STATS 726
Time Series
SEMESTER 2, 2021
1. Let Z be the set of integers, which means that Z = {. . . , −2, −1, 0, 1, 2, . . .}. Consider the following process:
Xt = εt − θεt−2,
where t ∈ Z, θ = 0.8 and {εt} is Gaussian white noise.
More precisely, we have {εt} ∼ IID N(0, 1).
Answer the following questions:
(a) Compute E(Xt) for t ∈ Z. [2 marks]
(b) Show that the autocovariance function of {Xt} has the expression:
Straightforward numerical calculations lead to
[10 marks]
(c) Decide if this process is invertible (or not). Explain your answer. [3 marks]
(d) Use the result obtained in part (c) in order to conclude that there exist constants πj such that and
[1 mark]
(e) In the equation above, find the values of πj for j ∈ {0, 1, 2, 3, 4}. [10 marks]
(f) Let be the best predictor of X4 that assumes knowledge of the infinite past X3, X2, X1, X0, X−1, . . .
Use the result in part (d) in order to calculate [8 marks]
[Total: 34 marks]
2. Consider again the process defined in Question 1:
Xt = εt − θεt−2,
where t ∈ Z, θ = 0.8 and {εt} ∼ IID N(0, 1).
Answer the following questions:
(a) Show that the best linear predictor of X4 given has the expression
where is a row vector and Γ−1 denotes the inverse of the matrix Γ. Write down the entries of the vector and of the matrix Γ. Your answer should contain either expressions that involve θ or numerical values. Justify your answer. Do not compute the inverse of the matrix Γ. [14 marks]
(b) The expression of is given by
where φ31, φ32 and φ33 are computed by using the steps of the Durbin-Levinson algo-rithm that are presented below. Copy to your answer the steps of the algorithm and replace ? with the correct quantities. Each quantity can be either an expression that involves θ or a numerical value.
For h = 1,
For h = 2,
For h = 3,
[10 marks]
(c) Use the result in part (b) in order to find Compare your answer with the value that you have obtained for in Question 1, part (f). [3 marks]
(d) An alternative solution for finding the expression of 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,
[14 marks]
(e) Using the values of φ31, φ32, φ33 that you computed in part (b), show that the expres-sion for given at the beginning of part (b) is equivalent to the expression for obtained from the innovations algorithm in part (d). [1 mark]
[Total: 42 marks]
3. Let n = 100. Suppose that we have the observations x1, x2, . . . , xn of a time series X1, X2, . . . , Xn. The observations x1, x2, . . . , xn gave the following sample statistics:
For all t, assume that E(Xt) = µ and {Xt − µ} can be modeled as the AR(2) process
Answer the following questions:
(a) Use the method of moments in order to find the estimates [12 marks]
Hint: The following result might be useful. Let a, b, c, d ∈ R such that ad − bc ≠ 0.
We have the following identity:
(b) We know that, for n large, we have approximate Normality of the estimators:
In the equation above, replace ? with the correct matrix. Compute the entries of the matrix; you may reuse expressions from part (a) where this is helpful. [6 marks]
(c) Apply the result from part (b) in order to construct 95% confidence intervals for φ1 and φ2. [6 marks]
[Total: 24 marks]