(y_t = Var(_t + _{t-1 0} + ₂t-2)

= Var(_t) +₀²Var(_t-1 + ₂Var(_t-2)

Tolociia di covarianze in quanto _t ~wn

= (t + ₀² + ₂²)^t2

(3^o) Var(y_t)= Var(^uy_t, + _t)

= ²Var(y_t-1) + Var(_t) + 2 ^{Cov(y_t-1, _t)}

= Var(qy) ^{pe sbloranemento}

di covarianze(¹1)

Var(qyt)

= ²Var(y_t) + Var(_t)

0 = ² ₀ + ²

0 = ²/¹-²

(4) Var(y_t)=Var(^ty_t-1 + _t + ^e_t+1)

= ²Var(y_t-1) + Var(_t) + ²Var(_t-1)

+2 ^{Cov(y_t, _t) + 2 Cov(y_t-1, t-2}

+ 2 ^{Cov(_t t-1)}

Abbrtuamo che: Var(y_t) = Var(⁰ _t) = ⁰

Var(_t) = ²

Cov(y_t, 1) = 0

Cov(y_t-1, _t-2) = Cov(y_t, _t) = E(y_{t t}) = ² puntos

Cov(_t, _t-1) = 0

⁰ = ²⁰ + ² + ^{2+ 2 Cov( 2=> 0}

= ^{1 + 2 + 2/1 2}

Area #2

d) E[R_T1|I_T] = E[ε_T1 + θε_t1I_T]

= E[ε_T1| I_T] + θE[ε_t1I_T]

= θε_t = 0.2 · 0.01

E[R_T2| I_T] = E[ε_T1 + θε_t1I_T] = 0

b) E[( R_T2 - R_Tt1) I_T] = E[(ε_tT1 + θε_Tt1 - θε_t)² | I_T]

= E[ ε_tT1| I_T] = σ² = 0.025

E[ ( R_T2 - R_T2t1)² | I_T] = E[ ( ε_tT1 + θε_tt1)² | I_T]

= E[ ε²_T1 | I_T ] + θ E[ ε²_Tt1 | I_T] + 2θ E[ ε_T2 ε_tt1 | I_T ]

= (1 + θ)² σ² = (1 + 0.2)² · 0.025

c) η̂₁ = δ₁ / x_o → η̂₁ = δ̂₁ / x_o = 0 / 1 + θ - 0.2 / 1 + 0.2²

η̂₂ = δ̂₂ / x_o → η̂₂ = δ̂₂ / x_o - 0 / (1 + θ)² = 0

5

c) a Var(_t) = a = ...

a 1 – φ₁

a d)

a r₁ = φ₁ y₀ ...

a₂ = φ₂ y₀ ...

ARMA 6

# 6

y y_t = φ₀ + φ₁ y_t-1 + ε_t + θ₁ ε_t-1 + θ₂ ε_t-2

E [r_t+1 | I_t ] = E [φ₀ + φ₁ r_t + ε_t+1 + θ₁ε_t+1 + θ₂ ε_t-1 | I_t ]

= 1.03 + 0.2 1.18, 3 + 0.4, 2.0 - 0.25 (-2.3) = ...

E [r_t+2 | I_t ] = E (φ₀ + φ₁ r_t+1 + ε_t+2 + θ₁ ε_t+1 + θ₂ ε_t | I_t)

= φ₀ + φ₁ E [r_t+1 | I_t ] + θ₂ ε_t

= 1.03 + 0.2 E [r_t+1 | I_t ] - 0.25 (2.6)

dal punto precedinato