STATS 726

Time Series (Exam)

SEMESTER TWO, 2022

1. Let Z be the set of integers, which means that Z = {. . . , −2, −1, 0, 1, 2, . . .}. Consider the following process:

Xt = φ2Xt−2 + εt ,

where t ∈ Z, 0 < φ2 < 1 and {εt} is Gaussian white noise.

More precisely, we have {εt} ∼ IID N(0, 1).

It is known that E(Xt) = 0 for all t ∈ Z, and the autocovariance function γ(k) of {Xt} has the following expression for k ≥ 0:

Use the results provided above in order to answer the following question.

For h ∈ {0, 1, 2, 3}, the best linear predictor of Xh+1 given X1, . . . , Xh has the expression

where φh 1, . . . , φh h are computed by using the steps of the Durbin-Levinson Algorithm that are presented below. Remark that  = 0.

Additionally, vh = E for h ∈ {0, 1, 2, 3}.

Copy to your answer the steps of the algorithm and replace ? with the correct quantities. Each quantity can be either an expression that involves φ2 or a numerical value.

For h = 1,

For h = 2,

For h = 3,

[Total: 15 marks]

2. Consider again the process defined in Question 1:

Xt = φ2Xt−2 + εt ,

where t ∈ Z, 0 < φ2 < 1 and {εt} ∼ IID N(0, 1).

Answer the following questions.

(a) With the notation from Question 1, we consider the following innovations:

Use these identities together with the expressions of  that you have obtained in Question 1 in order to find the entries of the matrix C, which is defined in the equation below. Copy to your answer the matrix C and replace ? with the correct quantities. Each quantity can be either an expression that involves φ2 or a numerical value. Justify your answer.

[12 marks]

(b) Let  With the notation from part (a), we have:

where I is the identity matrix. With the convention that Θ = C − I, we get

Use the result obtained in part (a) in order to find θt t−j for t ∈ {1, 2, 3} and j ∈ {0, . . . , t − 1}. Copy to your answer the matrix Θ and replace θt t−j for t ∈ {1, 2, 3} and j ∈ {0, . . . , t − 1} with the correct quantities. Each quantity can be either an expression that involves φ2 or a numerical value. Do not replace the symbols *. Justify your answer. [3 marks]

(c) An alternative solution for finding θt t−j for t ∈ {1, 2, 3} and j ∈ {0, . . . , t − 1} is to apply the Innovations Algorithm. Copy to your answer the steps of the algorithm that are presented below and replace ? with the correct quantities. Each quantity can be either an expression that involves θ or a numerical value.

For t = 1,

For t = 2,

For t = 3,

[15 marks]

(d) Confirm that the values of θt t−j for t ∈ {1, 2, 3} and j ∈ {0, . . . , t − 1} are the same as those obtained in part (b). [2 marks]

(e) Confirm that the values of Xe2 1 , Xe3 2 , Xe4 3 are the same as those obtained in Question 1.  [2 marks]

[Total: 34 marks]

3. Let {Xt} and {Yt} be two invertible MA(1) processes which are defined as follows:

Xt = Zt + αZt−1, α = 0.9, {Zt} ∼ WN(0, 1),              (1)

Yt = Vt + βVt−1, β = 0.5, {Vt} ∼ WN(0, 1).              (2)

We assume that Cov(Zs, Vt) = 0 for all s and t.

Furthermore, we define {St}:

St = Xt + Yt for all t.

It is known that E(St) = 0 for all t, and the autocovariance function γ+S(k) of {St} has the following expression for k ≥ 0:

Straightforward numerical calculations lead to

Use the results provided above in order to answer the following questions.

(a) It follows from the autocovariance function of {St} that we have the following repre-sentation:

St = εt + θεt−1, for all t,                    (3)

where θ ∈ R and εt ∼ WN(0, ).

For all lags k, let ρS(k) be the autocorrelation function of {St}, which is computed by employing (3). Use the identity

to find the value of θ such that the process in (3) is invertible. Then find the value of . [7 marks]

(b) For all t, we define the processes {Ft}, {Gt} and {Ht} as follows:

Ft = −αFt−1 + Vt ,

Gt = −βGt−1 + Zt ,

Ht = Ft + Gt .

Note that α is the same as in (1), {Vt} is the same as in (2), β is the same as in (2) and {Zt} is the same as in (1).

With the convention that B denotes the backshift operator, calculate

(1 + αB)(1 + βB)Ht .

Express the result by using {∈t} (see part (a)). [7 marks]

(c) Use the result in part (b) in order to demonstrate that {Ht} is an ARMA(p, q) process. Find the values of p and q. [3 marks]

(d) Let ρH(k) be the autocorrelation function of {Ht} at lag k. Use the result in part (c), to find the values of µ1 and µ2 in the following equation:

ρH(k) = µ1ρH(k − 1) + µ2ρH(k − 2) for k ≥ 2.

Justify your answer. [4 marks]

(e) Write down the characteristic equation for the homogeneous equation given in part (d) and then write down the general solution for the homogeneous equation given in part (d). [4 marks]

[Total: 25 marks]

4. Annie found the following question in a textbook:

“Let n = 100. Suppose that x1, . . . , xn are observations of a time series {Xt} that can be modeled as the causal AR(2) process

Xt = φ1Xt−1 + φ2Xt−2 + εt ,

where {εt} ∼ IID N(0, σ2). For h ∈ {0, 1, 2}, the sample autocovariances are computed under the assumption that the mean of {Xt} is known to be zero by applying the formula

and the following values are obtained:

Use these values to find the estimates of φ1, φ2 and σ2.”

Answer the following questions.

(a) Annie wants to obtain the Least Squares (LS) estimates for φ = [φ1 φ2] > by using the linear regression model

y = Xφ + ∈,

where the entries of the vector ∈ are IID N(0, σ2). To this end, she considers three different selections for the pair (yy, X). For each pair (yy, X), the answer should be expressed as follows:

In your answer, replace ? with the correct quantities, for each pair (yy, X). Do not compute explicitly the inverse of the matrix that appears in the answer.

The pairs (yy, X) considered by Annie are:

i Linear regression for the original time series

ii Linear regression for the time series padded with zeros after the last observation

iii Linear regression for the time series padded with zeros before the first observation and after the last observation

[20 marks]

(b) As Annie knows only the values of (0), (1) and (2), and she does not know the values of the observations x1, . . . , xn, she can use only one of the three linear regression models that are listed in part (a). Decide which is the model that Annie can use. Justify your answer. [3 marks]

(c) After doing all the calculations, Annie obtains  = 1.32 and  = −0.634. Use these values in order to find . [3 marks]

[Total: 26 marks]