STATS 726

STATISTICS

Time Series

SEMESTER TWO 2023

1. With the convention that B is the backshift operator, we define

where b is a complex number and is its complex conjugate.

We have b = reiη, where and i 2 = −1. Equivalently, we can write

b = r (cos(η) + isin(η)).

Elementary calculations lead to

β(B) = 1 − 0.8B + 0.4B2 .

Note that you do not need to prove that the expression of β(B) given above holds true.

All processes that we consider in this question are defined for t ∈ Z, where Z is the set of integers.

Let {Vt} ∼ WN(0, 1) and {Wt} ∼ WN(0, 4), with the property that Cov(Vs, Wt) = 0 for all s and t.

We define {Xt} and {Yt} as follows:

Xt = β(B)Vt ,

Yt = Xt + Wt .

Answer the following questions.

(a) Decide if the MA process {Xt} is invertible (or not). Justify your answer. [5 marks]

(b) Let γX(h) be the autocovariance function of {Xt}. Show that

[6 marks]

(c) Let γY (h) be the autocovariance function of {Yt}. Similarly, γW (h) is the autocovari-ance function of {Wt}. Show that

γY (h) = γX(h) + γW (h), for all h ≥ 0.

Use this result and the result from part (b) in order to write down the numerical values of γY (h) for h ≥ 0.            [8 marks]

(d) For h ∈ {1, 2, 3, 4}, the partial autocorrelation function αY (h) of {Yt} can be computed by using the steps of the Durbin-Levinson Algorithm that are presented below.

Note that, for h ∈ {0, 1, 2, 3}, vh = E, where denotes the best linear predictor of Yh+1 given Y1, . . . , Yh.

Copy to your answer the steps of the algorithm and replace ? with the correct numerical values. Use the results from part (c). Show your working.

For h = 1,

For h = 2,

For h = 3,

For h = 4,

[12 marks]

[Total: 31 marks]

2. Consider again {Vt}, {Wt}, {Xt} and {Yt}, which have been defined in Question 1.

Now we define the processes {Zt} and {St}, for t ∈ Z:

β(B)Zt = Wt ,

St = Zt + Vt ,

where β(B) is the same as in Question 1.

Answer the following questions.

(a) Decide if the AR process {Zt} is causal (or not). Justify your answer. [3 marks]

(b) Let ρZ(k) be the autocorrelation function of {Zt} at lag k. It is known that

ρZ(k) = 0.8ρZ(k − 1) − 0.4ρZ(k − 2) for k ≥ 1.

Write down the characteristic equation for the homogeneous equation given above. Then write down the general solution for the homogeneous equation. [7 marks]

(c) Show that the following identity is true for all t ∈ Z:

β(B)St = Yt .

[5 marks]

(d) By using the the autocovariance function of {Yt}, it is possible to find the following representation (for all t ∈ Z):

Yt = θ(B)εt,

where θ(B) = 1 − 0.19B + 0.07B2 and εt ∼ WN(0, 5.59).

Use this result together with the result from part (c) in order to conclude that {St} is an ARMA(p, q) process. Find the values of p and q. [5 marks]

Hint: In your answer you may use the fact that one of the roots of the polynomial 1 − 0.19z + 0.07z2 is , where i2 = −1.

(e) Let ρS(k) be the autocorrelation function of {St} at lag k. Use the result that you have obtained in part (d) in order to write down the general homogeneous difference equation for ρS(k). Use this equation and the result from part (b) for writing a short comment about the differences and similarities between ρZ(k) and ρS(k). [7 marks]

(f) In Figure 1 are displayed the autocorrelation function and the partial autocorrelation function for {Xt}, {Yt}, {Zt} and {St}, for lags 1, . . . , 5. For each panel (a), (b), (c), (d), identify the process whose autocorrelation and partial autocorrelation functions are represented in that panel. Justify your answer by using the results from Question 1 and from the previous parts of Question 2. [12 marks]

[Total: 39 marks]

Figure 1: The autocorrelation function and the partial autocorrelation function for {Xt}, {Yt}, {Zt} and {St}, for lags 1, . . . , 5.

3. Let n = 200. We have the observations x1, x2, . . . , xn of a zero-mean time series X1, X2, . . . , Xn. After applying the formulas from Chapter Introduction, the observa-tions x1, x2, . . . , xn gave the following values for the sample autocorrelation function:

Answer the following questions.

(a) Assume that the zero-mean time series X1, X2, . . . , Xn is actually an IID sequence. As n is large, this assumption leads to the conclusion that the sample autocorrelations (h), h > 0 are approximately IID N(, ). Replace with the correct numerical values. Justify your answer.

Then use this result in order to find the bounds of the interval where approximately 95% of sample autocorrelations should fall.

Compare the values of (1), (2) and (3) with the bounds that you have found. Based on these comparisons, write a short comment in which to discuss if we have evidence against the hypothesis that the observed time series is a realization of an IID process.   [8 marks]

(b) Assume that the zero-mean time series can be modeled as the AR(2) process

Xt = φ1Xt−1 + φ2Xt−2 + εt , where {εt} ∼ IID N(0, σ2).

Use the Yule-Walker equations in order to find the estimates and . Explain why it is not possible to find a numerical value for the estimate . [14 marks]

Hint: The following result might be useful. Let a, b, c, d ∈ R such that ad − bc ≠ 0.

We have the following identity:

(c) In connection with the estimation problem from part (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 (b) where this is helpful. [8 marks]

[Total: 30 marks]