Time Series Exam, 2009: Solutions - STAT

An ARMA(p,q) process φ(B)Y t = θ(B)ε t is causal if it can be written asa linear process∞∑Y t = ψ j ε t−j = ψ(B)ε t ,j=0where ∑ |ψ j | < ∞, and we set ψ 0 = 1. It is invertible if it can be writtenas∞∑ε t = π j Y t−j = π(B)Y t ,j=0where ∑ |π j | < ∞, and we set π 0 = 1.An ARMA(p,q) process φ(B)Y t = θ(B)ε t is causal iff φ(z) ≠ 0 within theunit disk D. If so, then the coefficients of ψ(z) satisfy ψ(z) = θ(z)/φ(z)for z ∈ D. The process is invertible iff θ(z) ≠ 0 for for z ∈ D. If so, thenthe coefficients of π(z) satisfy π(z) = φ(z)/θ(z) for for z ∈ D.(a) Since the autocovariance function of a MA(1) process Z t = ǫ t +iidϑǫ t−1 , ǫ t ∼ N(0,τ 2 ) is γ 0 = τ 2 (1 + ϑ 2 ), γ 1 = τ 2 ϑ and γ h = 0otherwise, the equality of the autocovariance functions follows immediately,substituting the pairs θ and σ 2 , then θ −1 and σ 2 θ 2 for ϑand τ 2 .(b) The corresponding polynomials areθ(z) = 1−0.5z,φ(z) = 0.5(1+1.5z −z 2 ) = 0.5(1−0.5z)(1+2z),so after cancelling the common term 1 − 0.5z and introducing therandom variables η t = 0.5ǫ t , we getY t = η t +2η t−1 , η tiid∼ N(0,0.25σ 2 )which is causal but not invertible: the polynomial 1+2z has the onlyroot z 1 = −1/2. Based on part (a), we can obtain a well-behavedequivalent ARMA representation by takingY t = η ′ t +0.5η ′ t−1, η ′ t5. A SARIMA(p,d,q)×(P,D,Q) s model is given byiid∼ N(0,σ 2 ).Φ P (B s )φ(B)(I −B) d (I −B s ) D Y t = α+Θ Q (B)θ(B)ε t ,where {ε t } is Gaussian white noise. The ordinary autoregressiveand movingaverage components are represented by the operators φ(B) and θ(B),respectively; the seasonal autoregressive and moving average componentsbyΦ P (B s ) and Θ Q (B s ), ofordersP and Q; and the ordinaryand seasonaldifference components by (I−B) d and (I−B s ) D of orders d and D. Thenotions of causality and invertibility is the same for the ordinary AR andMA component as before, and are defined similarly for the seasonal ARand MA component.Model identification: With the use of the plot of the series, the ACF, thePACF and (possibly smoothed) periodograms, we can detect someimportant characteristics of a time series: presence of trend or periodiccomponents (both implying nonstationarity).After removing nonstationarity by for example differencing, the ACFand the PACF may give indications of ARMA and seasonal ARMAcomponents:pure AR(p) model: ACF tailing off, PACF cuts off after lag p;pure MA(q): ACF cuts off after lag q, PACF tailing off;3