• All Questions are equally weighted and this worksheet contributes 15% to overall grade. (if included in best 2 out of 3)

• To get full marks you must show all working and justify your solution/discussion or interpret your solution where appropriate.

• If the question is a code question you must comment all code appropriately and present all outputs with correct labelling of plots and discussion/captions.

1. (174 & 274 attempt) Characteristic Polynomials and Stationarity. (Total marks for question 33%)

For each of the time series models below undertake the following steps:

• Write each time series in characteristic polynomial form using backshift operator (MA or AR or ARMA polynomials as required)

• Factorise the relevant polynomials and find the roots of the polynomials.

• You are given the following additional information regarding the relationship between stationarity and the characteristic polynomial roots.

– An AR(p) process is a stationary process if and only if the modulus of all the roots of the AR characteristic polynomial are greater than one.

– An MA(q) process is always stationary if the MA coefficients are finite.

– An ARMA(p,q) model admits a unique stationary solution when the AR characteristic poly- nomial and the MA characteristic polynomial have no common roots and the AR polynomial satisfies that it has no unit roots.

• Use this information to determine which of the below models is stationary.

(Make sure to justify your answer.)

(a)  ∇Yt +  (Yt + 5Yt − 1 ) = ϵt − ϵt − 1 − ϵt −2 +  (1 + B)∇Yt − Yt

(b) Yt −  − 2Yt − 1 = 2ϵt + ϵt − 1 − (1 − B2)ϵt

(c) Yt+1 + Yt − (1 − B)(1 + B)Yt+1 = ∇ϵt

(d)  ϕ(B )Yt  = ∇2 ϵt +  (1 + 2B)ϵt ,

where one of the roots of the polynomial ϕ(B ) lies inside the unit circle and all other roots lie outside the unit circle.

(e) Let {Yt} be a stationary process with ACF ρ(·) and let

M := E((Yt − (ϕk1Yt − 1 + . . . + ϕkkYt −k ))2 ) .

By differentiating M and setting the result to zero, show that the values ϕk1 , . . . , ϕkk  which minimise

M satisfy the Yule-Walker equations

ρ(j) = ϕk1ρ(j − 1) + . . . + ϕkkρ(j − k)

for j = 1, . . . , k .

2. (174 & 274 attempt) ARMA and ARIMA Models.

(Total marks for question 33%)

(a)  (ONLY 174 Attempt) Let {Yt} be a stationary ARMA(1, 1) model with mean µ  0 and white noise sequence {ϵt} ∼ WN(0, σ2 ).  Calculate E (ϵtYt+k), for all k  ≥ 0 and express your answer in

terms of Kronecker-Delta functions where possible.

(b)  (ONLY 274 Attempt)

Given Y0 = c where c is a real positive constant, find the variance Var (Yt) and covariance Cov (Yt, Yt+k) of a process {Yt} given by an ARIMA(0,1,0) model which can be written as

∇Yt = ϵt .

Comment on whether Yt  is weakly stationary.