Time series Exercise session: 2 June 2, 2017

keywords: Autocovariance functions, autocorrelation function, partial autocorrelation function. Estimation of the mean, of the autocovariance functions, of the autocorrelation function, of the partial autocorrelation function. Autocovariance functions Exercise 1 Let (Yt ) be a weak sense stationary centered process. For a square integrable 1/2 random variable Y , we write kY k = (E[Y 2 ]) 1. Show that the covariance matrix of (Yt−p+1 , . . . , Yt−1 , Yt ) is the same as the covariance matrix of (Yt−1 , . . . , Yt−p+1 , Yt−p ). 2. Deduce that kYt − PLC(Yt−1 ,...,Yt−p+1 ) (Yt )k = kYt−p − PLC(Yt−1 ,...,Yt−p+1 ) (Yt−p )k 3. Deduce that that rY (p) = corr(Ut , Vt ) where Ut = Yt − PLC(Yt−p+1 ,...,Yt−1 ) (Yt ) Vt = Yt−p − PLC(Yt−p+1 ,...,Yt−1 ) (Yt−p )

Exercise 2 Let (Xt ), (Yt ), (et ) be weakly stationary centered processes which are independent. Set: Zt = Xt Yt + et . What is the autocovariance function γz of Zt ? If (Xt ) is periodic (Xt+p = Xt ), what is the expression of its autocoviance function?

Exercise 3 Let (uk )k∈Z be a sequence of real numbers such that u0 = 1, u1 = u−1 = ρ and uk = 0 , |k| > 1. The aim of this exercise is to show on which condition on ρ, this sequence can be the autocovariance function of a weakly stationary process.

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1. Establish the recursion which links the determinants of Γk = [ui−j ]1≤i,j≤k associated to the sequence (uk )k∈Z . 2. Show that for all |ρ| ≤ 1/2 the matrices Γk are covariance matrices for all k. (We admit that a symmetric matrix is positive if all the principal determinants are positive). 3. Propose a weakly stationary process such that the autocovariance function is (uk )k∈N . Linear model and projection on the past of a process Exercise 4 Let (Yt )t∈Z be a weakly centered stationary process. Write Γk the covariance
0 matrix of the vector Yt−1 · · · Yt−k . 1. Give the expression of the linear regression (orthogonal projection) of Yt on the linear combination of Yt−1 , · · · , Yt−k , that we write pHyk−1,t−1 Yt . Deduce the variance of the prvision error σk2 . 2. Check that

det Γk+1 det Γk and express det Γk in terms of the prevision errors σk2 =

(1)

3. Let σ 2 be the variance of the inncation process (Yt )t∈Z , deduce that log σ 2 = lim

k→+∞

1 log det Γk k

(2)

(We admit that limk→+∞ σk2 = σ 2 ) 4. We consider the case of a MA(1): Yt = εt − θεt−1 , |θ| < 1

(3)

with (εt )t∈Z ∼ IID(0, 1) (i) Compute Γk et det Γk . (ii) Compute directly pHyk−1,t−1 Yt in the case where θ = 1. (iii) Consider again θ = 1. limk→+∞ Yt − pHyk−1,t−1 Yt = εt . Deduce that if θ = 1, (εt )t∈Z is the innovation processus of (Yt )t∈Z .

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