Time Series Analysis: TD4.

Time Series Analysis: TD4.

Olivier Wintenberger: olivier.wintenberger@upmc.fr

Exercise 1. Consider (Z_t) WN(σ²) and the AR(p) model (φ_p 6= 0) X_t=φ₁Xt−1+φ₂Xt−2+· · ·+φ_pXt−p+Z_t, t∈Z. 1. Discuss the existence of a causal solutionX_t=P

j≥0ψ_jZt−j withP

j≥0ψ_j²<∞.

2. Denote Xt = (X_t, . . . , Xt−p+1)^> ∈ R^p and show that (Xt) is an homogeneous Markov chain of the formXt=AXt−1+Zt,t≥0. Identify A andZt.

3. Show that z^pφ(z⁻¹) with φ(z) = 1−φ₁z− · · · −φ_pz^p is the characteristic polynomial ofA.

4. Deduce that the largest eigenvalueρ(A) of A has modulus smaller than 1 if and only if the roots ofφare outside the unit disc.

5. Show that

Xt=X

j≥0

A^jZt−j, t∈Z,

and that the series is normally convergent inL².

6. Deduce an algorithm to compute the coefficients ψ_j using the A and check that they decrease exponentially fast.

Exercise 2. We have the following theorem

Theorem 1 (Hannan). If (X_t) satisfies an ARMA(p, q) model φ(L)X_t = γ(L)Z_t with θ = (φ₁, . . . , φ_p, γ₁, . . . , γ_q)^> ∈ C and (Z_t) SWN(σ²), σ² > 0, then the QMLE is asymptotically

normal √

n(ˆθ_n−θ₀)−^d.→ N_p+q 0,Var(AR_p, . . . , AR₁, M A_q, . . . , M A₁)⁻¹ where (AR_t) and(M A_t) are the stationary AR(p) and AR(q) satisfying

φ(L)ARt=ηt, γ(L)M At=ηt, t∈Z, with the same SWN(1) (ηt).

Let us consider the ARMA(1,1) model

X_t=φXt−1+Z_t+γZt−1, withθ= (φ, γ)^>.

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1. Determine the causal representation of the time series (AR_t) and (M A_t) under the appropriate condition on θ.

2. Compute the matrix of variance-covariance Var(AR₁, M A₁).

3. Determine the assumption for inverting the matrix and interpret it.

4. Invert the matrix.

5. Deduce a consistent estimator of the asymptotic variance by plugging in the QMLE ˆθn. 6. Deduce a confidence region of asymptotic level 1−α for the coefficientθ.

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